Chingiz Hajiyev · Halil Ersin Soken

Sıtkı Yenal Vural

State Estimation
and Control
for Low-cost
Unmanned
Aerial Vehicles
State Estimation and Control for Low-cost
Unmanned Aerial Vehicles
Chingiz Hajiyev • Halil Ersin Soken
Sıtkı Yenal Vural

State Estimation and Control

for Low-cost Unmanned
Aerial Vehicles

123
Chingiz Hajiyev Halil Ersin Soken
Faculty of Aeronautics and Astronautics Institute of Space and Astronautical Science
Istanbul Technical University (ISAS)
Istanbul, Turkey Japan Aerospace Exploration
Agency (JAXA)
Sıtkı Yenal Vural Sagamihara, Kanagawa, Japan
Faculty of Aeronautics and Astronautics
Istanbul Technical University
Istanbul, Turkey

ISBN 978-3-319-16416-8 ISBN 978-3-319-16417-5 (eBook)

DOI 10.1007/978-3-319-16417-5

Library of Congress Control Number: 2015941902

Springer Cham Heidelberg New York Dordrecht London

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Preface

Over time, technological developments in aeronautics have a bigger impact in

all aspects of human life and, surely, unmanned aerial vehicles (UAVs) play an
important role in this process.
The main aim of this book is to make a contribution to progress in UAV
technology, especially in the sense of autonomous control, and that is achieved in
three stages: fault-free estimation of UAV parameters by robust adaptive Kalman
filters, using fault detection and isolation for the integration of UAV flight data, and
controlling the UAV by means of classical or fuzzy controllers.
The state estimation, control, and fault detection and isolation methods for low-
cost UAVs are illustrated by examples and computer simulations performed via the
software package MATLAB.
Most of the book is original and contributed by the authors.
This book is a research monograph in which the theory and application of the
proposed methods have been discussed together.
This monograph is composed of 11 chapters, as follows:
Chapter 1 is the introduction to the book. An introduction to UAVs along with
a brief history is given. It is discussed why UAVs are important and what are their
advantages over conventional manned planes
In Chap. 2, equations of motion for UAVs are given. The coordinate systems,
derivation of rigid body equations of motion, and linearized equations of motion for
UAVs are presented. The state-space representation of the linearized system is also
included in this chapter.
In Chap. 3, navigation systems for UAVs are given. In this context, inertial
navigation, air data and satellite radio navigation systems, Doppler and altimeter
radars, and magnetic sensors are described. Moreover, measurement faults for these
systems, when they are used in UAVs, are investigated and details for the fault
modeling are presented.
In Chap. 4, the optimal linear Kalman filter (OKF) algorithm is presented. The
OKF for UAV state estimation is shown. The stability of the OKF and necessity of
Kalman filter adaptation are discussed.

v
vi Preface

In Chap. 5, a robust Kalman filter (RKF) with filter gain correction for the case
of measurement malfunctions is introduced. By the use of a defined variable named
the measurement noise scale factor (MNSF), the faulty measurements are taken into
the consideration with a small weight and the estimations are corrected without
affecting the characteristics of the accurate measurements. The RKF algorithms with
single and multiple MNSFs (R-adaptation procedure) are proposed and applied for
the state estimation process of the UAV platform in the presence of measurement
faults. The results of these algorithms are compared for different types of sensor
faults and recommendations about their utilization are given. Stability analysis for
proposed RKFs is also included.
In Chap. 6, the process noise covariance adaptation procedure (Q-adaptation) for
UAV state estimation is introduced and the robust adaptive Kalman filter (RAKF)
algorithm with the R- and Q-adaptations against sensor/actuator failures is proposed.
Thus, the filter stands robust against the faults and, even in case of sensor/actuator
failure, continues providing accurate estimation results. The performance of the
proposed RAKF is investigated by simulations for the state estimation procedure
of UAVs. The presented RAKF ensures that the parameter estimation system of
the UAV is not influenced by sensor/actuator failures and, therefore, autonomous
missions can be performed successfully.
In Chap. 7, Global Positioning System (GPS)/Inertial Navigation System (INS)
measurements, which have high accuracy, and air data system (ADS) measurements,
which have low accuracy but high frequency, are integrated using the Kalman
filtering technique in order to obtain high-accuracy measurement data at high
frequency. It is shown that the designed system, which is based on the indirect
Kalman filtering technique, is successful in finding the wind speed data using the
known error dynamics of the system and the determined statistical values. The
fault detection and isolation (FDI) algorithms are developed by using the proposed
system and diagnostic tests are performed in order to evaluate the performance of
the system under different sensor measurement fault scenarios. For fusing the data
coming from different measurement sets, a federated Kalman filter is used. The best
data are produced by using the FDI algorithms and leaving the faulty data out before
fusing with the federated filter.
The stability of UAVs is investigated in Chap. 8 for the purpose of controller
design. The transfer functions for the longitudinal and lateral stability analysis of
UAVs are introduced and discussed in this chapter.
The design procedure for the flight control system of UAVs based on the
classical controllers is presented in Chap. 9. Classical control methods, such as
the proportional-integral-derivative (PID) controllers, are investigated. It is observed
that the PID controller responses are good at controlling the longitudinal and lateral
flight dynamics of small UAVs.
In Chap. 10, an optimal controller is designed using the linear quadratic regulator
(LQR) method. In order to better evaluate the effect of disturbances on the obtained
measurements, a Kalman filter is used. Initially, the controller is tested for a case
where disturbances are absent. Then, a Kalman filter is designed and the system
Preface vii

under disturbances is tested with the designed controller and filter. The results reveal
the effectiveness of the Kalman filter and LQR controller.
In Chap. 11, a fuzzy controller for UAV flight dynamics is investigated. Fuzzy
logic-based longitudinal and lateral controllers are designed. The fuzzy logic
controller works well, even though no optimization technique is used to optimize
the system, and it was designed without dynamic model knowledge. In fuzzy logic
controllers, a careful arrangement of membership functions may lead to better
results. The stability of the fuzzy controllers is investigated. The effectiveness of
different control methods for UAV flight control is compared in this chapter.
This monograph is useful for both researchers in academia and professional
engineers in the aerospace industry. It is also an important reference book for
graduate research students. Moreover, Chaps. 2, 3, 4, 8, 9, and 10 can also be used
by undergraduate students studying aerospace and control and systems engineering.
We would like to thank Mr. Volkan Kargin for his contribution to the simulations
given in Chap. 7 and M.Sc. student Demet Cilden who helped us with the corrections
and formatting of the book.
We hope that this book will be helpful to researchers, students, our colleagues,
and all the readers.

Istanbul, Turkey Chingiz Hajiyev

January, 2015 Halil Ersin Soken
Sıtkı Yenal Vural
Contents

1 Introduction to Unmanned Aerial Vehicles . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1
1.2 UAV Types and Applications . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4
1.3 A Brief History of UAVs . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6
1.4 Conclusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7
2 Equations of Motion for an Unmanned Aerial Vehicle .. . . . . . . . . . . . . . . . 9
2.1 Rigid Body Equations of Motion .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9
2.1.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9
2.1.2 Derivation of Rigid Body Equations of Motion .. . . . . . . . . . 10
2.2 Orientation and Position of an Aircraft.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 15
2.3 Small Perturbation Theory .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 16
2.4 Linearized Equations of Motion .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18
2.4.1 Equations in General . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 18
2.4.2 Characteristics of the Chosen Zagi UAV . . . . . . . . . . . . . . . . . . 19
2.4.3 Linearized Equations of Motion for the Zagi UAV . . . . . . . 20
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 23
3 Navigation Systems for Unmanned Aerial Vehicles. . . . . . . . . . . . . . . . . . . . . 25
3.1 Two Main Categories in Navigation .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 25
3.2 Inertial Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 26
3.3 Inertial Measurement Unit . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 27
3.3.1 Rate Gyros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 28
3.3.2 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 29
3.4 Air Data System .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 30
3.4.1 Air Data Measurements . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 31
3.4.2 Derivation of the True Airspeed Components .. . . . . . . . . . . . 33
3.5 Surface Radar .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 33

ix
x Contents

3.6 Altitude Measurements . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 35

3.6.1 Types of Flight Altitudes . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 35
3.6.2 Radio Altimeter .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 35
3.6.3 Barometric Altimeter .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 37
3.7 Speed-over-ground and Drift Angle Measurements via
the Doppler Method .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 38
3.8 Magnetic Measurements . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 40
3.9 Satellite Radio Navigation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 41
3.9.1 GPS Structure .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 42
3.9.2 Basic Concept of GPS . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 43
3.10 Vision-based Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 44
3.11 Simultaneous Localization and Mapping (SLAM) . . . . . . . . . . . . . . . . . 44
3.12 Measurement Fault Classification and Fault Modeling .. . . . . . . . . . . . 45
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 48
4 Estimation of Unmanned Aerial Vehicle Dynamics .. . . . . . . . . . . . . . . . . . . . 51
4.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 51
4.2 The Optimal Linear Discrete Kalman Filter . . . .. . . . . . . . . . . . . . . . . . . . 52
4.2.1 Optimal Kalman Filter (OKF) Equations.. . . . . . . . . . . . . . . . . 53
4.2.2 Derivation of Optimal Kalman Gain . . .. . . . . . . . . . . . . . . . . . . . 55
4.2.3 The Structure of the Kalman Filter . . . .. . . . . . . . . . . . . . . . . . . . 57
4.3 Stability of the Optimal Discrete Kalman Filter . . . . . . . . . . . . . . . . . . . . 59
4.4 OKF for UAV State Estimation .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 60
4.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 61
4.6 Necessity for Kalman Filter Adaptation.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 63
4.6.1 A Priori Uncertainty and Adaptation . .. . . . . . . . . . . . . . . . . . . . 63
4.6.2 Innovation-Based Adaptive Estimation .. . . . . . . . . . . . . . . . . . . 67
4.6.3 Residual-Based Adaptive Estimation . .. . . . . . . . . . . . . . . . . . . . 68
4.7 Conclusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 69
5 Estimation of Unmanned Aerial Vehicle Dynamics in the
Presence of Sensor Faults .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71
5.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 71
5.2 RKF with a Single Measurement Noise Scale Factor . . . . . . . . . . . . . . 73
5.3 RKF with Multiple Measurement Noise Scale Factors .. . . . . . . . . . . . 75
5.4 Comparison of the R-adaptation Techniques .. . .. . . . . . . . . . . . . . . . . . . . 76
5.4.1 Instantaneous Abnormal Measurements .. . . . . . . . . . . . . . . . . . 77
5.4.2 Continuous Bias of Measurements . . . .. . . . . . . . . . . . . . . . . . . . 79
5.4.3 Measurement Noise Increment.. . . . . . . .. . . . . . . . . . . . . . . . . . . . 81
5.4.4 Fault of Zero Output .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 82
5.5 Remark on Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 85
5.6 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 91
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93
Contents xi

6 Estimation of Unmanned Aerial Vehicle Dynamics in the

Presence of Sensor/Actuator Faults . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95
6.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95
6.2 Q-adaptation Using Multiple Adaptive Factors .. . . . . . . . . . . . . . . . . . . . 97
6.3 Integration Scheme for the Q- and R-adaptation Methods . . . . . . . . . 98
6.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 99
6.5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107
7 Fault Detection, Isolation, and Data Fusion for Unmanned
Aerial Vehicle Air Data Systems . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
7.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
7.2 Kalman Filter-Based Integrated ADS/GPS Navigation System. . . . 111
7.3 Federated Kalman Filter-Based Integrated ADS
and GPS/INS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 114
7.3.1 Data Fusion Methodology.. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 114
7.3.2 ADS and GPS/INS Data Fusion Based on FDI . . . . . . . . . . . 115
7.4 Sensor FDI Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 116
7.4.1 Statistical Test for Fault Detection .. . . .. . . . . . . . . . . . . . . . . . . . 116
7.4.2 Fault Isolation Algorithm . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 117
7.5 Simulation Results for Indirect Kalman Filter-Based
ADS and GPS/INS Data Fusion .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 118
7.5.1 Results with Fault Isolation . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 118
7.5.2 Results Without Fault Isolation . . . . . . . .. . . . . . . . . . . . . . . . . . . . 122
7.6 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 126
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 126
8 Stability Analysis for Unmanned Aerial Vehicles . . .. . . . . . . . . . . . . . . . . . . . 129
8.1 Trimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129
8.1.1 Trim Point .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 129
8.1.2 Linearization Around a Steady-State Condition . . . . . . . . . . 130
8.2 Derivation of the Transfer Functions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 130
8.2.1 State Equations . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 130
8.2.2 Transfer Functions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135
8.3 Longitudinal Stability Analysis. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 137
8.4 Lateral Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 138
8.5 Conclusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 139
9 Classic Controller Design for Unmanned Aerial Vehicles.. . . . . . . . . . . . . 141
9.1 Classical Proportional-Integral-Derivative (PID) Controller .. . . . . . 141
9.2 Classical Controller for the Longitudinal Motion . . . . . . . . . . . . . . . . . . 144
9.2.1 Pitch Angular Rate Controller (Inner Loop) . . . . . . . . . . . . . . 145
9.2.2 Altitude Controller (Outer Loop) . . . . . .. . . . . . . . . . . . . . . . . . . . 147
9.2.3 Speed Controller .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 154
9.3 Classical Controller for the Lateral Motion .. . . .. . . . . . . . . . . . . . . . . . . . 158
xii Contents

9.3.1 Roll Rate Controller . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 160

9.3.2 Yaw Damper . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 162
9.3.3 Roll Angle Loop .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 165
9.3.4 Heading Controller . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 166
9.4 Conclusion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 170
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 170
10 Linear Quadratic Regulator Controller Design . . . . .. . . . . . . . . . . . . . . . . . . . 171
10.1 Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 171
10.2 Linear Quadratic Optimal Controller .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 172
10.2.1 Lyapunov Stability Criteria . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 173
10.2.2 Linear Quadratic Optimal Control Using
Lyapunov Criterion .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 174
10.3 Altitude and Speed Controller Design Using the LQR Method.. . . 176
10.3.1 LQR Altitude Controller . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 176
10.3.2 LQR Speed Controller.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 177
10.4 LQR-Type Heading Controller . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 185
10.5 LQR Controller with the Kalman Estimator . . . .. . . . . . . . . . . . . . . . . . . . 188
10.5.1 Longitudinal LQR with the Kalman Estimator.. . . . . . . . . . . 190
10.5.2 Lateral LQR with the Kalman Estimator . . . . . . . . . . . . . . . . . . 195
10.6 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 200
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 200
11 Fuzzy Logic-Based Controller Design . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 201
11.1 Fuzzy Logic-Based Systems . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 201
11.1.1 Mamdani-Type Fuzzy Rules . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 202
11.1.2 Singleton-Type Fuzzy Rules . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 204
11.1.3 Takagi–Sugeno-Type Fuzzy Rules . . . . .. . . . . . . . . . . . . . . . . . . . 204
11.1.4 Fuzzy Inference Mechanism . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 204
11.2 Fuzzy Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 206
11.2.1 Fuzzy Logic-Based Altitude and Velocity Controllers.. . . 207
11.2.2 Lateral Fuzzy Logic Controller . . . . . . . .. . . . . . . . . . . . . . . . . . . . 210
11.3 Stability Analysis of the Fuzzy Controllers .. . . .. . . . . . . . . . . . . . . . . . . . 215
11.4 A Comparison of Flight Controllers for Unmanned
Aerial Vehicles (UAVs) . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 219
11.5 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 220
References .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 221

About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 223

Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 225
Chapter 1
Introduction to Unmanned Aerial Vehicles

1.1 Introduction

Everything started with a passion and a desire for flying like birds. Was looking
to the world from above only a dream for humanity? Maybe that was the only
thought of Hezarfen Ahmet Celebi when he jumped down from Galata Tower four
centuries ago, for his first flight over Istanbul. Ahmet Celebi is a legendary character
in Turkish history and we will probably never be sure of his flight’s reality, but
there is one thing that we are certain of; much has changed since that date. Flight
technology has come to a point that he could not imagine, if he ever existed. Today,
aircraft are ideal vehicles of transportation, perfect weapons of warfare, and helpers
of human beings, which makes life easier in many areas. Now, think that they are
not very expensive or hard to produce and do not even need a pilot. UAVs, namely,
unmanned aerial vehicles, were developed for making this purpose real. Although
they do not have a genius director, a human aboard, for controlling extraordinary
situations without risking human life, UAVs are preferred to manned aircraft for
their properties of being more economical, smaller, and lighter.
From the beginning, as can be understood from its name, a UAV is a vehicle
without a pilot or crew aboard, and when a battle operation, risky mission, or a
special case like redirecting a guided missile is discussed, that specialty makes them
greatly superior. For years, wars have been a potential death machine for humanity
and, in that situation, having an assistant which can reduce your army’s loss of life
becomes very important. As Holder states, the availability of using them during
dangerous missions, when rapid information is needed, manned aircraft cannot be
operated, and in other situations that they are required, gives UAVs an important
role in warfare [1]. Supporting that, possibly the best example of the usage of UAVs
in battle is the Iraq war. The eyes and assistants of the ground-based coalition
forces in civil war areas of Iraq were UAVs [2]. If the Fallujah operation, a hard,
key operation of the war, is examined, that truth can be seen.West explains the

© Springer International Publishing Switzerland 2015 1

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_1
2 1 Introduction to Unmanned Aerial Vehicles

position before the operation; since the consultation between the USA and Iraqi
governments finished, they agreed on an operation to Fallujah for taking control of
that part of Baghdad again. The American and Iraqi military forces, comprising
12,000 persons, must fight with the 4,000 rebels without a serious loss of life,
which threatened both them and about 200,000 civilians in the city. At that point,
finding the location of the rebels became important and the only unit of the army
which could do that perfectly was Marine Unmanned Aerial Squadron (VMU-1),
nicknamed “Watchdogs”, and their Pioneer UAVs [2]. At the end of the operation,
after 11 days, the city was cleared and the result of the mission was nearly 30 times
lower life loss for multinational forces. This precisely demonstrates how useful
UAVs can be in a war scenario.
Another situation that suits a UAV is risky operations. Perhaps the attacking
capability of a UAV is limited, but their excellence in discovery missions, which
are described as the most risky operations, cannot be questioned [1]. For example,
Newcome declares that a life-threatening type of job for pilots is carrying out
discovery flights in nuclear test areas, which occurred with high possibility during
the Cold War, the result of which could bring about a nuclear war immediately. For
that reason, these missions were described as being “dirty”, so giving the job to
UAVs was the best approach [3]. They really are excellent in risky missions and,
from this point of view, Holder aids Newcome as he describes another advantage of
a UAV; there is no need for a risky rescue mission with the potential for more loss
of life when a UAV is lost [1]. In 2000, after the volcanic activity of Mount Usu in
Japan, instead of manned aircraft, UAVs flew over the area by the decision of the
Ministry of Construction. The reason for using UAVs for the discovery of dangerous
situations was the active volcano, deemed too risky for human pilots [3]. Also, there
have been some strange suggestions made as a result of the suitability of UAVs in
“dirty” duties. For instance, according to Holder, making UAVs a clear target for
guided missiles and not causing the loss of a pilot’s life is one of the other ideas
which is examined for the usage of UAVs [1]. When the most valuable thing of a
pilot, his or her life, is in danger, UAVs are their lifebuoys.
In today’s world, it is a truth that making an artificial bird fly is very expensive
and, because of that, more economical UAVs may be attractive. Firstly, the education
process of a pilot is very long and, related to that, not cheap, so operating a UAV for
the same jobs can be better for economy, simply because it does not need a pilot.
A combat pilot candidate can learn everything about fighting and flying in 6 weeks,
a period nearly 17 times longer than the duration of the building process of a new
Predator UAV [3]. There is no need to waste time and money. Also, they have the
possibility of being used as part of a pilots’ fighting skill development process. That
secures both a better as well as a cheaper education. Being a moveable target for
fighter aircraft weapons is a popular use for drone aircrafts. Holder defines as an
example and, he continues, for educating their fighter pilots and improving their
combat talent, the Air Force and Navy of the US Army have used various kinds of
drone aircrafts for years.
Another direct economical advantage of UAV usage is the price of a UAV. When
the verification between manned and unmanned aircraft is done in a security area,
1.1 Introduction 3

this fact can be seen more clearly. In 2005, Farivar noted that aircraft used by
police-like organizations for surveillance and crime-fighting missions demanded
a lot of money to buy, replace, and repair [4]. Hence, more economical UAVs
that can replace manned planes became more interesting. Again, according to
Farivar, the person responsible for UAV technology development for Los Angeles
County Sheriff’s Department, Cmdr. Sid Heal, explains “the plane’s low cost makes
it highly desirable, particularly when compared with the current alternative of a
helicopter” and, he continues, “that helicopter costs $450 an hour—for every 10 h
of use I’ve bought a new one of these that I own.” Also, he claims “they shoot one
of these down and I’ll have funds for 100 of them the next day” [4]. There is one
thing that is certain; people want aircraft which are cheap on account to both their
construction and flight costs, and that is valid even for armies with large budgets.
Thirdly, being unmanned reduces the size of an aircraft and, when compared
with large planes, a small plane has many more advantages. Being unmanned does
not only mean not having a pilot; it also means not carrying many more devices
necessary for the crew. For example, life support equipment is an essential part
of manned aircraft. Holder indicates that, when manned aerial craft and UAVs
are considered, not requiring life support equipment for a pilot puts UAVs a step
ahead [1]. That equipment is big and heavy, so if the same conditions are under
discussion, using a small UAV with less chance of being spotted is more reasonable;
nobody wants his or her spy plane to be seen. Some other advantages that small size
brings to a plane are long flight times and ease of transportation and taking off.
As a result, UAVs are considered a part of many missions, such as fire fighting.
DeGaspari tells his readers that, unlike the limited vision of ground-based fire
fighting personnel, aerial vehicles have the chance to explore wide fire areas.
For meeting that requirement, the laborers of the Idaho National Engineering and
Environmental Laboratory (INEEL) are working on a fire fighter UAV [5]. Scott
Bauer, program manager for the lab’s UAV program at the INEEL, says that carrying
a small UAV with a trailer, sending it airborne with the help of a catapult, and
controlling it with a suitcase-sized remote control is plausible, where he describes a
small UAV as a vehicle with 3–3.5-m wingspan, 12-kg payload capacity, and 10 h
of flight time [5]. At the same time, there are other examples of UAVs which are
easier to transport because of their portability and, surprisingly, they can be ready to
fly in a few minutes. Security UAVs, which are used by law-keeping organizations
like the police, are one of these kinds of vehicles. According to Farivar regarding
the size of a security UAV, Dr. Chang explains that the plane is portable enough to
carry in a bag and, when it is required for an operation, mounting it and making it
fly in a short time is possible [4].
Making a heavy object fly is always harder than flying a lighter one and, in that
case, the property of an UAV being lighter can be examined. First, for purposes
like security keeping, traffic controlling, and having vision for the press, operating
a UAV over cities is safer. Farivar explains the situation by using a quotation from
Prof. Kroo, the security UAVs director of the Aircraft Aerodynamics and Design
Group at Stanford University. Prof. Ilan Kroo states that, “the miniaturization of
these aircrafts is also driven by a concern for public safety” and he adds, “if
4 1 Introduction to Unmanned Aerial Vehicles

something catastrophic happens with the airplane and it crashes in a populated

region, you want this thing to weigh a few ounces, not a few thousand pounds” [4].
At the same time, the advantages of lower mass go hand in hand with being small.
For example, a small and light UAV can be hand-launched. A javelin commercial
UAV with the ability of hand launching, a property related to its 6.8-kg weight,
has wide usage potential, such as live vision from the air for the press and law
enforcement organizations [1]. Another thing, lightness affects their invisibility.
Without the need for crew, the weight of the aircraft decreases nearly to its lower
limit, and that is something greatly desired by aerial early warning (AEW) system
developers because a light vehicle has greater opportunity to fly higher and for long
enough without being seen [6]. Lighter is always better.
In conclusion, being unmanned, more economical, smaller, and lighter makes
UAVs superior to manned craft; at that point, not having a genius director no longer
becomes important. The thing which is fascinating is that the visibility of UAVs’
supremacy is increasing day by day. Already, in many areas, UAVs have taken the
place of manned aircraft. Who knows, maybe one day, even passenger planes will
be UAVs and there will no longer be a need for pilots.

1.2 UAV Types and Applications

UAVs can be defined as aircraft without a human pilot aboard. According to the
International Civil Aviation Organization (ICAO), unmanned aircraft is classified
into two groups [7, 8]:
(A) Autonomous aircraft
Autonomous aircraft are considered not to be suitable for regulation due to
legal and liability issues
(B) Remotely piloted aircraft
Remotely piloted aircraft are subject to civil regulation under the ICAO and
under the relevant national aviation authority
There are also different names given to UAVs. They are called unmanned air
systems, unpiloted air systems, drones, unpiloted aerial vehicles, etc. The flight of
UAVs can be controlled using on-board computers autonomously (following the
online calculations or precoded control algorithms) or by remote control by a pilot
on the ground. UAV systems are complex systems that comprise:
• Control station used by system operators
• The UAV itself
• Communications system
UAVs can be used for many different military and civilian applications. These
include:
1.2 UAV Types and Applications 5

Civilian uses:
• Aerial photography
• Agricultural uses
• Meteorological purposes
• Traffic monitoring
• Logistics: UAVs for cargo operations
• Police authority
• Fire services
• Information services
• Research and development: as an experiment for further studies
Military uses:
• Reconnaissance
• Surveillance
• Target and decoy: functioning as an enemy missile or aircraft for aerial or
ground gunnery
• Combat: as fighter or bomber aircraft (in general) on high-risk missions
• Electronic intelligence
UAVs can be classified into groups depending on their size and capability for
specific missions. Endurance, range, weight, and maximum altitude characteristics
determine the capability of UAVs, and they are used to perform the classification.
However, these definitions are constantly changing as the technology advances and
new types of UAVs emerge. Different types of UAVs in use today are [7, 8]:
• Orbital UAVs – UAVs that may operate up to low Earth orbit. They can reach
speeds of Mach 25C and a primary mission example is launching spacecraft into
orbit.
• Hypersonic UAVs – UAVs that may reach supersonic (Mach 1–5) or hypersonic
(Mach 5C) speeds. They are suitable for flying at suborbital altitude.
• HALE (high altitude long endurance) – They operate over 15,000 m altitude and
for a period of 24 h or more. They are usually operated by the military.
• MALE (medium altitude long endurance) – They are operated at altitudes
between 5,000 and 15,000 m. They are generally used by the military.
• Tactical UAVs – Used by the military at smaller ranges. They are simpler systems
compared to MALE and HALE.
• Close-range UAVs – Used by mobile army groups. Their ranges are up to 100 km.
They are mostly used for surveillance, traffic monitoring, crop spraying, and
reconnaissance missions. Handheld UAVs may be regarded as a subcategory.
• Mini-UAVs – UAVs of up to 20 kg approximately. They may be hand-launched.
They are usually used by mobile army groups.
• Micro-UAVs – UAVs having a wingspan no greater than 150 mm. They are
required for operation in urban environments.
• Nano air vehicles – These are proposed to be on the same size scale as insects.
They can be used for ultrashort-range surveillance.
6 1 Introduction to Unmanned Aerial Vehicles

The listed types of aircraft can be either rotary wing aircraft (which are able to
land and take off vertically) or fixed-wing aircraft.
UAVs can also be classified according to their wing loading, engine type, and
power loading.
For calculating the wing loading, the total weight of the UAV is divided by the
wing area. Three classes have been created to classify UAVs according their wing
loading values. UAVs that have a wing loading value greater than 100 kg/m2 are
classified as having high loading. Those that have a wing loading value less than
100 kg/m2 but more than 50 kg/m2 are classified as having medium wing loading.
The other UAVs with wing loading values less than 50 kg/m2 are classified as having
low wing loading.
To complete different types of missions, different engines are used in UAVs.
These include turbofans, two-stroke engines, piston, rotary, turboprop, electric, and
propeller. Some examples of UAVs that use different engines are given below:
• Global Hawk – military UAV –turbofan
• Predator – military UAV – piston
• Dragon Eye – military UAV – electric

1.3 A Brief History of UAVs

“Unmanned aerial vehicle” is a term which came into general use in the early 1990s
and replaced the previously preferred expression of “remotely piloted vehicle”
(RPV) [3]. According to Newcome [3], a UAV is:
A powered, aerial vehicle that does not carry a human operator, uses aerodynamic forces
to provide vehicle lift, can fly autonomously or be piloted remotely, can be expendable or
recoverable, and can carry a lethal or nonlethal payload. Ballistic or semiballistic vehicles,
cruise missiles, and artillery projectiles are not considered unmanned aerial vehicles.

Although UAVs is a recent concept which has become popular following uses
especially in NATO operations during the Kosovo War and American operations
like Operation Enduring Freedom, it has a history as old as manned planes from
World War I. In fact, misunderstanding their role in military missions and not seeing
their accomplishments clearly are key reasons why military planners did not think of
them as a part of operations earnestly until the mid-1980s [9], which brought about a
reduced public interest in those vehicles up to that date. However, nowadays, UAVs
are used in many public applications, such as traffic monitoring, fire fighting, etc., as
well as their crucial function in military missions, like reconnaissance, attack, etc.
Historically, the “Kettering Bug” built by Charles Kettering for the USA Navy
in 1918 may be accepted as the first UAV. This prototype was successful enough to
make the army order additional ones, but it has never flown operationally because of
the termination of World War I. On the other hand, in the light of common opinion,
it is possible to show the “Lightning Bug”, which was substantially used by the USA
during the Vietnam War between 1964 and 1972, as the ancestor of modern UAVs.
References 7

During that period, more than 3,400 combat UAVs were flown over North Vietnam,
Laos, and China by Strategic Air Command 100th Strategic Reconnaissance Wing
of the USAF [9].
In the near future, it is expected that UAV technology will make progress
dramatically. If this development phase is examined from the point of view of
this study, then certainly, it is required to be the focus of researches about the
autonomy level of UAVs. The performance of autonomous missions, where planes
have the ability to make decisions without human intervention, is directly affected
by wrong decisions and not so precise positioning, attitude determination, etc. In
this sense, first, an accurate method for the estimation of UAV states and, then, an
efficient algorithm for controlling these states are what we need for increasing the
performance of autonomous UAV missions. Several algorithms presented in this
book aim to solve these problems.

1.4 Conclusion

As stated, the properties of being unmanned, more economical, smaller, and lighter
make UAVs superior to manned planes and, at that point, not having a genius
director no longer becomes important. The thing which is fascinating is that the
visibility of UAVs’ supremacy is increasing day by day. Already, in many areas,
UAVs have taken the place of manned aircraft. Maybe one day, even passenger
planes will be UAVs and there will no longer be a need for pilots. However, a
crucial challenge awaiting us before we can use UAVs in all areas of human life
is to increase the autonomy level of these vehicles, leading to a minimum of human
interaction. Several algorithms introduced in this book aim to solve this issue.

References

1. Holder WG, Holder B (2001) Unmanned air vehicles: an illustrated study of UAVs. Schiffer
Publishing, Atglen
2. West B (2005) Nowhere to hide. Pop Mech 182(2):54–59
3. Newcome LR (2004) Unmanned aviation: a brief history of unmanned aerial vehicles. AIAA,
Reston
4. Farivar C (2005) A flying crime fighter (some assembly required). New York Times (Article
dated: January, 13, 2005), p. G7
5. DeGaspari J (2005) Unmanned fire spies [Electronic version]. Mech Eng 127(2):20–22
6. Long MW (ed) (1992) Airborne early warning system concepts. Artech House Publishing,
Norwood
7. Austin R (2010) Unmanned aircraft systems: UAVs design, development and deployment.
Wiley, Chichester
8. Fahlstrom PG, Gleason TJ (2012) Introduction to UAV systems, 4th edn. Wiley, Hoboken
9. Cook KLB (2007) The silent force multiplier: the history and role of UAVs in warfare. IEEE
aerospace conference, Big Sky, MT, USA, March 2007
Chapter 2
Equations of Motion for an Unmanned Aerial
Vehicle

2.1 Rigid Body Equations of Motion

Before going into deeper analyses of the equations of motion, it will be useful to
review the axis systems that will be dealt with in further steps.

2.1.1 Coordinate Systems

2.1.1.1 Aircraft Body Axis System

The origin of the frame is located at the center of gravity of the aircraft. The x-axis
is directed toward the nose of the plane, while the y-axis is defined as coming out
of its right wing. The z-axis completes the system by the right-hand rule as it points
down through the bottom of the aircraft.

2.1.1.2 The Earth Axis System

The z-axis of the system shows the center of the Earth. The system lies on the
local horizontal plane of the Earth, as the orthogonal x- and y-axes are defined in
the directions of north and east, respectively. As a general manner, the Earth axis
system is assumed to be the inertial axis for aircraft-related studies. Since aircraft
rotation rates are relatively high in comparison to the rotation rate of the Earth, this
assumption gives acceptably low error for aircraft problems and can be utilized [1,
2] (Fig. 2.1).

© Springer International Publishing Switzerland 2015 9

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_2
10 2 Equations of Motion for a UAV

Fig. 2.1 Body and inertial axis system

2.1.2 Derivation of Rigid Body Equations of Motion

Under the assumptions that the aircraft is a rigid body and its mass remains constant
for a relatively short duration of time, Newton’s second law can be written as:
 !
!
!
 dVc
F Dm (2.1)
dt
i

where m is the mass of the aircraft, Vc is the velocity of the center of mass of
!
the airplane, F is the force acting on the plane, and subscript i indicates that the
derivation is taken with respect to the inertial frame.
If the equation is rewritten in order to fix the axis system to the aircraft body
frame, rotation of the plane with respect to the inertial frame must be taken into
!

consideration. The derivative of Vc , referred to a rotating body frame that has an
angular velocity of !

! b , can be shown by:
 !
!  !
! !

C!

dV c dV c
D ! b x Vc (2.2)
dt dt b
i b

!
Hence, F , the force that acts on the aircraft, becomes:
"  !
! #
!
 
!
F Dm
dV c !
C ! b x Vc (2.3)
dt b
b
2.1 Rigid Body Equations of Motion 11


!
where Vc , the velocity of the aircraft in the body frame, is formed by u, v, and w,
b
velocities in the x, y, and z directions, respectively, as:
!
 !
 !
 !

Vc D u i C v j C w k (2.4)
b

and !
! b , the angular velocity vector in the body frame, consists of p, r, and q, angular
rates about the x, y, and z axes (or roll, pitch, and yaw rates), respectively, as:

!
 !
 !
 !

!b D p i Cr j Cq k (2.5)

As a result,the three force equations for the aircraft in the body axis may be given
as:
2 3 2 3 ˇ ˇ 2 3
uP ˇ! !
 !
 ˇ
uP C qw  rv
Fx ˇ i j k ˇ
4 Fy 5 D m 4 vP 5 C m ˇˇ p q r ˇˇ D m 4 vP C ru  pw 5 (2.6a)
ˇ ˇ
Fz wP ˇ u v wˇ wP C pv  qu

Fx D m .Pu C qw  rv/
Fy D m .vP C ru  pw/ (2.6b)
Fz D m .wP C pv  qu/

On the other hand, in order to derive the three moment equations of the aircraft,
a more complicated methodology may be followed based on examining an arbitrary
small elemental mass, ım, which is at a distance from the center of mass of the
aircraft and rotates about it [2] (Fig. 2.2).

Fig. 2.2 Arbitrary small x

differential mass in the body
frame

r δm

δm

z
12 2 Equations of Motion for a UAV

First, the distance between the differential mass and the center of mass of the
aircraft is given by the following vector:

r D x!
!  !
 !

i Cy j Czk (2.7)
ım

Here, x, y, and z are the distances in the x, y, and z axes. Now, the velocity of that
differential mass with respect to the center of mass should be written as:
!
!
 d!
r
C!

!b x!
r
ım
V ım D ım (2.8)
dt
b

Since the aircraft is assumed to be rigid, !

r is constant in time and, so:
ım

!
d!
r
ım
D0 (2.9)
dt
b

Therefore:
ˇ ˇ
ˇ! !
 !
 ˇ
!
 ˇ i j k ˇ
ˇ ˇ
V ım D !

!b x!
r
ım Dˇ p q rˇ (2.10a)
ˇ ˇ
ˇ x y zˇ
!
 !
 !
 !

) V ım D .qz  ry/ i C .rx  pz/ j C .py  qx/ k (2.10b)

Furthermore, an expression for the angular momentum of differential mass,

!

ı H ım , can be derived as:
!
   
r x ım!
ı H ım D ! V ım (2.11)
ım

Hence, the three terms of the angular momentum can be obtained as:
2 3 ˇˇ !
 !
 !
 ˇ
ˇ
ıHx ˇ i j k ˇ
!
 ˇ ˇ
ı H ım D 4 ıHy 5 D ˇ x y z ˇ (2.12a)
ˇ ˇ
ıHz ˇ ım .qz  ry/ ım .rx  pz/ ım .py  qx/ ˇ
 
ıHx D p y2 C z2  ım  qxy ım  rxz ım
) ıHy D q x2 C z2 ım  ryz ım  pxy ım (2.12b)
 
ıHz D r x2 C y2 ım  pxz ım  qyz ım

After that, the expressions should be integrated over the whole aircraft to
determine the angular momentum equations of plane itself. As long as p, r, and
q are not dependent on mass, then:
2.1 Rigid Body Equations of Motion 13

• • • •
 2 2

Hx D ıHx D p y C z ım  q xy ım  r xz ım
• • • •
 
Hy D ıHy D q x2 C z2 ım  r yz ım  p xy ım (2.13)
• • • •
 2 2

Hz D ıHz D r x C y ım  p xz ım  q yz ım

It is known that moments of inertia are defined as:

•
 2 
Ixx D y C z2 ım (2.14a)
•
 
Iyy D x2 C z2 ım (2.14b)
•
 2 
Izz D x C y2 ım (2.14c)

and the products of inertia are:

•
Ixy D xyım (2.15a)

•
Iyz D yzım (2.15b)
•
Ixz D xzım (2.15c)

Consequently, if these moments of inertia and products of inertia terms are

substituted into Eq. (2.13), then:

Hx D pIxx  qIxy  rIxz

Hy D qIyy  rIyz  pIxy (2.16a)
Hz D rIzz  pIxz  qIyz

or in matrix notation:
2 3 2 32 3
Hx Ixx Ixy Ixz p
4 Hy 5 D 4  Ixy Iyy Iyz 5 4 q 5 (2.16b)
Hz  Ixz Iyz Izz r
14 2 Equations of Motion for a UAV

In general, the x–z plane of the aircraft is assumed to be a plane of symmetry and,
so:

Ixy D Iyz D 0 (2.17)

This simplifies the two parts of Eq. (2.16) and:

Hx D pIxx  rIxz
Hy D qIyy (2.18)
Hz D rIzz  pIxz

As the final stage, the time derivative of the angular momentum vector should
!
be determined. In a similar way to Eq. (2.2), the time derivative of H , referred to a
rotating body frame that has an angular velocity of !
! b , can be expressed as:
!
! !
!
!

C!

dH dH
D !b x Hb (2.19)
dt dt
i b

Since it is assumed that the mass and the mass distribution of the plane are
constant, the time moments of inertias and the products of inertias do not vary with
time:

IPxx D IPyy D IPzz D IPxz D 0 (2.20)

Apart from that point:

2 3
!
! pP Ixx  rPIxz
dH
D4 qP Iyy 5 (2.21)
dt
b rP Izz  pP Ixz

and it is obvious that:

ˇ ˇ
ˇ !
 !
  ˇ 2
! 3
ˇ i j k ˇ q .rIzz  pIxz /  rqIyy
!
 !
 ˇ ˇ
!b x Hb D ˇ p q r ˇ D 4 r .pIxx  rIxz /  p .rIzz  pIxz/ 5
ˇ ˇ
ˇ pIxx  rIxz qIyy rIzz  pIxz ˇ pqIyy  q .pIxx  rIxz /
(2.22)

These yields:
2   3
!
! !
! pP Ixx C qr Izz  Iyy .Pr C pq/ Ixz
dH dH !

D C!

