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Page 1: Feedback Control Systems- HOzbay

Introduction to

Feedback Control Theory

Hitay �Ozbay

Department of Electrical Engineering

Ohio State University

Page 2: Feedback Control Systems- HOzbay
Page 3: Feedback Control Systems- HOzbay

Preface

This book is based on my lecture notes for a ten�week second course

on feedback control systems� In our department the �rst control course

is at the junior level� it covers the basic concepts such as dynamical

systems modeling� transfer functions� state space representations� block

diagram manipulations� stability� Routh�Hurwitz test� root locus� lead�

lag controllers� and pole placement via state feedback� In the second

course� �open to graduate and undergraduate students� we review these

topics brie�y and introduce the Nyquist stability test� basic loopshap�

ing� stability robustness �Kharitanov�s theorem and its extensions� as

well as H��based results� sensitivity minimization� time delay systems�

and parameterization of all stabilizing controllers for single input�single

output �SISO� stable plants� There are several textbooks containing

most of these topics� e�g� � �� ��� � ���� But apparently there are

not many books covering all of the above mentioned topics� A slightly

more advanced text that I would especially like to mention is Feed�

back Control Theory� by Doyle� Francis� and Tannenbaum� ���� It is

an excellent book on SISO H��based robust control� but it is lacking

signi�cant portions of the introductory material included in our cur�

riculum� I hope that the present book �lls this gap� which may exist in

other universities as well�

It is also possible to use this book to teach a course on feedback con�

trol� following a one�semester signals and systems course based on ���

��� or similar books dedicating a couple of chapters to control�related

topics� To teach a one�semester course from the book� Chapter ��

should be expanded with supplementary notes so that the state space

methods are covered more rigorously�

Page 4: Feedback Control Systems- HOzbay

Now a few words for the students� The exercise problems at the end

of each chapter may or may not be similar to the examples given in

the text� You should �rst try solving them by hand calculations� if you

think that a computer�based solution is the only way� then go ahead

and use Matlab� I assume that you are familiar with Matlab� for

those who are not� there are many introductory books� e�g�� ��� � � ����

Although it is not directly related to the present book� I would also

recommend ��� as a good reference on Matlab�based computing�

Despite our best e�orts� there may be errors in the book� Please

send your comments to� ozbay���osu�edu� I will post the corrections

on the web� http���eewww�eng�ohio�state�edu��ozbay�ifct�html�

Many people have contributed to the book directly or indirectly� I

would like to acknowledge the encouragement I received from my col�

leagues in the Department of Electrical Engineering at The Ohio State

University� in particular J� Cruz� H� Hemami� �U� �Ozg�uner� K� Passino�

L� Potter� V� Utkin� S� Yurkovich� and Y� Zheng� Special thanks to

A� Tannenbaum for his encouraging words about the potential value

of this book� Students who have taken my courses have helped signi�c�

antly with their questions and comments� Among them� R� Bhojani and

R� Thomas read parts of the latest manuscript and provided feedback�

My former PhD students T� Peery� O� Toker� and M� Zeren helped my

research� without them I would not have been able to allocate extra

time to prepare the supplementary class notes that eventually formed

the basis of this book� I would also like to acknowledge National Sci�

ence Foundation�s support of my current research� The most signi�cant

direct contribution to this book came from my wife �Ozlem� who was

always right next to me while I was writing� She read and criticized the

preliminary versions of the book� She also helped me with the Matlab

plots� Without her support� I could not have found the motivation to

complete this project�

Hitay �Ozbay

Columbus� May ����

Page 5: Feedback Control Systems- HOzbay

Dedication

To my wife� �Ozlem

Page 6: Feedback Control Systems- HOzbay
Page 7: Feedback Control Systems- HOzbay

Contents

� Introduction �

��� Feedback Control Systems � � � � � � � � � � � � � � � � � �

��� Mathematical Models � � � � � � � � � � � � � � � � � � � � �

� Modeling� Uncertainty� and Feedback �

��� Finite Dimensional LTI System Models � � � � � � � � � � �

��� In�nite Dimensional LTI System Models � � � � � � � � � ��

����� A Flexible Beam � � � � � � � � � � � � � � � � � � ��

����� Systems with Time Delays � � � � � � � � � � � � � ��

���� Mathematical Model of a Thin Airfoil � � � � � � ��

�� Linearization of Nonlinear Models � � � � � � � � � � � � � ��

�� �� Linearization Around an Operating Point � � � � ��

�� �� Feedback Linearization � � � � � � � � � � � � � � � �

��� Modeling Uncertainty � � � � � � � � � � � � � � � � � � � ��

����� Dynamic Uncertainty Description � � � � � � � � � ��

����� Parametric Uncertainty Transformed to Dynamic

Uncertainty � � � � � � � � � � � � � � � � � � � � � ��

���� Uncertainty from System Identi�cation � � � � � � ��

��� Why Feedback Control� � � � � � � � � � � � � � � � � � � �

����� Disturbance Attenuation � � � � � � � � � � � � � � ��

Page 8: Feedback Control Systems- HOzbay

����� Tracking � � � � � � � � � � � � � � � � � � � � � � � ��

���� Sensitivity to Plant Uncertainty � � � � � � � � � � �

��� Exercise Problems � � � � � � � � � � � � � � � � � � � � � �

� Performance Objectives ��

�� Step Response� Transient Analysis � � � � � � � � � � � � �

�� Steady State Analysis � � � � � � � � � � � � � � � � � � � ��

� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ��

� BIBO Stability ��

��� Norms for Signals and Systems � � � � � � � � � � � � � � �

��� BIBO Stability � � � � � � � � � � � � � � � � � � � � � � � ��

�� Feedback System Stability � � � � � � � � � � � � � � � � � ��

��� Routh�Hurwitz Stability Test � � � � � � � � � � � � � � � �

��� Stability Robustness� Parametric Uncertainty � � � � � � ��

����� Uncertain Parameters in the Plant � � � � � � � � ��

����� Kharitanov�s Test for Robust Stability � � � � � � �

���� Extensions of Kharitanov�s Theorem � � � � � � � ��

��� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ��

� Root Locus ��

��� Root Locus Rules � � � � � � � � � � � � � � � � � � � � � � ��

����� Root Locus Construction � � � � � � � � � � � � � �

����� Design Examples � � � � � � � � � � � � � � � � � � �

��� Complementary Root Locus � � � � � � � � � � � � � � � � �

�� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ��

� Frequency Domain Analysis Techniques ��

��� Cauchy�s Theorem � � � � � � � � � � � � � � � � � � � � � ��

Page 9: Feedback Control Systems- HOzbay

��� Nyquist Stability Test � � � � � � � � � � � � � � � � � � � �

�� Stability Margins � � � � � � � � � � � � � � � � � � � � � � ��

��� Stability Margins from Bode Plots � � � � � � � � � � � � ��

��� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ��

Systems with Time Delays ��

�� Stability of Delay Systems � � � � � � � � � � � � � � � � � ��

�� Pad�e Approximation of Delays � � � � � � � � � � � � � � � ���

� Roots of a Quasi�Polynomial � � � � � � � � � � � � � � � ���

�� Delay Margin � � � � � � � � � � � � � � � � � � � � � � � � ��

�� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ���

� Lead� Lag� and PID Controllers ���

��� Lead Controller Design � � � � � � � � � � � � � � � � � � � ���

��� Lag Controller Design � � � � � � � � � � � � � � � � � � � � �

�� Lead�Lag Controller Design � � � � � � � � � � � � � � � � �

��� PID Controller Design � � � � � � � � � � � � � � � � � � � � �

��� Exercise Problems � � � � � � � � � � � � � � � � � � � � � �

� Principles of Loopshaping ���

��� Tracking and Noise Reduction Problems � � � � � � � � � � �

��� Bode�s Gain�Phase Relationship � � � � � � � � � � � � � ���

�� Design Example � � � � � � � � � � � � � � � � � � � � � � ���

��� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ���

� Robust Stability and Performance ���

���� Modeling Issues Revisited � � � � � � � � � � � � � � � � � ���

������ Unmodeled Dynamics � � � � � � � � � � � � � � � ���

������ Parametric Uncertainty � � � � � � � � � � � � � � ���

Page 10: Feedback Control Systems- HOzbay

���� Stability Robustness � � � � � � � � � � � � � � � � � � � � ���

������ A Test for Robust Stability � � � � � � � � � � � � ���

������ Special Case� Stable Plants � � � � � � � � � � � � ���

��� Robust Performance � � � � � � � � � � � � � � � � � � � � ���

���� Controller Design for Stable Plants � � � � � � � � � � � � ��

������ Parameterization of all Stabilizing Controllers � � ��

������ Design Guidelines for Q�s� � � � � � � � � � � � � ��

���� Design of H� Controllers � � � � � � � � � � � � � � � � � ��

������ Problem Statement � � � � � � � � � � � � � � � � � ��

������ Spectral Factorization � � � � � � � � � � � � � � � ���

����� Optimal H� Controller � � � � � � � � � � � � � � ���

������ Suboptimal H� Controllers � � � � � � � � � � � � ���

���� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ���

�� Basic State Space Methods ���

���� State Space Representations � � � � � � � � � � � � � � � � ���

���� State Feedback � � � � � � � � � � � � � � � � � � � � � � � ��

������ Pole Placement � � � � � � � � � � � � � � � � � � � ���

������ Linear Quadratic Regulators � � � � � � � � � � � � ���

��� State Observers � � � � � � � � � � � � � � � � � � � � � � � ���

���� Feedback Controllers � � � � � � � � � � � � � � � � � � � � ���

������ Observer Plus State Feedback � � � � � � � � � � � ���

������ H� Optimal Controller � � � � � � � � � � � � � � � ���

����� Parameterization of all Stabilizing Controllers � � ���

���� Exercise Problems � � � � � � � � � � � � � � � � � � � � � ���

Bibliography ��

Index ���

Page 11: Feedback Control Systems- HOzbay

Chapter �

Introduction

��� Feedback Control Systems

Examples of feedback are found in many disciplines such as engineering�

biological sciences� business� and economy� In a feedback system there

is a process �a cause�e�ect relation� whose operation depends on one or

more variables �inputs� that cause changes in some other variables� If

an input variable can be manipulated� it is said to be a control input�

otherwise it is considered a disturbance �or noise� input� Some of the

process variables are monitored� these are the outputs� The feedback

controller gathers information about the process behavior by observing

the outputs� and then it generates the new control inputs in trying to

make the system behave as desired� Decisions taken by the controller

are crucial� in some situations they may lead to a catastrophe instead

of an improvement in the system behavior� This is the main reason

that feedback controller design �i�e�� determining the rules for automatic

decisions taken by the feedback controller� is an important topic�

A typical feedback control system consists of four subsystems� a

process to be controlled� sets of sensors and actuators� and a controller�

Page 12: Feedback Control Systems- HOzbay

� H� �Ozbay

Actuators

Sensors

Processoutput

disturbance disturbance

desiredoutput

measurement noise

measured output

Plant

Controller

Figure ���� Feedback control system�

as shown in Figure ���� The process is the actual physical system that

cannot be modi�ed� Actuators and sensors are selected by process

engineers based on physical and economical constraints �i�e�� the range

of signals to be measured and�or generated and accuracy versus cost of

these devices�� The controller is to be designed for a given plant �the

overall system� which includes the process� sensors� and actuators��

In engineering applications the controller is usually a computer� or

a human operator interfacing with a computer� Biological systems can

be more complex� for example� the central nervous system is a very

complicated controller for the human body� Feedback control systems

encountered in business and economy may involve teams of humans as

main decision makers� e�g�� managers� bureaucrats� and�or politicians�

A good understanding of the process behavior �i�e�� the cause�e�ect

relationship between input and output variables� is extremely helpful

in designing the rules for control actions to be taken� Many engineer�

ing systems are described accurately by the physical laws of nature�

So� mathematical models used in engineering applications contain re�

latively low levels of uncertainty� compared with mathematical mod�

els that appear in other disciplines� where input�output relationships

Page 13: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory

can be much more complicated�

In this book� certain fundamental problems of feedback control the�

ory are studied� Typical application areas in mind are in engineering�

It is assumed that there is a mathematical model describing the dynam�

ical behavior of the underlying process �modeling uncertainties will also

be taken into account�� Most of the discussion is restricted to single

input�single output �SISO� processes� An important point to keep in

mind is that success of the feedback control depends heavily on the ac�

curacy of the process�uncertainty model� whether this model captures

the reality or not� Therefore� the �rst step in control is to derive a

simple and relatively accurate mathematical model of the underlying

process� For this purpose� control engineers must communicate with

process engineers who know the physics of the system to be controlled�

Once a mathematical model is obtained and performance objectives are

speci�ed� control engineers use certain design techniques to synthesize

a feedback controller� Of course� this controller must be tested by sim�

ulations and experiments to verify that performance objectives are met�

If the achieved performance is not satisfactory� then the process model

and the design goals must be reevaluated and a new controller should

be designed from the new model and the new performance objectives�

This iteration should continue until satisfactory results are obtained�

see Figure ����

Modeling is a crucial step in the controller design iterations�

The result of this step is a nominal process model and an uncertainty

description that represents our con�dence level for the nominal model�

Usually� the uncertainty magnitude can be decreased� i�e�� the con��

dence level can be increased only by making the nominal plant model

description more complicated �e�g�� increasing the number of variables

and equations�� On the other hand� controller design and analysis for

very complicated process models are very di�cult� This is the basic

trade�o� in system modeling� A useful nominal process model should

Page 14: Feedback Control Systems- HOzbay

� H� �Ozbay

ProcessEngineer

ControlEngineer

Math Model and

Design Specs

PhysicalProcess

No

Yes

No

Yes

Model and Specs

IterationsStop

SimulationResults

Satisfactory?

ExperimentalResults

Satisfactory?

Designed

ControllerFeedback

Reevaluate

Figure ���� Controller design iterations�

Page 15: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

u

u yp q

1 1y

System

Figure �� � A MIMO system�

be simple enough so that the controller design is feasible� At the same

time the associated uncertainty level should be low enough to allow the

performance analysis �simulations and experiments� to yield acceptable

results�

The purpose of this book is to present basic feedback controller

design and analysis �performance evaluation� techniques for simple SISO

process models and associated uncertainty descriptions� Examples from

certain speci�c engineering applications will be given whenever it is ne�

cessary� Otherwise� we will just consider generic mathematical models

that appear in many di�erent application areas�

��� Mathematical Models

A multi�input�multi�output �MIMO� system can be represented as shown

in Figure �� � where u�� � � � � up are the inputs and y�� � � � � yq are the out�

puts �for SISO systems we have p � q � ��� In this �gure� the direction

of the arrows indicates that the inputs are processed by the system to

generate the outputs�

In general� feedback control theory deals with dynamical systems�

i�e�� systems with internal memory �in the sense that the output at time

t � t� depends on the inputs applied at time instants t � t��� So� the

plant models are usually in the form of a set of di�erential equations

obtained from physical laws of nature� Depending on the operating

conditions� input�output relation can be best described by linear or

Page 16: Feedback Control Systems- HOzbay

� H� �Ozbay

Motor 3

Motor 2

Motor 1

Link 3

Link 2

Link 1

A three-link rigid robot A single-link flexible robot

Figure ���� Rigid and �exible robots�

nonlinear� partial or ordinary di�erential equations�

For example� consider a three�link robot as shown in Figure ����

This system can also be seen as a simple model of the human body�

Three motors located at the joints generate torques that move the three

links� Position� and�or velocity� and�or acceleration of each link can be

measured by sensors �e�g�� optical light with a camera� or gyroscope��

Then� this information can be processed by a feedback controller to pro�

duce the rotor currents that generate the torques� The feedback loop is

hence closed� For a successful controller design� we need to understand

�i�e�� derive mathematical equations of� how torques a�ect position and

velocity of each link� and how current inputs to motors generate torque�

as well as the sensor behavior� The relationship between torque and po�

sition�velocity can be determined by laws of physics �Newton�s law��

If the links are rigid� then a set of nonlinear ordinary di�erential equa�

tions is obtained� see ��� for a mathematical model� If the analysis and

design are restricted to small displacements around the upright equi�

librium� then equations can be linearized without introducing too much

error ���� If the links are made of a �exible material �for example�

Page 17: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory

in space applications the weight of the material must be minimized to

reduce the payload� which forces the use of lightweight �exible materi�

als�� then we must consider bending e�ects of the links� see Figure ����

In this case� there are an in�nite number of position coordinates� and

partial di�erential equations best describe the overall system behavior

�� ����

The robotic examples given here show that a mathematical model

can be linear or nonlinear� �nite dimensional �as in the rigid robot case�

or in�nite dimensional �as in the �exible robot case�� If the parameters

of the system �e�g�� mass and length of the links� motor coe�cients�

etc�� do not change with time� then these models are time�invariant�

otherwise they are time�varying�

In this book� linear time�invariant �LTI� models will be considered

only� Most of the discussion will be restricted to �nite dimensional

models� but certain simple in�nite dimensional models �in particular

time delay systems� will also be discussed�

The book is organized as follows� In Chapter �� modeling issues and

sources of uncertainty are studied and the main reason to use feedback

is explained� Typical performance objectives are de�ned in Chapter �

In Chapter �� basic stability tests are given� Single parameter control�

ler design is covered in Chapter � by using the root locus technique�

Stability robustness and stability margins are de�ned in Chapter � via

Nyquist plots� Stability analysis for systems with time delays is in

Chapter � Simple lead�lag and PID controller design methods are dis�

cussed in Chapter �� Loopshaping ideas are introduced in Chapter ��

In Chapter ��� robust stability and performance conditions are de�ned

and an H� controller design procedure is outlined� Finally� state space

based controller design methods are brie�y discussed and a parameter�

ization of all stabilizing controllers is presented in Chapter ���

Page 18: Feedback Control Systems- HOzbay
Page 19: Feedback Control Systems- HOzbay

Chapter �

Modeling� Uncertainty�

and Feedback

��� Finite Dimensional LTI System Models

Throughout the book� linear time�invariant �LTI� single input�single

output �SISO� plant models are considered� Finite dimensional LTI

models can be represented in time domain by dynamical state equations

in the form

�x�t� � Ax�t� � Bu�t� �����

y�t� � Cx�t� � Du�t� �����

where y�t� is the output� u�t� is the input� and the components of the

vector x�t� are state variables� The matrices A�B�C�D form a state

space realization of the plant� Transfer function P �s� of the plant is the

frequency domain representation of the input�output behavior�

Y �s� � P �s� U�s�

Page 20: Feedback Control Systems- HOzbay

�� H� �Ozbay

where s is the Laplace transform variable� Y �s� and U�s� represent the

Laplace transforms of y�t� and u�t�� respectively� The relation between

state space realization and the transfer function is

P �s� � C�sI �A���B � D�

Transfer function of an LTI system is unique� but state space realiza�

tions are not�

Consider a generic �nite dimensional SISO transfer function

P �s� � Kp�s� z�� � � � �s� zm�

�s� p�� � � � �s� pn���� �

where z�� � � � � zm are the zeros and p�� � � � � pn are the poles of P �s��

Note that for causal systems n � m �i�e�� direct derivative action is not

allowed� and in this case� P �s� is said to be a proper transfer function

since it satis�es

jdj �� limjsj��

jP �s�j ��� �����

If jdj � � in ����� then P �s� is strictly proper� For proper transfer

functions� ��� � can be rewritten as

P �s� �b�s

n�� � � � � � bnsn � a�sn�� � � � � � an

� d� �����

The state space realization of ����� in the form

A �

���n����� I�n�����n����an � � � �a�

�B �

���n�����

�C � bn � � � b� � D � d�

is called the controllable canonical realization� In this book� transfer

function representations will be used mostly� A brief discussion on state

space based controller design is included in the last chapter�

Page 21: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

��� In�nite Dimensional LTI System

Models

Multidimensional systems� spatially distributed parameter systems and

systems with time delays are typical examples of in�nite dimensional

systems� Transfer functions of in�nite dimensional LTI systems can

not be written as rational functions in the form ������ They are either

transcendental functions� or in�nite sums� or products� of rational func�

tions� Several examples are given below� for additional examples see

���� For such systems state space realizations ����� ���� involve in�nite

dimensional operators A�B�C� �i�e�� these are not �nite�size matrices�

and an in�nite dimensional state x�t� which is not a �nite�size vector�

See � � for a detailed treatment of in�nite dimensional linear systems�

����� A Flexible Beam

A �exible beam with free ends is shown in Figure ���� The de�ection

at a point x along the beam and at time instant t is denoted by w�x� t��

Ideal Euler�Bernoulli beam model with Kelvin�Voigt damping� �� ����

is a reasonably simple mathematical model�

���w

�t�� ��

��

�x�

�EI

��w

�x��t

��

��

�x�

�EI

��w

�x�

�� �� �����

where ��x� denotes the mass density per unit length of the beam� EI�x�

denotes the second moment of the modulus of elasticity about the elastic

axis and � � � is the damping factor�

Let �� � � � �� � � �� EI � �� and suppose that a transverse force�

�u�t�� is applied at one end of the beam� x � �� and the de�ection at the

other end is measured� y�t� � w��� t�� Then� the boundary conditions

for ����� are

��w

�x���� t� � �

��w

�x��t��� t� � ��

��w

�x���� t� � �

��w

�x��t��� t� � ��

Page 22: Feedback Control Systems- HOzbay

�� H� �Ozbay

x=0

w(x,t)u(t)

x=1x

Figure ���� A �exible beam with free ends�

��w

�x���� t� � �

��w

�x��t��� t� � ��

��w

�x���� t� � �

��w

�x��t��� t� � u�t��

By taking the Laplace transform of ����� and solving the resulting

fourth�order ODE in x� the transfer function P �s� � Y �s��U�s� is ob�

tained �see �� for further details��

P �s� ��

�� � �s��

�sinh � sin

cos cosh � �

where � � �s�����s� � It can be shown that

P �s� ��

s�

�Yn��

��� � �s� s�

���n

� � �s � s�

��n

�A � ����

The coe�cients �n� n� n � �� �� � � �� are the roots of

cos�n sinh�n � sin�n cosh�n and cosn coshn � �

for �n� n � �� Let j � k and �j � �k for j � k� then n�s alternate

with �n�s� It is also easy to show that n � �� � n� and �n � �

� � n�

as n���

����� Systems with Time Delays

In some applications� information �ow requires signi�cant amounts of

time delay due to physical distance between the process and the con�

troller� For example� if a spacecraft is controlled from the earth� meas�

Page 23: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

Reservoir

u(t)

v(t)

y(t)

Source

Controller

Feedback

τu(t- )

Figure ���� Flow control problem�

urements and command signals reach their destinations with a non�

negligible time delay even though signals travel at �or near� the speed

of light� There may also be time delays within the process� or the con�

troller itself �e�g�� when the controller is very complicated� computations

may take a relatively long time introducing computational time delays��

As an example of a time delay system� consider a generic �ow control

problem depicted in Figure ���� where u�t� is the input �ow rate at

the source� v�t� is outgoing �ow rate� and y�t� is the accumulation at

the reservoir� This setting is also similar to typical data �ow control

problems in high speed communication networks� where packet �ow

rates at sources are controlled to keep the queue size at bottleneck node

at a desired level� ���� A simple mathematical model is�

�y�t� � u�t� ��� v�t�

where � is the travel time from source to reservoir� Note that� to solve

this di�erential equation for t � �� we need to know y��� and u�t� for

t � ��� � ��� so an in�nite amount of information is required� Hence�

the system is in�nite dimensional� In this example� u�t� is adjusted

by the feedback controller� v�t� can be known or unknown to the con�

troller� in the latter case it is considered a disturbance� The output

Page 24: Feedback Control Systems- HOzbay

�� H� �Ozbay

V

h(t)

a

c

+b

α

β(t)

(t)

-b 0

Figure �� � A thin airfoil�

is y�t�� Assuming zero initial conditions� the system is represented in

the frequency domain by

Y �s� ��

s�e��sU�s�� V �s���

The transfer function from u to y is �s e��s� Note that it contains the

time delay term e��s� which makes the system in�nite dimensional�

����� Mathematical Model of a Thin Airfoil

Aeroelastic behavior of a thin airfoil� shown in Figure �� � is also rep�

resented as an in�nite dimensional system� If the air��ow velocity�

V � is higher than a certain critical speed� then unbounded oscillations

occur in the structure� this is called the �utter phenomenon� Flut�

ter suppression �stabilizing the structure� and gust alleviation �redu�

cing the e�ects of sudden changes in V � problems associated with this

system are solved by using feedback control techniques� ��� ���� Let

z�t� �� h�t�� ��t�� �t��T and u�t� denote the control input �torque ap�

plied at the �ap��

Page 25: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

+

-

B-1u(t) y(t)

Theodorsen’sfunction

T(s)

C

B

0

1

0(sI-A)

Figure ���� Mathematical model of a thin airfoil�

For a particular output in the form

y�t� � c�z�t� � c� �z�t�

the transfer function is

Y �s�

U�s�� P �s� �

C��sI �A���B�

�� C��sI �A���B� T�s�

where C� � c� c��� and A�B�� B� are constant matrices of appropriate

dimensions �they depend on V � a� b� c� and other system parameters

related to the geometry and physical properties of the structure� and

T�s� is the so�called Theodorsen�s function� which is a minimum phase

stable causal transfer function

Re�T�j �� �J�� r��J�� r� � Y�� r�� � Y�� r��Y�� r�� J�� r��

�J�� r� � Y�� r��� � �Y�� r�� J�� r���

Im�T�j �� ���Y�� r�Y�� r� � J�� r�J�� r��

�J�� r� � Y�� r��� � �Y�� r�� J�� r���

where r � b�V and J�� J�� Y�� Y� are Bessel functions� Note that

the plant itself is a feedback system with in�nite dimensional term T�s�

appearing in the feedback path� see Figure ����

Page 26: Feedback Control Systems- HOzbay

�� H� �Ozbay

��� Linearization of Nonlinear Models

����� Linearization Around an Operating Point

Linear models are sometimes obtained by linearizing nonlinear di�eren�

tial equations around an operating point� To illustrate the linearization

procedure� consider a generic nonlinear term in the form

�x�t� � f�x�t��

where f��� is an analytic function around a point xe� Suppose that

xe is an equilibrium point� i�e�� f�xe� � �� so that if x�t�� � xe then

x�t� � xe for all t � t�� Let �x represent small deviations from xe and

consider the system behavior at x�t� � xe � �x�t��

�x�t� � ��x�t� � f�xe � �x�t�� � f�xe� �

��f

�x

�x�xe

�x�t� � H�O�T�

where H�O�T� represents the higher�order terms involving ��x���� �

��x��� � � � �� and higher�order derivatives of f with respect to x eval�

uated at x � xe� As j�xj � � the e�ect of higher�order terms are

negligible and the dynamical equation is approximated by

��x�t� � A�x�t� where A ��

��f

�x

�x�xe

which is a linear system�

Example ��� The equations of motion of the pendulum shown in Fig�

ure ��� are given by Newton�s law� mass times acceleration is equal to

the total force� The gravitational�force component along the direction

of the rod is canceled by the reaction force� So the pendulum swings

in the direction orthogonal to the rod� In this coordinate� the accelera�

tion is ��� and the gravitational�force is �mg sin��� �it is in the opposite

Page 27: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

mgsin

mgcos

θ

mg

Length = l

Mass = m

θ

θ

Figure ���� A free pendulum�

direction to ��� Assuming there is no friction� equations of motion are

�x��t� � x��t�

�x��t� � �mg�

sin�x��t��

where x��t� � ��t� and x��t� � ���t�� and x�t� � x��t� x��t��T is the

state vector� Clearly xe � � ��T is an equilibrium point� When j��t�jis small� the nonlinear term sin�x��t�� is approximated by x��t�� So the

linearized equations lead to

�x��t� � �mg�x��t��

For an initial condition x��� � �o ��T� �where � � j�oj � ��� the

pendulum oscillates sinusoidally with natural frequencypmg

� rad�sec�

����� Feedback Linearization

Another way to obtain a linear model from a nonlinear system equation

is feedback linearization� The basic idea of feedback linearization is

illustrated in Figure ���� The nonlinear system� whose state is x� is

linearized by using a nonlinear feedback to generate an appropriate

input u� The closed�loop system from external input r to output x is

Page 28: Feedback Control Systems- HOzbay

�� H� �Ozbay

x = f(x,u). .u(t) x(t)

Linear System

r(t)u = h(u,x,r)

Figure ���� Feedback linearization�

linear� Feedback linearization rely on precise cancelations of certain

nonlinear terms� therefore it is not a robust scheme� Also� for certain

types of nonlinear systems� a linearizing feedback does not exist� See

e�g� �� �� for analysis and design of nonlinear feedback systems�

Example ��� Consider the inverted pendulum system shown in Fig�

ure ��� This is a classical feedback control example� it appears in

almost every control textbook� See for example � pp� ����� where

typical system parameters are taken as

m � ��� kg� M � � kg� � � ��� m� g � ��� m�sec�� J � m��� �

The aim here is to balance the stick at the upright equilibrium point�

� � �� by applying a force u�t� that uses feedback from ��t� and ���t��

The equations of motion can be written from Newton�s law�

�J � ��m��� � �m cos����x� �mg sin��� � � �����

�M � m��x � m� cos����� �m� sin��� ��� � u � �����

By using equation ������ �x can be eliminated from equation ����� and

hence a direct relationship between � and u can be obtained as�J

m�� �� m� cos����

M � m

��� �

m� cos��� sin��� ���

M � m� g sin���

� � cos���

M � mu� ������

Page 29: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

θ

x

Mass = M

Mass = m

Length = 2l

Force = u

Figure ��� Inverted pendulum on a cart�

Note that if u�t� is chosen as the following nonlinear function of ��t�

and ���t�

u � �M � m

cos���

�m� cos��� sin��� ���

M � m� g sin���

��J

m�� �� m� cos����

M � m� �� �� � � � r��

�������

then � satis�es the equation

�� � � �� � � � r� ������

where � and are the parameters of the nonlinear controller and r��t�

is the reference input� i�e�� desired ��t�� The equation ������ represents

a linear time invariant system from input r��t� to output ��t��

Exercise� Let r��t� � � and the initial conditions be ���� � ��� rad

and ����� � � rad�sec� Show that with the choice of � � � � the

pendulum is balanced� Using Matlab� obtain the output ��t� for the

parameters given above� Find another choice for the pair ��� �� such

that ��t� decays to zero faster without any oscillations�

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�� H� �Ozbay

��� Modeling Uncertainty

During the process of deriving a mathematical model for the plant�

usually a series of assumptions and simpli�cations are made� At the end

of this procedure� a nominal plant model� denoted by Po� is derived�

By keeping track of the e�ects of simpli�cations and assumptions made

during modeling� it is possible to derive an uncertainty description�

denoted by �P� associated with Po� It is then hoped �or assumed�

that the true physical system lies in the set of all plants captured by

the pair �Po��P��

����� Dynamic Uncertainty Description

Consider a nominal plant model Po represented in the frequency do�

main by its transfer function Po�s� and suppose that the !true plant"

is LTI� with unknown transfer function P �s�� Then the modeling un�

certainty is

#P �s� � P �s�� Po�s��

A useful uncertainty description in this case would be the following�

�i� the number of poles of Po�s� � #�s� in the right half plane is

assumed to be the same as the number of right half plane poles

of Po�s� �importance of this assumption will be clear when we

discuss Nyquist stability condition and robust stability�

�ii� also known is a function W �s� whose magnitude bounds the mag�

nitude of #P �s� on the imaginary axis�

j#P �j �j � jW �j �j for all �

This type of uncertainty is called dynamic uncertainty� In the MIMO

case� dynamic uncertainty can be structured or unstructured� in the

Page 31: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

sense that the entries of the uncertainty matrix may or may not be inde�

pendent of each other and some of the entries may be zero� The MIMO

case is beyond the scope the present book� see ��� for these advanced

topics and further references� For SISO plants� the pair fPo�s��W �s�grepresents the plant model that will be used in robust controller design

and analysis� see Chapter ���

Sometimes an in�nite dimensional plant model P �s� is approximated

by a �nite dimensional model Po�s� and the di�erence is estimated to

determine W �s��

Example ��� Flexible Beam Model� Transfer function ���� of the

�exible beam considered above is in�nite dimensional� By taking the

�rst few terms of the in�nite product� it is possible to obtain an ap�

proximate �nite dimensional model�

Po�s� ��

s�

NYn��

��� � �s� s�

���n

� � �s � s�

��n

�A �

Then� the di�erence is

jP �j �� Po�j �j � jPo�j �j��������

�Yn�N��

��� � j� � ��

���n

� � j� � ��

��n

�A������ �If N is su�ciently� large the right hand side can be bounded analytically�

as demonstrated in ���

Example ��� Finite Dimensional Model of a Thin Airfoil� Re�

call that the transfer function of a thin airfoil is in the form

P �s� �C��sI �A���B�

�� C��sI �A���B� T�s�

where T�s� is Theodorsen�s function� By taking a �nite dimensional

approximation of this in�nite dimensional term we obtain a �nite di�

Page 32: Feedback Control Systems- HOzbay

�� H� �Ozbay

mensional plant model that is amenable for controller design�

Po�s� �C��sI �A���B�

�� C��sI �A���B� To�s�

where To�s� is a rational approximation of T�s�� Several di�erent ap�

proximation schemes have been studied in the literature� see for example

��� where To�s� is taken to be

To�s� ������ sr � ������� sr � ��

����� sr � ��� ��� sr � ��� where sr �

s b

V�

The modeling uncertainty can be bounded as follows�

jP �j �� Po�j �j � jPo�j �j����R��j ��T�j �� To�j ��

��R��j �T�j �

����where R��s� � C��sI � A���B�� Using the bounds on approximation

error� jT�j �� To�j �j� an upper bound of the right hand side can be

derived� this gives W �s�� A numerical example can be found in ����

����� Parametric Uncertainty Transformed

to Dynamic Uncertainty

Typically� physical system parameters determine the coe�cients of Po�s��

Uncertain parameters lead to a special type of plant models where the

structure of P �s� is �xed �all possible plants P �s� have the same struc�

ture as Po�s�� e�g�� the degrees of denominator and numerator polyno�

mials are �xed� with uncertain coe�cients� For example� consider the

series RLC circuit shown in Figure ���� where u�t� is the input voltage

and y�t� is the output voltage�

Transfer function of the RLC circuit is

P �s� ��

LCs� � RCs � ��

Page 33: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

++R L

Cu(t) y(t)- -

Figure ���� Series RLC circuit�

So� the nominal plant model is

Po�s� ��

LoCos� � RoCos � ��

where Ro� Lo� Co are nominal values of R� L� C� respectively� Uncer�

tainties in these parameters appear as uncertainties in the coe�cients

of the transfer function�

By introducing some conservatism� it is possible to transform para�

metric uncertainty to a dynamic uncertainty� Examples are given below�

Example ��� Uncertainty in damping� Consider the RLC circuit

example given above� The transfer function can be rewritten as

P �s� � �o

s� � �� os � �o

where o � �pLC

and � � RpC��L� For the sake of argument� suppose

L and C are known precisely and R is uncertain� That means o is �xed

and � varies� Consider the numerical values�

P �s� ��

s� � ��s � �� � ��� � ����

Po�s� ��

s� � ��os � ��o � ����

Then� an uncertainty upper bound function W �s� can be determined by

plotting jP �j ��Po�j �j for a su�ciently large number of � � ��� � �����

Page 34: Feedback Control Systems- HOzbay

�� H� �Ozbay

abs(W)

