Optimal Controller Design for Twin Rotor MIMO System Ankesh Kumar Agrawal

Optimal Controller Design for Twin Rotor
MIMO System
Ankesh Kumar Agrawal
Department of Electrical Engineering
National Institute of Technology
Rourkela-769008, India
June, 2013
Optimal Controller Design for Twin Rotor MIMO
System
A thesis submitted in partial fulfilment of the requirements
for the award of degree
Master of Technology
in
Electrical Engineering
by
Ankesh Kumar Agrawal
Roll No: 211EE3333
Under The Guidance of
Prof. Subhojit Ghosh
National Institute of Technology
Rourkela-769008, India
2011-2013
Department of Electrical Engineering
National Institute of Technology, Rourkela
CERTIFICATE
This is to certify that the thesis titled “Optimal Controller Design for Twin Rotor MIMO
System”, by Ankesh Kumar Agrawal, submitted to the National Institute of Technology,
Rourkela for the award of degree of Master of Technology with specialization in Control &
Automation is a record of bona fide research work carried out by him in the Department of
Electrical Engineering, under my supervision. I believe that this thesis fulfills part of the
requirements for the award of degree of Master of Technology. The results embodied in this
thesis have not been submitted in parts or full to any other University or Institute for the award
of any other degree elsewhere to the best of my knowledge.
Place: Rourkela
Prof. Subhojit Ghosh
Date:
Dept. of Electrical Engineering
National Institute of Technology
Rourkela, Odisha, 769008, INDIA
i
ACKNOWLEDGEMENT
First of all I like to thanks from deep of my heart to my supervisors Prof. Subhojit Ghosh for the
confidence that they accorded to me by accepting to supervise this thesis. I express my warm
gratitude for their precious support, valuable guidance and consistent encouragement throughout
the course of my M. Tech.
My heartiest thanks to Prof. Bidyadhar Subudhi, an inspiring faculty and Prof. A. K. Panda,
Head of Dept. of Electrical Engineering Department, NIT, Rourkela.
Special thanks to Prasanna, Ankush, Zeeshan for their support, care and love. I would like to
extend special gratitude to my friends in NIT Rourkela, without whom, this journey would not
have been this enjoyable.
Finally, I dedicate this thesis to my family: my dear father, my dearest mother and my brother
who supported me morally despite the distance that separates us. I thank them from the bottom
of my heart for their motivation, inspiration, love they always give me. Without their support
nothing would have been possible. I am greatly indebted to them for everything that I am.
ii
ABSTRACT
Twin Rotor MIMO system (TRMS) is considered as a prototype model of Helicopter. The aim of
studying the TRMS model and designing the controller for controlling the response of TRMS is
that it provides a platform for controlling the flight of Helicopter. In this work, the non-linear
model of Twin Rotor MIMO system has been linearized and expressed in state space form. For
controlling action a Linear Quadratic Gaussian (LQG) compensator has been designed for a
multi input multi output Twin Rotor system. Two degree of freedom dynamic model involving
Pitch and Yaw motion has been considered for controller design. The two stage design process
consists of the design of an optimal Linear Quadratic Regulator followed by the design of an
observer (Kalman filter) for estimating the non-accessible state variable from noisy output
measurement. LQR parameter i.e. Q and R are varied randomly to get the desired response. Later
an evolutionary optimization technique i.e. Bacterial Foraging Optimization (BFO) algorithm has
been used for optimizing the Q and R parameter of Linear Quadratic Gaussian compensator.
Simulation studies reveal the appropriateness of the proposed controller in meeting the desired
specifications.
iii
CONTENTS
Certificate
i
Acknowledgement
ii
Abstract
iii
Contents
iv
List of Figures
vi
List of Tables
vii
List of Abbreviations
viii
Chapter 1 Introduction
1
1.1 Background
1
1.2 Literature Review
2
1.3 Objective
3
1.3 TRMS set Description
4
Chapter 2 TRMS model
6
2.1 TRMS mathematical model
6
2.2 Linearized model
9
Chapter 3 Controlling of TRMS
12
3.1 Controllers
12
3.2 Types of Controllers
13
3.3 Linear Quadratic Regulator (LQR)
14
3.3.1 Overview
14
3.3.2 Estimating optimal control gain K
15
3.3.3 Linear Quadratic Tracking problem
19
3.4 Kalman Filter
22
3.4.1 Overview
22
3.4.2 Requirement of Kalman Filter
23
3.4.3 Mathematical model of Kalman Filter
24
3.4.4 Design of Kalman Filter
25
3.5 Linear Quadratic Gaussian (LQG)
26
iv
3.5.1 Overview
26
3.5.2 Requirement of LQG compensator
27
3.5.3 Steps in the design of LQG compensator
27
3.5.4 Compensator for TRMS system
27
3.6 Results and Discussion
29
Chapter 4 Bacterial Optimization Algorithm based Controller Design
4.1 Bacterial Foraging Optimization Algorithm
33
33
4.1.1 Overview
33
4.1.2 Steps involved in BFO
33
4.1.3 BFO Algorithm
34
4.1.4 Problem Formulation
35
4.2 Results and Discussion
37
Chapter 5 Conclusions
40
5.1 Conclusions
40
References
41
v
List of Figures
1.1 TRMS mechanical unit
4
2.1 TRMS Phenomenological model
6
3.1 Feedback control Loop
12
3.2 Block diagram of Linear Quadratic Regulator
14
3.3 Block diagram of LQG controller along with plant
26
3.4 Reference signal applied to TRMS
29
3.5 Output of TRMS using LQR
30
3.6 Output of Kalman Filter
31
3.7 Output of TRMS using LQG
32
4.1 Variation of cost function with respect to number of iteration.
38
4.2 Comparison between output of TRMS using with and without BFOA
38
vi
List of Tables
2.1 TRMS system parameters
8
4.1 Parameters values
37
4.2 Comparison of TRMS response with and without using BFO
39
vii
List of Abbreviations
Abbreviation
Description
UAV
Unmanned Air Vehicle
TRMS
Twin Rotor Multi input Multi output system
DC
Direct current
PID
Proportional Integrator Derivative
LPV
Linear parameter varying
QFT
Quantitative Feedback theory
LMI
Linear matrix inequality
GA
Genetic Algorithm
BFO
Bacterial Foraging Optimization
SISO
Single input Single output
LQ
Linear Quadratic
LQT
Linear Quadratic Tracking
DOF
Degree of Freedom
EP
Evolutionary Programming
ES
Evolutionary Strategies
viii
Chapter 1
Introduction
INTRODUCTION
1.1 Background
Recent times have witnessed the development of several approaches for controlling the flight of
air vehicle such as Helicopter and Unmanned Air Vehicle (UAV). The modeling of the air
vehicle dynamics is a highly challenging task owing to the presence of high nonlinear
interactions among the various variables and the non-accessibility of certain states. The twin
rotor MIMO system (TRMS) is an experimental set-up that provides a replication of the flight
dynamics. The TRMS has gained wide popularity among the control system community because
of the difficulties involved in performing direct experiments with air vehicles. Aerodynamically
TRMS consist of two types of rotor, main and tail rotor at both ends of the beam, which is driven
by a DC motor and it is counter balanced by a arm with weight at its end connected at pivot. The
system can move freely in both horizontal and vertical plane. The state of the beam is described
by four process variables- horizontal and vertical angles which are measured by encoders fitted
at pivot and two corresponding angular velocities. For measuring the angular velocities of rotors,
speed sensors are coupled with DC motors.
The TRMS is basically a prototype model of Helicopter. However there is some
significant difference in aerodynamically controlling of Helicopter and TRMS. In Helicopter,
controlling is done by changing the angle of both rotors, while in TRMS it is done by varying the
speed of rotors. Several works have been reported on dynamic modeling and control of TRMS.
For instance, an intelligent control scheme for the design of hybrid PID controller has been
proposed in [1]. Other notable works include LPV Modeling and Control [2], QFT based control
[3], LMI based approach [4] and Single Neuron PID control [5]. Considering the unmodelled
dynamics and the presence of noise in the output measurement, in the present work, a state
feedback controller has been designed considering the effect of unmodelled dynamics and noisy
output data. The design of a state feedback controller demands the availability of all the state
variables in the output. However, for the TRMS since all the states are not accessible, an
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Chapter 1
Introduction
observer (Kalman filter) has been designed for estimating the unavailable state variables from
the noisy output measurement. The Kalman filter has been coupled with an optimal controller i.e.