! b x H b D 4 qP Iyy  pr .Izz  Ixx / C p2  r2 Ixz 5
dt dt  
i b rP Izz C pq Iyy  Ixx C .qr C pP / Ixz
(2.23)
2.2 Orientation and Position of an Aircraft 15

As a result, the three moment equations of the motion of the aircraft in the body
axis system can be written as:
 
L D pP Ixx C qr Izz  Iyy  .Pr C pq/ Ixz
 2 2

M D qP Iyy  pr .I
 zz  Ixx /C p  r Ixz
(2.24)
N D rP Izz C pq Iyy  Ixx C .qr C pP / Ixz

Here, L is the moment around the x-axis, M is the moment around the y–axis, and
N is the moment around the z-axis. Also, it is good to emphasize that the first terms
  
 pIxx ; qIyy ; rIzz , are related to the angular
of all three equation,  acceleration, while the
second terms, qr Izz  Iyy ; pr.Izz  Ixx / ; pq Iyy Ixx , denote
 gyro precision, and
  2 2
 
the third expressions, r C pq Ixz ; p  r Ixz ; qr C p Ixz , are coupling terms.
Therefore, in summary, the three force and three moment equations for an aircraft
are:

Fx D m .Pu C qw  rv/
Force Eq: )
Fy D m .vP C ru  pw/
Fz D m .wP C pv  qu/
 
L D pP Ixx C qr Izz  Iyy  .Pr C pq/ Ixz
Moment Eq: ) M D qP Iyy  pr .Izz  Ixx / C p2  r2 Ixz
 
N D rP Izz C pq Iyy  Ixx C .qr C pP / Ixz

2.2 Orientation and Position of an Aircraft

The equations of motion of an aircraft examined thus far are derived for the body
frame of the plane. However, the orientation and position of the aircraft cannot be
defined in a relatively moving frame and, instead, a fixed frame such as the Earth
axis system must be used.
Hence, it is necessary to introduce some methodology to relate these two axis
systems: the fixed inertial frame and the aircraft body frame. To achieve this, the
orientation of the aircraft can be described by three consecutive rotations, and these
angular rotations are called Euler angles [1], a concept developed by Leonhard Euler
to define the orientation of a rigid body in 3D Euclidean space.
For such a problem, the order of rotations is important and the 3-2-1 Euler angle
representation is used. That means:
• First, rotate about the z-axis through the yaw angle
• Then, as the second step, rotate about the y-axis through the pitch angle 
• And, finally, rotate about the z-axis through the roll angle of 
As a consequence, it is possible to associate flight velocity components relative
to the fixed reference frame and the velocity components in the body frame:
16 2 Equations of Motion for a UAV

Fig. 2.3 Euler angles, velocity components, and angular velocities for an unmanned aerial vehicle
(UAV)

2 dx 3 2 32 3
dt cc ssc  cs csc  ss u
4 dy 5 D 4 cs sss C cc css  sc 5 4 v 5 (2.25)
dt
dz
dt
 s sc cc w

Here, u, v, and w are velocity components in the body frame, dx , dy , and dz

dt dt dt
are
velocity components in the fixed inertial frame, and , , and  are Euler angles.
c() and s() are cosine and sine functions, respectively.
Per contra, by the use of Euler angles, angular rates (or angular velocities) of p,
q, and r may be related with the Euler rates of ,P P , and P (Fig. 2.3):
2 3 2 32 3
p 1 0 s P
4 q 5 D 4 0 c cs 5 4 P 5 (2.26a)
r 0 s cc P

or, inversely:
2 3 2 32 3
P 1 s tan  c tan  p
4 P 5 D 4 0 c s 5 4 q 5 (2.26b)
P 0 s sec  c sec  r

2.3 Small Perturbation Theory

Small perturbation theory is a technique for linearizing nonlinear equations. By

performing this mathematical method, it is possible to find approximate solutions
to problems that cannot be solved exactly.
2.3 Small Perturbation Theory 17

As long as the six equations of motion of the plane, as derived in the previous
section, are nonlinear and must be linearized for solving behavior problems of
aircraft, small perturbation theory must be used to progress. First of all, in order
to apply the method, it is assumed that the motion of the aircraft consists of small
perturbations about the steady flight condition [1]. Then, as a base for the theory,
small perturbation terms are added to the steady states:

u D u0 C u p D p0 C p ™ D ™0 C ™
v D v0 C v q D q0 C q  D 0 C  (2.27)
w D w0 C w r D r0 C r D 0 C 

If the reference flight condition is assumed to be symmetric, the propulsive forces

are assumed to remain constant and the x-axis is considered, as it coincides with the
direction of the aircraft’s velocity vector, then the perturbed states become:

u D u0 C u p D p ™ D ™0 C ™
v D v q D q  D  (2.28)
w D w r D r D

Here, u, v, and w are velocity components in the x, y, and z directions, p, q, and r

are angular velocities about the x, y, and z axes, and ™, , and are the pitch, roll,
and yaw angles. The subscript 0 indicates the steady states, where  is used for
perturbations.
Now, let’s handle the force equation in the x-direction as an example:

Fx D X D m .Pu C qw  rv/ (2.29)

Replace all of the states with perturbed ones:

X0 C X D m .Pu C qw  rv/ (2.30)

At level flight conditions, there is no acceleration and, so, X0 D 0. On the other

hand, if order of .q/ ; .w/ ; .r/ ; .v/ D ", neglect terms with the order of "2 .
Hence:

X D m .Pu/ (2.31)

If the same method is followed for all six equations of motion, linearized
equations may be obtained as follows:
2 3 2 3
X Pu
14
Y 5 D 4 vP  ru0 5 (2.32)
m
Z wP  qu0
18 2 Equations of Motion for a UAV

2 3 2 3
L Ixx pP C Izz rP
4 M 5 D 4 Iyy qP 5 (2.33)
N Izz rP C Ixz pP

Besides, X can be expressed by means of a Taylor series in terms of the

perturbation variables as:

@X @X @X @X
X D u C w C ıe C ıT (2.34)
@u @w @ıe @ıT

where ı e , ı T are control-related terms and stand for the elevator deflection and
change in thrust, respectively. Then, these two equations can be related as follows:

@X @X @X @X
m .Pu/ D u C w C ıe C ıT (2.35)
@u @w @ıe @ıT

If the same kind of association is built for all six linearized equations and, after
that, necessary simplifications are realized under the light of assumptions, the result
will be the longitudinal and lateral equations of motions for the aircraft, which is
also a consequence of small perturbation theory. To learn more about the technique,
[1] and [2] are good references.

2.4 Linearized Equations of Motion

In this final part of the chapter, first, linearized equations of motion of an airplane
are given in state space form and, then, by taking the specifications of the Zagi
unmanned aerial vehicle (UAV) [3] as a guideline, the equations are rewritten. Zagi
is a flying wing type radio-controlled aircraft that is popular amongst hobbyists.
At this point, it might be useful to indicate that we mainly take references [3–
5] as an essence to obtain scalar quantities of stability derivatives and, therefore,
stability coefficients. Hence, the values are accepted as true and used as they are in
general. Some small modifications are applied, like changing the initial conditions,
if necessary.

2.4.1 Equations in General

Linearized longitudinal equations of motion of the plane can be written in state

space form as:
2.4 Linearized Equations of Motion 19

2 3 2 32 3
Pu Xu Xw 0 g u
6 wP 7 6 07 6 7
6 7 6 Zu Zw u0 7 6 w 7
4 Pq 5 D 4 Mu C MwP Zu Mw C MwP Zw Mq C Mw u0 0 5 4 q 5
P 0 0 1 0 
2 3
Xıe XıT
 
6 Z ZıT 7 ıe
C4 6 ıe 7 (2.36a)
Mıe C MwP Zıe MıT C MwP ZıT 5 ıT
0 0

Following this, lateral equations of motion can be given as:

2 3 2 32 3 2 3
vP Yv Yp  .u0  Yr / g cos .0 / v 0 Yır
 
6 Pp 7 6 Lv 0 7 6 p 7 6 Lıa Lır 7
6 7 6 Lp Lr 76 7C6 7 ıa
4 Pr 5 D 4 Nv Np Nr 0 5 4 r 5 4 Nıa Nır 5 ır
P 0 1 0 0  0 0
(2.36b)

Here, ı e , ı a , and ı r are elevator, aileron, and rudder deflections, respectively,

ı T is the change in thrust, and Xu , Zu : : : Zıe : : : Yv , Lv : : : Lıa : : : are stability
derivatives.

2.4.2 Characteristics of the Chosen Zagi UAV

As an experimental platform, the Zagi UAV is chosen and, in the remainder

of the book, the proposed methods are demonstrated using the dynamics and
characteristics of this UAV (Fig. 2.4).

Fig. 2.4 The Zagi UAV used by the students of Brigham Young University [4]
20 2 Equations of Motion for a UAV

Table 2.1 Specifications of m 1.56 kg

the Zagi unmanned aerial
S 0.2589 m2
vehicle (UAV) [3]
b 1.4224 m
c 0.3302
0 m 1
0:1147 0 0:0015
B C
JD@ 0 0:0576 0 A kg:m2
 0:0015 0 0:1712

Table 2.2 Stability Longitudinal coefficients Lateral coefficients

coefficients of the Zagi UAV
[3] CL0 D 0:28 CY70 D 0
CD0 D 0:03 Cl0 D 0
Cm0 D 0 Cn0 D 0
CL˛ D 3:45 CYˇ D 0:98
CD˛ D 0:30 Clˇ D 0:12
Cm˛ D 0:38 Cnˇ D 0:25
CLq D 0 CYp D 0
CDq D 0 Clp D 0:26
Cmq D 3:6 Cnp D 0:022
CLıe D 0:36 CYr D 0
CDıe D 0 Clr D 0:14
Cmıe D 0:5 Cnr D 0:35
CYıa D 0
Clıa D 0:08
Cnıa D 0:06
CYır D 0:17
Clır D 0:105
Cnır D 0:032

In Table 2.1, m is the mass, S is the wing area, b is the span length, c is the mean
chord length, and J is the inertia matrix of the plane. Notwithstanding, the stability
coefficients of the Zagi UAV, which are used in order to obtain stability derivatives,
may be tabulated as in Table 2.1.

2.4.3 Linearized Equations of Motion for the Zagi UAV

In this last step, stability derivatives calculated for the Zagi UAV, in the light of
Tables 2.1 and 2.2 and related equations from [5], are substituted in the linearized
lateral and longitudinal equations of motion. However, before that process, the
2.4 Linearized Equations of Motion 21

equations are modified in such a way that observing the height of the UAV is also
possible and the sideslip angle is used instead of the starboard velocity v:
2 3 2 32 3
Pu Xu Xw 0 g 0 u
6 w 7 6 0 07 6 7
6 7 6 Z u Z w u 0 7 6 w 7
6 7 6 76 7
6 Pq 7 D 6 Mu C MwP Zu Mw C MwP Zw Mq C Mw u0 0 0 7 6 q 7
6 P7 6 76 7
4  5 4 0 0 1 0 0 5 4  5
hP 0 1 0 u0 0 h

2 3
Xıe XıT
6 ZıT 7
6 Zıe 7 
6 7 ıe
C 6 Mıe C MwP Zıe MıT C MwP ZıT 7
6 7 ıT
4 0 0 5
0 0
(2.37a)
2 3 2 3
2 3 Yˇ Yp u0  Yr g cos .0 / 2 3 Yır
ˇP 6 u0  7 ˇ 6
0
6 Pp 7 6 u0 u0 u0 7 6 w 7 6 u0 77
 
6 7 6L 0 7 6 7 C 6 Lıa Lır 7 ıa
4 Pr 5 D 6 ˇ
Lp Lr 74 5 6 7
4 Nˇ Np Nr 0 5 q 4 Nıa Nır 5 ır
P 0 1 0 0  0 0
(2.37b)

Here, h is the height of the plane and ˇ is the sideslip angle.

Therefore, as the calculated quantities are substituted into those equations,
specialized equations for the Zagi UAV will become:
2 3 2 32 3
Pu 0:3356 1:3181 0 9:80665 0 u
6 wP 7 6  1:7916 3:9003 9:8215 0 07 6 7
6 7 6 7 6 w 7
6 7 6 76 7
6 Pq 7 D 6 0:702 3:5375 11:392 0 0 7 6 q 7
6 P7 6 76 7
4  5 4 0 0 1 0 0 5 4  5
hP 0 1 0 9:8215 0 h

2 3
0:7436 6:8728
6 3:7855 0 7
6 7 
6 7 ıe
C 6 47:917 0 7
6 7 ıT
4 0 0 5
0 0
(2.38a)
22 2 Equations of Motion for a UAV

2 3 2 32 3
ˇP 0:1069 0:1962 1 0:9984 ˇ
6 Pp 7 6  1:2213 1:9155 1:0096 0 7 6 7
6 7 6 7 6 w 7
4 Pr 5 D 4 1:7255 0:0919 1:7198 0 5 4 q 5
P 0 1 0 0 
2 3
0 0:1855
 
6 8:348 0 7
C6 7 ıa (2.38b)
4 4:24 2:1272 5 ır
0 0

Note that it is also possible to show the equations in a joined way, as follows:
2 3 2 3
Pu 0:3356 1:3181 0 9:80665 0 0 0 0 0
6 wP7 6 0 7
6 7 6  1:7916 3:9003 9:8215 0 0 0 0 0 7
6 Pq 7 6 0:702 3:5375 11:392 0 0 0 0 0 0 7
6 7 6 7
6 P 7 6 7
6 7 6 0 0 1 0 0 0 0 0 0 7
6 7 6 7
6 hP 7 D 6 0 1 0 9:8215 0 0 0 0 0 7
6 7 6 7
6
6 ˇP 7
7 6
6 0 0 0 0 0 0:1069 0:1962 1 0:9984 7
7
6 Pp 7 6 0 0 0 0 0 1:2213 1:9155 1:0096 0 7
6 7 6 7
4 Pr 5 4 0 0 0 0 0 1:7255 0:0919 1:7198 0 5
P 0 0 0 0 0 0 1 0 0
2 3
0:7436 6:8728 0 0
6 3:7855 0 0 0 7
6 7
6 47:917 0 7
6 0 0 7 2 ı 3
6 7 e
6 0 0 0 0 76
6 7 6 ıT 7
7
C6 0 0 0 0 74
6 7 ıa 5
6 0 0 0 0:1855 7
6 7 ır
6 0 0 8:348 0 7
6 7
4 0 0 4:24 2:1272 5
0 0 0 0
(2.39)

By regarding the general representation of the state space form as:


X D AX C BU; (2.40)

where A is the system matrix, B is the control distribution matrix, U is the control
vector, and X is the state space vector, one can write the following:
References 23

2 3
0:3356 1:3181 0 9:80665 0 0 0 0 0
6  1:7916 3:9003 9:8215 0 0 0 0 0 0 7
6 0:702 3:5375 11:392 0 0 0 0 0 0 7
6 7
6 0 0 1 0 0 0 0 0 0 7
6 7
AD6
6 0 1 0 9:8215 0 0 0 0 0 77
6 0 0 0 0 0 0:1069 0:1962 1 0:9984 7
6 7
6 0 0 0 0 0 1:2213 1:9155 1:0096 0 7
4 5
0 0 0 0 0 1:7255 0:0919 1:7198 0
0 0 0 0 0 0 1 0 0
(2.41a)
2 3
0:7436 6:8728 0 0
6 3:7855 0 0 0 7
6 7
6 47:917 0 0 0 7
6 7
6 0 0 0 0 7
6 7
6 7
BD6 0 0 0 0 7 (2.41b)
6 7
6 0 0 0 0:1855 7
6 7
6 0 0 8:348 0 7
6 7
4 0 0 4:24 2:1272 5
0 0 0 0
 T
U D ıe ıT ıa ır (2.41c)

 T
X D u w q  h ˇ p r  (2.41d)

References

1. Nelson RC (1998) Flight stability and automatic control, 2nd edn. WCB/McGraw-Hill, Boston
2. Yechout TR (2003) Introduction to aircraft flight mechanics. AIAA Education Series, Reston
3. Matthews JS (2006) Adaptive control of micro air vehicles. M.Sc. thesis, Department of
Electrical and Computer Engineering, Brigham Young University, Provo, UT, USA
4. Christiansen RS (2004) Design of an autopilot for small unmanned aerial vehicles. M.Sc. thesis,
Brigham Young University, Provo, UT, USA
5. Vural Y (2007) Autopilot system design for a small unmanned aerial vehicle (in Turkish). M.Sc.
thesis, Department of Aeronautical Engineering, Istanbul Technical University, Istanbul, Turkey
Chapter 3
Navigation Systems for Unmanned Aerial
Vehicles

3.1 Two Main Categories in Navigation

It is possible to categorize navigation systems into two groups as follows [1]:

• Dead reckoning (DR) systems
• Positioning systems
The most developed version of navigation is the integration of two methods
optimally via a Kalman filter.
DR navigation systems determine the position of a vehicle by estimating the
range from a location using the velocity and direction data of the vehicle. They are
self-sufficient and independent from external systems. Some DR systems used for
aircraft are presented below:
• INS – inertial navigation system
• Doppler/attitude reference system
• Air data/attitude reference system
Positioning error increases with time for all DR navigation systems. Therefore, it
is required to correct the DR positioning error and update the system with accurate
positions from available positioning systems.
Positioning systems are navigation systems which generally locate on spacecraft
or the Earth itself and have transmitters. An aircraft’s receiver determines the
position by using transmitted signals via an auxiliary computer. Nowadays, the
Global Positioning System (GPS) is the most accurate and developed positioning
system. Other positioning systems are GLONASS, VOR/DME, TACAN, etc.

© Springer International Publishing Switzerland 2015 25

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_3
26 3 Navigation Systems for UAV

3.2 Inertial Navigation

An INS determines the movement and position of a vehicle according to a reference

inertial frame using the vehicle’s inertial reactions.
An INS has trios of accelerometers and gyroscopes which are perpendicular to
each other. Inertial navigation is done by measuring the acceleration of the vehicle
and integrating it twice to find the velocity and position. Therefore, position can be
found with respect to a specific center freely without any external signal and data.
Accelerometers are used to find the magnitude of acceleration according to a specific
reference coordinate system. Acceleration is a vector, so it has both magnitude
and direction. Gyroscopes are used to determine the direction of acceleration.
Gyroscopes are used to keep accelerometers on a specific coordinate system, called
the inertial space. The ring angles, which are the angles between the gyroscope’s
rings, determine the Euler angles, which show the angular orientation of the vehicle.
Thus, an INS provides both angular orientation and forward movement.
An INS has three main functions [2]:
1. Sensing
2. Computation
3. Output
Accelerometers and gyroscopes provide the sensing function. Acceleration and
angular velocity data from them are sent to the computer, which computes the
velocity, position, orientation or attitude, altitude, and range to the destination by
using these data.
If the real air velocity is supplied by the air data system, the INS can calculate
the velocity and direction of the wind. An output function sends the data to the flight
control system, engine control system, target sensors, and control display unit with
respect to features of the mission.
There are two types of INS:
1. Gimballed
2. Analytical (strapdown)
In gimballed systems (classical), accelerometers and gyroscopes are isolated
against rotation maneuvers, i.e., angular movements of the aircraft. Therefore, INS
devices stay constantly in a specific orientation according to the Earth or inertial
space. This case simplifies velocity and position calculations, and reduces the
requirement for gyroscopes, because, otherwise, gyroscopes have to measure great
rotation ratios.
In strapdown, acceleration and Euler angles are measured with respect to the
aircraft coordinate system, multiplied with a transfer matrix, and then transformed to
the main coordinate system. This procedure requires very fast computer processing,
so this system only started to come into use after the 1980s. These processes—angle
measurement and coordinate system transformation—must be done quickly by the
computer. The strapdown system does not require rings, engines, etc.
3.3 Inertial Measurement Unit 27

Typically, an INS is a self-contained system with high short-term stability and

is not influenced by interference. An INS is a standalone navigational system using
motion sensors to continuously keep track of the position, orientation, and velocity
of a vehicle [3].
The main advantages and disadvantages of an INS are as follows [3]:
Advantages of an INS:
An INS has many advantages in comparison with other navigation systems:
1. Position and velocity data are sudden and permanent. High data ratio and
bandwidth are obtained easily.
2. The device is satisfactorily self-sufficient, because it is based on acceleration
and angular change measurements. The system does not propagate radiation
and is not drawn to other systems.
3. Navigation air data supplied for whole latitudes, air conditions without ground
stations.
4. Position, total velocity, azimuth, vertical velocity, and altitude data are
obtained.
Disadvantages of an INS:
1. Position and velocity data corrupt over time. It is valid for both cases of
movement or stationary.
2. The device is expensive.
3. It must be tuned first. (First, the position must be entered in the system). So,
reference position data are required. Setting is difficult for moving vehicles.
4. The accuracy of the navigation data is related to the vehicle’s maneuvers.
Due to the characteristics of accelerometers and gyroscopes, velocity and
position data become corrupt over time. Also, system faults increase slowly over
time and they are infinite. Unless the fault is improved, an INS fault increases
significantly and the system will not be reliable. Therefore, an INS must be used
in conjunction with the other navigation systems to limit and damp the faults.

3.3 Inertial Measurement Unit

An INS contains an inertial measurement unit (IMU), including an accelerometer

and a gyroscope for all three axes, measuring the linear acceleration and angular
velocity of a vehicle with six degrees of freedom (6DOF). By processing the signals
from these sensors, it is possible to track the position and orientation of a vehicle.
Measurements of a vehicle’s specific forces and rotation rates are the basis of the
INS. Measurements can be obtained from triads of gyros and accelerometers that
create an IMU. Recent advancements in microelectromechanical systems (MEMS)
technology have enabled the production of low-cost inertial sensors. Therefore, the
application area of these sensors quickly expanded into the unmanned aerial vehicles
(UAVs) industry.
28 3 Navigation Systems for UAV

Inertial sensors are nonjammable, nonradiating, and self-consistent, so they can-

not be disturbed by any external factors and do not affect anything else surrounding
them. However, there can be errors corrupting useful data, even in the highest quality
MEMS inertial sensors which are used in UAVs. Regardless of whether the inertial
sensor error is caused by internal mechanical imperfections, electronics errors, or
other sources, the effect is errors in the indicated outputs of these devices. For the
gyros, the major errors are in measuring angular rates. For the accelerometers, the
major errors are in measuring acceleration [4]. For both instruments, the largest
errors are usually a bias instability [measured in deg/hr for gyro bias drift or micro g
(g) for the accelerometer bias] and scale factor stability [which is usually measured
in parts per million (ppm) of the sensed inertial quantity] [5].

3.3.1 Rate Gyros

Rate gyros measure angular velocities, in contrast to free gyros measuring attitude
angles. Rate gyros sense the vehicle’s angular rate relative to the inertial space [6].
These rate components are the craft angular rate relative to the Earth ! nb , an angular
rate as it moves about the spherical Earth ! en , and the angular rate of the Earth itself
! ie . The vector sum of these angular rates ! ib is given by:

!ibb D !ieb C !en

b
C !nb
b
(3.1)

There are three types of gyro technology used in today’s IMU systems:
• Ring laser gyro (RLG)
• Fiber optic gyro (FOG)
• MEMS
Most current RLG sensors are single DOF sensors requiring three mechaniza-
tions for an INS implementation. A single DOF RLG is shown schematically in Fig.
3.1. This figure illustrates a triangular version of the RLG. The gyro includes a laser
as a source, a closed-path cavity and mirrors at each intermediate corner in the path,
and an interferometer/photodetector. The operation of the gyro is based on optical
and electronic phenomena rather than mechanical phenomena [6, 7].
The Sagnac effect can cause a certain rate of rotation inducement with a small
difference in the time it takes light to traverse the ring in two directions. Small
separation can occur between the frequencies of the counterpropagating beams. Two
main patterns can be introduced: standing wave pattern within the ring and beat
pattern when those two beams are interfered outside the ring. The net shift of that
interference pattern follows the rotation of the unit in the plane of the ring. At very
slow rotation rates, RLGs can have the “lock-in” condition. However, they are more
accurate than mechanical gyroscopes in most cases. Also, the frequencies of the
counterpropagating laser modes get closer and become almost identical if the ring
laser is hardly rotating.
3.3 Inertial Measurement Unit 29

Mirror Sensitive Axis

Laser

Interferometer
Mirror

Fig. 3.1 Ring laser gyro

On the other hand, the FOG is a developing gyro technology. FOGs, in

comparison to RLGs, require no mechanical dither for their operation and, thus,
eliminate a troublesome noise source. They do not require high voltage for the laser
plasma, reducing power consumption, and, with the exception of a laser diode for the
light source, are composed of passive optical components and, thus, yield extremely
high reliability compared to any other currently available technology [6].
As a “rule-of-thumb”, an INS equipped with gyros whose bias stability is
0.01 deg/hr will see its navigational error grow at a rate of 1 nmi/hr of operation
[8]. Solid-state inertial sensors, such as MEMS devices, have potentially significant
cost, size, and weight advantages. The MEMS and interferometry FOG (IFOG)
technologies are expected to replace many of the current systems using RLGs
and mechanical instruments. The performance of MEMS sensors is continually
improving, and they are currently being used in UAVs. The use of MEMS will
result in low-cost, high-reliability, small-size, and light-weight inertial sensors being
integrated into systems.

3.3.2 Accelerometers

Accelerometers are devices that measure acceleration along the sensing axis. The
measurement of acceleration can be integrated into an onboard computer used to
derive the UAV’s velocity and position. All accelerometers use the principle of
sensing the force on a loosely suspended mass, from which the acceleration may
be calculated. In comparison to gyros, they have more developed technology. The
accelerometer converts acceleration into an electrical signal.
Dynamic acceleration can be measured using an accelerometer. Dynamic accel-
eration is due to any force except for the gravitational force applied on a rigid body
30 3 Navigation Systems for UAV

Fig. 3.2 Principle of an

accelerometer: (a) without
acceleration, (b) with
acceleration

and the static acceleration (or gravitational acceleration) is due to the gravitational
force. The output of an accelerometer can be analog or digital. In the analog case,
the output voltage is directly proportional to the acceleration. On the other hand, the
output of a digital accelerometer can be directly accessed using protocols such as
serial interfaces [9]. The principle of an accelerometer is illustrated in Fig. 3.2a, b.
If the body accelerates in the direction indicated in Fig. 3.2, the mass displace-
ment with respect to the body of the device supported on a bar by springs can
be recorded. This displacement is proportional to the measured acceleration on
the sensing axis. Displacement of the mass generates an electrical signal. For this
purpose, potentiometric, self-capacitance converters can be used.
The accelerometer senses the UAV’s inertial dynamic acceleration ai , but is
insensitive to the gravitational acceleration. The gravitational field vector gm , which
is the acceleration of mass attraction to the Earth, should be taken into account
to determine the total acceleration. Therefore, the acceleration of the UAV can be
determined as:

a D ai  gm (3.2)

3.4 Air Data System

All of the air data parameters that are relevant to flight performance are derived
by sensing the pressures, temperatures, and flow directions surrounding the vehicle.
Free-stream pressures and temperatures are required for the computation of static
air temperature, altitude, airspeed, and Mach number [3].
An air data system provides information on quantities such as pressure, altitude,
vertical speed, calibrated airspeed, true airspeed, Mach number, static air temper-
3.4 Air Data System 31

ature, and air density ratio [1]. It is, thus, one of the key avionic systems in its
own right and forms part of the essential core of avionic subsystems required of all
modern aircraft, civil or military. In this section, a short description of the air data
system is given.

3.4.1 Air Data Measurements

Air data quantities, such as pressure, altitude, vertical speed, calibrated airspeed,
true airspeed, true Mach number, etc., are derived from three basic measurements
by sensors connected to probes which measure [1]:
• Total (or pitot) pressure
• Static pressure
• Total (or indicated) air temperature
A basic air data system scheme is presented in Fig. 3.3.
The total pressure PT is measured by means of an absolute pressure transducer
connected to a pitot tube facing the moving airstream. This measures the impact
pressure QC exerted to bring the moving airstream to rest relative to the pitot tube
plus the static pressure PS of the free airstream, i.e.:

P T D QC C P S : (3.3)

The static pressure of the free airstream PS is measured by an absolute pressure

transducer. The transducer is connected to a suitable orifice located where the sur-
face pressure is nearly the same as the pressure of the surrounding atmosphere [1].

Total Temperature Probe

Pressure Altitude HP
Tm Vertical Speed
HP
Pitot static probe
Total Pressure Sensor/
PT Transducer Calibrated Airspeed
VC
PT
AIR DATA
COMPUTER
Mach number M
PS PS
True Airspeed VT

Static Pressure Sensor/ Static Air Temperature

TS
Transducer
ASI ALT VSI
Air Density Ratio ρ / ρ0
Reversionary Pneumatic Instruments

Fig. 3.3 Basic air data system

32 3 Navigation Systems for UAV

High-performance military aircraft and UAVs generally have a combined

Pitot/static probe which extends out in front of the aircraft, so as to be as far away
as practicable from aerodynamic interference effect and shock waves generated by
the aircraft structure. Some civil transport aircraft have pitot probes with separate
static pressure orifices located in the fuselage, generally somewhere between the
nose and the wing. The exact location of the static pressure orifices (and the pitot
tubes or probes) is determined by experience and experimentation.
From the measurements of the static pressure and total pressure, it is possible to
derive the following quantities [1]:
1. Pressure altitude HP ; this is derived from the static pressure PS , measured by
assuming a “standard atmosphere”.
2. Vertical speed HP ; this is basically derived by differentiating PS .
3. Calibrated airspeed VC ; this is derived directly from the impact pressure QC ,
which is, in turn, derived from the difference between the total and static
pressures (QC D PT  PS ).
4. Mach number M; this is the ratio of the true airspeed VT to the local speed of
sound A, that is, M D VT /A, and is derived directly from the ratio of the total
pressure to the static pressure PT /PS . (True airspeed is defined as the speed of the
aircraft relative to the air).
A flow diagram of the air data computation procedures, which are explained in
detail in [1], is given in Fig. 3.4.

H
ΔPT
Compute Pressure
Error Corrections
M

ΔPS PS Pressure Altitude

Compute Pressure
Altitude HP
PS Correct PS
PS
Static Pressure PT QC Calibrated Airspeed
Compute Compute Calibrated

Correct PT QC = PT − PS Airspeed VC

PT PT
PS PT / PS Mach Number M
Compute
Compute Mach No.
Total Pressure PT / PS
PS
Ti True Airspeed
Compute Static Air Compute True
Ti Temp Airspeed
VT

Total/Indicated Air Recovery Ratio r

Static Air Temperature
Temperature T

Fig. 3.4 Air data computation flow diagram

3.5 Surface Radar 33

3.4.2 Derivation of the True Airspeed Components

The information obtained from the air data system is as follows: true airspeed VT ,
angle of attack ˛, and sideslip angle ˇ. Using these data, the X, Y, and Z components
of the true airspeed can be obtained [10]:
q
VT D Vx2 C Vy2 C Vz2 ; (3.4)


Vz
˛ D arctan  ; (3.5)
Vx


Vy
ˇ D arcsin ; (3.6)
VT

where:

Vx D VT cos ˛ cos ˇ
Vy D VT sin ˇ (3.7)
Vz D VT sin ˛ cos ˇ

These speeds are derived according to the aircraft reference system. Later
on, these coordinates can be converted to a ground-based reference system for
navigational usage.

3.5 Surface Radar

A ground- or surface-based radar station may be used to provide line-of-sight

observations of a UAV during flight. These observations usually take the form of
measurements of a distance between the UAV and ground radar (range), elevation,
and bearing. Figure 3.5 illustrates these navigation parameters geometrically in the
Cartesian coordinate system.
The measurements of a UAV’s range (D), azimuth (˛), and elevation (ˇ) may
be expressed in terms of the Cartesian position coordinates of the UAV (x, y, z) as
follows:
p
D D x2 C y2 C z2 (3.8)
nyo
˛ D tan1 (3.9)
x
( )
1 z
ˇ D tan p (3.10)
x2 C y2
34 3 Navigation Systems for UAV

Fig. 3.5 Range, elevation,

and bearing observations of
an unmanned aerial vehicle
(UAV)

Measurement of angle and measurement of range methods are used at the same
time in the surface radar [11]. An integrated angle/range measurement method
is usually used in radar systems, and it determines the range D of the UAV,
azimuth angle ˛, and elevation angle ˇ. When this method is used, the coordinates
of the UAV is determined as the intersection point of the sphere state surface
(D D constant), cone state surface (ˇDconstant), and vertical plane suitable for
˛Dconstant state surface. UAV coordinates are determined with a single point
(ground station) by the help of this method and does not require difficult calcula-
tions. The following formulas are used to calculate the UAV’s Cartesian coordinates:

x D D cos ˇ cos ˛
y D D cos ˇ sin ˛ (3.11)
z D D sin ˇ

Range, azimuth, and elevation angles are determined by radar measurements.

A single radar station is sufficient to use this method.
Advantages of the angle/range integrated method:
• A single station is needed to measure the coordinates
• Coordinates are calculated as simple
• The method provides the required accuracy in whole measurement intervals
• Data processing is easy in the system
Disadvantage of the angle/range integrated method:
• The accuracy of the UAV coordinate measurement decreases with distance
away from the surface radar
3.6 Altitude Measurements 35

3.6 Altitude Measurements

3.6.1 Types of Flight Altitudes

An altimeter is a device that measures the flight altitude of aircraft on the Earth.
Three types of altitudes are considered during flight:
• Absolute altitude – flight altitude according to sea level
• Relative altitude – flight altitude with respect to the takeoff or landing site
• Real altitude – altitude over the flight area
Descriptions of the altitudes are given in Fig. 3.6.
The value of absolute altitude is required for setting corridors to flight routes and
flight tests of aircrafts and engines.
The relative altitude must be known for taking off and landing.
The real altitude is for arbitrary flight status.

3.6.2 Radio Altimeter

The measurement of flight altitude via radiotechnical methods depends on mea-

suring the time interval elapsed during the signal’s path from the aircraft to the
Earth [3]. Two antennas are installed in the aircraft in order to use a radiotechnical
method – transmitter A1 and receiver A2 . Antennas are often placed on the wings of
aircraft (Fig. 3.7a).
Transmitter antenna A1 sends out radio waves, which are reflected when they
reach the Earth’s surface. Some of the reflected waves are received by the aircraft’s
receiver antenna, A2 . The time duration between the signal’s transmission to Earth
and its return is determined as presented below:

Real altitude
Relative
altitude Absolute
altitude

Sea level

Take off
position

Fig. 3.6 Flight altitude types

36 3 Navigation Systems for UAV

Fig. 3.7 Radio altimeter: A1 ,

A2 – transmitter and receiver
antennas

2H
t1 D (3.12)
c

where H is the flight altitude and c D 3 108 m/s is the propagation velocity of radio
signals.
The registration of two signals’ receiving times is used to measure the propaga-
tion time of radio signals: the signal from the transmitter and the reflected signal.
The first signal reaches the receiver antenna after time t2 D l=c and the second after
time t1 D 2H=c. At that time, the real altitude of the flight will be equal to:

.c C l/
HD ; (3.13)
2
where  D t1 – t2 and l is the range between the transmitter and receiver antennas.
Expression (3.13) is used to steady radio altimeters or radio altimeters with
impulse teleportation.
When high altitudes are measured .H  1; 500 m/, the power of the transmitted
signal decreases significantly, so, for such altitudes, impulse teleportation must be
run after steady teleportation.
An altimeter’s receiver takes the reflected impulses from the aircraft and the
Earth and, after strengthening the impulses, they are passed to an electron gun (see
Fig. 3.7b). A circular movement with angular velocity ! is acted upon the beam to
find the first impulse towards the screen and the reflected second impulse. The angle
between the impulses ® (Fig. 3.7b) is calculated via the following formula:

2H!
' D ! D (3.14)
c
For increasing the measurement accuracy, the angular velocity ! of the beam
must be increased. In this case, the height of the device’s whole pointer must be
appropriate for the measurement interval.
3.6 Altitude Measurements 37

3.6.3 Barometric Altimeter

In airborne vehicles flying at low altitudes, it is very important to provide accurate

altitude information continuously. For this purpose, barometric (baro) altimeters
are commonly used nowadays. They can measure the atmospheric pressure value,
according to the relationship between the height and the air data, such as pressure
and temperature, and indirectly calculate the height. As is known [12], the baro
altimeter is an excellent sensor for vertical height measurement during straight-and-
level flight segments and provides short-term accurate vertical altitude information.
In general, barometers are initialized using the height, temperature, and pressure at
the starting point of the vehicle in which the barometer is installed. However, as
time elapses and as the aircraft moves to another place, the characteristics of the
atmosphere around the vehicle are changed, which produces large barometric errors
[13]. To surmount this problem, a baro altimeter must be aided by other altitude
sensors, such as GPS, radio altimeter, etc.
Altitude is determined via the barometric altitude measurement equation. Assum-
ing that air is an ideal gas and that the gradient of temperature as a function of
altitude (i.e., the atmospheric lapse rate) is known, then the altitude can be computed
via the following expression [14]:
2 3

 Tgrad R
T0 p g
hB D 41  5 (3.15)
Tgrad p0

where hB is the barometric altitude, T0 is the temperature at sea level, Tgrad is the
lapse rate, p0 is the pressure at sea level, p is the pressure measured at altitude hB , R
is the universal gas constant, and g is the local gravity.
The variation of atmospheric pressure with height is given in Fig. 3.8.

1
Atmospheric
Pressure 0.8
(x105 Pa)
0.6

0.4

0.2

0
0 2 4 6 8 10
Altitude (km)

Fig. 3.8 Variation of atmospheric pressure with altitude

38 3 Navigation Systems for UAV

The conversion of the measured air pressure to altitude given in Eq. (3.15) is
based on a theoretical standard atmosphere and the assumption that air is an ideal
gas. More precisely, the standard atmosphere is defined as follows [15]:
• Air is an ideal gas with gas constant R D 287 J= .Kg  K/
• The pressure at sea level is p0 D 29:92 in:  Hg
• The air temperature at sea level is T0 D 15 ı C
• The temperature gradient (lapse rate) is Tgrad D 0:0065 ı C=m
The pressure at sea level p0 is not constant but varies from day to day and location
to location. In practice, therefore, a value of p0 which makes the altitude estimated
using (1) correct is input into the altimeter. This value of p0 is known as the altimeter
setting.
The main source of error in barometric altitude measurement arises from dif-
ferences between the true and modeled atmospheric temperature and pressure. For
standalone barometry, altitude errors can be several hundred meters. For differential
barometry, the error increases with distance from the reference station and age of
the calibration data [16].

3.7 Speed-over-ground and Drift Angle Measurements via

the Doppler Method

Doppler radar provides speed-over-ground measurement of aircraft. In addition,

the aircraft drift angle between the velocity vector W and its component in the x
direction Wx can be measured by the Doppler method. A Doppler radar operates by
transmitting a narrow beam of microwave energy to the ground and measures the
frequency shift that occurs in the reflected signal as a result of the relative motion
between the aircraft and the ground [16].
Speed-over-ground (W) and drift angle (ˇ) can be measured by the use of the
Doppler method. The method depends on the transmission of definite frequency
signals and receiving the reflected signals with altered frequency. The change
infrequency is related to the flight velocity. The frequency change caused by the
Doppler effect is measured by a device calibrated by velocity units.
Assume that transmitter and receiver systems exist in the aircraft with W total
velocity. Radio signals with frequency f sent by the transmitter arrive at the Earth’s
surface and are then reflected and picked up by the aircraft.
The frequency of the signals observed by the aircraft’s receiver is f1 D f C 2FD ,
where FD D .W=/ cos is the Doppler frequency drift, œ is the wavelength, and ”
is the inclination angle of the beam.
The error of the method is increased by inaccurate filtration of the signal and the
roll angle of the aircraft. For instance, when D 70ı and the roll angle is 1ı , the
error of the velocity measurement is 4.7 %. Compensation methods presented below
are used to reduce the error [11]:
3.7 Speed-over-ground and Drift Angle Measurements via the Doppler Method 39

Fig. 3.9 Doppler velocity

measurement: scheme of a
measurement system with
four beams

1. Stabilization of antennas according to the Earth’s vertical axis

2. Consideration of corrections according to rolling analytically
3. Application of multibeam systems
The first two methods do not provide measurements of high accuracy. When
the third method is used in case of the determination of two beams symmetrically
according to the vertical, it is possible that the measurement error of the velocity is
decreased to 0.1 % in 1ı rounding.
In a Doppler velocity and drift angle measurement device, a four-beam antenna
is used (Fig. 3.9).
In such a system, four antennas send radio signals to areas 1, 2, 3, and 4 on Earth.
Therefore, the sum of Doppler frequencies of two pairs of channels, (1–3) and (2–4),
are, respectively:
  
fd.1C3/ D 4 W  cos cos . C ˇ/
   (3.16)
fd.2C4/ D 4 W  cos cos .  ˇ/

By calculating the sum and subtracting the divided frequencies, we can write:

fd.1C3/ C fd.2C4/ D k1 W cos ˇ

(3.17)
fd.1C3/  fd.2C4/ D k2 W sin ˇ

Here, k1 D .8=/ cos cos  and k2 D .8=/ cos sin .

The total flight velocity W and drift angle ˇ are determined via Eq. (3.17).
When the antenna system is rotated by a certain drift angle, the bisector of 2
angle of the system will be in the direction of the total flight velocity vector, so the
drift angle will be equal to zero. It is clear that the equations:

fd.1C3/ C fd.2C4/ D k1 W
(3.18)
fd.1C3/  fd.2C4/ D 0

are valid.
40 3 Navigation Systems for UAV

The second equation is the difference signal. This signal is used in schemes which
determine W and ˇ automatically.
The standard deviations of Doppler measuring devices is 0.2 % in respect of the
velocity and 0.1 % in respect of the drift angle.
Typically, a Doppler sensor operating over land is able to provide measurements
of velocity to an accuracy of about 0.25 % or better. Performance is degraded during
flights over water owing to poor reflectivity, scattering of the reflected signal, giving
rise to a bias in the measurement, wave motion, tidal motion, and water currents.
However, this navigation aid offers good long-term stability and a chance to bind
the position and velocity estimates provided by an inertial navigation system [16].

3.8 Magnetic Measurements

The variations in the Earth’s geomagnetic field can be internal, which are related to
the dynamo within the Earth’s core and crust. Inner dynamo variations, which are
long-term variations within the core, are called secular variations. On the other hand,
solar wind–geomagnetic dipole interaction variations are relatively short-duration
variations on the order of months, days, and hours. The Earth has a magnetic field
similar to that of a bar magnet, with poles located close to its axis of rotation. This
means that the direction of the horizontal component of the Earth’s magnetic field
lies close to true north, and the magnetic north, as determined by a magnetic field
sensor, or compass, can be used as a working reference [16]. The angle between
true north and the magnetic north is not constant. It varies with the position of the
observation on the Earth and slowly with time, although it is possible to compensate
for this. The direction of the Earth’s magnetic field at any point on its surface is
defined in terms of its orientation with respect to true north, known as the angle
of “magnetic declination”, and its angle with respect to the horizontal is called
the angle of “dip”. These angles are illustrated in Fig. 3.5. Magnetometers are
widely used in aerospace as attitude determination devices. These sensors provide
the direction of the Earth’s magnetic field and its magnitude; therefore, they are
useful for determining the attitude of flying vehicles. Moreover, their low mass and
low power consumption make them attractive for UAVs. For these reasons, most
UAVs have magnetometers as part of their basic sensor package.
In the last decade, the popularity of UAVs, due to their low cost and weight, has
increased significantly. That has brought about a search for lighter but more accu-
rate sensors. Under these circumstances, the three-axis magnetometer (TAM) has
become very attractive because of its advantages, such as providing continuously
available two-axis attitude measurements, relative low cost, and almost insignificant
power demand (Fig. 3.10).
In the absence of local magnetic disturbances, magnetometers sense the compo-
nents of the Earth’s magnetic field (H0 ) acting along its sensitive axis. A TAM may
be mounted in a UAV to sense the components of the Earth’s magnetic field along its
principal axes, (Hx , Hy , Hz ). Expressing a UAV’s attitude, with respect to the local
3.9 Satellite Radio Navigation 41

Fig. 3.10 Angle of

“magnetic declination” and
angle of “dip”

coordinate frame, can be done as a direction cosine matrix, Cbn . The relationship
between the magnetic measurements and the attitude of the UAV may be written as:
2 3 2 3
Hx H0 cos ı cos
4 Hy 5 D Cnb 4 H0 cos ı sin 5 ; (3.19)
Hz H0 sin ı

where ı is the angle of dip and ” is the angle of declination. If the angle of dip is
known, as well as the variation between true and magnetic norths, estimates of a
UAV’s attitude can be made from the magnetometer measurements.
The aircraft in which the magnetic sensor is mounted will almost always have a
magnetic field that cannot be distinguished from that of the Earth’s magnetic field
and, consequently, it is necessary to compensate for this effect as well. Therefore,
in order to estimate the components of the geomagnetic field vector for determining
the attitude of a UAV accurately, the magnetometer should be calibrated precisely.

3.9 Satellite Radio Navigation

The Global Positioning System (GPS) is a space-based global navigation satellite

system developed by the USA. The GPS provides reliable location and time
information in all weather conditions, at all times, and anywhere. The system uses a
42 3 Navigation Systems for UAV

total of 24 satellites. Four satellites are necessary to calculate the position of the user.
There are also other similar systems used or planned to be used, such as GLONASS
(Russia) and Galileo (European Union).
In a GPS system, the GPS receiver chooses the best available satellites to obtain
the measurements. The system is available as two services, a standard positioning
system (SPS) for civilians and a precise positioning service (PPS) for military users.
The order of accuracy of SPS is 10 m and that of PPS is around 3 m.

