Download - Feedback Control Systems- HOzbay
Introduction to
Feedback Control Theory
Hitay �Ozbay
Department of Electrical Engineering
Ohio State University
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�
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 ����
Dedication
To my wife� �Ozlem
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 � � � � � � � � � � � � � � ��
����� 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 � � � � � � � � � � � � � � � � � � � � � ��
��� 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 � � � � � � � � � � � � � � ���
���� 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 ���
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�
�
� 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
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
� 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�
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
� 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�
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 ���
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�
�
�� 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�
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� � ��
�� 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�
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
�� 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��
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 ����
�� 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
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
�� 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� ������
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�
�� 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
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�
�� 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 � ��
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 � � ��� � �����
�� 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 �� � �����
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�
�� 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�
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
�� 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�
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�
� 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
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��
� 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��� � ��
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
� 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 ��
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
�
� 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�
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 ��
� 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 � �
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�
�� 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�
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
�� 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�
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�
�
�� 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
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
�� 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��� � ��
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
�� H� �Ozbay
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
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
�� H� �Ozbay
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
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�
�� H� �Ozbay
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�
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����
�� H� �Ozbay
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 � ����
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�
�� H� �Ozbay
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�
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��
�� 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� � � � �
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 ���
�� 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
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� � ��
�� 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�
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�
�
�� 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� �����
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�
�� 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 �����
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�
�� 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 � ��
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
� 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 � ��
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
� 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�
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 �����
� 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
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�
� 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 � ��� � �
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��
� 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� ����
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 � ������
�� 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
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 � �� �� �
�� 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 � ���
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�
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
��
�� 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
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
�� 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�
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 � ���� �
�� H� �Ozbay
−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
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�
�� 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�
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 �
�
��� ��
�� 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
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�
�� 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
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�
�� 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�� ����� �
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 � � � � ����� ��
��� 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�
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�
���
��� 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�
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 �� ����
��� 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
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 ���
��� 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�
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
��� 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� � ����
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�
��� 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�
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��
��� 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��
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 � � � �����
��� 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�
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 � ���
��� 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 � �� � ����
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�
��� 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�
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�
��� 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 � �� �
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
���
��� 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�
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 ����
��� 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��
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 ��
��� 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 ��
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 �
��� 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
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
� � 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
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 ���
� � 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
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�
� � 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�
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
� � 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 ���
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 � �����
� � 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 � ����
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�
� �
��� 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 � ��
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�
��� 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
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�
��� 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
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
��� 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 �
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
��� 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�
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 �
��� 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�
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��
��� 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 � �����
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��
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�
���
��� 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
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
��� 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
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�
��� 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 ��
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 �
��� 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
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���
��� 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 � ������
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
��� 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
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
��� 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�
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�
�� 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
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
�� 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
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 ��� �
�� 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
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�
�� 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
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�� �������
�� 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� � �� �
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��
��� 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�
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�
��� 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
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��
��� 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�
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 � ����
��� 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��� �
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
��� 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 � � ���� ���� ��� ��
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�
��� 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��� �
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�
���
��� 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
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
��� 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
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 �
��� �
�
�
��� �
��� 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�
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 ���
��� 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 �����
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��
��� 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
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��
��� 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 �������
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 � ������
��� 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
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
��� 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�
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 ��������
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� ��� �
���
��� 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� ����
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� ����
��� 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� ����
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� ��������
��� 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� �����
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� ��
���
��� 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� �������� ���
��
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� ��� ���