Linear Quadratic Regulator (LQR) for tracking a desired trajectory. The combination of the
Kalman filter and LQR is commonly referred to as Linear Quadratic Gaussian (LQG)
Compensator. For an observer based state feedback control of a plant corrupted by state and
measurement noise, the control action and the appropriateness of the estimated states is heavily
dependent on the output and control weighting matrices. The selection of these parameters is not
trivial problem and hence is carried out by trial and error method. This involves maintaining a
trade-off between minimizing the control effort and improving the transient response. Thus an
optimization technique, i.e. Genetic Algorithm (GA) [6-8], is used for the selection of weighting
matrices of the LQR controller. But there are certain optimization problems for which GA is not
preferred, because of the selection of large number of parameters and high computational cost.
Thus Genetic Algorithm has limitation for use in real-time applications. Therefore in this work, a
new evolutionary optimization technique, i.e. Bacterial Foraging Optimization (BFO) algorithm
[9-11] is used for optimizing weighting matrices of LQR, which will overcome the limitation of
Genetic Algorithm. BFO is a globally optimization technique for distributed optimization.
Simulation results depict the appropriateness of the proposed controller in tracking a desired
trajectory with minimum control effort.
1.2 Literature Review
This section reflects the brief review about optimal control of Twin Rotor MIMO system. S.M.
Ahmad gives the dynamic modelling of TRMS [12]. The aerodynamic model and mathematical
model of TRMS is explained in this paper. The paper shows that mathematical model of TRMS
in non-linear, so linearization technique is explained by A. Bennoune and A. Kaddouri in
Application of the Dynamic Linearization Technique to the Reduction of the Energy
Consumption of Induction Motors [13] paper. The flight of TRMS is controlled by using various
techniques like, by designing hybrid PID controller [1] given by J.G. Juang and W.K. Liu or by
using LPV Modelling and Control [2] given by F. Nejjari and D. Rotondo. Some other
controlling techniques include QFT based control [3], LMI based approach [4] and Single
Neuron PID control [5]. The above discussed controllers are not optimal controllers. So in this
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Chapter 1
Introduction
work optimal controller, i.e. Linear Quadratic Regulator (LQR) which is proposed by L.S. Shieh
in “Sequential design of Linear Quadratic state Regulator [15] is designed for controlling the
flight of TRMS. While designing LQR all the states are assumed to be present. Since number of
output is less than number of states, so all the states of system cannot be measured directly. For
this Zhuoyi Chen has proposed a technique of designing Kalman Filter [19]. Kalman Filter is
used for estimating states of system depending upon input-output combination. The technique of
designing LQG compensator which is combination of LQR and Kalman Filter is proposed by
R.N. Paschall in [21]. While designing LQR controller, the parameter of LQR, i.e. state and
control weighting matrices are chosen by trial and error method. This involves maintaining a
trade-off between minimizing the control effort and improving the transient response. Thus
optimization technique is used for optimizing the LQR parameter. Several optimization
techniques like Genetic Algorithm proposed by Subhojit Ghosh in [6] can be used for
optimization. But due to its limitation another optimization technique proposed by Shiva
Boroujeny Gholami in Active noise control using bacterial foraging optimization algorithm [9].
BFO algorithm is used for optimization the state and control weighting matrices.
1.3 Objective
The main objective of designing a controller for Twin Rotor MIMO system is to provide a
platform through which flight of Helicopter can be controlled. The mathematical model of
TRMS is non-linear so the first objective of this work is to linearize the non-linear model linear.
The next objective is to design a optimal controller for TRMS system, which can control the
response of system. So Linear Quadratic Regulator (LQR) is designed, which will control the
response of system. The state of system is estimated by using Kalman Filter and it is combined
with Linear Quadratic Regulator resulting in a Linear Quadratic Gaussian (LQG) controller.
Bacterial foraging optimization (BFO) technique is used for optimizing the system.
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Chapter 1
Introduction
1.4 TRMS Experimental set-up
Aerodynamic model of TRMS is shown in Figure-1.1. It consists of two propellers which are
perpendicular to each other and joined by a beam pivoted on its base. The system can rotate
freely in both vertical and horizontal direction. Both propellers are driven by DC motor and by
changing the voltage supplied to beam, rotational speed of propellers can be controlled. For
balancing the beam in steady state, counterweight is connected to the system. Both propellers are
shielded so that the environmental effects can be minimized. The complete unit is attached to the
tower which ensures safe helicopter control experiments. The electrical unit is placed under the
tower which is responsible for communication between TRMS and PC. The electrical unit is
responsible for transfer of measured signal by sensors to PC and transfer of control signal via I/O
card.
Figure-1.1 TRMS mechanical unit
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Chapter 1
Introduction
Main rotor is responsible for controlling the flight of TRMS in vertical direction and Tail rotor is
responsible for controlling the flight of TRMS in horizontal direction. There is cross-coupling
between Main and Tail rotor.
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Chapter 2
TRMS MODEL
TRMS MODEL
2.1 TRMS Mathematical model
Figure-2.1 TRMS Phenomenological model
The mathematical model derived from phenomenological model shown in Figure-2.1 is nonlinear in nature that means at least one of the states (rotor current or position) is an argument of
non-linear function. In order to design the controller for controlling the flight of TRMS, the
mathematical model should be linearized.
According to model represented in Figure-2.1, the non-linear mathematical model of TRMS can
be represented as [12]-
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Chapter 2
TRMS MODEL
Mathematical equation in vertical plane is given as(
)
(
)
(
)
(
)
(
)
(
)
where
( )
(
)
(
)
(
(
)
)
The motor and the electrical control circuit is approximated as a first order transfer function, thus
the rotor momentum in Laplace domain is described as(
)
(
)
(
)
(
)
(
)
Mathematical equation in horizontal plane is given as(
)
where
(
)
(
(
(
)
)
)
(
)
(
)
Rotor momentum in Laplace domain is given as-
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Chapter 2
TRMS MODEL
The model parameters used in above (
)
(
) equations are chosen experimentally, which
makes the TRMS nonlinear model a semi-phenomenological model.
The boundary for the control signal is set to [ -2.5 to +2.5].
The following table gives the approximate value of parameter- [13].
Table- 2.1 TRMS system parameters
Parameter
Value
6.8*10-2 kg.m2
2*10-2 kg.m2
0.0135
0.0924
0.02
0.09
0.32 N-m
6*10-3 N-m-s/rad
1*10-3 N-m-s2/rad
1*10-1 N-m-s/rad
1*10-2 N-m-s2/rad
0.05 s/rad
1.1
0.8
1.1
1
1
1
2
3.5
-0.2
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Chapter 2
TRMS MODEL
2.2 Linearized model
The mathematical model given in equation (
)
(
) are non-linear and in order to design
controller for system, the model should be linearized. The first step in linearization technique
[14-15] is to find equilibrium point.
Equations (
)
(
) are combined to represent alternate model of TRMS. The alternate
model is given as-
(
(
(
)
(
)
(
)
)
(
(
((
(
)
)
)
)
(
)
(
))
(
)
(
)
(
)
(
)
(
)
)
Now let us assume -
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Chapter 2
Equations (
TRMS MODEL
)
(
(
) can be represented with state space variable as-
)
( )
(
( )
( )
(
(
)
)
( ))
(
)
(
)
(
)
(
)
(
)
(
)
(
)
Now Taylor series is applied to find equilibrium point. For this make all the derivative term of
equations (
)
(
) equal to zero and find equilibrium point, take
.
Thus equilibrium point will be-
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Chapter 2
TRMS MODEL
The non-linear equations (
)
(
) can be represented in state space form given as –
̇
(
)
(
)
where A, B, C can be found by applying Jacobean matrix method. Thus A, B, C are given as –
0