3.9.1 GPS Structure

The GPS consists of three parts: the space segment, the control segment, and the
user segment [3]. The structure of the GPS is shown in Fig. 3.11.
The U.S. Air Force develops, maintains, and operates the space and control
segments. GPS satellites broadcast signals from space, which each GPS receiver
uses to calculate its three-dimensional location (latitude, longitude, and altitude)
plus the current time.
The space segment is composed of 24–32 satellites in medium Earth orbit and
also includes the boosters required to launch them into orbit. The control segment
is composed of a master control station, ground antennas, and monitor stations. The
user segment is, in our case, composed of UAVs.

Fig. 3.11 Satellite radio navigation system structure

3.9 Satellite Radio Navigation 43

3.9.2 Basic Concept of GPS

A GPS receiver calculates its position by precisely timing the signals sent by the
GPS satellites. Each GPS satellite continually transmits messages, which include:
• The time the message was transmitted
• Precise orbital information (the ephemeris) of satellites
• The general system health
The receiver utilizes the messages it receives to determine the transit time of
each message and computes the distances to each satellite. These distances, along
with the satellites’ locations, are used to compute the position of the receiver. This
position is determined as the geographical coordinate system latitude, longitude,
and elevation. Many GPS units give derived information, such as the direction and
speed of the user, calculated from position changes.
A very small clock error, which is the discrete time scale of the user and the
satellite system multiplied by the very large speed of light—the speed at which satel-
lite signals propagate—results in a large positional error (clock bias). Therefore,
receivers use four or more satellites to solve for the four-parameter determination:
clock bias and receiver’s location as latitude, longitude, and elevation.
Today, satellite navigation systems are used for solving the navigation problems
of most of UAVs. In this technique, the position of the object is generally detected
with the help of distance measuring methods (distances from object to n  4
satellites are measured) (see Fig. 3.12).
Then, the coordinates of the object will be on the intersection of the sphere
surfaces with radiuses equal to the measured distances; see [3, 11, 17]. The
equation system that consists of n  4 nonlinear equations, which are obtained by
this approach, can be solved using a computer with the aid of iteration methods.
However, the distance equations used for formerly stated purposes have a term
other than the unknown coordinates of the user, which results from the discrete

Fig. 3.12 Scheme of locating an object using distance measuring

44 3 Navigation Systems for UAV

time scale of the user and the satellite system [3, 18]. In this case, this term and the
user coordinates are determined as the intersection point of the n different sphere
position surfaces (R D constant). In a simple case, the user coordinates and clock
bias can be obtained by solving the four nonlinear algebraic equations below:

.xi  x/2 C .yi  y/2 C .zi  z/2 D .Di  ıt/2 i D 1; 4 (3.20)

where xi , yi , and zi are the Descartes coordinates of the ith satellite, in the geocentric
axis system, 4 is the minimum number of satellites, x, y, and z are the Descartes
coordinates of the object (users) in the geocentric coordinate system, and ıt is the
distance measuring errors, resulting from the time differences between the user and
the satellite (clock bias).
UAVs are now able to work in different environments easily using a GPS receiver.
This allows the use of MALE UAVs (a medium-altitude, long-endurance UAV) and
HALE UAVs (a high-altitude, long-endurance UAV) in every part of the world.
However, in some places, by jamming or with local conditions, GPS receivers may
receive weak signals. Thus, depending solely on GPS may cause problems in the
navigation of UAVs. To overcome this problem, dead reckoning systems can be
used. A filter, such as a Kalman filter, may be used to integrate the measurements of
GPS with dead reckoning devices to obtain better measurements.

3.10 Vision-based Systems

Visual sensing can also provide data for position estimation. These kinds of systems
have the advantage of using visual sensors that actually sense the real physical data
to comprehend the results in an easier way. The data from the visual sensors can also
be combined with the data from INS, GPS, and magnetometers in order to supply a
better understanding to the operator. Improved digital imaging techniques make the
use of visual sensors in small UAVs navigation more practical and simpler. Also,
using methods for emulating flying insects is a new approach of developed systems
in visual sensing. One method is the use of a newly developed artificial compound
insect eye, which is a system that replicates the eye structure of an insect. This
system uses many small-sized lenses with hemispherical shapes. The camera chips
in this system are made of hard and soft composite materials, which allow stretching
and bending, therefore acting like an insect’s eye, as desired [19–21].

3.11 Simultaneous Localization and Mapping (SLAM)

SLAM algorithms are used to develop systems that are terrain aided. These are used
for map building. Using generated maps, these systems help other sensors. A few
of these systems are used on robotic systems that depend on visual sensors. Visual
motion estimation techniques that use stereovision and visual features tracking have
3.12 Measurement Fault Classification and Fault Modeling 45

been proposed in the context of ground rovers [22]. Stereovision and monocular
systems are also used in different studies [23]. These types of systems allow UAVs
to operate in all environments, including those in which they face GPS issues.

3.12 Measurement Fault Classification and Fault Modeling

A “fault” is to be understood as an unexpected change of any system function,

although it may not represent physical failure. Such a fault or malfunction hampers
or disturbs the normal operation of the system, thus causing an unacceptable
deterioration in performance or even leading to dangerous situations [23].
“A fault is an unpermitted deviation of at least one characteristic property
(feature) of the system from the acceptable, usual, standard condition” [24].
Based on this definition, a fault corresponds to an abnormal behavior of the
system, which may not affect its overall functioning, but may eventually lead to
a failure.
A fault may be small or hidden, and, therefore, difficult to detect, isolate, and
identify.
“A failure is a permanent interruption of a system’s ability to perform a required
function under specified operating conditions” [24].
Resulting from one or more faults, a failure of the device is, therefore, an event
that terminates the functioning of a device.
In this book, we use the term “fault” rather than “failure” to denote a malfunction
rather than a catastrophe. The concept of failure is known as complete breakdown of
a system component or function, whilst the concept of fault may be used to indicate
that a malfunction may be tolerable at its current step. A fault must be diagnosed as
early as possible to prevent any serious consequences.
A sensor fault occurs as soon as the measurement data deviate from the physi-
cally measured process by more than the noise uncertainty. Bias, noise increment,
or wrong scaling factors are also classified as a sensor fault. The possible sensor
faults are given in Fig. 3.13.
The mathematical representation of the above sensor faults is as follows [25]:
8
ˆ
ˆ xi .t/; 8t  t0  No: failure
ˆ
ˆ
ˆ
ˆ
ˆ xi .t/ C bi ; bP i .t/ D 0; bi .tFi / ¤ 0  Bias
<
xi .t/ C bi .t/; jbi .t/j D ci t; 0 < ci << 1; 8t  tFi  Drift
yi .t/ D
ˆ
ˆ xi .t/ C bi .t/; jbi .t/j  bi ; bP i .t/ 2 L1 ; 8t  tFi  Loss: of: accuracy
ˆ
ˆ
ˆ
ˆ xi .tFi / ; 8t  tFi  Sensor: freezing
:̂
ki .t/xi ; 0 < k  ki .t/  1; 8t  tFi  Calibration: error

Here, tFi is the time-off fault

 occurrence
on the ith sensor and bi is its accuracy

coefficient, such that bi 2 bi ; bi , where bi >0. Furthermore, ki 2 ki ; 1 , where
ki > 0, denotes the minimum sensor effectiveness, as can be seen. Except the
freezing case, the following general mathematical model can be represented:
46 3 Navigation Systems for UAV

y D Km x C B; (3.21)

where K  m is a positive-definite diagonal matrix whose elements

 are
slowly varying
within ki ; 1 and elements of vector B slowly vary within bi ; bi [26].
If the parameter is observed incase of measurement noise increment, then the
measurement equation can be presented in the form:

yi .t/ D xi .t/ C
i .t/vi .t/ (3.22)

where vi (t) is the random measurement noise and

i .t/ > 1 characterizes the
measurement noise increment. This type of sensor fault is presented in Fig. 3.13f.
The measurement noise corresponding to normal operation of the sensor in the
figure is shown as the red line, and the measurement noise increment is shown by
the black line.

a b
Actual Measured Actual Measured
y (t) y (t)

time time

c d
Actual Measured Actual Measured
y (t) y (t)

tF
tF

time time

e f
Measurement Noise Increment
Actual Measured
Measurement Noise
y (t)

v (t)

time

x(t)

Fig. 3.13 Several types of sensor faults: (a) sensor bias, (b) sensor drift, (c) loss of accuracy, (d)
frozen sensor (occurring after tF ), (e) calibration error, and (f) noise increment
3.12 Measurement Fault Classification and Fault Modeling 47

For example, all types of accelerometers and gyros exhibit biases, scale factor
and misalignment errors, and random noise to a certain extent [27]. The bias is a
constant error which manifests in the measurement results. Mostly, accelerometers
and gyros have bias errors. It is independent of the underlying specific force and
angular rate.
The scale factor error is the departure of the input–output gradient of the
instrument from unity following unit conversion by the IMU. The accelerometer
output error due to the scale factor error is proportional to the true specific force
along the sensitive axes, while the gyro output error due to the scale factor error is
proportional to the true angular rate about the sensitive axis [27].
Misalignment errors in all types of IMUs arise from the misalignment of the
sensitive axes of the inertial sensors with respect to the body frame orthogonal axes
because of manufacturing limitations.
To give an example regarding this subject, the reasons for INS sensor faults may
be listed as follows:
• Gyroscope drift faults caused by temperature changes, acceleration, magnetic
field, and vibration
• Gyroscope measurement faults with nonlinearity and asymmetry
• Accelerometer bias because of temperature changes and vibrations
• Accelerometer measurement faults with nonlinearity and asymmetry
• The input axes of gyroscopes and accelerometers are not perpendicular to each
other
• Initial state faults (must be considered that position and velocity errors are
proportional to time) etc.
Sensor faults may cause substantial performance degradation of a decision-
making system. For example, in a UAV flight control system, sensors are used
to measure directly the UAV states. Thus, the presence of faults in sensors may
deteriorate state estimates and, consequently, cause inaccurate flight control.
A fault monitoring system which is used to detect faults and diagnose their
location and significance within a system is called a “fault diagnosis system”. Such
a system normally consists of the following tasks [28]:
• Fault detection: to make a binary decision – either that something has gone wrong
or that everything is fine
• Fault isolation: to determine the location of the fault, e.g., which sensor or
actuator has become faulty
• Fault identification: to estimate the size and type or nature of the fault
A fault-tolerant flight control system is capable of controlling UAV dynamics
with satisfactory performance, even if there is one or more faults.
The fault-tolerant state estimation system of UAVs in this book is capable
of detecting and diagnosing faults in the UAV control system and adequately
compensate for failures.
48 3 Navigation Systems for UAV

References

1. Collinson RPG (1996) Introduction to avionics. Chapman & Hall, London

2. Lin C-F (1991) Modern navigation, guidance, and control processing. Prentice Hall, Engle-
wood Cliffs
3. Kayton M, Fried WR (1997) Avionics navigation systems. Wiley, New York
4. Prestero T (2001) Verification of a six-degree of freedom simulation model for the REMUS
autonomous underwater vehicle. M.Sc. thesis in Ocean and Mechanical Engineering, MIT,
Cambridge, MA
5. Stutters L, Liu H, Tiltman C, Brown DJ (2008) Navigation technologies for autonomous
underwater vehicles. IEEE T Syst Man Cybern Part C Appl Rev 38(4):581–589
6. Titterton DH, Weston JL (2004) Strapdown inertial navigation technology, vol 207, Progress
in astronautics and aeronautics. MIT Lincoln Laboratory, Lexington
7. Volk C, Lincoln J, Tazartes D (2008) Northrop Grumman’s family of fiber-optic based inertial
navigation systems, USA. http://www.nsd.es.northropgrumman.com. Date retrieved 19 Jan
2012
8. Schmidt GT (2007) INS/GPS technology trends. NATO SET-104 symposium, Antalya
9. Bekir E (2007) Introduction to modern navigation systems. World Scientific, Hackensack
10. Vasil’chenko KK et al (1996) Flight tests on aircraft. Mashinostroenie, Moscow (in Russian)
11. Hajiyev C (1999) Radio navigation. Istanbul Technical University, Istanbul (in Turkish)
12. Tang W, Howell G, Tsai Y-H (2005) Barometric Altimeter short-term accuracy analysis. IEEE
Aerospace and Electronic Systems Magazine, December, pp 24–26
13. Whang I-H, Ra W-S (2007) Barometer error identification filter design using sigma point
hypotheses. In: Proceedings of the international conference on control, automation and
Systems (ICCAS) 2007, Seoul, Korea, October 17–20 2007, pp 1410–1415
14. Jan S-S, Gebre-Egziabher D, Walter T, Enge P (2008) Improving GPS-based landing system
performance using an empirical barometric altimeter confidence bound. IEEE T Aero Elec Sys
44(1):127–146
15. Von Mises R (1959) Theory of flight. Dover, New York
16. Groves PD (2008) Principles of GNSS, inertial, and multisensor integrated navigation systems.
Artech House, Boston
17. Parkinson BW, Spilker JJ Jr (eds) (1996) Global positioning system: theory and applications,
vol II. AIAA, Inc., Washington, DC
18. Zarchan P, Musoff H (2000) Fundamentals of Kalman filtering: a practical approach. AIAA,
Inc., Washington, DC
19. Bento MF (2008) Unmanned air vehicles: an overview. Inside GNSS, Jan–Feb 2008
20. http://spectrum.ieee.org/robotics/robotics-hardware/insecteye-camera-offers-wideangle-
vision-for-tiny-drones, Jeremy Hsu, posted 01.05.2013 – accessed on 3 Dec 2014
21. Conta G, Doherty P (2009) Vision-based unmanned aerial vehicle navigation using geo-
referenced information. EURASIP J Adv Signal Process vol 2009, article ID 387308, Hindawi
Publishing Corporation, Cairo, Egypt
22. Mallet A, Lacroix S, Gallo L (2000) Position estimation in outdoor environments using pixel
tracking and stereovision. In: Proceedings of the IEEE international conference on robotics and
automation (ICRA), San Francisco, CA, April 2000, pp 3519–3524
23. Lemaire T, Berger C, Jung I-K, Lacroix S (2007) Vision-based SLAM: stereo and monocular
approaches. Int J Comput Vis 74(3):343–364
24. Isermann R (2006) Fault-diagnosis systems: an introduction from fault detection to fault
tolerance. Springer, Berlin/Heidelberg
25. Boskovic JD, Mehra RK (2002) Stable adaptive multiple model-based control design for
accommodation of sensor failures. In: Proceedings of the 2002 American control conference,
anchorage, AK, May 2002, pp 2046–2051
References 49

26. Sobhani-Tehrani E, Khorasani K (2009) Fault diagnosis of nonlinear systems using a

hybrid approach, vol 383, Lecture notes in control and information sciences. Springer
Science C Business Media, LLC, Dordrecht
27. Groves PD (2013) Principles of GNSS, inertial, and multisensor integrated navigation systems,
2nd edn. Artech House, Boston/London
28. Hajiyev C, Caliskan F (2003) Fault diagnosis and reconfiguration in flight control systems.
Kluwer Academic Publishers, Boston
Chapter 4
Estimation of Unmanned Aerial Vehicle
Dynamics

4.1 Introduction

In this chapter, we will consider the solution to a class of linear minimum-error-

variance sequential state estimation algorithms originally developed by Kalman in a
classical work [1]. The Kalman filter, also known as linear quadratic estimation
(LQE), is an algorithm that uses a series of measurements observed over time,
containing random noise and other inaccuracies, and produces estimates of unknown
variables that tend to be more precise than those based on a single measurement
alone. Mathematically, the Kalman filter is a system of first-order ordinary differen-
tial equations with quadratic nonlinearities, which are solved on digital computers.
Kalman filter algorithms, or one of their many extensions and variations, have
been applied in numerous practical situations, including navigation, space guidance,
motion control, and orbit determination. The Kalman filter is used for the following
tasks [2]:
1. Filtration (finding a better value of a measured parameter)
2. State estimation (calculating the state vector of an aircraft)
3. System identification (calculating the unknown parameters of the mathematical
model of an aircraft)
4. Prediction (predicting the state vector of an aircraft)
5. Smoothing (estimators that take into account both the past and the future)
6. Sensor fusion (integrating more than one source of information)
7. Fault detection and diagnosis (detection, isolation, and identification of the faults
that occur in an aircraft’s systems)
We shall begin our treatment of the Kalman filter with the discrete-time version
of the problem; that is, observation of a discrete dynamic system. For simple
problems, the discrete algorithms can be manipulated by hand and considerable
insight may be gained. The step-by-step processing of information lends itself to a
simple development.

© Springer International Publishing Switzerland 2015 51

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_4
52 4 Estimation of the UAV Dynamics

The equations of motion for an unmanned aerial vehicle (UAV) are given in
Chap. 2. In this chapter, the equations of motion are further examined and their
linearization is discussed in detail for the stability analysis. In this procedure, the
first step is investigating the trim conditions.

4.2 The Optimal Linear Discrete Kalman Filter

In this chapter, we introduce and briefly discuss the mathematical model and
the relations of the linear discrete-time Kalman filter and its application to UAV
dynamics.
Let us take a linear discrete dynamic system. The system’s dynamic state
equation defines the dynamics of the system, whereas the measurement equation
defines the generation mechanism of the measurement. The equations for a linear
system are written as follows:
State equation:

xkC1 D FkC1 xk C BkC1 uk C wk (4.1)

Measurement equation:

zk D Hk xk C vk (4.2)

where xk is the n-dimensional state vector of the system, FkC1 is the n  n

transfer matrix of the system, uk is the p-dimensional vector of the deterministic
control input, BkC1 is the n  p control distribution matrix, wk is the n-dimensional
random
h Gaussian
i noise vector (system noise) with zero mean and correlation matrix
E wk wTj D Qk ıkj is the statistic average operator, ı kj is the Kronecker symbol:

1; k D j;
ıkj D
0; k ¤ j:

zk is the s-dimensional measurement vector, Hk is the s  n-dimensional measure-

ment matrix of the system, and vk is thehs-dimensional
i measurement noise vector
with zero mean and correlation matrix E vk vjT D Rk ıkj .
The mean of the initial condition x0 is x0 and the correlation matrix is P0 .
Thereh is noicorrelation between the system noise wk and the measurement noise
vk : E wk vjT D 0; 8k; j.
It is desired to find the value of the state vector according to the sequence of the
yk measurements vectors. Linear filters theory based on the Kalman filter must be
used for that purpose.
4.2 The Optimal Linear Discrete Kalman Filter 53

Depending on the application, one might want to obtain an estimate of the state
at a certain time. If the state is estimated for some time in the future, the process is
called prediction. If the estimate is made using all measurements up to and including
the current moment, it is called filtering. If an estimate is made for some time in the
past using measurements up to the current moment, the process is called smoothing.
In this chapter, we limit ourselves to prediction and filtering.

4.2.1 Optimal Kalman Filter (OKF) Equations

Generally, the filtering process of Kalman filters can be examined in two distinct
phases and if the equations are presented in the following way [3]:
Time Update (Prediction) The phase where the estimations of the preceding step
is used for producing the estimations of the present step. It can be thought of as
preparation to the estimation.
Predicted state:

xQ k=k1 D Fkb
xk1=k1 C Bk uk1 (4.3)

Predicted estimate covariance:

Pk=k1 D Fk Pk1=k1 FkT C Qk1 (4.4)

Here, bxk1=k1 is the estimated state from the previous step, exk=k1 is the predicted
state of the current step, Pk=k1 is the predicted covariance matrix of the current step,
Qk1 is the covariance of process noise w in discrete form, Fk and Bk are system
dynamics and control distribution matrices, respectively, in discretized form, and
uk1 is the control vector . Subscript k=k  1 denotes that the computation is done
in the current step k by using measurements of step k  1: Similarly, the subscript
k 1=k 1 means that the variable is computed at step k 1 by taking measurements
of step k  1 into consideration.
At the same time, Qk1 should be introduced at that point. First, the continuous
process noise covariance Q is:
 
Q D E wwT (4.5)

where w is the process noise derived from a zero-mean Gaussian distribution and
E(.) is the operator of expected value.1

1
In mathematics, the expected value of a random variable is the sum of the probabilities of each
possible outcome multiplied by the numerical value of the outcome.
54 4 Estimation of the UAV Dynamics

Then, the discrete process noise covariance matrix is:

Zt
Qk1 D F ./ QF T ./ d (4.6)
0

Measurement Update The phase that the estimation of the present step is realized
by using prediction phase outputs and, also, measurements information operation,
where the predictions are improved.
Innovation or measurement residual:

eQ k D zk  Hk xQ k=k1 (4.7)

Innovation (or residual) covariance:

 
Pk D Hk Pk=k1 HkT C Rk (4.8)

Optimal Kalman gain:

 1
Kk D Pk=k1 HkT Hk Pk=k1 HkT C Rk (4.9)

Updated state estimate:

xk=k D xQ k=k1 C Kk eQ k
b (4.10)

Updated estimate covariance:

Pk=k D .I  Kk Hk / Pk=k1 (4.11)

Here, eek is the innovation (or measurement residual), zk is the measurement vector,
Hk is the measurement matrix, Kk is the optimal Kalman gain, xO k=k is the estimated
state, Pk/k is the covariance matrix of the present step, and Rk is the covariance of
measurement noise v in discrete form.
If discrete measurement noise covariance Rk is introduced, then similarly to
Qk1 :
 
Rk D E vk vkT (4.12)

where  k is the measurement white noise, which is assumed to be zero-mean

Gaussian white noise.
Consequently, the extended Kalman filter equations are summarized in Table 4.1.
4.2 The Optimal Linear Discrete Kalman Filter 55

Table 4.1 Optimal Kalman filter equations

State prediction xQk=k1 D Fkbxk1=k1 C Bk uk1
Covariance prediction Pk=k1 D Fk Pk1=k1 FkT C Qk1
Innovation eQk D zk  Hk xQk=k1
 
Innovation covariance Pk D Hk Pk=k1 HkT C Rk
 1
Optimal Kalman gain Kk D Pk=k1 HkT Hk Pk=k1 HkT C Rk
State estimation b
xk=k D xQk=k1 C Kk Qek
Covariance estimation Pk=k D .I  Kk Hk / Pk=k1

4.2.2 Derivation of Optimal Kalman Gain

The OKF uses a filter gain where the expected value of the square of the magnitude
of error in posterior state estimation is minimized. In other words, the filter runs
under some certain optimization law defined by the minimization rule of the
indicated vector and, so, it has the optimal gain. If this optimal gain is modified
in order to adapt the filter to the changing conditions, this means that the filter is no
longer optimal and can be called an adaptive filter.
To make the meaning of optimal gain for an OKF more understandable, the gain
derivation process can be described as the following [4].
First, it is known that the matrix Pk/k is the covariance of estimation and, so:
 
Pk=k D cov xk b
xk=k (4.13)

where cov() is the covariance operator, xk is the real value of the states in step k,
and b
xk=k is the estimation of the states in step k.
Then, by using the definition of state estimation xO k=k (Eq. 4.10):
  
Pk=k D cov xk  xQ k=k1 C Kk eQ k (4.14)

substituting ẽk (Eq. 4.7):

   
Pk=k D cov xk  xQ k=k1 C Kk zk  Hk xQ k=k1 (4.15)

and zk as:

zk D Hk xk C vk (4.16)

   
Pk=k D cov xk  xQ k=k1 C Kk Hk xk C vk  Hk xQ k=k1 (4.17)
56 4 Estimation of the UAV Dynamics

By collecting the necessary terms, the equation can also be written as:
   
Pk=k D cov .I  Kk Hk / xk  xQ k=k1  Kk vk (4.18)

Since the measurement noise vk is uncorrelated with other terms:

  
Pk=k D cov .I  Kk Hk / xk  xQ k=k1 C cov .Kk vk / (4.19)

And by the properties of vector covariance:

 
Pk=k D .I  Kk Hk / cov xk  xQ k=k1 .I  Kk Hk /T C Kk cov .vk / KkT (4.20)
   
xk=k1 stands for Pk=k1 and cov .vk / D E vk vkT D Rk :
As long as cov xk e

) Pk=k D .I  Kk Hk / Pk=k1 .I  Kk Hk /T C Kk Rk KkT (4.21)

This expression of updated covariance estimation is valid for every Kk and can
be used for obtaining the optimal Kalman gain.
The Kalman filter is a minimum mean square error estimator and, as previously
mentioned, to run the filter optimally, the minimum expected value of the square of
the magnitude of state estimation
hˇ is sought [4]. The estimation error is xk b
xk=k and
ˇ2 i  
the minimized quantity is E ˇxk b xk=k ˇ . However, since Pk=k D cov xk b xk=k ,
that is the same as minimizing the trace of covariance matrix Pk/k . It has been already
found as:

) Pk=k D .I  Kk Hk / Pk=k1 .I  Kk Hk /T C Kk Rk KkT

If the terms are expended:

Pk=k D Pk=k1  Kk Hk Pk=k1  Pk=k1 KkT HkT C Kk Pk KkT (4.22)

 
where Pk D Hk Pk=k1 HkT C Rk :
Therefore, for minimization:
 
@tr Pk=k  T
D 2 Hk Pk=k1 C 2Kk Pk D 0 (4.23)
@Kk

If Eq. (4.23) is solved for Kk , then:

 1
Kk D Pk=k1 HkT Pk 1 D Pk=k1 HkT Hk Pk=k1 HkT C Rk (4.24)
4.2 The Optimal Linear Discrete Kalman Filter 57

Initial Conditions

P(0)
P( k − 1/ k− 1)
calculation of P( k / k− 1)
Q( k)
P ( k / k− 1) Aprior data

calculation of P( k / k)
R ( k)
Initial Conditions
P (k / k)
z( k) x (0)
x ( k / k)
x (k )
System
Model
Measurement
Model ∑ K ( k) +
z( k / k− 1)
xˆ ( k − 1/ k− 1)
w(k) v ( k)
H ( k) φ (k, k − 1)
xˆ( k / k− 1)

Fig. 4.1 Structural diagram of the Kalman filter

As can be seen, this equation is the same as Eq. (4.9) and the gain derived so far
is called the optimal Kalman gain. Besides, if this expression of Kk is substituted
into Eq. (4.22), then the equation:

Pk=k D .I  Kk Hk / Pk=k1

can be found easily.

4.2.3 The Structure of the Kalman Filter

The structural diagram of the Kalman filter [5] is shown in Fig. 4.1.
The initial values x0 and P0 , correlation matrix of the system noise Q, and
correlation matrix of the measurement noise R must be known previously in order
for the filter to work.
According to formula (4.10), the estimation value is equal to the sum of
the extrapolation value xQ k=k1 and the correction term Kk ẽk . The extrapolation
(prediction) value is calculated by multiplying the estimation value found in the
previous step by the transfer matrix of the system. After that, correction is made
to the extrapolation value. Hence, the Kalman filter works on the principle of
correction of the prediction.
58 4 Estimation of the UAV Dynamics

y( k)
x(k 1)

xˆ ( k / k)
z( k / k 1)
x (k)

x ( k 1) ion
xˆ ( k 1/ k 1) polat y( k 1)
Extra
y( k 2)
K ( k) z ( k/ k 1)

x ( k 2) xˆ ( k / k 1)
y ( k 1)
Discrete Time

k 2 k 1 k k 1

Fig. 4.2 Time-dependent diagram of the value estimation mechanism in the Kalman filter

The time-dependent diagram showing the mechanism of value generation in

the Kalman filter is shown in Fig. 4.2. The filter algorithm includes the following
operations:
1. One step further prediction of the value (determination of the extrapolation value)
! xQ k=k1
2. Multiplication of xQ k=k1 with Hk from the left; in other words, prediction of
measurement
3. Determination of the difference between the measurement and the extrapolation
value (innovation sequence) ! eQ k D zk  Hk xQ k=k1
4. Multiplication of ẽk with matrix Kk from the left and addition with xQ k=k1 ! b
xk=k
5. Storage of the b
xk=k value and repetition of the loop
It is postulated that the Kalman filter is the best filter among the subset of all
linear filters and the best filter among the set of all filters when the noise processes
are Gaussian [6].
The important features of the Kalman filter are given below [2]:
1. The filter can be easily realized using a computer once the model of the dynamic
system is known
2. The value found by means of the filter is linear with respect to the measurement
3. The correlation matrix of the filtering error Pk/k is not related to the measuring of
zk since the filter is linear, and it can be calculated beforehand
4. Filtering algorithms can be easily applied to the multidimensional case
5. The Kalman filter is equivalent to the Wiener filter in steady state for stationary
dynamic filters
Adaptive filters become more important when the mathematical model of the
system is not known previously or it is changed during the operation process.
4.3 Stability of the Optimal Discrete Kalman Filter 59

This time evaluation operation is combined with the identification operation of the
parameters and/or the model structure.
Nonlinear equations must be linearized first.

4.3 Stability of the Optimal Discrete Kalman Filter

As seen above, the Kalman filter is a time-domain filter, because the design is done
in the time domain rather than the frequency domain. One of the advantages of the
Kalman filter is its ability to estimate time-variable parameters. When the Kalman
filter is used, the convergence of estimated values to the actual values of parameters
depends on the stability condition of the filter. The stability of conventional digital
filters is easily analyzed with z-transform methods [7]. We shall do the same for the
optimum Kalman filter.
It is assumed that the optimum discrete Kalman filter [Eqs. (4.3), (4.4), (4.7),
(4.8), (4.9), (4.10), and (4.11)] has reached a constant-gain condition. The estima-
tion equation (4.10) can also be expressed for convenience:
˚ 
b
xk=k D xQ k=k1 C Kk zk  Hk xQ k=k1 (4.25)

Replace xQ k=k1 with Fkb

xk1=k1 in Eq. (4.25). After simplifications, the following
equation is obtained:

xk=k D fFk  Kk Hk Fk gb
b xk1=k1 C Kk zk (4.26)

Taking the z-transform of both sides of Eq. (4.26) and retarding b

xk=k by one step
in the time domain is equivalent to multiplying b z
Xk=k by z1 in the z-domain. In the
z-domain, this means:

bz
Xk=k D fFk  Kk Hk Fk g z1bz
Xk=k C Kk Zkz (4.27)

After rearranging the terms, we have:

ŒzI  fFk  Kk Hk Fk g bz
Xk=k D zKk Zkz (4.28)

where z denotes the usual z-transform variable and Z zk refers to the z-transform of
the measurement vector.
It is well known from linear system theory that the bracketed quantity on the left
side of Eq. (4.28) describes the natural modes of the system. The determinant of the
bracketed n  n matrix gives the characteristic polynomial for the system; that is:

Characteristic polynomial D ŒzI  fFk  Kk Hk Fk g : (4.29)

60 4 Estimation of the UAV Dynamics

The roots of this polynomial provide information about the filter’s stability. If
all the roots lie inside the unit circle in the z-plane, the filter is stable; conversely,
if any root lies on or outside the unit circle, the filter is unstable. As a matter
of terminology, the roots of the characteristic polynomial are the same as the
eigenvalues of fFk  Kk Hk Fk g.

4.4 OKF for UAV State Estimation

When the Kalman filter algorithm is built for UAV state estimation, it is used for the
combined longitudinal and lateral dynamics of aircraft. Hence, the state vector to be
estimated is formed of n D 9 states as:
 T
x D u w q  h ˇ p r  ; (4.30)

while the control input vector is:

 T
u D ıe ıT ıa ır : (4.31)

Here, u and w are the perturbations of the velocity components in the x-

and z-directions, respectively, h is the perturbation of the height of the plane,
p, q, r are the perturbations of the angular velocities about the x, y, and z axes,
respectively, , , ˇ are the perturbations of the pitch, roll, and sideslip angles,
respectively, ı e , ı a , and ı r are the perturbations of the elevator, aileron, and
rudder deflections, respectively, and ı T is the change in the thrust.
Then, let us introduce the UAV process and observation models for the combined
dynamics in the state space form as follows:

xk D Fk xk1 C Bk uk1 C Gk wk ; (4.32)

zk D Hk xk C vk ; (4.33)

where Fk is the system dynamics matrix, Bk is the control distribution matrix, zk

is the measurement vector, Gk is the transition matrix of system noises, Hk is the
measurement matrix, which is a 9  9 identity matrix in this case, and wk and vk are
the white Gaussian system process and measurement noises, respectively:

E wk wTj D Qk ıkj ; (4.33)


E vk vjT D Rk ıkj ; (4.34)


E wk vjT D 0: (4.35)
4.5 Simulations 61

Here, Qk is the process noise covariance matrix, Rk is the measurement noise

covariance matrix, and ı kj is the Kronecker delta function.
After that, the optimal Kalman filter for such a combined UAV model can be
obtained by following the steps given in Table 4.1.

4.5 Simulations

In this section, simulation results for the state estimation of the Zagi UAV are given.
Simulation is realized in 500 steps for a period of 50 s with 0.1 s of sampling
time, t. For a joined state vector of longitudinal and lateral equations (4.30), the
measurement noise v is taken as:
 T
v D Nr  0:92 0:92 0:083 0:17 0:2 0:05 0:083 0:083 0:17 (4.36)

where Nr is a random number with zero mean and unit variance Gaussian distri-
bution. Afterwards, R, the covariance of measurement noise, a 9  9 matrix for all
steps, is given as:
2 3
0:922    0
6 :: : : :: 7
RD4 : : : 5 (4.37)
2
0    0:17

For simulating measurements and also introducing system dynamics into the
filter, the equations of motion of the UAV are put into discretized form. If the state
space form of equations is shown as:

XP D AX C BU (4.38)

the discretized equations become:

XiC1 D .I C tA/ Xi C tBUi (4.39)

Here, A is the system matrix, B is the control distribution matrix, X is the state vector
 T
as given above, and U is the control vector, given as u D ıe ıT ıa ır .
During simulations, the initial values of all states are assumed to be 0. On the
 T
other hand, the control vector is chosen as u D 0:2 0 0 0:2 rad.
Measurements are simulated by adding noise values to the real values of dynamic
variations as:

ZiC1 D XiC1 C v (4.40)

62 4 Estimation of the UAV Dynamics

Besides for showing the drawbacks of the OKF and the necessity for filter
adaptation in certain cases, errors are implemented in the measurements every 100th
step. These abrupt errors, which represent a failure condition at the measurement
channel, are formed by adding a constant term to the measurements of two states,
 and ˇ.
Here, the simulation results are given for two states, u and  in Figs. 4.3, 4.4,
4.5, 4.6, and 4.7. The results given in Figs. 4.5, 4.6, and 4.7 are for OKF in case
of faulty measurements. Also Table 4.2 presents the estimation errors for all states
at several time steps in case of faulty measurements. For any state, three graphs
are introduced in a combined fashion: a plot that shows the Kalman estimation
and simulated measurement values, a plot that gives the error of measurement
and Kalman estimation according to the real value found by discretized dynamic
equations (Eq. 4.39), and a plot that shows the variation of variance of Kalman
estimations.
The simulation results show that the OKF gives sufficiently good estimation
results in the case of normal operation of the estimation system (without faults).
However, the optimal filter fails in providing accurate estimations in case of
measurement faults, as expected. That signifies the importance of Kalman filter
adaptation, which will be discussed in the next section.

velocity "u" estimation

5
Kalman Estimation
0 Measurement Simulation
u(m/s)

-5

-10
0 5 10 15 20 25 30 35 40 45 50

5 Kalman Estimation Error

Simulated Measurement Error
error(m/s)

-5
0 5 10 15 20 25 30 35 40 45 50

1
Kalman Estimation Variance
variance

0.5

0
0 5 10 15 20 25 30 35 40 45 50
time(sec)

Fig. 4.3 u estimation of the optimal Kalman filter (OKF)

4.6 Necessity for Kalman Filter Adaptation 63

pitch angle "teta" estimation

2
Kalman Estimation
teta(rad)

1 Measurement Simulation

-1
0 5 10 15 20 25 30 35 40 45 50

1
Kalman Estimation Error
error(rad)

0.5 Simulated Measuremnt Error

-0.5
0 5 10 15 20 25 30 35 40 45 50

0.03
Kalman Estimation Variance
variance

0.02

0.01

0
0 5 10 15 20 25 30 35 40 45 50
time(sec)

Fig. 4.4  estimation of the OKF

4.6 Necessity for Kalman Filter Adaptation

4.6.1 A Priori Uncertainty and Adaptation

The Kalman filter examined in the previous section is valid in the presence of
whole prior statistical data. In practice, situations where a priori data are known
with estimations or which exist completely are rarely encountered. For this case,
the developed algorithms are not optimal and values estimated via these algorithms
may not converge. Therefore, another approach is required to synthesize estimation
algorithms.
A priori uncertainty degree can vary:
1. Exact a priori statistical uncertainty – In this case, the shape and parameters of
the probability distribution rules of the components of measured and estimated
random processes are not understood. However, allowable limited change areas
of appropriate components of random processes are given. In such a case, the
synthesis of estimation algorithms can be performed only on the basis of a
guaranteed approach [8].
64 4 Estimation of the UAV Dynamics

velocity "u" estimation

20
Kalman Estimation
10
u(m/s)

Measurement Simulation

-10
0 5 10 15 20 25 30 35 40 45 50

10
Kalman Estimation Error
error(m/s)

0 Simulated Measurement Error

-10

-20
0 5 10 15 20 25 30 35 40 45 50

1
Kalman Estimation Variance
variance

0.5

0
0 5 10 15 20 25 30 35 40 45 50
time(sec)

Fig. 4.5 u estimation of the OKF in case of faulty measurements

2. Partial prior statistical uncertainty – In this case, the probability distribution rules
of some parts of measured and estimated random processes’ components are
known. The number of parameters which have unknown probability characteris-
tics should not be too many. Increment in the number of these parameters causes
the quality of the problem solution to decrease. On account of this, when there
is a priori uncertainty in parameters, a distributions group is given instead of the
probability distribution rule of random processes. The estimation algorithm from
the distributions group must be chosen to provide optimal criterion. It means that
the estimation algorithm should be adaptive.
There are three approaches for the adaptation problem: parametric approach,
invariant principle-based approach, and structural approach. The parametric
approach is the most commonly used. The parametric adaptive estimation algorithm
is an algorithm that estimates the required components of random processes
at the end of the measurement data transaction and restores a priori statistical
characteristics of the dynamical system and measurements.
A self-tuning circuit is added to an ordinary Kalman filter in most parametric
adaptive estimation algorithms. When the Kalman filter designed, it is supposed
that the statistical characteristics of the system model, measurements, and noise
4.6 Necessity for Kalman Filter Adaptation 65

pitch angle "teta" estimation

40
Kalman Estimation
teta(rad)

20
Measurement Simulation
0

-20
0 5 10 15 20 25 30 35 40 45 50

20
Kalman Estimation Error
error(rad)

0
Simulated Measuremnt Error
-20

-40
0 5 10 15 20 25 30 35 40 45 50

0.03
Kalman Estimation Variance
variance

0.02

0.01

0
0 5 10 15 20 25 30 35 40 45 50
time(sec)

Fig. 4.6  estimation of the OKF in case of faulty measurements

pitch angle "teta" estimation

25
Kalman Estimation
Measurement Simulation

20

15
teta(rad)

10

0
8 8.5 9 9.5 10 10.5 11 11.5 12
time(sec)

Fig. 4.7 Focused graph for  estimation of the OKF in case of faulty measurements
66 4 Estimation of the UAV Dynamics

Table 4.2 Optimal Kalman filter performance in case of faulty measurements

Optimal filter in case of faulty measurements
Step 5 s. 20 s. 27.5 s. 40 s. 43 s.
u (m/s) 0.1602 3.0722 0.1089 2.9559 0.1362
w (m/s) 0.0259 0.8729 0.0326 0.9504 0.0499
q (rad/s) 0.0925 0.1222 0.0339 0.0325 0.0243
 (rad) 0.0196 2.2815 0.0235 2.2582 0.0412
h (m) 0.1429 2.5808 0.0627 2.5163 0.04
ˇ (rad) 0.008 10.042 0.0023 10.016 0.0021
p (rad/s) 0.0219 0.4861 0.0258 0.5356 0.0227
r (rad/s) 0.0189 0.3699 0.0212 0.3121 0.0029
 (rad) 0.0162 3.2214 0.0611 3.0618 0.0299

are known as being true. However, this supposition is not often accomplished, so
the applied filter is suboptimal. Another reason for the suboptimality of the filter
is simplifying the process calculations in the filter algorithm. In some cases, the
problem of concretizing the fault caused by suboptimality appears. For this reason,
noise covariances are estimated via the Kalman filter. This type of filter is called an
adaptive filter.
Some approaches to designing adaptive filters are presented below:
• Multiple model-based adaptive estimation (MMAE). In the MMAE approach,
adaptive filtering of a bank of Kalman filters runs in parallel under different
models for the statistical filter information matrices, i.e., the process noise matrix
Q and/or the update measurement noise matrix R.
• Unknown noise covariances of the Kalman filter are determined by the statistical
analysis of the innovation or residual series. This might be either as a direct
estimation of the covariance matrices or as an adaptation performed on the basis
of covariance scaling [9] (Fig. 4.8).
• Noise estimation is performed via probability methods. Estimation values of
unknown noise covariances are periodically used to renew the noise structure
in filtering algorithms.
• By processing the same values via an iteration procedure, the determination of
values of unknown covariances is based on the analysis of residual faults from
the previous data processing circuit. This method is generally used when huge
computational burden is required for a test.
Next, we review some of the existing common direct estimation techniques
for Kalman filter adaptation. Newly proposed scaling-based techniques will be
discussed in the next two chapters in detail. Readers are also referred to [9] for a
comparison between these different methods for Kalman filter adaptation.
4.6 Necessity for Kalman Filter Adaptation 67

Fig. 4.8 The Kalman filter (KF) adaptation by the statistical analysis of the innovation and
residual series: (a) innovation-based KF adaptation, (b) residual-based KF adaptation. For both
adaptation techniques, updating of the process and measurement noise covariances might be either
by estimation or by scaling

4.6.2 Innovation-Based Adaptive Estimation

Innovation-based adaptive estimation is the most widely used technique [10–15]. In

this method, the appropriateness of the a priori characteristic of the noise to its real
characteristic is determined by investigating the innovation sequence. In cases when
the real innovation sequence differs from the white noise characteristics, estimation
of the measurement noise (R) and system noise (Q) are realized. The analysis of
innovation sequence statistics is simply checking for the sequence to verify that it
has the expected zero value and the Gaussian white noise characteristic.
The covariance of the innovation sequence (ẽj ) is:

1 X
k
^
C ek D eQ j eQ T (4.41)
N jDkNC1 j

where N is the dimension of the “sliding window”.

