Non Linear System Analyis - Lyaponav Based Approach

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


Nonlinear Systems: An Introduction

Model:

��

� � � ��

Properties:

� Don’t follow the principle of superposition,i.e,

��


� 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



Autonomous Systems: the nonlinear function doesnot explicitly depend on time� .

��

� � � � � ��

Affine System:

�� Unforced System: input� �� ,

��



Example:Pendulum:

��

�The state equations are

��

��

��

��



Exercise

Identify the category to which the followingdifferential equations belong to? Why?

1. ��

2. ��

3. �� where� is an external input.

4. ��

5. ��


Linearization

� Concept of Equilibrium Point: Consider a system

��

where functions � �! are continuouslydifferentiable. The equilibrium point � � " �� " forthis system is defined as

� � " �� " � �


� 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.


Linearization

The method:Let us write the the general form of nonlinearsystem �� as:

# � �# � � � � � � � � � � � � $ �� %

# � �# � � � � � � � � � � � � $ �� % (1)

...

# � $# � � $ � � � � � � � � � $ �� %


Linearization

Let� " � &� � " � � " � % " '( be a constant inputthat forces the system �� to settle into aconstant equilibrium state

� " � & � � " � � " � $ " '( such that � � " �� " � �

holds true.


Linearization

We now perturb the equilibrium state by allowing:

) � ) " ) and * � * " * . Taylor’sexpansion yields

# )# � � � ) " ) � * " *

� � ) " � * " +

+ ) � ) " � * " ) ++ * � ) " � * " *


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


Linearization

Note that

# )# � � # ) "# �

# � ) # � � # � ) # �

because ) " is constant. Furthermore, � ) " � * " � � .Let

� ++ ) � ) " � * " and � ++ * � ) " � * "

Neglecting higher order terms, we arrive at thelinear approximation

# � ) # � � ) *


Linearization

Similarly, if the outputs of the nonlinear systemmodel are of the form

� � � � � � � � � � � � � � $ �� %

� � � � � � � � � � � � � � $ �� %

...

� : � � : � � � � � � � � � $ �� %

or in vector notation

; � < � ) � *


Linearization

Taylor’s series expansion can again be used to yieldthe linear approximation of the above outputequations. Indeed, if we let

; � ; " ;then we obtain

; � ) *


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.


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.


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.


Lyapunov Stability Theory

The concept of stability: Consider the nonlinearsystem

�� Let an equilibrium point of the system be G� ,

� G� � �


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.



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� )

��



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


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”.


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



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 ++ � $ $



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



Example

Oscillator with a nonlinear spring:

� � V �� W � �

Linearize this system,� � V ��

The characteristic equation of linearized system is

� � � V � � .



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.



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


The time derivative of is�

� � � ++ � �

�� ++ � �

��

� � W � � � � � � � V � � � � W �

� � V � ��

K �

It follows then that G� is asymptotically stable.



Disadvantage of Lyapunov based Approach

� There is no systematic way of obtainingLyapunov functions

� Lyapunov stability criterion provides onlysufficient condition for stability.


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


Nonlinear System AnalysisLyapunov Based Approach

Lecture 5 Module 1

Dr. Laxmidhar Behera

Department of Electrical Engineering,Indian Institute of Technology, Kanpur.



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.



Then we have

�� ( � � ( ��

� � � ( � � ( �

� � ( ( � � ( �

� � ( � ( �

� � � ( �

where

( � � Lyapunov Matrix Equation



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



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.



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[ � � ��



In addition, if � � is positive definite, then theequilibrium � � � is globally uniformlyasymptotically stable.



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


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� � � �



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 �

� ��



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



(this does not work always!):

� � � C� � � �

C� � ��

We see that � � B � for all � � , � � . Then

��

� � � � � � � � � � ��

� � � � � � � � ��



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



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


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 � � � �



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� � ��



Take its time derivative

��

� � � � � � � � � � W � � � � ��

� � � � � � � � � ��Y � � � � � �� Y �

� ��Y � �Y � � � � � � ��

� � � � � � �� C We can see, therefore, that stability is guaranteed if

�� K � or � � � � �� K C



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.



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.



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



Choose a Lyapunov Function Candidate

� C� �� > � C� : > �

Take its time derivative

� � �� > � > : >�>

� �> �� > � � �� ]^ � � : >January 4, 2003 Intelligent Control Lecture Series Page 54


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 : > � � � > � � .



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


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.



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


Taking f � � � �� e � � as a compositeLyapunov Function, we obtain

�f � � � � � � � � � �W e �

e � &� � C � � � � � � � � � �W e � '

� � � � � � � �Y

e � & � � � C � � � � � � � � � �W e � � '



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.


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


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Non Linear System Analyis - Lyaponav Based Approach