0


0

A
0

0

 4.70588

0

0
0
0
0
0
0
1
0  0.909
0
0
0
0

0
0
1
0
0
0
0 0.218181 0  0.5
0
0

0 1.358823 0
0
 0.088235 0
0
0
4.5  50
5
0
0
0
 0
 0
0 

 1
0


B 0
0.8
 0.35 0 


0
 0
 0
0 

0
0
0
1
1 0 0 0 0 0 0
C

0 1 0 0 0 0 0
By using A, B, C matrix TRMS system can be represented in state space form by using equation
(
) and (
).
After representing the system in state space form, the next approach is to design controller for
the system to achieve desired output.
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Chapter 3
Controller Design for TRMS
CONTROLLER DESIGN FOR TRMS
3.1 Controllers
Controller is a device, in the form of analogue circuits or digital circuits which monitors and
alters the parameter of system to attain desired output. Controllers are basically used if the
system does not meet desired performance specification, i.e. both stability and accuracy.
Controllers can be connected either in series with plant or parallel to the plant depending upon
requirement.
A simple feedback control along with controller is shown in Figure – 3.1.
Figure-3.1 Feedback control Loop
As shown in Figure-3.1 error signal ‘e’ is generated, which is difference between reference
signal ‘r’ and output signal ‘y’. The error signal decides the magnitude by which output signal
deviates from reference value. Depending upon error signal value parameter of controller ‘C’
will get changed and control input ‘u’ is applied to plant which will give satisfactory output.
For a plant with multiple input and multiple output, it requires multiple controllers. If the system
is SISO system with single input and single output than it requires single controller for
controlling purpose. Depending on the set-up of the physical (or non-physical) system, adjusting
the system's input variable (assuming it is MIMO) will affect the operating parameter, otherwise
known as the controlled output variable. The notion of controllers can be extended to more
complex systems. Natural systems and human made systems both requires controller for proper
operation.
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Chapter 3
Controller Design for TRMS
3.2 Types of Controllers
There are various types of controller which can be used for improving performance specification
of system. Basically all the controllers can be broadly classified in two categories, feedback and
feed forward controller. The input to a feedback controller is the same as what it is trying to
control - the controlled variable is "feedback" into the controller. However, feedback control
usually results in intermediate periods where the controlled variable is not at the desired setpoint. Feed-forward control can avoid the slowness of feedback control. By using feed-forward
control, the disturbances are measured and accounted for before they have time to affect the
system.
Controllers can be broadly classified asa) Proportional controller
b) Proportional – integral controller
c) Proportional – derivative controller
d) Proportional – integral- derivative controller
e) Pole placement controller
f) Optimal controller
The first four controllers are feedback controller and the fifth one is full state feedback
controller. Pole placement controller is a feedback controller which is used for placing the closed
loop poles to desired location in s plane. But pole placement can be used only for SISO system.
For MIMO system, problem of over-abundance of design parameters are faced. For such
systems, we did not know how to determine all the design parameters, because only a limited
number of them could be found from the closed loop pole locations. Optimal control provides the
technique by which all the design parameters can be found even for multi-input, multi-output
system. Also in pole placement technique some trial and error procedure with pole locations was
required because we don’t know priori which pole location will give satisfactory performance.
Optimal control allows us to directly formulate the performance objective of a control system
and get desired response. Also optimal control minimizes the time and cost required for
designing the system.
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Chapter 3
Controller Design for TRMS
3.3 Linear Quadratic Regulator (LQR)
3.3.1 Overview
The principle of optimal control is basically concerned with operating a dynamic system at
minimal cost. The system whose dynamics are given by set of linear differential equations and
cost by Quadratic function is called Linear Quadratic (LQ) problem. The setting of a controller
which governs either a machine or process are basically found by mathematical algorithm that
minimizes cost function consist of weighing factors. Mathematical algorithms are basically
objective function that must be minimized in design process.
The cost objective function for optimal control must be time integral of sum of control energy
and transient energy expressed as function of time. If system transient energy can be defined as
total energy of system when it is undergoing transient response, then control system should have
transient energy which decays to zero quickly. Maximum overshoot is defined by maximum
value of transient energy and time taken by transient response to decay to zero represent the
settling time. Thus acceptable value of settling time and maximum overshoot can be specified by
including transient energy in objective function. In same way, control energy should also be
included in objective function to minimize the control energy of system. Figure-3.2 shows the
block diagram of plant along with Linear Quadratic Regulator (LQR) [15-17]. Here output of
plant is controlled by varying the gain K of Linear Quadratic Regulator.
Figure-3.2 Block diagram of Linear Quadratic Regulator
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Chapter 3
Controller Design for TRMS
Consider a linear plant given by the following state equations –
̇( )
()
()
(
Here time-varying plant in equation (
)
) are taken because optimal control problem is
formulated for time-varying system. Control input vector for full state feedback regulator of the
plant is given by –
()
() ()
(
The control input given by equation (
energy is given by
)
) is linear, because the plant is also linear. The control
( ) ( ) ( ), where ( ) is a square and symmetric matrix called control
cost matrix. The expression for control energy is in quadratic form because the function
( ) ( ) ( ) contains quadratic function of ( ). The transient energy can be expressed as
( ) ( ) ( ), where
( ) is square and symmetric matrix called state weighing matrix. Thus
objective function can be represented as –
(
)
∫ (
where t and
( ) ( ) ( )
( ) ( ) ( ))
(
)
are initial and final time values respectively, where controlling process begins at
and ends at
. The main objective of optimal control problem is to find matrix ( )
such that objective function (
) given in equation (
) is minimized. The minimization
process is done in a way such that solution of plant’s state-equation (
( ). The main objective of design is to bring ( ) to zero at time
) is given by state vector
.
3.3.2 Estimating Optimal control gain K
The closed loop state equation is given by substituting equation (
) into equation (
), which
is given as –
̇( )
̇( )
( )) ( )
(
()
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(
)
(
)
Page 15
Chapter 3
(
) ( )
(
where
( )) is closed loop state dynamics matrix. The solution of equation (
(
where
given as –
()
Controller Design for TRMS
(
) is state transition matrix of closed loop system given by equation (
Equation (
) is
)
).
) indicates at any time ‘t’ state ( ) can be obtained by post multiplying the state at
some initial time, ( ) with
(
). On substituting equation (
) into equation (
), the
expression for objective function is given as –
(
)
()
∫
Equation (
(
(
)( ( )
( ) ( ) ( ))
(
) ()
(
)
(
)
(
)
) can be written as –
)
() (
) ()
where
(
)
∫
(
)( ( )
( ) ( ) ( ))
(
)
Linear optimal regulator problem given by equation (
)
(
) also called Linear Quadratic
Regulator problem because the objective function shown in equation (
function of initial state. By using the equation (
(
)
∫
( )( ( )
)
( )( ( )
)
( ̇ ( ))
(
( ) ( ) ( )) ( )
) ()
(
)
(
)
(
)
) partially with respect to time‘t’, we get –
Also partial differentiating equation (
(
), it is given as –
( ) ( ) ( )) ( )
Now on differentiating equation (
(
) and (
) is a quadratic
) with respect to‘t’ we get –
( )(
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(
)
) ()
() (
) ̇( )
Page 16
Chapter 3
Controller Design for TRMS
On combining equation (
(
)
( )(
() (
Equating equations (
is given as –
(
)
) and equation (
(
)
) and (
() (
)
(
)
) we get –
(
)
( )) ( )
()
()
)
) following matrix differential equation is obtained, which
)
()
( ()
( ) ( ) ( ))
The matrix Riccati equation for finite time duration is given by equation(
By solving the Riccati equation (
(
(
)
(
)
).
) optimal feedback gain matrix ( ) is given by –
()
There are large number of control problem where control time interval is infinite. By considering
infinite time interval optimal control problem gets simplified. The quadratic objective function
for infinite final time is given as –
()
where
∫ (
( ) ( ) ( )
( ) ( ) ( ))
(
)
( ) is the objective function of the optimal control problem for infinite time. For
infinite final time,
(
) is either constant or does not gives any energy to any limit. Thus
Thus Riccati equation for infinite final time is given by –
()
Since equation (
()
(
) is an algebraic equation, thus it is called Algebraic Riccati equation. The
condition for the solution of Riccati equation (
) to exist is either the system is
asymptotically stable or the system is controllable and observable with output
where
()
)
( ) ( ) and
()
( ) ( ),
( ) is positive definite matrix and symmetric. If the system is
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Chapter 3
Controller Design for TRMS
stabilizable and output ( )
exists with ( )
( ) ( ) is detectable then also solution to Riccati equation will
( ) ( ) and ( ) is positive definite matrix and symmetric.
In this system positive definite matrix
( ) and positive semi definite matrix
( ) are time
independent and are randomly chosen. While designing LQR value of Q and R are varied until
the output of system decays to zero at steady state.
For the present work Q and R matrix are given as –
1 0
0 0.1

0 0

Q  0 0
0 0

0 0
0 0

0 0
0 0
1 0
0 1
0 0
0 0
0 0
0
0
0
0 
0
0
0

0
0
0
0.001 0
0

0
0.4 0 
0
0 0.4
0
0
And
0.0395 0
R
1
 0
In matrix Q, the element
represents cross-coupling coefficient which needs to be
minimized, so its weight is taken to be minimum.
By applying LQR technique on system by using Q and R given above we calculate the optimal
control gain K of system.
The optimal control gain calculated is given as –
22.7462 1.5716  7.2280 3.9397  52.6533 0.1295 3.6697
K