If the real and a priori characteristics of the noise are different, R and Q must
be determined consecutively and must also be included in the filtering algorithm.
Thus, adaptive Kalman filters include filtering algorithms which are used to estimate
system states, system parameters, and, also, the noise characteristics.
In innovation-based approaches, the previously predicted R and Q matrices are
updated using the real-time measurements. The statistics matrices in the filter are
updated using the formulas given below:

^
b
Rk D Cek  Hk Pk=k1 HkT (4.42)

^
b
Qk D Kk Cek KkT (4.43)
68 4 Estimation of the UAV Dynamics

4.6.3 Residual-Based Adaptive Estimation

In the residual-based approach [16, 17], the measurement covariance matrix and/or
the system noise that are affected by the change in residual series are directly
updated.
The residual series can be described by:

k D zk  Hb
xk=k ; (4.44)

where zk is the measurement vector, H is the measurement matrix, and b

xk=k is the
estimation vector of the states.
The covariance matrix of the residual series is:

1 X
k
b
Ck D j  T (4.45)
N jDkNC1 j

where N is the “sliding window” dimension.

The algorithms given below are used to estimate the R and Q matrices:
R matrix estimation algorithm:

b
Rk D b
Ck C Hk Pk=k HkT (4.46)

Pk/k is the covariance matrix of the estimation error.

Q matrix estimation algorithm:
0 1
1 X
k
Qk D @
b xj xTj A C Pk=k  FPk1=k1 F T (4.47)
N jDkNC1

where Pk1=k1 is the covariance matrix of the estimation error in the previous step,
F is the system transformation matrix, and xk is the state correction series (the
difference between the filter estimation and the extrapolation estimation):

xk D b
xk=k b
xk=k1 (4.48)

In a stable state, only the first part of Eq. (4.47) is considered, in this case, taking
into account that xk D Kk k in Eq. (4.47) can be approximated by Eq. (4.43).
It is understood that, for some practical applications [(like an integrated inertial
navigation system (INS)/Global Positioning System (GPS)] [15], residual-based
adaptive estimation methods are more effective than innovation-based approaches.
Some of the disadvantages of the described adaptive estimation methods are
given below:
(a) In the estimation of unknown system and measurement noise covariance
matrices using the residual- or innovation-based approaches, the innovation
References 69

and residual vectors should be used for N cycles. This consumes a vast
amount of computing power and makes determining appropriate sliding window
dimension values a must.
(b) The measurement quantities, types, and distributions must be consistent in
order to be able to use the residual- or innovation-based estimation methods.
Otherwise, the covariance matrices of the system and measurement noise could
not be estimated using the residual or innovation matrices.
(c) Innovation-based estimators may cause a negative covariance matrix if the value
^
on the right-hand side of the formula b
Rk D Cek  Hk Pk=k1 HkT is bigger than that
on the left-hand side.

4.7 Conclusion

In this chapter, an OKF algorithm for the state estimation of a specific UAV’s
dynamics is presented and examined. Simulations realized by the use of a MATLAB
program shows that the OKF algorithm gives accurate estimations under normal
operation of the system (without faults). The OKF algorithm fails when faults occur
in the estimation system.
Mathematically, the Kalman filter is a system of first-order ordinary differential
equations with quadratic nonlinearities, which are solved on digital computers.
When implementing navigational and other algorithms on a maintenance digital
computer, incorrect results may occur due to equipment failure, malfunctioning
of the computer, and noise during information transmission. For these cases, the
adaptive Kalman filter can be used.
In accordance to the main motivation behind the study, supplying reliable
parameter estimations to the control system of an autonomous UAV and ensuring
that it completes its mission successfully is the objectives of this monograph.

References

1. Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASME J
Basic Eng 82:35–45
2. Hajiyev C, Caliskan F (2003) Fault diagnosis and reconfiguration in flight control systems.
Kluwer Academic Publishers, Boston
3. Zarchan P, Musoff H (2000) Fundamentals of Kalman filtering: a practical approach. Progress
in aeronautics and astronautics. AIAA, Reston
4. Sage AP, Melsa JL (1971) Estimation theory with applications in communication and control.
McGraw-Hill, New York
5. Hajiyev CM (1999) Radio navigation. Istanbul Technical University, Istanbul (in Turkish)
6. Anderson BDO, Moore JB (1979) Optimal filtering. Prentice Hall, Englewood Cliffs
7. Brown RG, Hwang PYC (1997) Introduction to random signals and applied Kalman filtering
with MATLAB exercises and solutions, 3rd edn. Wiley, New York
70 4 Estimation of the UAV Dynamics

8. Ogarkov MA (1990) Methods for statistical estimation of random processes parameters.

Energoatomizdat, Moscow (in Russian)
9. Hajiyev C, Soken HE (2015) Fault tolerant estimation of UAV dynamics via robust adaptive
Kalman filter. In: Dimirovski GM (ed) Complex systems: relationships between control,
communications and computing. Springer (in press), Berlin
10. Mehra RK (1970) On the identification of variances and adaptive Kalman filtering. IEEE Trans
Autom Control 15(2):175–184
11. Mehra RK (1971) On-line identification of linear dynamic systems with applications to Kalman
filtering. IEEE Trans Autom Control 16(1):12–21
12. ¯Èhra RK (1971) Identification and adaptive Kalman filtering. Mechanic 3:34–52
13. Kailath T (1972) A note on least squares estimation by the innovations method. SIAM J Control
10(3):477–486
14. Maybeck PS (1982) Stochastic models, estimation, and control, vols I and II. Academic Press,
New York
15. Salychev OS (1994) Special studies in dynamic estimation procedures with case studies
in inertial surveying, ENGO 699.26 lecture notes. Department of Geomatics Engineering,
University of Calgary, Canada
16. Mohamed AH, Schwarz KP (1999) Adaptive Kalman filtering for INS/GPS. J Geod 73:193–
203
17. Wang J, Stewart MP, Tsakiri M (1999) Adaptive Kalman filtering for integration of GPS with
GLONASS and INS. IUGG/IAG, Birmingham
Chapter 5
Estimation of Unmanned Aerial Vehicle
Dynamics in the Presence of Sensor Faults

5.1 Introduction

When there are sufficient measurements, the necessary states of an unmanned aerial
vehicle (UAV) can be estimated via a Kalman Filter. That is a desired procedure,
since it is important to know precisely parameters like velocity, altitude, attitude,
etc. When these states of the UAV are obtained without any problems, control of the
aircraft can be achieved successfully. On the other hand, it is a process dependent
upon the accuracy of measurements. If the measurements are not reliable due to any
kind of malfunction in the estimation system, the filter gives inaccurate results and
diverges over time. Since achieving fault tolerance in the design of a UAV flight
control system is important, the filter should be built robustly in order to avoid such
problems.
The Kalman filter approach to state estimation is quite sensitive to any mea-
surement malfunctions (abnormal measurements, sudden shifts in the measurement
channel, and other difficulties, such as decrease of instrument accuracy, an increase
in background noise, etc.). If the conditions of operation of the measurement
system do not correspond to the models used in the synthesis of the filter, then
these changes resulting from some possible failures at the measurement channels
significantly decrease the effectiveness of the estimation systems. When dealing
with the measurement faults in previous estimation steps, rather than the current
one, adaptive Kalman filters can be used to recover from the possible malfunctions.
A Kalman filter can be made adaptive and, hence, insensitive to the a priori
measurements or system uncertainties by using various different techniques. The
basic approaches to the adaptive Kalman filtering problem are multiple model-
based adaptive estimation (MMAE) [1–3], innovation-based adaptive estimation
(IAE) [4–6], and residual-based adaptive estimation (RAE) [7, 8]. While in the
first approach a bank of Kalman filters run in parallel under different models for
the filter’s statistical information, in the other approaches, the adaptation is done

© Springer International Publishing Switzerland 2015 71

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_5
72 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

directly to the covariance matrices of the measurement and/or system noises based
on the changes in the innovation or residual sequences.
In the methods described in [1–3], the faults are assumed to be known and the
Kalman filters are designed for the known sensor/actuator faults. As the MMAE
approach requires several parallel Kalman filters, and the faults should be known, it
can only be used in a limited number of applications.
Estimation of the covariance matrices by IAE and RAE requires the usage of
innovation vectors or residual vectors of m epoch. This increases the storage burden
and presents the determination of the width of the moving window, m, as another
problem. Furthermore, IAE and RAE estimators require that the number, type, and
distribution of measurements for all epochs within a window should be consistent. If
they are not, the covariance matrices of the measurement noises cannot be estimated
based on the innovation or residual vectors.
The adaptive Kalman filter presented in [9] has been applied to fuse position
signals from the Global Positioning System (GPS) and an inertial navigation system
(INS) for autonomous mobile vehicles. The extended Kalman filter (EKF) and the
noise characteristics have been modified using the fuzzy logic adaptive system. In
[10], a method for multisensor data fusion based on the adaptive fuzzy Kalman
filter is presented. This method is applied for fusing the position and orientation
signals from the dead reckoning (DR) system and the GPS for landing vehicle
navigation. The EKF and the characteristics of the measurement noise are modified
using the fuzzy adaptive system, which is based on a covariance matching technique.
It has been demonstrated that the fuzzy adaptive Kalman filter gives better results
(more accurate) than the EKF [9, 10]. In [11], a fuzzy logic-based adaptive
Kalman filter is used to build adaptive centralized, decentralized, and federated
Kalman filters for adaptive multisensor data fusion. The adaptation carried out
is in the sense of adaptively adjusting the measurement noise covariance matrix
of each local filter to fit the actual statistics of the noise profiles present in the
incoming measured data. A fuzzy inference system based on a covariance matching
technique is used as the adaptation mechanism in the paper. The simulation results
show that the architectures proposed by the authors are effective in situations
where there are several sensors measuring the same parameters, but each one has
different measurement dynamics and noise statistics. Although fuzzy logic-based
adaptive Kalman filter algorithms perform well under specific circumstances, they
are knowledge-based systems operating on linguistic variables, and these methods,
which are based on human experiences, are not widely applicable to vital systems
such as aircraft flight control systems.
Another concept is to scale the noise covariance matrix by multiplying it with a
time-dependent variable. One of the methods for constructing such an algorithm
is to use a single adaptive factor as a multiplier to the process or measurement
noise covariance matrices [12–15]. This algorithm, which may be called an adaptive
fading Kalman filter (AFKF), can be used when information about the dynamic
process or the a priori measurements are absent [16]. However, when the point
at issue is the current measurements, another technique to scale the measurement
noise covariance matrix and make the filter robust (insensitive to measurement
5.2 RKF with a Single Measurement Noise Scale Factor 73

faults) should be proposed. Therefore, if there is a malfunction in the measurement

system, a robust Kalman filter (RKF) algorithm can be utilized and, by the use
of a measurement noise scale factor (MNSF) as a multiplier of the measurement
noise covariance matrix, insensitiveness of the filter to the current measurement
faults can be satisfied. Consequently, via a correction applied to the filter gain, good
estimation behavior of the filter will be leveraged without being affected by faulty
measurements [17].
However, the estimation performance of the Kalman filter differs for each
variable when it is utilized for complex systems with multiple variables, and it
may be not sufficient to use a single measurement noise scale factor (SMNSF)
as a multiplier for the covariance matrices [18]. A single factor may not reflect
corrective effects for the faulty measurement to the estimation process accurately.
The technique that can be implemented to surmount this problem is to use multiple
measurement noise scale factors (MMNSF) to fix the relevant components of
the gain matrix individually. Unfortunately, thus far, any investigations about the
comparison of the RKF with single and multiple scale factors have not been
successful.
In this chapter, RKF algorithms with single and multiple MNSF, which require
less computational burden for scale factor evaluation than the existing algorithms,
are introduced and applied for the state estimation process of a UAV. The results of
these algorithms are compared for different types of measurement malfunctions and
recommendations about their utilization are given.

5.2 RKF with a Single Measurement Noise Scale Factor

Under normal operation conditions, where any kind of measurement malfunction

is not observed, the optimal Kalman filter (OKF) gives sufficiently good estimation
results. However, when the measurements are faulty due to malfunctions in the esti-
mation system, such as abnormal measurements, sudden shifts, or step-like changes
in the measurement channel, etc., filter estimation outputs become inaccurate.
Therefore, an RKF algorithm, which brings fault tolerance to the filter and
secures accurate estimation results in case of faulty measurements without affecting
the remaining good estimation characteristics, should be introduced.
The base of the RKF is the comparison of real and theoretical values of the
covariance of the innovation sequence [17]. When the operational condition of
the measurement system mismatches with the models used in the synthesis of
the filter, then the Kalman filter gain changes according to the differentiation in
the covariance matrix of the innovation sequence. Under these circumstances, the
covariance matrix of the innovation sequence differs as [19, 20]:

Pek D Hk Pk=k1 HkT C Sk Rk ; (5.1)

74 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

and, so, the Kalman gain becomes:

 1
Kk D Pk=k1 HkT Hk Pk=k1 HkT C Sk Rk : (5.2)

Here, Sk is the SMNSF.

Due to this approach, the Kalman gain is changed when the predicted observation
Hk xQ k=k1 is considerably different from the actual observation yk because of the
significant changes in the operational condition of the measurement system. In other
words, if the real value of the filtration error exceeds the theoretical error:
˚  ˚ 
tr eQ k eQ Tk  tr Hk Pk=k1 HkT C Rk ; (5.3)

the filter must be run robustly. Here, tr fg denotes the trace of the related matrix.
In order to determine the scale factor, Sk , let us substitute Eq. (5.1) into Eq. (5.3)
and use as input the scale factor calculated at the point where condition (5.3) is
satisfied:
˚  ˚ 
tr eQ k eQ Tk D tr Hk Pk=k1 HkT C Sk tr fRk g : (5.4)
˚ 
Then, in light of the equality tr eQ k eQ Tk D eQ Tk eQ k , Sk can be written as:
˚ 
eQ Tk eQ k  tr Hk Pk=k1 HkT
Sk D : (5.5)
tr fRk g

In case of malfunction in the measurement system, the adaptation of the Kalman

filter is performed via automatically correcting the Kalman gain. If the condition
in Eq. (5.3) is met, then it brings about an increase in the scale factor Sk . A larger
Sk causes a smaller Kalman gain (5.2) because of the covariance of the innovation
sequence (5.1), which is also increased in the robust case. Consequently, a small
Kalman gain value reduces the effect of the faulty innovation sequence on the
state estimation process. In all other cases, where the measurement system operates
normally, the SMNSF takes the value of Sk D 1 and, so, the filter runs optimally.
It must be noted that, due to the scale factor Sk , the covariance of the estimation
error of the RKF increases in comparison with the OKF. Therefore, the robust
algorithm is used only when the measurements are faulty and, in all other cases, the
procedure is run optimally with the regular Kalman filter. The process is controlled
by the use of a kind of statistical information. At that point, the two following
hypotheses may be introduced:
0 ; the system is operating normally
1 ; there is a malfunction in the estimation system
To detect failures, a statistical function may be defined as [21]:
 1
ˇk D eQ Tk Hk Pk=k1 HkT C Rk eQ k : (5.6)
5.3 RKF with Multiple Measurement Noise Scale Factors 75

This statistical function has a 2 distribution with M degrees of freedom, where

M is the dimension of the innovation vector.
If the level of significance, ˛, is selected as:
˚ 
P 2 > 2˛;M D ˛I 0 < ˛ < 1; (5.7)

the threshold value, 2˛,M , can be found. Hence, when the hypothesis 1 is correct,
the statistical value of ˇ k will be greater than the threshold value 2˛,M , i.e.:

0 W ˇk  2˛;M 8k
(5.8)
1 W ˇk > 2˛;M 9k:

5.3 RKF with Multiple Measurement Noise Scale Factors

As previously mentioned, it is possible to adapt the filter by using a single scale

factor as a corrective term on the filter gain [17, 19], but this is not a safe
procedure as long as the filter performance differs for each state for complex systems
with multiple variables [18, 22]. The preferred method is to use a matrix built
from multiple measurement noise scale factors to fix the relevant terms of the
measurement noise covariance matrix, and, consequently, the Kalman gain.
In order to determine the scale matrix, an innovation-based process may be
followed. It is known that the Kalman filter innovation sequence can be determined
by Eq. (5.3). Then, as the next step, real and theoretical values of the innovation
covariance matrix must be compared. When there is a measurement malfunction in
the estimation system, the real error will exceed the theoretical error. Hence, if a
scale matrix, Sk , is added into the algorithm as [23] follows:

1 X
k
eQ j eQ T D Hk Pk=k1 HkT C Sk Rk ; (5.9)
 jDkC1 j

then it can be determined using the formula:

0 1
1 X k
Sk D @ eQ j eQ T  Hk Pk=k1 HkT A R1
k : (5.10)
 jDkC1 j

Here,  is the width of the moving window.

In the case of normal operation, the scale matrix will be a unit matrix, Sk D I.
Here, I represents the unit matrix. Nonetheless, as  is a limited number due to the
quantity of measurements and computations performed using a computer, it implies
errors such as approximation errors and rounding off errors; the matrix Sk , found by
76 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

the use of Eq. (5.10), may not be diagonal and may have diagonal elements which
are “negative” or less than “one” (actually, that is physically impossible).
Therefore, in order to avoid such situations, it is suggested that the scale matrix
is composed by the following rule:
 
S D diag s1 ; s2 ; : : : ; sn ; (5.11)

where:

si D max f1; Sii g i D 1; n: (5.12)

Here, Sii represents the ith diagonal element of the matrix S. Apart from that point, if
the measurements are faulty, Sk will change and, so, affect the Kalman gain matrix:
 1
Kk D Pk=k1 HkT Hk Pk=k1 HkT C Sk Rk : (5.13)

Therefore, in case of any kind of malfunctions, related elements of the scale

matrix, which correspond to the faulty component of the measurement vector,
increase, and that brings about a smaller Kalman gain, which reduces the effect
of the innovation on the state update process. As a result, more accurate estimation
results can be obtained.
As long as the proposed RKF is not optimal due to the scale matrix Sk , the robust
algorithm is again operated only when the measurements are faulty and, in all other
cases, the procedure is run optimally with the regular OKF. The same statistical
information as the SMNSF-based algorithm, which is defined by Eqs. (5.6), (5.7),
and (5.8), is used as the supervision criteria for starting the adaptation of the RKF.

5.4 Comparison of the R-adaptation Techniques

The R-adaptation procedures with single and multiple measurement noise scale
factors are compared through simulations. Simulations are realized in 1,000 steps
for a period of 100 s with 0.1 s of sampling time, t. As an experimental platform,
the Zagi UAV is chosen and Kalman filter applications are performed with respect
to the dynamics and characteristics of this UAV.
During simulations, for testing the RKF algorithm, four kinds of measurement
malfunction scenarios are considered: instantaneous abnormal measurements, con-
tinuous bias, measurement noise increment, and fault of zero output. Also, in case
of measurement faults, simulations are also performed for the OKF to compare the
results with the RKF algorithms and understand the efficiency of the RKF in more
detail. Nonetheless, 2˛,s is taken as 16.919, and this value comes from a Chi-square
distribution when the number of degrees of freedom is 9 and the reliability level is
95 %.
5.4 Comparison of the R-adaptation Techniques 77

In this chapter, the first set of figures gives the OKF or RKF state estimation
results and the actual values are compared. The second set of figures shows the
error of the estimation process based on the actual state values of the UAV. The last
set of figures indicates the variance of the estimation that corresponds to the related
diagonal element of the covariance matrix of estimation errors.

5.4.1 Instantaneous Abnormal Measurements

Instantaneous abnormal measurements are simulated by adding a constant term

to the pitch angle, , with measurement every 20th second. It is obvious from
Figs. 5.1, 5.2, and 5.3 that both RKF algorithms (with SMNSF and MMNSF) give
more accurate estimation results than the OKF.
The results obtained by the regular OKF are not reliable when the obtained
measurements contain an error. However, the RKF with SMNSF and MMNSF
maintain their estimation characteristic for the whole process and affords precise
estimation outputs in case of abnormal measurements, as well as normal operational
conditions.

Fig. 5.1 Estimation result for the pitch angle  by the regular optimal Kalman filter (OKF) in case
of instantaneous abnormal measurements
78 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Fig. 5.2 Estimation result for the pitch angle  by the robust Kalman filter (RKF) with a single
measurement noise scale factor (SMNSF) in case of instantaneous abnormal measurements

The superiority of the RKF algorithms may also be seen in Table 5.1, which gives
the absolute estimation errors of the filters for the indicated time step. Note that, for
all the tables given hereafter, the highlighted results are obtained at time steps where
abnormal measurements are implemented.
When the measurements are faulty, the RKF with SMNSF compensates by
increasing its single scale factor and all of the measurements are taken into
consideration with a small weight for these time steps. Besides, the RKF with
MMNSF ensures the robustness of the filter by increasing related scale factors of the
matrix individually. Increment of the related scale factors brings about a decrement
in the related components of the Kalman gain, so as to reduce the corrective effect
of the innovation sequences of the faulty measurements on the state estimation
process. Progress may be understood better by examining the scale factor of the
RKF with SMNSF and one of the scalar measures (trace, determinant, and the
maximum eigenvalue) of the scale matrix of the RKF with MMNSF. The trace of
the scale matrix is examined in Table 5.2 as a scalar measure. Similar results have
been obtained when the fault is implemented in the other measurement channels.
5.4 Comparison of the R-adaptation Techniques 79

Fig. 5.3 Estimation result for the pitch angle  by the RKF with multiple measurement noise scale
factors (MMNSF) in case of instantaneous abnormal measurements

Even though both of the RKFs give similar estimation results in case of
instantaneous abnormal measurements, the RKF with MMNSF can be thought
of as a more advantageous algorithm, as it takes the faulty measurements into
account individually. Disregarding all of the measurements, as the RKF with
SMNSF does, affects the estimation procedure of all the states. However, since
the abnormal measurement is implemented in only one measurement channel,
leaving out the related measurement would be more significant, especially when
the filter mathematical model of the UAV, which is constructed with the decoupled
longitudinal and lateral dynamics, is taken into consideration.

5.4.2 Continuous Bias of Measurements

The continuous bias term is formed by adding a constant term to the measurement
of pitch angle, , in between the 30th and 60th seconds. As Fig. 5.4 shows,
again, the OKF fails in estimating states accurately. In contrast, the RKF with
SMNSF derogates estimation errors to a perceptible degree and gives relatively
better estimation outputs (Fig. 5.5). In this case, the RKF with MMNSF gives
80 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Table 5.1 Comparison of absolute estimation errors of regular the optimal Kalman filter (OKF),
robust Kalman filter (RKF) with a single measurement noise scale factor (SMNSF), and RKF
with multiple measurement noise scale factors (MMNSF) in case of instantaneous abnormal
measurements
Parameter Absolute values of Absolute values of Absolute values of
error for regular error for RKF with error for RKF with
OKF SMNSF MMNSF
40 s 70 s 40 s 70 s 40 s 70 s
Δu (m / s) 3.4509 0.0609 0.0738 0.1009 0.0746 0.0093
Δw(m / s) 1.0658 0.0715 0.0419 0.177 0.0148 0.009
Δq (rad /s ) 0.1541 0.0402 0.0011 0.1155 0.0905 0.0407
Δθ (rad ) 2.6624 0.0139 0.003 0.0504 0.0165 0.0073
Δh(m) 2.9367 0.0266 0.0203 0.1519 0.0492 0.0075
Δβ (rad ) 0.0318 0.0283 0.0448 0.0015 0.0137 0.0145
Δp (rad / s ) 0.006 0.0009 0.0047 0.0193 0.0894 0.0833
Δr (rad / s ) 0.0085 0.0587 0.0424 0.0241 0.0195 0.0049
Δφ (rad ) 0.04 0.0931 0.1455 0.0156 0.0037 0.0606

Table 5.2 MNSF investigation in case of instantaneous abnormal measurements

Parameter 20 s 40 s 45 s 50 s 60 s 90 s
Scale factor of 346 345 1 1 343 1
RKF with
SMNSF
Sk

Trace of scale 1,094 1,097 1 1 1,095 1

matrix of RKF
with MMNSF
trace ( Sk∗ )

superior estimation results to the RKF with SMNSF by reducing the effect of the
innovation term of  measurement and totally eliminating the estimation error
caused by the biased measurements of this channel (Fig. 5.6). As it disregards
only the related measurements, by the use of the innovative terms taken from all
other sensors operating correctly, it ensures accurate estimation outputs throughout
this period. It is possible to obtain the same results when the continuous bias is
implemented in the other measurement channels.
Comparison of the performance of the regular OKF, RKF with SMNSF, and
RKF with MMNSF in case of continuous bias of  measurement can be seen
in Table 5.3.
5.4 Comparison of the R-adaptation Techniques 81

Fig. 5.4 Estimation result for the pitch angle  by the regular OKF in case of continuous bias

In this table, the time means of absolute error values are given and they are
calculated for 30 s (between the 30th and 60th seconds for the fault case and between
the 65th and 95th seconds for the no fault case).
Again, it is possible to investigate the behavior of scale factors of both RKFs
(Table 5.4).

5.4.3 Measurement Noise Increment

In the third measurement malfunction scenario, measurement fault is characterized

by multiplying the variance of the noise of the velocity component u measurement
with a constant term between the 30th and 60th seconds. Figures 5.7, 5.8, and
5.9 show that the OKF outputs involve error, while the RKF algorithms achieve
estimation of the states accurately. In this case, it is not possible to detect
the superiority of the RKF with MMNSF by examining the estimation results,
although it is a known fact that individual increase of related scale factors, which
corresponds to the faulty measurements, is advantageous compared to disregarding
82 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Fig. 5.5 Estimation result for the pitch angle  by the RKF with SMNSF in case of continuous
bias

all measurements at the same time. Simulations give similar results when the noise
increment is implemented in the other measurement channels.
Table 5.5 shows a comparison of the mean of absolute estimation errors of the
regular OKF, RKF with SMNSF, and RKF with MMNSF in case of measurement
noise increment.
Table 5.6 presents the variation of the scale factor and the scalar measures of the
scale matrix.

5.4.4 Fault of Zero Output

In the last measurement malfunction scenario, it is assumed that height, h, cannot
be measured and the related sensor gives “0” as the output. This sort of fault is
easily simulated by taking the h measurement as “0” for the filter algorithm
in between the 30th and 60th seconds. As shown in Fig. 5.10, the OKF fails in
estimating the height and gives a result of “0” in light of the measurement. Similarly,
despite working robustly at the initial steps of the measurement fault implementation
5.4 Comparison of the R-adaptation Techniques 83

Fig. 5.6 Estimation result for the pitch angle  by the RKF with MMNSF in case of continuous
bias

Table 5.3 Comparison of the mean absolute estimation errors of the regular OKF, RKF with
SMNSF, and RKF with MMNSF in case of continuous bias of measurements (for 30 s)
Parameter Absolute values of Absolute values of Absolute values of
error for regular error for RKF with error for RKF with
OKF SMNSF MMNSF
Fault No fault Fault No fault Fault No fault
Δu (m / s) 23.091 0.0906 1.9103 0.0982 0.0846 0.1066
Δw(m /s) 6.9494 0.0605 0.3337 0.0603 0.0637 0.06
Δq (rad /s ) 0.7806 0.0535 0.2153 0.0518 0.0512 0.0515
Δθ (rad ) 6.0079 0.027 0.1473 0.0261 0.0225 0.0276
Δh(m) 12.9792 0.0792 2.6621 0.0855 0.082 0.086
Δβ (rad ) 0.0248 0.0227 0.0083 0.0212 0.023 0.0215
Δp (rad /s ) 0.0314 0.0327 0.0016 0.0311 0.0267 0.0332
Δr (rad / s ) 0.0238 0.0255 0.0078 0.0233 0.0219 0.0249
Δφ (rad ) 0.0424 0.0441 0.009 0.0375 0.0327 0.0403
84 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Table 5.4 MNSF investigation in case of continuous bias of measurements

Parameter 20 s 40 s 45 s 50 s 60 s 90 s

Scale factor of 1 290 303 294 297 1

RKF with
SMNSF Sk

Trace of scale 1 6,583 9,857 13,123 18,294 1

matrix of RKF
with MMNSF
trace ( Sk∗ )

Fig. 5.7 Estimation result for the velocity component u by the regular OKF in case of measure-
ment noise increment

(between the 30th and 40th seconds approximately), the RKF with SMNSF accepts
the h measurement as a true value and estimates it as “0” (Fig. 5.11).
Clearly, not taking healthy measurements into consideration throughout this
period affects the estimation procedure of the RKF with SMNSF. However, the RKF
5.5 Remark on Stability 85

Fig. 5.8 Estimation result for the velocity component u by the RKF with SMNSF in case of
measurement noise increment

with MMNSF performs well under these circumstances and, by disregarding only
the faulty h measurement and using innovative terms taken from all other sensors
operating correctly, accurately estimates all of the states (Fig. 5.12). This behavior
can also be seen from Table 5.7.
Nonetheless, the behavior of the RKFs can be observed by investigating the
variation of scale factor and the scalar measures of the scale matrix (Table 5.8).
As can be seen, the scale factor takes a higher value than one only at the initial
steps of the malfunction scenario, while the matrix scalar measures conserve that
increment for the whole process.

5.5 Remark on Stability

Indeed, the proposed RKFs are not so different from the linear Kalman filter from
the point of view of structure; they may be assumed to be a modification. In
this sense, a similar approach to that of a linear Kalman filter may be followed
86 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Fig. 5.9 Estimation result for the velocity component u by the RKF with MMNSF in case of
measurement noise increment

Table 5.5 Comparison of the mean absolute estimation errors of the regular OKF, RKF with
SMNSF, and RKF with MMNSF in case of measurement noise increment (for 30 s)
Parameter Absolute values of Absolute values of Absolute values of
error for regular error for RKF with error for RKF with
OKF SMNSF MMNSF
Fault No fault Fault No fault Fault No fault
Δu (m / s) 4.0183 0.0901 0.1571 0.0959 0.0836 0.112
Δw(m / s) 1.2699 0.0656 0.0302 0.0644 0.052 0.0656
Δq (rad / s ) 0.1462 0.056 0.0197 0.0562 0.0524 0.0557
Δθ (rad ) 0.4319 0.0287 0.0177 0.0266 0.0243 0.0275
Δh(m) 1.336 0.0869 0.1411 0.0913 0.0818 0.0905
Δβ (rad ) 0.0227 0.0226 0.0328 0.0222 0.0225 0.0224
Δp (rad / s ) 0.0295 0.0314 0.0125 0.0308 0.0283 0.0316
Δr (rad / s ) 0.0233 0.0231 0.03 0.0214 0.0198 0.0221
Δφ (rad ) 0.0455 0.0344 0.0443 0.0355 0.0388 0.0395
5.5 Remark on Stability 87

Table 5.6 MNSF investigation in case of measurement noise increment

Parameter 20 s 40 s 45 s 50 s 60 s 90 s

Scale factor of 1 279 1 254 6308 1

RKF with
SMNSF Sk

Trace of scale 1 9,457 3,973 18,241 4,935 1

matrix of RKF
with MMNSF
trace ( Sk∗ )

Fig. 5.10 Estimation result for height h by the regular OKF in case of fault of zero output

for the stability analysis of the RKFs. From [21], it is known that the following
characteristic polynomial of the system can be used for stability analyses of a linear
Kalman filter:

Characteristic polynomial D ŒzI  fFk  Kk Hk Fk g : (5.14)

where z denotes the usual z-transform variable.

88 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Fig. 5.11 Estimation result for height h by the RKF with SMNSF in case of fault of zero output

The roots of this polynomial provide information about the filter stability. If all
the roots lie inside the unit circle in the z-plane, the filter is stable; conversely,
if any root lies on or outside the unit circle, the filter is unstable. As a matter
of terminology, the roots of the characteristic polynomial are the same as the
eigenvalues of:

ŒfFk  Kk Hk Fk g (5.15)

Nonetheless, we assume that a similar investigation is achieved for the RKFs. Let’s
examine the case for the RKF with MMNSF. Hence, if we substitute the necessary
equality instead of Kalman filter gain, Eq. (5.15) becomes:
For the RKF with SMNSF:
n h  1 i o
Fk  Pk=k1 HkT Hk Pk=k1 HkT C Sk Rk Hk F k (5.16)
5.5 Remark on Stability 89

Fig. 5.12 Estimation result for height h by the RKF with MMNSF in case of fault of zero output

Table 5.7 Comparison of the mean absolute estimation errors of the regular OKF, RKF with
SMNSF, and RKF with MMNSF in case of fault of zero output (for 30 s)
Parameter Absolute values of Absolute values of Absolute values of
error for regular error for RKF with error for RKF with
OKF SMNSF MMNSF
Fault No fault Fault No fault Fault No fault
Δu (m /s) 3.45 0.1313 3.129 1.6054 0.1677 0.096
Δw(m /s) 1.1045 0.0709 0.744 0.3416 0.0621 0.0606
Δq (rad /s ) 0.1388 0.048 0.3018 0.2252 0.0554 0.0529
Δθ (rad ) 0.2646 0.0297 0.3057 0.2664 0.0357 0.0278
Δh(m) 80.984 0.0852 71.2394 17.6275 0.9192 0.0954
Δβ (rad ) 0.0223 0.0236 0.0306 0.019 0.0211 0.0246
Δp (rad / s ) 0.0317 0.0336 0.0245 0.0204 0.0316 0.0321
Δr (rad / s ) 0.0235 0.0215 0.0273 0.0214 0.0187 0.0221
Δφ (rad ) 0.0413 0.0384 0.041 0.0331 0.0404 0.041
90 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Table 5.8 MNSF investigation in case of fault of zero output

Parameter 20 s 40 s 45 s 50 s 60 s 90 s

Scale factor of 1 36.0551 1 1 1 1

RKF with
SMNSF
Sk

Trace of scale 1 125,149 158,130 192,833 275,062 1

matrix of RKF
with MMNSF
trace ( Sk∗ )

For the RKF with MMNSF:

n h  1 i o
Fk  Pk=k1 HkT Hk Pk=k1 HkT C Sk Rk Hk F k (5.17)

Now we should search for the roots of Eqs. (5.16) and (5.17). The realized
simulations for every malfunction case showed that both RKFs are stable in case
of a measurement malfunction. Nonetheless, the RKF with MMNSF is shown to be
more stable and, usually, only one root reaches near a point of the unit cycle limit.
In Fig. 5.13, the pole-zero map of the RKF with MMNSF is given.
The pole-zero map is obtained for the instantaneous abnormal measurement
malfunction case at the 20th second, where the malfunction is implemented. It is
not very different from that found for the OKF in the same sampling step. During
adaptation with MMNSF, related roots (that belong to the states, the innovation
channel of which is corrected via the scale matrix) moves through the unit cycle limit
but they do not approach the marginal stability border very closely. On the other
hand, for the RKF with SMNSF, where the whole innovation vector is disregarded in
case of malfunction, all of the roots move through the stability border (see Fig. 5.14)
and, usually, one of them takes a nearly marginal (jzj D 1) value, so the stability
characteristics worsen.
The simulation results indicate that the proposed RKFs are not at risk of con-
vergence; namely, they are stable. In order to guarantee the accuracy of simulation
outputs, some even harsher malfunction cases are tested (far greater malfunctions
are implemented). It is seen that the magnitude of the malfunction is not a factor
which may break down the stability of the filter and the roots of the RKFs stay
inside the unit cycle for every condition.
Moreover, the simulation results indicated the better stability characteristics of
the RKF with MMNSF compared to the RKF with SMNSF. For every malfunction
case, the roots of the RKF with MMNSF do not approach the stability limit (unit
cycle) as much those of the RKF with SMNSF.
5.6 Conclusion and Discussion 91

Fig. 5.13 Pole-zero map of the RKF with MMNSF for the case of instantaneous abnormal
measurement (at the 20th second)

5.6 Conclusion and Discussion

In this chapter, RKF algorithms with filter gain correction for the case of mea-
surement malfunctions at the current estimation step have been developed. By the
use of defined variables named scale factors, faulty measurements are taken into
consideration with small weight and the estimations are corrected without affecting
the characteristics of the accurate measurements. In the presented RKFs, the filter
gain correction is performed only in the case of malfunctions in the measurement
system.
RKF algorithms with single and multiple MNSF are proposed. In the first case,
the filter is adapted by using single scale factor as a corrective term on the filter
gain, and, in the second, a scale matrix composed of multiple factors is used to fix
the relevant terms of the Kalman gain matrix individually.
The proposed RKF algorithms with single and multiple MNSF are applied for the
state estimation process of a UAV platform. Algorithms are tested for four different
measurement malfunction scenarios and the results are compared with the outputs
of the OKF for the same cases: instantaneous abnormal measurements, continuous
bias of measurements, measurement noise increment, and fault of zero output. In all
92 5 Estimation of the UAV Dynamics in the Presence of Sensor Faults

Fig. 5.14 Pole-zero map of the RKF with SMNSF for the case of instantaneous abnormal
measurement (at the 20th second)

four scenarios, the OKF becomes faulty while the introduced RKF algorithms stand
robust to the measurement faults. Comparison of the simulation results show that
the performance of the RKF with MMNSF is better than the RKF with SMNSF. The
advantage of the RKF with MMNSF can be especially seen in cases of continuous
bias of measurements and fault of zero sensor output. Nonetheless, simulations for
the stability analyses indicated that both RKF algorithms are stable in case of a
malfunction, but the stability characteristics of the RKF with MMNSF are better
than the RKF with SMNSF.
The proposed approach does not require a priori statistical characteristics of
the faults and can be used for both linear and nonlinear systems. Furthermore,
the presented RKF algorithms are easily implemented in practice and their com-
putational burden is not heavy. These characteristics make the introduced RKF
algorithms extremely important from the point of view of supplying reliable
parameter estimations for the flight control system of an autonomous UAV.
References 93

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Chapter 6
Estimation of Unmanned Aerial Vehicle
Dynamics in the Presence of Sensor/Actuator
Faults

6.1 Introduction

An approach for designing reliable systems is to use high-quality components.

However, using high-quality components alone does not always reduce the likeli-
hood of system failures, and additional methods must be provided to tolerate faults
in the system.
A flight control system is essentially a multi-input/output feedback control
system with several nested feedback loops which are designed to meet different
performance specifications related to stability augmentation, automatic guidance
and navigation, and thrust management. The failure detection and isolation algo-
rithms used in flight computers are implemented with digital computers and are,
basically, signal-processing techniques. The reliability of the computer and physical
components is very important for the operation of the actual system. A failure may
occur in computer subsystems, lanes, sensors, control surfaces, and/or actuators.
Since we are examining the failure detection problem from the control engineers’
point of view, it is assumed that the failures do not occur in computing subsystems
and lanes, but in actuators, control surfaces, and sensors.
In [1], an approach is presented to detect and isolate an aircraft’s sensor/actuator
faults affecting the mean of the Kalman filter innovation. The effects of the sensor
and actuator faults in the innovation are investigated and a decision approach to
isolate the fault is proposed. Although it proposes to be an effective method for
fault isolation, this work examines the sensor and actuator faults affecting only the
mean of the innovation, so the method works under some certain conditions and it
is not appropriate for applying in case of various different fault types. Nonetheless,
the given measurement noise covariance adaptation procedure isdifficult to realize.

© Springer International Publishing Switzerland 2015 95

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_6
96 6 Estimation of the UAV Dynamics in the Presence of Sensor/Actuator Faults

Another approach is to scale the noisec ovariance matrix by multiplying it by a

time-dependent variable. An adaptive Kalman filter is used to tune the measurement
and process noise covariance matrices R and Q, respectively. The philosophy of
estimating (or scaling) the Q and R matrices follows one of these scenarios [2]:
• Fixing Q and varying R by trial and error until the realistic values are found, such
that the filter gives stable state estimates. In this case, the Q matrix should be
completely known.
• Varying the Q matrix if R is completely known and fixed to a certain value.
• Varying Q and R simultaneously; in this case, neither is known.
When the process noise covariance matrix is scaled (Q-adaptation), this means
that there is an actuator fault to be compensated. Otherwise, when the measurement
noise covariance matrix is scaled (R-adaptation), it is a sensor fault, which is dealt
with. In [3, 4], an adaptation procedure based on the single fading factor and
in [5], a scheme based on multiple fading factors are presented. The essence of
using multiple fading factors is the different effect of the fault on the estimation
performance of each estimated state. Especially for complex multivariable systems,
the effect of the fault on each state should be investigated carefully and, instead of
a single factor, a matrix formed of multiple fading factors (such that it weights the
adaptation differently for each state) should be used. Nonetheless, as a disadvantage,
these papers [3–5] take only the Q-adaptation procedure into consideration and do
not examine the R-adaptation methods. Per contra, the R-adaptation is investigated
for possible sensor faults in [6–9] with a considerably simpler technique, but the
Q-adaptation is not investigated. In these studies, it is shown that, if there is a
malfunction in the measurement system, the robust adaptive Kalman filter (RAKF)
algorithm can be utilized and, by the use of a measurement noise scale factor
(MNSF) as a multiplier of the measurement noise covariance, insensitiveness of
the filter to the current sensor faults can be ensured. In [10], a similar adaptation
algorithm is applied to the unscented Kalman filter for the nonlinear estimation
problem, just as for the R-adaptation.
Moreover, there are also published works in the literature which are interested
in both the R- and Q-adaptations [11–13]. However, in these papers, an isolation
scheme for the fault is absent; it is not described how to operate an RAFK in
the presence of sensor/actuator faults. These filters can detect whether there is a
fault in the system or not, but the type of fault cannot be determined. Instead
of using a fault isolation scheme, a simultaneous adaptation procedure for the R-
and Q-adaptations is performed in these studies. However, as indicated in [2], full
estimation of R and Q based on covariance matching is questionable, since these
two values rely on each other in the covariance matching equation. Consequently,
these filters cannot estimate the unmanned aerial vehicle (UAV) dynamics reliably
in the presence of sensor/actuator faults. This problem can be solved through the
combined fault isolation and robust adaptive filtration techniques. Besides, the
methods proposed in these papers are all based on the single adaptive factor and,
for complex multivariable systems, they may not be sufficiently accurate.
6.2 Q-adaptation Using Multiple Adaptive Factors 97

In this chapter, an RAKF is developed that incorporates the R- and Q-adaptation

procedures and adapts itself against sensor/actuator faults in order to ensure
robustness. The filter first detects the type of fault (either sensor or actuator) and
then isolates it. After that, the appropriate adaptation (R or Q) is applied such
that the estimation characteristic is not deteriorated. Unlike the existing studies,
the proposed adaptation methods for both the R- and Q-adaptations are simplistic
and, with a small modification, can be applied easily to the Kalman filter. Also in
this chapter, it is shown how to incorporate these two different adaptation processes
such that the fault is isolated and, so, the RAKF algorithm is a complete estimator
including the fault detection and isolation processes for the case of sensor/actuator
faults. The RAKF is applied for the state estimation procedure of a UAV and its
effectiveness is investigated by comparisons with the optimal Kalman filter (OKF)
in case of faults.

