(EE341L) Linear Control Systems Lab
LAB REPORT
Automatic Steering Control System
Faculty of Electrical Engineering
Ghulam Ishaq Khan Institute of Engineering Sciences and Technology
Instructions:
First and foremost, this is individual lab project, and carry 15 absolute marks.
The project covers 10% of your remaining weightage of lab performance. Like lab activity,
you may concern books, lectures, and seek help from your fellows, but ultimate approach
and idea to tackle the problem provided shall be solely yours. If anyone’s report was
found matching other student’s report, be in terms of ideas, calculations, plots, Simulink
model, or m files, lab reports of student(s) who copy and from whom it was copied shall
be straightaway cancelled and whole 15 marks deducted consequently.
Use MS word Insert Equation option for writing equation(s) in your report.
Unlike the course, since this is lab report, you are expected to extensively use the Apps
that we have used in post-mid labs. The Apps will help you save your energies and time.
Final deliverables are: MS word file, Simulink file, and MATLAB m file. Zip the three files
together and name the zip file by your Reg. no only (no names please!). For example,
2017XXX.rar.
Submission is due August 25, 12:00 midnight. Late submission is subject to deduction of
half absolute marks.
The Autopilot mode in Tesla Model-3 cars stems from a suite of advanced driver-assistance control systems
that promises to offer perfect lane centering, automatic lane changes, self-parking, semi-autonomous
navigation on limited access freeways, and the ability to summon the vehicle from a garage or parking spot.
Tesla claim that when the vehicle is on Autopilot, the respective mode will ensure perfect execution of the
maneuvers in strict compliance with the features promised. In consequence, as per the company claim the
Autopilot mode will help reduce accidents caused by driver negligence and fatigue from long-term driving,
besides offering utmost luxury in the form of effortless drive.
The lane changing and centering feature with absolute precision is intriguing and we want to explore it. We
aim to explore the underlying concept of automated car-steering control in the lane changing and centering
feature by considering a scenario shown below:
Figure 1. Lane changing and centering maneuver
Figure 2. Vehicle whose lateral motion to be controlled
The steering input controls the lateral motion of the vehicle (see Figure 1). The automated
vehicle steering control system uses information about the vehicle position relative to the center
of the current lane to determine the steering wheel angle. A lateral force on the vehicle (and,
hence, a lateral acceleration) is created as the wheels turn. The automated steering controller is
designed to steer the vehicle from the center of the current lane to the center of an adjacent lane.
Measurements of the vehicle’s lateral position during the maneuver will be computed from the
lateral acceleration measured by the accelerometer (see Figure 2).
The closed loop steering control system with accelerometer feedback is shown in Figure 3.
Figure 3. Automatic vehicle steering control system
0.1 10
The vehicle has a transfer function 𝐺𝑝 = 𝑠(𝑠+1), the steering actuator has transfer function 𝐺𝑎 = 𝑠+5, and
100
the accelerometer has transfer function 𝐻 = 𝑠2 +20𝑠+100.
The signals in the Figure. 2 are:
𝑥(𝑡): 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑝𝑜𝑠𝑡𝑖𝑜𝑛 (𝑢𝑛𝑖𝑡𝑠: 𝑙𝑎𝑛𝑒𝑠)
𝑒(𝑡): 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑝𝑜𝑠𝑡𝑖𝑜𝑛 𝑒𝑟𝑟𝑜𝑟 (𝑢𝑛𝑖𝑡𝑠: 𝑙𝑎𝑛𝑒𝑠)
𝑟(𝑡): 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑝𝑜𝑠𝑡𝑖𝑜𝑛 (𝑢𝑛𝑖𝑡𝑠: 𝑙𝑎𝑛𝑒𝑠)
𝑢(𝑡): 𝑠𝑡𝑒𝑒𝑟𝑖𝑛𝑔 𝑎𝑛𝑔𝑙𝑒 (𝑢𝑛𝑖𝑡𝑠: 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
𝑑(𝑡): 𝑤𝑖𝑛𝑑 𝑔𝑢𝑠𝑡 𝑑𝑖𝑠𝑡𝑢𝑟𝑏𝑎𝑛𝑐𝑒 (𝑢𝑛𝑖𝑡𝑠: 𝑑𝑒𝑔𝑟𝑒𝑒𝑠)
The objective of the design project is to design an automated steering control system; that is, to
design a suitable closed-loop controller. The selection of controller is based on the vehicle’s motion
during a lane change maneuver and on the effect of a lateral wind gust disturbance, d(t). Note
that there is a minus sign in the summing junction where d(t) enters because it is assumed to be
acting against the motion of the car.
