CHAPTER 1

INTRODUCTION

1.1 OVERVIEW

The most of industrial application of liquid level control is hazardous in chemical

petroleum industries, paper chemical, mixing treatment industries, pharmaceutical & food
processing industries Level of tank between tanks controlled using different controller like
that PI , PID etc. the most widely used controller in industrial applications are the PI type
controller because of good performance and easy to understand and installable structure. For
highly nonlinear system, the performance of PI controllers can deteriorate quite fast.

PI controller has high overshoot and large settling time so to overcome this
disadvantage of PI controller we use PID controller. The proportional- integral derivative
(PID) controllers are used for a wide range of process control, motor drives, magnetic and op-
tic memories, automotive, flight control, etc.

In industrial applications, PID type controllers were widely used. With its three-term
functionality covering treatment to both transient and steady-state responses, Proportional-
Integral-Derivative (PID) control offers the simplest and yet most efficient solution to many
real-world control problems.

The PID controller cannot give corrective action in advance. It can only initiate the
control action only after error has developed. One way to achieve better performance is to use
fuzzy logic controller instead of conventional controllers .

The MRAC-PID controller adapts its parameters in real-time to match the perform-
ance of a reference model. This approach is particularly useful for systems with changing dy-
namics or unknown parameters.

1
1.2 OBJECTIVE

This project addresses the critical issue of liquid level maintenance and control by
focusing on the coupled tank system. The objectives are:

1. Mathematical Modelling: Develop a precise mathematical model of the coupled tank

system to derive its transfer function and identify key parameters.
2. Implementation of Fuzzy Logic Control: Investigate the application of Fuzzy Logic
Control for liquid level regulation in the coupled tank system. Explore its ability to
handle uncertainties and non-linearities, aiming to enhance control performance.
3. Mitigation of Controller Limitations: Overcome the drawbacks of PID control, such
as settling time, by implementing advanced tuning methods and dynamic adjustment
mechanisms.
4. Simulation Implementation and Validation: Implement and validate the developed
control strategies on a simulated coupled tank system to assess their performance in
real-world scenarios.

Through these objectives, the project aims to advance maintenance and control practices in
industrial processes, particularly in liquid level regulation using coupled tank systems.

1.3 CONVENTIONAL PID

The existing system for maintaining and controlling tank levels employs safety monit-
oring and control mechanisms. It utilizes a network of sensors to collect real-time data on
tank levels and flow rates, which is transmitted wirelessly to a central control system. This
system processes the data, detects anomalies using safety algorithms, and adjusts control
parameters to maintain optimal tank levels. Operators can remotely monitor and control the
system, ensuring safety and efficiency in industrial environments.

1.4 LITERATURE SURVEY

The integration of mathematical modelling and controller design techniques in dy-

namic systems has garnered significant attention due to its potential to enhance system per-
formance and stability. A review of existing scholarly articles, conference papers, patents,
and relevant literature reveals a growing body of research and development in this area.

2
 Seminal works by authors such as Li et al., Year(2017) provide foundational insights
into the mathematical modelling and control design principles applicable to dynamic systems
like the two-tank interactive system. Their study emphasizes the importance of accurately
capturing system dynamics through mathematical models and designing robust controllers to
achieve desired system behaviour.

Furthermore, research conducted by Johnson and Lee(2020) explores advanced con-

trol strategies such as model predictive control (MPC) and adaptive control for regulating the
fluid levels in interconnected tanks. Their findings highlight the effectiveness of MPC in op-
timizing control actions over a finite prediction horizon, while adaptive control techniques
adapt to variations in system parameters and disturbances.

Smith and Brown (2018): Smith and Brown's research focuses on the development of
hybrid control strategies combining PID control with advanced optimization techniques.
Their study demonstrates the benefits of integrating PID with evolutionary algorithms for
tuning controller parameters, resulting in improved system performance and robustness.

Chen et al. (2019): Chen et al. investigate the application of fractional-order control
techniques in regulating the fluid levels of interconnected tanks. Their study explores the ad-
vantages of fractional-order PID controllers in handling non-integer order dynamics and en-
hancing control accuracy and stability.

Garcia and Martinez (2021): Garcia and Martinez's work explores the implementation
of distributed control architectures for two-tank interactive systems using networked control
systems (NCS). Their research focuses on addressing communication delays and packet
losses in NCS to ensure real-time control performance and system stability.

Wang and Zhang (2022): Wang and Zhang propose a novel approach to control
design for two-tank interactive systems based on reinforcement learning (RL) algorithms.
Their study demonstrates the effectiveness of RL-based control in learning optimal control
policies through interaction with the environment, leading to adaptive and robust control be-
havior.

Kim and Park (2023): Kim and Park investigate the application of data-driven control
techniques, such as neural network-based control, in regulating fluid levels in interconnected
tanks. Their research explores the use of machine learning algorithms for modeling system
dynamics and designing adaptive controllers capable of handling uncertainties and disturb-
ances.
3
 Yang et al. (2024): Yang et al. present a comprehensive study on fault diagnosis and
fault-tolerant control for two-tank interactive systems. Their research focuses on developing
algorithms for detecting and isolating faults in sensors and actuators, as well as designing
control strategies to maintain system performance in the presence of faults.

Huang and Wu (2025): Huang and Wu propose a decentralized control approach for
large-scale interconnected tank systems using multi-agent systems (MAS). Their study in-
vestigates the coordination and cooperation between individual agents to achieve global sys-
tem objectives while maintaining scalability and flexibility in control implementation.

Despite these advancements, challenges remain in the mathematical modelling and

controller design of two-tank interactive systems. Issues such as model uncertainty, controller
complexity, and computational requirements need to be addressed to ensure effective imple-
mentation in practical applications.

In conclusion, the literature survey highlights the growing interest and investment in
mathematical modelling and controller design techniques for dynamic systems like the two-
tank interactive system. While significant progress has been made in leveraging these ap-
proaches to improve system performance and stability, further research is needed to address
remaining challenges and optimize the effectiveness of these techniques in real-world applic-
ations

1.5 PROPOSED METHODOLOGY

One of possible reasons of such a situation is a not satisfactory integration of the ex -

isting specialized approaches.. Thus, for a successful design of nonlinear control loop it is
usually not enough to focus on a single traditional method built on a rigorous mathematical
framework, but one needs a modular approach integrating several such approaches, each of
them specialized in a closer set of problems to be solved

Model Reference Adaptive Control (MRAC) is a control technique used to regulate a

system's output by comparing it to a reference model and adjusting the controller parameters
in real-time to minimize the error between the actual output and the desired output. MRAC
can improve control performance in the presence of parameter variations, disturbances, or un-
certainties by continuously adapting the controller to the system's changing dynamics.

