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Nonlinear System AnalysisLyapunov Based Approach
Lecture 4 Module 1
Dr. Laxmidhar Behera
Department of Electrical Engineering,Indian Institute of Technology, Kanpur.
January 4, 2003 Intelligent Control Lecture Series Page 1
Overview
� Non-linear Systems : An Introduction
� Linearization
� Lyapunov Stability Theory
� Examples
� Summary
January 4, 2003 Intelligent Control Lecture Series Page 2
Nonlinear Systems: An Introduction
Model:
�� � ��� � � ��
� � � ��� � � ��
Properties:
� Don’t follow the principle of superposition,i.e,
�� � � � � � � � � � ��� � � �� � � ��� � � �� �
January 4, 2003 Intelligent Control Lecture Series Page 3
� Multiple equilibrium points
� Limit cycles : oscillations of constantamplitude and frequency
� Subharmonic, harmonic oscillations forconstant frequency inputs
� Chaos: randomness, complicated steady statebehaviours
� Multiple modes of behaviour
January 4, 2003 Intelligent Control Lecture Series Page 4
Nonlinear Systems: An Introduction
Autonomous Systems: the nonlinear function doesnot explicitly depend on time� .
�� � � � ��
� � � � � ��
Affine System:
�� � � � � � � �Unforced System: input� ��� � � ,
�� � � �
January 4, 2003 Intelligent Control Lecture Series Page 5
Nonlinear Systems: An Introduction
Example:Pendulum:
�� ��� � � � � �� ��� � � � ��
�The state equations are
�� � � � �
�� � � � �� ��� � � � � �� � � � �
��
�� �
January 4, 2003 Intelligent Control Lecture Series Page 6
Nonlinear Systems: An Introduction
Exercise
Identify the category to which the followingdifferential equations belong to? Why?
1. �� � � �
2. �� � � � � �
3. �� � � � where� is an external input.
4. �� � � � � � �
5. �� � � �
January 4, 2003 Intelligent Control Lecture Series Page 7
Linearization
� Concept of Equilibrium Point: Consider a system
�� � � � ��
where functions � �! are continuouslydifferentiable. The equilibrium point � � " �� " forthis system is defined as
� � " �� " � �
January 4, 2003 Intelligent Control Lecture Series Page 8
� What is linearization?Linearization is the process of replacing thenonlinear system model by its linear counterpartin a small region about its equilibrium point.
� Why do we need it?We have well stablished tools to analyze andstabilize linear systems.
January 4, 2003 Intelligent Control Lecture Series Page 9
Linearization
The method:Let us write the the general form of nonlinearsystem �� � � � �� as:
# � �# � � � � � � � � � � � � $ �� � �� � � �� %
# � �# � � � � � � � � � � � � $ �� � �� � � �� % (1)
...
# � $# � � $ � � � � � � � � � $ �� � �� � � �� %
January 4, 2003 Intelligent Control Lecture Series Page 10
Linearization
Let� " � &� � " � � " � % " '( be a constant inputthat forces the system �� � � � �� to settle into aconstant equilibrium state
� " � & � � " � � " � $ " '( such that � � " �� " � �
holds true.
January 4, 2003 Intelligent Control Lecture Series Page 11
Linearization
We now perturb the equilibrium state by allowing:
) � ) " ) and * � * " * . Taylor’sexpansion yields
# )# � � � ) " ) � * " *
� � ) " � * " +
+ ) � ) " � * " ) ++ * � ) " � * " *
January 4, 2003 Intelligent Control Lecture Series Page 12
Linearization
where+
+ ) � ) " � * " � ,,,- ./-0 / 1 1 1 - ./- 0 2
... ...
- . 2-0 / 1 1 1 - . 2- 0 2333
4444444440 5 687 5
++ * � ) " � * " � ,,,
- ./- 7 / 1 1 1 - ./- 7 9
... ...
