Advanced Stability Analysis

Advanced Stability Theory

Dr. Abdul Qayyum Khan

Room No. SE-302Department of Electrical Engineering,

Pakistan Institute of Engineering and Applied Sciences,P.O. Nilore Islamabad Pakistan

Email: [email protected]://www.pieas.edu.pk/aqayyum/

A. Q. Khan (DEE, PIEAS) EE-602 NCS 1 / 34

Non-Autonomous Systems I

Consider a general nonlinear system

x = f (x, t) (1)

with x = x is the Eq. point and is defined as

f (x, t) 0 t C t0

Note that

t0 the time at which the observation starts

x = x = 0 is assumedThe linear approximation is given as

x = A(t)


Extension of stability concepts to Non-autonomous

systems I

Stability of Equilibrium point:

An equilibrium point x = 0 is said to be stable at t = t0 if, for any R A 0,there exist r(R, t0) A 0, s.t. SSx(t0)SS < r , then SSx(t)SS < R for all t C t0.Otherwise the equilibrium point is unstable. Symbolically it can be writtenas

R A 0, r(R, t0) A 0, SSx(t0)SS < 0 t C t0, SSx(t)SS < R

or

R A 0, r(R, t0) A 0, x(t0) > Br t C t0, x(t) > BR



systems II

Asymptotic Stability (A.S.)

An equilibrium point x = 0 is asymptotically stable

if it is stable, and if in addition

there exists some r(t0) A 0 such that SSx(t0)SS < r(t0) implies thatx(t) 0 as t

Note that

The A.S. requires that there exists an attractive region for everyinitial time t0

The size of the region of attraction and speed of trajectoryconvergence may depend on t0.



systems III

Exponential Stability (E.S.)

An equilibrium point x = 0, is exponentially stable, if there exists twostrictly positive numbers and s.t. for sufficiently small x(t0)

t A t0, SSx(t)SS B SSx(0)SSe(tt0) (2)

where is known the rate of converges.

Global Asymptotic Stability (G.A.S.)

An equilibrium point x = 0 is globally asymptotically stable (G. A. S) if forall x(t0), x(t) 0 as t



systems IV

Example 1:

Consider

x = a(t)x(t)

and its solution

x(t) = x(t0)e R tt0 a(r)dr

Note that the above system is

stable if a(t) C 0 t C t0 asymptotically stable if R

t0

a(r)dr = exponentially stable if T A 0 s.t. t C 0 R

t+Tt

a(r)dr C , A 0

Example 2:



systems V

Consider

x = x(t)

(1 + t)2

and its solution

x(t) = x(t0)eR tt0 1(1+r)2 dr

Note that in the above system:

a(t) = 1(1+t)2

C 0 t C t0, hence stable

R

t0a(r)dr = R

t0

1

(1+r)2dr = 1 Not A.S.

Example 2:



systems VI

Consider

x = x(t)

(1 + t)

and its solution

x(t) = x(t0)e R tt0 1(1+r)dr


a(t) = 1(1+t)

C 0 t C t0, hence stable

R

t0a(r)dr = R

t0

1

(1+r)dr = A.S.

Example 3:



systems VII

Consider

x = tx(t)

and its solution

x(t) = x(t0)e R tt0 rdr


a(t) = t C 0 t C t0, hence stable R

t0

a(r)dr = R

t0rdr = A.S.

Rt+T

t0a(r)dr = R

t+Tt0

rdr = T2+2tT

2= E.S.

Example 4:



systems VIII

Consider

x = x(t)

1 + sin(x2(t))

and its solution

x(t) = x(t0)e R tt0 11+sin(x2(r))dr

The system is exponentially stable (show)


Uniform stability of Non-autonomous systems I

Time dependency is very important aspect of non-autonomoussystems

Initial observation time has important effect on the stability propertiesof systems

Uniformity in system behavior is desirable and often demanded inpractice. This motivates to define

1 Uniform stability

2 Uniform asymptotic stability

Non-autonomous systems with uniform properties are inherentlyrobust against disturbances


Uniform stability of Non-autonomous systems II

Uniform Stability

An equilibrium point x = 0

is locally uniformly stable at t = t0 if, for any R A 0, there existr(R, t0) = r(R) A 0 independent of t0, s.t. SSx(t0)SS < r , thenSSx(t)SS < R for all t C t0. Symbolically it can be written as

R A 0, r(R) A 0, SSx(t0)SS < r t C t0, SSx(t)SS < R

or

R A 0, r(R) A 0, x(t0) > Br t C 0 x(t) > BR

unstable if it is not stable


Uniform stability of Non-autonomous systems III

uniformly asymptotically stable, if it is stable r A 0 such that SSx(t0)SS < r x(t) 0 as t uniformly in

t0, i.e., for each R A 0, there exist T = T(R, r) A 0, such that

SSx(t0)SS < r SSx(t)SS < R,t C t0 +T(R, r)

globally uniformly stable if it uniformly stable r(R) can be chosen to satisfy limR r(R) , and for each pair

of positive numbers and c, there is T = T(, c) A 0 such that

SSx(t)SS < , t C t0 +T(, c), SSx(t0)SS < c


Uniform stability of Non-autonomous systems IV

Uniform convergence in t0?