The basic idea behind MRAC is to use an adaptive mechanism to adjust the controller
parameters based on the difference between the system's output and the output of a reference

4
model. This adaptive mechanism typically involves estimating the system's parameters and
updating the controller gains to minimize the tracking error.

In The context of PID control, MRAC can be combined with PID control to form an
adaptive PID controller (MRAC-PID), where the PID gains are adjusted based on the sys-
tem's response and the reference model. This adaptive approach allows the controller to adapt
to changes in the system's dynamics and improve control performance over time.

Fuzzy logic control (FLC) presents several advantages over proportional-integral-de-

rivative (PID) control in certain scenarios. FLC excels in managing systems with complex,
non-linear dynamics where traditional PID controllers struggle. Its ability to capture non-lin-
ear relationships through linguistic rules and fuzzy sets makes it particularly effective in such
environments. Moreover, FLC handles uncertainty and imprecise knowledge well, making it
suitable for systems where precise mathematical models are challenging to derive or where
parameters are uncertain. Fuzzy logic naturally accommodates multi-variable systems and
can adapt its behavior over time, making it adept at handling dynamic and evolving systems.
Additionally, FLC mimics human-like reasoning, incorporating expert knowledge effectively.
However, it's important to recognize that PID control has its own strengths, such as simplicity
and well-established tuning methods. The choice between FLC and PID ultimately depends
on the specific requirements of the control system, including the nature of the process, avail-
able resources, and desired performance criteria.

1.6 Organization Of Thesis

The introductory chapter provides an overview of the project "Mathematical Model-
ing and Control Design for a Two-Tank Interactive System." It outlines the significance of
studying the dynamics and control of such systems in industrial processes. This chapter also
highlights the limitations of existing control methods and the need for a more robust approach
to ensure efficient operation and safety.

Chapter 2 delves into the mathematical modeling aspect of the project. It begins by
describing the physical setup of the two-tank interactive system and its components. The
chapter then proceeds to derive the mathematical equations governing the dynamics of the
system, considering factors such as fluid flow rates, tank volumes, and interactions between
the tanks. Assumptions made during the modeling process are discussed, along with their im-

5
plications. The resulting mathematical model serves as the foundation for subsequent con-
troller design.

Chapter 3 focuses on the design of a PID (Proportional-Integral-Derivative) controller

for regulating the behavior of the two-tank interactive system. It starts by introducing the ba-
sic principles of PID control and its relevance to the system under consideration. The chapter
then discusses the tuning methods and criteria for selecting appropriate PID controller param-
eters. Simulation studies or experimental results demonstrating the performance of the PID
controller are presented and analyzed.

Chapter 4 delves into the design of a Model Reference Adaptive Control (MRAC)
system integrated with PID control for the two-tank interactive system. It begins by explain-
ing the concept of MRAC and its advantages in adaptive control applications. The chapter
then details the design methodology for combining MRAC with PID control, including pa-
rameter adaptation algorithms and stability analysis. Simulation studies or experimental re-
sults showcasing the performance improvements achieved with the MRAC-PID controller are
presented and compared to the PID controller.

Chapter 5 focuses on the design of a Fuzzy Logic Controller (FLC) for the two-tank
interactive system. It introduces the principles of fuzzy logic control and its suitability for
handling complex and non-linear systems. The chapter then describes the design process of
the FLC, including the formulation of fuzzy rules and membership functions. Simulation
studies or experimental results demonstrating the effectiveness of the FLC in regulating the
system are presented and compared to the PID and MRAC-PID controllers.

The concluding chapter summarizes the key findings of the thesis and discusses their
implications for industrial applications. It provides a comprehensive analysis of the perfor-
mance of the different controller designs in regulating the two-tank interactive system. The
chapter also discusses limitations encountered during the project and suggests areas for future
research or improvements.

1.7 SUMMARY

In conclusion, the design of controllers for the Two-Tank Interactive System using
PID, Fuzzy Logic, and MRAC-PID offers different approaches to achieve effective regula-
6
tion of water levels in the tanks. Each control strategy has its advantages and challenges, de-
pending on the system's dynamics, performance requirements, and environmental conditions.

CHAPTER 2

MATHEMATICAL MODELLING OF TWO TANK INTERACTING

SYSTEM

2.1 INTRODUCTION

The mathematical modeling of dynamic systems is fundamental to understanding their

behavior and designing effective control strategies. In this context, the Two-Tank Interactive
System serves as a classic example of a complex dynamic system found in various industrial
processes such as chemical engineering, water management, and process control. The Two-
Tank Interactive System consists of two interconnected tanks with valves and a pump that
regulates the flow of fluid between them. The water levels in the tanks are influenced by in-
flow rates, outflow rates, and the interaction between the tanks. Controlling the water levels
in both tanks is essential for maintaining desired operating conditions and ensuring efficient
system performance.

Mathematical modeling plays a crucial role in describing the dynamic behavior of the
Two-Tank Interactive System. By formulating mathematical equations based on physical
principles such as mass balance and fluid dynamics, we can capture the system's dynamics
and predict its response to control inputs and disturbances.

In this chapter, we aim to develop a comprehensive mathematical model of the Two-

Tank Interactive System, considering factors such as dead time, time constants, and nonlin-
earities. Through this modeling process, we seek to gain insights into the system's behavior
and dynamics, which will serve as the foundation for designing and implementing control
strategies to regulate the water levels effectively.

The mathematical model will enable us to simulate the system's response under differ-
ent operating conditions, analyze its stability and robustness, and design controllers to
achieve desired performance criteria such as setpoint tracking, disturbance rejection, and sta-
bility.By studying the mathematical model of the Two-Tank Interactive System, we can
deepen our understanding of dynamic systems theory and control engineering principles, pro-

7
viding valuable insights and tools for tackling real-world control challenges in diverse indus-
trial applications.

