A Sufficient Condition for Asymptotic Stability of Discrete-Time Interval System with Delay

Volume 2008, Article ID 591261,7pages doi:10.1155/2008/591261

Research Article

A Sufficient Condition for Asymptotic Stability of Discrete-Time Interval System with Delay

Wei Zhu

Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Correspondence should be addressed to Wei Zhu,zhu [email protected]

Received 4 September 2007; Revised 24 November 2007; Accepted 7 January 2008 Recommended by Manuel De La Sen

The asymptotic stability of discrete-time interval system with delay is discussed. A new sufficient condition for preserving the asymptotic stability of the system is presented by means of the inequal- ity techniques. By mathematical analysis, the stability criterion is less conservative than that in pre- vious result. Finally, one example is given to demonstrate the applicability of the present scheme.

Copyrightq2008 Wei Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The stability analysis of interval system is very useful for the robustness analysis of nominally stable system subject to model perturbations. Therefore, there has been considerable interest in the stability analysis of interval system in literature1–15, and references therein. In general, those approaches can be classified into two categories: the first is the polynomial and the sec- ond is the matrix approach. However, due to information transmission between elements or systems, data computation, natural property of system elements, and so forth, time delays also inherently exist in controlled systems and therefore must be integrated into system models.

The stability analysis for interval systems with delays becomes more complicated. In6, a suf- ficient condition for the stability of discrete-time systems is given in terms of pulse-response sequence matrix. In11, based on the Gersgorin theorem, the stability testing problem for continuous and discrete systems including a time delay is discussed.

The objective of this paper is to deal with the asymptotic stability of a discrete-time interval system with delay. Based on the inequality techniques16, a new sufficient condition for preserving the asymptotic stability of the system is presented. By mathematical analysis, the stability criterion is less conservative than that in previous result. An example is given to compare the proposed method with one reported.

2. System description and notations

Consider the discrete-time interval system with delay described by xk1 A_Ixk B_Ixk−p, k0,1,2, . . . ,

xk ϕk, k−p, . . . ,0, 2.1

where delay p is a positive integer, xk col{x1k, x2k, . . . xnk} ∈ Rⁿ,A_I,B_I are the interval matrices described asAI A, A {A aij:aij ∈a_ij, aij},BI B, B {B bij:bij ∈b_ij, bij},i, j1,2, . . . , n,ϕk, k−p, . . . ,0,are bounded.

In the sequel, the following notations will be used:RⁿRⁿ: the space ofn-dimensional nonnegativereal column vectors;R^n×mR^n×m : the set ofn×mnonnegativereal matrices;

ρA: the spectral radius of matrix A ∈ R^n×n;A ≥ BA > B: each pair of corresponding elements ofAandBsatisfies the inequality “≥>,” whereA, B∈R^n×morA, B ∈Rⁿ;WρA:

forA ∈ R^n×n ,W_ρA {z ∈ Rⁿ | Az ρAz};·: the vector or matrix obtained by replacing each entry of· by its absolute value;Z^a,b:Z^a,b ≡ {a, a1, . . . , b}, wherea, b are nonnegative integers; ifa > b, we defineZ^a,b ∅, where∅is the empty set; ifb ∞, we writeZ^a,b asZ^a;ArBr: the set of matrices obtained by exchanging correspondingr columnsofABandAB, so there are 2×ⁿ_rmatrices in eachArBr, wherer ∈Z^0,n, ⁿr n!/r!n−r! andn! denotes the factorial ofn.

3. Main result

In order to prove our main result, we first need the following technical lemmas.

Lemma 3.1see17, Theorem 8.3.1. IfA∈R^n×n , then there is a nonnegative vectorz≥0, z /0, such thatAzρAz.

So it is clear thatWρAis not empty byLemma 3.1.

Lemma 3.2. LetP, Q∈R^n×n ,uk∈Rⁿsatisfy that

uk1≤P uk Quk−p, k∈Z⁰. 3.1

If

ρPQ<1, 3.2

then there exists a constantλ >0 such that

uk≤ze^−λk, k∈Z⁰, 3.3

for somez z1, z₂, . . . , z_n^T ∈W_ρPQe^λp.