The specifications for the steering control design are that the vehicle completes the lane change
maneuver quickly and safely without causing the passengers discomfort. From a systems
engineering point of view, these specifications require that the step response of the vehicle’s
lateral position has a small rise and/or settling time and minimal overshoot. Furthermore, the
comfort of the passengers is closely related to the lateral acceleration during the lane change
maneuver. Specifically, passenger comfort requires that the lateral acceleration is small.
Equivalently, it can be shown that the lateral acceleration is proportional to the steering input
and, therefore, passenger comfort requires that the steering input is small.
The wind gust disturbance introduces a steady-state error which must be considered in the
control system design. To understand the impact of the wind gust, recall that the steering input
causes a lateral force on the vehicle. The wind gust disturbance creates a lateral acceleration
acting against the motion of the vehicle and reduces the effect of the steering input.
The specifications on the control system design can be divided into three categories:
1. Safety: The closed loop system must have less than 10% overshoot in the unit
step response.
2. Passenger comfort: The maximum steering input must be less than 4 degrees.
3. Disturbance rejection: The steady-state error for a unit disturbance must be
minimized.
For better understanding to help us in efficient design process, we divide the complete automated vehicle
steering control system into three phases with increased complexity.
In Phase 1, we focus on the analysis of uncontrolled lateral position x(t) in response to unitary step input
r(t) by considering the steering actuator 𝐺𝑎 (𝑠) as static system (that is, s=0). We then try to explore the
factors that govern the system response, and aim to establish relationship among the controlling factor,
controlled factor (x(t)), and the input r(t). Once the controlling and controlling factors are investigated and
relationship between the two is established, we proceed to Phase 2.
In Phase 2, we include an accelerometer in the feedback path as well as consider the dynamic nature of the
actuator 𝐺𝑎 (𝑠), giving realistic view of application under consideration. Since the inclusion of additional
systems (that is accelerometer and dynamic 𝐺𝑎 (𝑠)) undoubtedly alters the system response, we revert to
analysis stage and re-study the system in detail. At the end of Phase 2, we’ll have an ultimate closed-loop
transfer function of the system, but that transfer function too would be uncontrolled/uncompensated at that
stage. So from Phase 1 and Phase 2, we will have complete understanding of the application, the effect of
disturbance and varying input on the system response, and above all, we’ll be able to answer: why to control
the system output? We’ll be able to have only one question at the end of Phase 1 and Phase 2: how to control
the system output? Answer to this question will be addressed in final phase of the design process, Phase 3.
In Phase 3, based on design requirements, we’ll design a controller/compensator for the steering automation
system in order achieve the lane change maneuver quickly and with absolute precision. This will mark the
end of the design process.
Phase 1:
In this phase, the disturbance is not included, ideal feedback of the lateral position is assumed, and the
actuator dynamics are neglected (i.e. 𝐺𝑎 (𝑠) = 2).
Figure 4. Closed loop block diagram for Phase 1
We aim to analyze firstly the uncontrolled closed loop response of the system by equalizing 𝐺𝑐 (𝑠) to 1. To
begin with the analysis of the uncontrolled closed loop system,
1.1 Obtain its unit step response and note down settling time and steady sate error (through calculations or
using MATLAB tools). Compare those parameters with the specifications provided above and comment
briefly on whether or not Safety and Disturbance rejection have been achieved.