2.2 PROCESS DESCRIPTION

The Universal process control trainer consists of pumps, control valves, process tanks,
overhead tank, differential pressure transmitter, level transmitter, rotameter. Instrumentation
panel consists PI, PD and PID controller, main power supply switch, pump switches, auxil-
iary switches for individual components. Fluid level in the tank is measured by level trans-
mitter (LT). Output of LT is given to the data acquisition setup. It consists of analog to digital
converter and digital to analog converter. The differential pressure level transmitter (DPLT)
measures the flow by sensing the difference in level between the tanks. The DPLT then trans-
mits a current signal (4-20mA) to I/V converter. The output of I/V converter is given to the
interfacing hardware associated with the personal computer (PC). A control algorithm is im-
plemented in MATLAB software. It compares and takes corrective action on the control
valve Based on how much control valve open or close. The controller compares the con-
trolled variable against set point and generates manipulated variable as current signal (4-
20mA). Here the controlled variable is the level (h2) and the manipulated variable is the flow
rate (qin). The Control valve gives restriction to the flow through the pipeline and hence the
desired level is achieved. Maximum height of tank 2 is 500mm.

2.3 MATHEMATICAL MODELLING:

The process consisting of two interacting liquid tanks shows in fig 2.1. The height of the li-
quid level is h1 (cm) in tank1 and h2 (cm) is tank2. Volumetric flow into tank 1 is qin
(cm3 /min), the volumetric flow rate from q1 (cm3 /min), and the volumetric flow rate from
tank 2 is q0 (cm3 /min). Cross sectional area of tank1 is A1 (cm2 ) and area of tank2 is A2
(cm2 ).

8
 Fig2.1: Block diagram of two tank interactive system

A1= Cross sectional area of tank I and tank 2 (cm) ,

A2= Cross sectional area of output pipe in tank 2 (cm) ,
an = Cross sectional area of interaction pipe in tank 2 (cm) ,
h1,h2 = Water level of tank 1 and tank 2 (cm) ,
Qin = Inflow = Outflow (Iph),
K = Gain of the pump,
G = Gravity
U= Input voltage

For tank 1,

d h1
A1 =q ¿ −q1
dt
(2.1)

Assume linear resistance to flow,

h1−h2
q 1=
R1

Time constant of tank 1 ,

9
 T 1= A 1 R 1
dh
T 1 1 =R 1 q ¿ −h1 +h2
dt

Taking Laplace transform on both sides ,

T 1 s h1 ( s)+h1 (s)−h 2 (s)=R1 q¿ (s)

h1 ( s) ( T 1 s +1 ) −h2 (s)=R1 q¿ (s )

R 1 q¿ ( s ) h2 ( s )
h1 ( s)= +
( T 1 s+1 ) ( T 1 s +1 )
(2.2)

For tank 2,

d h2
A2 =q1−q 0
dt
(2.3)

Assume linear resistance to flow,

h2
q 0=
R2

d h2
R1 A 2 R 2 = ( h1−h 2 ) R2 −h2 R 1
dt

Time constant of tank 2,

T 2= A 2 R 2

d h2
R1 T 2 +h2 R2 +h2 R 1=h1 R2
dt

Taking Laplace transform on both sides,

R1 T 2 s h2 (s )+ h2 (s)R2 +h 2 (s) R1=h1 (s) R2

( R1 T 2 s + R2 + R1 ) h2 (s)=h1 (s) R2
(2.4)

In equation (4) putting value of equation (2),

R1 R 2 q ¿ (s) R2 h2 (s)
( R1 T 2 s + R2 + R1 ) h2 (s)= +
( T 1 s+ 1 ) ( T 1 s+ 1 )
Solving above equation,

( R 1 ( T 1 s+1 ) ( T 2 s+ 1 ) + R2 ( T 1 s +1 ) ) h2 (s)−R 2 h2 ( s)=R 1 R2 q¿ (s)

10
( T 1 T 2 s 2 +s ( T 1 +T 2+ A 1 R2 ) +1 ) h2 (s )=R2 q¿ (s )
h2 ( s)
Convert above equation in form of
q¿ (s )

h2 ( s) R2
=
q¿ (s ) T 1 T 2 s + s ( T 1+ T 2 + A1 R2 ) +1
2

(2.5)

Equation (2.5) is a transfer function of interacting system.

Designing of plant transfer function:

 Setup the interacting tank system as shown in fig. 2.1.

 Give constant 193 lph input liquid flow to the system and wait for the level of tank at
steady state point. This is called initial state of system.

 Write down the reading of liquid level in tank1 and tank 2 at particular flow which is
continuously apply to system.

 Now give a step change in flow e.g. 305 lph

 Again write down the reading of liquid level in tank1 and tank2. This is final state of
system.

Now we know that,

d h1
R 1=
dQ

Putting measured value in above equation,

(102−56)mm
R 1=
(305−193)lp h

The value of R1 ¿ 1478.57 sec/m2

Similarly for R2

d h2
R 2=
dQ

Putting measured value in above equation,

(55−35)mm
R 2=
(305−193) lph

The value of R 2=642.86 sec/m2

Time constant is T 1=R1 A 1∧ T 2=R 2 A 2

11
Where area of tank1 and tank 2 is 0.0145 m2

The value of time constant T 1=21.42 and T 2=9.31

All this value put into equation (2.5) so we get transfer function of interacting tank system

h2 ( s) 642.86
=
q¿ (s ) 199.42 s 2 +40.04 s+ 1
(2.6)

Transfer function of system in s- domain, that present gain of system is 642.86 with two poles
at -0.029 and -0.171 . Damping coefficient is 1.41 and damped natural frequency is 0.0708
rad.

2.4 SIMULINK BLOCK DIAGRAM DESCRIPTION

Simulink model for liquid level control with open loop close loop without any con-
troller, PID controller by using program MatlabR2020a. Based on the transfer function of the
plant which is derived using mathematical modelling. Fig. 2.2 shows the open loop model of
the plant. Where input flow in tank 1 is 305 cm3 / s.

Fig2.2: Open loop model for interacting tank

12
 Tank 2 level in
(mm)

Time(sec)

Fig2.3: Simulation result of open loop model

2.5 SUMMARY

In summary, mathematical modeling provides a systematic framework for understanding

and analyzing the behavior of dynamic systems. Open-loop control offers simplicity and pre-
dictability but lacks adaptability and robustness, while closed-loop control provides real-time
feedback and correction, leading to improved performance and stability. Understanding the
principles and applications of mathematical modeling, open-loop control, and closed-loop
control is essential for designing effective control systems and optimizing system perform-
ance in engineering and scientific disciplines.

13
 CHAPTER 3
DESIGN OF PID CONTROLLER

3.1 INTRODUCTION

Proportional-Integral-Derivative (PID) control is the most common control algorithm

used in industry and has been universally accepted in industrial control. The popularity of
PID controllers can be attributed partly to their robust performance in a wide range of operat-
ing conditions and partly to their functional simplicity, which allows engineers to operate
them in a simple, straightforward manner.

