Stability radius of linear normal distributed parameter systems with multiple directional perturbations

Proceedings of the 37th IEEE

Conference on Decision 8, Control

Tampa, Florida USA December 1998 _{WM10 14:40}

Stability Radius

of

Linear Normal Distributed Parameter Systems

with Multiple Directional Perturbations

Shin-Hao Lu and I-Kong Fong Department of Electrical Engineering

National Taiwan University Taipei, Taiwan 10617, Republic of China E-mail: sinhlu@acl2.ee.ntu.edu.tw ikfong@cc.ee.ntu.edu.tw Abstract

In this note, the stability robustness problem of linear time-invariant normal distributed parameter systems with multiple bounded or relative bounded directional perturbations is considered. The Lyapunov stability criterion is used to derive the system stability radius, i.e., the extent of perturbation within which the system can keep stability.

1. Introduction

In this paper we consider the robust stability prob- lem of a class of distributed parameter systems (DPS)

[l, 21. Just like the commonly discussed finite dimen- sional systems, DPS also have the stability and stability robustness problems. However, due to the intricacy of underlying mathematics, it is generally more difficult to study these problems for DPS. Various methods, such as [7], that can be successfully used in finite dimen- sional systems seem not directly applicable in DPS. In the literature, many authors have managed to study the stability robustness problem of DPS with unstruc- tured bounded perturbation [3], structural perturba- tion [ 6 ] , and time-varying perturbation [4].

Here we discuss via the Lyapunov stability approach a case of DPS with multiple structural perturbations, called the directional perturbations, which are opera- tors each multiplied by an unknown constant. It is shown that with this approach the bounded and rel- ative bounded operators can be treated together, and bounds on the unknown constants can be found to en- sure the system stability. More specifically, the bound of each unknown constant can be derived separately for each perturbation operator.

2. Problem Formulation

Let 2 be a Hilbert Space with the inner product func- tion (., .), and A0 : D ( A 0 )

c

2

-+

2 be a closed, linear unbounded operator densely defined on 2, with D ( A 0 ) denoting the domain of Ao. Assume that A0 is the in- finitesimal generator of a CO-Semigroup To(t). Thus, the mild solution of the system:

‘This research is supported by the National Science Council of the Republic of China under Grants NSC 86-2213-E-002-018

0-7803-4394-8198 $1 0.00 0 1998 IEEE 81

9

can be written as z ( t ) =

To(t)zo

[l, 21. We say that

A0 or T o ( t ) is exponentially stable if there exist M and

w

>

0 such that IITo(t)ll

5

Me-wt.

For an exponentially stable Ao, consider the following system with multiple directional perturbations:

(2)

“ z ( t ) = (A0

+

xzl

k i A i ) z ( t ) z ( 0 ) = zo, zo E

2

where for each i = 1,.

. .

,

N O , Ai is a known perturba- tion operator, which can be bounded or unbounded. If we let k = ( k 1 ,

...,

k ~E R N o , then we wish to find an ~ ) upper bound

E

of 11k1)2 =

4 5 -

such that when Ilk112

<

E ,

the system described by (2) is still exponen- tially stable.

3. Main Result

Theorem 1 Suppose A0 is a normal operator /2] gen- erating the CO-semigroup To(t), then the followings are true:

(1) (A0

+

A:) : D ( A 0 )

c

2

-+

2 is the infinitesimal generator of T,’(t)To(t), and A0

+

A: is exponentially stable provided A0 is.

(2) If A0 is exponentially stable, and Poz =

/ , T , * ( t ) T o ( t ) z d t , then POZ E D ( A o ) , POZ E D ( A G ) , and (A:

+

Ao)Poz = - z , where AoPo and A:Po both belong to C ( Z ) , the space of bounded linear operators defined o n 2.

Proof: Omitted for the sake of brevity.

For the perturbation operators Ai’s, we define the set of relative bounded perturbations of Ao.