10−2

10−1

100

101

102

0

0.5

1

1.5

2

2.5

3

omega

Figure ���� Uncertainty weight for a second order system�

The weight W �s� should be such that jW �j �j � jP �j � � Po�j �j for

all P � Figure ��� shows that

W �s� ������ ��� s � ��

�s� � ���� s � ��

is a feasible uncertainty weight�

Example ��� Uncertain time delay� A �rst�order stable system

with time delay has transfer function in the form

P �s� �e�hs

�s � ��where h � � � �����

Suppose that time delay is ignored in the nominal model� i�e��

Po�s� ��

�s � ���

As before� an envelop jW �j �j is determined by plotting the di�erence

jP �j �� Po�j �j for a su�ciently large number of h between �� � �����

Page 35: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

abs(W)

10−2

100

102

104

0

0.05

0.1

0.15

0.2

omega

Figure ����� Uncertainty weight for unknown time delay�

Figure ���� shows that the uncertainty weight can be chosen as

W �s� ������� ���� s � ��

���� s � ������� s � ���

Note that there is conservatism here� the set

Ph ��

P �s� �

e�hs

�s � ��� h � � � ����

is a subset of

P �� fP � Po � # � #�s� is stable� j#�j �j � jW �j �j � g�

which means that if a controller achieves design objectives for all plants

in the set P� then it is guaranteed to work for all plants in Ph� Since

the set P is larger than the actual set of interest Ph� design objectives

might be more di�cult to satisfy in this new setting�

Page 36: Feedback Control Systems- HOzbay

�� H� �Ozbay

����� Uncertainty from System Identi�cation

The ultimate purpose of system identi�cation is to derive a nominal

plant model and a bound for the uncertainty� Sometimes� physical laws

of nature suggest a mathematical structure for plant model �e�g�� a �xed�

order ordinary di�erential equation with unknown coe�cients� such as

the RLC circuit and the inverted pendulum examples given above�� If

the parameters of this model are unknown� they can be identi�ed by us�

ing parameter estimation algorithms� These algorithms give estimated

values of the unknown parameters� as well as bounds on the estimation

errors� that can be used to determine a nominal plant model and an

uncertainty bound �see ��� ����

In some cases� the plant is treated as a black box� assuming it is

linear� an impulse �or a step� input is applied to obtain the impulse

�or step� response� The data may be noisy� so the !best �t" may be

an in�nite dimensional model� The common practice is to �nd a low�

order model that explains the data !reasonably well�" The di�erence

between this low�order model response and the actual output data can

be seen as the response of the uncertain part of the plant� Alternatively�

this di�erence can be treated as measurement noise whose statistical

properties are to be determined�

In the black box approach� frequency domain identi�cation tech�

niques can also be used in �nding a nominal plant�uncertainty model�

For example� consider the response of a stable� LTI system� P �s�� to a

sinusoidal input u�t� � sin� kt�� The steady state output is

yss�t� � jP �j k�j sin� kt � � P �j k���

By performing experiments for a set of frequencies f �� � � � � Ng� it is

possible to obtain the frequency response data fP �j ��� � � � � P �j N�g��A precise de�nition of stability is given in Chapter �� but� loosely speaking� it

means that bounded inputs give rise to bounded outputs�

Page 37: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

Note that these are complex numbers determined from steady state re�

sponses� due to measurement errors and�or unmodeled nonlinearities�

there may be some uncertainty associated with each data point P �j k��

There are mathematical techniques to determine a nominal plant model

Po�s� and an uncertainty bound W �s�� such that there exists a plant

P �s� in the set captured by the pair �Po�W � that �ts the measurements�

These mathematical techniques are beyond the scope of this book� the

reader is referred to �� ��� �� for details and further references�

��� Why Feedback Control�

The main reason to use feedback is to reduce the e�ect of !uncertainty�"

The uncertainty can be in the form of a modeling error in the plant de�

scription �i�e�� an unknown system�� or in the form a disturbance�noise

�i�e�� an unknown signal��

Open�loop control and feedback control schemes are compared in

this section� Both the open�loop control and feedback control schemes

are shown in Figure ����� where r�t� is the reference input �i�e� desired

output�� v�t� is the disturbance and y�t� is the output� When H � ��

the feedback is in e�ect� r�t� is compared with y�t� and the error is fed

back to the controller� Note that in a feedback system when the sensor

fails �i�e�� sensor output is stuck at zero� the system becomes open loop

with H � ��

The feedback connection e�t� � r�t��y�t� may pose a mathematical

problem if the system bandwidth is in�nite �i�e�� both the plant and the

controller are proper but not strictly proper�� To see this problem�

consider the trivial case where v�t� � �� P �s� � Kp and C�s� � �K��p �

y�t� � �e�t� � y�t�� r�t�

which is meaningless for r�t� � �� Another example is the following

Page 38: Feedback Control Systems- HOzbay

�� H� �Ozbay

++

C

v(t)

+

H

H=1 : Closed Loop (feedback is in effect)H=0 : Open Loop

e(t)r(t)P

y(t)u(t)

-

Figure ����� Open�loop and closed�loop systems�

situation� let P �s� � �ss�� and C�s� � ����� then the transfer function

from r�t� to e�t� is

�� � P �s�C�s���� �s � �

which is improper� i�e�� non�causal� so it cannot be built physically�

Generalizing the above observations� the feedback system is said to

be well�posed if P ���C��� � ��� In practice� most of the physical

dynamical systems do not have in�nite bandwidth� i�e�� P �s� and hence

P �s�C�s� are strictly proper� So the feedback system is well posed in

that case� Throughout the book� the feedback systems considered are

assumed to be well�posed unless otherwise stated�

Before discussing the bene�ts of feedback� we should mention its ob�

vious danger� P �s� might be stable to start with� but if C�s� is chosen

poorly the feedback system may become unstable �i�e�� a bounded refer�

ence input r�t�� or disturbance input v�t�� might lead to an unbounded

signal� u�t� and�or y�t�� within the feedback loop�� In the remaining

parts of this chapter� and in the next chapter� the feedback systems are

assumed to be stable�

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Introduction to Feedback Control Theory ��

����� Disturbance Attenuation

In Figure ����� let v�t� � �� and r�t� �� In this situation� the mag�

nitude of the output� y�t�� should be as small as possible so that it is as

close to desired response� r�t�� as possible�

For the open�loop control scheme� H � �� the output is

Y �s� � P �s��V �s� � C�s�R�s���

Since R�s� � � the controller does not play a role in the disturbance

response� Y �s� � P �s�V �s�� When H � �� the feedback is in e�ect� in

this case

Y �s� �P �s�

� � P �s�C�s��V �s� � C�s�R�s���

So the response due to v�t� is Y �s� � P �s��� �P �s�C�s����V �s�� Note

that the closed�loop response is equal to the open�loop response multi�

plied by the factor ���P �s�C�s����� For good disturbance attenuation

we need to make this factor small by an appropriate choice of C�s��

Let jV �j �j be the magnitude of the disturbance in frequency do�

main� and� for the sake of argument� suppose that jV �j �j � � for

� $� and jV �j �j � � for outside the frequency region de�ned by

$� If the controller is designed in such a way that

j�� � P �j �C�j ����j � � � � $ ���� �

then high attenuation is achieved by feedback� The disturbance atten�

uation factor is the left hand side of ���� ��

����� Tracking

Now consider the dual problem where r�t� � �� and v�t� �� In this

case� tracking error� e�t� �� r�t� � y�t� should be as small as possible�

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� H� �Ozbay

In the open�loop case� the goal is achieved if C�s� � ��P �s�� But note

that if P �s� is strictly proper� then C�s� is improper� i�e�� non�causal�

To avoid this problem� one might approximate ��P �s� in the region of

the complex plane where jR�s�j is large� But if P �s� is unstable and if

there is uncertainty in the right half plane pole location� then ��P �s�

cannot be implemented precisely� and the tracking error is unbounded�

In the feedback scheme� the tracking error is

E�s� � �� � P �s�C�s����R�s��

Therefore� similar to disturbance attenuation� one should select C�s�

in such a way that j�� � P �j �C�j ����j � � in the frequency region

where jR�j �j is large�

����� Sensitivity to Plant Uncertainty

For a function F � which depends on a parameter �� sensitivity of F to

variations in � is denoted by SF� � and it is de�ned as follows

SF� �� lim���

#F �F

#���

�������o

��

F

�F

��

�������o

where �o is the nominal value of �� #� and #F represent the devi�

ations of � and F from their nominal values �o and F evaluated at �o�

respectively�

Transfer function from reference input r�t� to output y�t� is

Tol�s� � P �s�C�s� �for an open�loop system��

Tcl�s� �P �s�C�s�

� � P �s�C�s��for a closed�loop system��

Typically the plant is uncertain� so it is in the form P � Po�#P � Then

the above transfer functions can be written as Tol � Tol�o � #Tol and

Page 41: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

Tcl � Tcl�o � #Tcl � where Tol�o and Tcl�o are the nominal values when P

is replaced by Po� Applying the de�nition� sensitivities of Tol and Tcl

to variations in P are

STolP � limP��

#Tol�Tol�o#P �Po

� � ������

STclP � limP��

#Tcl�Tcl�o#P �Po

��

� � Po�s�C�s�� ������

The �rst equation ������ means that the percentage change in Tol is

equal to the the percentage change in P � The second equation ������

implies that if there is a frequency region where percentage variations

in Tcl should be made small� then the controller can be chosen in such

a way that the function �� � Po�s�C�s���� has small magnitude in that

frequency range� Hence� the e�ect of variations in P can be made small

by using feedback control� the same cannot be achieved by open�loop

control�

In the light of ������ the function �� � Po�s�C�s���� is called the

!nominal sensitivity function" and it is denoted by S�s�� The sensitivity

function� denoted by S�s�� is the same function when Po is replaced

by P � Po � #P � In all the examples seen above� sensitivity function

plays an important role� One of the most important design goals in

feedback control is sensitivity minimization� This is discussed further

in Chapter ���

��� Exercise Problems

�� Consider the �ow control problem illustrated in Figure ���� and as�

sume that the outgoing �ow rate v�t� is proportional to the square

root of the liquid level h�t� in the reservoir �e�g�� this is the case

if the liquid �ows out through a valve with constant opening��

v�t� � v�ph�t��

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� H� �Ozbay

Furthermore� suppose that the area of the reservoir A�h�t�� is

constant� say A�� Since total accumulation is y�t� � A�h�t��

dynamical equation for this system is

�h�t� ��

A��u�t� ��� v�

ph�t���

Let h�t� � h� � �h�t� and u�t� � u� � �u�t�� with u� � v�ph��

Linearize the system around the operating point h� and �nd the

transfer function from �u�t� to �h�t��

�� For the above mentioned �ow control problem� suppose that the

geometry of the reservoir is known as

A�h�t�� � A� � A�h�t� � A�

ph�t�

with some constants A�� A�� A�� Let the outgoing �ow rate be

v�t� � v� � w�t�� with v� � jw�t�j � �� The term w�t� can be

seen as a disturbance representing the !load variations" around

the nominal constant load v�� typically w�t� is a superposition of

a �nite number of sine and cosine functions�

�i� Assume that � � � and v�t� is available to the controller�

Given a desired liquid level hd�t�� there exists a feedback

control input u�t� �a nonlinear function of h�t�� v�t� and

hd�t�� linearizing the system whose input is hd�t� and output

is h�t�� For hd�t� � hd� �nd such u�t� that leads to h�t� � hd

as t�� for any initial condition h��� � h��

�ii� Now consider a linear controller in the form

u�t� � K �hd�t�� h�t�� � v�

�in this case� the controller does not have access to w�t���

where the gain K � � is to be determined� Let

A� � ��� A� � ��� A� � ���� � � ��

w�t� � �� sin����t�� v� � ��� hd�t� ��� h��� � ��

Page 43: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory

�note that time delay is non�zero in this case�� By using

Euler�s method� simulate the feedback system response for

several di�erent values of K � �� � ����� Find a value of K

for which

jhd � h�t�j � ���� � t � �� �

Note� to simulate a nonlinear system in the form

�x�t� � f�x�t�� t�

�rst select equally spaced time instants tk � kTs� for some

Ts � �tk�� � tk� � �� k � �� For example� in the above

problem� Ts can be chosen as Ts � ����� Then� given x�t���

we can determine x�tk� from x�tk��� as follows�

x�tk� �� x�tk��� � Ts f�x�tk���� tk��� for k � ��

This approximation scheme is called Euler�s method� More

accurate and sophisticated simulation techniques are avail�

able� these� as well as the Euler�s method� are implemented

in the Simulink package of Matlab�

� An RLC circuit has transfer function in the form

P �s� � �o

s� � �� os � �o�

Let �n � ��� and o�n � �� be the nominal values of the paramet�

ers of P �s�� and determine the poles of Po�s��

�i� Find an uncertainty bound W �s� for ��% uncertainty in

the values of � and o�

�ii� Determine the sensitivity of P �s� to variations in ��

�� For the �exible beam model� to obtain a nominal �nite dimen�

sional transfer function take N � � and determine Po�s� by com�

puting �n and n� for n � �� � � � � �� What are the poles and

zeros of Po�s�� Plot the di�erence jP �j �� Po�j �j by writing a

Page 44: Feedback Control Systems- HOzbay

� H� �Ozbay

Matlab script� and �nd a low�order �at most �nd�order� rational

function W �s� that is a feasible uncertainty bound�

�� Consider the disturbance attenuation problem for a �rst�order

plant P �s� � ����s � �� with v�t� � sin� t�� For the open�loop

system� magnitude of the steady state output is jP �j �j � � �i�e��

the disturbance is ampli�ed�� Show that in the feedback scheme

a necessary condition for the steady state output to be zero is

jC� j �j � � �i�e� the controller must have a pair of poles at

s � j ��

Page 45: Feedback Control Systems- HOzbay

Chapter �

Performance Objectives

Basic principles of feedback control are discussed in the previous chapter�

We have seen that the most important role of the feedback is to reduce

the e�ects of uncertainty� In this chapter� time domain performance

objectives are de�ned for certain special tracking problems� Plant un�

certainty and disturbances are neglected in this discussion�

��� Step Response Transient Analysis

In the standard feedback control system shown in Figure ����� assume

that the transfer function from r�t� to y�t� is in the form

T �s� � �o

s� � �� os � �o� � � � � o � IR

and r�t� is the unit step function� denoted by U�t�� Then� the output

y�t� is the inverse Laplace transform of

Y �s� � �o

�s� � �� os � �o�

s

Page 46: Feedback Control Systems- HOzbay

� H� �Ozbay

that is

y�t� � �� e��otp�� ��

sin� dt � �� t � ��

where d �� op

�� �� and � �� cos������ For some typical values of

�� the step response y�t� is as shown in Figure ��� Note that the steady

state value of y�t� is yss � � because T ��� � �� Steady state response

is discussed in more detail in the next section�

The maximum percent overshoot is de�ned to be the quantity

PO ��yp � yssyss

� ���%

where yp is the peak value� By simple calculations it can be seen that

the peak value of y�t� occurs at the time instant tp � �� d� and

PO � e��p��� � ���%�

Figure �� shows PO versus �� Note that the output is desired to reach

its steady state value as fast as possible with a reasonably small PO� In

order to have a small PO� � should be large� For example� if PO � ��%

is desired� then � must be greater or equal to ����

The settling time is de�ned to be the smallest time instant ts� after

which the response y�t� remains within �% of its �nal value� i�e��

ts �� minf t� � jy�t�� yssj � ���� yss � t � t�g�

Sometimes �% or �% is used in the de�nition of settling time instead of

�%� Conceptually� they are not signi�cantly di�erent� For the second�

order system response� with the �% de�nition of the settling time�

ts � �

� o�

So� in order to have a fast settling response� the product � o should be

large�

Page 47: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory

zeta=0.3zeta=0.5zeta=0.9

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

t*omega_o

Ste

p R

espo

nse

Figure ��� Step response of a second�order system�

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

100

zeta

PO

Figure ��� PO versus ��

Page 48: Feedback Control Systems- HOzbay

� H� �Ozbay

Im

Reo53

ζ=0.6oζω =0.5

-0.5x

Figure � � Region of the desired closed�loop poles�

The poles of T �s� are

r��� � �� o j op

�� ���

Therefore� once the maximum allowable settling time and PO are spe�

ci�ed� the poles of T �s� should lie in a region of the complex plane

de�ned by minimum allowable � and � o�

For example� let the desired PO and ts be bounded by

PO � ��% and ts � � sec�

For these design speci�cations� the region of the complex plane in which

closed�loop system poles should lie is determined as follows� The PO

requirement implies that � � ���� equivalently � � � �� �recall that

cos��� � ��� The settling time requirement is satis�ed if and only if

Re�r���� � ����� Then� the region of desired closed�loop poles is the

shaded area shown in Figure � �

Page 49: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

If the order of the closed�loop transfer function T �s� is higher than

two� then� depending on the location of its poles and zeros� it may be

possible to approximate the closed�loop step response by the response

of a second�order system� For example� consider the third�order system

T �s� � �o

�s� � �� os � �o��� � s�r�where r � � o�

The transient response contains a term e�rt� Compared with the envel�

ope e��ot of the sinusoidal term� e�rt decays very fast� and the overall

response is similar to the response of a second�order system� Hence� the

e�ect of the third pole r� � �r is negligible� Consider another example�

T �s� � �o�� � s��r � ���

�s� � �� os � �o��� � s�r�where � � �� r�

In this case� although r does not need to be much larger than � o� the

zero at ��r��� cancels the e�ect of the pole at �r� To see this� consider

the partial fraction expansion of Y �s� � T �s�R�s� with R�s� � ��s

Y �s� �A�

s�

A�

s� r��

A�

s� r��

A�

s � rwhere A� � � and

A� � lims��r�s � r�Y �s� �

�o�� or � � �o � r��

��

r � �

��

Since jA�j � � as �� �� the term A�e�rt is negligible in y�t��

In summary� if there is an approximate pole zero cancelation in the

left half plane� then this pole�zero pair can be taken out of the transfer

function T �s� to determine PO and ts� Also� the poles closest to the

imaginary axis dominate the transient response of y�t�� To generalize

this observation� let r�� � � � � rn be the poles of T �s�� such that Re�rk� �Re�r�� � Re�r�� � �� for all k � � Then� the pair of complex conjugate

poles r��� are called the dominant poles� We have seen that the desired

transient response properties� e�g�� PO and ts� can be translated into

requirements on the location of the dominant poles�

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�� H� �Ozbay

��� Steady State Analysis

For the standard feedback system of Figure ����� the tracking error is

de�ned to be the signal e�t� �� r�t�� y�t�� One of the typical perform�

ance objectives is to keep the magnitude of the steady state error�

ess �� limt�� e�t�� � ���

within a speci�ed bound� Whenever the above limit exists �i�e�� e�t�

converges� the �nal value theorem can be applied�

ess � lims��

sE�s� � lims��

s�

� � G�s�R�s��

where G�s� � P �s�C�s� is the open�loop transfer function from r to y�

Suppose that G�s� is in the form

G�s� �NG�s�

s� eDG�s�� ���

with � � � and eDG��� � � � NG���� Let the reference input be

r�t� � tk��U�t� �� R�s� ��

skk � ��

When k � � �i�e� r�t� is unit step� the steady state error is

ess �

��� � G������ if � � �

� if � � ��

If zero steady state error is desired for unit step reference input� then

G�s� � P �s�C�s� must have at least one pole at s � �� For k � �

ess �

� � �� if � � k � �eDG���NG���

if � � k � �

� if � � k�

Page 51: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

The system is said to be type k if ess � � for R�s� � ��sk� For the

standard feedback system where G�s� is in the form � ��� the system is

type k only if � � k�

Now consider a sinusoidal reference input R�s� ���o

s����o� The track�

ing error e�t� is the inverse Laplace transform of

E�s� ��

� � G�s�

� �o

s� � �o

��

Partial fraction expansion and inverse Laplace transformation yield a

sinusoidal term� Ao sin� ot� �� in e�t�� Unless Ao � �� the limit � ���

does not exist� hence the �nal value theorem cannot be applied� In

order to have Ao � �� the transfer function S�s� � �� � G�s���� must

have zeros at s � j o� i�e� G�s� must be in the form

G�s� �NG�s�

�s� � �o� bDG�s�where NG� j o� � ��

In conclusion� ess � � only if the zeros of S�s� �equivalently� the poles

of P �s�C�s� � G�s�� include all Im�axis poles of R�s�� Let ��� � � � � �k

be the Im�axis poles of R�s�� including multiplicities� Assuming that

none of the �i�s are poles of P �s�� to achieve ess � � the controller must

be in form

C�s� � eC�s��

DR�s�� � �

where DR�s� �� �s� ��� � � � �s� �k�� and eC��i� � � for i � �� � � � � k�

For example� when R�s� � ��s and jP ���j � �� we need a pole

at s � � in the controller to have ess � �� A simple example is PID

�proportional� integral� derivative� controller� which is in the form

C�s� �

�Kp �

Ki

s� Kd s

� ��

� � �s

�� ���

where Kp� Ki and Kd are the proportional� integral� and derivative

action coe�cients� respectively� the term � � � is needed to make the

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�� H� �Ozbay

controller proper� When Kd � � we can set � � �� in this case C�s��

� ���� becomes a PI controller� See Section ��� for further discussions

on PID controllers�

Note that a copy of the reference signal generator� �DR�s�

� is included

in the controller � � �� We can think of eC�s� as the !controller" to be

designed for the !plant"

eP �s� �� P �s��

DR�s��

This idea is the basis of servocompensator design� ��� ��� and repetitive

controller design� ��� � ��

��� Exercise Problems

�� Consider the second�order system with a zero

T �s� � �o ��� s�z�

s� � �� os � �o

where � � �� � �� and z � IR� What is the steady state error for

unit step reference input� Let o � � and plot the step response

for � � ��� � ���� ��� and z � ��� ��� ���� �

�� For P �s� � ��s design a controller in the form

C�s� �Kc�s � zc�

s� � �o

so that the sensitivity function S�s� � ���P �s�C�s���� has three

zeros at �� j o and three poles with real parts less than �����

This guarantees that ess � � for reference inputs r�t� � U�t�

and r�t� � sin��t�U�t�� Plot the closed�loop system response

corresponding to these reference inputs for your design�

Page 53: Feedback Control Systems- HOzbay

Chapter �

BIBO Stability

In this chapter� bounded input�bounded output �BIBO� stability of a

linear time invariant �LTI� system will be de�ned �rst� Then� we will

see that BIBO stability of the feedback system formed by a controller

C�s� and a plant P �s� can determined by applying the Routh�Hurwitz

test on the characteristic polynomial� which is de�ned from the numer�

ator and denominator polynomials of C�s� and P �s�� For systems with

parametric uncertainty in the coe�cients of the transfer function� we

will see Kharitanov�s robust stability test and its extensions�

��� Norms for Signals and Systems

In system analysis� input and output signals are considered as functions

of time� For example u�t�� and y�t�� for t � �� are input and output

functions of the LTI system F� shown in Figure ����

Each of these time functions is assumed to be an element of a func�

tion space� u � U � and y � Y � where U represents the set of possible

input functions� and similarly Y represents the set of output functions�

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�� H� �Ozbay

y(t)u(t) F

Figure ���� Linear time invariant system�

For mathematical convenience� U and Y are usually taken to be vector

spaces on which signal norms are de�ned� The norm kukU is a measure

of how large the input signal is� similarly for the output signal norm

kykY � Using this abstract notation we can de�ne the system norm�

denoted by kFk� as the quantity

kFk � supu���

kykYkukU � �����

A physical interpretation of ����� is the following� the ratio kykYkukU repres�

ents the ampli�cation factor of the system for a �xed input u � �� Since

the largest possible ratio is taken �sup means the least upper bound� as

the system norm� kFk can be interpreted as the largest signal ampli�c�

ation through the system F�

In signals and systems theory� most widely used function spaces are

L������ L������ and L������ Precise de�nitions of these Lebesgue

spaces are beyond the scope of this book� They can be loosely de�ned

as follows� for � � p ��

Lp���� �

f � ���� � IR� kfkpLp ��

Z �

jf�t�jpdt ��

L����� �

�f � ���� � IR� kfkL� �� sup

t�����

jf�t�j ����

Note that L����� is the space of all �nite energy signals� and L�����

is the set of all bounded signals� In the above de�nitions the real valued

function f�t� is de�ned on the positive time axis and it is assumed to be

piecewise continuous� An impulse� e�g�� ��t� to� for some to � �� does

not belong to any of these function spaces� So� it is useful to de�ne a

Page 55: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

space of impulse functions�

I� ��

�f�t� �

�Xk��

�k��t� tk�� tk � �� �k � IR�

�Xk��

j�kj ���

with the norm de�ned as kfkI� ��

�Xk��

j�kj�

The functions from L����� and I� can be combined to obtain another

function space

A� �� ff�t� � g�t� � h�t� � g � L����� � h � I�g �

For example�

f�t� � e��t sin����t�� ���t� �� � ���t� ��� � t � ��

belongs to A�� but it does not belong to L������ nor L������ The

importance of A� will be clear shortly�

Exercise� Determine whether time functions given below belong to any

of the function spaces de�ned above�

f��t� � �t � ���� � f��t� � �t � ���� � f��t� pt � �

f��t� �sin���t�

t� f��t� � �t� ����� � f��t� � �t� �����

��� BIBO Stability

Formally� a systemF is said to be bounded input�bounded output �BIBO�

stable if every bounded input u generates a bounded output y� and the

largest signal ampli�cation through the system is �nite� In the abstract

notation� that means BIBO stability is equivalent to having a �nite

system norm� i�e�� kFk ��� Note that de�nition of BIBO stability de�

pends on the selection of input and output spaces� The most common

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�� H� �Ozbay

de�nition of BIBO stability deals with the special case where the input

and output spaces are U � Y � L������ Let f�t� be the impulse

response and F �s� be the transfer function �i�e�� the Laplace transform

of f�t�� of a causal LTI system F�

Theorem ��� Suppose U � Y � L������ Then the system F is

BIBO stable if and only if f � A�� Moreover�

kFk � kfkA�� �����

Proof� The result follows from the convolution identity

y�t� �

Z �

f���u�t� ��d�� ��� �

If u � L�� � ��� then ju�t� ��j � kukL� for all t and � � It is easy to

verify the following inequalities�

jy�t�j �

����Z �

f���u�t� ��d�

�����

Z �

jf���j ju�t� ��jd�

� kukL�Z �

jf���jd� � kukL�kfkA��

Hence� for all u � �

kykL�kukL�

� kfkA�

which means that kFk � kfkA�� In order to prove the converse� consider

��� � in the limit as t�� with u de�ned at time instant � as

u�t� �� �

� � �� if f��� � ��

�� if f��� � ��

� if f��� � ��

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Introduction to Feedback Control Theory �

Clearly jy�t�j � kfkA�in the limit as t��� Thus

kFk � supu���

kykL�kukL�

� kfkA��

This concludes the proof�

Exercise� Determine BIBO stability of the LTI system whose impulse

response is �a� f��t�� �b� f��t�� where

f��t� ��

� � �� if t � �

� if � � t � t�

�t� ���k if t � t�

f��t� ��

� � �� if t � �

t�� if � � t � t�

� if t � t�

with t� � �� k � �� and t� � �� � � � � ��

Theorem ��� Suppose U � Y � L������ Then the system norm of

F is

kFk � sup���� ��IR

jF �� � j �j �� kFkH� �����

That is the system F is stable if and only if its transfer function has no

poles in C�� Moreover� when the system is stable� maximum modulus

principle implies that

kFkH� � sup��IR

jF �j �j �� kFkL� � �����

See� e�g�� ��� pp� ����� and ��� pp� ������ for proofs of this the�

orem� Further discussions can also be found in ���� The proof is based

on Parseval�s identity� which says that the energy of a signal can be com�

puted from its Fourier transform� The equivalence ����� implies that

when the system is stable� its norm �in the sense of largest energy amp�

li�cation� can be computed from the peak value of its Bode magnitude

plot� The second equality in ����� implicitly de�nes the space H� as

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the set of all analytic functions of s �the Laplace transform variable�

that are bounded in the right half plane C��

In this book� we will mostly consider systems with real rational

transfer functions of the form F �s� � NF �s��DF �s� where NF �s� and

DF �s� are polynomials in s with real coe�cients� For such systems the

following holds

kfkA��� �� kFkH� ��� �����

So� stability tests resulting from ����� and ����� are equivalent for this

class of system� To see the equivalence ����� we can rewrite F �s� in

the form of partial fraction expansions and use the Laplace transform

identities for each term to obtain f�t� in the form

f�t� �nX

k��

mk��X���

ck� t� epkt ����

where p�� � � � � pn are distinct poles of F �s� with multiplicities m�� � � � �mn�

respectively� and ck� are constant coe�cients� The total number of poles

of F �s� is �m� � � � � � mn�� It is now clear that kfkA�is �nite if and

only if pk � C�� i�e� Re�pk� � � for all k � �� � � � � n� and this condition

holds if and only if F � H��

Rational transfer functions are widely used in control engineering

practice� However� they do not capture spatially distributed parameter

systems �e�g�� �exible beams� and systems with time delays� Later in

the book� systems with simple time delays will also be considered� The

class of delay systems we will be dealing with have transfer functions

in the form

F �s� �N��s� � e��nsN��s�

D��s� � e��dsD��s�

where �n � �� �d � �� and N�� N�� D�� D� are polynomials with real

coe�cients� satisfying deg�D�� � maxfdeg�N��� deg�N��� deg�D��g� The

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Introduction to Feedback Control Theory ��

r

v

n

yue+

-

++

++H

C P

Figure ���� Feedback system�

equivalence ����� holds for this type of systems as well� Although there

may be in�nitely many poles in this case� the number of poles to the

right of any vertical axis in the complex plane is �nite �see Chapter ��

In summary� for the class of systems we consider in this book� a

system F is stable if and only if its transfer function F �s� is bounded

and analytic in C�� i�e� F �s� does not have any poles in C��

��� Feedback System Stability

In the above section� stability of a causal single�input�single�output

�SISO� LTI system is discussed� De�nition of stability can be easily

extended to multi�input�multi�output �MIMO� LTI systems as follows�

Let u��t�� � � � � uk�t� be the inputs and y��t�� � � � � y��t� be the outputs of

such a system F� De�ne Fij�s� to be the transfer function from uj to

yi� i � �� � � � � �� and j � �� � � � � k� Then� F is stable if and only if each