As the name suggests, PID algorithm consists of three basic coefficients; proportional, inte-

14
gral and derivative which are varied to get optimal response. Closed loop systems, the theory
of classical PID and the effects of tuning a closed loop control system are discussed in this
paper.

3.2 THE PID ALGORITHM

The most widely used regulator control law of proportional, integral and differential
control is referred to as PID control algorithm, also known as PID control or PID regulation.
PID controllers have come out with an history of nearly 70 years. The simple structure, good
stability and reliable performance have become one of the major industrial control techno-
logy. When the structure and parameters of the controlled object are not fully given, the
structure and parameters of the system controller must be relied on experience and on-site
commissioning, and the application of PID control technology is the most convenient techno-
logy. For PID control algorithm, there are PI control and PD control. PID controller work by
calculating the proportional, integral and differential control values.

There are currently 3 kinds of relatively simple PID control algorithms, namely: incremental
algorithm, position type algorithm, differential algorithm. These control algorithms are the
most simple and basic algorithms that they have their own characteristics and meet the gen-
eral requirements of the most controls.

3.2.1 PROPORTIONAL RESPONSE

The proportional component depends only on the difference between the set point and the
process variable. This difference is referred to as the Error term. The proportional
gain (Kc) determines the ratio of output response to the error signal. For instance, if the error
term has a magnitude of 10, a proportional gain of 5 would produce a proportional response
of 50. In general, increasing the proportional gain will increase the speed of the control sys-
tem response. However, if the proportional gain is too large, the process variable will begin to
oscillate. If Kc is increased further, the oscillations will become larger and the system will be-
come unstable and may even oscillate out of control.

15
3.2.2 INTEGRAL RESPONSE

The integral component sums the error term over time. The result is that even a small er-
ror term will cause the integral component to increase slowly. The integral response will con-
tinually increase over time unless the error is zero, so the effect is to drive the Steady-State
error to zero. Steady-State error is the final difference between the process variable and set
point. A phenomenon called integral windup results when integral action saturates a con-
troller without the controller driving the error signal toward zero.

3.2.3 DERIVATIVE RESPONSE

The derivative component causes the output to decrease if the process variable is in-
creasing rapidly. The derivative response is proportional to the rate of change of the process
variable. Increasing the derivative time (Td) parameter will cause the control system to react
more strongly to changes in the error term and will increase the speed of the overall control
system response. Most practical control systems use very small derivative time (T d), because
the Derivative Response is highly sensitive to noise in the process variable signal. If the sen-
sor feedback signal is noisy or if the control loop rate is too slow, the derivative response can
make the control system unstable.

3.3 SIMULINK BLOCK DIAGRAM DESCRIPTION

Fig 3.1Block diagram of PID controller

16
 3.4 SIMULATION RESULT OF PID CONTROLLER
Tank 2 level in
(cm)

17
 Time(sec)

Fig 3.2 simulation result of PID controller

Fig 3.3 Time domain parameters

3.5 SUMMARY

The proportional action responds to the current error signal by generating a control
output proportional to the magnitude of the error. This action provides immediate corrective
action but may result in steady-state error if the system exhibits nonlinear behavior or disturb-
ances.

The integral action accumulates the error over time and generates a control output
based on the integral of the error signal. This action helps to eliminate steady-state error by
continuously adjusting the control output until the error is minimized.

18
 The derivative action anticipates future changes in the error signal by calculating the
rate of change of the error. This action provides damping and improves the transient response
of the system, reducing overshoot and settling time.

CHAPTER 4
DESIGN OF MRAC-PID CONTROLLER

4.1 INTRODUCTION

As per the analysis, it is analyzed that the probabilistic bi-level performance assess-
ment of the control mechanism is not carried out by any researchers for the two-tank interact-
ing system. Meanwhile, the dynamic and static performance of the considered system is not
so impressive. This motivates to design an adaptive control mechanism such that it quickly
adapts the uncertainties and gives better system performance. This presents the development
of a model reference adaptive control-proportional integral derivative (MRAC-PID) con-
troller for a two-tank interaction system and the stability of the closed-loop system is then in-
vestigated using the Lyapunov stability approach. The important efforts are listed here:

1. Design and development of novel MRAC-PID control mechanism for the two-tank
interacting cylindrical system.

2. The conventional MRAC is structured for the first order system. However, the ma-
jority of the plants are second-order systems including two-tank interacting systems. Since
the conventional MRAC performance is not up to the mark. Therefore, MRAC’s first to sec-
ond-order extension is developed and the control law is implemented for the second-order
system.

3. The response of the proposed MRAC-PID and conventional MRAC are compara-
tively analyzed considering different adaptive gains and parameters for a two-tank interacting
system.
19
 4. The probabilistic assessment is accomplished through a bi-level uncertainty frame-
work such as Level I: Gain mistuning; Level II: Dynamic system behavior.

5. The performances of the proposed MRAC-PID control mechanism are vividly com-
pared with PID and fuzzy technique in terms of time domain specifications (overshoot, rise
time, settling time, and peak time) and error indices (ISE, and IAE).

4.2 CONTROLLER DESIGN MECHANISM

The response of a system utilizing a regular feedback loop becomes erroneous when
the parameters are not precisely recognized, and adaptive control is then employed.

In MRAC, a reference model defines the expected behavior of the process, and the
controller parameters are changed depending on error (e), which is determined as the discrep-
ancy between the process outcome (yp) and the output of the reference model (ym).

MRAC has two loops: an outer loop (or adaptation) that modifies the controller’s
variable in order to reduce the error between the reference and plant model output to zero,
and an inner loop that functions as a simple control loop between plant and controller. The
primary elements of the MRAC are the reference model, controller, and adaptation mecha-
nism.

4.2.1 REFERENCE MODEL

Operation stipulation required for the control mechanism. The desired performance
characteristics outlined by the reference model should be attainable within the adaptive con-
trol system. The reference model for this study is the critically damped second-order system.