- . 2- 7 / 1 1 1 - . 2- 7 9333
4444444440 5 687 5are the Jacobian matrices of
January 4, 2003 Intelligent Control Lecture Series Page 13
Linearization
Note that
# )# � � # ) "# �
# � ) # � � # � ) # �
because ) " is constant. Furthermore, � ) " � * " � � .Let
� ++ ) � ) " � * " and � ++ * � ) " � * "
Neglecting higher order terms, we arrive at thelinear approximation
# � ) # � � ) *
January 4, 2003 Intelligent Control Lecture Series Page 14
Linearization
Similarly, if the outputs of the nonlinear systemmodel are of the form
� � � � � � � � � � � � � � $ �� � �� � � �� %
� � � � � � � � � � � � � � $ �� � �� � � �� %
...
� : � � : � � � � � � � � � $ �� � �� � � �� %
or in vector notation
; � < � ) � *
January 4, 2003 Intelligent Control Lecture Series Page 15
Linearization
Taylor’s series expansion can again be used to yieldthe linear approximation of the above outputequations. Indeed, if we let
; � ; " ;then we obtain
; � ) *
January 4, 2003 Intelligent Control Lecture Series Page 16
Linearization
Example
Consider a first order system:�� � � � � � where � � � � � =
Linearize it about origin�� � � �
Its solution is : � �� � � =?>@ A . Whatever may be theinitial state � = , the state will settle at � ��� � � as
� � B , which is the only equilibrium point thatthis linearized system has.
January 4, 2003 Intelligent Control Lecture Series Page 17
Linearization
However, the solution of actual nonlinear system is� ��� � � =?> @ A
C � � = � =D>E@ A
For various initial conditions, the system has twoequilibrium points: � � � and � � C as can be seenin the following figure.
January 4, 2003 Intelligent Control Lecture Series Page 18
Linearization
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6 7 8
X(t)
Time(seconds)
Stable and Unstable Equilibrium Points
x(t)=x0e-t/(1-x0+x0e-t)
x0=1.5
x0=1.1 x0=1.01
x0=0.5
x0=-0.5
0 F = is a Stable while0 F � is an unstable equilibrium point.
January 4, 2003 Intelligent Control Lecture Series Page 19
Lyapunov Stability Theory
The concept of stability: Consider the nonlinearsystem
�� � � � Let an equilibrium point of the system be G� ,
� G� � �
January 4, 2003 Intelligent Control Lecture Series Page 20
We say that G� is stable in the sense of Lyapunov ifthere exists positive quantity H such that for every
I � I � H we have
J � �� = � G� JLK I � J � �� � G� JLK H
for all� B � = . we say that G� is asymptotically stableif it is stable and
J � ��� � G� J � as�
We call G� unstable if it is not stable.
January 4, 2003 Intelligent Control Lecture Series Page 21
Lyapunov Stability Theory
How to determine the stability or instability of G�
without explicitly solving the dynamic equations?
� Lyapunov’s first or indirect method: Start with anonlinear system
�� � � � Expand in Taylor series around G� (we alsoredefine � � � G� )
�� � � � � �
January 4, 2003 Intelligent Control Lecture Series Page 22
Lyapunov Stability Theory
where
� ++ �
4444NM0
is the Jacobian matrix of � � evaluated at G� and
� � � contains the higher order terms, i.e.,OPQ
R0 RTS =J � � � J
J � J� �
Then the nonlinear system �� � � � isasymptotically stable if and only if the linearsystem �� � � is stable, i.e., if all eigenvalues of
have negative real parts.January 4, 2003 Intelligent Control Lecture Series Page 23
Lyapunov Stability Theory
Advantage: Easy to applyDisadvantages:
– If some eigenvalues of are zero, then wecannot draw any conclusion about stability ofthe nonlinear system.
– It is valid only if initial conditions are “close”to the equilibrium G� . This method provides noindication as to how close is “close”.
January 4, 2003 Intelligent Control Lecture Series Page 24
Lyapunov Stability TheoryU Lyapunov’s second or direct method: Consider
the nonlinear system
�� � � �
Suppose that there exists a function, calledLyapunov function, � � with followingproperties:
1. � G� � �
2. � � B � , for � � G� :Positive definite
3.
�� � K � along trajectories of
�� � � � : Negative
January 4, 2003 Intelligent Control Lecture Series Page 25
Lyapunov Stability Theory
Definite
Then, G� is asymptotically stable. The methodhinges on the existence of a Lyapunov function,which is an energy-like function.