 2.5257 0.0494  1.0458 0.6041  3.3438  0.6577 0.1323
Now by using value of optimal control gain K in equation (
), optimal control input ‘u’ is
calculated. With the control input ‘u’, output of TRMS is regulated and response decays to zero
at steady state. Here optimal gain K is obtained by randomly varying Q and R matrix. This
involves maintaining a trade-off between minimizing the control effort and improving the
transient response. To overcome this, a optimization technique is used to optimize the value of Q
and R. Thus in this work, optimization algorithm, i.e. BFO algorithm is used for optimizing the
state and control weighting matrices.
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Chapter 3
Controller Design for TRMS
3.3.3 Linear Quadratic Tracking Problem
In Linear Quadratic Regulator problem the output of system decays to zero at steady state. In this
case no reference signal is applied to system. But if reference signal is applied then Linear
Quadratic Regulator problem becomes Linear Quadratic Tracking (LQT) problem. In Linear
Quadratic Tracking problem reference signal is applied to the system and output of system tracks
the reference signal.
Consider linear, time invariant plant given by equation(
). Now our aim is to design a tracking
system for plant (
( ), which is solution of equation –
̇()
) if desired state vector is given by
() ()
(
The desired state dynamics is given by homogeneous state equation, because
by the input signal ( ). Now by solving equations (
tracking error ( )
̇( )
()
(
()
) and (
)
( ) is unaffected
) we get the state equation for
( ).
) ()
()
(
)
The main objective is to find control input ( ), which makes the tracking error given by ( )
equal to zero in steady state. To achieve this by optimal control, our first aim is to find objective
function which is to be minimized. In tracking problem control input will depend on state vector
( ). Now combining equations (
[ ()
()
where
) and (
) and taking the state vector as
()
( ) ] , thus control input is given by following linear control law –
() ()
( )[ ( )
() ]
( ) is combined feedback gain matrix. The equations (
(
) and (
)
) can be
written as following combined state equation –
̇()
()
A
where A c t   
0
()
A d t   At 
A d t  
 Bt 
, B c t   

 0 
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(
)
(
)
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Chapter 3
Controller Design for TRMS
So now objective function can be expressed as –
(
)
∫ (
( )
( ) ( )
( ) ( ) ( ))
In tracking error problem final time
vector
(
)
cannot be taken as infinite, because the desired state
( ) will not go to zero in steady state, thus non-zero control input ( ) will be required
in steady state. The system represented by equation (
state dynamics given by equation (
represented by equation (
) is uncontrollable, because desired
) is unaffected by input
( ). Since the system
) is uncontrollable, thus unique solution of system is not
guaranteed. Thus for having a guaranteed positive definite and unique solution of the optimal
control problem, we have to exclude the uncontrollable desired state vector from objective
function by choosing combined state weighting matrix as follows –
Qt  0
Q c t   
0
 0
Thus changed objective function will be –
(
)
∫ ( ( ) ( ) ( )
Here in equation (
( ) ( ) ( ))
(
) ( ) is given by equation (
). Thus for existence of unique and
positive definite solution of optimal control problem, we choose
where
()
( ) and
( ) to be positive
( ) is given by –
semi definite and definite respectively. The optimal gain
()
)
()
(
)
(
)
(
)
is solution of the following equation –
()
()
()
is symmetric matrix which can be represented as –
M
MC   1
M 3
M2 
M 4 
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Chapter 3
where
(
and
) and (
()
[
Controller Design for TRMS
corresponds to plant and desired state dynamics. Now substitute equation
) into equation (
()
), the optimal feedback gain matrix is given as –
()
]
(
)
(
)
(
)
(
)
and optimal control input is given by –
()
()
()
Now substitute equation (
()
()
) and (
) into equation (
()
(
Optimal matrix
(
)
()
(
)
can be obtained by solving equation (
). Thus equation (
) we get –
) and this value is used in equation
) can be written as –
(
)
()
Where
Most of the time it is required to track a constant desired state vector given as,
corresponds to
(
) and (
. Thus both
and
[
()
]
()
(
)
, which
are constants in the steady state. Thus equations
) is the algebraic Riccati equation. From equation (
Now substituting equation (
()
)
) can be written as –
()
The equation (
()
(
()
) in equation (
()
[
]
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(
)
(
)
(
)
(
)
) we get –
) we get –
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Chapter 3
Controller Design for TRMS
Substituting equation (
̇( )
()
) into equation (
()
*
Thus from equation (
(
where
value of
()
(
+
)
) it is clear that tracking error can become zero in the steady state for
any non-zero constant desired state
()
)
) we get –
()
. The final optimal control input is given as –
()
(
)
( ) is feed forward gain matrix which will make ( ) zero in steady state for some
. By substituting equation (
) into equation (
), the state equation for tracking
is calculated as –
̇( )
()
[
( )]
(
)
Thus by using the same value of positive semi definite matrix Q and positive definite matrix R as
used in Linear Quadratic Regulator problem, optimal control gain K is calculated. Now by taking
specific reference value
optimal control input is calculated by using equation (
). In this
particular TRMS system there are two output i.e. pitch and yaw. So two reference signal are
taken, which are –
Thus output of TRMS, pitch will track
and yaw will track
at steady state.
3.4 Kalman Filter
3.4.1 Overview
The Kalman Filter, which is also known as Linear Quadratic Estimation (LQE), is basically an
algorithm which uses series of measurements observed over time, comprises of noise and other
inaccuracies and it produces estimates of unknown variables that seems to be more precise than
those based on single measurement. Kalman Filter [18-20] has large number of application in
technology. Some of applications are navigation and control of vehicles,
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Chapter 3
Controller Design for TRMS
guidance. Kalman Filter is widely applied concept in time analysis used in fields like signal
processing and econometrics.
3.4.2 Requirement of Kalman Filter
TRMS model is a stochastic system because due to the presence of process noise and
measurement noise, it cannot be modeled by using deterministic model. Thus a noisy plant is a
stochastic system, which can be modeled by passing white noise through appropriate linear
system. Consider a linear plant –
̇( )
()
()
() ()
(
)
()
()
()
()
(
)
where ( ) is measurement noise vector and
( ) is process noise vector and this may arise due
to modelling error such as neglecting high frequency and nonlinear dynamics. The correlation
matrices of non-stationary white noise,
(
)
(
)
where
() (
() (
( ) and ( ), and can be expressed as –
)
)
( ) and ( ) are time-varying power spectral density matrices of
(
)
(
)
( ) and ( ).
while designing the control system for stochastic plant, we cannot depend on full state feedback,
because state vector ( ) cannot be predicted. Thus for stochastic plant observer is for predicting
the state vector based upon measurement of output ( ) given in equation (
) and input ( ).
State observer cannot be used, because it would not take into account power spectral density of
process noise and measurement noise. And also there is designing problem for multi-input multioutput plants, thus it is used only for single output case. Thus we require the observer that takes
into account process and measurement noise into consideration and estimate the state vector ( )
of plant based upon statistical value of vector output and plant. Such observer is called Kalman
Filter.
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Chapter 3
Controller Design for TRMS
3.4.3 Mathematical model of Kalman Filter
Kalman filter which is an optimal observer, minimizes the statistical error of estimation error,
()
()
( ), where
( ) is estimated state vector. The state equation of Kalman Filter
is given as –
̇()
()
()
( )[ ( )
()
( )]
(
)
where L is Kalman Filter gain matrix. As optimal regulator minimizes the objective function
comprises of transient and steady state response and control energy, in the same way Kalman
Filter minimizes covariance of estimation error,
(
) from (
̇()
[
(
)
[ ( ) ( )]. Subtracting equation
) we get –
() ] ()
() ()
() ()
(
Thus after minimizing the covariance of estimation error
results for optimal covariance matrix,
(
)
), algebraic Riccati equation
(
)
(
)
(
)
(
)
where,
() ()
()
()
() ()
()
()
()
and ( ) is cross spectral density matrix between
( ) and ( ).
Kalman Filter gain matrix is given as –
where
is calculated by solving algebraic Riccati equation (
). The necessary and
sufficient condition for existence of a positive and semi-definite solution for L is that, [
stabilizable and [
] is
] is detectable.
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Chapter 3
Controller Design for TRMS
3.4.4 Design of Kalman Filter
While designing the Kalman Filter, the process noise spectral density matrix V and measurement
noise spectral density matrix Z are randomly chosen. These density matrices are varied until we
get desired response. The condition for checking the desired response is that the ratio between
the elements of the returned optimal covariance matrix of estimation error P and covariance of
simulated estimation error
( ) should be same.
For system TRMS the value of process spectral density V and measurement noise spectral
density matrix Z are taken as –
and
where, F=B
Thus the Kalman gain L, Returned optimal covariance matrix of estimation error P, eigen value
of Kalman Filte E of TRMS system is given as –
 1.0412
 .0.0534