For all R1 and R2 satisfying 0 < R2 < R1 B R0, T(R1,R2) A 0, such that,t0 A 0

SSx(t0)SS < R1 SSx(t)SS < R2 t C t0 +T(R1,R2)

This means that the trajectory, starting from within a ball BR1 , willconverge into a smaller ball BR2 after a time period T which isindependent of t0

Uniform A.S. always implies A.S.

A.S. does not always imply Uniform A.S.

Exponential stability always implies Uniform A.S.


Uniform stability of Non-autonomous systems V

Illustrative example 1:

Consider

x = x(t)

(1 + t)

The solution is

x(t) =1 + t0(1 + t)

x(t0) (3)

Note that

x(t) converges to zero as t . Hence A.S. The convergence of x depends on t0. For larger values of t0, the

convergence is slower and vice versa. Consequently, the stability isnot uniform.


Uniform stability of Non-autonomous systems VI

Illustrative example 2:

Consider

x = tx(t)

The solution is

x(t) = x(t0)eR t+Tt0 rdr = x(t0)e

T2+2tT

2

Note that

x(t) converges to zero exponentially as t . Hence ExponentialA.S.

The convergence of x is independent of t0. Consequently, the stabilityis uniform.


Lyapunov Analysis of Non-Autonomous Systems I

Intuitively stability concepts and definitions of Autonomous systemscan be extended to Non-autonomous systems

A bit complicated treatment is required

Lyapunov direct method

Stability can be concluded for non-autonomous system using aLyapunov function

more mathematical concepts are involved

The so-called La Salle theorem can not be used to study the stability

Barbalat theorem: A partial compensation

Stability discussion requires some basic concepts

Time varying p.d. function decrescent function


Lyapunov Analysis of Non-Autonomous Systems II

A scalar time-varying function V (x, t) is

1 locally p.d. if V (0, t) = 0 and there exists a time-invariant p.d.function V0(x) s.t. t C t0, V (x, t) C V0(x). Note also that V (x, t)is locally p.d. if V0(x) is locally p.d.

2 decrescent if V (0, t) = 0 and if there exists a time-invariant p.d.function VI (x) such that

t C 0 V (x, t) B VI (x)

In other words, a scalar time-varying function V (x, t) is descrescent ifit is dominated by a time invariant p.d.f.


Lyapunov Analysis of Non-Autonomous Systems III

Illustrative Example

Consider a scalar function

V (x, t) = (1 + sin2t)(x21 + x2

2)

Also two functions which are time-invariant in nature

V0(x) = x2

1 + x2

2

andVI (x) = 2(x

2

1 + x2

2)

Since

V (0, t) = 0 and t C 0 V (x, t) C V0(x)

V (x, t) Positive definite function


Lyapunov Analysis of Non-Autonomous Systems IV

V (0, t) = 0 and

t C 0 V (x, t) B VI (x)

V (x, t) decrescent function

Also note that the time derivative of V (x, t) along the trajectory ofx = f (x, t) is given as

dV

dt=V

t+V

xx =

V

t+V

xf (x, t)

Lyapunov theorem for non-autonomous systems


Lyapunov Analysis of Non-Autonomous Systems V

Stability: If, in a ball BR0 around the eq.point 0, there exists a scalarfunction V (x, t) with continuous partial derivatives s.t.

1 V A 02 V B 0

then 0 is stable in the sense of Lyapunov.

Uniform stability: If, furthermore,

3. V (x, t) is decrescentthen 0 is uniformly stable.

Uniform asymptotic stability: If condition 2 is strengthened asV < 0, then 0 is uniformly asymptotically stable.