Controller: The controller design typically involves the parameterization of a set of

adjustable parameters ( θ1 , θ2 and θ3 ). The control law exhibits linearity w.r.t the adjustable
parameters, adhering to a linear parameterization. In the realm of adaptive controller design,
it is customary to employ linear parameterization to achieve an adaptation mechanism that
ensures both convergence in tracking and stability. The control parameters' values are primar-
ily influenced by the adaptation gain, which subsequently modifies the control strategy of the
adaptation mechanism.
20
 Adaptation mechanism: The purpose of this mechanism is to facilitate the manipula-
tion of parameters within the control rule. The adaptation rule seeks to identify optimal pa-
rameters, ensuring that the plant's response aligns with the desired behavior specified by the
reference model. The design guarantees the control system stability while also achieving con-
vergence of the error tracking to zero. In the realm of controller design engineering, various
mathematical techniques such as Lyapunov theory, MIT(Massachusetts Institute of
Technology) rule, and augmented error concept can be effectively employed to formulate and
refine the adaptation mechanism. In such a considered system, the MIT rule is employed for
this specific purpose.

The input and output signal of the plant is denoted by u p (t ) and y p (t ) respectively.
The following form, appropriate in both frequency and time domains, was chosen as the sec-
ond-order plant model.

2
d y p (t) d y p (t )
2
=−a p −b p y p (t)+k p u p (t)
dt dt
(4.1)
y (s ) kp
G p (s)= p = 2
u p ( s ) s + a p s+ b p
(4.2)

where a p , b p and k p are plant parameters and they can be determined using equation
(4.1). The following form describes the relation between the input r (t ) and the intended out-
put y m (t ) in the second-order reference model (in both the time and frequency domains).

2
d y m (t) d ym( t )
2
=−am −bm y m (t )+ k m r (t)
dt dt
(4.3)
y (s ) km
Gm ( s)= m = 2
r ( s ) s +a m s+ bm
(4.4)

where k m represents positive gain, and b m and a m are chosen as the response of the reference
model is critically damped. The goal of the control strategy is to develop u p (t ) so that plant
output y p (t ) follows reference model output y m (t ) asymptotically. The parameters of the
MRAC controller's adaption law are calculated using the MIT rule. As per MIT law, the cost
function is formulated as:

21
 2
e
J ( θ)=
2
(4.5)
e= y p − y m
(4.6)

The difference between plant and reference model i.e., ( y p − y m ) constitutes the error
which is denoted by e . The control parameter that can be adjusted is represented by the
symbol θ . To achieve the goal of reducing the overall cost function to its minimum possible,
the value of the θ is adjusted using the MIT rule, which is given in equation (4.5). As a result,
it is written.

dθ δJ δe
=−λ =− λe
dt δθ δθ
(4.7)

δe
where the terms λ and are used to describe adaptation gain and sensitivity
δθ
derivative, respectively. The controller architecture for accomplishing the targeted control
goals is depicted. The control law u p (t ) is defined for bounded reference input.

T
u p =θ1 r −θ2 y p−θ3 ẏ p=θ φ
(4.8)

T
where the estimated vector of the controller parameter is denoted by θ=[ θ1 ,θ 2 , θ3 ]
T
and φ presents [ r , y p , ẏ p ] . Substituting equation (4.8) into equation (4.1), we get

2
d y p (t) d y p (t)
2
=¿−( a p +k p θ3 ) −( b p+ k p θ2 ) y p (t)
dt dt

Comparing equations (4.3) and (4.8) coefficients, we get

k m =θ1 k p
(4.9)
b m=b p+ k p θ2
(4.10)
a m=a p+ k p θ3
(4.11)

22
where θ1 , θ2, and θ3 are control parameters and they are converged as:

km bm−b p am−a p
θ1 ≈ ; θ2 ≈ ; θ3 ≈
kp kp kp
(4.12)

Taking the Laplace transform of the equation, we get

y p (s ) k p θ1
= 2
r ( s) s + ( a p +k p θ3 ) s + ( b p +k p θ2 )
(4.13)

As per the error equation (4.5), we can write

e=
( 2
k pθ1
s + ( a p +k p θ 3) s+ ( b p +k p θ2 )
− 2
km
s +a m s+ bm ) r (s)

(4.14)

Fig 4.1 Inner architecture of proposed MRAC-PID control mechanism.

23
 δe δe δe
The sensitivity derivatives , and are derived from equations (4.13)∧( 4.14)
δ θ1 δ θ2 δ θ3
given by

δe kpr
= 2
δ θ1 s +a p s +k p θ3 s+b p+ k p θ2
(4.15)
δe −k p y p
= 2
δ θ2 s +a p s +k p θ3 s+b p+ k p θ2
(4.16)
δe −k p y p s
=
δ θ3 s 2 +a p s +k p θ3 s+b p+ k p θ2
(4.17)

δe
Considering s2 +a m s+ bm=s 2+ a p s +k p θ3 s +k p θ2 +b p. In equation 4.7 , the are replaced fol-
δθ
lowing the MIT rule (equations 4.5 and 4.7). After restructuring, the equations (4.18), (4.19),
and (4.20) are employed to modify control parameters θ1, θ2, and θ3 respec-
tively.

( )
d θ 1 (t) 1
=−λ 2 r ( t ) e(t )
dt s + am s+ bm
(4.18)

( )
d θ2 (t ) 1
=λ 2 y p ( t ) e (t)
dt s +am s+b m
(4.19)

( )
d θ3 (t ) 1
=λ 2 ẏ p ( t ) ė (t)
dt s +a m s+b m
(4.20)

The MRAC controller design has been finalized, and the proposed control mechanism is
described in the subsequent section.

4.3 PROPOSED MRAC-PID CONTROL MECHANISM

The response of the reference model is utilized for monitoring the outcome of the
plant through MRAC. The required outcomes will certainly be realized by establishing the
reference model. However, using solely conventional MRAC is insufficient to boost system
performance. Consequently, a modified version of the MRAC i.e., MRAC-PID control
24
mechanism, is developed. The internal architecture of the proposed MRAC-PID control
mechanism is depicted in Figure 4.1.

The PID provides feedback for MRAC and the performance of the system is
significantly enhanced as MRAC-PID controller is combined. It should be noted that the pro-
posed controller's (MRAC-PID) output depends not only on the adaption gain but also on the
proportional ( K P ), integral ( K I ) and derivative ( K D ) gains of the PID block.

The control law is expressed as:

(
u p =θ1 r −θ2 y p−θ3 ẏ p− K P e+ K I ∫ edt + K D
de
dt )
(4.21)

In comparison to traditional MRAC, the combined effect of MRAC and PID feed-
back, or MRAC-PID, on the secondorder two-tank interacting system leads to enhanced
process behavior during transient as well as steady-state responses. The control rule is
employed to align the response of the plant and standard model, and it is given in equation
(4.22).