 0.1080

L   0.0047
  0.0839

 0.0435
 0.2505

 0.0010
 0.0001

 0.0001

P 0
 0.0001

 0
 0.0003

 0.0534
7.1475 
 0.0946

0.0375 
 0.7523

 0.6881
25.0448 
 0.0001
 3.7371


  1.8623  3.1630i 


  1.8623  3.1630i 


E   0.6253  1.9775i 
  0.6253  1.9775i 


 0.9830




 0.9908


 0.0001
0.0003 
0.0071  0.0001
0
 0.0008  0.0007 0.0250 
 0.0001 0.0005
0
0
0.0002  0.0008

0
0
0.0005
0
0
0.0003 
 0.0008
0
0
0.0004  0.0001  0.0057

 0.0007 0.0002
0
 0.0001 0.0043  0.0051
0.0250  0.0008 0.0003  0.0057  0.0051 0.1378 
0.0001
0
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0
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Chapter 3
Controller Design for TRMS
3.5 Linear Quadratic Gaussian (LQG)
3.5.1 Overview
Linear Quadratic Gaussian (LQG) [21-23] controller is an optimal controller. It deals with linear
system with additive white Gaussian noise and having incomplete state information and
undergoing control to quadratic cost. The solution of LQG control problem is unique and
consists of Linear dynamic feedback control law that can be easily implemented. Linear
Quadratic Gaussian controller is combination of Kalman Filter and Linear Quadratic Regulator.
LQG works on separation principle, it means that Kalman Filter and Linear Quadratic Regulator
can be designed and computed independently.
LQG controller application can be applied to Linear time invariant system along with Linear
time varying system. Here in this work Linear time invariant system is being considered.
Designing of system with LQG controller does not guarantee Robustness of system. The
robustness of system should be checked once the LQG controller has been designed. Figure-3.3
shows block diagram of LQG controller.
Figure-3.3 Block diagram of LQG controller along with plant
Here in Figure-3.3 it can be seen that Linear Quadratic Gaussian (LQG) controller composed of
Kalman Filter (which will estimate all the state of system), followed by Linear Quadratic
Regulator (LQR) (which is responsible for controlling the response of system). Along with
control input ‘u’ process noise ‘w’ is also applied to system. External white Gaussian noise is
added to plant because plant is stochastic with some unknown noise. Measurement noise ‘v’ is
also added to system and finally we get response as ‘y’.
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Chapter 3
Controller Design for TRMS
3.5.2 Requirement of LQG compensator
For TRMS system, while designing Linear Quadratic Regulator or Linear Quadratic Tracking
controller, we have assumed full state feedback. It means we have assumed that all the states of
system are available and can be measured directly. But since in our TRMS system, number of
output is less than number of states, thus all the state of system cannot be measured directly.
Thus for that type of system, an observer is designed, that will estimate all the state of system
based on input and output combination of system. Thus Kalman Filter is designed which will
estimate all the state of system, i.e. seven states from two output measurement and it is combined
with Linear Quadratic Regulator and combination will give Linear Quadratic Gaussian
controller.
3.5.3 Steps in the Design of LQG Compensator
a. Design optimal regulator (LQG) for linear plant assuming full state feedback, with a
quadratic objective function. Here we have assumed that all the state of system can be
measured directly. The regulator will generate the control input ( ) based upon state
vector ( ).
b. Design a Kalman Filter for the linear plant with control input
( ), measured output ( )
and combined with white noise ( ) and ( ). The Kalman Filter will give optimal
estimate of state vector
( ). The Kalman Filter designed in this work is full order
Kalman Filter.
c. Now combine Linear Quadratic Regulator (LQR) with Kalman Filter and the
combination will give Linear Quadratic Gaussian (LQG) controller, that will be
responsible for controlling the response of plant. This compensator will generate control
input ( ) based upon state estimated by Kalman Filter.
3.5.4 Compensator for TRMS system
The state-space representation of optimal compensator (LQG), for regulating the noisy plant with
state-space model is given by following state and output equation –
̇()
(
) ()
()
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(
)
Page 27
Chapter 3
()
Controller Design for TRMS
()
(
)
(
)
(
)
where L and K are Kalman Filter and optimal regulator gain matrices respectively.
Here optimal regulator gain matrix K is obtained by using following command –
(
)
where
1 0
0 0.1