�� � � +
+ �1 # �
# �
� ++ � � �
� ++ � � � ++ � � � 1 1 1 ++ � $ $
January 4, 2003 Intelligent Control Lecture Series Page 26
Lyapunov Stability Theory
Advantages:
� answers stability of nonlinear systems withoutexplicitly solving dynamic equations
� can easily handle time varying systems
�� � � � � �
� can determine asymptotic stability as well asplain stability
� can determine the region of asymtotic stability orthe domain of attraction of an equilibrium
January 4, 2003 Intelligent Control Lecture Series Page 27
Lyapunov Stability Theory
Example
Oscillator with a nonlinear spring:
� � V �� �W � �
Linearize this system,� � V �� � �
The characteristic equation of linearized system is
� � � V � � .
January 4, 2003 Intelligent Control Lecture Series Page 28
Lyapunov Stability Theory
The � V characteristic root corresponds to thedamping term but notice the existence of a � rootfrom the lack of a linear term in the spring restoringforce. The linearized version of the system cannotrecognize the existence of a nonlinear spring termand it fails to produce a non-zero characteristic rootrelated to the restoring force.
January 4, 2003 Intelligent Control Lecture Series Page 29
Lyapunov Stability Theory
Let’s look at Lyapunov based approach. Considerthe state space model
�� � � � �
�� � � � V � � � � W �
with equilibrium G� � � G� � � � . Let’s try for aLyapunov function
� � � CX ��Y �
C� � ��
We can see that � � B � for all � � , � � .January 4, 2003 Intelligent Control Lecture Series Page 30
Lyapunov Stability Theory
The time derivative of is�
� � � ++ � �
�� � ++ � �
�� �
� � W � � � � � � � V � � � � W �
� � V � ��
K �
It follows then that G� is asymptotically stable.
January 4, 2003 Intelligent Control Lecture Series Page 31
Lyapunov Stability Theory
Disadvantage of Lyapunov based Approach
� There is no systematic way of obtainingLyapunov functions
� Lyapunov stability criterion provides onlysufficient condition for stability.
January 4, 2003 Intelligent Control Lecture Series Page 32
References
1. Nonlinear Systems, Hassan K. Khalil, Third Edition,Prentice Hall.
2. Applied Nonlinear Control, J. J. E. Slotine and W. Li,Prentice Hall
3. Nonlinear System Analysis, M. Vidyasagar, SecondEdition, Prentice Hall
January 4, 2003 Intelligent Control Lecture Series Page 33
Nonlinear System AnalysisLyapunov Based Approach
Lecture 5 Module 1
Dr. Laxmidhar Behera
Department of Electrical Engineering,Indian Institute of Technology, Kanpur.
January 4, 2003 Intelligent Control Lecture Series Page 34
Lyapunov Stability Theory
Stability of Linear Systems
Consider a linear system in the form
�� � �Choose as Lyapunov function the quadratic form
� � � � ( �where is a symmetric positive definite matrix.
January 4, 2003 Intelligent Control Lecture Series Page 35
Lyapunov Stability Theory
Then we have
�� � � �� ( � � ( ��
� � � ( � � ( �
� � ( ( � � ( �
� � ( � ( �
� � � ( �
where
( � � Lyapunov Matrix Equation
January 4, 2003 Intelligent Control Lecture Series Page 36
Lyapunov Stability Theory
If the matrix is positive definite, then the system isasymptotically stable. Therefore, we could pick
� Z , the identity matrix and solve( � � Z
for and see if is positive definite
January 4, 2003 Intelligent Control Lecture Series Page 37
Lyapunov Stability Theory
Note: The usefulness of Lyapunov’s matrix equationfor linear systems is that it can provide an initialestimate for a Lyapunov function for a nonlinearsystem in cases where this is done computationally.Furthermore, it can be used to show stability of thelinear quadratic regulator design.