6.2 Q-adaptation Using Multiple Adaptive Factors

When there is an actuator fault in the system, it results in changes in the control
distribution matrix. In a similar manner to the R-adaptation procedure, in case
of an actuator fault, the real error of the innovation covariance will exceed the
theoretical covariance. Thus, the basic premise of the Q-adaptation is also to obtain
an appropriate multiplier matrix for the Q matrix such that the real and theoretical
values of the innovation covariance match. Hence, if a fading matrix (ƒk ) composed
of multiple fading factors is added into the algorithm as [14–17]:

1 X  
k
eQ j eQ Tj D Hk Fk Pk1=k1 FkT C ƒk Gk Qk GTk HkT C Rk (6.1)
 jDkC1

then the fading matrix can be determined as:

0 1
1 X k
 1
ƒk D Hk1 @ eQ j eQ T  Hk Fk Pk1=k1 Fk T HkT  Rk A Gk Qk Gk T HkT
 jDkC1 j
(6.2)

For a specific case where all the states are measured as .Hk D I/, as in the case
examined here, Eq. (6.2) reduces to:
0 1
1 X k
 1
ƒk D @ eQ j eQ T  Fk Pk1=k1 Fk T  Rk A Gk Qk Gk T (6.3)
 jDkC1 j
98 6 Estimation of the UAV Dynamics in the Presence of Sensor/Actuator Faults

In a similar manner to the R-adaptation, the obtained fading matrix should be

diagonalized, since the Q matrix must be a diagonal, positive-definite matrix:
 
ƒ D diag 1 ; 2 ; : : : ; n ; (6.4)

where:

i D max f1; ƒii g i D 1; : : : ; n: (6.5)

Here, ƒii represents the ith diagonal element of the matrix ƒ. Apart from that point,
if there is an actuator fault in the system, ƒk must be included in the estimation
process as:

Pk=k1 D Fk Pk1=k1 FkT C ƒk Gk Qk GTk (6.6)

6.3 Integration Scheme for the Q- and R-adaptation Methods

Due to the scale or fading matrix, the covariance of the estimation error of the RAKF
increases in comparison with the OKF. Therefore, the RAKF algorithm is used only
when the there is a fault and, in all other cases, the procedure is run optimally with
the regular OKF. The fault detection process is applied by checking a statistical
function at each estimation step. At that point, the two following hypotheses may be
introduced:
• 0 ; the system is operating normally
• 1 ; there is a malfunction in the estimation system
In order to detect the faults, the statistical function used is defined as:
 1
ˇk D eQ Tk Hk Pk=k1 HkT C Rk eQ k : (6.7)

This statistical function has a 2 distribution with M degrees of freedom (M is

the dimension of the innovation vector).
If the level of significance, ˛, is selected as:
˚ 
P 2 > 2˛;M D ˛I 0 < ˛ < 1; (6.8)

then the threshold value, 2˛,M , can be found. Hence, when the hypothesis 1 is
correct, the statistical value of ˇ k will be greater than the threshold value 2˛,M , i.e.:

0 W ˇk  2˛;M 8k
(6.9)
1 W ˇk > 2˛;M 9k:
6.4 Numerical Example 99

On the other hand, after detecting the fault in the system, the key point is
detecting the type of fault (either a sensor or an actuator fault). After that, the
appropriate adaptation (R- or Q-adaptation) may be applied [15–17]. The fault
isolation can be realized by an algorithm similar to that proposed for aircraft in
[18]. When a regular Kalman filter is used, the decision statistics change regardless
of whether the fault is in the sensors or in the actuators. If an RKF based on the
Doyle–Stein condition is used, it is easy to distinguish between sensor and actuator
faults. The Kalman filter that satisfies the Doyle–Stein condition is referred in [18],
as the RKF is insensitive to actuator failures and may also be used for fault isolation
in our study. After that, the appropriate adaptation (R- or Q-adaptation) may be
applied. The Doyle–Stein condition is as follows:

K.I C Hs K/1 D Bs .Hs Bs /1 : (6.10)

Here, K is the filter gain, I is a unit matrix, H is the system measurement matrix, Bs
is the control distribution matrix in continuous time, and s D .sI  Fs /1 , where
Fs is the system matrix in continuous time.
The RKF is very useful for isolating sensor and control surface failures, as it is
insensitive to the latter failures. If the Kalman filter process noise covariance matrix
is chosen as

Q .qr / D Q C q2r Bs VBTs (6.11)

then the filter becomes robust against the actuator faults. Here, Q is the process noise
covariance matrix for the nominal plant, qr is a parameter that approaches to infinity
as the filter’s robustness increases, and V is a positive-definite symmetric matrix.
If a sensor fault occurs, the R-adaptation is realized; else, the procedure is
continued with the Q-adaptation. The algorithm architecture given in Fig. 6.1
summarizes the integrated R- and Q-adaptation procedures. This new filter, where
the robust and adaptive Kalman filters are integrated, can be called the RAKF.

6.4 Numerical Example

In order to examine the effectiveness of the proposed RAKF, it was applied for the
state estimation of the Zagi UAV. The simulations are realized in 1,000 steps for a
period of 100 s with 0.1 s of sampling time, t.
In case of sensor/actuator fault, the simulations are also carried out for the OKF
so as to compare the results with the RAKF algorithm and understand better the
efficiency of the proposed algorithm.
Nonetheless, for the fault detection process, 2˛,M is taken as 21.7, and this value
comes from the Chi-square distribution when the number of degrees of freedom is
9 and the reliability level is 99 %.
100 6 Estimation of the UAV Dynamics in the Presence of Sensor/Actuator Faults

Fig. 6.1 System architecture for the robust adaptive Kalman filter (RAKF) algorithm

All the figures given in this section consist of three subfigures. The top
figure gives the OKF or RAKF state estimation results and the actual values for
comparison. The middle figure shows the error of the estimation process based on
6.4 Numerical Example 101

the actual state values of the UAV. The bottom figure indicates the variance of the
estimation that is obtained as the diagonal elements of the Kalman filter estimation
error covariance matrix.
During the simulations, for testing the fault detection and isolation procedures
and also the RAKF algorithm, two different faults are implemented to the systems.
Firstly, a sensor fault is formed by adding a constant term to the measurement of the
pitch angle (the constant term is nearly of the same magnitude as the measurement,
so the measurement’s magnitude is doubled for those instants of time), , in
between the 30th and 50th seconds. Furthermore, in order to check that fault
isolation can be achieved successfully and switching from one adaptation stage to
the other is realized properly, as another fault, the actuator fault is simulated by
taking the first column elements of the longitudinal control distribution matrix to
be nearly zero in between the 80th and 90th seconds. The graph for the statistical
values of ˇ k in case of using the OKF is shown in Fig. 6.2. As can be seen, except
for the periods where the sensor and actuator failures occur, ˇ k is lower than the
threshold value.
On the contrary, when one of these failures occurs, ˇ k grows rapidly and exceeds
the threshold value. Hence, the 1 hypothesis is judged to be true and it is shown
that, by using such a procedure, the fault can be detected. The results for the fault
isolation process are given in Fig. 6.3.

Fig. 6.2 Behavior of the statistics ˇ k in case of measurement failure (in between the 30th and 50th
seconds) and actuator failure (in between the 80th and 90th seconds) when the optimal Kalman
filter (OKF) is used
102 6 Estimation of the UAV Dynamics in the Presence of Sensor/Actuator Faults

Fig. 6.3 Behavior of the statistics ˇ k in case of measurement failure (in between the 30th and 50th
seconds) and actuator failure (in between the 80th and 90th seconds) when the robust Kalman filter
(RKF), which is insensitive to actuator failures, is used

As mentioned, the RKF, which is insensitive to actuator failures, is used for this
isolation process. The behavior of the statistics ˇ k in the case of sensor/actuator
failures shows that detecting the actuator failure is not possible when the RKF is
used, since it is insensitive against such faults; on the other hand, the sensor failure
is detected immediately. Hence, the fault isolation can be realized by using such a
procedure. The RKF, which is insensitive to actuator failures and sensitive to sensor
failures, is employed to isolate the sensor and actuator failures in the proposed
RAKF algorithm (see Fig. 6.1).
The results of the first fault scenario realized for testing the RAKF algorithm in
the presence of sensor/actuator failures are given in Figs. 6.4 and 6.5. It is clear that
the RAKF gives superior estimation results compared to the OKF in both cases.
The type of fault is precisely detected and the RAKF tunes itself according to the
appropriate adaptation rule. The RAKF algorithm first detects the fault by using the
fault detection process given with Eqs. (6.7), (6.8), and (6.9). Then, it isolates the
fault using the RKF based on the Doyle–Stein condition, such that it distinguishes
the sensor and actuator faults. As the final step, it adapts the R matrix in between
the 30th and 50th seconds and the Q matrix in between the 80th and 90th seconds.
6.4 Numerical Example 103

velocity "u" estimation

10 Kalman Estimation
u(m/s)

Actual Value
0

-10
0 10 20 30 40 50 60 70 80 90 100

5
error(m/s)

-5

-10
0 10 20 30 40 50 60 70 80 90 100
variance [(m/s)2]

0.5

0
0 10 20 30 40 50 60 70 80 90 100
time(sec)
Fig. 6.4 u velocity estimation via the OKF in case of actuator and bias sensor faults

As seen from the figures, it ensures accurate estimation results for the whole process,
while the OKF fails in both the sensor fault and the actuator fault conditions. This
proves us three facts:
– The fault detection process is working correctly and the fault in the system is
detected without any time delay
– By using the RKF based on the Doyle–Stein condition, the type of fault is
distinguished and the RAKF decides to use the right adaptation procedure against
the fault
– Both of the proposed adaptation methods are functioning properly
Nevertheless, the performance of both filters, the OKF and the RAKF, in case of
fault can be understood more clearly by examining Table 6.1.
In the table, the mean absolute values of error, which are calculated for 10 s,
are given for all three conditions: sensor fault, actuator fault, and no fault. The
tabulated results show that the estimations by the OKF worsen for all of the
longitudinal states in case of sensor/actuator fault, while the RAKF retains accurate
104 6 Estimation of the UAV Dynamics in the Presence of Sensor/Actuator Faults

velocity "u" estimation

10
Kalman Estimation
u(m/s)

Actual Value
0

-10
0 10 20 30 40 50 60 70 80 90 100

5
error(m/s)

-5

-10
0 10 20 30 40 50 60 70 80 90 100
variance [(m/s)2]

0.5

0
0 10 20 30 40 50 60 70 80 90 100
time(sec)
Fig. 6.5 u velocity estimation via the RAKF in case of actuator and bias sensor faults

Table 6.1 Comparison of the mean absolute values of error for the optimal Kalman filter (OKF)
and the robust adaptive Kalman filter (RAKF) in case of actuator and bias sensor faults
Mean absolute value of error for regular
Parameter OKF Mean absolute value of error for RAKF
Operating Operating
Sensor fault normally Actuator fault Sensor fault normally Actuator fault
u(m/s) 1.8799 0.1891 0.8116 0.0528 0.0578 0.1815
w(m/s) 0.6762 0.0931 1.1939 0.0649 0.0611 0.2678
q(rad/s) 0.0604 0.0621 0.1602 0.0521 0.0542 0.0686
 (rad) 0.6497 0.053 0.0796 0.0161 0.0132 0.0325
h(m) 0.5493 0.1059 0.1273 0.0644 0.0636 0.0932
ˇ(rad) 0.0319 0.0302 0.0338 0.0123 0.0155 0.0165
p(rad/s) 0.0514 0.0556 0.0538 0.0282 0.0257 0.0304
r(rad/s) 0.0508 0.045 0.0465 0.0115 0.0146 0.0163
(rad) 0.0758 0.0786 0.0805 0.0167 0.0192 0.0349
6.4 Numerical Example 105

velocity "u" estimation

10
Kalman Estimation
u(m/s)

Actual Value
0

-10
0 10 20 30 40 50 60 70 80 90 100

5
error(m/s)

-5

-10
0 10 20 30 40 50 60 70 80 90 100
variance [(m/s)2]

0.5

0
0 10 20 30 40 50 60 70 80 90 100
time(sec)
Fig. 6.6 u velocity estimation via the OKF in case of actuator and noise increment sensor faults

estimation behavior for the whole process. Because of the decoupled dynamics used
for the UAV, any fault in the longitudinal channel (e.g., sensor fault for the sensor
measuring  or the actuator fault affecting the longitudinal control derivatives)
does not have an effect on the lateral states as expected. This fact makes the
importance of using multiple adaptive factors for both adaptation procedures more
distinct.
As the second testing scenario for the RAKF algorithm, this time, the sensor
fault is characterized by multiplying the variance of the noise of the measurement
for the u velocity component with a constant term in between the 30th and 50th
seconds, while the actuator fault is the same as the first scenario. As Figs. 6.6 and
6.7 show, again, the OKF outputs involve error, while the RAKF algorithm achieves
estimation of the states accurately. Nonetheless, another consequence of the second
fault scenario is the good performance of the fault isolation scheme; regardless of the
type of sensor fault, the isolation scheme can distinguish the type of fault effectively
and the RAKF works properly.
In order to clarify the results, a comparison of the mean absolute values of error
is given in Table 6.2.
106 6 Estimation of the UAV Dynamics in the Presence of Sensor/Actuator Faults

velocity "u" estimation

10
Kalman Estimation
u(m/s)

Actual Value
0

-10
0 10 20 30 40 50 60 70 80 90 100

5
error(m/s)

-5

-10
0 10 20 30 40 50 60 70 80 90 100
variance [(m/s)2]

0.5

0
0 10 20 30 40 50 60 70 80 90 100
time(sec)
Fig. 6.7 u velocity estimation via the RAKF in case of actuator and noise increment sensor faults

Table 6.2 Comparison of the mean absolute values of error for the OKF and RAKF in case of
actuator and noise increment sensor faults
Mean absolute value of error for regular
Parameter OKF Mean absolute value of error for RAKF
Operating Operating
Sensor fault normally Actuator fault Sensor fault normally Actuator fault
u(m/s) 0.6235 0.2317 0.8352 0.0819 0.0635 0.1303
w(m/s) 0.2588 0.1482 1.1963 0.0687 0.0626 0.2621
q(rad/s) 0.0611 0.0673 0.1505 0.055 0.0508 0.0711
 (rad) 0.0653 0.0621 0.0688 0.0115 0.0155 0.0376
h(m) 0.1101 0.1 0.1323 0.0826 0.0733 0.0785
ˇ(rad) 0.0355 0.0392 0.0319 0.0163 0.0149 0.0134
p(rad/s) 0.0497 0.0465 0.0553 0.0308 0.026 0.0265
r(rad/s) 0.0456 0.0443 0.0449 0.0135 0.013 0.0112
(rad) 0.08 0.0694 0.0695 0.0262 0.0196 0.0359
References 107

6.5 Conclusion and Discussion

In this chapter, first, the Q-adaptation with multiple adaptive factors for actuator
fault compensation is presented. Then, the RAKF with the R- and Q-adaptations
against sensor/actuator failures is proposed.
The proposed RAKF algorithm with the multiple adaptive factors incorporates
the R- and Q-adaptation procedures and adapts itself against sensor/actuator faults
in order to ensure robustness. The filter first detects the type of fault (either sensor
or actuator) and then isolates it. After that, the appropriate adaptation (R- or Q-
adaptation) is applied such that the estimation characteristic is not deteriorated.
Unlike existing studies, the proposed adaptation methods for both the R- and Q-
adaptations are simplistic and can be applied easily with a small modification to the
Kalman filter.
In this chapter, it has been shown how to incorporate these two different
adaptation processes such that the fault is isolated and, so, the RAKF algorithm
is a complete estimator including fault detection and isolation processes for the
case of sensor/actuator faults. When a regular Kalman filter is used, the decision
statistics change regardless of whether the fault is in the sensors or in the actuators,
whereas when an RKF based on the Doyle–Stein condition is used, it is easy to
distinguish the sensor and control surface/actuator faults. The RAKF is applied for
the state estimation procedure of a UAV and its effectiveness is investigated by
comparisons with the OKF in case of faults. The proposed RAKF algorithm may
play an important role for UAV control systems since it gives accurate estimation
results despite the sensor or actuator faults. Regarding the harsh environment where
UAVs are generally used, encountering any kind of fault has a high probability and,
so, preferring the proposed RAKF algorithm instead of the regular OKF may bring
a significant advantage.

References

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innovation sequence and robust Kalman filtering. Aerosp Sci Technol 4:415–422. doi:10.1016/
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2. Almagbile A, Wang J, Ding W (2010) Evaluating the performances of adaptive Kalman filter
methods in GPS/INS integration. J Glob Position Syst 9:33–40. doi:10.5081/jgps.9.1.33
3. Hu C, Chen W, Chen Y, Liu D (2003) Adaptive Kalman filtering for vehicle navigation. J Glob
Position Syst 2(1):42–47
4. Ding W, Wang J, Rizos C, Kinlyside D (2007) Improving adaptive Kalman estimation in
GPS/INS integration. J Navig 60(3):517–529. doi:10.1017/S0373463307004316
5. Geng Y, Wang J (2008) Adaptive estimation of multiple fading factors in Kalman filter for
navigation applications. GPS Solutions 12(4):273–279. doi:10.1007/s10291-007-0084-6
6. Hajiyev C (2007) Adaptive filtration algorithm with the filter gain correction applied to
integrated INS/radar altimeter. Proc Inst Mech Eng Part G J Aerosp Eng 221(5):847–855.
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7. Hajiyev Ch, Soken HE (2009) Adaptive Kalman filter with the filter gain correction applied
to UAV flight dynamics. In: Proceedings of the 17th Mediterranean conference on control and
automation, Thessaloniki, Greece, 24–26 June 2009
8. Hajiyev Ch, Soken HE (2009) Adaptive Kalman filter with multiple fading factors for UAV
state estimation. In: Proceedings of the 7th IFAC international symposium on fault detection,
supervision and safety of technical processes (SafeProcess 2009), Barcelona, Spain, 30 June–3
July, pp 77–82
9. Hajiyev C, Soken HE (2010) Robust estimation of UAV dynamics in the presence of
measurement faults. J Aerosp Eng 25(1):80–89
10. Soken HE, Hajiyev C (2010) Pico satellite attitude estimation via robust unscented Kalman
filter in the presence of measurement faults. ISA Trans 49(3):249–256. doi:10.1016/j.isatra.
2010.04.001
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navigation. J Navig 61(4):705–721. doi:10.1017/S0373463308004827
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adaptation. Aircr Eng Aerosp Technol Int J 81:343–352. doi:10.1108/00022660910967336
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unknown random bias. Int J Adapt Control Signal Process 20(7):305–319. doi:10.1002/acs.
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Kalman filter. In: Proceedings of the special international conference on complex systems:
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Macedonia, 16–20 September 2011, pp 311–320
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Chapter 7
Fault Detection, Isolation, and Data Fusion for
Unmanned Aerial Vehicle Air Data Systems

7.1 Introduction

For unmanned aerial vehicles (UAVs) to carry out their assigned tasks and complete
missions successfully, appropriate air measurement data should be derived in case
of fault occurrence in sensor measurements. In order to produce appropriate data,
the following should be done: detecting and isolating the faults in the measurement
system, determining the most appropriate data, using other measurement data for
estimation purposes, and deriving the most appropriate data using the available
measurements (decision algorithm). There have been many studies using different
techniques investigating the subject [1–10]. Some of them are given below.
In [1], a method using the extended Kalman filtering (EKF) structure is proposed
and the efficacy of the method is shown using a space vehicle dynamic model which
includes redundant sensors to measure the parameters. Although using multiple
extended Kalman filters together in nonlinear systems for fault detection is a known
method, generally, the initial value of the covariance matrix is chosen by expertise;
however, in this study, a new method for determining the optimal covariance matrix
is proposed.
Two different fusion methods for a UAV that integrate inertial navigation system
(INS) and accurate dynamic model measurements are proposed in [2]. In order
to reduce the growing errors of the INS and obtain better measurements, the
measurements of a dynamic model are integrated using EKF. In that study, the
errors between the INS and flight vehicle model (FVM) measurements are estimated
with the filtering technique using the error model of the system. The first proposed
method is based on comparing speed and Euler angle measurements produced by the
INS and the FVM. In this first method, inertial measurement unit (IMU) acceleration
and angular speed errors, INS speed and Euler angle errors, and FVM speed and
Euler angle errors are determined and used to obtain better measurement data. In the
second method, the estimated values of acceleration and angular speed taken from

© Springer International Publishing Switzerland 2015 109

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_7
110 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

the FVM are used for the calibration of the IMU. The errors in the acceleration and
angular speed measurements are estimated using an error model of the FVM.
In [3], a sensor fault detection method using data fusion techniques is inves-
tigated. In this study, the EKF method is used to integrate data. An adaptive
method which updates the sensor noise covariance matrix according to residual
values covariance is used in EKF. This new method is called the adaptive modified
extended Kalman filtering (AMEKF) algorithm. The AMEKF system estimates the
system noise and sensor noise covariance matrices, which can affect normal filter
working conditions, and finds calibration errors.
A weak model-based method for the detection of sensor abrupt faults is
considered in [4]. In that study, some characteristics of the sensors are used to derive
the necessary signal for fault detection. The method is simulated using a small
aircraft and an environment model, which includes sensor noise, wind effect, and
turbulence. In the simulation, angle of attack (AOA) sensor faults are considered.
In [5], a new method is developed for a nonlinear flight model to detect input
and output sensor faults. In the study, a polynomial residual generator structure is
used. This approach, which is robust against faults, can filter the error signal and
decrease the effect of the additive noise and error that occurs due to linearization and
measurement. The input errors in sensors are separated as known input, disturbance,
and faults vectors. The designed system describes each input–output relation by
polynomials and uses residuals to detect and isolate faults. Different fault detection
methods are also used for the same fault conditions to better evaluate the proposed
method. For this purpose, nonlinear geometric approach (NLGA), unknown input
Kalman observers (UIKF), and neural networks (NN) are investigated.
In [6], the NN method is used for fault detection and isolation (FDI), and is
applied to a nonlinear UAV model. In the study, a radial basis function (RBF) type
neural NN is designed for modeling purposes. A novel residual generation technique
is also used to overcome wrong fault error situations. A nonlinear UAV model is
considered and longitudinal equations are used to simulate the fault situation in the
pitch angle gyro.
In [7], a similar system to that used in [4] is investigated. The characteristics of
the sensor signal are again used for FDI. The design is the same as that described in
[4], except for the dynamic dependency filter structure, which changed from linear
autoregressive to pooled autoregressive (PAR). The idea comes from changing the
linear structure of the filter in [4] to obtain better results. The results are again
positive, as in [4].
In [8], interacting multiple model (IMM) and unscented Kalman filter (UKF)
methods are used with nonlinear aircraft dynamics for fault detection purposes.
The differences between using the INS equations and a dynamic model are shown
for the situations without faults. The IMM-UKF structure is proved to estimate the
states of the system in the presence of one faulty sensor. In conclusion, the IMM-
UKF structure works properly if used together for fault detection purposes and is an
effective new way of finding the faults in the system.
In [9], an FDI system is proposed for a nonlinear model. There are widely
used FDI algorithms for linear systems; however, these methods are not sufficient
7.2 Kalman Filter-Based Integrated ADS/GPS Navigation System 111

for nonlinear systems and can cause errors in fault detection. For this reason, an
input estimation structure that estimates the input values using inputs–outputs of the
system and IMU-INS measurement values is proposed in that study.
An observer/Kalman filter identification method for fault diagnosis in a small
autonomous helicopter is proposed in [10]. The helicopter model under nominal
conditions is found using input–output test values in an observer/Kalman filter.
The differences between the designed model and the system outputs let the users
determine the faults in the system. The model of the system, which is the helicopter
model in this case, can be determined by Markov parameters and a Hankel matrix
which has the same rank as the system.
In the given studies, different Kalman filter techniques (EKF, UKF, IMM-UKF,
AMEKF), model-based methods such as using sensor signal characteristics, residual
generators using polynomials, and NN, are used for state estimation and FDI. The
methods vary due to the difficulty of the problem (multiple sensor fault detection,
redundancy of sensors, etc.) where the FDI algorithm is used (aircraft, helicopter,
etc.) and the seriousness of the fault. From these studies, we can understand that the
Kalman filter technique can detect and isolate faults using different dynamic models
and, in many studies, it is also used as a comparison test. In conclusion, this filtering
technique is a useful method in determining faults in sensors.
In the literature survey, it is seen that many of the studies on the subject include
fusing INS and IMU measurements with Global Positioning System (GPS) data.
However, there are few studies on obtaining the best data for flight control from the
air data system (ADS). In this study, the problem of obtaining the most appropriate
data from ADS measurements is considered and ADS data FDI, determining the best
data from the ADS, and estimation of the state subjects are investigated using high-
accuracy GPS/INS measurements and the Kalman filtering technique as a basis.

7.2 Kalman Filter-Based Integrated ADS/GPS Navigation

System

The indirect Kalman filtering technique is used in integrating two previously

mentioned navigation systems’ data. In this system, instead of estimating the system
states, the errors between the system measurements are estimated [11].
In the integrated system scheme in Fig. 7.1, the inputs of the filter are speed
measurement differences between the two different measurement systems. The
speed errors of the ADS are estimated by the filter.
The system error vector with the required parameters is as follows:
 T
x D Veadsx Veadsy Veadsz (7.1)

Here, Veadsx , Veadsy , Veadsz are the speed errors of the ADS, each in Cartesian
coordinates. In the ADS, the true airspeed error is mainly the result of the wind
112 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

Vadsx
Vadsy
Vadsz V̂adsx
Air Data V̂adsy
System +
− V̂adsz

Vgpsx V̂eadsx
Vgpsy
V̂eadsy
Vgpsz + V̂eadsz
Global Optimal
Positioning Kalman
System
− Filter

Fig. 7.1 Kalman filter-based air data system (ADS)/Global Positioning System (GPS) fusion
scheme

speed. Therefore, choosing the ADS errors as the vector parameters shown in Eq.
(7.1) is for the purpose of obtaining the true airspeed error, which is relatively high,
and, thus, the wind speed.
The uncorrelated model, which is used in the exponential correlation function
for the stagnant processes, will be used [11, 12]. This expression is in discrete and
matrix form. This model is very suitable for simulation purposes [11]:
2 3 2 32 3
Veadsx .k C 1/ 1  ˇVadsx T 0 0 Veadsx .k/
4 Veadsy .k C 1/ 5 D 4 0 1  ˇVadsy T 0 5 4 Veadsy .k/ 5
Veadsz .k C 1/ 0 0 1  ˇVadsz T Veadsz .k/
2 1 2 2
 3
ˇVadsx T  2 ˇ Vadsx T wVadsx
6 7
C 4 ˇVadsy T  12 ˇ 2 Vadsy T 2 wVadsy 5 (7.2)
 
ˇVadsz T  12 ˇ 2 Vadsz T 2 wVadsz

where ˇVadsx ; ˇVadsy ; ˇVadsz are the variables which are inverses of the true airspeed
correlation time, wVadsx ; wVadsy ; wVadsz are the Gauss distributed noises of the ADS
speed measurements, and T is the sampling period.
Using the ADS and GPS speed measurement differences as the measurements
observation vector in the Kalman Filter, the observation vector can be written as:

z1 .k/ D Veadsx C vVadsx  vVgpsx

z2 .k/ D Veadsy C vVadsy  vVgpsy (7.3)
z3 .k/ D Veadsz C vVadsz  vVgpsz
7.2 Kalman Filter-Based Integrated ADS/GPS Navigation System 113

In Eq. (7.3), Veadsx, Veadsy, Veadsz are the true airspeed measurement errors of
the ADS and, at the same time, are the wind speeds, and vVadsx ; vVadsy ; vVadsz and
vVgpsx ; vVgpsy ; vVgpsz are the zero-mean Gauss noises of the ADS and GPS measure-
ments, respectively. Namely, the difference between the GPS speed measurement
and the ADS true airspeed measurement gives us the ADS speed error and the
wind speed during flight. However, this wind speed information includes the random
noises of both systems. The measurement statements of Eq. (7.3) can be written in
matrix form as:
2 3 2 3 2 3 2 3
z1 .k/ VADSX  VGPSX 100 vVx
z.k/ D 4 z2 .k/ 5 D 4 VADSY  VGPSY 5 D 4 0 1 0 5x.k/ C 4 vVy 5 (7.4)
z3 .k/ VADSZ  VGPSZ 001 vVz
„ ƒ‚ …
H.k/

For integration of the ADS and INS/GPS data, the Kalman filtering equations
below are used:

b
x .k=k/ D b
x .k=k  1/ C K.k/.k/
.k/ D z.k/  H.k/b x .k=k  1/
K.k/ D P .k=k/ H T .k/R1 .k/
(7.5)
P .k=k/ D .I  K.k/H.k// P .k=k  1/
P .k=k  1/ D ˆ .k; k  1/ P .k  1=k  1/ ˆ .k; k  1/
CG .k; k  1/ Q .k  1/ G .k; k  1/

where b x .k=k/ is the estimation value, I is the identity matrix, K(k) is the gain value
of the optimal filter, P(k/k) is the estimation error correlation matrix, P .k=k  1/
is the extrapolation error correlation matrix, H(k) is the measurement matrix, (k)
is the innovation process, Q .k  1/ is the system measurement correlation matrix,
G .k; k  1/ is the noise transfer matrix, and ˆ .k; k  1/ is the system transfer
matrix:
2 3
1  ˇVadsx T 0 0
ˆ .k; k  1/ D 4 0 1  ˇVadsy T 0 5 (7.6)
0 0 1  ˇVadsz T

The initial conditions are:

b
x .0=0/ D x.0/
P .0=0/ D P.0/

To start the Kalman filter process, the x(0) and P(0) initial conditions, system
error correlation matrix Q(k), and measurement error correlation matrix R(k) must
first be determined. To perform the filtering, we need system and measurement error
models.
114 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

In Fig. 7.1, as we can see in the given integration scheme, the Kalman filter
outputs are the ADS speed error estimation values, which are actually the wind
speeds b
Veadsx ; b
Veadsy ; b
Veadsz . These estimated values are subtracted from the ADS
measurements to achieve the speeds b Vadsx ; b
Vadsy ; b
Vadsz . We can use the following
formulas for determining the speed values:

b
Vadsx D Vadsx  b
Veadsx
b b
Vadsy D Vadsy  Veadsy (7.7)
Vadsz D Vadsz  b
b Veadsz

7.3 Federated Kalman Filter-Based Integrated ADS

and GPS/INS Data

7.3.1 Data Fusion Methodology

A standard central Kalman filter with central measurements and a covariance matrix
can cause computational overload. In addition, any sensor fault can degrade the
overall performance of the filter. To overcome these disadvantages, a noncentral
Kalman filter is useful. A noncentral Kalman filter is a two-stage data-processing
technique and is composed of one basic filter and one or more local filters [13–15].
Firstly, the local filters process the data optimally to obtain the best result using each
set. Then, the central filter fuses local filter results and produces the most optimal
value. The described method is an effective technique for integrating multiple sensor
measurement sets. The federated Kalman filter has many advantages over the other
noncentral filtering techniques [11, 13–15].
The federated Kalman filter-based navigation sensor fusion scheme, which fuses
INS, GPS, and other navigation sensors, is given in Fig. 7.2.
The aforementioned federated Kalman filtering is a two-level data-processing
technique. Local filters that process data from the sensors are fused in a federated
filter. All the local filters estimate the same variables; thus, a correlation is present
between them. A global optimal estimation and error variance can be described by
the following equations:

1 1
Pm D P1 1
1  C P2 C    C PN (7.8)
Xem D Pm P1 1 1
1 X e1 C P 2 X e2 C    C P N X eN

In the formulas, Pi is the covariance matrix of the ith filter and Xei is the
estimation value of the ith filter.
7.3 Federated Kalman Filter-Based Integrated ADS and GPS/INS Data 115

Fig. 7.2 Federated Kalman filter-based navigation sensor fusion scheme

7.3.2 ADS and GPS/INS Data Fusion Based on FDI

In this study, INS/GPS measurements, which have high accuracy, and ADS
measurements, which have low accuracy but high frequency, are integrated using
the Kalman filtering technique in order to obtain high-accuracy measurement data
at high frequency. The designed system based on the indirect Kalman filtering
technique is shown to be successful in finding the wind speed data using the known
error or UAV dynamics of the system and determined statistical values.
In applications, generally, ADS and GPS or GPS/INS are integrated on the basis
of a Kalman filter. A federated Kalman filter-based navigation sensor fusion scheme
with FDI block to fuse the reliable data coming from the three ADS measurement
sets and integrated GPS/INS (faulty data are determined and isolated) is shown in
Fig. 7.3.
In the figure, it can be seen that the federated filter is composed of local filters
that process ADS navigation measurements and GPS/INS measurements (position
and velocity), together with the ADS data, FDI block for faulty data determination
and isolation, and a main filter that fuses the estimates of the local filters after the
elimination of faulty data. The main best variable estimates and error variance of
the filters can be found using the equations given below [7, 9]:

1 1
Pm D P1 1
1  C P2 C P3 (7.9)
Xem D Pm P1 1 1
1 X e1 C P 2 X e2 C P 3 X e3
116 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

Fig. 7.3 Kalman filter-based ADS and GPS/inertial navigation system (INS) data fusion with fault
detection and isolation (FDI)

The aforementioned algorithm can be used to derive the best data from the ADS
measurements.

7.4 Sensor FDI Algorithms

The sensor FDI algorithms are presented in this section.

7.4.1 Statistical Test for Fault Detection

Two hypotheses are introduced:

H0 : the system operates normally
H1 : a fault occurs in the system
Using an innovation approach is suitable for detecting sensor faults [16, 17].
To detect failures changing the mean of the innovation sequence, the following
statistical function can be used:

X
k
ˇ.k/ D Q T .j/.j/
 Q (7.10)
jDkMC1
7.4 Sensor FDI Algorithms 117

Q
where .j/ is the normalized innovation sequence of the Kalman filter and M is the
width of the sliding window.
The Kalman filter normalized innovation can be calculated as follows:
 1=2
Q
.k/ D H.k/P .k=k  1/ H T .k/ C R.k/ .k/ (7.11)

This statistical function has a 2 distribution with Ms degrees of freedom, where

s is the dimension of the state vector. If the level of significance, ˛, is selected as:
˚ 
P 2 > 2˛;Ms D ˛I 0<˛<1 (7.12)

then the threshold value, 2˛,Ms , can be found. Hence, when the hypothesis H1 is true,
the statistical value of ˇ(k) will be greater than the threshold value 2˛,Ms , i.e.:

H0 W ˇ.k/  2˛;Ms ; 8k
(7.13)
H1 W ˇ.k/ > 2˛;Ms ; 9k

7.4.2 Fault Isolation Algorithm

If the fault is detected, then it is necessary to determine what sensor set is faulty. For
this purpose, the s-dimensional sequence  Q is transformed into s one-dimensional
sequences to isolate the faulty sensor, and for each one-dimensional sequence
Q i .i D 1; 2; : : : ; s/, the corresponding monitoring algorithm is run. The statistics of
the faulty sensor set are assumed to be affected much more than those of the other
sensors. Let the statistics be denoted as  i (k). When max fi .k/=i D 1; 2; : : : ; sg D
m .k/ for i ¤ j and i .k/ ¤ j .k/, it is judged that the sensor set has failed.
.
Let the statistics that is a rate of sample and theoretical variances, b
i2

i2 , be
used to verify the variances of one-dimensional innovation sequences i .k/; i D Q
1; 2; : : : ; s. When  Q i  N .0;
i /, it is known that [18]:

i
 2M1 ; 8i; i D 1; 2; : : : ; s (7.14)

i2

where i D .M  1/b
i2 .
2
As
i D 1 for the normalized innovation sequence, it follows that:

i  2M1 ; 8i; i D 1; 2; : : : ; s: (7.15)

Using Eq. (7.15), it can be proved that any change in the mean of the normalized
innovation sequence can be detected [19]. When a fault affecting the mean or
variance of the innovation sequence occurs in the system, the statistics  i exceeds
118 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

the threshold value 2˛;M1 , depending on the level of significance ˛ and degrees of
freedom .M  1/.

7.5 Simulation Results for Indirect Kalman Filter-Based

ADS and GPS/INS Data Fusion

Three ADS sets are taken into consideration. The sets are composed of values taken
from an AOA sensor, sideslip angle sensor (AOS), and speed values determined
from true airspeed sensor (TAS) measurements. There are three AOA, two AOS, and
three TAS sensors. The first set is built using AOA sensor 1 (AOA1), AOS sensor 1
(AOS1), and TAS sensor 1 (TAS1). The second set includes AOA sensor 2 (AOA2),
AOS sensor 2 (AOS2), and TAS sensor 2 (TAS2). The third set is composed of AOA
sensor 3 (AOA3), AOS2 (as there are two different AOS sensors), and TAS sensor
3 (TAS3).
Three different cases are investigated in the simulations [20]:
1. TAS1 bias D 10 knots (C10 knots disturbance on the first set’s true airspeed data
after 80 s)
2. AOA3 bias D 5ı (constant C5ı bias on the third set’s angle of attack data after
80 s)
3. AOS1 bias D5ı (constant C5ı bias on the first set’s sideslip angle data after 80 s)
The faulty data are eliminated using the statistical test as described earlier. After
that, data from another set that are known to be true are used in place of the faulty
data. For example, when the TAS1 data is determined as faulty, TAS2 will be used
instead to make set 1 working. Of course, as there are only two sideslip angle
sensors, when one AOS value is wrong, there is only one other value to replace it.
In the second simulation, the AOA3 value is faulty and it is replaced by the AOA1
value as soon as the system detects the fault.
These sets are simulated with systems that isolate the faults and eliminate the
faulty data (Figs. 7.4, 7.5, 7.6, and 7.7) and the systems that do not perform isolation
(Figs. 7.8, 7.9, 7.10, and 7.11). In the figures, “beta” denotes the sideslip angle and
“KTAS” refers to the knots true airspeed.

7.5.1 Results with Fault Isolation

The case for the system with FDI when the TAS1 bias equal to 10 knots occurs at the
80th second is investigated. Using the statistical tests, faulty data can be determined
and isolated. The variables AOA and TAS estimated using the filters are given in
Figs. 7.4 and 7.5, respectively.
7.5 Simulation Results for Indirect Kalman Filter-Based ADS and GPS/INS. . . 119

Fig. 7.4 Angle of attack (AOA) estimation values with fault isolation [when true airspeed sensor
1 (TAS1) bias D 10 knots]

Fig. 7.5 TAS estimation values with fault isolation (when TAS1 bias D 10 knots)
120 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

Fig. 7.6 Wind velocity estimation values in the x-direction with fault isolation (when TAS1
bias D 10 knots)

In Fig. 7.6, the ADS speed error estimation in the x-direction (wind velocity
estimation in the x-direction) and real wind velocity in the x-direction are shown
together.
It can be seen that the wind velocity in the x-direction is accurately estimated by
the federated Kalman filter, even though the first and second sets do not give good
results at different time intervals. We can also tell that some of the measurement
values are eliminated by the system that includes the federated filter technique.
Secondly, the case where the bias in the third set’s angle of attack value is C5ı
after the 80th second is considered. In these simulations, it must be remembered that
the isolation algorithm is also working and that the faulty data are eliminated.
Again, for this set, fault isolation is carried out and AOA and TAS estimation
values are shown in Figs. 7.7 and 7.8, respectively.
As seen in Figs. 7.7 and 7.8, the estimated values of AOA and TAS via the
federated filter are close to the real values.
Thirdly, the case where the bias on the first set’s sideslip angle value is C5ı after
the 80th second is considered. In these simulations, it must be remembered that the
isolation algorithm is also working and that the faulty data are eliminated. Wind
speed values in the y-direction for this case are presented in Fig. 7.9.
As seen from Fig. 7.9, the wind velocity in the y-direction is accurately estimated
by the federated Kalman filter. Similar simulation results are obtained for the rest of
the parameters of the UAV ADS.
7.5 Simulation Results for Indirect Kalman Filter-Based ADS and GPS/INS. . . 121

Fig. 7.7 AOA estimation values with fault isolation (when AOA3 bias D C5ı )

Fig. 7.8 TAS estimation values with fault isolation (when AOA3 bias D C5ı )
122 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

Fig. 7.9 Wind speed values in the y-direction with fault isolation (when AOS1 bias D C5ı )

7.5.2 Results Without Fault Isolation

The federated filter is investigated for the same three cases. If isolation is not made,
the federated filter can give faulty results [20]. Again, firstly, the results when the
TAS1 bias is equal to 10 knots is given. Graphs that include the AOA, TAS, and
x-direction wind speed estimation values are given in Figs. 7.10, 7.11, and 7.12,
respectively.
As can be seen from Figs. 7.11 and 7.12, the TAS and wind speed estimation
values in the x-direction of the federated Kalman filter diverge. The cause is the
data coming from the first estimation set. The filter takes the values from the first set
into account; thus, the federated values are not very good for this reason (no faulty
data isolation is made).The federated filter estimated values of AOA are not affected
significantly by the TAS1 fault, as shown in Fig. 7.10.
Next, the case where the bias in AOA in set 3 is C5ı is considered. The estimated
AOA and TAS values for this case are given in Figs. 7.13 and 7.14, respectively.
As seen in Fig. 7.13, the estimated AOA values of the federated Kalman filter
also diverge. The values with errors, which are not isolated and used in calculations,
cause this divergence. The federated filter estimated values of TAS are not affected
significantly by the AOA3 fault, as shown in Fig. 7.14.
Wind speed estimation results in the y-direction when the AOS1 bias D 5ı and
fault isolation is absent are shown in Fig. 7.15.
7.5 Simulation Results for Indirect Kalman Filter-Based ADS and GPS/INS. . . 123

Fig. 7.10 AOA estimation values without fault isolation (when TAS1 bias D 10)

Fig. 7.11 TAS estimation values without fault isolation (when TAS1 bias D 10 knots)
124 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

Fig. 7.12 Wind velocity estimation values in the x-direction without fault isolation (when TAS1
bias D 10)

Fig. 7.13 AOA estimation values without fault isolation (AOA3 bias D C5ı )
7.5 Simulation Results for Indirect Kalman Filter-Based ADS and GPS/INS. . . 125

Fig. 7.14 TAS estimation values without fault isolation (AOA3 bias D C5ı )

Fig. 7.15 Wind speed estimation results in the y-direction without fault isolation (when AOS1
bias D 5ı )
126 7 Fault Detection, Isolation and Data Fusion for UAV Air Data System

As can be seen from the presented graphs, wind speed estimation values in the
y-direction of the federated Kalman filter diverge.