0 0

Q  0 0
0 0

0 0
0 0

0
0 0
0
0 0 
1 0
0
0 0

0 1
0
0 0
0 0 100000 0 0 

0 0
0
10 0 
0 0
0
0 0.1
0 0
0
0
0.0395 0
R
1
 0
0.0008 0.0002  0.0004 0.0007  1.5892 0.0034 0.0003
K

0.0039 0.0003  0.0010 0.0021  0.0030  0.0006 0.0007
s
Kalman Filter gain parameter L can be obtained by using –
[
]
(
)
The value of L,P,E is given in section 3.4.5.
The Eigen values of Linear Quadratic Gaussian (LQG) compensator, consists of Eigen values of
Linear Quadratic Regulator (LQR) and Eigen values of Kalman Filter. For system to be stable
Eigen values of Linear Quadratic Gaussian (LQG) compensator should be on left hand side of
imaginary axis. Ideally response of Linear Quadratic Gaussian (LQG) compensator should be
same that of Linear Quadratic Regulator (LQR). For that Eigen values Linear Quadratic
Regulator (LQR) should be dominating compared to Eigen values of Kalman Filter. It means that
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Chapter 3
Controller Design for TRMS
Eigen value of Kalman Filter should be far from imaginary axis as compared to Eigen values of
Linear Quadratic Regulator (LQG).
As Kalman Filter does not require control input signal, thus its Eigen values can be shifted
deeper into left half plane without requirement of large control input and it can be achieved free
of cost. But in some cases it is not possible to shift the Eigen value of Kalman Filter deeper into
left half plane by simply varying the noise spectral densities, so in that case proper choice of
Kalman Filter spectral densities will yield best recovery of full state feedback dynamics.
3.6 Results and Discussion
In this work two reference inputs signal
and
are applied for tracking the path
of TRMS as shown in Fig.3.4. The output of TRMS will track this corresponding reference
signal. The value of reference signal
and
can also be changed, depending upon
requirement, i.e. it can be either sinusoidal or it can be step or ramp input.
Figure-3.4 Reference signal applied to TRMS
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Chapter 3
Controller Design for TRMS
Now firstly TRMS response is controlled by Linear Quadratic Regulator. The output of TRMS in
2-DOF using LQR controller is shown in Fig.3.5.
Figure-3.5 Output of TRMS using LQR
The output of TRMS, i.e. pitch is able to track the desired reference
state error and yaw is able to track the desired reference
with no steady
with 4.6% steady state error.
The response of Yaw shows large Maximum peak overshoot because of cross coupling nature
between vertical plane and horizontal plane.
In using Linear Quadratic Regulator (LQR) as controller it has been assumed that all states are
directly measurable. But here in our case number of output is less than number of states, i.e.
number of output is two and number of states is seven. So it is not possible to measure all the
state of system directly. Thus for measuring all the state of system Kalman Filter is used.
Kalman Filter basically describes the states of plant. It shows the variation of plant parameter
with time. In this work full state Kalman Filter has been designed, which estimates all the state
of system depending upon input-output combination. The Fig.3.6 shows variation of TRMS state
using Kalman Filter.
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Chapter 3
Controller Design for TRMS
Figure-3.6 Output of Kalman Filter
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Chapter 3
Controller Design for TRMS
Fig.3.6 shows that some of the states of system, including one output i.e. yaw is unstable. Thus
with one output unstable, the complete plant is considered to be unstable, it means that output of
is system unbounded for bounded input. Thus to make the plant stable Linear Quadratic
Regulator (LQG) controller is combined with Kalman filter. Thus in this work, Linear Quadratic
Gaussian (LQG) controller is used for controlling the performance of TRMS. For arriving at the
simulated results of the standard LQG compensator, different combinations of spectral densities
are tried. The answers reported here correspond to the best combination. The Fig.3.7 shows the
output of TRMS using LQG controller.
Figure-3.7 Output of TRMS using LQG
In the Fig.3.7, there is no reference signal applied to system, so it is a regulating problem whose
steady state value is zero. Here, output of Linear Quadratic Regulator (LQR) is compared with
output of Linear Quadratic Gaussian (LQG) controller. As shown in Fig.3.7, the response of
LQR and LQG overlap to each other, and finally there steady state value is zero.
Fig.3.7 shows that Twin Rotor MIMO system can be made to give stable and accurate response
by using Linear Quadratic Gaussian (LQG) controller.
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Chapter 4
Bacterial Optimization Algorithm Based Controller Design
BACTERIAL OPTIMIZATION ALGORITHM BASED
CONTROLLER DESIGN
4.1 Bacterial Foraging Optimization Algorithm
4.1.1 Overview
Optimization techniques are used to minimize the input effort required and to maximize the
desired benefit. For over the last five decades, optimization algorithms like Genetic Algorithms
(GAs), Evolutionary Programming (EP), Evolutionary Strategies (ES), which draw their
inspiration from evolution and natural genetics, have been dominating the realm of optimization
algorithms. Genetic Algorithm has certain limitations like, it gives poor fitness function which
will generate bad chromosome blocks and the optimization response time obtained may not be
constant. In addition to this, it is not sure that this optimization technique will give global
optimum value. Thus BFO Algorithm is used as an optimization technique which overcomes all
limitations of GA [6-8]. Bacterial foraging optimization algorithm (BFO) is widely accepted
optimization algorithm for distributed optimization and control. BFO algorithm works on the