January 4, 2003 Intelligent Control Lecture Series Page 38
Lyapunov Stability Theory
LaSalle-Yoshizawa Theorem
Let � � � be an equilibrium point of �� � � � �� .Let � � be a continuously differentiable, positivedefinite and radially unbounded function such that
� � ++ � � � � � � � � �
where is a continuous function. Then, allsolutions of �� � � � �� are globally uniformlybounded and satisfy
O PQA S[ � � ��� � �
January 4, 2003 Intelligent Control Lecture Series Page 39
Lyapunov Stability Theory
In addition, if � � is positive definite, then theequilibrium � � � is globally uniformlyasymptotically stable.
January 4, 2003 Intelligent Control Lecture Series Page 40
Lyapunov Stability Theory
More Examples
We will discuss three examples that demonstrateapplications of Lyapunov’s method, namely
1. How to assess the importance of nonlinear termsin stability or instability.
2. How to estimate the domain of attraction of anequilibrium point.
3. How to design a control law that guaranteesglobal asymptotic stability
January 4, 2003 Intelligent Control Lecture Series Page 41
Lyapunov Stability Theory: Examples
Example .1
Consider the system
�� � � � � � � � � ��
�� � � � � � � � � � � �
with � � . Find the equilibrium of the system bysolving following equations:
� G� � G� � G� �� � �
G� � � � G� � � G� � � �
January 4, 2003 Intelligent Control Lecture Series Page 42
Lyapunov Stability Theory: Examples
Multiply the first equation by G� � , the second by G� �
and add them to get
G� � � G� �� � � � � �
Hence, the equilibrium point is G� � � G� � � � .The linearized system is
�� ��� �
� � � CC �
� �� �
January 4, 2003 Intelligent Control Lecture Series Page 43
Lyapunov Stability Theory: Examples
The characteristic equation is
# > �444444
� � � C
C � �444444
� � � � � C � � � � � \ �
Since the characteristic roots are purely imaginary,we can not draw any conclusion on the stability ofthe nonlinear system.Now we resort to Lyapunov based approach. Choosethe Lyapunov function � � to be the sum of thekinetic and potential energy of the linear system
January 4, 2003 Intelligent Control Lecture Series Page 44
Lyapunov Stability Theory: Examples
(this does not work always!):
� � � C� � � �
C� � ��
We see that � � B � for all � � , � � . Then
�� � � � � � � � � � � � �� � � � � � � � � � � � �
� � � � � � � � � � �� � � � � � � � � � � ��
� � � � � � � � ��
January 4, 2003 Intelligent Control Lecture Series Page 45
Lyapunov Stability Theory: Examples
Therefore, we see that b
� if K � , the system is asymptotically stable
� if B � , the system is unstablebthis result can not be obtained obtained by linearization
January 4, 2003 Intelligent Control Lecture Series Page 46
Lyapunov Stability Theory: Examples
Example .2
Suppose we want to determine the stability of theorigin � � � � of the nonlinear system (Show that thisis the equilibrium of the system.),
�� � � � � � � � � � � � � � � ��
�� � � � � � � � � � � � � � � � �� January 4, 2003 Intelligent Control Lecture Series Page 47
Lyapunov Stability Theory: Examples
The easiest way to show the stability is bylinearization. The linearized form of the system is
�� ��� �
�� C C
� C � C� �
� �
The characteristic equation is
� � � � � � �We can see that the system is stable. Wait! since thisresult is based on linearization, it says that if initialcondition is “close” to the equilibrium point � � � �
January 4, 2003 Intelligent Control Lecture Series Page 48
Lyapunov Stability Theory: Examples
then the solution will tend to the equilibrium as
� .To find how close is “close” we need to get anestimate of the domain of attraction. We can do thisby using Lyapunov theory.
