The Study of Partial Connective Stability for Singular Linear Large-Scale Interconnected Systems

(1)

Canadian Research & Development Center of Sciences and Cultures 12/20/2009 Http://www.cscanada.org Http://www.cscanada.net

E-mail: [email protected]; [email protected]

The Study of Partial Connective Stability for Singular Linear Large-Scale Interconnected

Systems

SUN Shui-Ling

¹

PENG Ping

²

Abstract: The connective stability is an important content among the large-scale interconnected systems. The many results recently have been given for the normal systems. In this paper, we shall discuss the connective stability of singular linear large-scale dynamical systems by means of singular Lyapunovf equation and vector Lyapunovf function. We give a simple form of a stable domain of connective parameters.

Key words: Singular system; Large-scale systems; Stability; Connective stability;

Vector Lyapunovf function

1 Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou, Guangdong, China.

E-mail: [email protected]

2Department of Computer Science, Guangdong Polytechnic Normal University. Guangzhou, Guangdong, China.

E-mail: [email protected]

* Received 11 June 2009; accepted 2 September 2009

1. INTRODUCTION

With the development of modern control theory and the permeation into other application area such as aviation, spaceflight, energy, power, oil, chemical industry and communication and so forth, many researchers discover a kind of dynamical system which possesses more widely representative in form, that is singular system, which is of the following form.

() ( ( ), ) .

t t X f t X

I =

Where ^I is a n×n matrix, it is usually singular. X(t)∈Rn is a n−vector, ^f is a n−vector function. The concept of the singular system is first proposed in the 1970s, within short thirty years, it has become an independent branch of the modern control theory. Especially later decade more and more scholars are interested in studying of the singular system theory, and many important results have been obtained as (ZHANG, 1997; CHEN & LIU, 1998; YANG & ZHANG, 2004; Campbell, 1992). There exist singular systems models in many area of the social production, known Dynamic Leontief Input Output Model, Hopfield neural network model, multi-robot Subjective Coordinate Work Dynamic

(2)

The connective stability is an important content among the large-scale dynamical interconnected systems. The many results recently have been given for the normal systems (SUN & CHEN, 2003; SUN

& PENG, 2008; CHEN, 1998; LIU, 1989). In this paper by means of the singular Lyapunovf equation and vector Lyapunovf function we discuss the connective stability of linear singular large-scale system and give a connective parameters stable domain. An example to illustrating its efficiency is given.

2. BASIC CONCEPTION

We consider singular linear time-invariant large-scale dynamical system described by equations of the form

⋅ = + ∑

≠

=

r e A X t t

X A t X I

i j

j ij ij j

i i ii

i

( ) ( )

1,

( )

(1) i=1,2,L,r

Where I_i and A_ii are all n_i×n_i constant matrices, A_ij is a constant n_i×n_j matrix.

) , , 2 , 1 ,

( i j r

R

X

_i

∈

ⁿⁱ

= L

. We denote

= ∑

= r

i

n

i

n

1 ，

X

^T

= ( X

₁^T

, X

₂^T

, L , X

^T_r

) ∈ R

ⁿ ^，

)

, , ,

( I

₁

I

₂

I

_r

diag

Block

I = − L

， and we assume that I is a singular matrix， and we assume for any initial time t₀∈R⁺ and any initial state

x

₀

= x ( t

₀

)

, there is one (and only one) solution

) ,

; ( t t

₀

x

₀

x

x =

of the system (1).

A matrix

E = ( e

ij

)

r_×r is called an interconnection matrix associated with E, if eij =0 imply

= 0

eij and if eij =1 imply e_ij =1or 0e_ij = . The elements eijof the fundamental interconnection matrix E are defined as

⎩ ⎨

= ⎧

i j

ij

X cann t act on X

X on act can e X

' ,

0 , 1

It is denoted asE∈E(LIU, 1989).

The isolated subsystems of (1) are

. , , 2 , 1 ), ( )

( t A X t i r

X

I

i

⋅

i

=

_ii _i

= L

(2)

The system (2) is called regular (YANG, ZHANG, 2004), if there exists a constant s0for every )

, , 2 , 1

(i r

i = L , such that 0 ) det(s₀I_i−A_ii ≠

The system (2) is called impulsive-free (YANG, ZHANG, 2004), if for any s∈C(C domain of complex number), it always holds that

deg det( sI

_i

− A

_ii

) = rank ( I

_i

)

,

( i = 1 , 2 , L , r )

.