Definition 1 [5] Let A o : D ( A o ) c 2

-+

2 be a n un- bounded operator defined o n 2. T h e set of relative bounded perturbation operators with respect t o A0 is de- fined as FU(Ao) = { A : D ( A ) c 2

+

2

I

D ( A 0 ) c

D ( A ) , 3 a , P

2

0 such that Vz E D ( A ~ ) , l l A z l l I

allAo~ll

+

PII~llI.

From the definition of P,(Ao), we note that C ( 2 ) C

P,,(Ao). A rich amount of examples of relative bounded

operators can be found in 151. It is noted that a relative bounded operator can be unbounded by itself.

Theorem 2 Let A0 : D(A0)

c

2 2 be an expo- nentially stable normal operator, and the CO -semigroup

To(t) generated b y A0 satisfy IITo(t)II

5

M e p w t , where

M , w

>

0 are constants. Assume that there are per- turbations Ai, i = 1 , . .

.

, N O , satisfying A: E Pu(Ag), and (A0

+

llzl

kiAi) : D(A0)

c

2

+

2 generates a Co-semigrou,p T N ~ ( ~ ) . Let

7 , n y = -1,n = 3

Y = - 2 , n = 3

-

where 0

<

E

<

1, and

E

= ( ~ ~ l ( E i ) - 2 ) - ~ . If llkllz

5

k , then the perturbed system (2) is still exponentially stable.

k1 k2

0.072 0.9 0.0717 0.138 1.8 0.1376

Proof: Omitted for the sake of brevity.

In this Theorem, the bound

E

is based on xi’s, for which the formula involves solving infimum over all z with llzll = 1. This is not always easy to do for given A0 and Ai’s. To provide simplified but convenient conditions, we give the following two Corollaries.

Corollary 1 Under the assumption of Theorem 2. If an estimation of the relative bounded coeficients ai and such that IIAfzll

5

aillA;TzII

+

Pillzll exists, then a lower bound of xi is (4) (1 -

€1

(1

+

Ro)ai

+

%Pi - ki

2

where Ro = llAoPoll and 0

<

6

<

1.

Proof: Omitted for the sake of brevity.

Corollary 2; Under the assumption of Theorem 2. If

A0 is further assumed to be self-adjoint, i.e., AT, = Ao, and a pair o,f relative bounded coeficients ai and Pi is

known, then another lower bound of

Ti

is

- (1 - E )

k i 2 M

ai

+

,Pi ( 5 )

where 0

<

E

.:

1

Proof: Omitted for the sake of brevity.

4. Example

The fol1owin;y is an example about diffusion equations. Consider the following system defined on 2 = L2(0,1):

at - -

(s

az + y I ) z

+

k l ( % )

+

ICzv(x) Jb’h(x)z(t,z)dz z ( t , 0 ) = Z ( t , 1) = 0

{

z ( 0 , z ) = zo

where h ( z )

2

0 with llh(x)II = 1, ~ l w ( x ) ~ ~ = 1, and y

<

0. P u t this system into the framework of our dis- cussion, we have Aoz = (&

+

y I ) z with D(A0) =

{ z E 2

I

z and

%

absolutely continuous,

9

E 2 ,

z ( t , 0 ) = z ( t , 1) = 0). It is easy to verify that AT, = Ao,

i.e., A0 is self-adjoint, and since y

<

0, the semigroup To(t) satisfies IITo(t)ll

5

e y t . The perturbation opera- tors are Alz = & z , with D(A1) = { z E 2

I

z abso- lutely continuous, E 2 , z(t,O) = z ( t , l ) = 0) and A2z = U(.)

J;

h ( x ) z ( t , x)dx..

First we check that AI E Pu(AT,) with a1 = and

P1 =

w,

where n is any positive integer larger than unity [5]. Also A; = -A1 and D ( A ; ) = D(A1). Therefore we have

Thus by Corollary 2, a lower bound of E l can be ob-

tained as

For E = 0.1, we give the resulting values of E l ,

&,

and

k in the following table:

-

y = - l O , n = 4

I

0.54

I

9

I

0.539

References

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