Let’s try a Lyapunov function candidate
� � � C� � � �
C� � ��
January 4, 2003 Intelligent Control Lecture Series Page 49
Lyapunov Stability Theory: Examples
Take its time derivative
�� � � � � �� � � � �� �
� � � � � � � � � � W � � � � �� � � � � � � � � � � � � � �
� � � � � � � � � ��Y � � � � � �� � � � � � � � �� � � � � �� ��Y �
� ��Y � �Y � � � � � � �� � � � � � � ��
� � � � � � �� � � � � � �� � C We can see, therefore, that stability is guaranteed if
�� � K � or � � � � �� K C
January 4, 2003 Intelligent Control Lecture Series Page 50
Lyapunov Stability Theory: Examples
This means that the domain of attraction of theequilibrium is a circular disk of radius 1. As long asthe initial conditions are inside the disk, it isguaranteed that the solution will end up at the stableequilibrium. In case, the initial conditions lie outsidethe disk then convergence is not guaranteed.
January 4, 2003 Intelligent Control Lecture Series Page 51
Lyapunov Stability Theory: Examples
Note: It should be mentioned that the above disk isan estimate of the domain attraction based on theparticular Lyapunov function we selected. Adifferent Lyapunov function could have produced adifferent estimate of the domain of attraction.
January 4, 2003 Intelligent Control Lecture Series Page 52
Lyapunov Stability Theory: Examples
Example .3Trajectory Tracking
Consider a single link manipulator
�� ��� �
�� � �� ]^ � � � � _
Now we want to find a control so that � tracks adesired trajectory � ` . Define> � � � � ` . Then aboveequation may be written as
� � �� > �> � �� ]^ � � � _ error dynamic equation
January 4, 2003 Intelligent Control Lecture Series Page 53
Lyapunov Stability Theory: Examples
Choose a Lyapunov Function Candidate
� C� �� � �> � C� : > �
Take its time derivative
� � �� � �> � > : >�>
� �> �� � �> � � �� ]^ � � : >January 4, 2003 Intelligent Control Lecture Series Page 54
Lyapunov Stability Theory: Examples
We can choose our control input as
� � � �� ]^ � � � : > � a �> . Then
� � � � a �> � assume � a B �
K �
This implies that �> � and� > � . But, it does notimply that> � . Substituting the control into errordynamic equation, we get
�� �� > � a �> : > � �This implies that : > � � � > � � .
January 4, 2003 Intelligent Control Lecture Series Page 55
Lyapunov Stability Theory: Examples
Example .4Back-Stepping
Consider the system�� � � � � � � � �W � �
�� � � �
Start with the scalar system
�� � � � � � � � �W � �Design the feedback control � � � � � � to stabilizethe origin � � � � .January 4, 2003 Intelligent Control Lecture Series Page 56
Lyapunov Stability Theory: Examples
We cancel the nonlinear term � � � by taking
� � � � � � � � � � � � � �
Thus we obtain�� � � � � � � � �W
Taking Lyapunov function candidate as
� � � � �� � � � , its time derivative satisfies
� � � � � � � � �Y K � b � �dcHence, the origin is globally asymptotically stable.
January 4, 2003 Intelligent Control Lecture Series Page 57
Lyapunov Stability Theory: Examples
Now to back step, we use the change of variablee � � � � � � � � � � � � � � � �
So the transformed system
�� � � � � � � � �W e �
�e � � � � C � � � � � � � � � �W e � January 4, 2003 Intelligent Control Lecture Series Page 58
Lyapunov Stability Theory: Examples
Taking f � � � �� � � � �� e � � as a compositeLyapunov Function, we obtain
�f � � � � � � � � � �W e �
e � &� � C � � � � � � � � � �W e � '
� � � � � � � �Y
e � & � � � C � � � � � � � � � �W e � � '
January 4, 2003 Intelligent Control Lecture Series Page 59
Lyapunov Stability Theory: Examples
Choosing control input as
� � � � � � C � � � � � � � � � �W e � � e �
� � � � � C � � � � � � � � � � �W � �
� � � � � � � � � we get �
f � � � � � � � �Y � e � �Hence with this control law, the origin is globallyasymptotically stable.
January 4, 2003 Intelligent Control Lecture Series Page 60
References1. Nonlinear Systems, Hassan K. Khalil, Third Edition, Prentice Hall.
2. Applied Nonlinear Control, J. J. E. Slotine and W. Li, Prentice Hall
3. Nonlinear System Analysis, M. Vidyasagar, Second Edition, PrenticeHall
January 4, 2003 Intelligent Control Lecture Series Page 61