(3)

91-100

3. MAIN RESULTS

We always assume that for every isolated subsystem (2) is regular and impulsive-free.

If there exists an positive definite matrixB_i, satisfying the following singular Lyapunovf equation

i

T i ii i T i i i T

ii

B I I B A I I

A + = −

(3) We take singular Lyapunovf function as follows

)]

( [ )]

(

[ I i X i t I i X i t B i I i X i t

V

_i

=

^T (4)

Then we get

)]

( [ ) ( )

( B

_i

I i X i t

²

V i I i X i t

m

≤

λ

)

2

( ) ( B

_i

I i X i t λ

M

≤

(5)

Where

|| ⋅ ||

represents Euclidean norm,

λ

_m

(B )

and

λ

_M

(B )

represent respectively minimum and maximum eigenvalue of the positive definite matrixB. And the total derivative of

V

_i

[ I

_i

X

_i

( t )]

with respect to t along solutions of the systems (2) is

] [ ] [ )]

(

[ I i X i t

₍₂₎

I

_i

X

_i ^T

I

_i

X

_i

V ⋅ i = −

(6)

Lemma (CHEN & LIU, 1998): For any a pair of

X , Y ∈ R

ⁿ, a positive definite matrix B and a constant numberμ>0, the following inequality holds

BY Y BX X BX Y BY

X

^T

+

^T

≤ μ

^T

+ μ

⁻¹ ^T (7)

Definition (CHEN, 1998): E is the fundamental interconnection matrix of the system (1). The system (1) is called connectively asymptotically stable, if the equilibrium state X^* =0 of the system (1) is always asymptotically stable for anyE∈E. Otherwise, the system (1) is called connectively unstable.

Theorem 1: The equilibrium state X^* = 0 of the system (1) is connectively asymptotically stable if the following conditions are satisfied

(ⅰ) We assume that every isolated subsystem (2) is asymptotically stable , in which there exists a positive definite matrix B_i, such that it satisfies (5) and (6);

(ⅱ) for singular large-scale dynamical systems (1), there are positive numbers δ_ij, such that

) ( )

( t X t

X

A

_ij _j

≤ δ

_ij ^I_j _j

) , , , 2 , 1 ,

( i j = L r j ≠ i

(8)

(ⅲ) all successively principal minors C_k (k=1,2,L,r) of the r×r matrix C=(c_ij)_r_×_r satisfy

(4)

0 )

1 ( ) 1 (

1

1 11

>

−

=

−

kk k

k k

c c

C

L M M

L

(9)

where

⎪ ⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

− ≠ + =

−

=

j B i

B i e

r

j i B i

B c

j m

M ij ij M

i M

ij

) , (

) ( )

1 (

) , (

) ( 1

2

λ λ δ λ

λ

（10）

i,j=1,2,_L,r

Proof: Let Y_i = IiXi(t), (i=1,2,Lr,) for any E=(e_ij)_r_×_r∈E , taking singular vector Lyapunovf function as follows

)], ( [ )], ( [ { )]

(

[ IX t V

₁

I

₁

X

₁

t V

₂

I

₂

X

₂

t

V = L , V

_r

[ I r X r ( t )]}

^T

The total derivative of Vi[IiXi(t)] with respect to t along solutions of the systems (1) is

) 1

)](

( [IiXi t V⋅i

)]

( [ )]

(

[ I i X i t

^T

B i I i X i t I i X i t

^T

B i I i X ⋅ i t

⋅ +

=

+ ∑

=

= ≠

r e ij A ij X j t B i I i X i t i t

ii X A

i j j

T ,

1

( )] [ ( )]

) (

[

+ ∑

+

= ≠

r e ij A ij X j t i t

ii X i A B i t i X I

i j j T

,

1

( )]

) ( [ )]

(

[

) ( ) (

)]

(

[ X

_i

t

^T

A

_ii^T

B i I i + I

_i^T

B i A

_ii

X

_i

t

=

+ ∑

≠

=

r e A X t B I X t

i j

j i i i

T j ij , ij

1

( )] [ ( )]