After analyzing the uncontrolled response, now include the proportional controller 𝐺𝑐 (𝑠) having gain K.
1.2 Express the closed loop poles in terms of the proportional controller, K.
1.2 Similarly, express the damping and natural frequency as a function of K.
1.3 Thereafter, establish relationship between the K and %overshoot, and based on the relation explain
how the variations in K will affect the Safety and Disturbance rejection.
1.4 Moreover, from the relation between K and %overshoot, determine the value of K that will lead to the
worst-case Safety parameter in the form of having maximum overshoot of 10%? Note down this value of
K.
1.5 Since you have already expressed damping ratio and natural frequency in terms of K, now use these
equations to find the settling time (through calculations or using MATLAB tool). Does the settling time
depend on K value in the application under consideration?
1.6 What is the roots sensitivity of the proportional controlled system? Predict the shift in the pole location
as K is increased 9 times from that 1.4. Comment on the unit step transient and steady-state response of the
closed-loop system as K is increased (through calculations or using MATLAB tool).
Phase 2:
In Phase 1, we analyzed the system behavior when the actuator dynamics were neglected (i.e. 𝐺𝑎 (𝑠) = 2),
and the feedback path was ideal (i.e. no sensor included), and consequently we concluded that how the
proportional controller, K, value affects the output response. We also analyzed that how the vehicle Safety,
and Disturbance Rejection vary as we change K.
10
In Phase 2, we include dynamic model of the actuator (i.e. 𝐺𝑎 (𝑠) = ) and re-analyze the system behavior
𝑠+5
as we did in Phase 1.
2.1 After including the dynamic actuator, determine the unity-feedback closed-loop transfer function of the
system (through calculations or using MATLAB tool).
2.2 Express settling time in terms of K. Did the settling time change after including the dynamic actuator?
2.3 Determine the upper-limit on K value.
2.4 Let K=10, determine dominant closed loop poles, obtain damping ratio, and natural frequency.
Comment on %overshoot and settling time as K is increased from 10 to 20 (through calculations or using
MATLAB tool). How does it affect the Safety factor?
2.5 By introducing the dynamics of the actuator, we get a limitation on the value of K. Determine maximum
permissible value of K to have stable output? Will the system be stable if we take K=40?
2.6 Now let’s introduce the accelerometer, 𝐻(𝑠), in the feedback path (Figure 5) with 𝐻(𝑠) given on Page
4. What is the system order after including 𝐻(𝑠)? Determine the value of K to have stable output? Will the
system be stable if we take K=20?
As you might have noticed that after including the accelerometer, the vehicle’s Safety factor is no longer
optimum, even if we set the proportional gain K at its maximum possible value.
2.7 Comment on how the settling time and %overshoot change after including the accelerometer? And can
we reduce the settling time and %overshoot to 1s and 5% respectively if we were to change the proportional
controller value, K, only?
Figure 5. Ultimate closed loop diagram of Automatic Steering Control System
Phase 3:
Till this point, we have deduced that by using only the proportional controller, the settling time and
%overshoot cannot be further reduced.
3.1 Suggest a suitable controller and provide its transfer function that could help us decrease the settling
time and %overshoot to less than 1s and 2%, respectively in order to ensure best-possible Safety and
Disturbance Rejection.
3.2 Also, verify your designed closed-loop controller for unitary step input by implementing the complete
closed-loop controlled automatic steering control system in Simulink.
That will be an ultimate closed-loop controller that will ensure lane changing and centering maneuver with
absolute precision while keeping Safety as well as Passenger Comfort factor at top priority.
Optional: This part is optional and ignoring it won’t degrade your final grad. However, one will get 3
absolute credit marks if one successfully accomplishes this part as well besides providing the required stuff.
Tune and then optimize your designed controller if we incorporate disturbance 𝑑(𝑡) into the system as
unitary step-input (as shown in Figure 3.). Comment on how the optimized controller will help to reduce
the implementation cost of the designed controller.