7.6 Conclusion and Discussion

In UAVs, having highly accurate ADS parameters is needed for the purposes of
control. To obtain the required speed data, fusing the measurements of the system
is a must. In this study, GPS/INS measurements, which have high accuracy, and
ADS measurements, which have low accuracy but high frequency, are integrated
using the Kalman filtering technique in order to obtain high-accuracy measurement
data at high frequency. The designed system based on the indirect Kalman filtering
technique is shown to be successful in finding the wind speed data using the known
error or UAV dynamics of the system and determined statistical values.
In this chapter, the problem of obtaining the most appropriate data from ADS
measurements is considered and ADS data FDI, determining the best data from
the ADS, and estimation of the state subjects are investigated using high-accuracy
GPS/INS measurements and the Kalman filtering technique as a basis.
FDI algorithms are developed and diagnostic tests performed for faulty cases.
The mentioned tests are based on the statistical properties of the normalized
innovation process of the Kalman filter.
To fuse the data originating from the different measurement sets, a federated
Kalman filter is presented. The mentioned algorithms can be used to derive the
best data from the ADS measurements. The best data are produced using the FDI
algorithm and leaving the faulty data out before fusing with the federated filter. As a
result, the air data parameters are estimated by the federated Kalman filter very well
and the estimated values converge to the real values.

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Chapter 8
Stability Analysis for Unmanned Aerial Vehicles

8.1 Trimming

The equations of motion for unmanned aerial vehicles (UAVs) are given in Chap. 2.
In this chapter, the equations of motion are further examined and their linearization
is discussed in detail for the stability analysis. In this procedure, the first step is to
investigate the trim conditions.

8.1.1 Trim Point

A nonlinear system can be described as:

xP D f .x; u/ ; (8.1)

where u represents the system control inputs and x represents the states of the
system. The trim condition for the system can be found as:

f .b u/ D 0:
x;b (8.2)

If the system is in equilibrium with the given states and control inputs, the
variables do not change. Stable (trimmed) flight condition for a UAV is an example
of such a situation. In this case, we can say that a subspace of the UAV states is in
equilibrium. In the literature, this case is called the trim condition. Wings-level,
fixed-altitude turn, and climb–descent maneuvers are different examples of trim
conditions.

© Springer International Publishing Switzerland 2015 129

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_8
130 8 Stability Analysis for UAV

8.1.2 Linearization Around a Steady-State Condition

The states of a UAV can be calculated for different trim conditions using a
minimization function. The linearization of the equations is necessary for controller
design.
We can find the linearized equations of motion for a UAV starting with the
equations given in Chap. 2.
Around the equilibrium point, we have:
b
xP D f .b
x;b
u/ (8.3)

A change around the equilibrium point can be represented as:

xQ D x b
x; (8.4)

P
xPQ D xP  b
x; (8.5)

xPQ D f .x; u/  f .b u/ ;
x;b (8.6)

xPQ D f .x C xQ  xQ ; u C uQ  uQ /  f .b u/ ;
x;b (8.7)

xPQ D f .b u C uQ /  f .b
x C xQ ;b u/ :
x;b (8.8)

With Taylor series approximation, we can find:

@f .b
x;b
u/ @f .b
x;b
u/
xPQ D f .b u/ C
x;b xQ C uQ C     f .b u/ ;
x;b (8.9)
@x @u
Then, the changes in states can be given with the following equation:

@f .b
x;b
u/ @f .b
x;b
u/
xPQ Š xQ C uQ : (8.10)
@x @u
@f @f
In order to linearize the equations, we need to solve for @x and @u at the
equilibrium point.

8.2 Derivation of the Transfer Functions

8.2.1 State Equations

The longitudinal and lateral motions can be handled separately. The variables that
we choose to describe the longitudinal and lateral motions of the UAV are h, V, ˛,
u, w, q,  and ˇ, v, p, r, ·, , respectively.
8.2 Derivation of the Transfer Functions 131

The longitudinal equations of motion for the UAV can be found   by taking
the lateral variables as zero and using definitions of ˛ D arctan wu and Vt D
p
u2 C w2 , where Vt is the true airspeed.
The differentiation matrix for the states can be found as:
2




 3
@Pu @Pu @Pu @Pu @Pu
; ; ; ;
6 @u 7
6

@w 
@q 
@ 
@h  7
6 7
6 @wP @wP @wP @wP @wP 7
6 ; ; ; ; 7
6 @u 7
6

@w
@q
@
@h 7
6 7
@f 6 @Pq @Pq @Pq @Pq @Pq 7
6 ; ; ; ; 7
D 6 @u @w @q @ @h 7 (8.11)
@x 6 ! ! ! ! !7
6 7
6 @P @P @P @P @P 7
6 ; ; ; ; 7
6 7
6 @u @w @q @ @h 7
6 ! ! ! ! ! 7
6 7
4 @hP @hP @hP @hP @hP 5
; ; ; ;
@u @w @q @ @h

Similarly, the differentiation matrix for the control inputs can be derived and the
longitudinal state and control matrices—Alon and Blon , respectively—can be found.
The equations for the longitudinal stability coefficients are given below:

uS Swcx˛ u Sccxq uq Sprop cprop u

Xu D Œcxo C cx˛ ˛ C cxıe ıe  C  (8.12)
m 2m 2mV m
uS SVcx˛ u Sccxq wq Sprop cprop w
Xw D q C Œcxo C cx˛ ˛ C cxıe ıe  C 
m 2m 2mV m
(8.13)
SVcxq qc
Xq D w C (8.14)
2m

V 2 cxıe S
Xıe D (8.15)
2m
Sprop cxp ıa h
Xıt D (8.16)
m
uS Scz˛ w Scczq uq
Zu D q C Œczo C cz˛ ˛ C czıe ıe  C (8.17)
m 2m mV
wS Scz˛ u Scczq uq
Zw D Œczo C cz˛ ˛ C czıe ıe  C (8.18)
m 2m mV
Sczq Vc
Zq D u C (8.19)
2m
132 8 Stability Analysis for UAV

Sczıe V 2
Zq D (8.20)
2m

uSc Swccm˛ u Sc2 cmq uq

Mu D Œcmo C cm˛ ˛ C cmıe ıe  C (8.21)
Jy 2Jy 2Jy V

wSc Sccm˛ w Sc2 cmq wq

Mw D Œcmo C cm˛ ˛ C cmıe ıe C C (8.22)
Jy 2Jy 2Jy V
Scmqe Vc
Mq D (8.23)
2Jy

Scmıe V 2 c
Mıe D (8.24)
2Jy

The derivation procedure for the lateral equations of motions and, therefore,
the lateral state and control matrices—Alat and Blat , respectively—are similar. The
equations for the lateral stability coefficients are given below:

vbS  Sv   Scyˇ p

Yv D cyp p C cyr r C cyo C cyˇ ˇ C cyıa ıa C cyır r C u2 C w2
4mVair m 2m
(8.25)
SVair b
Yp D w C cyp (8.26)
4m
SVair b
Yr D u C cyr (8.27)
4m

SVair 2
Yıq D cyıa (8.28)
2m

SVair 2
Yır D cyır (8.29)
2m

vb2 S  Sbv  
Lv D cpp p C cpr r C cpo C cpˇ ˇ C cpıa ıa C cpır ır
8mVair 2m
Sbcpˇ p 2
C u C w2 (8.30)
4m

SV 2 air b
Lp D 1 q C cpp (8.31)
8m

SV 2 air b
Lr D 2 q C cpr (8.32)
8m
8.2 Derivation of the Transfer Functions 133

SbVair 2
Lıq D cpıa (8.33)
4m

SbVair 2
Lır D cpır (8.34)
4m

vb2 S  Sbv  
Nv D crp p C crr r C cro C crˇ ˇ C crıa ıa C crır ır
8mVair 2m
Sbcrˇ p 2
C u C w2 (8.35)
4m

SV 2 air b
Np D 3 q C crp (8.36)
8m

SV 2 air b
Nr D 4 q C crr (8.37)
8m

SbVair 2
Nıq D crıa (8.38)
4m

SbVair 2
Nır D crır (8.39)
4m
The next step is to give the linearized state equations for the UAV. The equations
in the state space form are:
0 1 2 30 1 2 3
uP .Xu/ .Xw/ .Xq/ .g cos .// 0 u Xıe Xıt
B C 6 7B C 6 7
B wP C 6 .Zu/ .Zw/ .Zq/ .g sin .// 0 7 B C 6 07 " #
B C 6 7 B w C 6 Zıe 7 ı
B C 6 7B C 6 7 e
B qP C D 6 Mu Mw Mq 0 0 7 B q C C 6 Mıe 07
B C 6 7B C 6 7 ıt
B P C 6 0 0 1 0 7 B C
05@ A 4 0 6 0 7
@ A 4 5
hP  sin ./  cos ./ 0 t 0 h 0 0
(8.40)

t D u cos ./ C w sin ./ (8.41)

0 1 2 30 1 2 3
vP .Yv/ v u g cos ./ 0 v Yıa Yır
B C 6 7B C 6 7
B pP C 6 .Lv/ .Lp/ .Lr/ 0 07 B C 6 Lıa Lır 7 " #
B C 6 7Bp C 6 7 ı
B C 6 7B C 6 7 a
B rP C D 6 .Nv/ .Np/ .Nr/ 0 07Br C C 6 Nıa Nır 7
B C 6 7B C 6 7 ır
B 'P C 6 0 1 tan ./ 0 07 B C 60 0 7
@ A 4 5@' A 4 5
P 0 0 sec ./ 0 0 0 0
(8.42)
134 8 Stability Analysis for UAV

Here, ten states are used to describe the lateral and longitudinal motions of the UAV.
The heading angle equation is also added to the system.
Finally, the linearized equations of motion for the Zagi UAV are found as below.
These equations will be used when we design the various controller algorithms that
are presented in the next chapters of this book. The values given in the matrices Alon ,
Alat , Blon , and Blat are for the trim conditions. The longitudinal equations are:

xP D Ax C Bu (8.43)
2 3
0:3356 1:3181 1:9276 9:6610 0
6 7
6 1:7916 3:9003 9:8215 1:7035 07
6 7
6 7
Alon D 6 0:7020 3:5375 11:3920 0 07 (8.44)
6 7
6 0 0 1:0000 0 07
4 5
0:1736 0:9848 0 17:4865 0
2 3
0:7436 6:8728
6 7
6 3:7855 0 7
6 7
6 7
Blon D 6 47:9170 0 7 (8.45)
6 7
6 0 0 7
4 5
0 0
 
ıe
ulon D (8.46)
ıt
 T
Xlon D u w q  h (8.47)

On the other hand, the lateral equations are:

2 3
1:0502 1:9276 9:8215 9:6610 0
6 7
6 1:2213 1:9155 1:0096 0 07
6 7
6 7
Alat D 6 1:7255 0:0919 1:7198 0 07 (8.48)
6 7
6 0 1:0000 0:1763 0 07
4 5
0 0 1:0154 0 0
2 3
0 1:8218
6 8:348 0 7
6 7
6 7
Blat D 6 4:24 2:1272 7 (8.49)
6 7
40 0 5
0 0
8.2 Derivation of the Transfer Functions 135

 
ıa
ulat D (8.50)
ır
 T
xlat D v p r ' ‰ (8.51)

8.2.2 Transfer Functions

The effect of elevator and thrust inputs on the states can be represented with the
transfer functions that can be derived using the Laplace transformation as:

Y.s/
D CŒsI  A1 B C D (8.52)
U.s/
Y.s/ Cadj .sI  A/ B  jsI  Aj
D (8.53)
U.s/ jsI  Aj

where Y(s) is the output and U(s) is the control input. The determinant jsI  Aj
gives the characteristic equation. MATLAB is used to calculate all the transfer
functions. As an example, if the output variable is chosen as u, the forward velocity,
then we can calculate ıeu and ıtu . First, the “state_space_long D ss(Alon, Blon,
[1,0,0,0,0], 0)” command gives us the state space model for the system and then
the “tf(state_space_long)” command can be used to derive the equations for the
forward velocity [1–3]:

u 0:7436s3  98:75s2  179:1s  1779

D 4 (8.54)
ıe s C 15:63s3 C 88:02s2 C 62:64s C 87:23

u 6:873s3 C 105:1s2 C 544:2s  41:42

D 4 (8.55)
ıt s C 15:63s3 C 88:02s2 C 62:64s C 87:23

In a similar way, the output matrix C (given as [1,0,0,0,0] in the “ss” function)
can be changed to obtain the desired transfer functions and the effect of control
inputs on the rest of the longitudinal states can be found:

w 3:785s3 C 516:3s2 C 179:1s C 828:5

D 4 ; (8.56)
ıe s C 15:63s3 C 88:02s2 C 62:64s C 87:23

w 12:31s2  92:89s C 8:219

D 4 ; (8.57)
ıt s C 15:63s3 C 88:02s2 C 62:64s C 87:23

q 47:92s3 C 189:1s2 C 168:1s

D 4 ; (8.58)
ıe s C 15:63s3 C 88:02s2 C 62:64s C 87:23
136 8 Stability Analysis for UAV

q 4:825s2 C 68:38s
D 4 ; (8.59)
ıt s C 15:63s C 88:02s2 C 62:64s C 87:23
3

 47:92s2 C 189:1s C 168:1

D 4 ; (8.60)
ıe s C 15:63s3 C 88:02s2 C 62:64s C 87:23

 4:825s C 68:38
D 4 ; (8.61)
ıt s C 15:63s C 88:02s2 C 62:64s C 87:23
3

h 3:6s3 C 346:5s2 C 62:64s C 2433

D 5 ; (8.62)
ıe s C 15:63s4 C 88:02s3 C 62:64s2 C 87:23s

h 1:193s3  6:124s2 C 81:35s C 1106

D 5 : (8.63)
ıt s C 15:63s4 C 88:02s3 C 62:64s2 C 87:23s

If the same process is repeated for the lateral equations of motion, we can obtain the
transfer functions for the lateral motion as:

v 25:55s2 C 36:49s C 195:2

D 4 ; (8.64)
ıa s C 4:686s3 C 26:32s2 C 44:27s  1:977

v 1:822s2 C 35:2s C 157:5 C 154:4

D 4 ; (8.65)
ır s C 4:686s3 C 26:32s2 C 44:27s  1:977

p 8:348s3 C 27:4s2 C 211:9s  33:36

D 4 ; (8.66)
ıa s C 4:686s3 C 26:32s2 C 44:27s  1:977

p 10:86s3 C 30:15s2 C 176:5s  27:48

D 4 ; (8.67)
ır s C 4:686s3 C 26:32s2 C 44:27s  1:977

r 4:24s3 C 13:34s2 C 47:08s C 189:2

D 4 ; (8.68)
ıa s C 4:686s3 C 26:32s2 C 44:27s  1:977

r 2:127s3  8:454s2 C 22:05s C 155:9

D 4 ; (8.69)
ır s C 4:686s3 C 26:32s2 C 44:27s  1:977

' 9:096s2 C 29:76s C 220:2

D 4 ; (8.70)
ıa s C 4:686s3 C 26:32s2 C 44:27s  1:977

' 10:48s2 C 28:66s C 180:4

D 4 ; (8.71)
ır s C 4:686s3 C 26:32s2 C 44:27s  1:977
8.3 Longitudinal Stability Analysis 137

4:305s3 C 13:55s2 C 47:81s C 192:1

D ; (8.72)
ıa s5 C 4:686s4 C 26:32s3 C 44:27s2  1:977s

2:16s3  8:585s2 C 22:39s C 158:3

D : (8.73)
ır s5 C 4:686s4 C 26:32s3 C 44:27s2  1:977s

In the next sections, we investigate the characteristic equations for the stability
analysis.

8.3 Longitudinal Stability Analysis

Before designing the controller, we need to investigate the stability conditions

for the UAV by checking the modes for the longitudinal and lateral characteristic
equations. Using the derivations given in previous sections of this chapter, the
characteristic equation for the longitudinal motion can be given as follows:
2  
s C 15:043s C 78:0719 s2 C 0:587s C 1:1174 D 0 (8.74)

As we can see, there are two equations that describe the longitudinal motion
of the UAV. These two equations represent two different modes: the phugoid mode,
which is easy to control but takes a long time to dampen, and the short-period mode,
which makes the plane show fluctuation motions with high frequency as the name
refers and has a high damping ratio.
In the short-period mode, the changes in the attack angle and pitch angle are
significant, whereas the change in the forward velocity is negligible. On the other
hand, in the phugoid mode, the changes in u and pitch angle are significant, and
this mode can be described as the change in the kinetic and potential energies of the
aircraft. Next, we investigate the roots of the characteristic equation [2–5].
The roots of the equation are 7.5215 C 4.6367i, 7.5215  4.6367i,
0.2935 C 1.0155i, and 0.2935  1.0155i. There are two roots which are close to
the imaginary axis and describe the phugoid motion and two roots which are far
away from the imaginary axis and describe the short-period motion of the UAV. The
angular frequency (w), damping factor (—), and period (T) values are calculated and
given in Table 8.1. The UAV is longitudinally stable in the given condition because
of the location of the roots of the characteristic equation.

Table 8.1 Short-period and phugoid mode characteristic values

Motion Natural frequency Damping factor Period Time to half amplitude
Short period 8.8538 rad/s 0.8513 1.3553 s 0.0917 s
Phugoid 1.0566 rad/s 0.0815 5.9664 s 2.3509 s
138 8 Stability Analysis for UAV

As can be seen from the calculations and the values given in the table, the roots
are on the left of the imaginary axis for the longitudinal motion of the UAV and the
damping factor and natural frequency values are in the desired zone.

8.4 Lateral Stability Analysis

The characteristic equation of the lateral motion is investigated and the stability
condition is checked. The characteristic equation is as follows:
2 
s C 2:5638s C 20:9734 .s  0:0435/ .s C 2:1658/ D 0 (8.75)

As can be seen, the lateral characteristic equation is also of the fourth order. It
represents three different modes, which are the roll, Dutch roll, and spiral modes [2,
4, 6].
The roll mode is mainly a pure rolling motion. If the aircraft is not stable, this
motion will lead to an increase in rolling in the direction in which the aircraft
turns. The roll mode is represented by one root and gives a first-order nonoscillatory
response; it is generally stable at low attack angles.
The spiral mode also gives a first-order response and it is represented by one first-
order root. This mode includes both rolling and yawing motions. If it is not stable,
the rolling motion increases while yawing, and the aircraft starts a spiral motion.
However, instability in the spiral mode is tolerable under some specific conditions.
The Dutch roll mode is triggered by perturbations in the sideslip angle and
includes yaw, roll, and sideslip changes combined. If the directional stability is not
high enough to lead the aircraft into the spiral mode, the Dutch roll mode occurs
and the aircraft shows oscillatory yaw and roll motions.
The roots that symbolize each motion are found from the characteristic equation
and given in Table 8.2.
The results show that the spiral mode root is positive and this mode is unstable.
The roll mode is stable because of the root at 2.1658. The Dutch roll mode

Table 8.2 Roll mode, Dutch roll mode and spiral mode characteristic values
Natural
Motion frequency Damping factor Roots Time constant
Roll N/A N/A 2.1658 0.4616 s
mode
Spiral N/A N/A 0.0435 (unstable) 16 s (for double amplitude)
mode
Dutch 4.58 rad/s 0.27 1.282 C 4.397i 0.808 s
roll mode 1.282  4.397i
References 139

damping factor is 0.27 and its natural frequency is 4.58 rad/s. For stable modes,
the times to get back to 36.8 % of the amplitude are given. For the spiral mode, the
time to double amplitude is calculated and given in the table.

8.5 Conclusion

The linearized equations of motion of UAVs are used to find the transfer functions
that relate the actuator inputs to the state outputs. The UAV is analyzed for
lateral and longitudinal stability by means of these transfer functions. First, the
characteristic equations are obtained using the transfer functions and the stability
condition is investigated regarding the roots of the polynomials. The short-period
and phugoid modes for longitudinal motion and the roll, Dutch roll, and spiral
modes for lateral motion are discussed. It is shown that the UAV is stable in the
analyzed flight condition.

References

1. Tewari A (2002) Modern control design with Matlab and Simulink. Wiley, Chichester
2. Pamadi BN (2003) Performance, stability, dynamics, and control of airplanes. AIAA education
series, Reston
3. Etkin B, Reid LD (1996) Dynamics of flight: stability and control. Wiley, New York
4. Yechout TR, Morris SL, Bossert DE, Hallgren WF (2003) Introduction to aircraft flight
mechanics. AIAA education series, Reston
5. Blakelock JH (1991) Automatic control of aircraft and missiles. Wiley, New York
6. Nelson RC (1998) Flight stability and automatic control. McGraw-Hill, New York
Chapter 9
Classic Controller Design for Unmanned Aerial
Vehicles

9.1 Classical Proportional-Integral-Derivative (PID)

Controller

After investigating the characteristic equations of an unmanned aerial vehicle

(UAV), we can now design the control system. In this book, different methods are
proposed to design a controller for UAVs; the first of these is the classical controller.
In this control structure, proportional-integral-derivative (PID) type controllers are
used, and longitudinal and lateral equations are investigated using the root locus
method. Longitudinal and lateral controllers are designed separately.
Three different types of basic controllers and their combinations are present in
the classical controller scheme. All three types have different characteristics and
transfer functions. These basic controllers may be named after the type of gains
used within them:
1. Proportional gain
2. Integral gain
3. Differential gain
Proportional gain has an output which is the multiple of the error between the
desired and actual conditions of the state. Therefore, we can describe that:

c.t/ D Ke.t/ (9.1)

where c(t) is the output, e(t) is the error for making the actual state reach the desired
condition as e.t/ D xd .t/  xa .t/, and K is the proportional gain. Here, xd (t) is the
desired state condition and xa (t) is the actual (or observed) state.
Integral gain takes the integral of the error input and can be described by:

© Springer International Publishing Switzerland 2015 141

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_9
142 9 Classic Controller Design for the UAV

Zt
1
c.t/ D e.t/dt: (9.2)
i
0

Using the Laplace transformation, the transfer function can be represented as:

1
c.s/ D e.s/: (9.3)
i s

Here,  i is called the integral time.

Differential gain takes the derivative of the error input and can be described by
the following equations:

de.t/
c.t/ D d (9.4)
dt

c.s/ D d se.s/: (9.5)

where  d is the differential time.

Differential gain only takes the derivative of the error and affects the future state,
but cannot remove the constant error in theory. Therefore, it is not used in a control
system by itself.
Different controller types can be designed using proportional, integral, and
derivative gains in combination.
We can use proportional gain on its own to design a P-type controller. The system
can be given by c.t/ D Ke.t/, and its transfer function is C.s/ E.s/ D K. A P-type
controller normally results in a stable system. Increasing the gain value can decrease
the steady-state error; however, excessive increase can also lead the system to an
unstable state.
In some cases, it is possible to build a controller using only the integral gain, for
1
which the transfer function is given as C.s/
E.s/ D i s . It removes the steady-state error
but can provide a very slow response.
If we integrate P- and I-type gains, what we have will be a combination of these
two controllers, called a PI controller. For this type of controller, the controller input
is:
Zt
K
c.t/ D Ke.t/ C e.t/dt (9.6)
i
0

and the transfer function is:


C.s/ 1
DK 1C : (9.7)
E.s/ i s
9.1 Classical Proportional-Integral-Derivative (PID) Controller 143

PI control fuses two control effects and makes the steady-state error zero. The
integral action adds a root at zero to the system transfer function and removes
the error at the steady state. On the other hand, the integral action also affects the
overall system response, so P-type control is used to obtain the required response,
compensating for the unwanted effects of the integral gain.
Another possible combination is to use the derivative effect (D) together with the
proportional gain. Then, the controller is called a PD controller and its response can
be given by:

de.t/
c.t/ D Ke.t/ C Kd (9.8)
dt
where the transfer function is:

C.s/
D K .1 C d s/ : (9.9)
E.s/

In case we use the PD-type controller, the controlled system acts quickly because
of the differential effect; however, the steady-state error remains. If we just want
systems to reach the desired state in a short time, then this type of controller may be
used in some cases.
The PID-type controller has all three control type effects and it gives the response
according to the control law of:

Zt
K de.t/
C.t/ D Ke.t/ C e.t/dt C Kd : (9.10)
i dt
0

The transfer function can be given by:


C.s/ 1
DK C 1 C d s : (9.11)
E.s/ i s

The PID-type controller makes the system respond quickly and removes the
steady-state error. By choosing the appropriate gains, we can obtain the required
response [1].
In our design, PID effects are used together to control the pitch angle, speed,
heading, and altitude. A comparison is made to choose the best controller for the
system. The controller coefficients are determined using the root locus method (by
choosing the gain), carrying out simulations, and using optimization methods such
as determining the signal response specifications through MATLAB.
144 9 Classic Controller Design for the UAV

9.2 Classical Controller for the Longitudinal Motion

The longitudinal controller includes a speed controller, a pitch angle controller in

the inner loop, and an altitude controller in the outer loop. The scheme for the pitch
angle controller is given in Fig. 9.1.
The equations for a UAV are written in the state space form and used in the blocks
in MATLAB as shown in Fig. 9.2.

u
w
deltae1
q
theta
h
deltat
Vt
all
Longitudinal Dynamics2

Qref de Theta
ue 10 -K-
-K-
s+10
Krg1 Scope2-de
Kamp Elevator Actuator

s
s+1
Washout filter

Fig. 9.1 P-type controller for pitch angle control

1
u
2 3
x' = Ax+Bu w
1 q
y = Cx+Du
deltae1
State-Space1 4
theta

2 5
deltat 6 h
all

Fig. 9.2 Simulink block representing the longitudinal equations

9.2 Classical Controller for the Longitudinal Motion 145

If we use the transfer function that relates ı e to q and include the elevator actuator
function, we can choose the appropriate Kq gain for the inner loop using the root
locus analysis. In the outer loop, a washout filter is added to the system to exclude
the unnecessary signal inputs. For controlling , a P-type controller is used. K can
also be chosen using the root locus analysis.

9.2.1 Pitch Angular Rate Controller (Inner Loop)

The feedback for the pitch angular rate and pitch angle is usually done together
to obtain the damping factor and angular frequency of the required values in the
inner loop for the short-period mode and in the outer loop for the phugoid mode.
However, as we can see in the linearized longitudinal equations, the values for the
short-period mode are already in the required zone and feedback is not needed in
the inner loop. We can check this from Military Specification MIL-F-8785C or
normal aircraft standard FAR. Also, there are studies on the subject claiming that,
for small UAVs, different standards should be used to evaluate the characteristics of
the system [2–4]:

w1;2 D 8; 8358rad=s; 1 D 0:8513; T1;2 D 1:3553s; ta D 0:0917rad=s

w3;4 D 1; 0566rad=s; 2 D 0:0815; T3;4 D 5:9664s; tb D 2:3509rad=s:

The short-period mode damping factor is around 0.8513 and no feedback is

needed in the inner loop. Hence, instead of choosing a Kq , just the inner loop gain
K is used to improve the characteristic values of the phugoid mode. The inner loop
may be investigated for different flight conditions and changing characteristics. The
transfer function that relates  to the ue input can be found as:

 479:2s3 C 1; 891s2 C 1; 681s

D 6 (9.12)
ue s C 26:63s5 C 269:9s4 C 1; 187s3 C 1; 656s2 C 1; 586s C 872:3

Here, the elevator actuator function, which has the time constant of 0.1 s, is modeled
as:
10
TFelevator D (9.13)
s C 10

The root locus of the system for negative values of gain is shown in Fig. 9.3.
With the K feedback, we can observe from the root locus that, for gain values of
approximately 0.5, the short-period mode damping values do not decrease much
(0.665), while the phugoid characteristics are improved (to a damping factor of
0.7). In other words,  feedback improves the phugoid characteristics. The transfer
function of the system, which includes the washout filter, and its step response is as
follows (Fig. 9.4):
146 9 Classic Controller Design for the UAV

Fig. 9.3 Root locus for controlling the  angle

Fig. 9.4 Closed-loop response for the  feedback loop

9.2 Classical Controller for the Longitudinal Motion 147

 L
D (9.14)
ref Lref

L D 2; 492s8 C 7; 619s7 C 94; 320s6 C 584; 500s5 C 18; 160; 00s4 C 3; 062; 000s3
C3; 225; 000s2 C 2; 244; 000s3 C 762; 500
(9.15)

Lref D 103 0:001s11 C 0:05226s10 C 1:197s9 C 15:8s8 C 131:2s7 C 695:4s6

C 2; 279s5 C 4; 372s4 C 5; 669s3 C 4881s2 C 2; 768s C 760:9
(9.16)

As we can see, the system is stable and the steady-state error is very low.
For the feedback loop, an equivalent feedback system transfer function can be
given as:

G.s/
TFfeedback D (9.17)
1 C G.s/ .H.s/  1/

where G(s) is the multiplication of the elevator actuator transfer function (/ı e ) and
K , and H(s) is the washout filter transfer function [5].
The position constant where the zn are the zeros and the pn are the poles can be
given as:

k  zn
Kp D (9.18)
pn

The steady-state error can be calculated using the position constant as:

1
e1 D (9.19)
1 C Kp

Using the steady-state error and position constant formulas, we can choose the
gain for the feedback loop to reach a low steady-state error value. In our case, the
steady-state error that we obtain with the root locus-based design is 0.003, which is
sufficiently small. Besides, we will build an outer loop controller for the altitude in
the next section. Thus, we do not change the inner loop controller structure and we
also do not use the method given by Eqs. (9.17), (9.18), and (9.19).

9.2.2 Altitude Controller (Outer Loop)

For the altitude controller, the performance of a PID-type controller is examined.

As the first step of the design procedure, the transfer function that relates the pitch
148 9 Classic Controller Design for the UAV

angle to the altitude is found. The transfer function from the output pitch angle to the
reference input is calculated and then combined with h/, which can be calculated
by multiplying h/ı e and /ı e . As a result, we find the transfer function of h/ ref .
Then, a root locus study is made to investigate the system:

18:72 .s 104:5/ .sC10/ .sC7:34/ .sC2:6/ .sC1:34/ .sC1/ .sC0:88/

TF1 D
s .sC12:26/ .sC11:33/ .sC2:6/ .sC1:35/ .sC0:74/ .s2 C0:59sC1:12/
(9.20)
2  2  2  2 
s C0:58sC1:11 s C0:587C1:117 s C15:04sC78 s C15:04sC77:9
TF2 D
.s2 C0:59sC1:12/.s2 C15:04sC75/.s2 C15:04sC78/.s2 C10:7sC65/
(9.21)


h
D TF2  TF1 (9.22)
ref

The critical gain value for stability is 0.299, as we can see in Fig. 9.5. A PID
controller can be used for such a system; however, this system is already stable
and it has a root at zero, which gives the integrator effect. Thus, instead of a PID
controller, a PD controller is designed. The PD controller gives a quick response
and is appropriate for the requirements.

Fig. 9.5 Root locus used in the design of the altitude controller (outer loop)
9.2 Classical Controller for the Longitudinal Motion 149

Firstly, the coefficient for the PID controller, which was planned to be used, was
calculated using the Ziegler–Nichols technique, such that the integral of the error
is minimum [6]. The critical values of the gain and angular frequency are Khpu D
0:299 and !u D 2:35rad=s. According to the Ziegler–Nichols rules, the P gain
must be 0.6 times the critical gain, and the integral and differential gains should be
calculated accordingly [6, 7]:

2
Tu D D 2:72s (9.23)
!u

Khp D 0:6  Khpu D 0:1794 (9.24)

Khd D 0:6  Khpu  0:125  Tu D 0:06 (9.25)

0:6  Khpu
Khi D D 0:134 (9.26)
0:5  Thu

The PID values, calculated using the Ziegler–Nichols method, are as given above.
The gains are used in the system together with the saturators, which will keep the
reference pitch angle at the required levels. The results are satisfactory. Both the
pitch angle and the altitude reach the desired values in a short period of time.
In order to better control the speed and suppress the overshoot of the response,
the gains are recalculated. The responses of the controlled and uncontrolled systems
are given in Figs. 9.6, 9.7, and 9.8. Both the results found using the Ziegler–Nichols
gains values and tryouts are given. The improvement in the response of the system
can be seen when the gains are found by tryouts. For the tryouts, this response is
obtained by taking Khp D 0:58, Khi D 0:25, and Khi D 0:2.
The response of the altitude system is improved. However, the integral effect is
already contained in the system equations, and we prefer to use a PD-type controller,
not a PID-type controller. The gains of the PD-type controller are determined using
the Ziegler–Nichols values as a starting point. The proportional gain value Khp is
chosen as 0.1794, and tryouts are done using the previously determined Khd . The
final value of Khd that is used in the PD controller is 0.132. In this case, a zero at
1.33 is added to the system and a nonvibratory quick response is obtained. The
step response of the system with the PD controller is given in Fig. 9.9.
The response of the designed altitude controller for the case where the reference
input is 100 m and the speed is held constant at 20 m/s using the speed controller
is given in Fig. 9.10. Moreover, the response of the speed controller, which tries to
hold the speed at 20 m/s, is given in Fig. 9.11. As we can see, the PD controller is
working efficiently as an altitude controller for our system. Nonetheless, in cases
where other disturbances are present, a PID-type controller may be used instead.
In Fig. 9.12, the Simulink diagram for the final structure of the altitude controller
is given. As can be seen, the control is obtained using the K in the inner loop for
pitch angle control and the PD-type controller in the outer loop for altitude control
150 9 Classic Controller Design for the UAV

Fig. 9.6 Response of the uncontrolled system (h/ ref )

Fig. 9.7 Response of the altitude control system with a PID controller using Ziegler–Nichols gains
9.2 Classical Controller for the Longitudinal Motion 151

Fig. 9.8 Response of the altitude control system with a PID controller using the gains found in
tryouts

Fig. 9.9 Step response of the PD controller

152 9 Classic Controller Design for the UAV

Fig. 9.10 Response of the PD-type altitude controller to a 100-m reference input

Fig. 9.11 Response of the system when the speed is trying to be held at 20 m/s (while the altitude
controller is working with the reference input)
 u
w
deltae1
q
theta
Scope5-q
h
deltat
Vt
all
Uzunlamasina Denklemler

Saturation Qref de Theta

ue 10
-K- -K-
s+10
Krg1 Scope2-de
Kamp Elevator Actuator

du/dt -K-

Step3 Derivative1 Kd
9.2 Classical Controller for the Longitudinal Motion

-K- s

Kp s+1
Washout filter

-K-
Krg

Scope8-theta

Fig. 9.12 Simulink diagram for the altitude controller

153
154 9 Classic Controller Design for the UAV

and saturators. The saturators are used for limiting the elevator angle and reference
pitch angle values. Although not shown in the diagram, the ıt input is determined
using the speed controller.

9.2.3 Speed Controller

A controller that controls the speed using the thrust input, ıt, is designed. A P-type
controller is used. Both the results of the step response and the simulations show that
a P-type speed controller which uses ıt as the input is satisfactory. In order to find
the gain value K, the root locus method is used. The Vt /ıt transfer function does not
normally exist in the previously found functions; however, we can obtain it using
the equation below:

P C WW
VP t Vt D U U P C V VP (9.27)

We can find the Vt /ıt transfer function if we assume that, for a given linearization
point, Vut and Vwt are almost constant and v is zero. The other values can be determined
using the calculated values for the speed values, etc.
A sixth row to include the Vt transfer function is added to the system. The new
Alon matrix, denoted Alon _ vt , can be given as:
2 3
0:3356 1:3181 1:9276 9:6610 0 0
6  1:7916 3:9003 9:8215 1:7035 0 0 7
6 7
6 7
6 0:7020 3:5375 11:3920 0 0 07
Alon_vt D 6 7 (9.28)
60 0 1:0000 0 0 07
6 7
4  0:1736 0:9848 0 17:4865 0 0 5
 1:234 3:5261 9:1667 3:8967 0 0

As explained previously, the measured value can be chosen as Vt and then the
related transfer function can be calculated. The transfer function for the thrust
actuator is modeled with a first-order transfer function with a time constant of 0.5 s:

Vt 1:146s4 C 9:433s3 C 58:81s2 C 318:8s C 423:2

D (9.29)
ıt s5 C 15:63s4 C 88:02s3 C 62:64s2 C 87:24s
2
ActuatorTF D (9.30)
sC2

The open-loop transfer function can be obtained by combining these two transfer
functions:

Vt 2:292s4 C 18:87s3 C 117:6s2 C 637:6s C 846:4

D 6 (9.31)
ut s C 17:63s5 C 119:3s4 C 238:7s3 C 212:5s2 C 174:5s
9.2 Classical Controller for the Longitudinal Motion 155

Fig. 9.13 The root locus diagram for Vt /ıt

The root locus for the obtained Vt /ıt transfer function is as in Fig. 9.13. As can be
seen, the system is stable for gains smaller than 0.126 (K < 0:126). Firstly, the PID
gains proposed by the Ziegler method are checked. The determined critical values
are Kpu D 0.126 and ! u D 1.07 rad/s. The calculation for the gains are as follows:

2
Tu D D 5:8721s; (9.32)
!u

Kp D 0:6  Kpu D 0:6  0:126 D 0:0756; (9.33)

Kd D 0:6  Kpu  0:125  Tu D 0:0257; (9.34)

0:6  Kpu
Ki D D 0:0555: (9.35)
0:5  Tu

In Fig. 9.14, we see that the overshoot values and settling time are not good.
Hence, we designed a simpler controller using only the proportional gain value. The
system is already stable because of the roots of the transfer function. The response of
the system when Kvt D 0:013 is given in Fig. 9.15. We may obtain better responses
by increasing the gain value; however, the change in ıt to obtain the better responses
affects the overall system response. The response of the system when the gain value
is chosen as 0.025 is given in Fig. 9.16.
It is clear that the response quickens. However, the response of the overall control
system is also affected. In Figs. 9.17 and 9.18, the results for the altitude change
156 9 Classic Controller Design for the UAV

Fig. 9.14 PID controller response for the gains calculated using the Ziegler–Nichols method

Fig. 9.15 Response of the speed controller system to a 15-m/s reference input with only P gain
(Kvt D 0:013)
9.2 Classical Controller for the Longitudinal Motion 157

Fig. 9.16 Response of the speed controller system to a 15-m/s reference input with only P gain
(Kvt D 0:025)

Fig. 9.17 Change in altitude when the speed controller changes the UAV’s speed by 15 m/s
(Kvt D 0:015)
158 9 Classic Controller Design for the UAV

Fig. 9.18 Change in altitude when the speed controller changes the UAV’s speed by 15 m/s
(Kvt D 0:025)

when the speed value is set to a certain value are given. It can be understood that,
when the gain used for controlling Vt is increased, the change in altitude is also
increased. Changes of around 10 m in altitude over a short period, as we see for the
case Kvt D 0:025, is an undesired response for mini-UAV operations (Fig. 9.18).
Thus, a gain value that gives a quick response for controlling Vt and results in lower
altitude changes must be used.
In Fig. 9.19, the step response for the controllerwith Kvt D 0:015 is given. This is
the final gain value that we choose for the speed controller. For speed changes below
10 m/s, it may be increased to Kvt D 0:02, depending on the desired response.
The overall structure for the speed controller is given in Fig. 9.20.

9.3 Classical Controller for the Lateral Motion

The equations that we derived for the lateral motion were given in Chap. 8. In
addition to the classical states, we also used the heading equation, which was given
as P D r sec . In this section, we design a control system using the derived
equations and the matrices Alat and Blat .
For the lateral classical controller design, we prefer a method that consists of
inner loops for increasing the stability. A yaw damper is used to reduce the rudder
9.3 Classical Controller for the Lateral Motion 159

Fig. 9.19 Step response of the system for Kvt D 0:015

u
w
deltae1
q
theta
h
deltat
Vt
all
Longitudinal Dynamics1
dt
ue 2
-K-
s+2
KVt Throttle Actuator Scope-dt

Fig. 9.20 Diagram of the speed controller

reference value to zero when the controller for the heading is working. Changes in
the roll rate (p), roll angle (·), and heading angle (‰) are used to determine the
required input to the aileron. This method is a classical and effective method for
designing the lateral controller. However, during the design process, the coupling
160 9 Classic Controller Design for the UAV

Fig. 9.21 Lateral control system scheme

between the aileron and rudder should be kept in mind to overcome the difficulties;
washout filter, aileron and rudder equations should be handled together, and actuator
equations should be included in the state equations while the controller is designed
(Fig. 9.21).