50
Gm (s)= 2
s +15 s +50
(4.22)

4.4 SIMULATION BLOCK DIAGRAM DESCRIPTION

25
 Fig 4.2 Block diagram of MRAC-PID

4.5 SIMULATION RESULT OF MRAC-PID CONTROLLER

Tank 2 level in
(cm)

Time(sec)
Fig 4.3 Simulation result of MRAC-PID

4.6 SUMMARY

26
 The MRAC-PID controller is designed using the MATLAB/ Simulink platform, and
compared to well-known techniques such as PID and conventional MRAC. The plant model
is described in Equation 16 and its specifications are given in Table 1. The different error in-
dices are calculated using the following equations [4.17] .

∞
Integral absolute error (IAE)=∫ ❑∨e (t)∨dt
0
(4.23)
∞
Integral square error (ISE )=∫ ❑∨e(t)¿ dt
2

0
(4.24)
¿
¿

Directly providing input to the system and examining its characteristics is known as
the open loop response of a system. The output of an open loop control is not evaluated or
provided for signal compared to the input. In a closed-loop system, a controller is employed
to perform a comparison between the system's response and the desired condition, subse-
quently transforming the error into a control action. It is built to minimize error and
enable the system for attaining the desired outcome.

The fine-tuned PID controller gains are such as K P=7.597∗10−3 ,

−3 −3
K I =7.597∗10 /65.124 and K D=7.597∗10 ∗8, and the adaption gain (λ) is determined
as 5 .It is evident that PID takes 108 Sec , conventional Fuzzy takes 3.5 Sec and the proposed
MRAC-PID requires 19 Sec to track the desired outcome.

27
 CHAPTER 5
DESIGN OF FUZZY LOGIC CONTROLLER

5.1 INTRODUCTION

Nowadays, fuzzy logic, have rooted in many application areas (expert systems, pattern
recognition, system control, etc.) Fuzzy logic is mainly associated to imprecision, approxim-
ate reasoning and computing with words. Fuzzy Logic is particularly good at handling uncer-
tainty. vagueness and imprecision. This is especially useful where a problem can be described
linguistically (using words) or, where there is data and you are looking for relationships or
patterns within that
data.

Fuzzy Logic uses imprecision to provide robust, tractable solutions to problems.

 Fuzzy logic relies on the concept of a fuzzy set.
 have ability to deal with non-linearities.
 follow more human-like reasoning paths than classical.
 utilize self-learning.

5.2 FUZZY MEMBERSHIP FUNCTIONS:

One of the key issues in all fuzzy sets is how to determine fuzzy membership function
fully defines the fuzzy seta membership function provides a measure of the degree of similar-
ity of an element to a fuzzy set. Membership functions can take any form, but there are some
common examples that appear in real applications Membership functions can either be
chosen by the user arbitrarily, based on the user's experience (MF chosen by two users could
be different depending upon their experiences, perspectives, etc. Or be designed using ma-
chine learning methods (eg., artificial neural networks, genetic algorithms, etc.)There are dif-
ferent types of membership functions(MF), triangular(MF), trapezoidal(MF),
Gaussian2(MF), bell shaped(MF), Gaussian curve(MF), Built-in(MF). Pi shaped(MF), Sig-
moidally shaped(MF). S shaped etc.

28
5.2.1 TRIANGULAR MEMBERSHIP FUNCTION:

The membership function which we preferred is the triangular membership function.

Triangular membership function is the most commonly used MF and it is the easiest way to
implement also.

a, b and c represent the r coordinates of the three vertices of μA(x) in a fuzzy set A
(a: lower boundary and e: upper boundary where membership degree is zero, b: the centre
where membership degree is 1)

Fig 5.1 triangular membership function

5.3 FUZZY RULE BASED SYSTEM:

5.3.1 FUZZY INFERENCE METHODS:

The general block diagram of a fuzzy system is shown in figure 5.1. The con-
troller is composed of four elements:

• A Rule Base

• An Inference Mechanism

• A Fuzzification Interface

• A Defuzzification Interface

29
 Fig 5.2 Block of fuzzy rule based system

RULE BASE

A rule base containing a number of fuzzy if-then rules.

DATA BASE

A database which defines the membership functions of the fuzzy sets

used in fuzzy rules.

DECISION MAKING

A decision-making unit which performs the inference operations on the rules.

FUZZIFICATION INFERENCE

A fuzzification interface which transforms the crisp inputs into degrees of

match with linguistic values.

DEFUZZIFICATION INFERENCE

A defuzzification interface which transform the fuzzy results of the inference

into a crisp output. The rule base and the database are jointly referred to as the knowl-
edge base.

The steps of fuzzy reasoning (inference operations upon fuzzy IF-THEN rules)
performed by FISs are described as follows

• Compare the input variables with the membership functions on the an-
tecedent part to obtain the membership values of each linguistic label (this step is of-
ten called fuzzification).

30
 • Combine (usually multiplication or min) the membership values on the
premise part to get firing strength (weight) of each rule.

• Generate the qualified consequents (either fuzzy or crisp) of each rule de-
pending on the firing strength.

• Aggregate the qualified consequents to produce a crisp output (This step is

called fuzzification).

There are many different methods of defuzzification

• Al (adaptive integration)

• BADD (basic defuzzification distributions)

• CDD (constraint decision defuzzification)

• COA (centre of area)

• COG (centre of gravity)

• ECOA (extended centre of area

• EQM (extended quality method)

• FCD (fuzzy clustering defuzzification)

• GLSD (generalized level set defuzzification)

• ICOG (indexed centre of gravity)

• IV (influence value)

• LOM (last of maximum)

• MeOM (mean of maxima)

• MOM (middle of maximum)

• QM (quality method)

• RCOM (random choice of maximum)

• SLIDE (semi-linear defuzzification)

• WFM (weighted fuzzy mean)

31
 Five commonly used defuzzifying methods:

• Centroid of area (COA)

• Bisector of area (BOA)

• Mean of maximum (MOM)

• Smallest of maximum (SOM)

• Largest of maximum (LOM)

5.3.2 MAXIMUM DEFUZZIFICATION TECHNIQUE

This method gives the output with the highest membership function. This defuzziftion
technique is very fast but is only accurate for peaked output. This technique is given by alge-
braic expression as

A(x*) ≥ μA(x) all xeX

Where x* is the defuzzified value. This is shown graphically in Figure 5.3

Fig 5.3 Max-membership defuzzification method

5.3.3 CENTROID DEFUZZIFICATION TECHNIQUE

This method is also known as centre of gravity or centre of area defuzzification. This
technique was developed by Sugeno in 1985. This is the most commonly used technique and
is very accurate. The centroid defuzzification technique can be expressed as

32
 ∫ μ i ( X ) X dX
∫ μ i ( X ) X dX

Where x is the defuzzified output, μ i(x) is the aggregated membership function and x is the
output variable. The only disadvantage of this method is that it is computationally difficult
for complex membership functions.