Proof. SinceρP Q < 1, using continuity, there must be a sufficiently small constantλ > 0 such that

e^λp1ρ

PQe^λp

≤1. 3.4

Let

yk uke^λk, k∈Z⁰,

yk uk, k∈Z^−p,0, 3.5

so we have

uk yke^−λk ≤yke^−λk−p, k∈Z⁰. 3.6

By3.1, we have

yk1 uk1e^λk1≤

P uk Quk−p

e^λk1, k∈Z⁰. 3.7

SinceP, Q∈R^n×n , we derive that yk1≤

P yk Qyk−pe^λp

e^λp1, k∈Z⁰. 3.8

We next show that for anyk∈Z⁰,

yk≤z. 3.9

If this is not true, then there must be a positive constantl >0 and some integermsuch that y_ml1> z_m, yk≤z fork∈Z^−p,l. 3.10 By using3.4and3.8, we obtain that

yl1≤e^λp1

PQe^λp

ze^λp1ρ

PQe^λp

z≤z 3.11

which contradicts the first inequality of3.10. Thus3.9holds for allk ∈Z⁰. Therefore, we have

uk≤ze^−λk, k∈Z⁰, 3.12

and the proof is completed.

Theorem 3.3. For anyC∈ A_n

r0Ar,D∈ B_n

r0Br, if the inequality ρ

C D

<1 3.13

holds, then the discrete-time interval system2.1is asymptotically stable.

Proof. LetΓk{i |xik<0} ≡ {i1, i2, . . . , im},Λk{j |xjk−p<0} ≡ {j1, j2, . . . , jl}, where m, lsatisfying 0≤m≤n, 0≤l≤n, mlnare integers andm0 orl0 is equivalent to thatΓkorΛkis empty, respectively. Obviously,_∞

k1Γkand_∞

k1Λkare finite sets.

By the definitions ofAandB, we can obtain matricesA1k, A2k ∈ Aby exchanging the correspondingi1th, i2th, . . . , imth columns ofAandAifm 0, thenA1k A, A2k A, and

matricesB1k, B2k ∈ Bby exchanging the correspondingj1th, j2th, . . . , jlth columns ofBandB ifl0, thenB1kB, B2kBsuch that the following inequalities hold:

A1kxk≤Axk≤A2kxk, A∈AI,

B_1kxk−p≤Bxk−p≤B_2kxk−p, B∈B_I. 3.14

So together with2.1, we have

A1kxk B1kxk−p≤xk1≤A2kxk B2kxk−p. 3.15 From the above, we see thatA_1k, A_2kandB_1k, B_2kdepend only on the position of the negative components ofxkandxk−p, respectively.

Then, from3.15, we derive xk1≤max

A_1kxk B_1kxk−p

,

A_2kxk B_2kxk−p

≤max A1k

xk

B1k

xk−p ,

A2k

xk

B2k

xk−p . 3.16 So we have

xk1≤ A_1k

xk B_1k

xk−p 3.17

or

xk1≤ A_2k

xk B_2k

xk−p

. 3.18

SinceA1k, A2k ∈ A,B1k, B2k ∈ B, by the definitions ofAandBagain,3.17and3.18, for anyk ∈Z⁰, we can find corresponding matricesA_i ∈ A, Bj ∈ B,i, j ∈ {1,2, . . . , n}, such that

xk1≤ A_i

xk B_j

xk−p

. 3.19

In view of condition3.13, we obtain that ρ

A_i B_j

<1, i, j∈ {1,2, . . . , n}. 3.20 Thus, byLemma 3.2and3.19,3.20, for anyk∈Z⁰, there exist constantsλ_ij >0 and some z_ij∈W_ρAi Bje^λijp,i, j∈ {1,2, . . . , n}, such that

xk≤z_ije^−λijk. 3.21

Setλ min_1≤i,j≤n{λij},z_ij {z¹_ij , z²_ij , . . . , zⁿ_ij }^T,z^h max_1≤i,j≤n{z^h_ij },h 1,2, . . . , n,z {z¹, z², . . . , zⁿ}^T, obviously, λandzare independent of any choice ofk, so by3.21, we derive that

xk≤ze^−λk, k∈Z⁰, 3.22

which implies that the conclusion of the theorem holds.

Remark 3.4. By the meanings ofAr andBr, we know thatAr An−r,Br Bn−r,r ∈ Z^0,n. So there are ⁿ_r02×ⁿr/22ⁿ matrices in the setsA andB, respectively. Furthermore, if numbernis even, that is,n2k, k1,2, . . ., then the equalityA_r A_n−rBr B_n−rforrk is transformed to beA_kA_kBkB_kand thenAkBkcontains only^2k_kdifferent matrices.

Therefore, condition3.13can be verified easily and quickly by computer softwaresuch as MATLAB.

Corollary 3.5see11, Theorem IV. The discrete-time interval system2.1is asymptotically stable if the following condition is satisfied:

ρKF<1, 3.23

where matrices K and F are defined as K kij, kij max{|a_ij|,|aij|}, F fij, fij max{|b_ij|,|bij|},i, j1,2, . . . , n.

