a sufficient condition for asymptotic stability of discrete …the stability analysis for interval...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2008, Article ID 591261, 7 pagesdoi:10.1155/2008/591261

Research ArticleA Sufficient Condition for Asymptotic Stability ofDiscrete-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 sufficientcondition 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.

Copyright q 2008 Wei Zhu. This is an open access article distributed under the Creative CommonsAttribution 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 nominallystable system subject to model perturbations. Therefore, there has been considerable interest inthe stability analysis of interval system in literature ([1–15], 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 orsystems, data computation, natural property of system elements, and so forth, time delays alsoinherently exist in controlled systems and therefore must be integrated into system models.The stability analysis for interval systems with delays becomes more complicated. In [6], a suf-ficient condition for the stability of discrete-time systems is given in terms of pulse-responsesequence matrix. In [11], based on the Gersgorin theorem, the stability testing problem forcontinuous and discrete systems including a time delay is discussed.

The objective of this paper is to deal with the asymptotic stability of a discrete-timeinterval system with delay. Based on the inequality techniques [16], a new sufficient conditionfor 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 tocompare the proposed method with one reported.

mailto:[email protected]

2 Discrete Dynamics in Nature and Society

2. System description and notations

Consider the discrete-time interval system with delay described by

x(k + 1) = AIx(k) + BIx(k − p), k = 0, 1, 2, . . . ,

x(k) = ϕ(k), k = −p, . . . , 0,(2.1)

where delay p is a positive integer, x(k) = col{x1(k), x2(k), . . . xn(k)} ∈ Rn, AI , BI are theinterval matrices described as AI � [A,A] = {A = (aij) : aij ∈ [aij , aij]}, BI � [B, B] = {B =

(bij) : bij ∈ [bij , bij]}, i, j = 1, 2, . . . , n, ϕ(k), k = −p, . . . , 0, are bounded.In the sequel, the following notations will be used: Rn(Rn

+): the space of n-dimensional(nonnegative) real column vectors; Rn×m(Rn×m

+ ): the set of n × m (nonnegative) real matrices;ρ(A): the spectral radius of matrix A ∈ Rn×n; A ≥ B(A > B): each pair of correspondingelements of A and B satisfies the inequality “≥ (>),” where A,B ∈ Rn×m or A,B ∈ Rn; Wρ(A):for A ∈ Rn×n

+ , Wρ(A) � {z ∈ Rn+ | Az = ρ(A)z}; [·]+: the vector (or matrix) obtained by

replacing each entry of [·] by its absolute value; Z[a,b]: Z[a,b] ≡ {a, a + 1, . . . , b}, where a, bare nonnegative integers; if a > b, we define Z[a,b] = ∅, where ∅ is the empty set; if b = ∞,we write Z[a,b] as Z[a]; Ar(Br): the set of matrices obtained by exchanging corresponding rcolumn (s) of A (B) and A (B), so there are 2 × ( nr ) matrices in each Ar(Br), where r ∈ Z[0,n],( nr ) = n!/r!(n − r)! and n! denotes the factorial of n.

3. Main result

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

Lemma 3.1 (see [17, Theorem 8.3.1]). If A ∈ Rn×n+ , then there is a nonnegative vector z ≥ 0, z /= 0,

such that Az = ρ(A)z.

So it is clear that Wρ(A) is not empty by Lemma 3.1.

Lemma 3.2. Let P,Q ∈ Rn×n+ , u(k) ∈ Rn

+ satisfy that

u(k + 1) ≤ Pu(k) +Qu(k − p), k ∈ Z[0]. (3.1)

If

ρ(P +Q) < 1, (3.2)

then there exists a constant λ > 0 such that

u(k) ≤ ze−λk, k ∈ Z[0], (3.3)

for some z = (z1, z2, . . . , zn)T ∈Wρ(P +Qeλp).

Proof. Since ρ(P + Q) < 1, using continuity, there must be a sufficiently small constant λ > 0such that

eλ(p+1)ρ(P +Qeλp

)≤ 1. (3.4)

Wei Zhu 3

Let

y(k) = u(k)eλk, k ∈ Z[0],

y(k) = u(k), k ∈ Z[−p,0],(3.5)

so we have

u(k) = y(k)e−λk ≤ y(k)e−λ(k−p), k ∈ Z[0]. (3.6)

By (3.1), we have

y(k + 1) = u(k + 1)eλ(k+1) ≤(Pu(k) +Qu(k − p)

)eλ(k+1), k ∈ Z[0]. (3.7)

Since P,Q ∈ Rn×n+ , we derive that

y(k + 1) ≤(Py(k) +Qy(k − p)eλp

)eλ(p+1), k ∈ Z[0]. (3.8)

We next show that for any k ∈ Z[0],

y(k) ≤ z. (3.9)

If this is not true, then there must be a positive constant l > 0 and some integer m such that

ym(l + 1) > zm, y(k) ≤ z for k ∈ Z[−p,l]. (3.10)

By using (3.4) and (3.8), we obtain that

y(l + 1) ≤ eλ(p+1)(P +Qeλp)z = eλ(p+1)ρ

(P +Qeλp

)z ≤ z (3.11)

which contradicts the first inequality of (3.10). Thus (3.9) holds for all k ∈ Z[0]. Therefore, wehave

u(k) ≤ ze−λk, k ∈ Z[0], (3.12)

and the proof is completed.