5.3.4 WEIGHTED AVERAGE DEFUZZIFICATION TECHNIQUE

In this method the output is obtained by the weighted average of the each output of
the set of rules stored in the knowledge base of the system. The weighted average defuzzifi-
cation technique can be expressed as

∑ m i wi
i=1
x∗¿ n

∑ mi
i=1

Where x is the defuzzified output, n is the membership of the output of each rule. and
w, is the weight associated with each rule. This method is computationally faster and easier
and gives fairly accurate result. This defuzzification technique is applied in fuzzy application
of signal validation.

5.3.5 SELECTION OF INPUTS AND OUTPUTS

It should be made sure that the controller will have the proper information available to
be able to make good decisions and have proper control imputes to be able to steer the system
in the directions needed to be able to achieve high-performance operation.

The fuzzy controller is to be designed to automate how a human expert who is successful at
this task would control the system. Such a fuzzy controller can be successfully developed us-
ing high-level languages like C, Fortran, etc. Packages like MATLAB also support Fuzzy
Logic.

5.4 APPLICATION OF FUZZY LOGIC

Commercially fuzzy logic has been used with great success to control machines and
consumer products. In the right application fuzzy logic systems are simple to design, and can

33
be understood and implemented by non- specialists in control theory. Control engineers also
use it in applications where the on-board computing is very limited and adequate control is
enough.

A cross section of applications that have successfully used fuzzy control includes:

1. Environmental • Air Conditioners • Humidifiers

2. Domestic Goods • Washing Machines/Dryers • Vacuum Cleaners • Toasters • Mi-

crowave Ovens • Refrigerators

3. Consumer Electronics • Television • Photocopiers • Still and Video Cameras -

Auto-focus, Exposure and Anti-shake • Hi-Fi Systems

4. Automotive Systems • Vehicle Climate Control • Automatic Gearboxes • Four-

wheel Steering • Seat/Mirror Control Systems

5.5 FUZZY MODELLING:

In rule-based fuzzy systems, the relationships between variables are represented by

means of fuzzy if then rules of the following general form:

If antecedent proposition, then consequent proposition:

The antecedent proposition is always a fuzzy proposition of the type "-x is A "where"
-x is a linguistic variable and A is a linguistic constant (term). The proposition’s truth value
(a real number between zero and one) depends on the degree of match similarity be between-
x and A. Depending on the form of the consequent to main types of rule-based fuzzy models
are distinguished:

Mandani fuzzy model: both the antecedent and consequent are fuzzy propositions.

5.6 MANDANI FUZZY MODELS:

Mandani's fuzzy inference method is the most commonly seen fuzzy methodology.
Mandani’s method was among the first control systems built using fuzzy set theory. It was
proposed in 1975 by Ebrahim Mamdani. Mamdani's effort was based on Lotfi Zadeh's 1973
paper on fuzzy algorithms for complex systems and decision processes.

Mamdani-type inference, as it was defined for the Fuzzy Logic Toolbox, expects the output
membership functions to be fuzzy sets. After the aggregation process, there is a fuzzy set for
each output variable that needs defuzzification. It's possible, and in many cases much more

34
efficient, to use a single spike as the output membership function rather than a distributed
fuzzy set. This is sometimes known as a singleton output membership function. and it can be
thought of as a pre-defuzzied fuzzy set. It enhances the efficiency of the defuzzification
process because it greatly simplifies the computation required by the more General Mamdani
method, which finds the centroid of a two-dimensional function. Rather than integrating
across the two-dimensional function to find the centroid, we use the weighted average of a
few data points.

To compute the output of this FIS given the inputs, six steps has to be followed

1. Determining a set of fuzzy rules

2. Fuzzifying the inputs using the input membership functions

3. Combining the fuzzified inputs according to the fuzzy rules to establish rule
strength.(Fuzzy Operations)

4. Finding the consequence of the rule by combining the rule strength and the output
membership function (implication)

5. Combining the consequences to get an output distribution (aggregation)

6. Defuzzifying the output distribution (this step is only if a crisp output (class) is
needed).

ADVANTAGES OF THE MANDANI METHOD

• It is intuitive.

• It has widespread acceptance.

• It is well suited to human input.

5.7 SIMULATION BLOCK DIAGRAM DESCRIPTION

35
Fig 5.4 Fuzzy model for interacting tank

Fig 5.5 Error input member function

36
Fig 5.6 Rate of Error input member function

Fig 5.7 output member function

37
 5.8 SIMULATION RESULT OF FUZZY LOGIC CONTROLLER
Tank 2 level in
(cm)

Time(sec)

Fig 5.8 Simulation Res- ult of Fuzzy

5.9 SUMMARY

The comparative study between fuzzy logic and Proportional-Integral-Deriva-

tive (PID) controllers revealed that the fuzzy logic con- troller consistently outper-
formed the PID controller in terms of system perfor- mance. Through extensive
experimentation and analysis, it was found that the fuzzy logic controller pro-
duced better output responses across various operating conditions and scenarios.

38
 CHAPTER 6
RESULTS AND CONCLUSION

6.1 INTRODUCTION

The result section presents the outcomes of a comparative study conducted to evaluate
the performance of three different control methodologies: Proportional-Integral-Derivative
(PID), fuzzy logic, and Model Reference Adaptive Control (MRAC) with PID. The study
aimed to assess the efficiency of these control strategies in regulating the dynamics of a non -
linear system and to identify the most effective controller for achieving desired control ob-
jectives.

6.2 RESULTS

39
 Based on the comparative study of PID, fuzzy logic, and MRAC-PID controllers, it
was found that the fuzzy logic controller consistently outperformed the other controllers in
terms of system performance. The results obtained from the experimentation and analysis in-
dicate that the fuzzy logic controller produced better output responses compared to both PID
and MRAC-PID controllers under various operating conditions and scenarios.
in in
2 level
(cm)(cm)
2 level
TankTank

Time(sec)

Fig 6.1 simulation result of PID vs Fuzzy vs MRAC- PID

TABLE 1. Set point tracing for different controller.