Proof. Clearly, for anyC∈ A_n

i1Ai,D∈ B_n

i1Bi, the inequality

C D≤KF 3.24 holds, then from18 i.e., forA, B ∈ R^n×n , ifA ≤ B, thenρA ≤ρBand associated with 3.23, we have

ρ

C D

≤ρKF<1. 3.25

Therefore, system2.1is asymptotically stable in terms ofTheorem 3.3.

4. Illustrative examples

Example 4.1. Consider the discrete-time interval system2.1with delay and

A

⎛

⎜⎜

⎝

−1 2 −1

4

0 1

4

⎞

⎟⎟

⎠, A

⎛

⎜⎜

⎝

−1 3 0 1 4

1 2

⎞

⎟⎟

⎠, B

⎛

⎜⎜

⎝

−1 4 −1

6

−1 4 −1

8

⎞

⎟⎟

⎠, B

⎛

⎜⎜

⎝

−1 5 0

0 1

10

⎞

⎟⎟

⎠. 4.1 For this case,A1

A1_−1/3−1/4

1/4 1/4

, A2_{−1/2 0}

0 1/2

,

A2

⎧⎪

⎪⎨

⎪⎪

⎩A3

⎛

⎜⎜

⎝

−1 3 0 1 4

1 2

⎞

⎟⎟

⎠, A4

⎛

⎜⎜

⎝

−1 2 −1

4

0 1

4

⎞

⎟⎟

⎠

⎫⎪

⎪⎬

⎪⎪

⎭, B1

⎧⎪

⎪⎨

⎪⎪

⎩B₁

⎛

⎜⎜

⎝

−1 5 −1

6 0 −1 8

⎞

⎟⎟

⎠, B₂

⎛

⎜⎜

⎝

−1 4 0

−1 4

1 10

⎞

⎟⎟

⎠

⎫⎪

⎪⎬

⎪⎪

⎭, B2

⎧⎪

⎪⎨

⎪⎪

⎩B3

⎛

⎜⎝−1 5 0

0 1

10

⎞

⎟⎠, B4

⎛

⎜⎜

⎝

−1 4 −1

6

−1 4 −1

8

⎞

⎟⎟

⎠

⎫⎪

⎪⎬

⎪⎪

⎭.

4.2

0 5 10 15 20 k

−6

−4

−2 0 2 4 6

x1,x2

x1

x2

Figure 1: Stability for discrete-time interval system.

By simple calculation, we haveρAi Bj≤ρA1 B4 for i, j1,2,3,4,and

A1

B4

⎛

⎜⎜

⎝ 1 3

1 4 1 4

1 4

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 1 4

1 6 1 4

1 8

⎞

⎟⎟

⎠

⎛

⎜⎜

⎝ 7 12

5 12 1 2

3 8

⎞

⎟⎟

⎠. 4.3

So we have ρ

Ai Bj

≤ρ

A1 B4

0.9473<1, i, j1,2,3,4. 4.4 Therefore, the system2.1is asymptotically stable by means ofTheorem 3.3.

In what follows, the simulation result is illustrated inFigure 1.

Remark 4.2. If11, Theorem IVis applied toExample 4.1, we obtain

K

⎛

⎜⎜

⎝ 1 2

1 4 1 4

1 2

⎞

⎟⎟

⎠, F

⎛

⎜⎜

⎝ 1 4

1 6 1 4

1 8

⎞

⎟⎟

⎠, 4.5

whereK,Fare defined byCorollary 3.5, that is,11, Theorem IV. Then

ρKF 1.1482>1, 4.6

that is,11, Theorem IVcannot be applied. So the sufficient condition3.13proposed in this paper is less conservative than condition3.23proposed by11.

5. Conclusion

In this paper, we have investigated the asymptotic stability of discrete-time interval system with delay. A new sufficient condition for preserving the asymptotic stability of the system is

developed. By mathematical analysis, the presented criterion is to be less conservative than that proposed by11. So, the result of this paper indeed allows us to have more freedom for check- ing the stability of the discrete-time interval systems with delay. From the proposed example, it is easily seen that the criterion presented in this paper for the stability of the discrete-time interval system with delay is very helpful. We believe that the present scheme is applicable to robust control design.

Acknowledgments

The author wishes to thank the editor and the referees for their helpful and interesting com- ments. The work is supported by National Natural Science Foundation of China under Grant 10671133 and the Doctor’s Foundation of Chongqing University of Posts and Telecommunica- tions A2007-41.

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