Fij�s� is a stable transfer function �i�e�� it has no poles in C�� for all

i � �� � � � � �� and j � �� � � � � k�

The standard feedback system shown in Figure ��� can be considered

as a single MIMO system with inputs r�t�� v�t�� n�t� and outputs e�t��

u�t�� y�t�� The feedback system is stable if and only if all closed�loop

transfer functions are stable� Let the closed�loop transfer function from

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input r�t� to output e�t� be denoted by Tre�s�� and similarly for the

remaining eight closed�loop transfer functions� It is an easy exercise to

verify that

Tre � S Tve � �HPS Tne � �HS

Tru � CS Tvu � S Tnu � �HCS

Try � PCS Tvy � PS Tny � �HPCS

where dependence on s is dropped for notational convenience� and

S�s� �� �� � H�s�P �s�C�s�����

In this con�guration� P �s� and H�s� are given and C�s� is to be de�

signed� The primary design goal is closed�loop system stability� In en�

gineering applications� the sensor model H�s� is usually a stable transfer

function� Depending on the measurement setup� H�s� may be non�

minimum phase� For example� this is the case if the actual plant output

is measured indirectly with a certain time delay� The plant P �s� may

or may not be stable� If P �s� is unstable� none of its poles in C� should

coincide a zero of H�s�� Otherwise� it is impossible to stabilize the feed�

back system because in this case Tvy and Try are unstable independent

of C�s�� though S�s� may be stable� Similarly� it is easy to show that

if there is a pole zero cancelation in C� in the product H�s�P �s�C�s��

then one of the closed�loop transfer functions is unstable� For example�

let H�s� � �� P �s� � �s � ���s � ���� and C�s� � �s � ����� In this

case� Tru is unstable�

In the light of the above discussion� suppose there is no unstable

pole�zero cancelation in the product H�s�P �s�C�s�� Then the feedback

system is stable if and only if the roots of

� � H�s�P �s�C�s� � � �����

are in C�� For the purpose of investigating closed�loop stability and

controller design� we can de�ne PH�s� �� H�s�P �s� as !the plant seen

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Introduction to Feedback Control Theory ��

by the controller�" Therefore� without loss of generality� we will assume

that H�s� � � and PH �s� � P �s��

Now consider the �nite dimensional case where P �s� and C�s� are

rational functions� i�e�� there exist polynomials NP �s�� DP �s�� NC�s�

and DC�s� such that P �s� � NP �s��DP �s�� C�s� � NC�s��DC�s� and

�NP � DP � and �NC � DC� are coprime pairs�

�A pair of polynomials �N�D� is said to be coprime if N and D do not

have common roots�� For example� let P �s� � �s���s�s��s��� � in this case

we can choose NP �s� � �s � ��� DP �s� � s�s� � s � ��� Note that

NP �s� � ��s� �� and DP �s� � �s�s� � s� �� would also be a feasible

choice� but there is no other possibility� because NP and DP are not

allowed to have common roots� Now it is clear that the feedback system

is stable if and only if the roots of

��s� �� DP �s�DC�s� � NP �s�NC�s� � � �����

are in C�� The polynomial ��s� is called the characteristic polynomial�

and its roots are the closed�loop system poles� A polynomial is said to

be stable �or Hurwitz stable� if its roots are in C�� Once a controller is

speci�ed for a given plant� the closed�loop system stability can easily

be determined by constructing ��s�� and by computing its roots�

Example ��� Consider the plant P �s� � �s���s �s��s��� � with a controller

C�s� � � �� �s�� ���s�� �� � The roots of the characteristic polynomial

��s� � s� � ���s� � ����s� � �� ��s� �����

are ���� j��� and ����� j��� � So� the feedback system is stable�

The roots of ��s� are computed in Matlab� Feedback system stability

analysis can be done likewise by using any computer program that solves

the roots of a polynomial�

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Since P �s� and C�s� are �xed in the above analysis� the coe�cients

of ��s� are known and �xed� There are some cases where plant para�

meters are not known precisely� For example�

P �s� ���� s�

s�s � ��

where � is known to be in the interval �� � �max�� yet its exact value

is unknown� The upper bound �max represents the largest uncertainty

level for this parameter� Let the controller be in the form

C�s� ��

�s � ��

The parameter is to be adjusted so that the feedback system is stable

for all values of � � �� � �max�� i�e�� the roots of

��s� � s�s � ���s � � � ��� s� � s� � �� � �s� � �� � ��s � �

are in C� for all � � �� � �max�� This way� closed�loop stability is

guaranteed for the uncertain plant� For each �xed pair of parameters

��� �� the roots of ��s� can be computed numerically� Hence� in the

two�dimensional parameter space� the stability region can be determined

by checking the roots of ��s� at each ��� � pair� For each �xed �

the largest feasible � � � can easily be determined by a line search�

Figure �� shows �max for each � For � ���� the feedback system is

unstable� When � ��� the largest allowable � increases non�linearly

with � The exact relationship between �max and will be determined

from the Routh�Hurwitz stability test in the next section�

The numerical approach illustrated above can still be used even if

there are more than two parameters involved in the characteristic poly�

nomial� However� in that case� computational complexity �the number

of grid points to be taken in the parameter space for checking the roots�

grows exponentially with the number of parameters�

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Introduction to Feedback Control Theory �

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

70beta versus alpha_max

beta

alph

a_m

ax

Figure �� � versus �max

��� RouthHurwitz Stability Test

The Routh�Hurwitz test is a direct procedure for checking stability of

a polynomial without computing its roots� Consider a polynomial of

degree n with real coe�cients a�� � � � � an�

��s� � a�sn � a�s

n�� � � � �� an� where a� � ��

The polynomial ��s� is stable if and only if the number of sign changes

in the �rst column of the Routh table is zero� The Routh table is

constructed as follows�

a� a� a� a� � � �

a� a� a� a � � �

R��� R��� R��� R��� � � �

R��� R��� R��� R��� � � ����

������

���

Rn����

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The �rst two rows of this table are determined directly from the coef�

�cients of the polynomial� For k � the kth row is constructed from

the k � �st and k � �nd rows according to the formula

Rk�� �Rk������Rk���� �Rk������Rk����

Rk����k � � � � ��

We should set a� � � for � � n for constructing the rd� �th� and

remaining rows� If n is even� the length of the �rst row is �� � n� ��� and

the length of the second row is �� � ����� if n is odd then �� � n��� � and

�� � ��� Then� for k � the length of kth row� �k� can be determined as

�k � �k�� � �� Therefore the Routh table has a block upper triangular

form� Another important point to note is that if Rk���� is zero� then the

kth row cannot be determined from the above formula because of the

division� A zero element in the �rst column of the Routh table indicates

existence of a root in C�� so the polynomial is unstable in that case�

Suppose that there is no zero element in the �rst column� Then the

number of roots of ��s� in C� is equal to the number of sign changes

in the �rst column of the Routh table�

Example ��� Consider the polynomial ��s� � s� � �� � �s� � �� ���s � �� for which the Routh table is

� � � �

� � �

R��� �

R���

where R��� � �� and R��� � �� � ��� ��� � ���� So� for stability of

��s� we need � � �� � ��� and ������� �� � � � �� which imply

� ���� These inequalities are in agreement with Figure �� � It is now

clear that the exact relationship between and �max is

�max � �� � � � for � ����

Page 65: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

Exercise� The unstable plant

P �s� ��

�s� ���s� � �s � ��

is to be compensated by the controller

C�s� �K�s � ��

�s � ��

The gain K will be adjusted for closed�loop system stability� By using

the Routh�Hurwitz test� show that the feedback system is stable if and

only if � K � �� and ����K �K�� � �� So the gain K should be

selected from the range � K �p

������ ��� � ���

��� Stability Robustness

Parametric Uncertainty

The Routh�Hurwitz stability test determines stability of a character�

istic polynomial ��s� with �xed coe�cients� If there are only a few

uncertain parameters �due to plant uncertainty� or free parameters �of

controller� in the coe�cients of ��s�� then it is still possible to use the

Routh�Hurwitz test to determine the set of all admissible parameters

for stability� This point was illustrated with the above examples� If the

number of variable parameters is large� then the analysis is cumbersome�

In this section� we will see simple robust stability tests for plants with

uncertain coe�cients in the transfer function�

����� Uncertain Parameters in the Plant

Recall that P �s� is a mathematical model of the physical system� De�

pending on the con�dence level with this model� each coe�cient of

NP �s� and DP �s� can be assumed to lie in an interval of the real line�

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For example� taking only one mode of a �exible beam� its transfer func�

tion �from force input to acceleration output� can be written as

P �s� �Kp

s� � �� os � �o�

Suppose Kp � ��� � �� �� � � ��� � �� �� and o � �� � ���� Then�

NP �s� � q�� where q� � ��� � �� �� and DP �s� � r�s� � r�s� r�� where

r� � � � ��� r� � � � ���� r� � ��� � �����

Generalizing this representation� we assume that P � NP

DPwhere

NP �s� � q�sm � q�s

m�� � � � �� qm � qk � q�k � q�k � � k � �� ����m�

DP �s� � r�sn � r�s

n�� � � � �� rn r� � r�� � r�� � � � � �� ���� n�

Since P �s� is proper� n � m� The set of all possible plant transfer

functions �determined from the set of all possible coe�cients of NP �s�

and DP �s�� is denoted by Pq�r� More precisely�

Pq�r �

NP �s�

DP �s��q�s

m � q�sm�� � � � �� qm

r�sn � r�sn�� � � � �� rn�qk � q�k � q�k �

r� � r�� � r�� �

where r�� � r�� � � � � � r

�m� r

�m� q

�� � q

�� � � � � � q

�n � q

�n are given upper and lower

bounds of the plant parameters� In the literature� the class of uncertain

plants of the form Pq�r is called interval plants� Total number of uncer�

tain parameters in this description is �n�m���� In the parameter space

IR�n�m���� the set of all possible parameters is a !multi�dimensional

box�" for example� when n � m � � � � the set is a rectangle� when

n�m�� � the set is a three dimensional box� and for �n�m��� � �

the set is a polytope�

The feedback system formed by a �xed controller C � NC

DCand an

uncertain plant P � Pq�r is said to be robustly stable if all the roots of

��s� � DC�s�DP �s� � NC�s�NP �s�

are in C� for all P � NP

DP� Pq�r�

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Introduction to Feedback Control Theory �

For each qk� k � �� � � � �m� we assume that it can take any value

within the given interval� independent of the values of the other coe��

cients� Same assumption is made for r�� � � �� � � � � n� For example� if

two parameters are related by� say� r� � x � � and r� � � � x � �x��

where x � � � ��� then a conservative assumption would be r� � �� � ��

and r� � � � ������� In the �r�� r�� parameter space� the set Rr��r� is a

rectangle that includes the line Lr��r� �

Rr��r� � f�r�� r�� � r� � �� � ��� r� � � � ������gLr��r� � f�r�� r�� � r� � x� �� r� � � � x� �x�� x � � � ��g�

If the closed�loop system is stable for all values of �r�� r�� in the set

Rr��r� � then it is stable for all values of �r�� r�� in Lr��r� � However� the

converse is not true� i�e�� there may be a point in Rr��r� for which the

system is not stable� while the system might be stable for all points in

Lr��r� � This is the conservatism in transforming a dependent parameter

uncertainty to an independent parameter uncertainty�

����� Kharitanov�s Test for Robust Stability

Consider a feedback system formed by a �xed controller C�s� � NC�s�DC �s�

and an uncertain plant P �s� � NP �s�DP �s�

� Pq�r� To test robust stability�

�rst construct the characteristic polynomial

��s� � DC�s�DP �s� � NC�s�NP �s�

with uncertain coe�cients� Note that from the upper and lower bounds

of the coe�cients of DP �s� and NP �s� we can determine �albeit in a

conservative fashion� upper and lower bounds of the coe�cients of ��s��

Example ��� Let NC�s� � �s���� DC�s� � �s���s��� and DP �s� �

r�s� � r�s

� � r�s � r�� NP �s� � q�� with r� � � � ����� r� � � � �����

r� � � � ��� r� � �� � ���� and q� � � ��� Then ��s� is in the form

��s� � �s� � �s � ���r�s� � r�s

� � r�s � r�� � �s � ���q��

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�� H� �Ozbay

� r�s� � �r� � �r��s

� � �r� � �r� � �r��s�

� �r� � �r� � �r��s� � ��r� � �r� � q��s � ��r� � �q��

� a�s� � a�s

� � a�s� � a�s

� � a�s � a�

where a� � � � ����� a� � � � ����� a� � �� � ������ a� � � � ������

a� � � � ���� a� � �� � ���� We assume that the parameter vector

a�� � � � � a�� can take any values in the subset of IR� determined by the

above intervals for each component� In other words� the coe�cients vary

independent of each other� However� there are only �ve truly free para�

meters �r�� � � � � r�� q��� which means that there is a dependence between

parameter variations� If robust stability can be shown for all possible

values of the parameters ak in the above intervals� then robust stability

of the closed�loop system can be concluded� but the converse is not

true� In that sense� by converting plant �and�or controller� parameter

uncertainty into an uncertainty in the coe�cients of the characteristic

polynomial� some conservatism is introduced�

Now consider a typical characteristic polynomial

��s� � a�sN � a�s

� � � � � � aN

where each coe�cient ak can take any value in a given interval a�k � a�k ��

k � �� �� � � � � N � independent of the values of aj � j � k� The set of

all possible characteristic equations is de�ned by the upper and lower

bounds of each coe�cient� It will be denoted by Xa� i�e�

Xa �� fa�sN � � � � � aN � ak � a�k � a�k � � k � �� �� � � � � Ng�

Theorem ��� Kharitanov s Theorem� � All polynomials in Xa are

stable if and only if the following four polynomials a��s�� � � � � a��s� are

stable�

a��s� � a�N � a�N��s � a�N��s� � a�N��s

� � a�N��s� � a�N��s

� � � � �a��s� � a�N � a�N��s � a�N��s

� � a�N��s� � a�N��s

� � a�N��s� � � � �

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Introduction to Feedback Control Theory ��

a��s� � a�N � a�N��s � a�N��s� � a�N��s

� � a�N��s� � a�N��s

� � � � �a��s� � a�N � a�N��s � a�N��s

� � a�N��s� � a�N��s

� � a�N��s� � � � �

In the literature� the polynomials a��s�� � � � � a��s� are called Kharit�

anov polynomials� By virtue of Kharitanov�s theorem� robust stability

can be checked by applying the Routh�Hurwitz stability test on four

polynomials� Considering the complexity of the parameter space� this

is a great simpli�cation� For easily accessible proofs of Kharitanov�s

theorem see �� pp� ���� and �� pp� ���������

����� Extensions of Kharitanov�s Theorem

Kharitanov�s theorem gives necessary and su�cient conditions for ro�

bust stability of the set Xa� On the other hand� recall that for a �xed

controller and uncertain plant in the set Pq�r the characteristic polyno�

mial is in the form

��s� � DC�s� �r�sn � r�s

n�� � � � � � rn�

� NC�s� �q�sm � q�s

m�� � � � � � qm�

where rk � r�k � r�k �� k � �� � � � � n� and q� � q�� � q�� �� � � �� � � � �m�

Let us denote the set of all possible characteristic polynomials corres�

ponding to this uncertainty structure by Xq�r� As seen in the previous

section� it is possible to de�ne a larger set of all possible characteristic

polynomials� denoted by Xa� and apply Kharitanov�s theorem to test ro�

bust stability� However� since Xq�r is a proper subset of Xa� Kharitanov�s

result becomes a conservative test� In other words� if four Kharitanov

polynomials �determined from the larger uncertainty set Xa� are stable�

then all polynomials in Xq�r are stable� but the converse is not true� i�e��

one of the Kharitanov polynomials may be unstable� while all polyno�

mials in Xq�r are stable� There exists a non�conservative test for robust

stability of the polynomials in Xq�r� it is given by the result stated below�

called the � edge theorem ��� or generalized Kharitanov�s theorem ���

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�� H� �Ozbay

First de�ne N��s�� � � � �N��s�� four Kharitanov polynomials corres�

ponding to uncertain polynomial NP �s� � q�sm�� � ��qm� and similarly

de�ne D��s�� � � � �D��s�� four Kharitanov polynomials corresponding to

uncertain polynomial DP �s� � r�sn � � � � � rn� Then� for all possible

combinations of i� � f�� �� � �g� and �i�� i�� � f��� �� ��� ��� ��� �� ��� ��gde�ne �� polynomials� which depend on a parameter ��

e�����s� �� � Ni��s�NC�s� � ��Di��s� � ��� ��Di� �s��DC�s��

Similarly� for all possible combinations of i� � f�� �� � �g� and �i�� i�� �f��� �� ��� ��� ��� �� ��� ��g de�ne the next set of �� � dependent poly�

nomials�

e������s� �� � Di��s�DC�s� � ��Ni��s� � ��� ��Ni��s��NC�s��

Theorem ��� ��� Assume that all the polynomials in Xq�r have the

same degree� Then� all polynomials in Xq�r are stable if and only if

e��s� ��� � � � � e���s� �� are stable for all � � � � ���

This result gives a necessary and su�cient condition for robust sta�

bility of polynomials in Xq�r� The test is more complicated than Khar�

itanov�s robust stability test� it involves checking stability of � poly�

nomials for all values of � � � � ��� For each ek�s� �� it is easy to

construct the Routh table in terms of � and test stability of ek for all

� � � � ��� This is a numerically feasible test� However� since there

are in�nitely many possibilities for �� technically speaking one needs

to check stability of in�nitely many polynomials� For the special case

where the controller is �xed as a �rst order transfer function

C�s� �Kc�s� z�

�s� p�� where Kc� z� p� are �xed and z � p�

the test can be reduced to checking stability of �� polynomials only�

Using the above notation� let N��s�� � � � �N��s� and D��s�� � � � �D��s�

be the Kharitanov polynomials for NP �s� � �q�sm � � � � � qm� and

Page 71: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

DP �s� � �r�sn � � � � � rn�� respectively� De�ne Pq�r as the set of all

plants and assume that � � r�� � r�� �� i�e�� the degree of DP �s� is �xed�

Theorem ��� ���� The closed�loop system formed by the plant P and

the controller NC

DC� where NC�s� � Kc�s � z� and DC�s� � �s � p�� is

stable for all P � Pq�r if and only if the following �� polynomials are

stable�

NC�s�Ni��s� � DC�s�Di��s�

where i� � f�� �� � �g and i� � f�� �� � �g�

This result is called the �� plant theorem� and it remains valid for

slightly more general cases in which the controller NC�DC is

NC�s� � Kc�s� z�UN�s�RN �s� ������

DC�s� � s��s� p�UD�s�RD�s� ������

where Kc� z� p are real numbers� � � � is an integer� UN �s� and UD�s�

are anti�stable polynomials �i�e�� all roots in C��� and RN �s� and RD�s�

are in the form R�s��� where R�s� is an arbitrary polynomial� ��� When

the controller is restricted to this special structure� � edge theorem re�

duces to checking stability of the closed loop systems formed by the con�

troller NC�DC and �� plants P �s� � Ni��s��Di��s�� for i� � f�� �� � �g�and i� � f�� �� � �g�

For the details and proofs of � edge theorem and �� plant theorem

see �� pp� �� ������ and �� pp� ��� ���

��� Exercise Problems

�� Given a characteristic polynomial ��s� � a�s� � a�s

� � a�s � a�

with coe�cients in the intervals ��� � a� � ���� � � a� � ��

Page 72: Feedback Control Systems- HOzbay

�� H� �Ozbay

� � a� � � � � a� � �� Using Kharitanov�s test� show that we

do not have robust stability� Now suppose a� and a� satisfy

a� � � � �x� a� � � � x� where � � � x � � �

Do we have robust stability� Hint� Use the Routh Hurwitz test

here� Kharitanov�s test does not give a conclusive answer�

�� Consider the standard feedback control system with an interval

plant P � Pq�r�

Pq�r �

P �s� �

q�s� � q�s

� � q�s � q�r�s� � r�s� � r�s� � r�s � r�

where q� � ���� � ������ q� � ���� � � ���� q� � �� � ����

q� � ��� � q�� r� � ��� � ����� r� � ���� � ������ r� � �� � ����

r� � �� � ���� r� � ���� � ����� By using the �� plant theorem

�nd the maximum value of q such that there exists a robustly

stabilizing controller of the form

C�s� �K �s � ��

s

for the family of plants Pq�r� Determine the corresponding value

of K�

Page 73: Feedback Control Systems- HOzbay

Chapter �

Root Locus

Recall that the roots of the characteristic polynomial

��s� � DP �s�DC�s� � NP �s�NC�s�

are the poles of the feedback system formed by the controller C �

NC�DC and the plant P � NP �DP � In Chapter we saw that in or�

der to achieve a certain type of performance objectives� the dominant

closed�loop poles must be placed in a speci�ed region of the complex

plane� Once the pole�zero structure of G�s� � P �s�C�s� is �xed� the

gain of the controller can be adjusted to see whether the design spe�

ci�cations are met with this structural choice of G�s�� In the previous

chapter we also saw that robust stability can be tested by checking sta�

bility of a family of characteristic polynomials depending on a parameter

�e�g� � of the ��edge theorem�� In these examples� the characteristic

polynomial is an a�ne function of a parameter� The root locus shows

the closed�loop system poles as this parameter varies�

Numerical tools� e�g�� Matlab� can be used to construct the root

locus with respect to a parameter that appears nonlinearly in the char�

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�� H� �Ozbay

−4 −3 −2 −1 0 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Figure ���� Root locus as a function of h � � � �����

acteristic polynomial� For example� consider the plant

P �s� ���sh������ sh�� � ��

s ��sh����� � sh�� � �������

as an approximation of a system with a time delay and an integrator�

Let the controller for the plant P �s� be

C�s� � �� ����

s�

Then the characteristic polynomial

��s� � �h�

��s� � ���s� � �� s � ���� �

h

�s �s� � �� s� ����

is a nonlinear function of h� due to h� terms� Figure ��� shows the

closed�loop system pole locations as h varies from a lower bound

hmin � � to an upper bound hmax � ���� The �gure is obtained by

computing the roots of ��s� for a set of values of h � hmin � hmax��

As mentioned above� the root locus primarily deals with �nding the

roots of a characteristic polynomial that is an a�ne function of a single

parameter� say K�

��s� � D�s� � KN�s� �����

Page 75: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

where D�s� and N�s� are �xed monic polynomials �i�e�� coe�cient of the

highest power is normalized to ��� In particular� ��edge polynomials

are in this form� For example� recall that

e��s� �� � N��s�NC�s� � ��D��s� � ��� ��D��s��DC�s�

� �N��s�NC�s� � D��s�DC�s�� � ��D��s�� D��s��DC�s��

By de�ning K � ���� �i�e� � � K

K���� N � �N�NC � D�DC� and

D � �N�NC � D�DC�� it can be shown that the roots of e��s� �� over

the range of � � � � �� are the roots of ��s� de�ned in ����� over

the range of K � � � ���� If N and D are not monic� the highest

coe�cient of D can be factored out of the equation and the ratio of the

highest coe�cient of N to that of D can be absorbed into K�

As another example� consider a �xed plant P � NP �DP and a PI

controller with �xed proportional gain Kp and variable integral gain Ki

C�s� � Kp �Ki

s�

The characteristic equation is

��s� � s�DP �s� � KpNP �s�� � KiNP �s� �

which is in the form ����� with K � Ki� D�s� � s �DP �s� �KpNP �s��

and N�s� � NP �s��

The most common example of ����� is the variable controller gain

case� when the controller and plant are expressed in the pole�zero form

as

P �s� � KP�s� zi�� � � � �s� zim�

�s� pi�� � � � �s� pin�

C�s� � KC�s� zj�� � � � �s� zjm�

�s� pj�� � � � �s� pjn�

the characteristic equation is as ����� with K � KPKC � and

D�s� � �s� zi�� � � � �s� zim��s� zj�� � � � �s� zjm�

N�s� � �s� pi�� � � � �s� pin��s� pj�� � � � �s� pjn�

Page 76: Feedback Control Systems- HOzbay

�� H� �Ozbay

To simplify the notation set m �� im � jm� and n �� in � jn� and

enumerate poles and zeros of G�s� � P �s�C�s� in such a way that

G�s� � K�s� z�� � � � �s� zm�

�s� p�� � � � �s� pn�� K

N�s�

D�s��

then assuming K � � the characteristic equation ����� is equivalent to

� � G�s� � � �� � � KG��s� � � ��� �

where G��s� � N�s��D�s�� which is equal to G�s� evaluated at K � ��

The purpose of this chapter is to examine how closed�loop system poles

�roots of the characteristic equation that is either in the form ������ or

��� �� change as K varies from � to ��� or from � to ���

��� Root Locus Rules

The usual root locus �abbreviated as RL� shows the locations of the

closed�loop system poles as K varies from � to ��� The roots of D�s��

p�� � � � � pn� are the poles of the open�loop system G�s�� and the roots of

N�s�� z�� � � � � zm� are the zeros of G�s�� Since P �s� and C�s� are proper�

G�s� is proper and hence n � m� So the degree of the polynomial ��s�

is n and it has exactly n roots�

Let the closed�loop system poles� i�e�� roots of ��s�� be denoted by

r��K�� � � � � rn�K�� Note that these are functions of K� whenever the

dependence on K is clear� they are simply written as r�� � � � � rn� The

points in C that satisfy ��� � for some K � � are on the RL� Clearly� a

point r � C is on the RL if and only if

K � � �

G��r�� �����

The condition ����� can be separated into two parts�

jKj ��

jG��r�j �����

Page 77: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

� K � �� � ����� ������ � � G��r�� � � �� �� �� � � � � �����

The phase rule ����� determines the points in C that are on the RL� The

magnitude rule ����� determines the gain K � � for which the RL is at

a given point r� By using the de�nition of G��s�� ����� can be rewritten

as

���� ������ �nXi��

� �r � pi��mXj��

� �r � zj�� ����

Similarly� ����� is equivalent to

K �

Qni�� jr � pijQmj�� jr � zj j � �����

����� Root Locus Construction

There are several software packages available for generating the root

locus automatically for a given G� � N�D� In particular� the related

Matlab commands are rlocus and rlocfind� In many cases� approx�

imate root locus can be drawn by hand using the rules given below�

These rules are determined from the basic de�nitions ������ ����� and

������

�� The root locus has n branches� r��K�� � � � � rn�K��

�� Each branch starts �K �� �� at a pole pi and ends �as K � ��

at a zero zj � or converges to an asymptote� Rej�� � where R��and �� is determined from the formula

�n�m��� � ���� ������� � � �� � � � � �n�m� ���

� There are �n�m� asymptotes with angles ��� The center of the

asymptotes �i�e�� their intersection point on the real axis� is

�a ��Pn

i�� pi�� �Pm

j�� zj�

n�m�

Page 78: Feedback Control Systems- HOzbay

�� H� �Ozbay

�� A point x � IR is on the root locus if and only if the total number

of poles pi�s and zeros zj �s to the right of x �i�e�� total number of

pi�s with Re�pi� � x plus total number of zj �s with Re�zj� � x�

is odd� Since G��s� is a rational function with real coe�cients�

poles and zeros appear in complex conjugates� so when counting

the number of poles and zeros to the right of a point x � IR we

just need to consider the poles and zeros on the real axis�

�� The values of K for which the root locus crosses the imaginary

axis can be determined from the Routh�Hurwitz stability test�

Alternatively� we can set s � j in ����� and solve for real and

K satisfying

D�j � � KN�j � � ��

Note that there are two equations here� one for the real part and

one for the imaginary part�

�� The break points �intersection of two branches on the real axis�

are feasible solutions �satisfying rule &�� of

d

dsG��s� � �� �����

� Angles of departure �K �� �� from a complex pole� or arrival

�K � ��� to a complex zero� can be determined from the phase

rule� See example below�

Let us now follow the above rules step by step to construct the root

locus for

G��s� ��s � �

�s� ���s � ���s � � � j���s � �� j���

First� enumerate the poles and zeros as p� � �� � j�� p� � �� � j��

p� � ��� p� � �� z� � � � So� n � � and m � ��

Page 79: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

�� The root locus has four branches�

�� Three branches converge to the asymptotes whose angles are ����

���� and ����� and one branch converges to z� � � �

� Center of the asymptotes is � � ���� � �� � � �

�� The intervals ��� � ��� and � � �� are on the root locus�

�� The imaginary axis crossings are the feasible roots of

� � � j�� � � � � � j�� � ���� � K�j � � � � ������

for real and K� Real and imaginary parts of ������ are

� � � � � ��� � K � �

j ���� � � �� � K� � ��

They lead to two feasible pair of solutions �K � ���� � � �� and

�K � ����� � � ������

�� Break points are the feasible solutions of

s� � �s� � ���s� � ���s� ��� � ��

Since the roots of the above polynomial are ����� j���� and

����� j����� there is no solution on the real axis� hence no

break points�

� To determine the angle of departure from the complex pole p� �

���j� let # represent a point on the root locus near the complex

pole p�� and de�ne vi� i � �� � � � � �� to be the vectors drawn from

pi� for i � �� � � � � �� and from z� for i � �� as shown in Figure ����

Let ��� � � � � �� be the angles of v�� � � � � v�� The phase rule implies

��� � �� � �� � ���� �� � ����� ������

As # approaches to p�� �� becomes the angle of departure and the

other �i�s can be approximated by the angles of the vectors drawn

Page 80: Feedback Control Systems- HOzbay

� H� �Ozbay

x

x

o x

x

Im

Re

v

vv

-4+j2

-5 -3

1

54

1

-4-j2

3

2v

v

Figure ���� Angle of departure from �� � j��

from the other poles� and from the zero� to the pole p�� Thus

�� can be solved from ������ where �� � ���� �� � tan������

�� � ���� � tan��� �� �� and �� � ��� � tan��� �� �� That yields

�� � �����

The exact root locus for this example is shown in Figure �� � From

the results of item &� above� and the shape of the root locus it is

concluded that the feedback system is stable if

� � K � �����

i�e�� by simply adjusting the gain of the controller� the system can be

made stable� In some situations we need to use a dynamic controller to

satisfy all the design requirements�

����� Design Examples

Example ��� Consider the standard feedback system with a plant

P �s� ��

���

�s � ���s � ��

Page 81: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

−8 −6 −4 −2 0 2−6

−4

−2

0

2

4

6

Real Axis

Imag

Axis

RL for G1(s)=(s+0.3)/(s^4+12s^3+47s^2+40s−100)

Figure �� � Root Locus for G��s� � �s����s����s����s���j���s���j�� �

and design a controller such that

� the feedback system is stable�

� PO � ��%� ts � � sec� and ess � � when r�t� � U�t�

� ess is as small as possible when r�t� � tU�t��

It is clear that the second design goal �the part that says that ess should

be zero for unit step reference input� cannot be achieved by a simple

proportional controller� To satisfy this condition� the controller must

have a pole at s � �� i�e�� it must have integral action� If we try an

integral control of the form C�s� � Kc�s� with Kc � �� then the root

locus has three branches� the interval �� � �� is on the root locus� three

asymptotes have angles f���� ���������g with a center at �a � ���

and there is only one break point at �� � �p�� See Figure ���� From

the location of the break point� center� and angles of the asymptotes�

it can be deduced that two branches �one starting at p� � ��� and the

other one starting at p� � �� always remain to the right of the point

��� On the other hand� the settling time condition implies that the

real parts of the dominant closed�loop system poles must be less than

Page 82: Feedback Control Systems- HOzbay

� H� �Ozbay

or equal to ��� So� a simple integral control does not do the job� Now

try a PI controller of the form

C�s� � Kc�s� zc�

sKc � ��

In this case� we can select zc � �� to cancel the pole at p� � �� and

the system e�ectively becomes a second�order system� The root locus

for G��s� � ��s�s � �� has two branches and two asymptotes� with

center �a � �� and angles f��������g� the break point is also at ���

The branches leave �� and �� and go toward each other� meet at ���

and tend to in�nity along the line Re�s� � ��� Indeed� the closed�loop

system poles are

r��� � �� p��K where K � Kc���� �

The steady state error� when r�t� is unit ramp� is ��K� So K needs

to be as large as possible to meet the third design condition� Clearly�

Re�r���� � �� for all K � �� that satis�es the settling time requirement�

The percent overshoot is less than ��% if � of the roots r��� is greater

than ���� A simple algebra shows that � � ��pK� hence the design

conditions are met if K � ���� �� i�e� Kc � �� Thus a PI controller

that solves the design problem is

C�s� � ��s � ��

s�

The controller cancels a stable pole �at s � ��� of the plant� If

there is a slight uncertainty in this pole location� perfect cancelation

will not occur and the system will be third�order with the third pole

at r� �� ��� Since the zero at zo � �� will approximately cancel

the e�ect of this pole� the response of this system will be close to the

response of a second�order system� However� we must be careful if the

pole zero cancelations are near the imaginary axis because in this case

small perturbations in pole location might lead to large variations in

the feedback system response� as illustrated with the next example�

Page 83: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory

−4 −3 −2 −1 0 1 2−3

−2

−1

0

1

2

3

Real Axis

Imag

Axis

rlocus(1,[1,3,2,0])