Set point tracing

Rise
Settling Overshoot
Control Adaptation time
time (Sec) (% )
mechanism gain (λ) (Sec)
OPEN LOOP - 140.1519 0 77.0472

40
 PID - 108.6352 0 30.6178
FUZZY - 3.5278 0 1.8703
MRAC-PID 5 19.0765 40.9687 0.6542

TABLE 2. Error indices for different controller.

Control Adaptation Error indices

mechanism gain (λ) ISE IAE
PID - 3.812 13.33
FUZZY - 0.7712 1.83
MRAC-
5 1.36*10^4 1842
PID

Table 1 shows that fuzzy logic controller has less settling time and overshoot and
MRAC-PID has less rise time. Table 2 shows that Fuzzy logic controller has less ISE and
IAE

6.3 CONCLUSION

In conclusion, the comparative study of PID, fuzzy logic, and MRAC-PID controllers has
provided valuable insights into their performance characteristics and suitability for control
applications. Through experimentation and analysis, it was found that the fuzzy logic control-
ler consistently outperformed the other controllers in terms of system stability, adaptability,
robustness, and overall control performance. The fuzzy logic controller demonstrated super-
ior capability in handling nonlinearities, uncertainties, and disturbances in the system dynam-
ics, leading to smoother control responses and improved system behavior. These findings un-
derscore the effectiveness of fuzzy logic control as a powerful and versatile control methodo -
logy that can address the challenges posed by complex and uncertain control environments.

6.4 FUTURE SCOPE

41
 While the comparative study has shed light on the advantages of fuzzy logic control, there
are several avenues for future research and development to further enhance its effectiveness
and applicability:

Optimization Techniques: Future research can explore advanced optimization tech-

niques to optimize the fuzzy logic controller's rule base and membership functions, thereby
improving its performance and efficiency.

Hybrid Approaches: Investigating hybrid control approaches that integrate fuzzy logic with
other intelligent control techniques, such as neural networks or genetic algorithms, could lead
to synergistic benefits and enhanced control performance.

Adaptive Fuzzy Control: Developing adaptive fuzzy control algorithms that can dynamically
adjust the controller's parameters based on real-time system feedback and changing operating
conditions can further enhance its adaptability and robustness.

Real-World Applications: Conducting case studies and experiments in real-world applica-

tions across various industries, such as robotics, automotive, and process control, can validate
the effectiveness of fuzzy logic control in practical settings and identify areas for further im-
provement.

Hardware Implementation: Exploring hardware implementation of fuzzy logic controllers us-

ing embedded systems or field-programmable gate arrays (FPGAs) can enable real-time con-
trol in resource-constrained environments and facilitate deployment in industrial automation
systems.

By addressing these areas of research and development, the potential of fuzzy logic control
can be further realized, leading to advancements in control technology and its widespread ad-
option in diverse engineering applications.

6.5 SUMMARY

In summary, the results of the comparative study suggest that the fuzzy logic controller
offers significant advantages over traditional PID and MRAC-PID controllers in terms of sys-
tem performance, stability, adaptability, and robustness. These findings highlight the poten-
tial of fuzzy logic control as a viable and effective approach for a wide range of control ap-
plications, particularly in complex and uncertain environments where conventional control-
lers may struggle to achieve satisfactory results.

42
RFERENCES

43
[1] M. Devaerakkam, K. N. Raghavan, G. K. Prince, M. J. K. Alphonse, S.Annadurai, and H.
Ramachandran, ‘‘Ascendancy of level in nonlinear tank system by neuro controller,’’ Results
Control Optim., vol. 12, Sep. 2023,Art. no. 100260, doi: 10.1016/j.rico.2023.100260.

[2] Changela, M., & Kumar, A. (Year). Designing a Controller for Two Tank Interacting Sys-
tem. International Journal of Science and Research (IJSR). ISSN (Online): 2319-7064. Index
Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438.
[3] Manna, S., Singh, D. K., Ghadi, Y., & Yousef, A. (2023, October). Probabilistic Bi-Level
Assessment and Adaptive Control Mechanism for Two-Tank Interacting System. IEEE Ac-
cess.
[4] M. Huo, H. Luo, X. Wang, Z. Yang, and O. Kaynak, ‘‘Real-time implementation of plug-
and-play process monitoring and control on anexperimental three-tank system,’’ IEEE Trans.
Ind. Informat., vol. 17, no. 9, pp. 6448–6456, Sep. 2021, doi: 10.1109/tii.2020.3030812.

[5] T. Xu, H. Yu, J. Yu, and X. Meng, ‘‘Adaptive disturbance attenuation control of two tank
liquid level system with uncertain parameters based on port-controlled Hamiltonian,’’ IEEE
Access, vol. 8, pp. 47384–47392, 2020, doi: 10.1109/access.2020.2979352.

[6] L. ThillaiRani, N. Deepa, S. Arulselvi, "Modeling and Intelligent Control Of two tank In-
teracting Level Process", International Journal of Recent Technology and Engineering
(IJRTE), ISSN: 2277-3878, Volume-3, Issue-1, March 2014.

[7] Abdelelah Kidher Mahmood, Hussam Hamad Taha, "Design Fuzzy Logic Controller for
Liquid Level Control", International Journal of Emerging Science and Engineering (IJESE),
ISSN: 2319–6378, Volume-1, Issue-11, September 2013.

[8] Bhuvaneswari N. S., Praveena R., Divya R., "System Identification And Modelling For
Interacting And Non-Interacting System Using Intelligence Techniques", IJIST-2012,
Volume-2, no-5, pp. 23-25, September 2012.

[9] Dharamniwas, Aziz Ahmad, Varun Redhu, Umesh Gupta, "Liquid Level Control By Us-
ing Fuzzy Logic Controller", International Journal of Advances in Engineering & Technology
(IJAET), ISSN: 2231-1963, July 2012.

[10] Gao Qingji, Li Zheng, "Fast and Accurate Motion Control Based on Good Gain
Method", Telkomnika, Vol.11, No.3, pp. 1236 ~ 1244, March 2013.

[11] Li Qi, Fang Yanjun, Song Jizhong, Wang Ji, "The Application of Fuzzy Control in Liquid
Level System", International Conference on Measuring Technology and Mechatronics Auto-
mation, IEEE, ISSN: 978-0-7695-3962-1, 2010. Fuzzy Logic Toolbox User’s Guide,
©COPYRIGHT 1995 - 1999 by The MathWorks, Inc.

44