principle of behaviour of Escherichia coli bacteria. BFO algorithm is capable of optimizing realtime problems, which arise in several application domains. Bacterial foraging optimization
algorithm is based on nature inspired optimization algorithm. The key idea behind BFOA is
grouping foraging strategy of Escherichia coli bacteria in multi optimal function optimization.
Bacterial search takes place in a manner that it maximizes the energy intake per unit time. Each
bacterium communicates with other bacteria by sending signals.
4.1.2 Steps involved in BFO
The complete Bacterial foraging optimization algorithm is divided into 4 basic steps. They are –
a) Chemotaxis – In this step, movement of an Escherichia coli bacterium through
swimming and tumbling via flagella is simulated. Escherichia coli bacteria basically can
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Chapter 4
Bacterial Optimization Algorithm Based Controller Design
move in two ways. The bacteria can swim for a period of time in one direction or may
tumble and alternate between these two modes for entire life time.
b) Swarming – In this step, some of the bacterium will attract each other and move with
higher density.
c) Reproduction – In this step, the least healthy bacteria (bacteria with highest cost
function) will die, and while other healthier bacteria will split into two and will take their
place.
d) Elimination and Dispersal – In this step, there is sudden or gradual changes takes place
due to which some of the bacteria die due to various reasons, like, rise of temperature
may kill bacteria that are in region with high concentration of nutrient gradients. In this,
events take place in such a way that either group of bacteria gets killed or dispersed to
new position. To simulate this in BFOA some amount of bacteria are liquidated with
small probability at random time.
4.1.3 BFO Algorithm
For a given objective function, BFO algorithm involves the execution of the following steps –
1) Let S be the number of bacteria used for the searching algorithm. Here each bacterium
represents a string of filter weights.
2) The number of parameters to be optimized is represented as ‘p’.
3) Swimming length represented as
, after each tumbling of bacteria it is taken in
Chemotaxis loop.
4) Number of iterations which is to be undertaken in Chemotaxis loop is given as
5) The maximum amount of reproduction is given as
6)
.
.
represent the maximum amount of Elimination dispersal iteration that bacteria
undergoes.
7)
represent the probability by which elimination dispersal process will take place.
8) P gives the position of each bacterium in bacteria population.
9)
( ) represent the size of step taken for each bacterium in random direction.
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Chapter 4
Bacterial Optimization Algorithm Based Controller Design
4.1.4 Problem Formulation
In this section, the controller design process has been framed as an optimization problem, with
controller parameters as the process variables. Modelling of bacterial population is carried out
through Chemotaxis, Reproduction and Elimination and Dispersal steps.
Let us assume j=k=l=0, where j,k,l represents Chemotaxis, Reproduction, Elimination-dispersal
iteration respectively. By updating
, P is automatically updated.
1) Elimination dispersal iterations:
2) Reproduction iterations: k=k+1
3) Chemotaxis iterations: j=j+1
a) For i=1,2,3,…S, perform the bacterium iterations as follows –
b) Determine cost function –
(
)
where
∫ ((
and
)
(
)
(
)
(
(
) )
are TRMS outputs, i.e. pitch and yaw, and
and
)
are control
inputs.
(
c) Let
), save value of cost function, because we may find any better
value.
d) Tumble. Now generate a random vector
( ), such that each element of
( ),
m=1,2….p, is between [-1,1].
e) The next position is given by –
(
)
(
)
()
()
(
() ()
)
This will give the step of size ( ) for bacterium i, in direction of tumble.
f) Compute the next cost function (
) corresponding to position
(
).
g) Swimming
I.
Let counter for swim length is defined as m=0
II.
While
i.
following condition takes place –
Let m=m+1
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Chapter 4
ii.
Bacterial Optimization Algorithm Based Controller Design
Now check cost function condition that if (
(
Use
iv.
Else if
then
=
), this will give another step of size ( ) in the direction given in
equation (
iii.
)<
).
(
) to find new cost function (
).
, while statement will terminate.
h) Now switch to next bacterium (
) and check if
then go to step ‘b’ to
process the next bacterium.
4) Now check if
switch to step 3. Now since the life of bacteria is not over hence
continue Chemotaxis iteration.
5) Reproduction
a) For the given value of k and l and i=1,2,…S , let
((
Equation (
order of
b)
)) for j = 1,2,….
(
)
) gives the health of bacteria i. now sort the bacteria in ascending
.
⁄ bacteria which have highest
will die out and rest of
will split out
and takes their place.
6) If
, switch to step 2, because the number of given reproduction step has not yet
completed, so start the Chemotaxis step.
7) Now Elimination-dispersal: for each i=1,2… S with probability
elimination and
dispersal of each bacterium takes place. Thus it keeps the number of total bacteria in
population constant. To perform this operation simply place one to random location in
optimization domain.
For the present work, the parameter
consists of the tuning parameters for the LQR
compensator i.e., the elements of the matrices Q and R.
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Chapter 4
Bacterial Optimization Algorithm Based Controller Design
4.2 Results and Discussion
For the present work, the application of Bacterial foraging optimization algorithm (BFO), aims at
obtaining an optimal compensator for the TRMS system. For this, the positive semi-definite state
weighting matrix Q and positive definite control input weighting matrix R is given by –
1 0 0 0 0 0 0
0 q2 0 0 0 0 0