Global stability: If the ball BR0 is replaced by the whole state space,and condition 1, strengthened condition 2, condition 3, and thecondition

4. V (x, t) is radially unboundedthen 0 is globally uniformly asymptotically stable.


Comparison functions I

Moving from autonomous to non-autonomous system a difficultyfaced

the solution of x = f (x, t) depends on both t and t0 Definition of Uniform stability and UAS are introduced

More transparent definitions of US and UAS are possible in terms ofcomparison functions, known as class K and KL

Class K function:

A continuous function R+ R+ or [0,a) [0,) is said to belongto class K if it is strictly increasing

(0) = 0 (r) A 0 r A 0

It is said to belong to class K if a = and (r) as r A. Q. Khan (DEE, PIEAS) EE-602 NCS 22 / 34

Comparison functions II

Class KL function:

A continuous function [0,a) [0,) [0,) is said to belong toclass KL if for each fixed s, the mapping (r , s) belongs to class K with respect

to r

for each fixed r, the mapping (r , s) is decreasing with respect to sand

(r , s) 0 s

Relation between p.d. and decrscent functions and class K functions


Comparison functions III

Lemma: A function V (x, t) is locally (globally) p.d. function if and ifthere exists a function of class K s.t. V (0, t) = 0 and

V (x, t) C (SSx SS) t C 0 and x > BR0 (4)A function is locally decrescent if and only if there exists a class Kfunction s.t. V (0, t) = 0 and

V (x, t) B (SSx SS) t C 0, x > BR0 (5)For the above definitions to hold globally, BR0 is replaced by the wholestate space.

Proof:

Part 1. Sufficiency: Since (SSx SS) is p.d.f. and as V (x, t) C (SSx SS), itsuffices p.definiteness of V (x, t)


Comparison functions IV

Necessity: Let there exists V0(x) Time invariant and p.d.function s.t.

V (x, t) C V0(x)

Let us define(p) = inf

pBSSxSSBRV0(x)

Then (0) = 0; is continuous and non-decreasing. BecauseV0(x) is a continuous and non-zero. Because V0(x) C 0 andcontinuous and (p) A 0 with p A 0 is class-K function

Part 2. Define V1(x) as p.d. and time invariant and also define(p) = sup

0BSSx SSBpV1(x)

This implies equation (5).


Comparison functions V

In the light of the above lemma, the following theorem is presented.

Theorem: Assume that, in the neighborhood of the origin there exists ascalar function V (x, t) with continuous first order derivatives and aclass-K function such that, x x 0

1. V (x, t) C (SSx SS) A 02. V (x, t) B 0then the origin is Lyapunov stable. If, furthermore, there is a scalarfunction of class KL s.t.3. V (x, t) B (SSx SS)Then the origin is uniformly stable. If condition 1 and 3 are satisfied andcondition 2 is replaced by


Comparison functions VI

4. V (x, t) B (SSx SS) < 0.with being another class K function, then 0 is uniformly asymptoticallystable. If condition 1,3,and 4 are satisfied in the whole state space, and

limx(SSx SS)

Then 0 is globally uniformly asymptotically stable.

Example:Investigate the stability of the equilibrium point of the

following dynamical system

x1 = x1 e2tx2 (6)

x2 = x1 x2 (7)


Comparison functions VII

LetV (x) = x21 + (1 + e2t)x22

Also defineV2 = x

2

1 + x2

2

V2 = x2

1 + 2x2

2

Note that

V (x) C V2(x) positive definite functionV (x) B V1(x) decrescent function

Now

V = 2 x21 x1x2 + x22(1 + 2e2t)B 2 x21 x1x2 + x22 = (x1 x2)2 x21 x22 = (SSx SS)

Note thatA. Q. Khan (DEE, PIEAS) EE-602 NCS 28 / 34

Comparison functions VIII

V (x) is positive definite, decrescent (SSx SS) = x2

1+ x2

2is radially unbounded

Hence 0 is GAS


Why is decrescent condition important? I

Intuitively, for system stability, it is sufficient to prove that

V A 0

V B 0

However, the above is Incorrect for time varying systems

A Counter Example:

Let

g(t) some scalar function Figure shows the plot of g2(t) w.r.t t It coincides with et~2 except some peaks which occurs at each

integer value of t.

The width of each peak corresponding to abscissa n is assumed tobe smaller than 1

2n


Why is decrescent condition important? II

The infinite integral of g2 satisfies

S

0

g2(r)dr < S

0

erdr +Qn=1

1

2n= 2


Why is decrescent condition important? III

Consider a system described by

x =g(t)g(t)x

and choose a Lyapunov function

V (x, t) = x2

g2(t) 3 St

0

g2(r)dr

Note that

V (x) A x2 is positive definite V = x2


Why is decrescent condition important? IV

Intuitively, this analysis shows that the system is asymptotically stable.

Now consider the general solution

x =g(t)g(t0)x(t0)

This shows that the system is NOT A.S.

Why?

V (x, t) is not decrescent



Lecture 26

Advanced Stability Analysis

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