[ 2

By (7), we get

) 1

)](

( [IiXi t V⋅i

)]

( [ )]

[ )]

(

[IiX t (2) I X ⁽t B IiXi t

Vi i

T i i

i +

≤ ⋅

∑ ∑

+

= ≠ = ≠

r r

t X j A e B t i X A e

i j

j T i j j i i ij j

j j , ij

1

( )] [

1,

( ) ]

[

2

( ) ( )

)

( t B I X t

X

I

_i _i

+ λ

_M _i _i _i

−

≤

(5)

91-100

2 ,

1

( ) ||

||

)

( ∑

+

= ≠

r e A X t B i

i j

j ij ij j

λ

M

2 ,

1 2

2

+ ( ) + ( )( ∑ )

−

≤

= ≠

r e Y B i

i Y B Y

i j

j ij ij j

M i M

i

λ λ δ

∑

≠

=

− +

+

−

≤ ^r

i j j

j ij ij i

M i i

M B Y B r e Y

, 1

2 2

2 ( )( 1)

)]

( 1

[

λ λ δ

By the hypothesis (ⅲ) of the theorem 1, it follows From

[ − 1 + λ

_M

( B

_i

)] < 0

, and by the (5), we have

)]

( ) [

( ) ( )] 1

(

[ V IX t

B t B

X i I

V _i _i _i

i M

i M i

i

λ

+

≤−

⋅

)]

( ) [

(

) ( )

1 (

, 1

2

t X I r V

B B e

r

j j i j

j

j m j

i M ij ij

∑ −

+

= ≠

λ

λ δ

that is

) , , 2 , 1 ( )]

( [ )]

( [

1

r i

t X I V c t

X I

V

^r _j _j _j

j ij i

i i

≤ ∑ =

L

=

⋅

The total derivative of V[IX(t)] with respect to t along solutions of the systems (1) holds following inequality

)]

( [ )]

(

[ IX t

₍₁₎

CV IX t

V ⋅ ≤

Since matrix

C

satisfies the hypothesis (ⅲ) of theorem 1, all eigenvalues of matrix C have negative real parts . It follows from the Comparison principle that

) (

0 )]

(

[IX t → t →+∞

V

Therefore

) , , 2 , 1 ( ) (

0 )]

(

[IiXi t t i r

Vi → →+∞ = L

and

) , , 2 , 1 ( ) (

0 )

(t t i r

Xi

Ii → →+∞ = L (11)

Since every isolated subsystem (2) is regular and impulsive-free, so there exist nonsingular matrices Qi

P_i, , such that

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

−

0 0

)

0

1 (

1 i

i

I Q i I i

P

(12)

and

(6)

⎟⎟ ⎠

⎜⎜ ⎝

=

₍₂₎

0

_i

ii

i

A Q i I

P

(13)

denote

) , , 2 , 1 ) (

( ) ) (

( ) (

2

1 1

i r

t x

t t x

X t X Q

i i i

i

⎟⎟ ⎠ = L

⎞

⎜⎜ ⎝

= ⎛

−

=

where

I

_i⁽¹⁾

, I

_i⁽²⁾ are respectively

r

_i

× r

_i and

( n

_i

− r

_i

) × ( n

_i

− r

_i

)

unit matrices, xi1(t)∈R^rⁱ ，

i

i r

i

t R

n

x

2

( ) ∈

⁻ , A_ii⁽¹⁾ is a

r

_i

× r

_i constant matrix.

By the (11) and

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

= ⎛

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛

⎟⎟ ⎠

⎞

⎜⎜ ⎝

= ⎛

−

0 ) ( )

( ) ( 0 0 ) 0

(

¹

2 1 )

1 (

1

x t

t x

t I x

t X I

P

ⁱ

i i i

i i

i

We get

) , , 2 , 1 ( ) (

0 )

1(t t i r

x_i → →+∞ = L

By (8) and (11) we have

) , , , 2 , 1 , ( ) ( 0 )

( t t i j r j i

X

A

_ij _j

→ → +∞ = L ≠

Since the system (1) can be written as

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎛

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟ ⎠

⎜ ⎞

⎝

⎛

⋅ =

⋅

) (

) ( 0

0 )