Theorem 3.3. For any C ∈ A � ⋃nr=0Ar , D ∈ B � ⋃n

r=0Br , if the inequality

ρ([C]+ + [D]+

)< 1 (3.13)

holds, then the discrete-time interval system (2.1) is asymptotically stable.

Proof. Let Γk = {i | xi(k) < 0} ≡ {i1, i2, . . . , im}, Λk = {j | xj(k − p) < 0} ≡ {j1, j2, . . . , jl} , wherem, l satisfying 0 ≤ m ≤ n, 0 ≤ l ≤ n, m + l = n are integers and m = 0 or l = 0 is equivalent tothat Γk or Λk is empty, respectively. Obviously,

⋃∞k=1Γk and

⋃∞k=1Λk are finite sets.

By the definitions of A and B, we can obtain matrices A1k,A2k ∈ A by exchanging thecorresponding i1th, i2th, . . . , imth columns of A and A (if m = 0, then A1k = A,A2k = A), and


matrices B1k, B2k ∈ B by exchanging the corresponding j1th, j2th, . . . , jlth columns of B and B(if l = 0, then B1k = B, B2k = B) such that the following inequalities hold:

A1kx(k) ≤ Ax(k) ≤ A2kx(k), A ∈ AI,

B1kx(k − p) ≤ Bx(k − p) ≤ B2kx(k − p), B ∈ BI.(3.14)

So together with (2.1), we have

A1kx(k) + B1kx(k − p) ≤ x(k + 1) ≤ A2kx(k) + B2kx(k − p). (3.15)

From the above, we see that A1k,A2k and B1k, B2k depend only on the position of the negativecomponents of x(k) and x(k − p), respectively.

Then, from (3.15), we derive

[x(k + 1)

]+ ≤ max{[A1kx(k) + B1kx(k − p)

]+,[A2kx(k) + B2kx(k − p)

]+}

≤ max{[A1k

]+[x(k)

]+ +[B1k

]+[x(k − p)

]+,[A2k

]+[x(k)

]+ +[B2k

]+[x(k − p)

]+}.

(3.16)

So we have

[x(k + 1)

]+ ≤[A1k

]+[x(k)

]+ +[B1k

]+[x(k − p)

]+ (3.17)

or

[x(k + 1)

]+ ≤[A2k

]+[x(k)

]+ +[B2k

]+[x(k − p)

]+. (3.18)

Since A1k,A2k ∈ A, B1k, B2k ∈ B, by the definitions of A and B again, (3.17) and (3.18),for any k ∈ Z[0], we can find corresponding matrices Ai ∈ A, Bj ∈ B, i, j ∈ {1, 2, . . . , n}, suchthat

[x(k + 1)

]+ ≤[Ai

]+[x(k)

]+ +[Bj]+[

x(k − p)]+. (3.19)

In view of condition (3.13), we obtain that

ρ([Ai

]+ +[Bj]+)

< 1, i, j ∈ {1, 2, . . . , n}. (3.20)

Thus, by Lemma 3.2 and (3.19), (3.20), for any k ∈ Z[0], there exist constants λij > 0 and somezij ∈Wρ([Ai]

+ + [Bj]+eλijp), i, j ∈ {1, 2, . . . , n}, such that

x(k) ≤ zije−λijk. (3.21)

Set λ = min1≤i,j≤n{λij}, zij = {z(1)ij , z(2)ij , . . . , z

(n)ij }

T , z(h) = max1≤i,j≤n{z(h)ij }, h = 1, 2, . . . , n, z =

{z(1), z(2), . . . , z(n)}T , obviously, λ and z are independent of any choice of k, so by (3.21), wederive that

x(k) ≤ ze−λk, k ∈ Z[0], (3.22)

which implies that the conclusion of the theorem holds.

Wei Zhu 5

Remark 3.4. By the meanings of Ar and Br , we know that Ar = An−r , Br = Bn−r , r ∈ Z[0,n].So there are (

∑nr=02 × ( nr ))/2= 2n matrices in the sets A and B, respectively. Furthermore, if

number n is even, that is, n = 2k, k = 1, 2, . . ., then the equality Ar = An−r (Br = Bn−r) for r = kis transformed to be Ak = Ak (Bk = Bk) and thenAk (Bk) contains only ( 2k

k ) different matrices.Therefore, condition (3.13) can be verified easily and quickly by computer software (such asMATLAB).