Figure ���� Root locus for Example ����

Example ��� A �exible structure with lightly damped poles has trans�

fer function in the form

P �s� � ��

s��s� � �� �s � ����

By using the root locus� we can see that the controller

C�s� � Kc�s� � �� �s � ����s � ����

�s � r���s � ��

stabilizes the feedback system for su�ciently large r and an appropriate

choice of Kc� For example� let � � �� � � ��� and r � ��� Then the

root locus of G��s� � P �s�C�s��K� where K � Kc �� � is as shown in

Figure ���� For K � ��� the closed�loop system poles are�

f����� j��� � ����� j���� � ���� j���� � �����g�

Since the poles ���� j���� are canceled by a pair of zeros at the same

point in the closed�loop system transfer function T � G�� � G���� the

dominant poles are at ����� and ����� j���� �they have relatively

large negative real parts and the damping ratio is about �����

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� H� �Ozbay

−12 −10 −8 −6 −4 −2 0 2

−4

−3

−2

−1

0

1

2

3

4

5

Real Axis

Imag

Axis

Figure ���� Root locus for Example ��� �a��

Now� suppose that this controller is �xed and the complex poles of

the plant are slightly modi�ed by taking � � ���� and � � ���� The

root locus corresponding to this system is as shown in Figure ���� Since

lightly damped complex poles are not perfectly canceled� there are two

more branches near the imaginary axis� Moreover� for the same value

of K � ���� the closed�loop system poles are

f����� j��� � ����� j���� � ���� j��� � �����g�

In this case� the feedback system is unstable�

Example ��� An approximate transfer function of a DC motor �

pp� ������ � is in the form

Pm�s� �Km

s �s � ���m�� �m � ��

Note that if �m is large� then Pm�s� � Pb�s�� where

Pb�s� �Kb

s�

is the transfer function of a rigid beam� In this example� the general

class of plants Pm�s� will be considered� Assuming that pm � ���m

and

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Introduction to Feedback Control Theory �

−12 −10 −8 −6 −4 −2 0

−4

−3

−2

−1

0

1

2

3

4

Real Axis

Imag

Axis

Figure ���� Root locus for Example ��� �b��

Km are given� a �rst�order controller

C�s� � Kc�s� zc�

�s� pc�������

will be designed� The aim is to place the closed�loop system poles

far from the Im�axis� Since the order of G��s� � Pm�s�C�s��KmKc

is three� the root locus has three branches� Suppose the desired closed

loop poles are given as p�� p� and p�� Then� the pole placement problem

amounts to �nding fKc� zc� pcg such that the characteristic equation is

��s� � �s� p���s� p���s� p��

� s� � �p� � p� � p��s� � �p�p� � p�p� � p�p��s� p�p�p��

But the actual characteristic equation� in terms of the unknown con�

troller parameters� is

��s� � s�s� pm��s� pc� � K�s� zc�

� s� � �pm � pc�s� � �pmpc � K�s�Kzc

where K �� KmKc� Equating the coe�cients of the desired ��s� to the

coe�cients of the actual ��s�� three equations in three unknowns are

obtained�

pm � pc � p� � p� � p�

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� H� �Ozbay

−10 −8 −6 −4 −2 0 2−6

−4

−2

0

2

4

6

Real Axis

Imag

Axis

Figure ��� Root locus for Example �� �a��

pmpc � K � p�p� � p�p� � p�p�

Kzc � p�p�p�

From the �rst equation pc is determined� then K is obtained from the

second equation� and �nally zc is computed from the third equation�

For di�erent numerical values of pm� p�� p� and p� the shape of the

root locus is di�erent� Below are some examples� with the corresponding

root loci shown in Figures �������

�a� pm � ������ p� � p� � p� � �� ��

K � ���� pc � ����� zc � ������

�b� pm � ����� p� � ��� p� � ��� p� � � ��

K � ���� pc � ����� zc � ��� �

�c� pm � ��� p� � ���� p� � �� � j�� p� � ��� j� ��

K � � pc � ��� zc � ��� � �

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Introduction to Feedback Control Theory

−6 −5 −4 −3 −2 −1 0−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real Axis

Imag

Axis

Figure ���� Root locus for Example �� �b��

−16 −14 −12 −10 −8 −6 −4 −2 0 2−6

−4

−2

0

2

4

6

Real Axis

Imag

Axis

Figure ���� Root locus for Example �� �c��

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� H� �Ozbay

Example ��� The plant ����� with a �xed value of h � � will be

controlled by using a �rst�order controller in the form ������� The

open�loop transfer function is

P �s�C�s� � Kc�s� � s� ��s� zc�

s�s� � s � ��s� pc�

and the root locus has four branches� The �rst design requirement is

to place the dominant poles at r��� � ����� The steady state error for

unit ramp reference input is

ess �pc

Kczc�

Accordingly� the second design speci�cation is to make the ratio Kczc�pc

as large as possible�

The characteristic equation is

��s� � s�s� � s� ��s� pc� � Kc�s� � s � ��s� zc��

and it is desired to be in the form

��s� � �s � ������s� r���s� r��

for some r��� with Re�r���� � �� which implies that

��s�

����s��� �

� ��d

ds��s�

����s��� �

� �� ���� �

Conditions ���� � give two equations�

�������� � pc�� �� �Kc���� � zc� � �

�� �Kc � ����� �������� � pc� � ��Kc���� � zc� � �

from which zc and pc can be solved in terms of Kc� Then� by simple

substitutions� the ratio to be maximized� Kczc�pc� can be reduced to

Kczzpc

� ���Kc� ����

�������Kc� ����

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Introduction to Feedback Control Theory �

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1.5

−1

−0.5

0

0.5

1

1.5

Real Axis

Imag

Axis

Figure ����� Root locus for Example ����

The maximizing value of Kc is ������ it leads to pc � ������� and

zc � ����� � For this controller� the feedback system poles are

f����� � j�� � ������ j�� � ������ �����g�

The root locus is shown in Figure �����

��� Complementary Root Locus

In the previous section� the root locus parameter K was assumed to

be positive and the phase and magnitude rules were established based

on this assumption� There are some situations in which controller gain

can be negative as well� Therefore� the complete picture is obtained by

drawing the usual root locus �for K � �� and the complementary root

locus �for K � ��� The complementary root locus rules are

�� ��� �

nXi��

� �r � pi��mXj��

� �r � zj�� � � �� �� �� � � � ������

jKj �

Qni�� jr � pijQmj�� jr � zj j � ������

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�� H� �Ozbay

−20 −15 −10 −5 0 5 10 15 20 25−10

−8

−6

−4

−2

0

2

4

6

8

10

Real Axis

Imag

Axis

Figure ����� Complementary root locus for Example �� �

Since the phase rule ������ is the ���� shifted version of ����� the

complementary root locus is obtained by simple modi�cations in the

root locus construction rules� In particular� the number of asymptotes

and their center are the same� but their angles ���s are given by

�� ���

�n�m�� ����� � � �� � � � � �n�m� ���

Also� an interval on the real axis is on the complementary root locus if

and only if it is not on the usual root locus�

Example ��� In the Example �� given above� if the problem data is

modi�ed to pm � ��� p� � ��� and p��� � �� j� then the controller

parameters become

K � ��� pc � ��� zc � ���

Note that the gain is negative� The roots of the characteristic equation

as K varies between � and �� form the complementary root locus� see

Figure �����

Example ��� �Example ��� revisited�� In this example� if K increases

from �� to ��� the closed�loop system poles move along the

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Introduction to Feedback Control Theory ��

−4 −2 0 2 4 6 8−5

−4

−3

−2

−1

0

1

2

3

4

5

Real Axis

Imag

Axis

Figure ����� Complementary and usual root loci for Example ����

complementary root locus� and then the usual root locus as illustrated

in Figure �����

��� Exercise Problems

�� Let the controller and the plant be given as

C�s� �K�s� � ���s� ��

s�P �s� �

�s � ��

�s� � ���s � ����

Draw the root locus with respect to K without using any numer�

ical tools for polynomial root solution� Show as much detail as

possible�

�� Consider the feedback system with

C�s� �K

�s � ��P �s� �

�s� ���s � ���

�a� Find the range of K for which the feedback system is stable�

�b� Let r�� r�� r� be the poles of the feedback system� It is desired

to have

Re�rk� � �x for all k � �� �� �

Page 92: Feedback Control Systems- HOzbay

�� H� �Ozbay

for some x � �� so that the feedback system is stable� De�

termine the value of K that maximizes x�

Hint� Draw the root locus �rst�

� �a� Draw the complementary root locus for

G��s� ��s � �

�s� ���s � ���s � � � j���s � �� j���

and connect it to the root locus shown in Figure �� �

�b� Let r�� � � � � r� be the roots of � � KG��s� � �� It is desired

to have Re�ri� � ��� i � �� � � � �� for the largest possible

� � �� Determine the value of K achieving this design goal�

and show all the corresponding roots on the root locus�

�� For the plant

P �s� ��

s�s � ��

design a controller in the form

C�s� � K�s� zc�

�s� pc�

such that

�i� the characteristic polynomial can be factored as

��s� � �s� � �� ns � �n��s � r�

where � � ��� � ��� � n � � and r � ��

�ii� the ratio Kzc�pc is as large as possible�

Draw the root locus for this system�

�� Consider the plant

P �s� ��s � ��

�s� � �s � ���

Page 93: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

Design a controller in the form

C�s� �K�s� z�

�s� � as � b�� where K� z� a� b are real numbers

such that

� the feedback system is stable with four closed�loop poles sat�

isfying r� � 'r� � r� � 'r��

� the steady state tracking error is zero for r�t� � sin�t�U�t��

Draw the root locus for this system and show the location of the

roots for the selected controller gain�

�� Consider the plant

P �s� ���� �s�

s �� � �s�

where � is an uncertain parameter� Determine the values of � for

which the PI controller

C�s� � � ���

s

stabilizes the system� Draw the closed�loop system poles as �

varies from � to ��� Find the values of � such that the dominant

closed�loop poles have damping coe�cient � � ����� What is the

largest possible � and the corresponding ��

Useful Matlab commands are roots� rlocus� rlocfind� and sgrid�

Page 94: Feedback Control Systems- HOzbay
Page 95: Feedback Control Systems- HOzbay

Chapter �

Frequency Domain

Analysis Techniques

Stability of the standard feedback system� Figure ����� is determined

by checking whether the roots of � � G�s� � � are in the open left half

plane or not� where G�s� � P �s�C�s� is the given open�loop transfer

function� Suppose that G�s� has �nitely many poles� then it can be

written as G � NG�DG� where NG�s� has no poles and DG�s� is a

polynomial containing all the poles of G�s�� Clearly�

� � G�s� � � �� F �s� ��DG�s� � NG�s�

DG�s�� ��

The zeros of F �s� are the closed�loop system poles� while the poles of

F �s� are the open�loop system poles� The feedback system is stable

if and only if F �s� has no zeros in C�� The Nyquist stability test

uses Cauchy�s Theorem to determine the number of zeros of F �s� in

C�� This is done by counting the number of encirclements of the origin

by the closed path F �j � as increases from �� to ��� Cauchy�s

theorem �or Nyquist stability criterion� not only determines stability

��

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�� H� �Ozbay

Im

Re

Γ

o

o

x

x

x x

x

Im

Re

ΓFs

F(s) planes-planeΓF( )s

Figure ���� Mapping of contours�

of a feedback system� but also gives quantitative measures on stability

robustness with respect to certain types of uncertainty�

��� Cauchy�s Theorem

A closed path in the complex plane is a positive contour if it is in the

clockwise direction� Given an analytic function F �s� and a contour

(s in C� the contour (F is de�ned as the map of (s under F ���� i�e�

(F �� F �(s�� See Figure ���� The contour (F is drawn on a new

complex plane called F �s� plane� The number of encirclements of the

origin by (F is determined by the number of poles and zeros of F �s�

encircled by (s� The exact relationship is given by Cauchy�s theorem�

Theorem ��� �Cauchy s Theorem� Assume that a positive contour

(s does not go through any pole or zero of F �s� in the s�plane� Then�

(F �� F �(s� encircles the origin of the F �s� plane

no � nz � np

Page 97: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

times in the positive direction� where nz and np are the number of zeros

and poles �respectively� of F �s� encircled by (s�

For a proof of this theorem see e�g�� � pp� �������� In Figure ����

(s encircles nz � � zero and np � � poles of F �s�� the number of positive

encirclements of the origin by the contour (F is no � ��� � ��� i�e�� the

number of encirclements of the origin in the counterclockwise direction

is � � �no�

��� Nyquist Stability Test

By using Cauchy�s theorem� stability of the feedback system can be

determined as follows�

�i� De�ne a contour (s encircling the right half plane in the clockwise

direction�

�ii� Obtain (F � � � (G and count its positive �clockwise� encircle�

ments of the origin� Let this number be denoted by no�

�iii� If the number of poles of G�s� in (s is np� then the number of

zeros of F �s� in (s �i�e�� the number of right half plane poles of

the feedback system� is nz � no � np�

In conclusion� the feedback system is stable if and only if

nz � � �� no � �np� which means that the map (G encircles

the point ��� in G�s� plane� np times in the counterclockwise direction�

This stability test is known as the Nyquist stability criterion�

If G�s� has no poles on the Im�axis� (s is de�ned as�

(s �� limR��

IR � CR

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�� H� �Ozbay

jR

Im

Re

-jR

x

x

ωj

ε R

o

-jωo

Figure ���� De�nition of (s�

where

IR �� fj � increases from �R to � RgCR �� fRej� � � decreases from

��

�to���g�

When G�s� has an imaginary axis pole� say at j o� a small neighborhood

of j o is excluded from IR and a small semicircle� C��j o�� is added to

connect the paths on the imaginary axis�

C��j o� �� fj o � �ej� � � increases from���

to��

�g�

where �� �� See Figure ����

Since (s is symmetric around the real axis and G�s� � G�s�� the

closed path (G is symmetric around the real axis� Therefore� once

(�G �� G�(�

s � is drawn for

(�s �� (s � fs � C � Im�s� � �g�

Page 99: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

the complete path (G is obtained by

(G � (�G � (�

G �

Moreover� in most practical cases� G�s� � P �s�C�s� is strictly proper�

meaning that

limR��

G�Rej�� � � � ��

In such cases� (�G is simply the path of G�j � as increases from � to

��� excluding Im�axis poles� This path of G�j � is called the Nyquist

plot� in Matlab it is generated by the nyquist command�

Example ��� The open�loop system transfer function considered in

Section ����� is in the form

G�s� � K�s � �

�s� ���s � ���s � � � j���s � �� j��� �����

and we have seen that the feedback system is stable forK � � ���� � ����� ��

In particular� for K � ��� the feedback system is stable� This result

can be tested by counting the number of encirclements of the critical

point� ��� by the Nyquist plot of G�j �� To get an idea of the gen�

eral shape of the Nyquist plot� �rst put s � j in ������ this gives

G�j � � NG�j��DG�j��

� Then multiply NG�j � and DG�j � by the complex

conjugate of DG�j � so that the common denominator for the real and

imaginary parts of G�j � is real and positive�

G�j � ������ �� � � ��� � � ��� � j � � � �� � � ���� �

� � � � � � ����� � ���� � � ����

Clearly� the real part is negative for all � � and the imaginary part

is zero at � � � and at � � ���� � Note that

Im�G�j ��

�� � for � � � ����

� � for � ���� �

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−4 −3 −2 −1 0 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real Axis

Imag

Axi

s

Figure �� � Nyquist plot of ����s����s����s����s���j���s���j�� �

Also� G�j�� � � and G�j���� � � ����� � These numerical values

can be used for roughly sketching the Nyquist plot� The exact plot

is shown in Figure �� � where the dashed line is G�j � for � � and

the arrows show the increasing direction of from �� to ��� Since

(s contains one pole in C� and (G encircles �� once in the counter

clockwise direction� the feedback system is stable� By looking at the

real axis crossings of (G� it is deduced that the feedback system remains

stable for all open�loop transfer functions �G�s� as long as �� � � �

�� ���� � ����� � This is consistent with the results of Section ������

Example ��� The Nyquist plot for the Example �� �part b� can be

obtained as follows� The open�loop transfer function is

G�s� ����� �s � �� �

s �s � ���� �s � �����

The system is strictly proper� so G�Rej�� � � for all � as R � �� In

this case� there is a pole on the imaginary axis� at s � �� so (s should

exclude it� The Nyquist plot will be obtained by drawing G�s� for

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Introduction to Feedback Control Theory ��

�i� s � �ej� with �� � and � varying from � to �� � and for �ii� s � j

where varying from � to ��� Simple substitutions give

G��ej�� � ����

�e�j�

and as � varies from � to �� this segment of the Nyquist plot follows the

quarter circle which starts at � ��� and ends at �j � ��� � For the second

segment of the Nyquist plot let s � j and as before� separate real and

imaginary parts of G�j ��

G�j � � ����� ���� � �� � j���� � � ������

������ ��� � � ��

Note that as � �� where � � �� �� we have

G�j�� � ������ j����

��

Also� the real and imaginary parts of G�j � are negative for all � ��

so this segment of the Nyquist plot follows a path which remains in the

third quadrant of the complex plane� Hence� the critical point� ��� is

not encircled by (G� This implies that the feedback system is stable�

because G�s� has no poles encircled in (s� The second segment of the

Nyquist plot is shown in Figure ����

��� Stability Margins

Suppose that for a given plant P and a controller C the feedback system

is stable� Stability robustness� with respect to certain special types of

perturbations in G � PC� can be determined from the Nyquist plot�

Three di�erent types of perturbations will be studied here

� gain scaling

� phase shift

� combined gain scaling and phase shift�

Throughout this section it is assumed that G�s� is strictly proper�

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�� H� �Ozbay

−2 −1.5 −1 −0.5 0−30

−20

−10

0

10

20

30

Real Axis

Imag

Axi

s

Figure ���� Nyquist plot of � ���s�� ��s�s�� ���s�� �� �

Case �� G has no poles in the open right half plane�

In this case� the closed�loop system is stable if and only if G�j � does

not encircle the critical point� ��� A generic Nyquist plot corresponding

to this class of systems is shown in Figure ���� �only the segment � �

is shown here��

The number of encirclements of the critical point does not change

for all Nyquist plots in the form kG�j � as long as

� � k ��

��

where �� � ��� � �� is the Re�axis crossing point closest to ���

Similarly� the critical point is not encircled for all e�j�G�j � provided

� � � � � minf� � � G�j c� � c is such that jG�j c�j � � g�

Here we assumed that � G�j c� � ��� � ��� Also� if jG�j �j � � for

all � then we de�ne � �� �� In the light of these observations� the

following de�nitions are made�

Page 103: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

Im

Re

β

α

-1

ϕ

Figure ���� Stability margins for G�s� with no pole in C��

GM� the gain margin is �� ��� log����� dB�

PM� the phase margin is � �in radians��

There might be some cases in which negative real axis crossings

occur� not only between � and �� but also between �� and ��� yet the

feedback system is stable� For example� consider the Nyquist plot shown

in Figure ���� The total number of encirclements of the critical point is

zero �once in counterclockwise and once in the clockwise direction� and

this does not change for all kG�j � when

k � �� ��

��� � �

���

����

The nominal value of the gain k � � falls in the second interval� so the

upper and lower gain margins are de�ned as

GMupp ��

��� � and GMlow �

��� ��

Page 104: Feedback Control Systems- HOzbay

�� H� �Ozbay

Re-1

Im

α

α

21

α

3

Figure ���� Upper and lower gain margins�

In such cases we can de�ne the relative gain margin as

GMrel ��GMupp

GMlow�����

�� log�������

� dB�

For good stability robustness to perturbations in the gain� we want

GMupp to be large and GMlow to be small�

Exercise� Draw the root locus for

G�s� �K�s � ������

�s � ��������s � �����s� � ���s � ������� �����

and show that for K � ���� � the system is stable� Draw the Nyquist

plot for this value of K and determine the upper� lower� and relative

gain margins and the phase margin�

Answer� GMupp � � dB and PM � ���

Figure ��� suggests that the distance between the critical point and

the Nyquist path can be dangerously small� yet the gain and phase

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Introduction to Feedback Control Theory ��

margins may be large� In this situation� simultaneous perturbations

in gain and phase of G�j � may change the number of encirclements�

and hence may destabilize the system� Therefore� the most meaningful

stability margin is the so�called vector margin� which is de�ned as the

smallest distance between G�j � and ��� i�e�

VM � � inf�jG�j �� ����j� ��� �

An upper bound for the VM can be obtained from the PM�

� inf�j� � G�j �j � j� � G�j c�j � � sin�

���

Simple trigonometric identities give the last equality from the de�nition

of � in Figure ���� Note that� if � is small� then is small� A rather

obvious upper bound for the VM can be determined from Figure ����

� minf j�� ��j � j�� ��j g�

Recall that the sensitivity function is de�ned as S�s� � ���G�s�����

So� by using the notation of Chapter � �system norms� we have

�� � sup�jS�j �j � kSkH� � �����

In other words� vector margin is the inverse of the H� norm of the

sensitivity function� Hence� the VM can be determined via ����� by

plotting jS�j �j and �nding its peak value ��� The feedback system

has !good" stability robustness �in the presence of mixed gain and

phase perturbations in G�j �� if the vector margin is !large"� i�e�� the

H� norm of the sensitivity function S is !small�" Since G�s� is assumed

to be strictly proper G��� � � and hence

lim���

jS�j �j � ��

Thus� kSk� � � and � � whenever G�s� is strictly proper�

Page 106: Feedback Control Systems- HOzbay

�� H� �Ozbay

Case �� G has poles in the open right half plane�

Again� suppose that the feedback system is stable� The only di�erence

from the previous case is the generic shape of the Nyquist plot� it en�

circles �� in the counterclockwise direction as many times as the number

of open right half plane poles of G� Otherwise� the basic de�nitions of

gain� phase� and vector margins are the same�

� the gain margin is the smallest value of k � � for which the

feedback system becomes unstable when G�j � is replaced by

kG�j ��

� the phase margin is the smallest phase lag � � for which the

feedback system becomes unstable when G�j � is replaced by

e�j�G�j ��

� the vector margin is the smallest distance between G�j � and

the critical point ���

The upper� lower and relative gain margins are de�ned similarly�

Exercise� Consider the system ����� with K � ���� Show that the

phase margin is approximately ���� the gain margin �i�e�� the upper

gain margin� is ����� �� dB and the relative gain margin is ��� ���� dB� The vector margin for this system is �� ��

��� Stability Margins from Bode Plots

The Nyquist plot shows the path of G�j � on the complex plane as

increases from � to ��� This can be translated into Bode plots�

where the magnitude and the phase of G�j � are drawn as functions

of � Usually� logarithmic scale is chosen for the frequency axis� and

the magnitude is expressed in decibels and the phase is expressed in

degrees� In Matlab� Bode plots are generated by the bode command�

The gain and phase margins can be read from the Bode plots� but the

Page 107: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

vector margin is not apparent� We need to draw the Bode magnitude

plot of S�j � to determine the vector margin�

Suppose that G�s� has no poles in the open right half plane� Bode

plots of a typical G�j � are shown in Figure ��� The gain and phase

margins are illustrated in this �gure�

GM � ��� log�� jG�j p�jPM � � � � G�j c�

where c is the gain crossover frequency� �� log�� jG�j c�j � � dB� and

p is the phase crossover frequency� � G�j p� � ������ The upward

arrows in the �gure indicate that the gain and phase margins of this

system are positive� hence the feedback system is stable� Note how

the arrows are drawn� on the gain plot the arrow is drawn from the

magnitude of G at p to � dB and on the phase plot from ����� to

� G�j c�� If either GM� or PM� or both� are negative then the closed�

loop system is unstable� The gain and phase margins can be obtained

via Matlab by using the margin command�

Example ��� Consider the open�loop system

G�s� ������ �s� � s � � �s � ���� �

s �s� � s � � �s � ��������

which is designed in example � of Section ������ The Bode plots given

in Figure ��� show that the phase margin is about �� and the gain

margin is approximately �� dB�

Exercise� Draw the Bode plots for G�s� given in ����� with K �

���� �� Determine the gain and phase margins from the margin com�

mand of Matlab� Verify the results by comparing them with the sta�

bility margins obtained from the Nyquist plot�

Page 108: Feedback Control Systems- HOzbay

�� H� �Ozbay

0 dB

-180 oPM

GM

Gain

Phaseωp

cω ω

ω

20log|G(j )|ω

G(j )ω

Figure ��� Gain and phase margins from Bode plots�

10−2

10−1

100

101

−40

−30

−20

−10

0

10

20

Frequency (rad/sec)

Gai

n dB

10−1

100

−420

−360

−300

−240

−180

−120

Frequency (rad/sec)

Pha

se d

eg

Figure ���� Bode plots of � ��� �s���s��� �s�� ��� �s �s���s��� �s�� ����� �

Page 109: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

��� Exercise Problems

�� Sketch the Nyquist plots of

�a� G�s� �K�s � ��

s�s� ��

�b� G�s� �K�s � �

�s� � ���

and compute the gain and phase margins as functions of K�

�� Sketch the Nyquist plot of

G�s� �K�s � ���s� � �� s � ��

s�s � ���s� � ��

and show that the feedback system is stable for all K � �� By

using root locus� �nd K� which places the closed�loop system poles

to the left of Re�s� � �� for the largest possible � � �� What is

the vector margin of this system�

� Consider the system de�ned by

G�s� �K�s� � s � ��s� zc�

s�s� � s� ��s� pc�

where K � �� pc � zc and zc is the parameter to be adjusted�

�a� Find the range of zc for which the feedback system is stable�

�b� Design zc so that the vector margin is as large as possible�

�c� Sketch the Nyquist plot and determine the gain and phase

margins for the value of zc computed in part �b��

�� For the open�loop system

G�s� �K�s � �

�s� ���s � ���s� � �s � ���

it was shown that the feedback system is stable for all

K � � � � ����� ��

Page 110: Feedback Control Systems- HOzbay

��� H� �Ozbay

�a� Find the value of K that maximizes the quantity

minfGMupp � �GMlow���g��b� Find the value of K that maximizes the phase margin�

�c� Find the value of K that maximizes the vector margin�

Page 111: Feedback Control Systems- HOzbay

Chapter �

Systems with Time Delays

Mathematical models of systems with time delays were introduced in

Section ������ If there is a time delay in a linear system� then its trans�

fer function includes a term e�hs where h � � is the delay time� For

this class of systems� transfer functions cannot be written as ratios of

two polynomials of s� The discussion on Routh�Hurwitz and Kharit�

anov stability tests� and root locus techniques do not directly extend to

systems with time delays� In order to be able to apply these methods�

the delay element e�hs must be approximated by a rational function

of s� For this purpose� Pad�e approximations of delay systems will be

examined in this chapter�

The Nyquist stability criterion remains valid for a large class of delay

systems� so there is no need for approximations in this case� Several

examples will be given to illustrate the e�ects of time delay on stability

margins� and the concept of delay margin will be introduced shortly�

The standard feedback control system considered in this chapter is

shown in Figure ��� where the controller C and plant P are in the form

C�s� �Nc�s�

Dc�s�

���

Page 112: Feedback Control Systems- HOzbay

��� H� �Ozbay

+r(t)C(s) P (s)

u(t-h)e(t)0

y(t)

v(t)

-+ u(t)

P(s)

-hse-

Figure ��� Feedback system with time delay�

and

P �s� � e�hsP��s� where P��s� �Np�s�

Dp�s�

with �Nc� Dc� and �Np� Dp� being coprime pairs of polynomials� The

open�loop transfer function is

G�s� � e�hsG��s��

where G��s� � P��s�C�s� corresponds to the case h � ��

The response of a delay element e�hs to an input u�t� is simply

u�t� h�� which is the h units of time delayed version of u�t�� Hence by

de�nition� the delay element is a stable system� In fact� the Lp� ���

norm of its output is equal to the Lp� ��� norm of its input for any

p� Thus� the system norm �de�ned as the induced Lp� � �� norm� of

e�hs is unity�

The function e�hs is analytic in the entire complex plane and has

no �nite poles or zeros� The frequency response of the delay element is

determined by its magnitude and phase on the Im�axis�

je�jh� j � � for all

� e�jh� � �h �

The magnitude identity con�rms the norm preserving property of the

delay element� Moreover� it also implies that the transfer function e�hs

is all�pass� For � � the phase is negative and it is linearly decreasing�

Page 113: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

The Bode and Nyquist plots of G�j � are determined from the iden�

tities

jG�j �j � jG��j �j ����

� G�j � � �h � � G��j � � ����

This fact will be used later� when stability margins are discussed�

��� Stability of Delay Systems

Stability of the feedback system shown in Figure �� is equivalent to

having all the roots of

��s� �� Dc�s�Dp�s� � e�hsNc�s�Np�s� �� �

in the open left half plane� C�� Strictly speaking� ��s� is not a poly�

nomial because it is a transcendental function of s� The functions

of the form �� � belong to a special class of functions called quasi�

polynomials� Recall that the plant and the controller are causal sys�

tems� so their transfer functions are proper� deg�Nc� � deg�Dc� and

deg�Np� � deg�Dp�� In fact� most physical plants are strictly proper�

accordingly assume that deg�Np� � deg�Dp��

The closed�loop system poles are the roots of

� � G�s� � � �� � � e�hsG��s� � � �

or the roots of

��s� � D�s� � e�hsN�s� ����

where D�s� � Dc�s�Dp�s� and N�s� � Nc�s�Np�s�� Hence� to determ�

ine closed�loop system stability� we need to check that the roots of ����

are in C�� Following are known facts �see �� ����

Page 114: Feedback Control Systems- HOzbay

��� H� �Ozbay

�i� if rk is a root of ����� then so is rk� �i�e�� roots appear in complex

conjugate pairs as usual��

�ii� there are in�nitely many poles rk � C� k � �� �� � � �� satisfying

��rk� � ��

�iii� and rk�s can be enumerated in such a way that Re�rk��� � Re�rk��

moreover� Re�rk� � �� as k ���

Example �� If G�s� � e�hs�s� then the closed�loop system poles rk�

for k � �� �� � � �� are the roots of

� �e�h�ke�jh�k

�k � j ke�j�k� � � ����

where rk � �k � j k for some �k� k � IR� Note that e�j�k� � � for

all k � �� �� � � �� The equation ���� is equivalent to the following set of

equations

e�h�k � j�k � j kj ����

��k � ��� � h k � � ��k � j k� k � �� �� � � � ���

It is quite interesting that for h � � there is only one root r � ���

but even for in�nitesimally small h � � there are in�nitely many roots�

From the magnitude condition ����� it can be shown that

�k � � �� j kj � �� ����

Also� for �k � �� the phase � ��k � j k� is between ��� and ��

� � therefore

��� leads to

�k � � �� h j kj � �

�� ����

By combining ���� and ����� it can be proven that the feedback system

has no roots in the closed right half plane when h � �� � Furthermore�

the system is unstable if h � �� � In particular� for h � �

� there are two

Page 115: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

roots on the imaginary axis� at j�� It is also easy to show that� for

any h � � as k ��� the roots converge to

rk � �

h

��ln�

�k�

h� j�k�

��

As h� �� the magnitude of the roots converge to ��

As illustrated by the above example� property �iii� implies that for

any given real number � there are only �nitely many rk �s in the region

of the complex plane

C� �� fs � C � Re�s� � �g�

In particular� with � � �� this means that the quasi�polynomial ��s�

can have only �nitely many roots in the right half plane� Since the e�ect

of the closed�loop system poles that have very large negative real parts

is negligible �as far as closed�loop systems� input�output behavior is

concerned�� only �nitely many !dominant" roots rk� for k � �� � � � �m�

should be computed for all practical purposes�

��� Pad e Approximation of Delays

Consider the following model�matching problem for a strictly proper

stable rational transfer function G��s�� approximate G�s� � e�hsG��s�

by bG�s� � Pd�s�G��s�� where Pd�s� � Nd�s��Dd�s� is a rational ap�

proximation of the time delay� We want to choose Pd�s� so that the

input�output behavior of bG�s� matches the input�output behavior of

G�s�� To measure the mismatch� apply the same input u�t� to both

G�s� and bG�s�� Then� by comparing the respective outputs y�t� andby�t�� we can determine how well bG approximates G� see Figure ���

Page 116: Feedback Control Systems- HOzbay

��� H� �Ozbay

e

dP (s) G (s)

0

G (s)0

-hs

G(s)