( ) ( 0 0

0

2 ) 1 2 ( ) 1 (

2 ) 1

1 (

t x

t x I A t

x t

i i i ii i

xi Ii

+

⎟⎟

⎠

⎞

⎜⎜

⎝

∑ ⎟⎟⎛

⎠

⎞

⎜⎜

⎝ + ⎛

≠

= ( )

) (

2 1 ,

1 (3) (4)

) 2 ( ) 1 (

t x

t x A A

A e A

j r j

i j

j ij ij

ij ij ij

Where

) , , , 2 , 1 ,

)

(

4 ( ) 3 (

) 2 ( ) 1 (

1

i j r i j

A A

A Q A

A P

ij ij

ij ij j ij

i

⎟ ⎟ = ≠

⎠

⎞

⎜ ⎜

⎝

= ⎛

− L

We get

)]

( [

)

( (4) 2

, 1

) 1( ) 3 (

2 r A t

x A e t

x ij j

i j j

j t ij

i =−

∑

ij +

x

≠

=

Then

) , 2 , 1 ( ) (

0 )

2

( t t i r

x

_i

→ → +∞ = L

And

(7)

91-100 )

, , 2 , 1 ( ) (

0 )

( t t i r

i

X → → +∞ = L

) , , 2 , 1 ( ) (

0 )

(t t i r

Xi → →+∞ = L

Finally we have ) (

0 )

(t → t→+∞

X

That is, the equilibrium point X^* =0 of (1) is connectively asymptotically stable.

Theorem 2: We assume that

(a) There exists a positive definite matrix B_i and a

Constant number μi >0 such that total derivative of (4) with respective to t along the solution of (2) satisfies

2 )

2

(

( )

)]

(

[ I i X i t i I i X i t

V ⋅ i = μ

(14)

(b) For singular large-scale systems (1), there exists positive constant numbers δ_ij, such that

)

( )

( t I X t

X

A

_ij _j

≤ δ

_ij _j _j (15) (i,j=1,2,L,r,i≠ j)

(c) All successively principal minors C_k of the r×r matrix C =(c_ij)_r_×_r satisfy )

, , 2 , 1 (

0 k r

Ck > = L ,

Then the equilibrium point X^* = 0 of the system (1) is connectively unstable, where

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

− ≠

−

− =

=

j B i

i B e r

j B i

B c

j m

i M j ij i M

i M i

ij

) , (

) ( )

1 (

) , (

) (

2

λ λ δ λ

λ μ

(16)

Proof: Let

Y

_i

= I

_i

X

_i

( t ) , i = 1 , 2 , L r ,

and forE=(e_ij)=E , taking vector singular Lyapunovf function as follows

)], ( [ )], ( [ { )]

(

[ IX k V

₁

I

₁

X

₁

k V

₂

I

₂

X

₂

k

V = L , V

_r

[ I r X r ( k )]}

^T

The total derivative of

V

_i

[ I

_i

X

_i

( t )]

with respect to t along solutions of the systems (1) is

) 1

)]

(

( [ I X t V

i _i _i

⋅

=

[ I i X i ( t )]

^T

B i [ I i X i ( t )] [ I i X i ( t )]

^T

B i [ I i X ⋅ i ( t )]

⋅ +

By (7), we have

(8)

)]

( [ )]

(

[I X t ₍₂₎ IX t B IX t V⋅_i _i _i − _i _i ^T _i _i _i

≥

− ∑ ∑

≠

= = ≠

r i j j

r i j

j ij ij j

i T j ij

ij

A X t B e A X t

e

,

1 1,

] ) ( [

)]

( [

≥ μ

_i

I i X i ( t )

²

− λ

_M

( B

_i

) I i X i ( t )

²

,

1

( ) ||

||

)

( ∑

−

= ≠

r i j

j ij ij j

i

M

B e A X t

λ

2 ,

1 2

2− ( ) − ( )( ∑ )

≥ = ≠

r i j

j ij ij j

M i M i

iY

λ

Bi Y

λ

Bi e

δ

Y

μ

− ∑

−

≥ = ≠

r i j

j ij ij j

i M i i M

i B Y B r e Y

, 1

2 2

2 ( )( 1)

)]

(

[

μ λ λ δ

By the condition (c) of the theorem 2, it follows form[μ_i−λ_M(B_i)]>0, and by the (5), we have