9.3.1 Roll Rate Controller

We can start the design with the p loop that has the least effect on the controller
[8]. p/ı a can be obtained using the lateral equations. The aileron actuator transfer
20
function is chosen as 20Cs .

p 8:348s3 C 27:4s2 C 211:9s  33:36

D 4 (9.36)
ıa s C 4:686s3 C 26:32s2 C 44:27s  1:977

Combining the given transfer functions, we can find p/ua (roll angle
change/control input):

p 167s3 C 548s2 C 4; 238s  667:2

D 5 (9.37)
ua s C 24:69s4 C 120s3 C 570:7s2 C 883:4s  39:54

The root locus drawn for this system is given in Fig. 9.22.
9.3 Classical Controller for the Lateral Motion 161

Fig. 9.22 Root locus diagram for p/ua

What we see from the root locus is that the root of the roll mode equation must
be moved to the left on the real axis to increase the stability of the system and
the damping factor of roll angle change. The roots of the characteristic equation
previously determined are:

sspiral D 0:0435; sroll D 2:1658; sdutchroll D 1:2819C4:3966i; 1:2819  4:3966i:

From the root locus, we understand that the root may be moved to 4.03 by
choosing a gain value of 0.165. At the same time, this gain value faintly decreases
the Dutch roll mode damping factor. We also need to check the value of the spiral
mode root. The spiral mode root is unstable; however, when the gain is changed, the
root becomes 0.09 and the time for double amplitude can be found as:

log.2/
TD D 7:5s; (9.38)
0:09
which is good enough to ensure a normal working condition. Thus, the gain value
Kp can be chosen as 0.165.
The second loop to be closed is r/ı r (change in yaw angle by rudder input).
162 9 Classic Controller Design for the UAV

9.3.2 Yaw Damper

Starting from this step of the lateral controller design, we need to define output
matrix C and gain matrix K by taking the coupling effect in lateral motion and the
washout filter used in the r/ı r loop into account.
The state matrix (Alat ) of the lateral system is:
2 3
1:0502 1:9276 9:8215 9:6610 0
6  1:2213 1:9155 1:0096 0 07
6 7
6 7
Alat D 6 1:7255 0:0919 1:7198 0 07 (9.39)
6 7
40 1:0000 0:1763 0 05
0 0 1:0154 0 0

The new state matrix can be found as:

2 3
Alat Blat 0
6 0 0 0 0 0  1=Ta 0 0 7
XD6 7
4 0 0 0 0 0 0  1=Tr 0I 5 (9.40)
0 0 1 0 0 00  1=Tw

Here, ı a , aileron input, ı r , rudder input, and xw , washout, are added into the lateral
state matrix. Ta , Tr , and Tw are the time constants for the aileron actuator, rudder
actuator, and washout filter, respectively. The new lateral state matrix is:
2 3
1:0502 1:9276 9:8215 9:6610 0 0 1:8218 0
6  1:2213 1:9155 1:0096 0 0 8:3479 10:8560 0 7
6 7
6 1:7255 0:0919 1:7198 0 0 4:2400 2:1272 0 7
6 7
6 7
60 1:0000 0:1763 0 0 0 0 0 7
XD6 7 (9.41)
60 0 1:0154 0 0 0 0 0 7
6 7
60 0 0 0 0 20:0000 0 0 7
6 7
40 0 0 0 0 0 20:0000 0 5
0 0 1:0000 0 0 0 0 5

In this case, the control distribution matrix B becomes:

2 3
0 0
60 0 7
6 7
60 0 7
6 7
6 7
B D 60 0 7 (9.42)
6 7
6 20 0 7
6 7
4 0 20 5
0 0
9.3 Classical Controller for the Lateral Motion 163

The values in the new B matrix are the rudder and aileron actuator constants. The
control inputs and states in this case are:
 
ua
ulat D (9.43)
ur
 T
xlat D v p r ' ‰ ıa ır xw (9.44)

At this point, if we can define the necessary C and K matrices, then the feedback
can be achieved by calculating:

Xlat D X  BKC (9.45)

The matrix C includes feedback from xw , p, and ®. In B, one input from the
aileron and one input from the rudder are present. K must be determined at each
step using the appropriate feedback channel.
For example, for the first step, the gain matrix is:
 
0:1650 0 0
KD ; (9.46)
0 00

and C can be determined as:

2 3
01000000
C D 4 0 0 1 0 0 0 0 5 5 : (9.47)
00010000

where xw , p, and ® are chosen as the outputs. In this way, for the first step, the
following matrix can be calculated:
2 3
00 00 00 00
60 0 00 00 0 07
6 7
60 0 00 00 0 07
6 7
6 7
60 0 00 00 0 07
BKC D 6 7 (9.48)
60 0 00 00 0 07
6 7
6 0 3:3000 0 0 00 0 07
6 7
40 0 00 00 0 05
00 00 00 00

The system must be updated using the newly calculated A matrix and finding r/ı r
by means of the xw state. The new lateral state matrix with the first feedback is:
164 9 Classic Controller Design for the UAV

2 3
1:0502 1:9276 9:8215 9:6610 0 0 1:8218 0
6  1:2213 1:9155 1:0096 0 0 8:3479 10:8560 0 7
6 7
6 1:7255 0:0919 1:7198 0 0 4:2400 2:1272 0 7
6 7
6 7
60 1:0000 0:1763 0 00 0 0 7
X1 D 6 7 (9.49)
60 0 1:0154 0 00 0 0 7
6 7
60 3:3000 0 0 0 20:0000 0 0 7
6 7
40 0 0 0 00 20:0000 0 5
0 0 1:0000 0 00 0 5

Now, we can find the required transfer functions as:

r 42:54s4  1; 020s3  7; 151s2 C 5; 162s C 6; 2350

D 6 ;
ır s C 44:69s5 C 641:3s4 C 3; 613s3 C 14; 810s2 C 3; 1510s  2; 992
(9.50)
r 42:54s5 1; 020s4 7; 151s3 C5; 162s2 6; 2350s
D 7 :
ur s C49:69s C864:8s5 C6; 820s4 C32; 875s3 C105; 560s2 C154; 558s14;960
6
(9.51)

r/ur is calculated by adding the washout filter transfer function to r/ı r .

The roots of the characteristic equation can be calculated as 20.0642,
18.2896, 1.1988 C 4.5611i, 1.1988  4.5611i, 5.0000, 4.0297, and 0.0910.
The purpose of using feedback in this loop is to increase the Dutch roll mode
damping factor. As can be understood from the root locus graphic (Fig. 9.23), for

Fig. 9.23 Root locus diagram for the yaw angle loop
9.3 Classical Controller for the Lateral Motion 165

negative gains, a gain value of 0.405 makes the damping factor 0.43 and does not
affect the spiral mode root much. Therefore, Kr can be chosen as 0.4. After this
step, we must find ®/ı a , which is the equation that is necessary for ® feedback.

9.3.3 Roll Angle Loop

Similar steps to those introduced in the previous section can be used to find the
equation that relates ® to ı a and ua :

' 181:9s4 C5; 143s3 C4; 768s2 C186; 000sC440; 400

D 7
ua s C49:69s C881:7s5C7; 227s4 C35; 730s3 C103; 500s2 C129; 600s14; 960
6
(9.52)

The roots of the characteristic equation are: 19.0107 C 2.3780i, 19.0107

 2.3780i, 2.1175 C 4.5109i, 2.1175  4.5109i, 3.7698 C 1.1211i, 3.7698
 1.1211i, and 0.1061.
Gain values are determined using the root locus diagram (Fig. 9.24).

Fig. 9.24 Root locus diagram for the roll angle feedback loop
166 9 Classic Controller Design for the UAV

A gain value of K'  1 means that the spiral mode root becomes more stable
and the Dutch roll mode damping factor also stays high. In this case, the roll mode
root stays at a value of 5.5. This choice of K gives us a nonvibratory response. Also,
choosing a high proportional gain value decreases the steady-state error value.

9.3.4 Heading Controller

In order to design the heading controller, we should use a last gain matrix to
determine the equation that relates heading to the aileron input. The last gain matrix
we found is:
 
0:1650 0 1
Klast D (9.53)
0 0:4 0

Recalculating the lateral A matrix and choosing the appropriate output, we can find:

86:11s5C2;424s4 C16;340s3C54;840s2 C191;700sC384;200

D
ua s8C49:69s7C881:7s6 C7;409s5C40;870s4 C151;100s3 C315;600s2 C425;400s
(9.54)

The open-loop roots of the last loop are: 19.117 C 1.92i, 19.117  1.92i,
1.5285 C 4.79i, 1.5285  4.79i, 5.6534, 1.3728 C 2.4857i, and 1.3728
 2.4857i.
The proportional gain value is chosen as 0.5 (Fig. 9.25). With this value, the roll
mode root value is improved for better response and the spiral mode is stabilized.
Also, the Dutch roll mode damping factor becomes 0.3. The response of the system
is given in Figs. 9.26, 9.27, 9.28, 9.29, and 9.30.
In the heading controller, a P-type controller is chosen because of its simple
design and quick response. As we can see from the figures, the controller directs the
aircraft to the reference heading value quickly, without significant change in the roll
angle.
In this system, the rudder and aileron actuator time constants are taken as 0.05 s
20
and their transfer functions are positive and negative sC20 , respectively.
The washout filter in the rudder loop has a time constant of T D 0.2 s, and its
s
transfer function is sC5 .
9.3 Classical Controller for the Lateral Motion 167

Fig. 9.25 Root locus value for K‰ D 0.5

Fig. 9.26 Step response of the system to unit heading reference input
168 9 Classic Controller Design for the UAV

0.35
X: 26
Y: 0.35
0.3

0.25
Heading angle (rad)

0.2

0.15

0.1

0.05

0
0 50 100 150 200 250
t(s)

Fig. 9.27 Response of the system to a 0.35 radians reference heading input

0.08

0.07

0.06

0.05
r (rad/s)

0.04

0.03

0.02

0.01 X: 9
Y: 0.0004563
0

−0.01
0 50 100 150 200 250
t(s)

Fig. 9.28 Change in r for a heading input of 0.35 radians

9.3 Classical Controller for the Lateral Motion 169

0.09

0.08

0.07

0.06
Phi – roll angle (rad)

0.05

0.04

0.03

0.02

0.01

0
0 50 100 150 200 250
t(s)

Fig. 9.29 Change in · for a heading input of 0.35 radians

0.03

0.02
P change in roll angle(rad/s)

0.01
X: 23
Y: −6.287e-005
0

−0.01

−0.02

−0.03
0 50 100 150 200 250
t(s)

Fig. 9.30 Change in p (rad/s) for a heading input of 0.35 radians

170 9 Classic Controller Design for the UAV

9.4 Conclusion

In this chapter, a flight control system is designed for UAVs using classical control
methods. Separate discussions for controlling different parameters, such as altitude
and heading, are given, and controller design procedures are explained in detail.
This method is the most common method for UAV control, and good results for
controlling both the lateral and longitudinal motion parameters also prove that.

References

1. Golnaraghi F, Kuo BC (2009) Automatic control systems, 9th edn. Wiley, Englewood Cliffs
2. Yechout TR, Morris SL, Bossert DE, Hallgren WF (2003) Introduction to aircraft flight
mechanics. AIAA education series, Reston
3. Williams W (2003) UAV handling qualities : : : ..you must be joking. Aerospace Sciences
Corporation Pty. Ltd., Elizabeth, Australia. Available online at: http://aerosciences.com.au/
hidden/UAV%20Handling%20Qualities%20Paper%20v1.pdf
4. Capello E, Guglieri G, Marguerettaz P, Quagliotti F (2012) Preliminary assessment of flying and
handling qualities for mini-UAVs. J Intell Robot Syst 65(1–4):43–61. doi:10.1007/s10846-011-
9565-5
5. Pamadi BN (2003) Performance, stability, dynamics, and control of airplanes. AIAA education
series, Reston
6. Nelson RC (1998) Flight stability and automatic control. McGraw-Hill, New York
7. Haugen F (2004) PID control. Tapir Academic Press, Trondheim
8. Stevens BL, Lewis FL (1992) Aircraft control and simulation. Wiley, Chichester
Chapter 10
Linear Quadratic Regulator Controller Design

10.1 Introduction

The first steps to design a control system for an UAV is defining the dynamic
model and finding the aerodynamic coefficients. The nonlinear model can then
be linearized to design a linear controller for the UAV [1]. The characteristic
values of different motions that show how the aircraft normally behaves must
also be investigated at this stage. Finally, the controller and filter can be designed
considering the required specifications.
In this chapter, the optimal control technique linear quadratic regulator (LQR)
is chosen to design a controller for the longitudinal motion of a small fixed-wing
type UAV. The Kalman filter technique is then applied to see how the controller
is affected by disturbances. The effectiveness of the controller with and without
the Kalman filter is also shown through simulations. Linear quadratic control is
an optimal control technique that is used for controlling the aircraft. This control
technique aims to decrease the energy that is used to control the aircraft. This
technique can be applied together with a filtering technique (Kalman filter, etc.)
in cases where some of the states are not available for measurement or when the
measurements are noisy.
Studies that include LQR control design for different types of aircraft have
been carried out in the literature [2]. In [3], an LQR method is used to control
the trajectory and mission paths of the autonomous helicopter. Nonlinear motion
dynamics are linearized at certain operating points and a linear model is obtained
by Taylor series expansion. By using the LQR methodology, the attitude of the
autonomous Puma helicopter is controlled. The study by Öner et al. [4] shows that
the LQR controller is quite effective in the vertical flight mode for all possible
yaw angles. In [5], the design procedure for a gain-scheduled LQR controller for
an autonomous airship is presented. Two types of control subsystems (lateral and
longitudinal) have been designed from the nonlinear six degrees of freedom (6DOF)

© Springer International Publishing Switzerland 2015 171

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_10
172 10 LQR Controller Design

airship model to fulfill different goals (yaw as well as speed and position control).
In references [2–5], the LQR controller is not applied together with a filtering
technique.
In some studies, filtering techniques are used to analyze the effects of LQR design
and increase the effectiveness of the controller. The paper by Santoso et al. [6]
depicts the application of linear quadratic optimal control to the longitudinal flight
motion of a UAV which has elevon control only. The LQR controller is followed by
a Kalman filter-based estimator for unmeasurable states. The LQR controller is then
combined with the Kalman estimator using the separation principle to investigate the
feasibility of altitude control. The simulation results show improvements compared
with classical design counterparts in the sense that the combined approach offers
more design flexibility and is able to tolerate noisy environments.
Some advanced techniques, including robust and adaptive control techniques,
are also used in UAV control. For example, in one study by Johnson et al., the H-
infinity technique is used to design a velocity and altitude controller that follows a
determined model [7]. However, in this study, these advanced techniques are only
investigated to design a better controller and are not used.
In this chapter, an altitude controller is designed for a small UAV using the
optimal LQR control method with the Kalman estimator. The effectiveness of the
controller with the gains found by the LQR method is investigated under the effect
of disturbances and with the Kalman filter.

10.2 Linear Quadratic Optimal Controller

In order to design an optimal controller for the UAV, we use the linear quadratic
controller method in this chapter. This method depends on minimizing criteria using
the feedback u D kx.t/. The performance index that is used for minimizing both
the control effort and the states can be given as [8]:

Z1
1  
JD xT Qx C uT Ru dt (10.1)
2
0

kxk D xT .t/x.t/ and x is an n  1 state vector. The given norm that depends
on the states can be used as a measure of the system response. The index also
includes u inputs to control the system. The Q and R matrices in J are called the
weight matrices. The weight matrices allow us to define the effect of each state and
control input in the controller response when we use the feedback for designing the
controller.
10.2 Linear Quadratic Optimal Controller 173

Q is an n  n real, symmetric, positive (or semipositive) definite matrix. Q can be

given as:
2 3
q1 0    0
6 0 q2 07
6 7
QD6 : :: 7 (10.2)
4 :: : 05
0 0 0 qn

The weight given to the ith element of the matrix Q is a measure of the control effort
to control the related state. The bigger the element, the bigger the control effort used
for that state.
The matrix R is a p  p, real, symmetric, and positive-definite matrix. Choosing
a positive-definite R is a must for an optimal solution.
If we can find the feedback u D kx.t/ that minimizes the J index for the system
given by xP D AxCBu, then we can solve the optimal control problem. In order to do
this, we need to relate the Lyapunov stability criteria with the quadratic performance
index.

10.2.1 Lyapunov Stability Criteria

The Lyapunov stability criteria deal with the behavior of the system response. When
the system leaves a steady state, whether it returns to a stable point or not can be
measured using the Lyapunov criteria. If a nonvibratory system response takes the
system very close to a point where it is stable for every x, then the system is called
Lyapunov stable. If the nonvibratory response of a system from a point x goes near
point xe (equilibrium point) for every x, then the system is called Lyapunov stable.
If the system response goes to xe for every x, then the system is called Lyapunov
asymptotically stable.
In order to control this criterion, the norm values kxk D xT .t/x.t/ that produce
a hypercubic space near the equilibrium point can be used. If the system response
stays in an area defined as S, it shows that the system is stable. In conclusion, using
the norm value, the total energy in the system can be calculated and the stability
condition can be checked. This method is called the Lyapunov method.
Generally, Lyapunov-type scalar functions can be defined as:

V.x/ D xT Px (10.3)

where P is a real, symmetric, positive- or negative-definite matrix. The positive

definiteness of the matrix P defines the status of the Lyapunov function. If a
174 10 LQR Controller Design

Lyapunov function can be built for the system and the function has partial derivatives
for every x variable, the stability condition can be checked using these partial
derivatives.
The positive definiteness requirements are:

V.x/ > 0 (10.4)

V.0/ D 0 (10.5)

The following conditions are checked in order to understand whether the system
is stable or not:
P
(a) Asymptotically stable if V.x/ < 0 and x ¤ 0
(b) Generally asymptotically stable if the requirements in (a) are met and V.x/ >
1 when kxk  > 1
Thus, a quadratic Lyapunov function can be defined and used for checking
system stability. This leads us to the linear quadratic optimal control method.

10.2.2 Linear Quadratic Optimal Control Using Lyapunov

Criterion

When we add u D kx.t/, the performance index J becomes:

Z1
1  
JD xT Qx C xT K T RKx dt (10.6)
2
0

A Lyapunov stability function V(x(t)) can be defined similar to the J performance

index:
Z1
1  T 
V .x.t// D x Qx C xT K T RKx dt (10.7)
2
0

The derivative of the Lyapunov function can be found, keeping in mind that
x(1) D 0:

1 
VP .x.t// D  xT Qx C xT K T RKx (10.8)
2
The Lyapunov function can also be written as:

1 T
V.x/ D x Px (10.9)
2
10.2 Linear Quadratic Optimal Controller 175

The derivative of this function is:

P 1 T 
V.x/ D x Px C xT PPx (10.10)
2
Combining the equations found for the derivative of V(x), the following equation
can be solved:

.A  BK/T P C P .A  BK/ C K T RK C Q D 0 (10.11)

The value of V(x) at 0 is equal to the performance index value for the system
trajectory starting from 0. Thus, the J index can be calculated as:

1 T
JD x .0/Px.0/ (10.12)
2
The values of matrix P that minimize the performance index J can then be calculated
as:
 
@ xT .0/Px.0/
D0 (10.13)
@kij

If the values of P that are optimal make the matrix P positive-definite, this
guarantees a stable loop defined by the Lyapunov function and we can calculate
the optimal gain. We may obtain the same result using the Riccati equation, which
is:

R D  T ; (10.14)

from the definition of the matrix R. Using the given equation for R and deriving P
to find the optimal values, we obtain the optimal control input:

@   1 T   1 
K   T BT P K   T BT P D 0 (10.15)
@kij

This equation is calculated using the Lyapunov-type formula. Deriving for K, we

can find u, the input that brings the system optimally under control:
 1
K D  T BT P (10.16)

K D R1 BT P (10.17)

u.t/ D KR1 BT Px.t/ (10.18)

176 10 LQR Controller Design

The Riccati matrix equation can be found by using the previously determined value
of K [8].

AT P C PA  PBR1 BT P C Q D 0 (10.19)

For designing a controller, both the first formula and the given Riccati equation can
be used. If the Riccati equation is to be used, the longitudinal matrices Alon , Blon and
lateral matrices Alat , Blat must first be determined. Then, the equation for finding the
matrix P can be determined or MATLAB functions can be used to find the required
values.
As an advantage, the LQR design can be done easily. However, a disadvantage
of this method is expecting that the values of the states are known exactly at every
moment. However, in general, these values are not known exactly and a filtering
technique is necessary. In this study, Kalman filtering is used to estimate the states
that are needed in the optimal controller.

10.3 Altitude and Speed Controller Design Using the LQR

Method

10.3.1 LQR Altitude Controller

Using the LQR method, a longitudinal controller for altitude and speed and a lateral
controller for heading is designed. The structure of the system allows us to use the
gain values found using the Riccati equation directly. The formulas given below
describe the motion of the vehicle around the stable point. Here, the xQ represents the
divergence from the required point and xd represents the reference input:

xQ D x  xd (10.20)

xPQ D xP  xdP (10.21)

xPQ D Ax C Bu  xdP (10.22)

xPQ D Ax C Bu  xdP C Axd  Axd (10.23)

xPQ D AQx C Bu C Axd  xdP (10.24)

As we can understand, if Axd D xdP holds using only u D kx.t/ without adding
anything into the state matrices, it will be enough to control the system optimally.
With choosing xd D [0; 0; 0; 0; hd], where hd is the reference altitude input, we can
10.3 Altitude and Speed Controller Design Using the LQR Method 177

reach the reference point only using state feedback in the given system. In short,
in cases where we have to control the state which has no direct effect on the other
states (where the A matrix columns related to the state are zeros), we can use the
given method directly.
The scheme of the controller is given in Fig. 10.1. In the scheme, hd values are
multiplied by K1 D [0;0;0;0;1] and added to u D kLQR x.t/ to find the real control
input that is used in the state model. The longitudinal states h, u, w, q, and  are
scoped.
The controller is used to take the altitude value to the reference while holding the
other values constant. As we can see in Figs. 10.2, 10.3, 10.4, 10.5, 10.6, and 10.7,
the changes in the values other than the altitude goes to zero and the states hold their
values.
The weight matrices (Q, Q1 , and R) used in the controller are given below. The
change in U observed while the altitude value is changing can be controlled by
changing the corresponding weight matrix value. For this purpose, two different Q
matrices (Q and Q1 ) are used and the results are given. In R, the value for the elevator
is chosen to be high in order to use the elevator effort more effectively:
 
100 0
RD (10.25)
0 1
2 3
1000 0
60 1 0 0 0 7
6 7
6 7
Q D 60 0 1 0 0 7 (10.26)
6 7
4 0 0 0 100000 0 5
0000 10
2 3
10 0 0 0 0
60 1 0 0 0 7
6 7
6 7
Q1 D 6 0 0 1 0 0 7 (10.27)
6 7
4 0 0 0 100000 0 5
0 000 10

10.3.2 LQR Speed Controller

To design an LQR-type speed controller, an integrator is added to the system. For

this system, which includes forward velocity and integrator states, a gain value using
the given LQR method is obtained. In Fig. 10.8, the system built in Simulink for
simulation purposes can be seen. Two controllers are needed for the longitudinal
control in order to control both speed and altitude. A MUX function is used to add
the integrator state to the system and the u input values are fed through actuator
functions and the integrator. After adding the integrator, the gain value Klon is
multiplied by system states and supplied to the system model as inputs.
 178

simout1

u_tilde To Workspace1
1
Matrix
Gain1 Kazanc Matrisi
x' = Ax+Bu
20 K*u K* u
y = Cx+Du u_tilde
h_d Saturation1 Plant

w_tilde simout2

delta_e To Workspace2
delta_t
q_tilde

-K-

Gain theta_tilde
10

simout

To Workspace

Fig. 10.1 Linear quadratic regulator (LQR) altitude controller

LQR Controller Design
10.3 Altitude and Speed Controller Design Using the LQR Method 179

Fig. 10.2 LQR altitude controller response for a 20-m hd input (with Q)

Fig. 10.3 Change in U for a 20-m altitude input (with Q)

180 10 LQR Controller Design

2.5

2
Pitch angle (degrees)

1.5

0.5
X: 19
Y: 0.0002329
0

-0.5
0 5 10 15 20 25 30 35 40
t(s)

Fig. 10.4 Change in pitch angle for a 20-m hd input (with Q)

Fig. 10.5 LQR altitude controller response for a 20-m hd input (with Q1 )
10.3 Altitude and Speed Controller Design Using the LQR Method 181

Fig. 10.6 Change in U for a 20-m altitude input (with Q1 )

4
Pitch angle (degrees)

1
X: 25
Y: 0.002847
0

-1
0 5 10 15 20 25 30 35 40
t(s)

Fig. 10.7 Change in pitch angle for a 20-m hd input (with Q1 )

 182
10

Fig. 10.8 Simulink scheme of the forward velocity controller

LQR Controller Design
10.3 Altitude and Speed Controller Design Using the LQR Method 183

Fig. 10.9 Response of the LQR speed controller to a 10-m/s reference input

The designed speed controller takes the aircraft to the reference speed while
holding the other states constant. The changes in different states when a 10-m/s
change in U is commanded are given in Figs. 10.9, 10.10, and 10.11.
The integrator is added to the system as the first state. The error of the speed is
fed to the system as a sixth state in the longitudinal equations. The new A and B
matrices built by adding the integrator are given below:
2 3
0 1:0000 0 0 0 0
60 0:3356 1:3181 1:9276 9:6610 07
6 7
6 7
60 1:7916 3:9003 9:8215 1:7035 07
Alon D6 7 (10.28)
60 0:7020 3:5375 11:3920 0 07
6 7
40 0 0 1:0000 0 05
0 0:1736 0:9848 0 17:4865 0
2 3
0 0
6  0:7436 6:8728 7
6 7
6 7
6 3:7855 0 7
Buzu D 6 7 (10.29)
6 47:9170 0 7
6 7
40 0 5
0 0
184 10 LQR Controller Design

Fig. 10.10 Change in altitude when the LQR speed controller takes the system to 10 m/s

-0.02

-0.04

-0.06
Pitch angle (rad)

-0.08

-0.1

-0.12

-0.14

-0.16
X: 18
Y: -0.1832
-0.18

-0.2
0 10 20 30 40 50 60 70 80 90 100
t(s)

Fig. 10.11 Change in pitch angle when the LQR speed controller takes the system to 10 m/s
10.4 LQR-Type Heading Controller 185

It is observed in the tryouts that the reference speed value can be reached quickly
without affecting the other states (altitude, pitch angle, etc.) when we choose the
diagonal elements of Qlon as 1000, 10, 1, 1, 100, and 30000. The diagonal elements
of Rlon are chosen as 100 and 100 to give the same effort to both elevator and thrust
control inputs:
2 3
1000 0 0 0 0 0
60 10 0 0 0 0 7
6 7
6 7
60 0 100 0 7
Quzu D6 7 (10.30)
60 0 010 0 7
6 7
40 0 0 0 100 0 5
0 0 0 0 0 30000
 
100 0
Ruzu D (10.31)
0 100

10.4 LQR-Type Heading Controller

For the lateral controller, an integrator may also be added to the system; however, the
structure of the system tells us that, for controlling the heading using the reference
input xd D [0; 0; 0; 0; d], the normal lateral matrices are sufficient. For the case
where heading is the input, the following equation holds:

xdP D Alat xd (10.32)

Thus, as stated, we can use the lateral state and control distribution matrices to
calculate the result of the Riccati equation and the LQR gain values. The Q and R
matrices for the lateral LQR controller are chosen as follows:
2 3
10 0 0 0
6 0 10 0 0 0 7
6 7
6 7
Q D 6 0 0 100 0 0 7 (10.33)
6 7
4 0 0 0 10000 0 5
00 0 0 1000
 
100 0
RD (10.34)
0 100

The scheme of the lateral LQR heading controller is given in Fig. 10.12. As we
can understand from the results (Figs. 10.13, 10.14, and 10.15), the LQR controller
is working optimally and takes the UAV to the desired heading with little changes
occurring in ı a and ı r . The vibrations in states may be lessened by choosing
appropriate Q and R weight matrices.
 186

Matrix Matrix
Gain1 Gain
x' = Ax+Bu
0.35 K*u K*u
y = Cx+Du beta_tilde
Psi_d Saturation1 Plant

p_tilde

delta_a
delta_r
r_tilde

-K-

Gain Phi
Psi
10

Fig. 10.12 Lateral LQR controller: MATLAB Simulink scheme

LQR Controller Design
10.4 LQR-Type Heading Controller 187

0.4

0.35
X: 26
Y: 0.35
0.3
Heading angle(rad)

0.25

0.2

0.15

0.1

0.05

0
0 50 100 150 200 250 300
t(s)

Fig. 10.13 LQR controller response for a d D 0.35 radians input

0.16

0.14

0.12
r(change in yaw angle(rad/s))

0.1

0.08

0.06

0.04

0.02
X: 37
Y: -2.034e-007
0

-0.02
0 50 100 150 200 250 300
t(s)

Fig. 10.14 Change in r (yaw angle change, rad/s) when a d D 0.35 radians input is given
188 10 LQR Controller Design

5
Phi(roll angle(rad))

1
X: 30
Y: 0.0004451
0

-1
0 50 100 150 200 250 300
t(s)

Fig. 10.15 Change in roll angle (·) when a d D 0.35 radians input is given

10.5 LQR Controller with the Kalman Estimator

In this section, it is assumed that the measured states are correct at any time and the
designed LQR is fed with them to control the UAV. However, noise is always present
and the system may not give accurate results under normal operating conditions. For
this purpose, a Kalman filter that has the capability to estimate the real values of the
states is incorporated in the study. The results of the system with a Kalman filter
and LQR and only LQR are then compared to show the effectiveness of the Kalman
filter. The system model is discretized using the Euler method. Firstly, the nominal
model is tested with 20 m altitude and 20.02ı heading inputs, with the optimal
gain values calculated previously. Then, the system with noise is simulated. The
response of the system with the controller under the disturbances is tested with and
without the Kalman filter and the effectiveness of the designed LQR and Kalman
filter system is shown [9].
The Kalman filter uses state equations (state space matrices) and initial values
to calculate the residue and gain values and to estimate the real signal value. The
steps of the Kalman filter can be explained using the linear discretized state and
measurement equations:

X .k C 1/ D AX.k/ C Bu.k/ C Gw.k/ (10.35)

y.k/ D HX.k/ C v.k/ (10.36)

10.5 LQR Controller with the Kalman Estimator 189

In the state equation, X(k) is the state vector of the system, A is the system
transition matrix, u(k) is the input vector, B is the control distribution matrix, w(k)
is the random Gaussian noise vector (system noise) with zero mean and known
covariance structure, and G is the transition matrix of the system noise. In the
measurement equation, y(k) is the measurement vector, H is the measurement
matrix, and v(k) is the measurement noise vector with zero mean and known
covariance structure.
There is no correlation between the system noise w(k) and the measurement noise
v(k). The covariance matrices for the w(k) and v(k) vectors are given by:

E w.k/wT .j/ D Q.k/ı.kj/


E v.k/v T .j/ D R.k/ı.kj/

Here, E is the expected value operator and ı(kj) is the Kronecker symbol.
The optimum linear Kalman filter that estimates the state vector of the system
(10.35) is expressed using the following recursive equations system:
Equation of the extrapolation value:

Xe .k=k  1/ D AXe .k  1=k  1/ C BKLQR .k  1/ .Xd  Xe .k  1=k  1//

(10.37)

The innovation sequence:

.k/ D Z.k/  HXe .k=k  1/ (10.38)

Equation of the estimation value:

Xe .k=k/ D Xe .k=k  1/ C K.k/.k/ (10.39)

Gain matrix of the optimum linear Kalman filter:

 1
K.k/ D P .k=k/ H T R1 .k/ D P .k=k  1/ H T HP .k=k  1/ H T C R.k/
(10.40)

The covariance matrix of the filtering error is:

P .k=k/ D .I  K.k/H/ P .k=k  1/ (10.41)

The covariance matrix of the extrapolation error is:

P .k=k  1/ D AP .k  1=k  1/ AT C BDu .k  1/ BT C GQ .k  1/ GT (10.42)

where Xd is the desired vector and I is the identity matrix.

190 10 LQR Controller Design

The Kalman filter tries to estimate the real signal from the signal with distur-
bance, which has a Gaussian distribution, using the described steps and decreasing
the value between two signals [10].

10.5.1 Longitudinal LQR with the Kalman Estimator

The model of longitudinal motion can be discretized using the Euler approach for
a sampling time of dt. Thus, the new discretized A and B matrices to be used in the
filtering approach can be found by A1 D A  dt C I and B1 D B  dt.
The discretized UAV model can be given as follows:
2 30 1
0:997 0:013 0:02 0:097 0 u.k/
6  0:018 0:961 0:098 0:017 0 7 B w.k/ C
6 7B C
6 7B C
X .k C 1/ D 6 0:007 0:035 0:886 0 0 7 B q.k/ C
6 7B C
40 0 0:01 1 0 5 @ .k/ A
 0:0017 0:01 0 0:175 1 h.k/
2 3
0:007 0:07
6 0:04 0 7
6 7 
6 7 ıe .k/
C 6 0:48 0 7 C Gw.k/ (10.43)
6 7 ıt .k/
40 0 5
0 0

y.k/ D HX.k/ C v.k/ (10.44)

In our case, disturbance with Gaussian white noise characteristics generated by

MATLAB commands is applied to the real values found using the model of the
longitudinal dynamics of the UAV. The Kalman filtering technique is then applied
and its effectiveness is shown. In the real scenario, disturbances in measurements
and processes are normal and have an effect on the controller. Thus, using a filtering
technique is very important.
The values of the states can be calculated using Eq. (10.43), which includes
the control rule. The gain value K is the one found for the longitudinal LQR
controller case. The disturbances on the states must, of course, be first determined
and applied to the system. Finally, the Kalman filter can be applied to the system
with the disturbances to develop an effective controller. To do this, some MATLAB
code is written. The results are given in Figs. 10.16, 10.17, 10.18, 10.19, 10.20,
10.21, 10.22, 10.23, and 10.24. The Kalman filter that works as an optimal observer
estimates the new values of the states correctly and decreases the error [10, 11].
As can be seen from Figs. 10.16, 10.17, 10.18, 10.19, 10.20, 10.21, 10.22, 10.23,
and 10.24, in case of disturbances, using a Kalman filter to estimate the values of
the longitudinal states clearly increases the effectiveness of the LQR controller.
10.5 LQR Controller with the Kalman Estimator 191

Fig. 10.16 Change in U velocity (5,000 steps D 500 s)

Fig. 10.17 Change in U velocity (zoomed, 5,000 steps D 500 s)

192 10 LQR Controller Design

Change in altitude
25
LQR without disturbances
LQR(with disturbances)
20 LQR(with Kalman filter

15
altitude,h

10

-5
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
step

Fig. 10.18 Change in h in the LQR controller (5,000 steps D 500 s)

Change in altitude

20.15
LQR without disturbances
LQR(with disturbances)
20.1 LQR(with Kalman filter

20.05
altitude,h

20

19.95

19.9

19.85

1710 1715 1720 1725 1730 1735 1740 1745

step

Fig. 10.19 Change in h in the LQR controller (zoomed, 5,000 steps D 500 s)
10.5 LQR Controller with the Kalman Estimator 193

Altitude errors with Kalman filter

0.4
Errors in altitude
0.3

0.2
error,kalman-normal

0.1

-0.1

-0.2

-0.3

-0.4
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
step

Fig. 10.20 Differences between Kalman filter estimation and normal values for altitude

Forward velocity errors with Kalman filter

1.5
Errors in forward velocity
1

0.5
error,kalman-normal

-0.5

-1

-1.5

-2

-2.5

-3
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
step

Fig. 10.21 Differences between Kalman filter estimation and normal values for forward velocity
194 10 LQR Controller Design

Altitude error without Kalman filter

0.8
Errors in altitude
0.6

0.4
error,disturbance-normal

0.2

-0.2

-0.4

-0.6

-0.8
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
step

Fig. 10.22 Differences between readings with disturbances and normal values for altitude

Forward velocity errors without Kalman filter

5
Errors in forward velocity
4

3
error,disturbances-normal

-1

-2

-3

-4
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
step

Fig. 10.23 Differences between readings with disturbances and normal values for forward
velocity
10.5 LQR Controller with the Kalman Estimator 195

Diagonal element of covariance matrix

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Fig. 10.24 Change in the fifth diagonal element of the covariance matrix (P1 )

10.5.2 Lateral LQR with the Kalman Estimator

A similar study for the lateral controller is accomplished. For this system, heading
reference is the input. If we choose Xd D [0; 0; 0; 0; d] and add the necessary LQR
gain input, we can build a model for the system. The MATLAB code for the system
is similar to that for the longitudinal case.
The change in r (yaw angle change, rad/s) and heading angle are given in the
graphs. The simulation results are given in Figs. 10.25, 10.26, 10.27, 10.28, 10.29,
10.30, 10.31, and 10.32. The results show the effectiveness of the Kalman filter.
The differences between the normal and noisy values for heading and yaw angles
are also given.
The presented simulation results (Figs. 10.25, 10.26, 10.27, 10.28, 10.29, 10.30,
10.31, and 10.32) show that, in case of disturbances, using a Kalman filter to
estimate the values of the lateral states clearly increases the effectiveness of the
LQR controller.
196 10 LQR Controller Design

Heading angle
25

20

15
heading,psi

LQR without disturbances

LQR(with disturbances)
LQR(with Kalman filter
10

0
0 50 100 150 200 250 300 350 400 450 500
step

Fig. 10.25 Heading controller response to a 20ı input (500 steps D 50 s)

Heading angle
20.3

20.2
LQR without disturbances
20.1 LQR(with disturbances)
LQR(with Kalman filter
20
heading,psi

19.9

19.8

19.7

19.6

19.5

19.4

390 395 400 405 410 415 420 425 430 435
step

Fig. 10.26 Heading controller response to a 20ı input (zoomed, 500 steps D 50 s)
10.5 LQR Controller with the Kalman Estimator 197

Change in roll angle

35
LQR without disturbances
30 LQR(with disturbances)
LQR(with Kalman filter

25
change in roll angle,r

20

15

10

-5
0 50 100 150 200 250 300 350 400 450 500
step

Fig. 10.27 Change in r (500 steps D 50s)

Roll angle change errors with Kalman filter

0.15
Errors in roll angle change

0.1

0.05
errors,kalman-normal

-0.05

-0.1

-0.15

-0.2
0 50 100 150 200 250 300 350 400 450 500
step

Fig. 10.28 Differences between Kalman filter estimations and normal system for r
198 10 LQR Controller Design

Heading angle errors with Kalman filter

0.3
Heading angle errors
0.25

0.2

0.15
errors,kalman-normal

0.1

0.05

-0.05

-0.1

-0.15

-0.2
0 50 100 150 200 250 300 350 400 450 500
step

Fig. 10.29 Differences between Kalman filter estimations and normal system for heading angle

Roll angle change errors without Kalman filter

3
Errors in roll angle change
2.5

2
errors,disturbance-normal

1.5

0.5

-0.5

-1
0 50 100 150 200 250 300 350 400 450 500
adim

Fig. 10.30 Differences between readings with disturbances and normal system for r
10.5 LQR Controller with the Kalman Estimator 199

Heading angle errors without Kalman filter

0.8
Errors in heading angle
0.6

0.4
errors,disturbance-normal

0.2

-0.2

-0.4

-0.6

-0.8
0 50 100 150 200 250 300 350 400 450 500
step

Fig. 10.31 Differences between readings with disturbances and normal system for heading

Diagonal element of covariance matrix

0.03

0.025

0.02

0.015

0.01

0.005

0
0 50 100 150 200 250 300 350 400 450 500

Fig. 10.32 Change in the diagonal element (fifth element) of the P2 covariance matrix
200 10 LQR Controller Design

10.6 Conclusion and Discussion

Optimal controllers using the LQR method to control the longitudinal and lateral
flight dynamics of small UAVs have been designed and examined in this chapter.
The effects of disturbances on measurements are taken into account and the
effectiveness of the Kalman filter in obtaining correct measurements and achieving
the desired control level are shown using the controller designed for the system.
A Kalman estimator is designed and added to the system and the response of the
controller with and without disturbances and the Kalman filter is tested. The results
reveal the effectiveness of the Kalman filter and the LQR controller.
The simulations show that, in case of disturbances, using a Kalman filter to
estimate the values of the states clearly increases the effectiveness of the LQR
controller. In conclusion, it is found that the LQR method with a Kalman estimator
is effective in controlling the longitudinal and lateral flight dynamics of UAVs and
can be used for such applications.

References

1. Jang JN (2003) Longitudinal stability augmentation system design for the dragon fly UAV
using a single GPS receiver. In: Proceedings of the AIAA guidance, navigation, and control
conference and exhibit, Austin, TX, August 2003. AIAA paper 2003-5592
2. Kinoshita T, Imado F (2006) A study on the optimal flight control for an autonomous UAV.
In: Proceedings of the IEEE 2006 international conference on mechatronics and automation
(ICMA), Luoyang, China, June 2006, 43(38)
3. Franko S (2009) LQR-based trajectory control of full envelope, autonomous helicopter. In:
Proceedings of the world congress on engineering (WCE 2009), London, UK, July 1–3 2009,
vol I
4. Öner KT, Çetinsoy E, S{r{moğlu E, Hancer C, Ayken T, Ünel M (2009) LQR and SMC
stabilization of a new unmanned aerial vehicle. World Acad Sci Eng Technol 34:373–378
5. Masar I, Stöhr E (2011) Gain-scheduled LQR-control for an autonomous airship. In: Proceed-
ings of the 18th international conference on process control, Tatranská Lomnica, Slovakia,
14–17 June 2011, pp 197–204
6. Santoso F, Liu M, Egan GK (2007) Linear quadratic optimal control synthesis for an UAV. In:
Proceedings of the 12th Australian international aerospace congress (AIAC12), Melbourne,
Australia, 16–22 March 2007
7. Johnson MD, Calise AJ, Johnson EN (2003) Evaluation of an adaptive method for launch vehi-
cle flight control. In: Proceedings of the AIAA guidance, navigation, and control conference
and exhibit, Austin, TX, August 2003
8. Anderson BDO, Moore JB (1990) Optimal control: linear quadratic methods. Prentice Hall,
Englewood Cliffs
9. Hajiyev C, Vural SY (2010) LQR controller with Kalman estimator applied to UAV longitu-
dinal dynamics. International workshop on unmanned air vehicles UVW2010, Istanbul, 10–12
June 2010
10. Hajiyev C (1999) Radio navigation. Istanbul Technical University, Istanbul (in Turkish)
11. Sage AP, Melsa JL (1971) Estimation theory with applications to communications and control.
McGraw-Hill, New York
Chapter 11
Fuzzy Logic-Based Controller Design

11.1 Fuzzy Logic-Based Systems

Fuzzy logic-based systems can be defined as knowledge rule-based systems. An

if–then rule-based knowledge system is the basic structure of a fuzzy logic system.
For example:
If speed is low, then apply more thrust.
In a fuzzy logic system, the if–then rules may be designed using expert
knowledge and results from experiences. Three different types of fuzzy logic-based
system can be given as follows:
1. Basic fuzzy logic system
2. Takagi–Sugeno (TSK) fuzzy logic system
3. A fuzzy system that has fuzzification and defuzzification systems
TSK gives the output as a function, while the other systems’ inputs and outputs
are fuzzy logic based. In systems that include fuzzification and defuzzification,
multiple inputs can be turned into a single output. The general scheme of a fuzzy
logic controller is given in Fig. 11.1.
In short, starting from a knowledge base, a nonlinear input–output linking
procedure (a controller) can be found using a fuzzy logic-based controller. Fuzzy
logic-based systems are widely used in automobile control systems, subway control
systems, and production control systems. For flight control systems, there have been
studies based on using solely fuzzy logic systems, as well as using fuzzy logic and
other types of controllers together [1, 2].
In fuzzy logic-based systems, the relation between the inputs and outputs can be
made with:
IF antecedent proposition THEN consequent proposition
type “IF–THEN” fuzzy rules. The antecedent proposition is always a fuzzy
proposition, like “x A”. Here, x is a linguistic variable and A is a linguistic term.

© Springer International Publishing Switzerland 2015 201

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5_11
202 11 Fuzzy Logic-Based Controller Design

Fig. 11.1 General scheme of a fuzzy logic-based system

The accuracy of the proposition is a value between 0 and 1, and depends on the
relation between x and A.
For example, the relationship between throttle and vehicle speed can be defined
by a fuzzy rule as IF throttle openness is high THEN vehicle speed is high. A
constructed fuzzy model can be named by the fuzzy rules included in it. The output
structure in a fuzzy proposition determines the fuzzy rule type. Four different types
are present:
1. Mamdani-type fuzzy rules
2. Singleton-type fuzzy rules
3. Takagi–Sugeno-type fuzzy rules
4. Tsukamoto-type fuzzy rules

11.1.1 Mamdani-Type Fuzzy Rules

This type of rule includes semiqualified knowledge about the variables, and they
can be given as:
IF x Ai THEN y Bi .
In this rule, x is the linguistic input and the Ai s are antecedent linguistics terms.
Likewise, y is the linguistic output and the Bi s are consequent linguistic terms. The
linguistic variables x and y and the linguistic terms Ai and Bi in the rules are fuzzy
sets that are defined in their own definition sector.
For example, the membership functions for the antecedent and consequent fuzzy
sets can be given as:

x 2 X Rp (11.1)

y 2 Y Rq (11.2)
11.1 Fuzzy Logic-Based Systems 203

.x/ W X > Œ0; 1 (11.3)

.y/ W Y > Œ0; 1 (11.4)

The fuzzy sets Ai are fuzzy spaces in the antecedent space, where related
consequent propositions are real. The linguistic variables Ai and Bi are generally
chosen between the terms little, very little, moderate, big, very big, etc. If we define
these sets as A and B, we can define the Ai and Bi as subsets Ai 2 A and Bi 2 B.
For example, a heater system with constant gas flow where the heat received
depends on the given O2 input may be described as a fuzzy model:
The input is the O2 flow and the output is heating power. The sets for input and
output linguistic terms can be given as:

A D fLow; Moderate; Highg (11.5)

B D fLow; Highg (11.6)

The relationship between the inputs and outputs can be given as:
Rule 1: IF O2 flow is LOW THEN the heating power is low
Rule 2: IF O2 flow is MODERATE THEN the heating power is high
Rule 3: IF O2 flow is HIGH THEN the heating power is low
The linguistic terms are defined in the fuzzy system by membership functions.
For the exemplified inputs and outputs, the membership functions given in Fig. 11.2
can be used.