G(s)

u(t)

y(t)

y(t)

error-

+

Figure ��� Model�matching problem�

The model�matching error �MME� will be measured by

MME �� supu���

ky � byk�kuk� �����

where ky � byk� denotes the energy of the output error due to an input

with energy kuk�� The largest possible ratio of the output error energy

over the input energy is de�ned to be the model�matching error� From

the system norms de�ned in Chapter �� MME � MMEH� � MMEL� �

MMEH� � kG� bGkH� �����

MMEL� � sup�jG�j �� bG�j �j

� sup�jG��j �j j�e�jh� � Pd�j ��j� �����

It is clear that if MMEL� is small� then the di�erence between the

Nyquist plots of G�j � and bG�j � is small� This observation is valid

even if G��s� is unstable� Thus� for a given G��s� �which may or may

not be stable�� we want to �nd a rational approximation Pd�s� for the

delay term e�hs so that the approximation error MMEL� is smaller

than a speci�ed tolerance� say � � ��

Pad�e approximation will be used here�

e�hs � Pd�s� �Nd�s�

Dd�s��

Pnk������kckh

kskPnk�� ckh

ksk�

Page 117: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

The coe�cients are

ck ���n� k� n

�n k �n� k� � k � �� �� � � � � n�

For n � � the coe�cients are c� � �� c� � ���� and

Pd�s� �

��� hs��

� � hs��

��

For n � � the coe�cients are c� � �� c� � ���� c� � ����� and the

second�order approximation is

Pd�s� �

��� hs�� � �hs�����

� � hs�� � �hs�����

��

Now we face the following model order selection problem� Given h

and G��s�� what should be the degree of the Pad�e approximation� n� so

that the the error MMEL� is less than or equal to a given error bound

� � ��

Theorem �� ���� The approximation error satis�es

j�e�jh� � Pd�j ��j �

�� eh��n ��n�� � �neh

� � �neh

In the light of Theorem ��� we can solve the model order selection

problem using the following procedure�

�� From the magnitude plot of G��j � determine the frequency x

such that

jG��j �j � �

�for all � x

and initialize n � ��

�� For each n � � de�ne

n � maxf x � �n

ehg

Page 118: Feedback Control Systems- HOzbay

��� H� �Ozbay

and plot the function

E� � ��

��jG��j �j � eh��n ��n�� for � �n

eh

�jG��j �j for n � � �neh

� De�ne

E�n� ��

�maxfE� � � � � � x�g�

If E�n� � �� stop� this value of n satis�es the desired error bound�

MMEL� � �� Otherwise� increase n by �� and go to Step ��

�� Plot the approximation error function

jG��j �j j�e�jh� � Pd�j ��j

and verify that its peak value is less than ��

Since G��s� is strictly proper� the algorithm will pass Step eventually

for some �nite n � �� At each iteration� we have to draw the error

function E� � and check whether its peak is less than �� Usually� for

good results in stability and performance analysis� � is chosen to be in

the order of ���� to ����� In most cases� as � decreases� x increases�

and that forces n to increase� On the other hand� for very large values

of n� the relative magnitude c��cn of the coe�cients of Pd�s� become

very large� in which case numerical di�culties arise in analysis and

simulations� Also� as time delay h increases� n should be increased to

keep the level of the approximation error � �xed� This is a fundamental

di�culty associated with time delay systems�

Example �� Let N�s� � �s���� D�s� � �s���s��� and h � ���� The

magnitude plot of G� � N�D shows that if � � ���� then x � ��� see

Figure � � By applying the algorithm proposed above� the normalized

error E�n� is obtained� see Figure ��� Note that E��� � �� which means

that the choice n � � guarantees MMEL� � ����

Page 119: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

max(|Go|)|Go| delta/2

10−2

10−1

100

101

102

10−2

10−1

100

omega

Figure � � Magnitude plot of G��j ��

0 1 2 3 4 5−140

−120

−100

−80

−60

−40

−20

0

20

approximation order: n

20*lo

g10(

|E(n

)|)

Figure ��� Detection of the smallest n�

Page 120: Feedback Control Systems- HOzbay

��� H� �Ozbay

Exercise� For the above example� draw the root locus of

D�s�Dd�s� � KN�s�Nd�s� � �

for n � �� � and � Show that in the region Re�s� � ���� predicted

roots are approximately the same for n � � and n � � for the values of

K in the interval � � K � ��� Determine the range of K for which the

roots remain in the left half plane for these three root loci� Comment

on the results�

��� Roots of a QuasiPolynomial

In this section� we discuss the following problem� given N�s�� D�s� and

h � �� �nd the dominant roots of the quasi�polynomial

��s� � D�s� � e�hsN�s� �

For each �xed h � �� it can be shown that there exists �max such that

��s� has no roots in the region C�max� see ��� for a simple algorithm to

estimate �max� based on Nyquist criterion� Given h � � and a region

of the complex plane de�ned by �min � Re�s� � �max� the problem is

to �nd the roots of ��s� in this region�

A point r � � � j in C is a root of ��s� if and only if

D�� � j � � �e�h�e�jh�N�� � j �

Taking the magnitude square of both sides of the above equation� ��r� �

� implies

A��x� �� D�� � x�D�� � x� � e��h�N�� � x�N�� � x� � �

where x � j � The term D�� � x� stands for the function D�s� evalu�

ated at � � x� The other terms of A��x� are calculated similarly� For

each �xed �� the function A��x� is a polynomial in the variable x� By

symmetry� if x is a zero of A���� then ��x� is also a zero�

Page 121: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

If A��x� has a root x� whose real part is zero� set r� � � � x��

Next� evaluate the magnitude of ��r��� if it is zero� then r� is a root of

��s�� Conversely� if A��x� has no root on the imaginary axis� then ��s�

cannot have a root whose real part is the �xed value of � from which

A���� is constructed�

Algorithm� Given N�s�� D�s�� h� �min and �max�

Step �� Pick � values ��� � � � � �M between �min and �max such that

�min � ��� �i � �i�� and �M � �max� For each �i perform the

following�

Step �� Construct the polynomial Ai�x� according to

Ai�x� �� D��i � x�D��i � x�� e��h�iN��i � x�N��i � x�

Step �� For each imaginary axis roots x� of Ai� perform the following

test� Check if j���i � x��j � �� if yes� then r � �i � x� is a root

of ��s�� if not discard x��

Step �� If i � M � stop� else increase i by � and go to Step ��

Example �� Now we will �nd the dominant roots of

� �e�hs

s� �� ��� �

We have seen that ��� � has a pair of roots j� when h � ��� � ����

Moreover� dominant roots of ��� � are in the right half plane if h �

���� and they are in the left half plane if h � ���� So� it is expected

that for h � ���� � ���� the dominant roots are near the imaginary axis�

Take �min � ���� and �max � ���� with M � ��� linearly spaced �i�s

between them� In this case

Ai�x� � ��i � e��h�i � x��

Page 122: Feedback Control Systems- HOzbay

��� H� �Ozbay

−0.5 −0.3 −0.1 0.1 0.3 0.510

−4

10−3

10−2

10−1

100

101

sigma

F(s

igm

a)

Detection of the Roots

Figure ��� Detection of the dominant roots�

Whenever e��h�i � ��i � Ai�x� has two roots

x� � jqe��h�i � ��i � � �� ��

For each �xed �i satisfying this condition� let r� � �i � x� �note that

x� is a function of �i� so r� is a function of �i� and evaluate

F ��i� ��

����� �e�hr�

r�

���� �If F ��i� � �� then r� is a root of ��� �� For �� di�erent values of

h � ���� � ���� the function F ��� is plotted in Figure ��� This �gure

shows the feasible values of �i for which r� �de�ned from �i� is a root

of ��� �� The dominant roots of ��� �� as h varies from ��� to ���� are

shown in Figure ��� For h � ��� all the roots are in C�� For h � ���

the dominant roots are in C�� and for h � ��� they are at j��

Page 123: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1−1.5

−1

−0.5

0

0.5

1

1.5

Real(r)

Imag

(r)

Locus of dominant roots for 1.2<h<2.0

Figure ��� Dominant roots as h varies from ��� to ����

��� Delay Margin

Consider a strictly proper system with open�loop transfer function G�s� �

e�hsG��s�� Suppose that the feedback system is stable when there is

no time delay� i�e�� the roots of

� � e�hsG��s� � � �����

are in C� for h � �� Then� by continuity arguments� it can be shown

that the feedback system with time delay h � � is stable provided h is

small enough� i�e�� the roots of ����� remain in C� for all h � � � h���

for su�ciently small h� � �� An interesting question is this� given G��s�

for which the feedback system is stable� what is hmax� the largest value

of h�� At the critical value of h � hmax� all the roots of ����� are in the

open left half plane� except �nitely many roots on the imaginary axis�

Therefore� the equation

� � e�j�hmaxG��j � � � �����

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��� H� �Ozbay

holds for some real � Equating magnitude and phase of �����

� � jG��j �j�� � � hmax � � G��j � �

Let c be the crossover frequency for this system� i�e�� jG��j c�j � ��

Recall that the quantity

� � �� � � G��j c��

is the phase margin of the system� Thus� from the above equations

hmax ��

c�

If there are multiple crossover frequencies c�� � � � � c� for G��j �

�i�e�� jG��j ci�j � � for all i � �� � � � � ��� then the phase margin is

� � minf �i � � � i � �g where

�i � �� � � G��j ci���

In this case� hmax is given by

hmax � minf �i ci

� � � i � �g�

Now consider G�s� � e�hsG��s�� for some h � � � hmax�� In the

light of the above discussion� the feedback system remains stable for all

open�loop transfer functions e��sG�s� provided

� � �hmax � h��

DM� The quantity �hmax�h� is called the delay margin of the system

whose open�loop transfer function is G�s��

The delay margin DM determines how much the time delay can be

increased without violating stability of the feedback system�

Page 125: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

−4 −3 −2 −1 0 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Real Axis

Ima

g A

xis

h=0.05

h=0.16

h=0.5

Figure �� E�ect of time delay �Nyquist plots��

Example �� In Example ���� the Nyquist plot of

G��s� �����s� �

�s� ���s � ���s � �s � ���

was used to demonstrate stability of this feedback system� By using the

margin command of Matlab it can be shown that the phase margin

for this system is ����� ���� rad and the crossover frequency is c �

���� rad�sec� Therefore� hmax � ��������� �� ���� sec� This means that

the feedback system with open�loop transfer function G�s� � e�hsG��s�

is stable for all values of h � � � ������ and if h � ����� it is unstable�

The Nyquist plots with di�erent values of h ������ ���� and ���� are

shown in Figure ��

Example �� Consider the open�loop transfer function

G��s� ���

s�s � ���

Page 126: Feedback Control Systems- HOzbay

��� H� �Ozbay

The feedback system is stable with closed�loop poles at

r��� � � j� �

The Bode plots of G� are shown in Figure ��� The phase margin is

�� �� �� rad and the crossover frequency is c � ���� rad�sec� hence

hmax � �� ������� � ���� sec� The Bode plots of G�s� � e�hsG��s�

are also shown in the same �gure� they are obtained from the relations

���� and ����� The magnitude of G is the same as the magnitude of

G�� The phase is reduced by an amount h at each frequency � The

phase margin for G is

� � �� ��� ���� h

For example� when h � �� the phase margin is ����� rad� which is

equivalent to ��� This is con�rmed by the phase plot of Figure ���

Example �� Consider a system with multiple crossover frequencies

G��s� ����s� � ���s � ��

s��s� � ���s � ���

Nyquist and Bode plots of G��j � are shown in Figures �� and ����

respectively� The crossover frequencies are

c� � ��� rad�sec� c� � ���� rad�sec� c� � ��� rad�sec�

with phase values

� G��j c�� � ����� � � G��j c�� � ���� � � G��j c�� � �����

Therefore� �� � ���� �� � ���� and �� � � �� The phase margin is

� � minf��� ��� ��g � �� � ����

Page 127: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

10−1

100

101

102

−60

−40

−20

0

20

Ma

gn

itud

e in

dB

Bode Plots of Go and G

h=0 h=0.1h=0.3h=1.2

10−1

100

101

102

−180

−165

−150

−135

−120

−105

−90

Ph

ase

in d

eg

ree

s

omega

Figure ��� E�ect of time delay �Bode plots��

However� the minimum of �i� ci is achieved for i � �

�� c�

� ���� sec��� c�

� ��� sec��� c�

� ���� sec�

Hence hmax � ���� sec� Nyquist and Bode plots of G�s� � e�hsG��s�

with h � hmax are also shown in Figures �� and ���� respectively�

From these �gures� it is clear that the system is stable for all h � hmax

and unstable for h � hmax�

Page 128: Feedback Control Systems- HOzbay

��� H� �Ozbay

h=0 h=0.14 unit circle

−3 −2 −1 0 1−1.5

−1

−0.5

0

0.5

1Nyquist Plot of Go, and G with h=0.14

Figure ��� Nyquist plots for Example ���

10−1

100

101

102

−20−15−10

−505

101520

Ma

gn

itud

e in

dB

Bode Plots of Go and G

h=0 h=0.14

10−1

100

101

102

−180−150−120

−90−60−30

0

Ph

ase

in d

eg

ree

s

omega

Figure ���� Bode plots for delay margin computation�

Page 129: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

��� Exercise Problems

�� Consider the feedback system de�ned by

C�s� ����

�s � ������� P��s� �

s� � s � �

with possible time delay in the plant P �s� � e�hsP��s��

�a� Calculate hmax�

�b� Let h � �� Using the algorithm given in Section ��� �nd the

minimal order of Pad�e approximation such that

MMEL� �� kG�G�Pdk� � � ��

����

�c� Draw the locus of the dominant roots of

� � G��s�e�hs � �

for h � ���� � �����

�d� Draw the Nyquist plot of G�s� for h � ����� ����� ����� and

verify the result of the �rst problem� What is the delay

margin if h � ��

�� Assume that the plant is given by

P �s� � e�hs�

s� � s � �

��

where h is the amount of the time delay in the process�

�a� Design a �rst�order controller of the form

C�s� � Kc�s � z��

�s � ��

such that when h � � the closed�loop system poles are equal�

r� � r� � r�� What is hmax�

�b� Let z� be as found in part �a�� and h � ���� Estimate the

location of the dominant closed�loop system poles for Kc

as determined in part �a�� How much can we increase Kc

without violating feedback system stability�

Page 130: Feedback Control Systems- HOzbay

��� H� �Ozbay

� Compute the delay margin of the system

G�s� �Ke��s�s� � ��� s� ������s � ��

s��s� � ����s� �����

when �a� K � ����� �b� K � ������

�� Consider the feedback system with

C�s� � K and P �s� �e�hs�s � ��

s �s� ���

For a �xed K � �� let hmax�K� denote the largest allowable time

delay� What is the optimal K maximizing hmax�K�� To solve this

problem by hand� you may use the approximation

tan���x� � ����x� ����x� for � � x � �� �

Page 131: Feedback Control Systems- HOzbay

Chapter �

Lead� Lag� and PID

Controllers

In this chapter �rst�order lead and lag controller design principles will

be outlined� Then� PID controllers will be designed from the same

principles� For simplicity� the discussion here is restricted to plants

with no open right half plane poles�

Consider the standard feedback system with a plant P �s� and a con�

troller C�s�� Assume that the plant and the controller have no poles in

the open right half plane and that there are no imaginary axis pole zero

cancelations in the product G�s� � P �s�C�s�� In this case� feedback

system stability can be determined from the Bode plots of G�j �� The

design goals are�

�i� feedback system is stable with a speci�ed phase margin� and

�ii� steady state error for unit step �or unit ramp� input is less than

or equal to a speci�ed quantity�

The second design objective determines a lower bound for the DC

���

Page 132: Feedback Control Systems- HOzbay

��� H� �Ozbay

gain of the open�loop transfer function� G���� In general� the larger the

gain of G�s�� the smaller the phase margin� Therefore� we choose the

DC gain as the lowest possible value satisfying �ii�� Then� additional

terms� with unity DC gain� are appended to the controller to take care

of the �rst design goal� More precisely� the controller is �rst assumed to

be a static gain C��s� � Kc� which is determined from �ii�� Then� from

the Bode plots of G��s� � C��s�P �s� the phase margin is computed� If

this simple controller does not achieve �i�� the controller is modi�ed as

C�s� � C��s�C��s�� where C���� � ��

The DC gain of C��s� is unity� so that the steady state error does not

change with this additional term� The poles and zeros of C��s� are

chosen in such a way that the speci�ed phase margin is achieved�

Example ��� For the plant

P �s� ��

s�s � ����

design a controller such that the feedback system is stable with a phase

margin of ���� and the steady state error for a unit ramp reference input

is less than or equal to ��� � Assuming that the feedback system is stable�

the steady state error for unit ramp reference input is

ess ��

lims��

sG�s����

Hence� the steady state error requirement is satis�ed if C��s� � Kc � ��

For G��s� � KcP �s� the largest phase margin is achieved with the

smallest feasible Kc� Thus Kc � � and for this value of Kc the phase

margin� determined from the Bode plots of G�� is approximately ��� So�

additional term C��s� is needed to bring the phase margin to ����

Several di�erent choices of C��s� will be studied in this chapter� Be�

fore discussing speci�c controller design methods� we make the following

de�nitions and observations�

Page 133: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

Wb/WcWb/WoWc/Wo

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

zeta

Rat

io

Figure ���� b and c as functions of ��

Consider a feedback system with closed�loop transfer function �from

reference input r to system output y� T �s� � G�s���G�s� satisfying

jT �j �j�� �p

� � dB for all � b

� �p� � dB for all � b

The bandwidth of the system is b rad�sec� Usually� the bandwidth� b�

and the gain crossover frequency c �where jG�j c�j � �� are of the

same order of magnitude� For example� let

G�s� � �o

s�s � �� o�then T �s� �

�os� � �� os � �o

By simple algebraic manipulations it can be shown that both b and c

are in the form

b�c � o

rq� � x�b�c � xb�c

where xc � ��� for c� and xb � ��� ���� for b� See Figure ����

Page 134: Feedback Control Systems- HOzbay

��� H� �Ozbay

Note that as � o increases� b increases and vice versa� On the other

hand� � o is inversely proportional to the settling time of the step re�

sponse� Therefore� we conclude that the larger the bandwidth b� the

faster the system response� Figure ��� also shows that

���� c � b � �� c�

So� we expect the system response to be fast if c is large� However�

there is a certain limitation on how large the system bandwidth can be�

This will be illustrated soon with lead lag controller design examples�

and also in Chapter � when we discuss the e�ects of measurement noise

and plant uncertainty�

For the second�order system considered here� the phase margin� � �

�� � � G�j c��� can be computed exactly from the formula for c�

� ��

�� tan���

c�� o

� � tan�����

qp� � ��� � �����

which is approximately a linear function of �� see Figure ���� Recall that

as � increases� the percent overshoot in the step response decreases� So�

large phase margin automatically implies small percent overshoot in the

step response�

For large�order systems� translation of the time domain design ob�

jectives �settling time and percent overshoot� to frequency domain ob�

jectives �bandwidth and phase margin� is more complicated� but the

guidelines developed for second�order systems usually extend to more

general classes of systems�

We now return to our original design problem� which deals with

phase margin and steady state error requirements only� Suppose that

G� � PC� is determined� steady state error requirement is met� phase

margin is to be improved by an additional controller C��

Page 135: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

zeta

Pha

se M

argi

n (d

eg)

Figure ���� Phase margin versus ��

��� Lead Controller Design

A controller in the form

C��s� � Clead�s� ��� � ��s�

�� � �s�where � � �� � � ��

is a lead controller� � and � are to be determined�

Asymptotic Bode plots of Clead�j � are illustrated in Figure �� �

The phase is positive for all � �� Hence the phase of G��j � is

increased with the addition of the lead term C��j �� The largest phase

increase is achieved at the frequency

o ��

�p��

The exact value of is given by

sin�� ��� �

� � ��

Note that � ��� � ����� See Figure ��� for a plot of versus ��

Page 136: Feedback Control Systems- HOzbay

��� H� �Ozbay

Magnitude (dB)

Phase

ω

ω

ω

φ

o

o0

0 dB10log(α )

Figure �� � Bode plots of a lead controller�

100

101

102

103

0

15

30

45

60

75

90

alpha

phi (

in d

eg.)

Figure ���� Phase lead versus ��

Page 137: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

The basic idea behind lead controller design is to increase the phase

of G�� to increase the phase margin� Therefore� the largest phase lead

should occur at the crossover frequency� i�e� o should be the crossover

frequency for G � G�C�� Since the magnitude of G� is increased by

�� log����� at o� the crossover frequency for G � G�C� is the frequency

at which the magnitude of G� is ��� log������ Once � is determined�

o can be read from the Bode magnitude plot G� and then

� ��

op��

How is � selected from the desired phase margin� To understand the

e�ect of additional term C� on the phase margin� �rst let c be the

crossover frequency for G� and let

�old � � � � G��j c� �����

be the phase margin of the system with PC � G�� Then de�ne

�des � � � � G��j o� � � C��j o� �����

as the desired phase margin of the system with PC � G�C� �in this

case the crossover shifts from c to o�� Since � C��j o� � � equations

����� and ����� yield

� �des � �old � �� G��j c�� � G��j o��� ��� �

Equation ��� � is the basis for the lead controller design method sum�

marized below�

Step �� Given Bode plots of G��j �� determine the phase margin �old

and crossover frequency c� Desired phase margin �des � �old is

also given�

Step �� Pick a small safety angle �saf � �� to ��� and de�ne

� �des � �old � �saf �

Page 138: Feedback Control Systems- HOzbay

��� H� �Ozbay

Step �� Calculate � from the formula

� �� � sin��

�� sin���

Step �� From the Bode magnitude plot of G��j � �nd the frequency o

for which

�� log�� jG��j o�j � ��� log�����

Step �� Check if

�saf � �� G��j c�� � G��j o���

If yes� go to Step �� If no� go to Step � and increase �saf by �� �if

this leads to � ��� stop iterations� desired phase margin cannot

be achieved by a �rst�order additional lead term��

Step �� De�ne

� ��

op��

Draw the Bode plots of G�C�� and check the phase margin�

The potential problem with the test in Step � is the following� As

increases� � and o increase� On the other hand� if the phase of G��j �

decreases very fast for � c� then the di�erence

�� G��j c�� � G��j o��

can be larger than �saf as o increases� Therefore� lead controllers are

not suitable for such systems�

Example ��� The system designed in Example ��� did not satisfy the

desired phase margin of �des � ���� The Bode plots of G��s� � �s�s�� �� �

shown in Figure ���� give �old � �� and c � ��� rad�sec� By de�ning

Page 139: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

G1 G1*Clead

G1 G1*Clead

10−2

10−1

100

101

−40

−20

0

20

40

60

80

omega

Mag

nitu

de (

dB)

10−2

10−1

100

101

−180

−165

−150

−135

−120

−105

−90

omega

Pha

se (

deg)

Figure ���� Bode plots of G� and G � G�Clead�

�saf � �� additional phase is calculated as � ��������� � ���� This

leads to � � ��� and ��� log����� � ��� dB� From Figure ���� we

see that the magnitude of G��j � drops to��� dB at o � ��� rad�sec�

Also note that �� safety is su�cient� Indeed�

�� � �� G��j c�� � G��j o�� � ����� � �������� � �����

Example ��� For the system considered in the previous example� in�

troduce a time delay in the plant�

G��s� � P �s� ��e�hs

s�s � ����

and study the e�ect of time delay on phase margin� Let h � ��� sec�

Then� the Bode plots of G� are modi�ed as shown in Figure ���� In

this case� the uncompensated system is unstable� with phase margin

�old �� � ��� Try to stabilize this system by a lead controller with

�des � ��� and �saf � ��� For these values � ��� �same as in

Page 140: Feedback Control Systems- HOzbay

� � H� �Ozbay

10−2

10−1

100

101

−40

−20

0

20

40

60

80

omega

Mag

nitu

de (

dB)

10−2

10−1

100

101

−360−330−300−270−240−210−180−150−120

−90

omega

Pha

se (

deg)

Figure ���� Bode plots of G� for Example �� �

Example ����� hence � � ��� and o � ��� rad�sec� But in this case�

the phase plot in Figure ��� shows that

�� G��j c�� � G��j o�� �� ������ � �������� � �� � �saf �

The lead controller designed this way does not even stabilize the

feedback system� because

� G��j o� � � ����� � ����� �

which means that the phase margin is ����� The main problem here is

the fact that the phase decreases very rapidly for � c � ��� rad�sec�

On the other hand� in the frequency region � � � �� �� the phase

is approximately the same as the phase of G� without time delay� If

the crossover frequency can be brought down to � ��� to �� range�

then a lead controller can be designed properly� There are two ways to

reduce the crossover frequency� �i� by reducing the gain Kc� or �ii� by

Page 141: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory � �

ω

ω10/ τ

α-20log( )

Phase

Magnitude (dB)

Figure ��� Bode plots of a lag controller�

adding a lag controller in the form

Clag�s� ��� � �s�

�� � ��s�� � � � � � �� �����

The �rst alternative is not acceptable� because it increases the steady

state error by reducing the DC gain of G�� Lag controllers do not change

the DC gain� but they reduce the magnitude in the frequency range of

interest to adjust the crossover frequency�

��� Lag Controller Design

As de�ned by ����� a lag controller is simply the inverse of a lead

controller�

C��s� � Clag�s� ��� � �s�

�� � ��s�with � � � and � � ��

Typical Bode plots of Clag�j � are shown in Figure ���

Page 142: Feedback Control Systems- HOzbay

� � H� �Ozbay

The basic idea in lag controller design is to reduce the magnitude of

G� to bring the crossover frequency to a desired location� To be more

precise� suppose that with the phase lag introduced by the controller

C�� the overall phase is above ����� for all � c� and at a certain

!desired" crossover frequency d � c the phase satis�es

�des � � � � G��j d� � � C��j d��

Then� by choosing

� � jG��j d�j � �

the new crossover frequency is brought to d and the desired phase

margin is achieved� for this purpose we must choose ��� � d� Usually

� is chosen in such a way that

d � ����

where d is determined from the phase plot of G��j � as the frequency

at which

� G��j d� � � � �des � ��

�note that � C��j d� � ����� Thus� lag controller design procedure

does not involve any trial and error type of iterations� In this case�

controller parameters � and � are determined from the Bode plots of

G� directly�

Example ��� We have seen that the time delay system studied in Ex�

ample �� was impossible to stabilize by a �rst�order lead controller�

Bode plots shown in Figure ��� illustrate that for a stable system with

�des � ���� the crossover frequency should be d � ��� rad�sec �the

Page 143: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

G1 G1*Clag

0.0001 0.001 0.01 0.1 1 10−40−20

020406080

100120

omega

Mag

nitu

de (

dB)

G1 G1*Clag

0.0001 0.001 0.01 0.1 1 10−240

−210

−180

−150

−120

−90

omega

Pha

se (

deg)

Figure ���� Bode plots of G� and G�Clag for Example ����

phase of G� is ����� at ��� rad�sec�� The magnitude of G� at d is

about dB� that means � � �� dB� Choosing � � ��� d � ��

the lag controller is

C��s� � Clag�s� �� � �� s

� � ���� s�

The Bode plots of the lag compensated and uncompensated systems

are shown in Figure ���� As expected� the phase margin is about ����

Although ��� is not a very large phase margin� considering that the

uncompensated system had � �� phase margin� this is a signi�cant

improvement�

��� Lead�Lag Controller Design

Lead�lag controllers are combinations of lead and lag terms in the form

C��s� ��� � ����s�

�� � ��s�

�� � ��s�

�� � ����s�

Page 144: Feedback Control Systems- HOzbay

� � H� �Ozbay

G1 G1*C2 G1*C2*C3

0.0001 0.001 0.01 0.1 1 10−240

−210

−180

−150

−120

−90

omega

Pha

se (

deg)

0.0001 0.001 0.01 0.1 1 10−40−20

020406080

100120

omega

Mag

nitu

de (

dB)

Figure ���� Bode plots of G�� G�C�� and G�C�C��

where �i � � and �i � � for i � �� �� and ���� ��� are determined

from lead controller design principles� ���� ��� are determined from lag

controller design guidelines�

Example ��� �Example ��� continued� Now� for the lag compensated

system� let C��s� be a lead controller� such that the phase margin of

the lead lag compensated system G�C�C� is �des � ��� This time

assuming �saf � ���� additional phase lead is � ���������� � ����

That gives �� � ����� and ��� log������ � �� dB� The new crossover

frequency is o � ���� rad�sec �this is the frequency at which the

magnitude of G�C� is �� dB�� So� �� � �������p

����� � ���� and the

lead controller is

C��s� �� � ���� s

� � ���� s�

The Bode plots of the lead lag compensated system� shown in Figure ����

verify that the new system has �� phase margin as desired�

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Introduction to Feedback Control Theory � �

��� PID Controller Design

If a controller includes three terms in parallel� proportional �P�� integral

�I�� and derivative �D� actions� it is said to be a PID controller� Such

controllers are very popular in industrial applications because of their

simplicity� PID controllers can be seen as extreme cases of lead�lag

controllers� This point will be illustrated shortly�

General form of a PID controller is

CPID�s� � Kp �Ki

s� Kds�

Let e�t� be the input to the controller CPID�s�� Then the controller

output u�t� is a linear combination of e�t�� its integral� and its derivative�

u�t� � Kpe�t� � Ki

Z t

e���d� � Kd �e�t��

Unless Kd � � �i�e�� the derivative term is absent� the PID controller

is improper� so it is physically impossible to realize such a controller�

The di�culty is in the exact implementation of the derivative action� In

practice� the derivative of e�t� can be approximated by a proper term�

CPID�s� �� Kp �Ki

s�

Kds

� � �s

where � is a small positive number�

When Kd � �� we have a PI controller

CPI �Ki�� � �s�

swhere � � Kp�Ki�

Typical Bode plots of a PI controller are shown in Figure ����� This

�gure shows that the DC gain of the open�loop transfer function is

increased� hence the steady state error for step �or ramp� reference input

is decreased� In fact� the system type is increased by one� By choosing

Kp � � the magnitude can be decreased near the crossover frequency

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� � H� �Ozbay

ω

10/ τ ω

20log(Kp)

0 dB

-90

0

Phase (deg)

Magnitude (dB)

Figure ����� Bode plots of a PI controller�

and� by adjusting � and Kp� desired phase margin can be achieved�

This is reminiscent of the lag controller design� The only di�erence

here is that the DC gain is higher� and the phase lag at low frequencies

is larger� As an example� consider G� of Example ���� a PI controller

can be designed by choosing � � �� � Kp�Ki and � � �� � �Kp

This PI controller achieves ��� phase margin� as does the lag controller

designed earlier�

Exercise�

�a� Draw the Bode plots for the system G�CPI� with the PI controller

�Kp � ���� and Ki � Kp���� determined from the lag controller

design principles and verify the phase margin�

�b� Now modify the PI controller �i�e�� select di�erent values of Kp

and Ki from lag controller design principles� so that the the cros�

sover frequency d is greater or equal to ��� rad�sec� and the

phase margin is greater or equal to ���

Page 147: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

A PD �proportional plus derivative� controller can be implemented

in a proper fashion as

CPD�s� �Kp�� � Kd

Kps�

�� � �s�

where � � � is a �xed small number� Usually �� Kd�Kp� Note that by

de�ning � � � and Kd

Kp� �� � PD controller becomes a lead controller

with � � �� � � � and an adjustable DC gain CPD��� � Kp� Thus� we

can design PD controllers using lead controller design principles�

By cascading a PI controller with a PD controller� we obtain a PID

controller� To see this� let

CPD�s� ��Kp� � Kd�s�

�� � �s�and CPI�s� � Kp� �

Ki�

s

then

CPID�s� � CPI�s�CPD�s� � �Kp �Ki

s� Kds� �� � �s���

where Kp � �Kp�Kp� � Ki�Kd��� Ki � Ki�Kp� and Kd � Kd�Kp��

Therefore� PID controllers can be designed by combining lead and lag

controller design principles� like lead�lag controllers�

��� Exercise Problems

�� Design an appropriate controller �lead� or lag� or lead�lag� for the

system

G��s� �Ke��s�s� � ��� s� ������s � ��

s��s� � ���s� �����

so that the phase margin is �� when K � �����

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� � H� �Ozbay

�� What is the largest phase margin achievable with a �rst�order lead

controller for the plant

G��s� ��e�hs

s�s � ����

�a� when h � ��

�b� when h � �� �

�c� when h � ����

Page 149: Feedback Control Systems- HOzbay

Chapter �

Principles of Loopshaping

The main goal in lead�lag controller design methods discussed earlier

was to achieve good phase margin by adding extra lead and�or lag

terms to the controller� During this process the magnitude and the

phase of the open loop transfer function G � PC is shaped in some de�

sired fashion� So� lead�lag controller design can be considered as basic

loopshaping� In this chapter� we consider tracking and noise reduction

problems involving minimum phase plants and controllers�

��� Tracking and Noise Reduction Problems

Control problems studied in this chapter are related to the feedback

system shown in Figure ���� Here yo�t� is the plant output� its noise

corrupted version y�t� � yo�t� � n�t� is available for feedback control�

We assume that there are no right half plane pole zero cancelations in the

product P �s�C�s� � G�s� �otherwise the feedback system is unstable�

and that the open�loop transfer function G�s� is minimum phase�

� �

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��� H� �Ozbay

r(t) u(t)