) 1

)]

(

( [ I X t V

i _i _i

⋅

)]

( ) [

( )

( Vi IiX t B

i B

i i

M i M

λ λ

≥ μ −

[ ()]

) (

) ( )

1 (

, 1

2

t X I B V

B e

r

j j j r

i j

j m j

i M ij ij

∑ −

− = ≠

λ

λ δ

That is

[ ( )] [ ( )]

) 1 1

(

c V I X t

t X I

V

^r _j _j _j

j ij i

i i

≥ ∑

⋅

=

( i = 1 , 2 , L , r )

The total derivative of V[IX(t)] with respect to t along solutions of the systems (1) holds following inequality

V ⋅ [ IX ( t )]

₍₁₎

≥ CV [ IX ( t )]

By the condition (c), it follows from the Comparison principle that the equilibrium point X^* = 0 of (1) is connectively unstable.

4. FOR EXAMPLE

Consider following singular linear large-scale system which includes two subsystems with 5 order

⎪⎩

⎪ ⎨

⎧

+

⋅ = = +

⋅

2 22 1 21 2 21

2

2 12 12 1 1 11

1

X A X A e X I

X A e X A X I

(17)

(9)

91-100

Where

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛ 1 0

0 1

I1 ,

⎟ ⎟

⎟

⎠

⎞

⎜ ⎜

⎜

⎝

⎛

=

0 0 0

0 1 0

0 0 1

I2 ,

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛

−

= −

1 1

A11 ,

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

−

=

1 0 0

0 1 1

A22 ,

⎟ ⎠

⎜ ⎞

⎝

⎛

= −

0 2 1 2 1

A12 ,

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

−

=

0 0

2 1 3 2

3 2 2 1

A21

We get positive definite matrices

⎟⎟⎠

⎜⎜ ⎞

⎝

=⎛

2 1 0

0 2 1

B1 ,

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

=

3 2 0 0

0 2 1 0

0 0 2 1 B2

Taking

⎟

⎠

⎜ ⎞

⎝

=

⎛

0 1

1

E 0 , letδ₁₂ =δ₂₁ =δ ,

Then for∀E∈E, we have followed matrix

⎟ ⎠

⎜ ⎞

⎝

⎛

−

= −

2 1

3 1 δ C δ

Forδ < 6 2, matrix C satisfies the condition (iii) of theorem 1, therefore, the equilibrium point

* = 0

X of the system (17) is connectively asymptotically stable.

5. CONCLUSION

The item e_ij in the paper is taken to be 0 or 1, in practice, the conclusions above are adapted to the condition 0≤e_ij(t)≤1, that is to say, for any moment

t

, under the different intensity of connective among the sub-systems, the conclusion is still holds.

REFERENCES

CHEN Chao-tian. (1998). Partial Connective Stability of Large-Scale Systems. Journal of Guangdong Polytechnic Normal University, (4), pp1-8.

(10)

Stability of Singular Linear System. Journal of South China University of Technology, Vol.26, No.3. pp 1-6.

LIU Yong-qing. (1989). Theory and Application of Large-Scale Dynamic Systems, Vol.2.

S.L. Campbell. (1992). Singular Systems of Differential Equations. Pttman Advanced Publishing Program, Sanfracisco. London: Melbourne, 11.

SUN Shui-ling, CHEN Chao-tian. (2003). A Critical Method on Connective Stability of Linear Discrete Large-Scale Systems. Journal of Hohai University, (9), Vol. 30, No.4 pp589-592.

SUN Shui-ling, CHEN Chao-tian. (2004). Connective Stability of a Kind of Nonlinear Discrete Large-Scale Systems. Journal of Lanzhou University of Technology, (8), Vol.31, No.5, pp124-127.

SUN Shui-ling, PENG Ping. (2008). Connective Stability of Nonlinear Discrete Large-Scale Systems, 2008 Proceeding of Information Technology and Environmential System Sciences, part 1, pp1164-1169.

ZHANG Qing-lin. (1997). The Decentralized Control and Robust Descript Control of Singular Large-Scale Systems. Xi’an: The West-North Industry University Press, 11.

YANG Dong-mei, ZHANG Qing-ling. (2004). Singular System. Beijing: Science Press, 5.