Low Mod High Low High

1 1

0 1 2 3 0 25 50 75 100

O2 flow (m3/s) Heating power (W)

Fig. 11.2 Membership functions for O2 flow and W (heating power)

204 11 Fuzzy Logic-Based Controller Design

11.1.2 Singleton-Type Fuzzy Rules

Singleton-type fuzzy rules are a special type of Mamdani rules. The output fuzzy
set is a single set.
The singular membership function can be given as:

1; x D x
.x/ D (11.7)
0; IF NOT

The singular type rule can be described as:

IF x is Ai THEN y D bi , i D 1, 2, 3, : : : : : : ,r.

11.1.3 Takagi–Sugeno-Type Fuzzy Rules

In Mamdani-type IF–THEN rules, the antecedent and consequent parts define the
system with fuzzy propositions. In the consequent part of Takagi–Sugeno (T-S) type
fuzzy models, a mathematical function is present. Thus, this type of model is built
with both linguistic and mathematical expressions.
The rules in T-S models are as given below:
IF x is Ai THEN yi D fi (x), i D 1, 2, : : : .., r.
Unlike the Mamdani-type model, the input x is exact. All the f (x) functions are
of the same type; however, the parameters within them are different. For simplicity,
a linear function can be used as an output function.

11.1.4 Fuzzy Inference Mechanism

Every Mamdani-type rule as:

IF x Ai THEN y Bi , i D 1, 2, : : : , r
can be shown as a fuzzy relation:
Ri : (X  Y) ! [1,0]
This relation can be calculated using fuzzy relation intersection (Mamdani
method). When IF–THEN rules can be defined as Ai ! Bi , a fuzzy relation is
used [2].
When unification is used, IF–THEN rules can be defined as A and B must both
be true at the same time. The relation is symmetric and reversible. If ^ the minimum
operator is chosen for intersection, the membership function for Ri D Ai  Bi can be
defined using the membership function of A (Ai ) and the membership function of
B (Bi ) as:
11.1 Fuzzy Logic-Based Systems 205

0.3
0.3

1
0
min 0 max
0.3 and 0.5–0.3

0.3
1
0
min 0 max

0.5
0.5

0
min 0 max

Fig. 11.3 Fuzzy IF–THEN rule application example

Ri .x; y/ D Ai .x/ ^ Bi .y/ D min .Ai .x/; Bi .y// (11.8)

The minimum operator is applied for all possible x–y combinations in the
Cartesian space of X and Y. The fuzzy R relation that defines the model is given
by the unification of r fuzzy rules that are in the Mamdani rule base:
r
R D [ Ki (11.9)
iD1

If we choose the maximum operator for unification, the membership function for
the R relation becomes:

R .x; y/ D max .Ai .x/ ^ Bi .y// D max .min .Ai .x/; Bi .y/// (11.10)
lir lir

For an example, we can take the case that is shown in Fig. 11.3, where one rule
is presented:
IF “altitude is low (0.3 shows low altitude membership value)” and “vertical
velocity (w) is moderately low (0.5 shows low mod. membership value)” THEN
elevator angle output is “low mod”
We can use min.–max. relations to obtain the efficacy factor of the output. In this
case, the lowest value of the inputs is taken and the area of the membership function
for low or moderate elevator angle output is calculated. Mostly, more than one rule
is effective and, in this case, every single output value can be used to find an average
value:
206 11 Fuzzy Logic-Based Controller Design

mu.1/output.1/ C mu.2/output.2/ C    C mu.n/output.n/

result D (11.11)
mu.1/ C mu.2/ C    C mu.n/

The total value can be calculated by multiplying mu by the outputs. If we

choose an area-centered method for the output, the average of the areas can be
calculated.

11.2 Fuzzy Controllers

As observed in the other controller designs, the classical methods require knowledge
of the system model. However, a fuzzy logic controller may be designed using
previously known input–output relations and expert knowledge, without actually
having the system model.
A Mamdani-type controller is based on using knowledge-based rules that are
useful to take the system to previously determined requirements. Mamdani proposed
that, starting from requirements and expert knowledge, a controller based on IF–
THEN rules and membership functions for input and outputs can be designed. In
this study, Mamdani-type controllers are used [3].
In this case, the error and the derivative of the error are chosen as inputs. This
type of controller is similar to a proportional-derivative (PD) controller. For altitude,
velocity, and heading controllers, a PD-type fuzzy logic controller is designed.
The PD controller scheme used in this study is shown in Fig. 11.4. Error (e(t))
and the change in error (e(t)  e(t  1)) is used to calculate the output u. Triangular-
type membership functions and PD-type inputs are used in the controller.
The rule base for both Vt–dt and h–™ref are as shown in Table 11.1. For an
inner loop controller, the one that was previously used in the classical method is
used (P controller) and the outer loop controllers are designed based on fuzzy logic
rules.
The rules are defined as below:

-K-
-K- u fcn y
K1
K3
FLC PLANT

-K-
1
K2
z
Unit Delay

Fig. 11.4 Proportional-derivative (PD) type fuzzy logic controller scheme

11.2 Fuzzy Controllers 207

Table 11.1 Rule base that e/e(t)  e(t  1) NB NK S PK PB

depends on e(t) and
e(t)  e(t  1) NB NB NB NO NK S
NK NB NO NK S PK
S NO NK S PK PO
PK NK S PK PO PB
PB S PK PO PB PB

1. IF the error in altitude is negatively high and the change in the error of altitude is
negatively high THEN the required output for pitch angle is negatively high
A total of 25 rules are defined. In these rules, the value of the error itself is
considered to be important. Then, the coefficients K1 and K2 may be changed
for different operating conditions. Also, K3 can be changed for the required
output. Decreasing the input and output coefficients means widened membership
functions for the input functions and centered membership functions for the output
membership functions [2].
An area-centered structure is used as an output defuzzifier.

11.2.1 Fuzzy Logic-Based Altitude and Velocity Controllers

Firstly, longitudinal fuzzy controllers are designed. For every controller, error and
error change membership functions are defined. For these calculations, MATLAB
software is used [4]. The membership functions for the altitude controller are given
in Figs. 11.5, 11.6, 11.7, and 11.8.
The membership functions for the change in error are similar; however, the range
is between 100 and C100. The error ranges in altitude, change of error in altitude,
and the output pitch angle are 400/C400 m, 100/C100, and C0.1745/0.1745
radians (10ı ), respectively. Seven output membership functions are defined.
The same rules and functions are used for the velocity controller. The range
of error and change in error membership functions in the velocity controller are
20/C20 m/s and 20/C20, respectively. For the dt output, the range is between
1 and C1. In the dt membership functions, zero is showing the actual dt input.
When the controller can be designed using expert knowledge and considering
different conditions, no input–output relationship may be necessary. However,
optimizing the system requires a detailed study on membership functions and, in
this study, the coefficients are used to develop a better system.
The results show that a Mamdani-type PD controller with homogenously dis-
tributed membership functions gives good results when the coefficients are deter-
mined carefully. The advantages of the fuzzy logic controller are that it can
be designed without knowing the dynamics of the system and it can be easily
changed using coefficients instead of the membership functions themselves. Using
208 11 Fuzzy Logic-Based Controller Design

Fig. 11.5 Fuzzy altitude controller input functions for error

Fig. 11.6 Change in error membership functions for the altitude controller
11.2 Fuzzy Controllers 209

Fig. 11.7 h–™ref fuzzy logic controller output functions

Fig. 11.8 h–™ref fuzzy controller surface diagram

210 11 Fuzzy Logic-Based Controller Design

a more complicated optimization technique like genetic algorithms to change the

membership functions may also increase the performance of the controller.
The results when inputs to the altitude and velocity controllers are 100 m and
14 m/s, respectively, are given in Figs. 11.9, 11.10, 11.11, and 11.12.
The results of the altitude controller are found using the coefficients K1 D 1,
K2 D 1, and K3 D 46.46. The K3 value is calculated using the signal constraint
method when the other coefficients are 1. The mentioned method optimizes the
response using the gradient descent method. The results of the iterations using the
gradient method are given in Table 11.2.
K3 D 46.4599 is the optimal value found for the output coefficient. When the
inputs are low, the system gives a more vibrating response and the output coefficient
must be changed in such cases. For example, an input of 5 m for K3 can be
taken as 30 to achieve a better result, as seen in Figs. 11.13, 11.14, 11.15, 11.16,
and 11.17.

11.2.2 Lateral Fuzzy Logic Controller

In the lateral controller design (heading controller), the rules and membership
functions types are the same as for the longitudinal controller. The input and
output functions are defined in radians. The range for error and the change in error

Fig. 11.9 Results from the fuzzy logic controller using a 100-m altitude input
11.2 Fuzzy Controllers 211

Fig. 11.10 Change in velocity when the altitude input is 100 m

10

6
de(degrees)

X: 49.2
Y: 1.661
2

-2
0 20 40 60 80 100 120 140
t(s)

Fig. 11.11 Change in elevator angle when the altitude input is 100 m
212 11 Fuzzy Logic-Based Controller Design

0.3

0.25

0.2
Pitch angle (radian)

0.15

0.1

X: 50
0.05 Y: 0.02881

-0.05
0 20 40 60 80 100 120 140
t(s)

Fig. 11.12 Change in pitch angle when the altitude input is 100 m (in radians)

Table 11.2 Results of the signal constraint method

Iteration SCount f (x) Max. limit Directional step length First derivative Optimality
0 1 0 538.6
1 6 0 251.9 0.635 0 1
2 9 0 0.3245 1 0 1
3 12 0 0.1203 1 0 942
4 15 0 0.04807 1 0 724
5 18 0 0.01452 1 0 389
6 21 0 0.002461 1 0 40.1
7 24 0 0.000126 1 0 2.42

are 5/C5 radians and 1/C1, respectively. For the output functions, 20/C20ı
(0.349/C0.349 radians) is chosen as the range. Again, the output coefficient is
used to scale the membership functions.
The membership functions are given in Figs. 11.18 and 11.19, and the heading
controller surface diagram is presented in Fig. 11.20.
The rules for the lateral controller are similar to those used for the longitudinal
controller [3].
The same types of membership functions are also used for the change in error.
11.2 Fuzzy Controllers 213

Fig. 11.13 K3 output coefficient calculation using the signal constraint method

Fig. 11.14 14-m/s velocity input result of the fuzzy logic controller
214 11 Fuzzy Logic-Based Controller Design

Fig. 11.15 Change in altitude for a 14-m/s input

12

10

4
de(degrees)

2 X: 41.2
Y: -0.2067
0

-2

-4

-6

-8
0 20 40 60 80 100 120 140
t(s)

Fig. 11.16 Change in elevator angle for a 14-m/s input

11.3 Stability Analysis of the Fuzzy Controllers 215

0.25

0.2

0.15
Pitch angle(radians)

0.1

0.05
X: 58
Y: -0.0007956
0

-0.05

-0.1
0 20 40 60 80 100 120 140
t(s)

Fig. 11.17 Change in pitch angle for a 14-m/s input

The results of the lateral fuzzy logic controller are given in Figs. 11.21, 11.22,
and 11.23.
The heading controller and lateral fuzzy logic controller schemes are shown in
Figs. 11.24 and 11.25, respectively.
The results show that the lateral controller also works well if good coefficients
can be determined.

11.3 Stability Analysis of the Fuzzy Controllers

Performing a stability analysis for a nonlinear fuzzy logic-based system is a difficult

task. Methods like Lyapunov and Popov analytic methods need well-defined system
definitions and can make the stability analysis viable only for simplified models. The
controllers constructed become conservative. Consequently, Lyapunov, absolute
stability, and circle criterion (and similar) methods can only determine if a system
is stable under some conditions. For fuzzy logic controllers, in general, simulations
are used to determine if the system is stable with the designed controller or not
[1, 2].
In this study, simulations are also used to evaluate system stability.
216 11 Fuzzy Logic-Based Controller Design

Fig. 11.18 Error membership functions for the lateral controller

Fig. 11.19 Heading output membership functions

11.3 Stability Analysis of the Fuzzy Controllers 217

Fig. 11.20 Heading controller surface diagram

0.4

0.35
X: 70.6
Y: 0.3498
0.3
Heading angle(radians)

0.25

0.2

0.15

0.1

0.05

0
0 50 100 150 200 250
t(s)

Fig. 11.21 Response of the fuzzy logic controller to a 0.35 radians (20.02ı ) input
218 11 Fuzzy Logic-Based Controller Design

0.2

0.15

0.1
P(roll angle change-rad/s)

0.05
X: 36.6
Y: -0.000292
0

-0.05

-0.1

-0.15

-0.2
0 50 100 150 200 250
t(s)

Fig. 11.22 Change in 'P for a 0.35 radians heading input

0.015

0.01
Rudder input(dr-rad)

0.005
X: 46
Y: 1.234e-005
0

-0.005

-0.01

-0.015
0 50 100 150 200 250
t(s)

Fig. 11.23 Change in dr (rudder input) for a 0.35 radians heading input
11.4 A Comparison of Flight Controllers for Unmanned Aerial Vehicles (UAVs) 219

Kfl
Phi angle
+
− Kfl +
−
Heading Ref1
Saturation
Heading Controller

1
z
Scope1-Heading error Unit Delay

Heading feedback

Fig. 11.24 Heading controller scheme

−K

Kphi
+ + −K
+ − −
−
Saturation1 Kp
Heading Ref1 Saturation
Heading controller
μa

20
Aileron actuator
s+20
X>
deltaa R
1 [
z −20 Phl
Unit Delay s+20 deltar Psl
Scope2- roll error
Saturation2 all >
Rudder actuator
Lateral Dynamics
+
− Kr
Rudderref s
−K
s+5
Washout filter

Fig. 11.25 Lateral fuzzy logic controller scheme

11.4 A Comparison of Flight Controllers for Unmanned

Aerial Vehicles (UAVs)

Three types of flight control systems (classical control methods, optimal control
methods, and fuzzy logic technique) for a small UAV are compared in this section.
Classical control system design is the most common system for these kinds of
vehicles, and it has been proven to be a good way of controlling aircraft. On the
other hand, there are several advantages of other control systems. Linear quadratic
regulator (LQR) controllers can be designed quickly and fuzzy logic controllers
can be designed without having previous knowledge of the system dynamics. The
obtained results may be utilized in the design of UAV flight control systems. The
response values of the controllers are given in Tables 11.3, 11.4, and 11.5.
220 11 Fuzzy Logic-Based Controller Design

Table 11.3 Proportional- PID controller Settling time Rise time

integral-derivative (PID)
controller response data Altitude (h) 12.13 s 2.17 s
Total vel. (Vt ) 81.5 s 35.63 s
Heading ( ) 24 s 12.76 s

Table 11.4 Linear quadratic LQR controller Settling time Rise time
regulator (LQR) controller
response data Altitude (h) 6s 3.05 s
Total vel. (Vt ) 7.82 s 1.6 s
Heading ( ) 14.3 s 6.7625 s

Table 11.5 Fuzzy logic Fuzzy logic controller Settling time Rise time
controller response data
Altitude (h) K3 D 10 37 s 15 s
Total vel. (Vt ) K3 D 30 37 s 15.32 s
Heading ( ) 60 s 26 s

All the given values are read from simulations for similar inputs. The PID
controller works well as altitude and heading controllers, but the effect of altitude
and velocity on each other makes it necessary to use a slow-response controller for
the velocity. The LQR controller responses are quick; however, the control inputs
are high, even though the cost matrices are adjusted carefully. Besides, the values
that are fed to the LQR controller must be filtered, as previously described in the
Kalman filtering section. The fuzzy logic controller works well, even though no
optimization technique is used to optimize the system and it was designed without
dynamic model knowledge. In the fuzzy logic controller, a careful arrangement of
membership functions may lead to better results.
In conclusion, it is found that all three types of controllers are suitable for use
as small UAV controllers. It is observed that, although the PID controller responses
are good, an LQR controller and a fuzzy logic controller with membership function
optimization can also be used as a controller in UAV systems.

11.5 Conclusion and Discussion

In this chapter, fuzzy logic-based longitudinal and lateral controllers have been
designed and different UAV flight control methods were compared. The fuzzy logic
controller works well, even though no optimization technique is used to optimize
the system and it was designed without dynamic model knowledge. In the fuzzy
logic controller, a careful arrangement of membership functions may lead to better
results. For fuzzy logic controllers, in general, simulations are used to determine if
the system is stable with the designed controller or not. In this case, the simulations
showed that the considered system is stable.
References 221

Comparison of different UAV flight control methods shows that the PID con-
troller responses are good and, also, LQR and fuzzy logic controllers with mem-
bership function optimization are effective in controlling the UAV. Designing better
controllers without neglecting the effect of each motion on one another, simulating
the system under different inputs and conditions, and trying to design more efficient
and robust controllers may be potential research topics.

References

1. Klir GJ, Yuan B (1995) Fuzzy sets and fuzzy logic theory and applications. Prentice Hall, Upper
Saddle River
2. Passino KM, Yurkovich S (1998) Fuzzy control. Addison-Wesley, Menlo Park
3. Vural SY, Hajiyev C (2014) A comparison of longitudinal controllers for autonomous UAV. Int
J Sustain Aviat 1(1):58–71
4. Hines JW (1997) MATLAB supplement to fuzzy and neural approaches in engineering. Wiley,
New York
About the Authors

Chingiz Hajiyev was born in Kelbecer (Azerbaijan Republic). In 1981, he

graduated from Moscow Aviation Institute (Russia) with an honors diploma. He
received his Ph.D. and D.Sc. (Eng.) degrees in Process Control in 1987 and 1993,
respectively.
From 1987 to 1994, he worked as a Scientific Worker, Senior Scientific Worker,
and Chief of the Information-Measurement Systems Dept. at the Azerbaijanian
Scientific and Production Association (ASPA) “Neftgazavtomat”. Between 1995
and 1996, he was a Leading Scientific Worker at the Institute of Cybernetics
of the Academy of Sciences, Azerbaijan Republic. He was also a Professor at
the Department of Electronically-Calculated System Design, Azerbaijan Technical
University, where he had been teaching in 1995–1996.
Since 1996, he has been with the Department of Aeronautics and Astronautics,
Istanbul Technical University (Istanbul, Turkey), where he is currently a Professor.
He has authored more than 400 scientific publications, including 11 books, 10
book chapters, and about 250 international journals and international conference
papers.
He is a full member of the International Academy of Navigation and Motion
Control (Saint Petersburg, Russia). He was awarded a grant from the International
Science Foundation (USA) (1993). He is a member of the Editorial Boards of a
number of international journals.
His research interest includes system identification, fault detection and isolation,
fault-tolerant flight control, spacecraft attitude determination and control, and
integrated navigation systems.

Halil Ersin Soken was born in Istanbul, Turkey. He graduated “Summa cum Laude”
from the Astronautical Engineering Department of Istanbul Technical University in
2007. In 2008, he completed his second degree in Aeronautical Engineering. He
earned his Ph.D. in Space and Astronautical Science from The Graduate University

© Springer International Publishing Switzerland 2015 223

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5
224 About the Authors

for Advanced Studies (Japan) in 2013. During his Ph.D. studies, he was a visiting
researcher at Aalborg University (Denmark) for 3 months.
Dr. Söken worked as a research assistant at Istanbul Technical University (2010)
and Japan Aerospace Exploration Agency – JAXA (2010–2013). Currently, he is
a research associate at the Institute of Space and Astronautical Science (ISAS) of
JAXA.
His research interest is Kalman filter-based estimation techniques and satellite
attitude determination and control, specifically for small satellites.

S. Yenal Vural was born in Adana, Turkey. He is a Ph.D. student at the Aeronautical
Engineering Department at Istanbul Technical University (ITU). He received his
B.Sc. degree in Mechanical Engineering from ITU in 2002 and his M.Sc. degree in
Aerospace Engineering from the same university in 2008.
His research interests include unmanned aerial vehicles (UAVs), flight control,
Kalman filtering, and fault-tolerant flight control.
Index

A coordinate system, 9–10, 26

Abnormal measurements, 71, 73, 76–80, 90, flight control, 72, 95
91, 92 systems, 51
Absolute Air data system (ADS), vi, 26, 30–33,
altitude, 35 109–126
pressure, 31 Altimeter, v, 35–38
transducer, 31 Altitude
Acceleration, 15, 17, 26, 27, 28, 29, 30, 47, controller, 144, 147–154, 172, 176–180,
109, 110 207, 208, 210
Accelerometer, 26–30, 47 sensors, 37
Actuator, vi, 47, 72, 95–107, 139, 145, 147, Angle of
154, 160, 162, 163, 166, 177 attack (AOA), 33, 110, 118–125
fault, 72, 95–107 declination, 41
Adaptive dip, 41
estimation, 67–69, 71 Angle sensor (AOS), 118, 122, 125
factor, 72, 96–98, 105, 107 Angular
filter, 55, 58, 66 rate, 16, 28, 47, 145–147
fuzzy Kalman filter, 72 speed, 109, 110
Kalman filtering, 67, 71 velocity, 10, 11, 14, 16, 17, 26, 28, 36, 60
modified extended Kalman filtering, 110 AOA. See Angle of, attack (AOA)
multi sensor data fusion, 72 AOS. See Angle sensor (AOS)
ADS. See Air data system (ADS) Asymptotically stable, 173, 174
ADS/GPS fusion, 112 Atmospheric pressure, 37
Aerodynamic Attitude angles, 28
coefficients, 171 Azimuth, 27, 33, 34
forces, 6
Aileron actuator, 160, 162, 163, 166
Air B
speed, 30–33, 111, 112, 113, 118, 119, 131 Bank of Kalman filters, 66, 71
temperature, 30, 31, 38 Barometric altimeter, 37–38
Aircraft Bearing, 33, 34
body axis system, 9 Biased measurements, 80
body frame, 11, 15, 16 Broadcast signals, 42

© Springer International Publishing Switzerland 2015 225

C. Hajiyev et al., State Estimation and Control for Low-cost Unmanned Aerial
Vehicles, DOI 10.1007/978-3-319-16417-5
226 Index

C of measurement noise, 54, 61

Calibrated airspeed, 30–32 prediction, 55
Characteristic of process noise, 53
equation, 135, 137, 138, 139, 141, 161,
164
polynomial, 59, 87, 88 D
Chi-square distribution, 76, 99 Damping factor, 137–139, 145, 161, 164, 166
Circle criterion, 215 Data fusion, 72, 109–126
Clock bias, 43, 44 Dead reckoning (DR), 25, 44, 72
Closed loop response, 146 navigation, 25
Complex multivariable systems, 96 positioning error, 25
Computational burden, 66, 73, 92 Decision
Computer subsystems, 95 approach, 95
Consequent fuzzy set, 202 -making system, 47
Consequent proposition, 201, 203 statistics, 99, 107
Continuous Defuzzification, 201
bias, 79–84 Desired
process noise covariance, 53 control level, 200
time, 99 transfer function, 135
Control Digital filter, 59
derivatives, 105 ¦2 Distribution, 75, 98, 117
distribution matrix, 22, 52, 53, 60, 61, 99, Doppler
101, 162, 185, 189 attitude reference system, 25
inputs, 52, 60, 129, 131, 135, 160, 163, method, 38–40
172, 175, 185, 220 radar, 38
input vector, 60 Doyle–Stein condition, 99, 102, 103, 107
surface failures, 99 Drift angle, 38–40
surfaces, 95, 99, 107 DR system. See Dead reckoning (DR) systems
system, vi, 26, 47, 69, 71, 72, 92, 95, 107, Dutch roll mode, 138, 139, 161, 164, 166
141, 150, 151, 158, 160, 170, 171, Dynamic(s)
201, 219 acceleration, 29, 30
system design, 141, 158, 171, 219 of aircraft, 60
vector, 22, 53, 61 model, vii, 109, 110, 171, 220
Controller of the system, vi, 52, 115, 126, 207
design, 141–221
input, 142, 208
output, 209 E
response, vi, 156, 172, 179, 180, 187, 196, Earth axis system, 9–10, 15
200, 220 Earth’s geomagnetic field, 40
scheme, 141, 206, 215, 219 Eigenvalues, 60, 78, 88
structure, 147 EKF. See Extended Kalman filter (EKF)
types, 142 Elevation, 33, 34, 43
Convergence of estimated values, 59 Elevator
Correlation matrix of actuator, 145, 147, 153
the filtering error, 58 actuator function, 145, 147
matrix of the system noise, 57 angle, 154, 205, 211, 214
the measurement noise, 57 deflection, 18
Covariance Error covariance, 101
estimation, 56 Estimation
of estimation, 55 algorithm, 51, 63–64, 68
matching technique, 72 characteristic, 73, 77, 97, 107
matrix, 53, 54, 61, 68, 69, 72, 73, 75, error, 56, 68, 77, 78, 79, 80, 82, 83, 86, 89,
77, 96, 99, 101, 109, 110, 114, 189, 98, 100, 101, 113, 114, 120
195, 199 system, vi, 47, 62, 71, 74, 75, 98
Index 227

Estimator, 51, 56, 69, 72, 97, 107, 172, G

188–200 Galileo, 42
Euler angles, 15, 16, 26, 109 Gaussian
Extended Kalman filter (EKF), 54, 72, 109, distribution, 53, 61, 190
110, 111 noise, 52, 58, 189
Extrapolation, 57, 58, 68, 113, 189 white noise, 54, 67, 190
Global
optimal estimation, 114
F Global positioning system (GPS), vi, 25, 37,
Failure detection, 74, 95, 102, 116 41–45, 68, 72, 111–116, 118–126
Fault receiver, 42–44
classification, 45–47 structure, 42
detection, v, vi, 47, 51, 97, 98, 99, 101, 102, GLONASS, 25, 42
103, 107, 109–126 GPS. See Global positioning system (GPS)
diagnosis, 47, 111 Gravitational acceleration, 30
identification, 47 Gyroscope, 26, 27, 28, 47
isolation, 47, 95, 96, 99, 102, 105, 117–126 drift, 47
modeling, 45–47
Fault-tolerant
flight control system, 47 H
state estimation, 47 HALE. See High altitude long endurance
Federated Kalman filter, vi, 114–116, 120, 122, (HALE)
126 Heading
Feedback control system, 95 angle, 134, 159, 195, 198
Fiber optic gyro (FOG), 28, 29 controller, 166–169, 185–188, 196, 206,
Flight 210, 215, 217, 219, 220
controller, 219–220 High altitude long endurance (HALE), 5, 44
control system, vi, 47, 71, 72, 92, 95, 170, Hypersonic, 5
201, 219
dynamics, vi, vii, 200
vehicle model (FVM), 109, 110 I
FOG. See Fiber optic gyro (FOG) IAE. See Innovation-based adaptive estimation
Frequency domain, 59 (IAE)
Frozen sensor, 46 IF-THEN rules, 201, 204, 205
Fuzzification, 201 IMM. See Interacting multiple model (IMM)
Fuzzy Indirect Kalman filter, 118–126
controller, v, vii, 206–219 Inertial
inference mechanism, 204–206 dynamic acceleration, 30
Kalman filter, 72 frame, 10, 15, 16, 26
logic adaptive system, 72 measurement unit (IMU), 27–30, 47, 109,
logic based controller, 201–221 110, 111
logic based system, 201–206, 215 navigation, 26–27
logic controller, 206, 207, 209–215, 217, navigation system (INS), vi, 25–29, 40, 44,
219, 220, 221 47, 68, 72, 109, 110, 111, 113–116,
logic system, 201 118–126
model, 202, 203, 204 reactions, 26
relation, 204 sensors, 27, 28, 29, 47
relation intersection, 204 space, 26, 28
rules, 201–205 Innovation
sets, 203 approach, 116
space, 203 based adaptive estimation, 67, 71
system, 201, 203 based estimation, 69
FVM. See Flight, vehicle model (FVM) based estimator, 69
228 Index

Innovation (cont.) L
based KF adaptation, 67 Lapse rate, 37, 38
channel, 90 Lateral
covariance, 54, 55, 75, 97 characteristic, 137, 138
matrix, 75 classical controller, 158–169
process, 113, 126 controller, vii, 141, 159, 162, 185, 195,
sequence, 58, 67, 73, 74, 75, 78, 116, 210, 212, 215, 216, 220
117, 189 dynamics, 60, 79, 219
Innovation-based adaptive estimation (IAE), equations, 18, 19, 61, 132, 134, 136, 141,
67, 71, 72 160
Input flight dynamics, 200
axes, 47 fuzzy logic controller, 210–215, 219
error, 110 LQR controller, 185, 186
function, 207, 208 LQR heading controller, 185
Kalman observers, 110 matrices, 176, 185
–output gradient, 47 motion, 130, 136, 138, 139, 158–169
–output relationship, 110, 206, 207 stability, vi, 132, 138–139
–output test, 111 state matrix, 162, 163
values, 111, 177 states, 105, 132, 195
vector, 60, 189 variables, 131
Integrated Latitude, 27, 42, 43
ADS and INS/GPS, 113 Lift, 6
ADS/GPS navigation system, 111–114 Likelihood of system failures, 95
INS/GPS system, 68, 72, 111 Linear
R and Q-adaptation, 99 acceleration, 27
system, 111 controller, 171
Interacting multiple model (IMM), 110, discrete dynamic system, 52
111 discrete Kalman filter, 52–59
filters, 52, 58
filter theory, 52
K Kalman filter, v, 85, 189
Kalman quadratic
estimation, 62, 197, 198 estimation (LQE), 51
estimator, 172, 188–200 optimal control, 172–176
filter, v, vi, vii, 25, 44, 51–70, 71–78, 80, regulator (LQR), vi, vii, 171–200, 219,
85, 87, 88, 95, 96, 97, 99, 100, 101, 220, 221
102, 104, 107, 109–126, 171, 172, system, 51, 92, 110
188, 189, 190, 193, 195, 197, 198, Linearization, 52, 110, 129, 130, 154
200 Linearized
filter application, 76 equations, v, 18–23, 130, 134, 139
filter gain, 73, 88 longitudinal equations, 18, 145
filtering algorithm, 110 Local filter, 72, 114, 115
filtering equations, 113 Longitude, 42, 43
filtering structure, 109 Longitudinal
filtering technique, vi, 71, 111, 115, 126, channel, 105
176, 190, 220 characteristic equation, 137
filter innovation sequence, 75 control, 101, 105
filter normalized innovation, 117 control derivatives, 105
gain, 54–57, 74, 75, 76, 78, 91 control distribution matrix, 101
gain matrix, 76, 91 controller, 144, 176, 210
observers, 110 dynamics, 144, 159, 190
Kronecker equations, 18, 20, 110, 131, 134, 144, 183
delta function, 61 flight dynamics, 200
symbol, 52, 189 fuzzy controllers, 207
Index 229

LQR controller, 190 noise covariance matrix, 61, 72, 73, 75, 96
LQR with Kalman estimator, 190–195 noise scale factor (MNSF), vi, 73–76,
matrices, 176 78–80, 96
motion, 137, 138, 139, 144–158, 170, 171, noise vector, 52, 189
190 residual, 54
stability, 131, 137–139 Medium altitude long endurance (MALE), 5,
states, 103, 131, 135, 177, 190 44
Loop transfer function, 154 Membership function, vii, 202–208, 212, 216,
Loss of accuracy, 45, 46 220
LQR Microelectromechanical systems (MEMS)
altitude controller, 176–180 inertial sensors, 27–29
control design, 171 technology, 27
gain, 185, 195 Minimum
heading controller, 185 error variance, 51
Lyapunov mean square error estimator, 56
asymptotically stable system, 173, operator, 204, 205
174 MMAE. See Multiple model-based adaptive
criteria, 173–174 estimation (MMAE)
function, 173–175 MMNSF. See Multiple, measurement noise
method, 173 scale facto (MMNSF)
stability criteria, 173–174 MNSF. See Measurement, noise scale factor
stability function, 174 (MNSF)
Model-based adaptive estimation, 66, 71, 72
Moment of inertia, 13, 14
M Monitoring, 5, 6, 47, 117
Mach number, 30–32 Moving
Magnetic measurements, 40–41 frame, 15
MALE. See Medium altitude long endurance window, 72, 75
(MALE) Multi-input/output feedback control, 95
Malfunction, vi, 45, 69, 71, 73–76, 81, 82, 85, Multiple
90–92, 96, 98 adaptive factor, 97–98, 105, 107
Mamdani fading factor, 96
method, 204 measurement noise scale facto (MMNSF),
rule base, 205 73, 75–92
rules, 204, 205 model, 66, 71, 72, 110
type controller, 206 sensor fault detection, 111
type fuzzy rule, 202–203 Multiple model-based adaptive estimation
type IF-THEN rule, 204 (MMAE), 66, 71, 72
type PD controller, 207
Mathematical model, 45, 52, 58, 79
Maximum eigenvalue, 78 N
Mean Navigation
absolute values of error, 103–106 sensor, 114, 115
chord length, 20 systems, v, 25–47, 111
of the innovation, 95, 116, 117 NLGA. See Nonlinear equation, geometric
Measurement approach (NLGA)
channel, 62, 71, 73, 78, 79, 80, 82 Noise increment, 45, 46, 76, 81–82, 84–87, 91,
data, vi, 45, 64, 109, 115, 126 105, 106
equation, 37, 46, 52, 188, 189 Nonlinear equation
faults, v, vi, 45–47, 62, 71, 76, 81, 82, 92 estimation, 96
matrix, 54, 60, 68, 99, 113, 189 fuzzy logic based system, 215
noise, 46, 52, 54, 56, 57, 60, 61, 66, 67, 69, geometric approach (NLGA), 110
72, 73, 76, 81, 84–87, 91, 95, 96, motion dynamics, 171
189 system, 92, 109–111, 129
230 Index

Normalized innovation, 117, 126 Pitch

Normal operation angle, 15, 77–79, 81–83, 101, 110, 137,
operating conditions, 188 143–145, 147–149, 154, 180, 181,
184, 185, 207, 212, 215
angular rate, 145–147
O Pitot probes (tube), 32
Observation Plant, 99, 178, 186, 206
model, 60 Pole-zero map, 90–92
Observer, 110, 111, 190 Position
OKF. See Optimal, Kalman filter (OKF) constant, 147
On-board computer, 4, 29 Positioning system, 25
Open loop, 154, 166 Posterior state estimation, 55
Optimal Precise positioning service (PPS), 42
control, 171–176, 219 Prediction, 51, 53–55, 57
controller, vi, 172–176, 200 Pressure, 30–32, 37, 38
criterion, 64 Probability
gain, 55, 175, 188 characteristics, 64
Kalman filter (OKF), 53–55, 60–67, 69, 73, distribution rule, 63, 64
74, 76, 77, 79–84, 86, 87, 89–92, methods, 66
97–105 Process
Kalman gain, 54–57 noise, 53, 66
linear discrete Kalman filter, 52–59 noise covariance, vi, 53, 54, 61, 72, 96, 99
LQR control method, 172 Proportional gain, 141–143, 149, 155, 166
observer, 190 Proportional-integral-derivative (PID)
Optimality, 66, 212 controller, vi, 141–143, 149–151,
Optimum discrete Kalman filter 156, 220
Kalman filter, 59 Pure rolling motion, 138
linear Kalman filter, 189
Orbit determination, 51
Q
Orthogonal, 9, 47
Q-adaptation, vi, 96–99, 107
Output
Quadratic Lyapunov function, 174
fuzzy set, 203
linguistic term, 202, 203
matrix, 135, 162 R
variable, 135 R-adaptation, vi, 76–85, 96–99
Radar altimeter, v
Radio
P altimeter, 35–37
Parallel Kalman filters, 72 navigation system, v, 42
Parameter estimation, vi, 69, 92 RAE. See Residual, based adaptive estimation
Parametric adaptive estimation, 64 (RAE)
P controller, 206 RAKF. See Robust, adaptive Kalman filter
PD controller, 143, 148, 149, 151, 152, 206, (RAKF)
207 Random
Perturbation Gaussian noise, 52, 189
method, 16 noise, 47, 51, 113
term, 17 process, 63, 64
theory, 16–18 Rate gyros, 28–29
variable, 18 Real altitude, 35, 36
Phugoid mode, 137, 139, 145 Real time measurement, 67
PI controller, 142 Recursive equations, 189
PID controller. See Proportional-integral- Redundancy
derivative (PID) controller of sensors, 111
Index 231

Regulator Simultaneous
design, 171–200 adaptation procedure, 96
Relative altitude, 35 localization and mapping (SLAM), 44
Residual Single
based adaptive estimation (RAE), 68–69, adaptive factor, 72, 96
71, 72 fading factor, 96
sequence, 72 measurement noise scale factor (SMNSF),
series, 66–68 73–80, 82–90, 92
vector, 69, 72 Singleton type fuzzy rules, 202, 204
Riccati equation, 175, 176, 185 SLAM. See Simultaneous, localization and
Rigid body, v, 9–15, 29–30 mapping (SLAM)
Ring laser gyro (RLG), 28, 29 Sliding window, 67–69, 117
Robust Small perturbation theory, 16–18
adaptive Kalman filter (RAKF), v, vi, SMNSF. See Single, measurement noise scale
96–107 factor (SMNSF)
algorithm, 74, 76 Smoothing, 51, 53
controller, 221 Speed
Robust Kalman filter (RKF), vi, 73–76, 78–92, controller, 144, 149, 154–159, 176–185
99, 102, 103, 107 -over-ground, 38–40
algorithm, vi, 73, 76–78, 81, 91, 92 Spiral mode, 138, 139, 161, 165, 166
Robustness, 78, 97, 107 SPS. See Standard positioning system (SPS)
Roll Stability
angle, 15, 38, 159–161, 165–166, 188 analysis of robust the Kalman filter, 87
mode, 138–139, 161, 164, 166 analysis of the fuzzy controllers, 215–219
motion, 138 augmentation, 95
rate, 160–161 characteristics, 90, 92
Root locus, 141, 143, 145–148, 154, 155, 160, coefficient, 18, 20, 131, 132
161, 164, 167 condition, 59, 137–139, 174
Rudder actuator condition of filter, 59
deflection, 19, 60 of conventional digital filters, 59
criteria, 173
derivatives, 18–20
S of the filter, v, 59, 90
Satellite radio navigation, v, 41–44 function, 174
Scalar measure, 78, 82, 85 limit, 90
Scale factor, 28, 47, 73, 74, 78, 80, 81, 82, 85, of the optimal discrete Kalman filter, 59–60
87, 91 Standard positioning system (SPS), 42
Self-tuning circuit, 64 State
Sensor correction series, 68
bias, 46 equation, 52, 130–135, 160, 188, 189
data fusion, 72 estimates, 47, 54, 96
drift, 46 estimation, v, vi, 47, 51, 55, 56, 60–61, 69,
failures, 102 71, 73, 74, 77, 78, 91, 97, 99, 100,
faults, 45, 46, 47, 71–92, 96, 99, 101–106, 107, 111
110, 111, 114, 116 feedback, 177
fusion, 51, 114, 115 matrices, 162, 163, 176
Sensor/actuator model, 177
failures, vi, 102, 107 prediction, 55
faults, 72, 95–107 space, v, 18, 22, 60, 133, 135, 144, 188
Sequential state estimation, 51 space form, 18, 22, 60, 133, 144
Signal vector, 51, 52, 59–61, 117, 172, 189
constraint method, 212, 213 Static
input, 145 acceleration, 30
processing techniques, 95 pressure, 31, 32
232 Index

Steady state error, 142, 143, 147, 166 True airspeed, 30, 31, 32, 33, 111, 112, 113,
Strapdown system, 26 118, 131
Suboptimality of the filter, 66 True airspeed sensor (TAS), 118–123, 125
Suborbital, 5 TSK fuzzy logic system. See Takagi–Sugeno
Subway control systems, 201 (TSK) fuzzy logic system
Supersonic, 5 Tsukamoto type fuzzy rules, 202
Surface
failure, 99
radar, 33–34 U
System UAV. See Unmanned aerial vehicle (UAV)
dynamics and control, 53 UIKF. See Unknown input Kalman observers
dynamics matrix, 60 (UIKF)
failures, 95 Unit
identification, 51, 223 circle, 60, 88
measurement matrix, 99 Unknown input Kalman observers (UIKF),
model, 64, 177, 188, 206 110
noise, 52, 57, 60, 67, 68, 72, 110, 189 Unmanned aerial vehicle (UAV)
process and measurement noises, 60 dynamics, 47, 51–69, 71–92, 115, 126
states, 67, 111, 177 flight control, vi, vii, 47, 92, 219–221, 224
theory, 59
transfer function, 143, 147
transformation matrix, 68 V
uncertainties, 71 Variance
of the estimation, 77, 101, 104
of Kalman estimation, 62
T of the noise, 81, 105
TACAN, 25 Velocity
Takagi–Sugeno (TSK) fuzzy logic system, 201 controller, 182, 207–210
TAS. See True airspeed sensor (TAS) Vertical velocity, 27, 205
Taylor series, 18, 130, 171 Vision based system, 44
Temperature gradient, 38 VOR/DME, 25
Theoretical error, 74, 75
Three-axis magnetometer, 40
Threshold, 75, 98, 101, 117, 118 W
Thrust actuator, 154 Wash-out filter, 144, 145, 147, 153, 160, 162,
Time 164, 166, 219
constant, 138, 145, 154, 162, 166 Weight matrices, 172, 177, 185
delay, 103 White noise, 54, 67, 190
domain, 59 Wiener filter, 58
to double amplitude, 139
to half amplitude, 137
Total Y
pressure, 31, 32 Yaw
velocity, 27, 38 angle, 15, 17, 161, 164, 171, 187, 195
Trace of damper, 158, 162–165
the covariance matrix, 56
scale matrix, 78, 80, 84, 87, 90
Transfer Z
function, vi, 130–137, 139, 141–143, 145, Zagi UAV, 19–23, 61, 76, 99, 134
147–148, 154, 155, 160, 164, 166 z-domain, 59
matrix, 52, 57, 113 Zero-mean, 52–54, 61, 113, 189
matrix of the system, 52, 57 Ziegler–Nichols method, 149, 156
Transformation matrix, 68 z-transform, 59, 87