+

+-

C(s) P(s) y(t)o

y(t) n(t)+

Figure ���� Feedback system considered for loopshaping�

A proper transfer function in the form

G�s� � e�hsNG�s�

DG�s�

�where �NG� DG� is a pair of coprime polynomials� is said to be min�

imum phase if h � � and both NG�s� and DG�s� have no roots in the

open right half plane� The importance of this assumption will be clear

when we discuss Bode�s gain�phase relationship�

The reference� r�t�� and measurement noise� n�t�� belong to

r � R � fR�s� � Wr�s�Ro�s� � krok� � �gn � N � fN�s� � Wn�s�No�s� � knok� � �g

where Wr and Wn are �lters de�ning these classes of signals� Typically�

the reference input is a low�frequency signal� e�g�� when r�t� is unit step

function R�s� � ��s and hence jR�j �j is large at low frequencies and

low at high frequencies� So� Wr�s� is usually a low�pass �lter� The

measurement noise is typically a high�frequency signal� which means

that Wn is a high�pass �lter� It is also possible to see Wr and Wn as

reference and noise generating �lters with arbitrary inputs ro and no

having bounded energy �normalized to unity��

Example ��� Let r�t� be

r�t� �

�t�� for � � t � �

� for t � ��

Page 151: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

For � � � � �� this signal approximates the unit step� There exist a

�lter Wr�s� and a �nite energy input ro�t� that generate r�t�� To see

this� let us choose Wr�s� � K�s� with some gain K � �� and de�ne

ro�t� �

���p� for � � t � �

� for t � ��

Verify that ro has unit energy� Also check that when K � ��p�� output

of the �lter Wr�s� with input ro�t� is indeed r�t��

There are in�nitely many reference signals of interest r in the set R�

The �lter Wr emphasizes those signals ro for which jRo�j �j is large at

the frequency region where jWr�j �j is large� and it de�emphasizes all

inputs ro such that the magnitude jRo�j �j is mainly concentrated to

the region where jWr�j �j is small� Similar arguments can be made for

Wn and the class of noise signals N �

Tracking error e�t� � r�t� � yo�t� satis�es

E�s� � S�s�R�s� � T �s�N�s� where

S�s� ��

� � G�s�and T �s� �

G�s�

� � G�s�� �� S�s� �

For good tracking and noise reduction we desire

jS�j �R�j �j � � �

jT �j �N�j �j � � � �

i�e�� jS�j �j should be small in the frequency region where jR�j �j is

large and jT �j �j should be small whenever jN�j �j is large� Since

S�s� � T �s� � �

it is impossible to make both jS�j �j and jT �j �j small at the same

frequency � Fortunately� in most practical cases jN�j �j is small when

jR�j �j is large� and vice versa�

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��� H� �Ozbay

We can formally de�ne the following performance problem� given

a positive number �r� design G�s� so that the largest tracking error

energy� due to any reference input r from the set R� is less than or

equal to �r� A dual problem for noise reduction can be posed similarly

for a given �n � �� These problems are equivalent to�

sup

r � R

n � �

kek� � �r �� kWrSk� � �r �����

sup

n � N

r � �

kek� � �n �� kWnTk� � �n� �����

The performance objective stated via ����� is satis�ed if and only if

jS�j �j � �rjWr�j �j � � ��� �

Clearly� the system has !good tracking performance" if �r is !small�"

Similarly� the e�ect of the noise n� on the system output yo� is !small"

if the inequality

jT �j �j � �njWn�j �j � �����

holds with a !small" �n� Hence� we can see �r and �n as performance

indicators� the smaller these numbers the better the performance�

Assume that Wr is a low�pass �lter with magnitude

jWr�j �j�� �r for � � � low

� � for � low

for some low indicating the !low�frequency" region of interest� Then�

��� � is equivalent to having

j� � G�j �j � � � � low

Page 153: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

which means that jG�j �j � � in the low�frequency band� In this case�

jS�j �j � �jG�j �j � ����� and hence� by choosing

jG�j �j � �� �rjWr�j �j � � low

the design condition ��� � is automatically satis�ed� In conclusion� G

should be such that

jG�j �j � � � ���r jWr�j �j � � low� �����

For the high�pass �lter Wn� we assume

jWn�j �j�� � for � � � high

� �n for � high

where high � low� Then� the design condition ����� implies

jG�j �jj� � G�j �j � � � � high

i�e�� we want jG�j �j � � in the high�frequency band� In this case� the

triangle inequality gives jT j � jGj��� jGj��� and we see that if

jG�j �j�� jG�j �j �

�njWn�j �j � � high

then G satis�es ������ Thus� in the high�frequency band jG�j �j should

be bounded as

jG�j �j � �� � ���n jWn�j �j��� � � high� �����

In summary� low and high�frequency behaviors of jGj are character�

ized by ����� and ������ The magnitude of G should be much larger

than � at low frequencies and much smaller than � at high frequencies�

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��� H� �Ozbay

Therefore� the gain crossover frequencies are located in the transition

band � low � high�� In the next section� we will see a condition on the

magnitude of G around the crossover frequencies�

��� Bode�s Gain�Phase Relationship

There is another key design objective in loopshaping� the phase margin

should be as large as possible� Now we will try to understand the

implications of this design goal on the magnitude of G� Recall that the

phase margin is

� � minf� � � G�j ci� � i � �� � � � � �g

where ci�s are the crossover frequencies� i�e�� jG�j ci�j � �� Without

loss of generality assume that � � �� in other words gain crossover occurs

at a single frequency c �since G � PC is designed here� we can force

G to have single crossover�� For good phase margin � G�j c� should be

signi�cantly larger than ��� For example� for ��� to ��� phase margin

the phase � G�j c� is desired to be ����� to �����

It is a well known mathematical fact that if G�s� is a minimum phase

transfer function� then its phase function � G�j � is uniquely determined

from the magnitude function jG�j �j� once the sign of its DC gain G���

is known �i�e�� we should know whether G��� � � or G��� � ��� In

particular � G�j c�� hence� the phase margin can be estimated from

the asymptotic Bode magnitude plot� Assume G��� � �� then for any

o � � the exact formula for the phase is �see ��� p� � ��

� G�j o� ��

Z �

��M���W ���d�

where � � ln� � o� is the normalized frequency appearing as the in�

tegration variable �note that � oe�� and

M��� �d

d�jG�j oe

��j

Page 155: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

−5 −4 −3 −2 −1 0 1 2 3 4 50

1

2

3

4

5

6

normalized frequency

inte

grat

ion

wei

ght

Figure ���� W ��� versus ��

W ��� � ln�coth�j�j�

���

The phase at o depends on the magnitude at all frequencies ��� � ��� So� it is not an easy task to derive the whole phase function

from the magnitude plot� But we are mainly interested in the phase at

the crossover frequency o � c� The connection between asymptotic

Bode plots and the above phase formula is this� M��� is the slope of

the log�log magnitude plot of G at the normalized frequency �� The

weighting function W ��� is independent of G� As shown in Figure ����

its magnitude is very large near � � � �in fact W ��� � �� and it decays

to zero exponentially� For example� when j�j � � we can treat W ��� as

negligibly small� W � �� � �����

Now assume that the slope of the log�log Bode magnitude plot is

almost constant in the frequency band � ce�� � ce��� say �n� ��

dB per decade� Then M��� � �n in the normalized frequency band

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��� H� �Ozbay

j�j � � and hence�

� G�j c� �� �n�

Z �

��W ���d� �� �n

Z �

��W ���d� � �n�

�the error in the approximation of the integral of W ��� is less than �

percent of the exact value ������ Since Bode�s work� ���� the approx�

imation � G�j c� �� �n�� has been used as an important guideline for

loopshaping� if the Bode magnitude plot of G decreases smoothly with a

slope of ��� dB�dec� near � c� then the phase margin will be large

�close to ����� From the above phase approximations it is also seen

that if the Bode magnitude of G decreases with a slope ��� dB�dec� or

faster� near � c� then the phase margin will be small� if not negative�

In summary� we want to shape the Bode magnitude of G so that

around the crossover frequency its slope is almost constant at��� dB�dec�

This assures good phase margin�

��� Design Example

We now have a clear picture of the desired shape of the magnitude plot

for G � PC� Loopshaping conditions derived in the previous sections

are illustrated in Figure �� � where Glow and Ghigh represent the lower

and upper bounds given in ����� and ����� respectively� and the dashed

lines indicate the desired slope of jGj near the crossover frequency c�

It is clear that the separation between low and high should be at

least two decades� so that c can be placed approximately one decade

away from low and high frequencies� and ��� dB�dec slope can be as�

signed to jGj around the crossover frequency� Otherwise� the transition

frequency band will be short� in this case� if Glow is too large and Ghigh

is too small� it will be di�cult to achieve a smooth ��� dB�dec slope

around a su�ciently large neighborhood of c�

Another important point in the problem setup is this� since C �

Page 157: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

0 dBω

ωhighωc ω

Glow

highG

low

|G|

Figure �� � Loopshaping conditions�

G�P must be a proper function� jG�j �j should decay to zero at least

as fast as jP �j �j does� for ��� Therefore� Ghigh� � should rollo�

with a slope ��� �� dB�dec� where � is greater or equal to the relative

degree of P � NP �DP i�e��

� � deg�DP �s��� deg�NP �s���

This is guaranteed if Wn�s� is improper with relative degree ���The example we study here involves an uncertain plant

P �s� � Po�s� � #P �s� where Po�s� � �o

s� � �� os � �o

#P �s� is an unknown stable transfer function� o � � and � � �� � ���

In this case� the standard feedback system with uncertain P � depicted

in Figure ���� is equivalent to the feedback system shown in Figure ����

The feedback signal y�t� is the sum of two signals� yo�t� and y��t�� The

latter signal is unknown due to plant uncertainty #P � Using frequency

Page 158: Feedback Control Systems- HOzbay

��� H� �Ozbay

r(t) u(t)P(s)C(s)

+-

o + (s)∆y(t)

Figure ���� Standard feedback system with uncertain plant�

r(t)

+-

C(s) u(t) P(s)

∆P(s)P(s)o

o

++

y(t)

y(t)o

-1

δy(t)

Figure ���� Equivalent feedback system�

domain representations�

Y��s� ��

Po�s�#P �s�Yo�s��

Now de�ning

kn�s�No�s� �� #P �s�Yo�s� and Wn�s� �kn�s�

Po�s�

we see that Figure ��� is equivalent to Figure ��� with P � Po and

N � WnNo�

Assuming that the feedback system is stable� yo is a bounded func�

tion� that implies no is bounded because #P is stable� The scaling factor

kn�s� is introduced to normalize No� Large values of jkn�j �j imply a

large uncertainty magnitude j#P �j �j� Conversely� if jkn�j �j is small�

then the contribution of the uncertainty is small at that frequency� In

summary� the problem is put in the framework of Figure ��� by taking

out the uncertainty and replacing y��t� with n�t�� The noise n�t� is from

the set N � which is characterized by the �lter Wn�s� � kn�s��Po�s��

where jkn�j �j represents the !size" of the uncertainty�

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Introduction to Feedback Control Theory ���

Let us assume that jkn�j �j � � for � �� o �� high� i�e�� the

nominal model Po�j � represents the actual plant P �j � very well in

the low�frequency band� Note that� jPo�j �j � � for � high� So

we can assume that in the high�frequency region the magnitude of the

uncertainty can be relatively large� e�g��

kn�j � � ����j��� � o

� � j���� � o� � �� o�

In this numerical example the uncertainty magnitude is ���� near �

�� o� and it is about ��� for � ��� o� To normalize the frequency

band� set o � �� Now the upper bound �determined in ������

Ghigh� � ���n

�n � jWn�j �j ��njPo�j �j

�njPo�j �j� jkn�j �j �

can be computed for any given performance indicator �n� Here we take

�n � ���� arbitrarily�

For the tracking problem� suppose that the steady state error� ess�

must satisfy

� ess � �� for unit ramp reference input�

� jessj � ��� for all sinusoidal inputs whose periods are �� seconds

or larger�

The �rst condition implies that

lims��

sG�s� � lims��

s�

s� ��

thus we have a lower bound on the DC behavior of G�s�� Recall that for

a sinusoidal input of period � the amplitude of the steady state error is

jessj � jS�j��

��j �

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��� H� �Ozbay

and this quantity should be less than or equal to ��� for � � �� sec�

Therefore� by using the notation of Section ���� we de�ne

���r jWr�j �j � �� � � �� � low�

where low � ����� �� ��� rad�sec� In conclusion� a lower bound for

jG�j �j can be de�ned in the low�frequency region as

jG�j �j � maxf �

� �� � ��� g �� Glow� ��

The low�frequency lower bound� Glow� �� and the high�frequency

upper bound Ghigh� � are shown as dashed lines in Figure ���� The

desired open�loop shape

G�s� ������ � ���s������ � s���������

s�� � ��s���� � s��������� � s�������

is designed from following guidelines�

� as � �� we should have jG�j �j � �� � the term � �s takes care

of this requirement�

� c should be placed near � rad�sec� say between ��� rad�sec and

� rad�sec� and the roll�o� around a large neighborhood of c

should be ��� dB�dec�

� to have c in this frequency band we need fast roll�o� near ���� rad�sec� the second�order term in the denominator takes care

of that� its damping coe�cient is chosen relatively small so that

the magnitude stays above �� log���� dB in the neighborhood of

� ��� rad�sec�

� for � low the slope should be constant at ��� dB�dec� so the

zeros near � ����� rad�sec are introduced to cancel the e�ect

of complex poles near � ��� rad�sec�

Page 161: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

|G| G_low G_high

10−2

10−1

100

101

102

−80

−60

−40

−20

0

20

40

60

omega

Gai

n (d

B)

Figure ���� Glow� �� Ghigh� � and jG�j �j�

� in the frequency range c � � �� rad�sec the magnitude should

start to rollo� fast enough to stay below the upper bound� so we

use double poles at s � �

For this example� verify that the gain margin is �� dB and the phase

margin is ��� It is possible to tweak the numbers appearing in ����

and improve the gain and�or phase margin without violating upper and

lower bound conditions� In general� loopshaping involves several steps

of trial and error to obtain a !reasonably good" shape for jG�j �j� The

!best" loop shape is di�cult to de�ne�

Once G�s� is determined� the controller is obtained from

C�s� �G�s�

P �s��

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��� H� �Ozbay

In the above example P �s� � Po�s� ���o

s����os���o� so the controller is

C�s� � G�s��s� � �� os � �o�

�o� �����

where G�s� is given by ����� The last term in ����� cancels the com�

plex conjugate poles of the plant� As we have seen in Chapter �� it is

dangerous to cancel lightly damped poles of the plant �i�e�� poles close

to Im�axis�� If the exact location of these poles are unknown� then we

must be cautious� By picturing how the root locus behaves in this case�

we see that the zeros should be placed to the left of the poles of the

plant� An undesirable situation might occur if we overestimate o and

underestimate � in selecting the zeros of the controller� in this case� two

branches of the root locus may escape to the right half plane as seen

in Chapter �� Therefore� we should use the lowest estimated value for

o� and highest estimated value for �� in C�s� so that the associated

branches of the root locus bend towards left�

��� Exercise Problems

�� For the above example let G�s� be given by ���� and de�ne � �

����� o � �� b� �� � � � � b o � o � ��o � where j� j � ����� and

j��o j � ����

�a� Draw Bode plots of

bG�s� � G�s� o �s� � �b�b os � b �o�b �o �s� � �� os � �o�

for di�erent values of � and ��o � illustrating the changes in

gain and phase margins�

�b� Let � � ����� and ��o � ����� and draw the root locus�

�c� Repeat part �b� with � � ������ and ��o � �����

Page 163: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

�� In this problem� we will design a controller for the plant

P �s� ��� � ��ns� n � �s� n���

s �� � ��ds� d � �s� d���

by using loopshaping techniques with the following weights

���r jWr�j �j �k

� low � ��� rad�sec

���n jWn�j �j ����� k �p� � ����� �

high � �� rad�sec�

where k � � is a design parameter to be maximized�

�a� Find a reasonably good desired loopshape G�s� for �i� k �

���� �ii� k � �� and �iii� k � �� What happens to the upper

and lower bounds �Ghigh and Glow� as k increases� Does

the problem become more di�cult or easier as k increases�

�b� Assume that �n � �� �%� �d � ��� �%� n � � �%�

d � � �% and let k � �� Derive a controller from the

result of part �a�� For this system� draw the Bode plots with

several di�erent values of the uncertain plant parameters and

calculate the gain� phase� and vector margins�

� Let P �s� � ��s�� ���s�� ��� � Design a controller C�s� such that

�i� the system is stable with gain margin � �� dB and phase

margin � ����

�ii� the steady state error for a unit step input is less than or

equal to �����

�iii� and the loop shaping conditions

kWrSk� � �

kWnTk� � �

hold for Wr�s� � ���s� � �s� � �s � ���� and

Wn�s� � ����� s��

Page 164: Feedback Control Systems- HOzbay
Page 165: Feedback Control Systems- HOzbay

Chapter �

Robust Stability and

Performance

���� Modeling Issues Revisited

In this chapter� we consider the standard feedback control system shown

in Figure ����� where C is the controller and P is the plant�

As we have seen in Section ���� there are several approximations in�

volved in obtaining a mathematical model of the plant� One of the reas�

ons we desire a simple transfer function �with a few poles and zeros� for

C(s) P(s)r(t)

v(t)

y(t)+++-

e(t) u(t)

Figure ����� Standard feedback system�

���

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��� H� �Ozbay

the plant is that it is di�cult to design controllers for complicated plant

models� The approximations and simpli�cations in the plant dynamics

lead to a nominal plant model� However� if the controller designed for

the nominal model does not take into account the approximation er�

rors �called plant uncertainties�� then the feedback system may become

unstable when this controller is used for the actual plant� In order to

avoid this situation� we should determine a bound on the uncertainty

and use this information when we design the controller� There are

mainly two types of uncertainty� unmodeled dynamics and paramet�

ric uncertainty� Earlier in Chapter � we saw robust stability tests for

systems with parametric uncertainty �recall Kharitanov�s test� and its

extensions�� In this chapter� H��based controller design techniques are

introduced for systems with dynamic uncertainty� First� in this section

we review modeling issues discussed in Section ���� give additional ex�

amples of unmodeled dynamics� and illustrate again how to transform

a parametric uncertainty into a dynamic uncertainty�

������ Unmodeled Dynamics

A practical example is given here to illustrate why we have to deal with

unmodeled dynamics and how we can estimate uncertainty bounds in

this case� Consider a �exible beam attached to a rigid body �e�g�� a large

antenna attached to a space satellite�� The input is the torque applied

to the beam by the rigid body and the output is the displacement at

the other end of the beam� Assume that the magnitude of the torque is

small� so that the displacements are small� Then� we can use a linearized

model� and the input�output behavior of this system is given by the

transfer function

P �s� � P��s� �K�

s��

�Xk��

�k �k

s� � ��k ks � �k

Page 167: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

where k�s represent the frequency of natural oscillations �modes of the

system�� �k�s are the corresponding damping coe�cients for each mode

and �k�s are the weights of each mode� Typically� � � �k � � for

some � � �� �k � � and k � � as k � �� Obviously� it is not

practical to work with in�nitely many modes� When �k�s converge to

zero very fast� we can truncate the higher�order modes and obtain a

�nite dimensional model for the plant� To illustrate this point� let us

de�ne the approximate plant transfer function as

PN �s� �K�

s��

NXk��

�k �k

s� � ��k ks � �k�

The approximation error between the !true" plant transfer function and

the approximate model is

#PN �s� � P��s�� PN �s� ��X

k�N��

�k �k

s� � ��k ks � �k�

As long as j#PN �j �j is !su�ciently small" we can !safely" work with

the approximate model� What we mean by !safely" and how small is

su�ciently small will be discussed shortly� First try to determine the

peak value of the approximation error� assume � � � � �k ��p�

for all

k � N � then

j#PN �j �j ��X

k�N��

���� �k �k

�k � � � j��k k

���� � �Xk�N��

j�kj�kp

Now suppose j�kj � �kp

�k�� for all k � N � Then� we have the follow�

ing bound for the worst approximation error�

sup�j#PN �j �j �

�Xk�N��

k�� �Z �

x�N

x��dx ��

N�

For example� if we take the �rst �� modes only� then the worst ap�

proximation error is bounded by ��� � Clearly� as N increases� the worst

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��� H� �Ozbay

approximation error decreases� On the other hand� we do not want to

take too many modes because this complicates the model� Also note

that if j�kj decays to zero faster� then the bound is smaller�

In conclusion� for a plant P �s�� a low�order model denoted by P��s�

can be determined� Here � represents the parameters of the low�order

model� e�g�� coe�cients of the transfer function� Moreover� an upper

bound of the worst approximation error can be obtained in this process�

jP �j �� P��j �j � ��� ��

The pair �P��s�� ��� �� captures the uncertain plant P �s�� The para�

meters represented by � may be uncertain too� But we assume that

there are known bounds on these parameters� This gives a nominal

plant model Po�s� �� P�o�s� where �o represent the nominal value of

the parameter vector �� See next section for a speci�c example�

������ Parametric Uncertainty

Now we transform a parametric uncertainty into a dynamic uncertainty

and determine an upper bound on the worst approximation error� As

an example� consider the above �exible beam model and take

P��s� �K�

s��

�� ��

s� � ��� �s � ���

Here � � K� � �� � �� � ��� Suppose that an upper bound ��� � is

determined as above for the unmodeled uncertainty using bounds on

�k�s and �k�s� for k � �� In practice� � is determined from the physical

parameters of the beam �such as inertia and elasticity�� If these para�

meters are not known exactly� then we have parametric uncertainty as

well� At this point� we can also assume that the parameters of P��s�

are determined from system identi�cation and parameter estimation al�

gorithms� These techniques give upper and lower bounds� and nominal

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Introduction to Feedback Control Theory ���

values of the uncertain parameters� For our example� let us assume

that K� � � and �� � ���� are known exactly and �� � ��� � ���� with

nominal value �� � ����� � � � � ��� with nominal value � � ���

De�ne

�o � � � ��� � ���� � ���

and the nominal plant transfer function

Po�s� � P�o�s� ��

s��

��

s� � s � ����

The parametric uncertainty bound is denoted by ��� ��

jP��j �� Po�j �j � ��� � �

for all feasible �� Figure ���� shows the plot of jP��j � � Po�j �j for

several di�erent values of �� and � in the intervals given above� The

upper bound shown here is ��� � � jWp�j �j�

Wp�s� ����s � �����

�s� � � s � ����

Parametric uncertainty bound and the bound for unmodeled dy�

namic uncertainty can be combined to get an upper bound on the overall

plant uncertainty

#P �s� � P �s�� Po�s� where j#P �j �j � j��� �j� j��� �j

As in Section ���� let W �s� be an overall upper bound function such

that

jW �j �j � j��� �j� j���j �j � j#P �j �j �

Then� we will treat the set

P � fP � Po � #P �s� � j#P �j �j � jWa�j �j and

P and Po have the same number of poles in C�gas the set of all possible plants�

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��� H� �Ozbay

10−3

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

omega

mag

nitu

de

Uncertainty Bound

Figure ����� Parametric uncertainty bound�

���� Stability Robustness

������ A Test for Robust Stability

Given the nominal plant Po�s� and additive uncertainty bound W �s��

characterizing the set of all possible plants P� we want to derive

conditions under which a �xed controller C�s� stabilizes the feedback

system for all plants P � P� Since Po � P� the controller C

should stabilize the nominal feedback system �C�Po�� This is a ne�

cessary condition for robust stability� Accordingly� assume that �C�Po�

is stable� Then� we compare the Nyquist plot of the nominal system

Go�s� � C�s�Po�s�� and the Nyquist plots of all possible systems of the

form G�s� � C�s�P �s�� where P � P� By the assumption that Po

and P have the same number of poles in C�� for robust stability G�j �

should encircle �� as many times as Go�j � does� Note that for each

� we have

G�j � � Go�j � � #P �j �C�j ��

Page 171: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

Im

Re

|WC |

G0

G

-1

Figure ��� � Robust stability test via Nyquist plot�

This identity implies that G�j � is in a circle whose center is Go�j �

and radius is jW �j �C�j �j� Hence� the Nyquist plot for G lies within

a tube around the nominal Nyquist plot Go�j �� The radius of the tube

is changing with � as shown in Figure ��� �

By the above arguments� the system is robustly stable if and only if ��

is outside this tube� in other words�

jGo�j � � #P �j �C�j �� ����j � �

for all and for all admissible #P �j �� Since the magnitude of #P �j �

can be as large as �jW �j �j � �� where � � � and its phase is not

restricted� we have robust stability if and only if

j� � Go�j �j � jW �j �C�j �j � � for all �

which is equivalent to

jW �j �C�j ��� � Po�j �C�j ����j � � for all �

Page 172: Feedback Control Systems- HOzbay

��� H� �Ozbay

i�e�

kWCSk� � � ������

where S � �� � PoC��� is the nominal sensitivity function�

It is very important to understand the di�erence between the set

of plants P� �where #P �s� can be any transfer function such that P

and Po have the same number of right half plane poles and j#P �j � �

jW �j �j� and a set of uncertain plants described by the parametric

uncertainty� For example� consider the uncertain time delay example of

Section �����

P �s� �e�hs

�s � ��h � � � ����

with Po�s� � ��s��� and W �s� � � ���� ���� s���

�� � s����� �� s��� � It was noted that

the set

Ph �� fP �s� �e�hs

�s � ��� h � � � ����g

is a proper subset of

P �� fPo � # � #�s� is stable� j#�j �j � jW �j �j � g�

Hence� if we can prove robust stability for all plants in P� then we

have robust stability for all plants in Ph� The converse is not true� In

fact� we have already determined a necessary and su�cient condition

for robust stability for all plants in Ph� for a �xed controller C� from

the Bode plots of the nominal system Go � PoC determine the phase

margin� � in radians� and crossover frequency� c in radians per second�

then the largest time delay the system can tolerate is hmax � �� c�

That is� the closed�loop system is robustly stable for plants in Ph if

and only if ��� � hmax� The controller may be such that this condition

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Introduction to Feedback Control Theory ��

holds� but ������ does not� So� there is a certain degree of conservatism

in transforming a parametric uncertainty into a dynamic uncertainty�

This should be kept in mind�

Now we go back to robust stability for the class of plants P� If a

multiplicative uncertainty bound Wm�s� is given instead of the additive

uncertainty bound W �s�� we use the following relationship�

P � Po � #P � Po�� � P��o #P �

i�e�� multiplicative uncertainty bound satis�es

jP��o �j �#P �j �j � jWm�j �j for all �

Then� in terms of Wm� the condition ������ can be rewritten as

kWmPoC�� � PoC���k� � �� ������

Recall that the sensitivity function of the nominal feedback system

�C�Po� is de�ned as S � �� � PoC��� and the complementary sens�

itivity is T � ��S � PoC�� �PoC���� So� the feedback system �C�P �

is stable for all P � P if and only if �C�Po� is stable and

kWmTk� � �� ���� �

For a given controller C stabilizing the nominal feedback system� it

is possible to determine the amount of plant uncertainty that the system

can tolerate� To see this� rewrite the inequality ���� � as

jWm�j �j � jT �j �j�� for all �

Hence� the largest multiplicative plant uncertainty that the system can

tolerate at each is jT �j �j���

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��� H� �Ozbay

0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

K

alph

a(K

)

Figure ����� � versus K�

Example ��� Let

Po�s� ����s � ����

s�s� �����s� � �s � ���

W �s� � �cW �s� � �����s � �����

�s � ���

The system has !good" robustness for this class of plants if � is !large�"

We are now interested in designing a proportional controller C�s� � K

such that k�cWCSk� � � for the largest possible �� First� we need to

�nd the range of K for which �C�Po� is stable� By the Routh�Hurwitz

test� we �nd that K must be in the interval ����� � ���� Then� we de�ne

����K� � sup�

����� cW �j �K

� � KPo�j �

����� �By plotting the function ��K� versus K � ����� � ���� see Figure �����

we �nd the largest possible � and the corresponding K� �max � ��� is

achieved at K � ������

Page 175: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

H1

H2

+

++

-

Figure ����� Feedback system with small gain kH�H�k� � ��

������ Special Case Stable Plants

Suppose that the nominal plant� Po�s�� and the additive uncertainty�

#�s�� are stable� Then� we can determine a su�cient condition for ro�

bust stability using the small gain theorem stated below� This theorem

is very useful in proving robust stability of uncertain systems� Here it

is stated for linear systems� but the same idea can be extended to a

large class of nonlinear systems as well�

Theorem ��� �Small Gain� Consider the feedback system shown

in Figure ��� where H� and H� are stable linear systems� If

jH��j �H��j �j � � for all � ������

then the feedback system is stable�

The proof of this theorem is trivial� by ������ the Nyquist plot of

G � H�H� remains within the unit circle� so it cannot encircle the

critical point� By the Nyquist stability criterion� the feedback system

is stable�

Now consider the standard feedback system �C�P �� Suppose that the

nominal feedback system �C�Po� is stable� Then the system is robustly

stable if and only if S � �� � �Po � #P �C��� and CS are stable� It

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��� H� �Ozbay

is a simple exercise to show that

S � S�� � #PCS����

Since �C�Po� is stable� S� CS and PoS are stable transfer functions�

Now� applying the small gain theorem� we see that if

j#P �j �C�j �S�j �j � jW �j �C�j �S�j �j � � for all �

then S� CS and PS are stable for all P � P� which means that

the closed�loop system is robustly stable�

���� Robust Performance

Besides stability� we are also interested in the performance of the closed�

loop system� In this section� performance will be measured in terms of

worst tracking error energy� As in Chapter �� de�ne the set of all

possible reference inputs

R � fR�s� � Wr�s�Ro�s� � krok� � �g�

Let e�t� �� r�t�� y�t� be the tracking error in the standard unity feed�

back system formed by the controller C and the plant P � Recall that

the worst error energy �over all possible r in R�� is

supr�R

kek� � sup�jWr�j ��� � P �j �C�j ����j � kWr�� � PC���k��

If a desired error energy bound� say �r� is given� then the controller

should be designed to achieve

kWr�� � PC���k� � �r ������

for all possible P � Po � #P � P� This is the robust performance

condition� Of course� the de�nition makes sense only if the system is

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Introduction to Feedback Control Theory ��

robustly stable� Therefore� we say that the controller achieves robust

performance if it is a robustly stabilizing controller and ������ holds for

all P � P� It is easy to see that this inequality holds if and only if

jW��j �S�j �j � j� � #P �j �C�j �S�j �j

for all and all admissible #P � where S�s� � �� � Po�s�C�s���� is

the nominal sensitivity function and W��s� �� ���r Wr�s�� Since the

magnitude of #P �j � is bounded by jW �j �j and its phase is arbitrary

for each � the right hand side of the above inequality can be arbitrarily

close to �but it is strictly greater than� ��jW �j �C�j �S�j �j� Hence

������ holds for all P � P if and only if

jW��j �S�j �j� jW��j �T �j �j � � for all ������

where T � ��S is the nominal complementary sensitivity function and

W��s� � Wm�s� is the multiplicative uncertainty bound jWm�j �j �jP��o �j �#P �j �j� In the literature� W��s� is called the performance

weight and W��s� is called the robustness weight�

In summary� a controller C achieves robust performance if �C�Po�

is stable and ������ is satis�ed for the given weights W��s� and W��s��

Any stabilizing controller satis�es the robust performance condition for

some specially de�ned weights� For example� de�ne W��s� � ��s�S�s� and

W��s� � ��s�T �s� with any ( and ) satisfying j(�j �j� j)�j �j � �� then

������ is automatically satis�ed� Hence� we can always rede�ne W� and

W� to assign more weight to performance �larger W�� by reducing the

robustness weight W� and vice versa� But� we should emphasize that

the weights de�ned from an arbitrary stabilizing controller may not be

very meaningful� The controller should be designed for weights given

a priori� More precisely� the optimal control problem associated with

robust performance is the following� given a nominal plant Po�s� and

two weights W��s�� W��s�� �nd a controller C�s� such that �C�Po� is

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��� H� �Ozbay

stable and

jW��j �S�j �j� jW��j �T �j �j � b� � �����

holds for the smallest possible b�� It is clear that the robust performance

condition is automatically satis�ed for the weights cW��s� � b���W��s�

and cW��s� � b���W��s�� So� by trying to minimize b� we maximize

the amount of allowable uncertainty magnitude and minimize the worst

error energy in the tracking problem�

Example ��� Let Po�s� � �s���s���s��s���s��� � Here we want to �nd the

optimal proportional controller C�s� � K such that

J�K� �� sup�

�jS�j �j� jT �j �j� � b�for the smallest possible b�� subject to the condition that the feedback

system �C�Po� is stable� Applying the Routh�Hurwitz test we �nd that

K must be positive for closed�loop system stability� Using the Bode

plots� we can compute J�K� for each K � �� See Figure ���� where

the smallest b� � ���� is obtained for the optimal K � Kopt � ����

Magnitude plots for jS�j �j and jT �j �j are shown in Figure ��� for

Kopt � ���� K� � ��� and K� � ������

In general� the problem of minimizing b� over all stabilizing control�

lers� is more di�cult than the problem of �nding a controller stabilizing

�C�Po� and minimizing � in

jW��j �S�j �j� � jW��j �T �j �j� � �� � � ������

A solution procedure for ������ is outlined in Section ����� In the liter�

ature� this problem is known as the mixed sensitivity minimization� or

two�block H� control problem�

Page 179: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

0 5 10 152

3

4

5

6

7

8

9

10

11

K

J(K

)

J(K) versus K

Figure ����� J�K� versus K�

10−1

100

101

102

10−2

10−1

100

omega

Mag

nitu

de

Kopt=7.54 (−), K1=4.67 (−−), K2=10.41 (−.)