Corollary 3.5 (see [11, Theorem IV]). The discrete-time interval system (2.1) is asymptotically stableif the following condition is satisfied:

ρ(K + F) < 1, (3.23)

where matrices K and F are defined as K = (kij), kij = max{|aij |, |aij |}, F = (fij), fij =

max{|bij |, |bij |}, i, j = 1, 2, . . . , n.

Proof. Clearly, for any C ∈ A =⋃ni=1Ai, D ∈ B =

⋃ni=1Bi, the inequality

[C]+ + [D]+ ≤ K + F (3.24)

holds, then from [18] (i.e., for A,B ∈ Rn×n+ , if A ≤ B, then ρ(A) ≤ ρ(B)) and associated with

(3.23), we have

ρ([C]+ + [D]+

)≤ ρ(K + F) < 1. (3.25)

Therefore, system (2.1) is asymptotically stable in terms of Theorem 3.3.

4. Illustrative examples

Example 4.1. Consider the discrete-time interval system (2.1) with delay and

A =

⎛

⎜⎜⎝

−12−1

4

014

⎞

⎟⎟⎠ , A =

⎛

⎜⎜⎝

−13

0

14

12

⎞

⎟⎟⎠ , B =

⎛

⎜⎜⎝

−14−1

6

−14−1

8

⎞

⎟⎟⎠ , B =

⎛

⎜⎜⎝

−15

0

0110

⎞

⎟⎟⎠ . (4.1)

For this case,A1 ={A1 =

( −1/3 −1/41/4 1/4

), A2 =

( −1/2 00 1/2

)},

A2 =

⎧⎪⎪⎨

⎪⎪⎩A3 =

⎛

⎜⎜⎝

−13

0

14

12

⎞

⎟⎟⎠ , A4 =

⎛

⎜⎜⎝

−12−1

4

014

⎞

⎟⎟⎠

⎫⎪⎪⎬

⎪⎪⎭,

B1 =

⎧⎪⎪⎨

⎪⎪⎩B1 =

⎛

⎜⎜⎝

−15−1

6

0 −18

⎞

⎟⎟⎠ , B2 =

⎛

⎜⎜⎝

−14

0

−14

110

⎞

⎟⎟⎠

⎫⎪⎪⎬

⎪⎪⎭,

B2 =

⎧⎪⎪⎨

⎪⎪⎩B3 =

⎛

⎜⎝−1

50

0110

⎞

⎟⎠ , B4 =

⎛

⎜⎜⎝

−14−1

6

−14−1

8

⎞

⎟⎟⎠

⎫⎪⎪⎬

⎪⎪⎭.

(4.2)


0 5 10 15 20k

−6

−4

−2

0

2

4

6

x1,x

2

x1x2

Figure 1: Stability for discrete-time interval system.

By simple calculation, we have ρ([Ai]+ + [Bj]

+) ≤ ρ([A1]+ + [B4]

+) for i, j = 1, 2, 3, 4, and

[A1

]+ +[B4

]+ =

⎛

⎜⎜⎝

13

14

14

14

⎞

⎟⎟⎠ +

⎛

⎜⎜⎝

14

16

14

18

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

712

512

12

38

⎞

⎟⎟⎠ . (4.3)

So we have

ρ([Ai

]+ +[Bj]+) ≤ ρ

([A1

]+ +[B4

]+) = 0.9473 < 1, i, j = 1, 2, 3, 4. (4.4)

Therefore, the system (2.1) is asymptotically stable by means of Theorem 3.3.In what follows, the simulation result is illustrated in Figure 1.

Remark 4.2. If [11, Theorem IV] is applied to Example 4.1, we obtain

K =

⎛

⎜⎜⎝

12

14

14

12

⎞

⎟⎟⎠ , F =

⎛

⎜⎜⎝

14

16

14

18

⎞

⎟⎟⎠ , (4.5)

where K, F are defined by Corollary 3.5, that is, [11, Theorem IV]. Then

ρ(K + F) = 1.1482 > 1, (4.6)

that is, [11, Theorem IV] cannot be applied. So the sufficient condition (3.13) proposed in thispaper is less conservative than condition (3.23) proposed by [11].

5. Conclusion

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

Wei Zhu 7

developed. By mathematical analysis, the presented criterion is to be less conservative than thatproposed by [11]. 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-timeinterval system with delay is very helpful. We believe that the present scheme is applicable torobust 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 Grant10671133 and the Doctor’s Foundation of Chongqing University of Posts and Telecommunica-tions A2007-41.

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