0 0 1 0 0 0 0 


Q  0 0 0 1 0 0 0 
0 0 0 0 1 0 0 


0 0 0 0 0 q6 0
0 0 0 0 0 0 1 


And
r1 0
R

 0 1
Here q2, q6, r1 are assumed to be varying and needs to be optimized by using Bacterial foraging
optimization algorithm (BFO). Values of q2, q6, r1 are given in Table-4.1.
Table-4.1
Parameter
Initial Value
Final Value
q2
0.05
0.1
q6
0.08
0.4
r1
0.12
0.395
Figure-4.1 depicts the iterative variation of cost function given by equation (4.1), cost function
decreases with number of iteration and attain minimum value of 1.1023e+5 at twelfth iteration.
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Chapter 4
Bacterial Optimization Algorithm Based Controller Design
Figure 4.1 Variation of cost function (
) with respect to number of iteration.
Figure-4.2 shows comparison of output of TRMS of BFO based LQG with standard LQG.
Figure-4.2 Comparison between output of TRMS using with and without BFOA
As shown in Figure-4.2, response of system i.e. pitch and yaw, after using Bacterial Foraging
Optimization (BFO) Algorithm is better than the standard LQG. Table-4.2 displays the
performance specification of LQG compensator with random selection of spectral densities
(standard LQG) along with BFO based selection.
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Chapter 4
Bacterial Optimization Algorithm Based Controller Design
Table – 4.2 Comparison of TRMS responses with and without using BFO
BFO based LQG
Standard LQG
Performance Specification
Pitch
Yaw
Pitch
Yaw
Rise Time
25.4113
0.1579
27.1669
0.1658
Settling Time
38.2387
46.8260
38.7667
46.8942
Maximum Peak Overshoot
0.0035
1.6464e+003
0.0034
1.8249e+003
57.4db
36db
54db
33.3db
Phase Margin
Gain Margin
As given in Table-4.2 the performance specification of system improves after applying Bacterial
foraging optimization algorithm (BFO). The response of system becomes faster with decrease in
rise time, system response settles to steady state value in less time and with decrease in
maximum peak overshoot, stability of system gets improved.
Phase margin and Gain margin of system is calculated by using the mat lab command,
A = a-b*k-l*c;
sys = c*inv(s*eye(7)-(A))*b;
bode (sys);
where a, b, c is plant matrix, k is LQR gain and l is Kalman gain.
The improvement in performance specification of system takes place because after using
Bacterial foraging optimization algorithm (BFO), we obtain optimal value of state and control
weighting matrix. However, this comes at the cost of high bandwidth, which ultimately leads to
low noise attenuation at high frequencies.
National Institute of Technology, Rourkela
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Chapter 5
Conclusions
CONCLUSIONS
5.1 Conclusions
In this work, a state feedback controller (LQG) based on noisy output measurement has been
designed for an experimental set-up (TRMS), that replicates the aircraft flight dynamics, i.e.
pitch and yaw. The simulation results corresponding the cases related to full state feedback and
observer (Kalman Filter) based feedback with state estimation reflects the appropriateness of the
proposed approach in meeting the desired specifications. In using Linear Quadratic Gaussian
(LQG) compensator for controlling the flight dynamics of TRMS, the elements of the positive
semi definite matrix ‘Q’ and positive definite matrix ‘R’ are randomly chosen. To overcome the
hit and trial approach, an evolutionary optimization technique i.e. Bacterial Foraging
Optimization (BFO) algorithm has been applied and their results has been compared with Linear
Quadratic Gaussian (LQG) controller result.
National Institute of Technology, Rourkela
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