S T

Figure ���� jS�j �j and jT �j �j�

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�� H� �Ozbay

Exercise� Show that if � � �p�b�� then ������ implies ������

Sometimes robust stability and nominal performance would be suf�

�cient for certain applications� This is weaker �easier to satisfy� than

robust performance� The nominal performance condition is

kW�Sk� � � ������

and the robust stability condition is

kW�Tk� � �� �������

���� Controller Design for Stable Plants

������ Parameterization of all Stabilizing

Controllers

The most important preliminary condition for optimal control problems

such as ����� and ������ is stability of the nominal feedback system

�C�Po�� Now we will see a description of the set of all C stabilizing the

nominal feedback system �C�Po��

Theorem ��� Let Po be a stable plant� Then the set of all controllers

stabilizing the nominal feedback system is

C �

C�s� �

Q�s�

�� Po�s�Q�s�� Q�s� is proper and stable

� �������

Proof� Recall that the closed�loop system is stable if and only if

S � �� � PoC���� CS and PoS are stable transfer functions� Now� if

C � Q���PoQ��� for some stable Q� then S � ���PoQ�� CS � Q and

PoS � Po�� � PoQ� are stable� Conversely� assume �C�Po� is a stable

feedback system� then in particular C�� � PoC��� is stable� If we let

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Introduction to Feedback Control Theory ��

Q � C�� � PoC���� then we get C � Q�� � PoQ���� This completes

the proof�

For unstable plants� a similar parameterization can be obtained�

but in that case� derivation of the controller expression is slightly more

complicated� see Section ����� � A version of this controller parameter�

ization for possibly unstable MIMO plants was obtained by Youla et al��

� �� So� in the literature� the formula ������� is called Youla paramet�

erization� A similar parameterization for stable nonlinear plants was

obtained in ���

������ Design Guidelines for Q�s�

An immediate application of the Youla parameterization is the robust

performance problem� For a given stable plant Po� with performance

weight W� and robustness weight W�� we want to �nd a controller C

in the form C � Q��� PoQ���� where Q is stable� such that

jW��j �S�j �j� jW��j �T �j �j � � � �

In terms of the free parameter Q�s�� sensitivity and complementary

sensitivity functions are�

S�s� � �� Po�s�Q�s� and T �s� � �� S�s� � Po�s�Q�s��

Hence� for robust performance� we need to design a proper stable Q�s�

satisfying the following inequality for all

jW��j ���� Po�j �Q�j ��j� jW��j �Po�j �Q�j �j � � � �������

Minimum phase plants

In the light of �������� we design the controller by constructing a proper

stable Q�s� such that

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�� H� �Ozbay

� Q�j � � P��o �j � when jW��j �j is large and jW��j �j is small�

� jQ�j �j � � when jW��j �j � � and jW��j �j is large�

Obviously� if jW��j �j and jW��j �j are both large� it is impossible to

achieve robust performance� In practice� W� is large at low�frequencies

�for good tracking of low�frequency reference inputs� and W� is large

at high frequencies for robustness against high�frequency unmodeled

dynamics� This design procedure is similar to loop shaping when the

plant Po is minimum phase�

Example ��� Consider the following problem data�

Po�s� ��

s� � �s� �� W��s� �

s� W��s� �

s�� � ��� s�

���

Let Q�s� be in the form

Q�s� �s� � �s � �

�eQ�s�

where eQ�s� is a proper stable transfer function whose relative degree is

at least two� The design condition ������� is equivalent to

*� � �� j ���eQ�j ��

j j� jj �� � j ��� � eQ�j �

��j � � � �

So we should try to select eQ�j � �� � for all � low and j eQ�j �j � �

for all � high� where low can be taken as the bandwidth of the �lter

W��s�� i�e low �p

� rad�sec� and high can be taken as the bandwidth

of W��s���� i�e� high � �� rad�sec� Following these guidelines� let

us choose eQ�s� � �� � � s��� with ��� being close to the midpoint

of the interval �p

� � ���� say � � ���� For eQ�s� � �� � s����� the

function *� � is as shown in Figure ����� *� � � ���� � � for all

hence� robust performance condition is satis�ed� Figure ���� shows

J��� �� max� *� � as a function of ��� � We see that for all values of

Page 183: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

��� � �� � ��� robust performance condition is satis�ed and the best

value of � that minimizes J��� is � � ������ as expected� this is close

to � � ����

10−2

10−1

100

101

102

0.25

0.3

0.35

0.4

0.45

0.5

0.55

omega

Phi

Figure ����� *� � versus for � � ����

0 5 10 150

0.5

1

1.5

2

2.5

3

1/tau

J(ta

u)

Figure ����� J��� �� max� *� � versus ��� �

Page 184: Feedback Control Systems- HOzbay

�� H� �Ozbay

Non�minimum phase plants

When Po�s� has right half plane zeros or contains a time delay� we must

be more careful� For example� let Po�s� � e�hs �s�z��s���s���� with some

h � � and z � �� For good nominal performance we are tempted to

choose Q�s� �� Po�s��� � e�hs �s

���s����

�s�z� � There are three problems

with this choice� �i� Q�s� is improper� �ii� there is a time advance term

in Q�s� that makes it non�causal� and �iii� the pole at z � � makes

Q�s� unstable� The �rst problem can be avoided by multiplying Q�s�

by a strictly proper term� e�g�� eQ�s� � �� � � s��� with � � �� as

demonstrated in the above example� The second and third problems

can be circumvented by including the time delay and the right half

plane zeros of the plant in T �s� � Po�s�Q�s��

We now illustrate this point in the context of robust stability with

asymptotic tracking� here we want to �nd a stabilizing controller satis�

fying kW�Tk� � � and resulting in zero steady state error for unit step

reference input� as well as sinusoidal reference inputs with frequency

o� As seen in Section ��� G�s� � Po�s�C�s� must contain poles at

s � � and at s � j o for asymptotic tracking� At the poles of G�s� the

complementary sensitivity T �s� � G�s����G�s���� is unity� Therefore�

we must have

T �s���s����j�o � � � ����� �

But the controller parameterization implies T �s� � Po�s�Q�s�� So� ro�

bust stability condition is satis�ed only if jW����j � � and jW�� j o�j ��� Furthermore� we need Po��� � � and Po� j o� � �� otherwise

����� � does not hold for any stable Q�s�� If Po�s� contains a time

delay� e�hs� and zeros z�� � � � � zk in the right half plane� then T �s� must

contain at least the same amount of time delay and zeros at z�� � � � � zk�

because Q�s� cannot have any time advance or poles in the right half

Page 185: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

plane� To be more precise� let us factor Po�s� as

Po�s� � Pap�s�Pmp�s� � Pap�s� � e�hskYi��

zi � s

zi � s�������

and Pmp�s� � Po�s��Pap�s� is minimum phase� Then� T �s� � Po�s�Q�s�

must be in the form

T �s� � Pap�s� eT �s�

where eT � PmpQ is a stable transfer function whose relative degree is

at least the same as the relative degree of Pmp�s��

eT �s���s����j�o �

Pap�s�

��s����j�o

and kW�eTk� � �� Once eT �s� is determined� the controller parameter

Q�s� can be chosen as Q�s� � eT �s��Pmp�s� �note that Q is proper and

stable� and then the controller becomes

C�s� �Q�s�

�� Po�s�Q�s��

eT �s��Pmp�s�

�� Pap�s� eT �s�� �������

The controller itself may be unstable� but the feedback system is stable

with the choice ��������

Example ��� Here� we consider

Po�s� ��s� �

�s� � �s � ���

So� Pap�s� � ��s��s and Pmp�s� � ��s���

�s���s��� � We want to design a con�

troller such that the feedback system is robustly stable for W��s� �

����� � s��� and steady state tracking error is zero for step reference

inputs as well as sinusoidal reference inputs whose periods are ��� i�e��

o � � rd�sec�

Page 186: Feedback Control Systems- HOzbay

�� H� �Ozbay

There are four design conditions for eT �s��

�i� eT �s� must be stable�

�ii� eT ��� � �Pap������ � �� eT � j�� � �Pap� j����� � ��� j����

�iii� relative degree of eT �s� is at least one�

�iv� j eT �j �j � ��� � ������

� for all �

We may assume that

eT �s� �a�s

� � a�s � a��s � x��

�s�s� � ��

�s � x��eT��s�

for some x � �� the coe�cients a�� a�� a� are to be determined from

the interpolation conditions imposed by �ii�� and eT��s� is an arbitrary

stable transfer function� Interpolation conditions �ii� are satis�ed if

a� � x�

a� � Im� ���� � j�����x � j��� �

a� � x� �Re� ���� � j�����x � j��� � �

Let us try to �nd a feasible eT �s� by arbitrarily picking eT��s� � � and

testing the fourth design condition for di�erent values of x � �� By

using Matlab� we can compute kW�eTk� for each �xed x � �� See

Figure ������ which shows that the robustness condition �iv� is satis�ed

for x � ������ � ������ The best value of x �in the sense that kW�Tk�is minimum� is x � �����

Exercises�

�� For x � ����� compute eT �s� and the associated controller ��������

verify that the controller has poles at s � �� j�� Plot jW��j �T �j �jversus and check that the robustness condition is satis�ed�

�� Find the best value of x � ������ � ����� which maximizes the

vector margin�

�� De�ne eT��s� �� � �a�s��a�s�a��

�s����s���s��� � and pick x � ����� Plot jT �j �jversus for � � ���� ���� ����� ������ Find kW�Tk� for the same

Page 187: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory �

0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

9

x

|| W

2 T

||

Figure ������ kW�eTk� versus x�

values of � and compare these results with the lower bound

maxfjW����j � jW�� j��jg � kW�Tk��

Smith predictor

For stable time delay systems there is a popular controller design method

called Smith predictor� which dates back to ����s� We will see a con�

nection between the controller parameterization ������� and the Smith

predictor dealing with stable plants in the form Po�s� � P��s�e�hs

where h � � and P��s� is the nominal plant model for h � �� To ob�

tain a stabilizing controller for Po�s�� �rst design a controller C��s�

that stabilizes the non�delayed feedback system �C�� P��� Then let

Q� �� C��� �P�C����� note that by construction Q��s� is stable� If we

use Q � Q� in the parameterization of stabilizing controllers ������� for

Po�s� � P��s�e�hs� we obtain the following controller

C�s� �Q��s�

�� e�hsP��s�Q��s��

C��s�

� � P��s�C��s���� e�hs�� �������

Page 188: Feedback Control Systems- HOzbay

�� H� �Ozbay

The controller structure ������� is called Smith predictor� ���� When

C is designed according to �������� the complementary sensitivity func�

tion is T �s� � e�hsT��s� where T��s� is the complementary sensitivity

function for the non�delayed system� T��s� � P��s�C��s���P��s�C��s�

� Therefore�

jT �j �j � jT��j �j for all � This fact can be exploited to design

robustly stabilizing controllers� For a given multiplicative uncertainty

bound W�� determine a controller C� from P� satisfying kW�T�k� � ��

Then� the controller ������� robustly stabilizes the feedback system with

nominal plant Po�s� � P��s�e�hs and multiplicative uncertainty bound

W��s�� The advantage of this approach is that C� is designed independ�

ent of time delay� it only depends on W� and P�� Also� note that the

poles of T �s� are exactly the poles of the non�delayed system T��s��

���� Design of H� Controllers

A solution for the H� optimal mixed sensitivity minimization problem

������ is presented in this section� For MIMO �nite dimensional sys�

tems a state space based solution is available in Matlab� the relevant

command is hinfsyn �it assumes that the problem data is transformed

to a !generalized" state space form that includes the plant and the

weights�� see ��� The solution procedure outlined here is taken from

���� It allows in�nite dimensional plant models and uses SISO transfer

function representations�

������ Problem Statement

Let us begin by restating the H� control problem we are dealing with�

given a nominal plant model Po�s� and two weighting functions W��s�

and W��s�� we want to �nd a controller C stabilizing the feedback sys�

tem �C�Po� and minimizing � � � in

+� � �� jW��j �S�j �j� � jW��j �T �j �j� � �� �

Page 189: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

where S � �� � PoC��� and T � � � S� The optimal controller is

denoted by Copt and the resulting minimal � is called �opt�

Once we �nd an arbitrary controller that stabilizes the feedback sys�

tem� then the peak value of the corresponding +� � is an upper bound

for ��opt� It turns out that for the optimal controller we have +� � � ��opt

for all � This fact implies that if we �nd a stabilizing controller for

which +� � is not constant� then the controller is not optimal� i�e�� the

peak value of +� � can be reduced by another stabilizing controller�

Assumptions and preliminary de�nitions

Let fz�� � � � � zkg be the zeros and fp�� � � � � p�g be the poles of Po�s� in

the open right half plane� Suppose that Po�s� � e�hsP��s�� where P��s�

is rational and h � �� De�ne the following proper stable functions

Mn�s� �� e�hskYi��

zi � s

zi � s� Md�s� ��

�Yi��

pi � s

pi � s� No�s� ��

Po�s�Md�s�

Mn�s��

Note that Mn�s� and Md�s� are all�pass and No�s� is minimum phase�

Since P��s� is a rational function� No�s� is in the form

No�s� �nNo�s�

dNo�s�

for some stable polynomials nNo�s� and dNo�s�� Moreover� jPo�j �j �

jNo�j �j for all � As usual� we assume Po�s� to be strictly proper�

which means that deg�nNo� � deg�dNo��

For example� when

Po�s� � e��s��s� ����s � �

�s� ���s� � �s � ����

we de�ne

Mn�s� � e��s��� s��

�� � s��� Md�s� �

��� s�

�� � s��

Page 190: Feedback Control Systems- HOzbay

��� H� �Ozbay

No�s� ����s � ����s � �

�s � ���s� � �s � ����

We assume that W��s� is in the form W��s� � nW��s��dW��s��

where nW��s� and dW��s� are stable polynomials with deg�nW�� �

deg�dW�� � n� � �� We will also use the notations

M��s� ��dW���s�dW��s�

E�s� ��nW���s�dW��s�

� ��dW���s�nW��s�

Recall that since W��s� � Wm�s� we have jW��j �j � jW �j��jjPo�j��j �

where jW �j �j is the upper bound of the additive plant uncertainty mag�

nitude� So we let W��s� � W �s�No�s��� with W �s� � nW �s��dW �s��

where nW �s� and dW �s� are stable polynomials and we assume that

deg�nW � � deg�dW � � ��

������ Spectral Factorization

Let A�s� be a transfer function such that A�j � is real and A�j � � �

for all � for some � � �� Then� there exists a proper stable function

B�s� such that B�s��� is also proper and stable and jB�j �j� � A�j ��

Construction of B�s� from A�j � is called spectral factorization� There

are several optimal control problems whose solutions require spectral

factorizations� H� control is one of them�

For the solution of mixed sensitivity minimization problem de�ne

A�j � ��

�jW �j �j� � jW��j �j�

�jNo�j �j� � jW �j �j�

��

�����

Since W��s�� W �s� and No�s� are rational functions� we can write

A�s� � nA�s��dA�s� where nA�s� and dA�s� are polynomials�

nA�s� � dW��s�dW���s�dNo�s�dNo��s�dW �s�dW ��s�

Page 191: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

dA�s� � nW���s�nW��s�nNo�s�nNo��s�dW �s�dW ��s�� dW���s�dW��s�dNo�s�dNo��s�nW �s�nW ��s�� ���nW���s�nW��s�dNo�s�dNo��s�nW �s�nW ��s��

By symmetry� if ,p � C is a root of dA�s�� then so is �,p� Suppose �

is such that dA�s� has no roots on the Im�axis� Then we can label the

roots of dA�s� as ,p�� � � � � ,p�nb with ,pnb�i � �,pi� and Re�,pi� � � for all

i � �� � � � � nb� Finally� the spectral factor B�s� � nB�s��dB�s� can be

determined as

nB�s� � dW��s� dNo�s� dW �s�

dB�s� �pdA���

nbYi��

��� s�,pi��

Note that B�s� is unique up to multiplication by �� �i�e� B�s� and

�B�s� have the same poles and zeros� and the same magnitude on the

Im�axis�� Another important point to note is that both B�s� and B�s���

are proper stable functions�

������ Optimal H� Controller

The optimal H� controller can be expressed in the form

Copt�s� �W��s�

�opt

B�s�Md�s�E�s�

Lopt�s� � �optM��s�Mn�s�No�s�B�s�������

where Lopt�s� � nL�s��dL�s�� for some polynomials nL�s� and dL�s�

with deg�nL� � deg�dL� � �n� � � � ��� The function Lopt�s� is such

that Dopt�s� and bDopt�s� do not have any poles at the closed right half

plane zeros of Md�s� and E�s��

Dopt�s� ��Lopt�s� � �optM��s�Mn�s�No�s�B�s�

B�s�Md�s�E�s��������

bDopt�s� ��Lopt�s�� ��Lopt��s�

B�s�Md�s�E�s��������

Next� we compute Lopt�s� and �opt from these interpolation conditions�

Page 192: Feedback Control Systems- HOzbay

��� H� �Ozbay

First� note that the zeros of E�s� can be labeled as ,z�� � � � � ,z�n� � with

,zn��i � �,zi and Re�,zi� � � for all i � �� � � � � n�� Now de�ne

�i ��

�pi i � �� � � � � �

,zi�� i � � � �� � � � � � � n� �� n

where pi� i � �� � � � � � are the zeros of Md�s� �i�e�� unstable poles of

the plant�� For simplicity� assume that �i � �j for i � j� Total

number of unknown coe�cients to be determined for nL�s� and dL�s�

is �n� Because of the symmetric interpolation points� it turns out that

nL�s� � dL��s� and hence bDopt�s� �� Let us introduce the notation

dL�s� �n��Xi��

visi � � s � � � sn��� v �������

nL�s� �n��Xi��

vi��s�i � � s � � � sn��� Jn v �������

where v �� v� � � � vn���T denotes the vector of unknown coe�cients of

dL�s� and Jn is an n � n diagonal matrix whose ith diagonal entry is

����i��� i � �� � � � � n� For any m � � de�ne the n �m Vandermonde

matrix

Vm ��

��� � �� � � � �m������

��� � � � ���

� �n � � � �m��n

��� �and the n� n diagonal matrix F whose ith diagonal entry is F ��i� for

i � �� � � � � n where

F �s� �� �M��s�Mn�s�No�s�B�s��

Finally� �opt is the largest value of � for which the n� n matrix

R �� VnJn � F Vn

Page 193: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

is singular� By plotting the smallest singular value of R �in Matlab�s

notation min�svd�R��� versus � we can detect �opt� The corresponding

singular vector vopt satis�es

Roptvopt � � �������

where Ropt is the matrix R evaluated at � � �opt� The vector vopt

de�nes Lopt�s� via ������� and �������� The optimal controller Copt�s�

is determined from Lopt�s� and �opt�

The controller ������ can be rewritten as

Copt�s� �W��s�

dopt�opt

� � Hopt�s������ �

Hopt�s� ��Dopt�s�

dopt� � �������

where Dopt�s� is given by �������� and

dopt �� Dopt��� �W���� jW ���j���optq

�� jW����j����opt�

Clearly Hopt�s� is strictly proper� moreover it is stable if Lopt�s� is

stable� Controller implementation in the form ����� � is preferred over

������ because when Mn�s� contains a time delay term� it is easier to

approximate Hopt�s� than Dopt�s��

Example ��� We now apply the above procedure to the following

problem data�

Po�s� �e�hs

�s� ��� W��s� �

��� � s�

�� � ��s�� W �s� � ��� �

Our aim is to �nd �opt and the corresponding H� optimal controller

Copt�s�� We begin by de�ning

Mn�s� � e�hs � Md�s� ���� s�

�� � s�� No�s� �

��

�� � s�

M��s� ���� ��s�

�� � ��s�� E�s� �

����� ��� � ������ � ��s�

�� � ��s���� � s��

Page 194: Feedback Control Systems- HOzbay

��� H� �Ozbay

Next� we perform the spectral factorization for

dA�s� � �������� ����� ������ ��������s� � ��� ��������s��

Since dB�s�dB��s� � dA�s� we de�ne dB�s� � �a � bs � cs��� where

a �p

������� ���� c �p

�� �������� b �p

����� ������� � �ac

that leads to

B�s� ��� � ��s��� � s�

�a � bs � cs��and F �s� �

�e�hs���s� ��

�a � bs � cs���

Let �min be the smallest and �max be the largest � values for which this

spectral factorization makes sense� i�e�� a � �� b � �� c � �� Then�

�opt lies in the interval ��min � �max�� A simple algebra shows that

�min � ��� ��� and �max �p

������ �� ��� For � � ���� �� � ��� the

zeros of E�s� are ,z��� � jq

������������� and the only zero of Md�s� is

p� � �� Therefore� we may choose �� � � and �� � jq

������������� � Then�

the �� � matrix R is

R �

�� ��

� ��

� �� �

� ��

���F ���� �

� F ����

� �� ��

� ��

��

The smallest singular value of R versus � plots are shown in Figure �����

for three di�erent values of h � ���� ���� ����� We see that the optimal �

values are �opt � ����� ��� � ����� respectively� �since the zeros of E�s�

are not distinct at � � ��� the matrix R becomes singular� independent

of h at this value of �� this is the reason that we discard � � ����

The optimal controller for h � ��� is determined from �opt � ���� and

the corresponding vopt � ���� � � �������T� which gives dL�s� �

����� � � ������ s�� Since nL�s� � dL��s� we have Lopt�s� �� �� �����s���� � ����s�� Note that Lopt�s� is stable� so the resulting Hopt�s�

is stable as well� The Nyquist plot of Hopt�j � is shown in Figure ������

from which we deduce that Copt�s� is stable�

Page 195: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

h=0.1 h=0.5 h=0.95

0 2 4 6 8 10 120

0.5

1

1.5

min

(svd

(R))

gamma

Figure ������ min�svd�R�� versus ��

−5 0 5 10 15 20 25 30−15

−10

−5

0Nyquist Plot of Hopt, h=0.1

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−2

−1.5

−1

−0.5

0Nyquist Plot of Hopt − Zoomed

Figure ������ Nyquist plot of Hopt�j � for h � ����

Page 196: Feedback Control Systems- HOzbay

��� H� �Ozbay

Example ��� Here we consider a �nite dimensional plant model

Po�s� ���� s�

�� � s��� � s�

with the weights W��s� and W �s� being the same as in the previous

example� In this case� the plant is stable and W��s� is �rst order� so

n � n� � � � �� which implies that Lopt�s� � �� The matrix R is �� �

and �opt is the largest root of

R � � � F ��� � � �������

where � is equal to �� of the previous example and

F �s� � ���� ��s���� s�

�� � s��a � bs � cs��

a� b� c being the same as above� The root of equation ������� in � ����� �� � ��� is �opt �� ��� � After internal pole zero cancelations within

the optimal controller� we get

Copt�s� �� �� �s � ���s � ��

�s � ����s � �����

������ Suboptimal H� Controllers

In this section� we characterize the set of all suboptimal H� controllers

stabilizing the feedback system �C�Po� and achieving +� � � �� for a

given suboptimal performance level � � �opt� All suboptimal controllers

have the same structure as the optimal controller�

Csub�s� �W��s�

� dsub

� � Hsub�s�

Hsub�s� ��Dsub�s�

dsub� �

Dsub�s� ��Lsub�s� � �M��s�Mn�s�No�s�B�s�

B�s�Md�s�E�s�

dsub �� Dsub��� �

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Introduction to Feedback Control Theory ��

In this case Lsub�s� is in the form

Lsub�s� � L��s�� � Q�s��bL��s�

� � bL�s�Q�s��������

where Q�s� is an arbitrary proper stable transfer function with

kQk� � � and L��s� � nL��s��dL��s�� bL�s� � nL���s��dL��s�� with

deg�nL�� � deg�dL�� � n� By using ������� in Dsub�s� we get

Dsub�s� � D��s�� bD�s�bL�s�Q�s�

� � bL�s�Q�s�

D��s� ��L��s� � �M��s�Mn�s�No�s�B�s�

B�s�Md�s�E�s�bD�s� ��L��s�� ��L���s�B�s�Md�s�E�s�

In particular� for Q�s� � � we have Dsub�s� � D��s��

Note that� in this case there are ��n � �� unknown coe�cients in

nL� and dL�� The interpolation conditions are similar to the optimal

case� D��s� and bD�s� should have no poles at the closed right half

plane zeros of Md�s� and E�s�� These interpolation conditions give

��n� � �� � �n equations� We need two more equations to determine

nL� and dL�� Assume that nW��s� has a zero -z that is distinct from

��i for all i � �� � � � � n� Then� we can set dL��-z� � � and nL��-z� � ��

The last two equations violate the symmetry of interpolation conditions�

so nL��s� � dL���s� and hence bD�s� � �� Let us de�ne

dL��s� � � s � � � sn� v ������

nL��s� � � s � � � sn� w �������

-zn�� � � -z � � � -zn�

where v � v� � � � vn�T and w � w� � � � wn�T are the unknown coe��

cients of dL��s� and nL��s�� respectively� The set of ��n� �� equations

corresponding to the above mentioned interpolation conditions can be

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��� H� �Ozbay

g=0.57g=0.6 g=0.7 g=5

g=0.57g=0.6 g=0.7 g=5

−5 0 5 10 15 20 25 30−15

−10

−5

0Nyquist Plot of Hsub, h=0.1

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−2

−1.5

−1

−0.5

0Nyquist Plot of Hsub − Zoomed

Figure ���� � Nyquist plot of Hsub�j � for h � ����

written as������n���n��

����� �

�����Vn�� F Vn��

F Vn��Jn�� Vn��Jn��

����n��� -zn��

-zn�� ����n���

������w

v

�� �������

For � � �opt equations ������� yield unique vectors w and v� Then� by

������ and ������� we de�ne nL��s� and dL��s�� and thus parameterize

all suboptimal H� controllers via Lsub�s�� ��������

Example �� For the H� optimal control problem studied in Ex�

ample ���� we have determined Copt�s� for h � ���� Now we �nd

suboptimal controllers for the same plant with a performance level

� � ���� � �opt� In this case� we construct the set of equations �������

and solve for v and w� these vectors along with an arbitrarily selected

proper stable Q�s�� with kQk� � �� give Lsub�s�� which de�nes Csub�s��

For Q � �� i�e� Lsub � L�� the Nyquist plots of Hsub�s� are shown in

Figure ���� for � � ���� ���� ��� ��

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Introduction to Feedback Control Theory ���

���� Exercise Problems

�� Consider the set of plants

P ���P �s� � Po�s� � #�s� �

#�s� is stable and j#�j �j �p

� � �where the nominal plant transfer function is Po�s� � �

s�� �

We want to design a controller in the form C�s� � Ks such that

�i� the nominal feedback system is stable with closed�loop sys�

tem poles r� and r� satisfying Re�ri� � �� for i � �� �� and

�ii� the feedback system is robustly stable�

Show that no controller exists that can satisfy both �i� and �ii��

�� A second�order plant model is in the form

P �s� ���� �o

s� � �� os � o�

Let o � �� � ��� and � � ���� � �� � with nominal values o � ��

and � � ����� i�e� Po�s� �� ��s���s���� � Find an appropriate W �s�

so that

jP �j �� Po�j �j � jW �j �j � �

De�ne W��s� � W �s�Po�s��� and W��s� � ���s

�����s � By using

the Youla parameterization �nd a controller achieving robust per�

formance� jW��j �S�j �j� jW��j �T �j �j � � � �

� Let

Po�s� � e��sP��s� � P��s� ��s� ��

�s � ���s� � �s � ��

and W��s� � ������s����� Design a robustly stabilizing controller

achieving asymptotic tracking of step reference inputs� Write this

controller in the Smith predictor form

C�s� �C��s�

� � P��s�C��s���� e��s�

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��� H� �Ozbay

and determine C��s� in this expression�

�� Consider the H� optimal control problem data

Po�s� ��z � s�

�z � s��s � ��� W��s� �

��� � s�

�� � ��s�W �s� � ���

and plot �opt versus z � �� Compute H� optimal controllers for

z � ���� �� ��� What are the corresponding closed�loop system

poles�

�� Consider Example ���� of Section ����� � Find the H� optimal

controllers for h � ��� sec and h � ���� sec� Are these controllers

stable�

�� �i� Find L��s� and bL�s� of the suboptimal controllers determined

in Example ��� of Section ������ for h � ��� sec with

� � ���� ���� ��� ��

�ii� Select Q�s� � � and obtain the Nyquist plot of Hsub�s� for

h � ��� sec and � � ����� ���� ��� Plot *� � versus

corresponding to these suboptimal controllers�

�iii� Repeat �ii� for Q�s� � e�s�s����s��� �

Page 201: Feedback Control Systems- HOzbay

Chapter ��

Basic State Space

Methods

The purpose of this chapter is to introduce basic linear controller design

methods involving state space representation of a SISO plant� The

coverage here is intended to be an overview at the introductory level�

Interested readers are referred to new comprehensive texts on linear

system theory such as �� ��� ���� ��� ����

���� State Space Representations

Consider the standard feedback system shown in Figure ����� where the

plant is given in terms of a state space realization

Plant �

��x�t� � Ax�t� � Bu�t�

yo�t� � Cx�t� � Du�t�������

Here A� B� C and D are appropriate size constant matrices with real

entries and x�t� � IRn is the state vector associated with the plant�

���

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��� H� �Ozbay

+

++

+ +-

y(t)C(s) P(s)

u(t)

v(t)

r(t)

y(t) n(t)

o

Figure ����� Standard feedback system�

We make the following assumptions� �i� D � �� which means that

the plant transfer function P �s� � C�sI�A���B�D� is strictly proper�

and �ii� u�t� and yo�t� are scalars� i�e�� the plant is SISO� The multiple

output case is considered in the next section only for the speci�c output

yo�t� � x�t��

The realization ������ is said to be controllable if the n� n control�

lability matrix U is invertible�

U ��

�B

��� AB��� � � � ��� An��B

�������

�U is obtained by stacking n vectors AkB� k � �� � � � � n � �� side by

side�� Controllability of ������ depends on A and B only� so� when U

is invertible� we say that the pair �A�B� is controllable� The realization

������ is said to be observable if the pair �AT� CT� is controllable �the

superscript T denotes the transpose of a matrix��

Unless otherwise stated� the realization ������ will be assumed to

be controllable and observable� i�e�� this is a minimal realization of the

plant �which means that n� the dimension of the state vector� is the

smallest among all possible state space realizations of P �s��� In this

case� the poles of the plant are the roots of the polynomial det�sI �A��

From any minimal realization fA�B�C�Dg� another minimal realization

fAz� Bz� Cz � Dg can be obtained by de�ning z�t� � Zx�t� as the new

state vector� where Z is an arbitrary n�n invertible matrix� Note that

Az � ZAZ��� Bz � ZB and Cz � CZ��� Since these two realizations

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Introduction to Feedback Control Theory ��

represent the same plant� they are said to be equivalent� In Section ����

we saw the controllable canonical state space realization

Ac ��

���n����� I�n�����n����an � � � �a�

�Bc ��

���n�����

����� �

Cc �� bn � � � b� � D �� d� ������

which was derived from the transfer function

P �s� �NP �s�

DP �s��

b�sn�� � � � � � bn

sn � a�sn�� � � � � � an� d�

So� for any given minimal realization fA�B�C�Dg there exists an in�

vertible matrix Zc such that Ac � ZcAZ��c � Bc � ZcB� Cc � CZ��c �

Exercise� Verify that Zc � UcU�� where Uc is the controllability

matrix� ������� of the pair �Ac� Bc��

Transfer function of the plant� P �s�� is obtained from the following

identities� the denominator polynomial is

DP �s� � det�sI �A� � det�sI �Ac� � sn � a�sn�� � � � �� an�

and when d � �� the numerator polynomial is

NP �s� � Cadj�sI �A�B � b�sn�� � � � � � bn�

where adj�sI �A� � �sI �A���det�sI �A�� which is an n� n matrix

whose entries are polynomials of degree less than or equal to �n� ���

���� State Feedback

Now consider the special case yo�t� � x�t� �i�e�� the C matrix is identity

and all internal state variables are available to the controller� and as�

sume r�t� � n�t� �� In this section� we study constant controllers of

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��� H� �Ozbay

the form C�s� � K � kn� � � � � k��� The controller has n inputs� �x�t�

which is n � � vector� and it has one output� therefore K is a � � n

vector� The plant input is

u�t� � v�t� �Kx�t�

and hence� the feedback system is represented by

�x�t� � Ax�t� � B�v�t� �Kx�t�� � �A�BK�x�t� � Bv�t��

We de�ne AK �� A�BK as the !A�matrix" of the closed�loop system�

Since �A�B� is controllable� it can be shown that the state space system

�AK � B� is controllable too� So� the feedback system poles are the roots

of the closed�loop characteristic polynomial �c�s� � det�sI��A�BK���

In the next section� we consider the problem of �nding K from a given

desired characteristic polynomial �c�s��

������ Pole Placement

For the special case A � Ac and B � Bc� it turns out that �Ac �BcK�

has the same canonical structure as Ac� and hence�

�c�s� � det�sI � �Ac �BcK�� � sn � �a� � k��sn�� � � � �� �an � kn��

Let the desired characteristic polynomial be

�c�s� � sn � ��sn�� � � � �� �n �

Then the controller gains should be set to ki � ��i � ai�� i � �� � � � � n�

Example ���� Let the poles of the plant be ��� �� � j�� then

DP �s� � s� � s� � s� � �s� Suppose we want to place the closed�loop

system poles to ��� ��� �� j�� i�e�� desired characteristic polynomial

is �c�s� � s� � s� � ��s� � � s � ��� If the system is in controllable

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Introduction to Feedback Control Theory ���

canonical form and we have access to all the states� then the controller

C�s� � k�� k�� k�� k�� solves this pole placement problem with k� � ���

k� � ��� k� � �� and k� � ���

If the system �A�B� is not in the controllable canonical form� then

we apply the following procedure�

Step �� Given A and B� construct the controllability matrix U via

������� Check that it is invertible� otherwise closed�loop system

poles cannot be placed arbitrarily�

Step �� Given A� set DP �s� � det�sI � A� and calculate the coe��

cients a�� � � � � an of this polynomial� Then� de�ne the controllable

canonical equivalent of �A�B��

Ac �

���n����� I�n�����n����an � � � �a�

�Bc �

���n�����

��

Step �� Given desired closed�loop poles� r�� � � � � rn� set

�c�s� � �s� r�� � � � �s� rn� � sn � ��sn�� � � � �� �n�

Step �� Let eki � �i�ai for i � �� � � � � n and eK � ekn� � � � � ek��� Then�

the following controller solves the pole placement problem�

C�s� � K � eKUcU��

where Uc is the controllability matrix associated with �Ac� Bc��

Step �� Verify that the roots of �c�s� � det�sI � �A � BK�� are

precisely r�� � � � � rn�

Example ���� Consider a system whose A and B matrices are

A �

��� � � �

� �� �

� � �

��� B �

��� �

��� �

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��� H� �Ozbay

First we compute U � B��� AB

��� A�B� and check that it is invertible�

U �

��� � � �

� �� �

� � ��

��� � U�� �

��� � � �

� �� ��

��� ���� ����

��� �Next� we �nd DP �s� � det�sI � A� � s� � s� � �s � �� The roots

of DP �s� are ��� � �� � and ���� so the plant is unstable� From the

coe�cients of DP �s�� we determine Ac and compute Uc�

Ac �

��� � � �

� � �

�� � ��

��� � Bc �

��� �

��� � Uc �

��� � � �

� � ��

� �� �

��� �

Suppose we want the closed�loop system poles to be at ��� �� j��

then �c�s� � s� � �s� � �s� �� Comparing the coe�cients of �c�s� and

DP �s� we �nd eK � � �� �� Finally� we compute

K � eKUcU�� � �� � ��� � ���� �

Verify that the roots of det�sI � �A�BK�� are indeed ��� �� j��

When the system is controllable� we can choose the roots of �c�s�

arbitrarily� Usually� we want them to be far from the Im�axis in the left

half plane� On the other hand� if the magnitudes of these roots are very

large� then the entries of K may be very large� which means that the

control input u�t� � �Kx�t� �assume v�t� � � for the sake of argument�

may have a large magnitude� To avoid this undesirable situation� we

can use linear quadratic regulators�

������ Linear Quadratic Regulators

In the above setup� assume v�t� � � and x��� � xo � �� By a state feed�

back u�t� � �Kx�t� we want to regulate a �ctitious output ey�t� � eQx�t�

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Introduction to Feedback Control Theory ��

to zero �the ��n vector eQ assigns di�erent weights to each component

of x�t� and it is assumed that the pair �AT� eQT� is controllable�� but

we do not want to use too much input energy� The trade�o� can be

characterized by the following cost function to be minimized�

JLQR�K� ��

Z �

�jey�t�j� � jer u�t�j�� dt � keyk�� � ker uk�� ������

where er � � determines the relative weight of the control input� whener is large the cost of input energy is high� The optimal K minimizing

the cost function JLQR is

K� �� er��BTX ������

where X is an n� n symmetric matrix satisfying the matrix equation

ATX � XA� er��XBBTX � eQT eQ � � �

which can be solved by using the lqr command of Matlab� The sub�

script � is used in the optimal K because JLQR is de�ned from the

L�� ��� norms of ey and eru� �������

In this case� the plant is P �s� � �sI � A���B and the controller is

C�s� � K� so the open�loop transfer function is G�s� � C�s�P �s� �

K�sI � A���B� For K � K�� it has been shown that �see e�g�� ����

G�j � satis�es j� � G�j �j � �� for all � which means that VM � ��

PM � ���� GMupp � � and GMlow � �� �

Example ���� Consider A and B given in Example ���� and let eQ �

� � � ��� er � �� Then� by using the lqr command of Matlab

we �nd K � �� � ���� � ������ For er � �� and er � ���� the

results are K � ��� � �� � ���� and K � ���� � ��� � ������

respectively� Figure ���� shows the closed�loop system poles� i�e� the

roots of �c�s� � det�sI � �A�BK���� as er�� increases from � to ���

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��� H� �Ozbay

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Real Axis

Imag

Axi

s

Figure ����� Closed�loop poles for an optimal LQR controller�

The �gure coincides with the root locus plot of an open�loop system

whose poles are ��� ��������� and zeros are ���� j���� Recall

that the roots of det�sI �A� are ��� � ��� and �� � Also� check that

the roots of eQadj�sI�A�B are ��� j���� To generalize this observation�

let -p�� � � � � -pn be the poles and -z�� � � � � -zm be the zeros of eQ�sI�A���B�

Now de�ne pi � -pi if Re� -pi� � � and pi � �-pi if Re� -pi� � �� for

i � �� � � � � n� De�ne z�� � � � � zm similarly� Then� the plot of the roots of

�c�s�� as er�� increases from � to ��� is precisely the root locus of a

system whose open�loop poles and zeros are p�� � � � � pn� and z�� � � � � zm�

respectively�

Exercise� For the above example� �nd G�s� � K�sI � A���B� draw

the Nyquist plot� verify the inequality j� �G�j �j � � and compute the

stability margins VM� PM� GMupp� GMlow for er � ���� ���� ���

Another interpretation of the LQR problem is the following� De�ne

+��s� �� eQ�sI � �A�BK����� +��s� �� erK�sI � �A�BK���� and

eJ��K� ��

Z �

���k+��j �k� � k+��j �k��d �����

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Introduction to Feedback Control Theory ���

where we have used the notation kV k ��pjv�j� � � � � � jvnj� for any

��n complex vector V � v�� � � � � vn�� Then� the problem of minimizing

JLQR�K� is equivalent to minimizing eJ��K� over all state feedback gains

K resulting in a stable �c�s� � det�sI � �A�BK���

A slightly modi�ed optimal state feedback problem is to �nd K such

that the roots of �c�s� are in the open left half plane and

eH� ��

Z ��

���jW��j �S�j �j� � jW��j �T �j �j��d ������

is minimized for

jW��j �j � j eQ�j I �A���Bj and jW��j �j � er ������

where S�s� � ���G�s����� T �s� � ��S�s� and G�s� � K�sI�A���B�

It turns out that K� is also the optimal K minimizing eH�� see ���

pp� ��� ����

���� State Observers

In the previous section� we assumed that the state variables are available

for feedback� We now return to our original SISO setting� where yo�t� �

Cx�t� is the single output of the plant� The basic idea in state space

based control is to generate bx�t�� which is an estimate of x�t� obtained

from y�t�� and then use bx�t� in state feedback as if it were the actual

x�t��

A state observer is used in generating the state estimate bx�t��

�bx�t� � Abx�t� � Bu�t� � L�y�t�� Cbx�t�� �������

where L is the n � � observer gain vector to be designed� The ob�

server ������� mimics the plant ������ with an additional correction

term L�y�t�� by�t��� where by�t� � Cbx�t��

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��� H� �Ozbay

The estimation error e�t� �� x�t� � bx�t� satis�es

�e�t� � �x�t�� �bx�t� � �A� LC�e�t��

Therefore� e�t� is the inverse Laplace transform of �sI��A�LC����e����

where e��� is the initial value of the estimation error� We select L in

such a way that the roots of �o�s� � det�sI� �A�LC�� are in the open

left half plane� Then� e�t� � � as t � �� In fact� the rate of decay of

e�t� is related to the location of the roots of �o�s�� Usually� these roots

are chosen to have large negative real parts so that bx�t� converges to

x�t� very fast� Note that

�o�s� � det�sI � �A� LC�� � det�sI � �AT � CTLT���

Therefore� once the desired roots of �o�s� are given� we compute L as

L � KT� where K is the result of the pole placement procedure with

the data �c � �o� A � AT� B � CT� For arbitrary placement of the

roots of �o�s�� the pair �AT� CT� must be controllable�

Exercise� Let C � � � � �� and consider the A matrix of Ex�

ample ����� Find the appropriate observer gain L� such that the roots

of �o�s� are ��� �� j��

���� Feedback Controllers

������ Observer Plus State Feedback

Now we use state feedback in the form u�t� � v�t��Kbx�t�� where bx�t� is

generated by the observer �������� The key assumption used in �������

is that both y�t� and u�t� are known exactly� i�e�� there are no disturb�

ances or measurement noises� But v�t� is a disturbance that enters

the observer formula via u�t�� Also� y�t� � yo�t� � n�t�� where yo�t�

is the actual plant output and n�t� is measurement noise� We should

Page 211: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

not ignore the reference input r�t� either� Accordingly� we modify the

observer equations and de�ne the controller as

Controller �

��bx�t� � bAbx�t�� L�r�t� � y�t��

u�t� � �Kbx�t� � v�t�

where bA �� �A�BK�LC�� The input to the controller is �r�t��y�t��

and the controller output is �Kbx�t�� Transfer function of the feedback

controller is

C�s� �NC�s�

DC�s�� K�sI � bA���L� �������

In this case� the state estimation error e�t� � x�t� � bx�t� satis�es

�e�t� � �A� LC�e�t� � Bv�t� � L�r�t� � n�t��� �������

Moreover� the state x�t� can be determined from e�t��

�x�t� � �A�BK�x�t� � BKe�t� � Bv�t�� ����� �

Equations ������� and ����� � determine the feedback system behavior�

closed�loop system poles are the roots of �c�s� � det�sI � �A � BK��

and �o�s� � det�sI��A�LC��� If the state space realization fA�B�Cgof the plant is minimal� then by proper choices of K and L closed�loop

system poles can be placed arbitrarily�

In general� the controller ������� has n poles� these are the roots of

DC�s� � det�sI � bA�� The zeros of the controller are the roots of the

numerator polynomial NC�s� � K adj�sI � bA� L� where adj��� denotes

the adjoint matrix which appears in the computation of the inverse�

Therefore� the controller is of the same order as the plant P �s�� unless

K and L are chosen in a special way that leads to pole zero cancelations

within C�s� � NC�s��DC�s��

Page 212: Feedback Control Systems- HOzbay

��� H� �Ozbay

Exercise� By using equations ������� and ����� � show that the closed�

loop transfer functions are given by

S�s� � ���KM��s�B� �� � KM��s�B�

T �s� � CM��s�B KM��s�L

C�s�S�s� � ���KM��s�B� KM��s�L

P �s�S�s� � CM��s�B �� � KM��s�B�

where M��s� �� �sI � �A � BK����� M��s� �� �sI � �A � LC�����

S�s� � �� � P �s�C�s���� and T �s� � �� S�s��

������ H� Optimal Controller

We have seen that by appropriately selecting K and L a stabilizing

controller can be constructed� As far as closed�loop system stability

is concerned� designs of K and L are decoupled� Let us assume that

K is determined from an LQR problem as the optimal gain K � K�

associated with the problem data eQ and er� Now consider the state

estimation error e�t� given by �������� Suppose r�t� � �� v�t� � � and

n�t� � �eqw�t� � � for some constant eq � �� Then� the transfer function

from v to e is *��s� � �sI � �A � LC����B and the transfer function

from w to e is *��s� � eq�sI � �A� LC����L� De�ne

bJ��L� ��

Z �

���k*��j �Tk� � k*��j �Tk��d � �������

Comparing ������� with ������ we see that the problem of minimizingbJ��L� over all L resulting in a stable �o�s� � det�sI � �A � LC�� is

an LQR problem with the modi�ed data A � AT� B � CT� eQ � BT�er � eq� K � LT� Hence� the optimal solution is L � L��

L� � eq��Y CT �������

Page 213: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

the n� n symmetric matrix Y satis�es

AY � Y AT � eq��Y CTCY � BBT � ��

We can use the lqr command of Matlab �with the data as speci�ed

above� to �nd Y and corresponding L��

The controller C�opt�s� � K��sI � �A � BK� � L�C����L� is an

H� optimal controller� see ��� Chapter ���� As eq � �� the optimal H�

controller C�opt�s� approaches the solution of the following problem�

�nd a stabilizing controller C�s� for P �s� � C�sI � A���B� such thateH�� ������� is minimized� where S�s� � �� �G�s����� T �s� � �� S�s��

G�s� � C�s�P �s� and the weights are de�ned by ������� Recall that in

������ we try to minimize the peak value of

+� � �� jW��j �S�j �j� � jW��j �T �j �j� �������

whereas here we minimize the integral of +� ��

Example ���� Consider the plant

P �s� �s� �

s� � s� � �s � �

with controllable canonical realization

A �

��� � � �

� � �

�� � ��

��� B �

��� �

��� C � �� � �� �

Let eQ � � � � ��� er � � and eq � ���� Using the lqr command

of Matlab we compute the optimal H� controller as outlined above�

Verify that for this example the controller is

C�opt�s� ����� �s � � �s � ���

�s � �� �s� � �� � s � ������

Page 214: Feedback Control Systems- HOzbay

��� H� �Ozbay

and the closed�loop system poles are f���� j��� � ��� �� j�� ����g�

Exercise� Find the poles and zeros of P �s� and eQ�sI �A���B� Draw

the root locus �closed�loop system pole locations� in terms of er�� andeq��� Obtain the Nyquist plot of G�j � � C�opt�j �P �j � for the above

example� Compute the vector margin and compare it with the vector

margin of the optimal LQR system�

������ Parameterization of all Stabilizing

Controllers

Given a strictly proper SISO plant P �s� � C�sI�A���B� the set of all

controllers stabilizing the feedback system� denoted by C� is parameter�

ized as follows� First� �nd two vectors K and L such that the roots of

�c�s� � det�sI � �A�BK�� and �o�s� � det�sI � �A�LC�� are in the

open left half plane �the pole placement procedure can be used here��

For any proper stable Q�s� introduce the notation

CQ�s� �� KM�s�L ����KM�s�B���� CM�s�L� Q�s�

�� CM�s�B Q�s�������

�KM��s�L � ��� CM��s�L� Q�s�

� � KM��s�B � CM��s�B Q�s��������

where M��s� �� �sI � �A � LC����� M�s� �� �sI � bA��� and bA ��

A�BK � LC� Then� the set C is given by

C � f CQ�s� � Q is proper and stableg �������

where CQ�s� is de�ned in ������ or �������� For Q�s� � �� we obtain

C�s� � C��s� � K�sI � bA���L� which coincides with the controller

expression �������� The parameterization ������� is valid for both stable

and unstable plants� In the special case where the plant is stable� we can

choose K � LT � ���n� which leads to bA � A and CM�s�B�s� � P �s��

and hence� ������� becomes the same as �������� A block diagram

of the feedback system with controller in the form ������ is shown in

Page 215: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

C

A

B

-L

-K

B

A

C

Q(s)

y

y

v

u++

+

++

++

++

++

+

-

o

n

r

Plant

1/s

1/s

C (s)Q

P(s)

Controller

+

+

x

x

Figure ��� � Feedback system with a stabilizing controller�

Figure ��� � The parameterization ������� is a slightly modi�ed form of

the Youla parameterization� ���� An extension of this parameterization

for nonlinear systems is given in ���

���� Exercise Problems

�� Consider the pair

A �

��� � � �

� �� �

� � �

��� B �

��� �

��� �

What are the roots of DP �s� �� det�sI � A� � � � Show that

�A�B� is not controllable� However� there exists a �� vector K

such that the roots of �c�s� � det�sI��A�BK�� are fr�� r�� � gwhere r� and r� can be assigned arbitrarily� Let r��� � �� j� and

�nd an appropriate gain K� In general� the number of assignable

poles is equal to the rank of the controllability matrix� Verify

that the controllability matrix has rank two in this example� If

the non�assignable poles are already in the left half plane� then the

pair �A�B� is said to be stabilizable� In this example� the pole at

Page 216: Feedback Control Systems- HOzbay

��� H� �Ozbay

� is the only non�assignable pole� so the system is stabilizable�

�� Verify that the transfer function of the controller shown in Fig�

ure ��� is equal to CQ�s� given by ������ and ��������

� Show that the closed�loop transfer functions corresponding to the

feedback system of Figure ��� are�

S�s� � ���KM��s�B� �� � KM��s�B � CM��s�B Q�s��

T �s� � CM��s�B �KM��s�L � ��� CM��s�L� Q�s��

C�s�S�s� � ���KM��s�B� �KM��s�L � ��� CM��s�L� Q�s��

P �s�S�s� � CM��s�B �� � KM��s�B � CM��s�B Q�s��

where M��s� � �sI��A�BK���� and M��s� � �sI��A�LC�����

�� Given

P �s� ��s� z�

�s � z��s� ��z � � and z � �

obtain the controllable canonical realization of P �s��

Let eQ � ��z ��� er � � and eq � ����

�i� Compute theH� optimal controllers for z � ���� ���� ���� ���

Determine the poles and zeros of these controllers and obtain

the Nyquist plots� What happens to the vector margin as

z � ��

�ii� Plot the corresponding +� � for each of these controllers�

What happens to the peak value of +� � as z � ��

�iii� Fix z � � and let K � K�� L � L�� Determine the closed�

loop transfer functions S� T � CS and PS whose general

structures are given in Problem � Note that

P �s� �P �s�S�s�

S�s��

T �s�

C�s�S�s��

CM��s�B

��KM��s�B�

Verify this identity for the H� control problem considered

here�

Page 217: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

�� Di�culty of the controller design is manifested in parts �i� and

�ii� of the above exercise problem� We see that if the plant has a

right half plane pole and a zero close by� then the vector margin

is very small� Recall that

VM�� � kSk�

and S�s� � ���KM��s�B� �� � KM��s�B � CM��s�B Q�s���

�i� Check that� for the above example� when z � ��� we have

S�s� � �s����s���S��s� where

S��s� ��s � ���s � ����

s� � ��� s � �

�s � ������s� ������ �s� ����Q�s�

s� � ����� s � ����

Clearly kSk� � kS�k�� Thus�

VM�� � supRe�s���

jS��s�j � jS������j �� �� �

which means that VM � ������ for all proper stable Q�s��

i�e�� the vector margin will be less than ������ no matter

what the stabilizing controller is� This is a limitation due to

plant structure�

�ii� Find a proper stable Q�s� which results in VM � �����

Hint� write S��s� in the form

S��s� � S��s�� S��s�Q�s�

where S��s� and S��s� are proper and stable and S��s� has

only one zero in the right half plane at z � ���� Then set

Q�s� � Q��s� ���

�� � �s��S��s�� S������

S��s�

where � is relative degree of S��s� and � � �� Check that

Q�s� de�ned this way is proper and stable� and it leads to

kS�k� � �� for su�ciently small �� Finally� the controller is

obtained by just plugging in Q�s� in ������ or ��������

Page 218: Feedback Control Systems- HOzbay
Page 219: Feedback Control Systems- HOzbay

Bibliography

�� Ak.cay� H�� G� Gu� and P� P� Khargonekar� !Class of algorithms for

identi�cation in H�� continuous�time case�" IEEE Transactions

on Automatic Control� vol� � ���� �� pp� ��������

�� Anantharam� V�� and C� A� Desoer� !On the stabilization of nonlin�

ear systems�" IEEE Trans� on Automatic Control� vol� �� �������

pp� �������

� Antsaklis� P� J�� and A� N� Michel� Linear Systems� McGraw�Hill�

New York� ����

�� Balas� G� J�� J� C� Doyle� K� Glover� A� Packard� and R� Smith�

��Analysis and Synthesis Toolbox User�s Guide� MathWorks Inc��

Natick MA� �����

�� Barmish� B� R�� New Tools for Robustness of Linear Systems�

Macmillan� New York� �����

�� Bay� J� S�� Fundamentals of Linear State Space Systems� McGraw�

Hill� New York� �����

� B�elanger� P� R�� Control Engineering� Saunders College Publishing�

Orlando� �����

�� Bellman� R� E�� and K� L� Cooke� Dierential Dierence Equa�

tions� Academic Press� New York� ��� �

���

Page 220: Feedback Control Systems- HOzbay

��� H� �Ozbay

�� Bhattacharyya� S� P�� H� Chapellat� and L� H� Keel� Robust Control�

The Parametric Approach� Prentice�Hall� Upper Saddle River NJ�

�����

��� Bode� H� W�� Network Analysis and Feedback Ampli�er Design�

D� Van Nostrand Co�� Inc�� Princeton NJ� �����

��� Chen� C� T�� Linear System Theory and Design� rd ed�� Oxford

University Press� New York� �����

��� Curtain� R� F�� !A synthesis of time and frequency domain methods

for the control of in�nite�dimensional systems� a system theoretic

approach�" Control and Estimation in Distributed Parameter Sys�

tems� H� T� Banks ed�� SIAM� Philadelphia� ����� pp� �������

� � Curtain� R� F�� and H� J� Zwart� An Introduction to In�nite�

Dimensional Linear System Theory� Springer�Verlag� New York�

�����

��� Davison� E� J�� and A� Goldenberg� !Robust control of a general

servomechanism problem� the servo compensator�" Automatica�

vol� �� ������ pp� �������

��� Desoer� C� A�� and M� Vidyasagar Feedback Systems Input�Output

Properties� Academic Press� New York� ����

��� Devilbiss� S� L�� and S� Yurkovich� !Exploiting ellipsoidal para�

meter set estimates in H� robust control design�" Int� J� Control�

vol� �� ������� pp� �������

�� Dorf� R� C�� and R� H� Bishop� Modern Control Systems� �th ed��

Addison Wesley Longman� Menlo Park� �����

��� Doyle� J� C�� B� A� Francis� and A� Tannenbaum� Feedback Control

Theory� Macmillan� New York� �����

��� Etter� D� M�� Engineering Problem Solving with MATLAB� �nd

ed�� Prentice�Hall� Upper Saddle River NJ� ����

Page 221: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ���

��� Foias� C�� H� �Ozbay� and A� Tannenbaum� Robust Control of In�

�nite Dimensional Systems� LNCIS ���� Springer�Verlag� London�

�����

��� Francis� B�� O� A� Sebakhy and W� M� Wonham� !Synthesis of mul�

tivariable regulators� the internal model principle�" Applied Math�

ematics � Optimization� vol� � ������ pp� ������

��� Franklin� G� F�� J� D� Powell� and A� Emami�Naeini� Feedback

Control of Dynamic Systems� rd ed�� Addison�Wesley� Reading

MA� �����

� � Hanselman� D� � and B� Little�eld� MasteringMatlab �� Prentice�

Hall� Upper Saddle River NJ� �����

��� Hara� S�� Y� Yamamoto� T� Omata� and M� Nakano� !Repetitive

control system� a new type servo system for periodic exogenous

signals�" IEEE Transactions on Automatic Control� vol� ������

pp� �������

��� Helmicki� A� J�� C� A� Jacobson and C� N� Nett� !Worst�

case�deterministic identi�cation in H�� The continuous�time

case�" IEEE Transactions on Automatic Control� vol� �������

pp� ��������

��� Hemami� H�� and B� Wyman� !Modeling and control of con�

strained dynamic systems with application to biped locomotion

in the frontal plane�" IEEE Trans� on Automatic Control� vol� ��

������ pp� ����� ��

�� Isidori� A�� Nonlinear Control Systems An Introduction� �nd ed��

Springer�Verlag� Berlin� �����

��� Kamen� E� W�� and B� S� Heck� Fundamentals of Signals and Sys�

tems� Prentice�Hall� Upper Saddle River NJ� ����

Page 222: Feedback Control Systems- HOzbay

��� H� �Ozbay

��� Kataria� A�� H� �Ozbay� and H� Hemami� !Point to point motion of

skeletal systems with multiple transmission delays�" Proc� of the

��� IEEE International Conference on Robotics and Automation�

Detroit MI� May �����

�� Khajepour� A�� M� F� Golnaraghi� and K� A� Morris� !Application

of center manifold theory to regulation of a �exible beam�" Journal

of Vibration and Acoustics� Trans� of the ASME� vol� ��� ������

pp� ��������

�� Khalil� H�� Nonlinear Systems� �nd ed�� Prentice�Hall� Upper

Saddle River NJ� �����

�� Kosut� R� L�� M� K� Lau� and S� P� Boyd� !Set�membership iden�

ti�cation of systems with parametric and nonparametric uncer�

tainty�" IEEE Transactions on Automatic Control� vol� �������

pp� ��������

� Lam� J�� !Convergence of a class of Pad�e approximations for delay

systems�" Int� J� Control� vol� �� ������� pp� ���������

�� Lenz� K� and H� �Ozbay� !Analysis and robust control techniques

for an ideal �exible beam�" in Multidisciplinary Engineering Sys�

tems Design and Optimization Techniques and their Applications�

C� T� Leondes ed�� Academic Press Inc�� ��� � pp� �������

�� Lu� W�M� !A state�space approach to parameterization of stabiliz�

ing controllers for nonlinear systems�" IEEE Trans� on Automatic

Control� vol� �� ������� pp� ���������

�� M�akil�a� P� M�� J� R� Partington� and T� K� Gustafsson� !Worst�

case control�relevant identi�cation�" Automatica� vol� � �������

pp� ��������

� Ogata� K�� Modern Control Engineering� rd ed�� Prentice�Hall�

Upper Saddle River NJ� ����

Page 223: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

�� Oppenheim� A� V� and A� S� Willsky� with S� H� Nawab� Signals

and Systems� �nd ed�� Prentice�Hall� Upper Saddle River NJ� ����

�� �Ozbay� H�� !Active control of a thin airfoil� �utter suppression and

gust alleviation�" Preprints of �th IFAC World Congress� Sydney

Australia� July ��� � vol� �� pp� ��������

��� �Ozbay� H�� and G� R� Bachmann� !H��H� Controller design for

a �D thin airfoil �utter suppression�" AIAA Journal of Guidance

Control � Dynamics� vol� � ������� pp� ������

��� �Ozbay� H�� S� Kalyanaraman� A� �Iftar� !On rate�based congestion

control in high�speed networks� Design of an H� based �ow con�

troller for single bottleneck�" Proceedings of the American Control

Conference� Philadelphia PA� June ����� pp� � ��� ���

��� �Ozbay� H�� and J� Turi� !On input�output stabilization of singular

integro�di�erential systems�" Applied Mathematics and Optimiza�

tion� vol� � ������� pp� ������

� � Peery� T� E� and H� �Ozbay� !H� optimal repetitive controller

design for stable plants�" Transactions of the ASME Journal of

Dynamic Systems� Measurement� and Control� vol� ��� ������

pp� �������

��� Pratap� R�� Getting Started with MATLAB �� Oxford University

Press� Oxford �����

��� Rohrs C� E�� J� L� Melsa� and D� G� Schultz� Linear Control Sys�

tems� McGraw�Hill� New York� ��� �

��� Russell� D� L�� !On mathematical models for the elastic beam with

frequency�proportional damping�" in Control and Estimation in

Distributed Parameter Systems� H� T� Banks ed�� SIAM� Phil�

adelphia� ����� pp� ��������

Page 224: Feedback Control Systems- HOzbay

��� H� �Ozbay

�� Rugh� W� J� Linear System Theory� �nd ed�� Prentice�Hall� Upper

Saddle River NJ� �����

��� Smith� O� J� M�� Feedback Control Systems� McGraw�Hill� New

York� �����

��� Stepan� G�� Retarded Dynamical Systems Stability and Character�

istic Functions� Longman Scienti�c / Technical� New York� �����

��� Toker� O� and H� �Ozbay� !H� optimal and suboptimal control�

lers for in�nite dimensional SISO plants�" IEEE Transactions on

Automatic Control� vol� �� ������� pp� ������

��� Ulus� C�� !Numerical computation of inner�outer factors for a class

of retarded delay systems�" Int� Journal of Systems Sci�� vol� ��

������ pp� �������

��� Van Loan� C� F�� Introduction to Scienti�c Computing A Matrix�

Vector Approach Using Matlab� Prentice�Hall� Upper Saddle

River NJ� ����

� � Youla� D� C�� H� A� Jabr� and J� J� Bongiorno Jr�� !Modern Wiener

Hopf design of optimal controllers� part II�" IEEE Transactions on

Automatic Control� vol� �� ������ pp� ��� ��

��� Zhou� K�� with J� C� Doyle� and K� Glover� Robust and Optimal

Control� Prentice�Hall� Upper Saddle River NJ� �����

Page 225: Feedback Control Systems- HOzbay

Index

ts �settling time�� �

A�� ��

H�� �

H� control� ���� ���

H� optimal controller� ��

I�� ��

L������ ��

L������ ��

L������ ��

�� plant theorem� ��

� edge theorem� ��

additive uncertainty bound� ��

airfoil� ��� ��

all�pass� ���� ��

bandwidth� ��

BIBO stability� ��

Bode plots� ��

Bode�s gain�phase formula� ���

Cauchy�s theorem� ��

characteristic equation� ��� ��

characteristic polynomial� ��� � �

���

communication networks� �

complementary sensitivity� �� �

��

controllability matrix� ���

controller parameterization� ���

���

convolution identity� ��

coprime polynomials� ��� ����

���

DC gain� ���

delay margin �DM�� ���

disturbance attenuation� ��

dominant poles� �

estimation error� ���

feedback control� �

feedback controller� ���

feedback linearization� �

�nal value theorem� ��

�exible beam� ��� ��� � ���

�ow control� � � �� �

�utter suppression� ��

gain margin �GM�� �

lower GMlow� �

relative GMrel� ��

���

Page 226: Feedback Control Systems- HOzbay

��� H� �Ozbay

upper GMupp� �

generalized Kharitanov�s theorem�

��

gust alleviation� ��

high�pass �lter� ���� ��

improper function� ��� � �

impulse� ��

interval plants� ��

inverted pendulum� ��

Kharitanov polynomials� ��

Kharitanov�s theorem� ��

lag controller� � �

lead controller� ���

lead�lag controller� �

linear quadratic regulator �LQR��

���� ���

linear system theory� ���

linearization� ��

loopshaping� � �

low�pass �lter� ���� ���

LTI system models� �

�nite dimensional� �

in�nite dimensional� ��

MIMO� �

minimum phase� ���� ���� ���

��� ��

mixed sensitivity minimization�

���� ��

multiplicative uncertainty bound�

��� ��

Newton�s Law� ��

Newton�s law� ��

noise reduction� ���

nominal performance� ��

nominal plant� ��

non�minimum phase� ��

Nyquist plot� ��� ���

Nyquist stability test� �

open�loop control� �

optimal H� controller� ���

Pad�e approximation� ���

PD controller� �

pendulum� ��

percent overshoot� �� ���

performance weight� ��

phase margin �PM or ��� � �

���

PI controller� � �

PID controller� ��� � �� �

PO �percent overshoot�� �

pole placement� ���

proper function� ��

quasi�polynomial� �� � ���

repetitive controller� ��

RLC circuit� ���

robust performance� �������� ���

��

Page 227: Feedback Control Systems- HOzbay

Introduction to Feedback Control Theory ��

robust stability� ��� ���� ��

robust stability with asymptotic

tracking� ��

robustness weight� ��

root locus� ��

root locus rules� ��

complementary root locus�

magnitude rule� �

phase rule� �� ��

Routh�Hurwitz test� �

sensitivity� �

sensitivity function� �� ��

servocompensator� ��

settling time� �� ���

signal norms� ��

SISO systems� �

small gain theorem� ���

Smith predictor� ��

spectral factorization� ���

stable polynomial� ��

state equations� �

state estimate� ���

state feedback� �� � ���

state observer� ���

state space realization� �� ��� ���

controllable� ���

controllable canonical� ���

��

equivalent� ��

minimal� ���

observable� ���

stabilizable� ���

state variables� �

steady state error� ��

step response� �

strictly proper function� ��� ���

���

suboptimal H� controller� ���

system identi�cation� ��

system norm� ��� �

system type� ��

Theodorsen�s function� ��� ��

time delay� ��� ��� � ���� ���

�� ��� ��

tracking� ���

tracking error� ��� ���� ���� ��

transfer function� ��

transient response� �

transition band� ���

uncertainty description

dynamic uncertainty� ��� ���

parametric uncertainty� ���

���

unit step� �� ��

unmodeled dynamics� ���

vector margin� ��

vector margin �VM or �� ��

well�posed feedback system� ��

Youla parameterization� ��� ���