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

Authors: Wei Zhu
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

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 inequality
techniques. By mathematical analysis, the stability criterion is less conservative
than that in previous result. Finally, one example is given to demonstrate
the applicability of the present scheme.


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## Body

## 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 literature ([1–15], and references therein). In general, those
approaches can be classified into two categories: the first is the polynomial
and the second 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. In [6], a
sufficient condition for the stability of discrete-time systems is given in
terms of pulse-response sequence matrix. In [11], 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 techniques [16], 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(2.1)x(k+1)=AIx(k)+BIx(k−p),k=0,1,2,…,x(k)=φ(k),k=−p,…,0,  where delay p is a positive
integer, x(k)=col{x1(k),x2(k),…xn(k)}∈Rn, AI, BI are the
interval matrices described as AI≜[A¯,A¯]={A=(aij):aij∈[a¯ij,a¯ij]}, BI≜[B¯,B¯]={B=(bij):bij∈[b¯ij,b¯ij]}, i,j=1,2,…,n,    φ(k),k=−p,…,0, are bounded.In the sequel, the following notations will be used:Rn(R+n): the space of n-dimensional (nonnegative) real column
vectors;
Rn×m(R+n×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 corresponding elements of A and B satisfies the
inequality  “≥(>),"  where A,B∈Rn×m or A,B∈Rn;
Wρ(A): for A∈R+n×n, Wρ(A)≜{z∈R+n∣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,b are 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 r column (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]).
IfA∈R+n×n, then there is a nonnegative vector z≥0,z≠0,  such that  Az=ρ(A)z.So it is clear
thatWρ(A) is not empty by
Lemma 3.1.Lemma 3.2.
LetP,Q∈R+n×n, u(k)∈R+n satisfy that (3.1)u(k+1)≤Pu(k)+Qu(k−p),k∈Z[0].  If(3.2)ρ(P+Q)<1,  then there
exists a constant λ>0 such that(3.3)u(k)≤ze−λk,k∈Z[0],  for some z=(z1,z2,…,zn)T∈Wρ(P+Qeλp).Proof.
Sinceρ(P+Q)<1, using continuity, there must be a sufficiently small
constant λ>0 such that(3.4)eλ(p+1)ρ(P+Qeλp)≤1.
Let(3.5)y(k)=u(k)eλk,k∈Z[0],y(k)=u(k),k∈Z[−p,0],   so we have(3.6)u(k)=y(k)e−λk≤y(k)e−λ(k−p),k∈Z[0].  By (3.1), we have(3.7)y(k+1)=u(k+1)eλ(k+1)≤(Pu(k)+Qu(k−p))eλ(k+1),k∈Z[0]. Since P,Q∈R+n×n, we derive that(3.8)y(k+1)≤(Py(k)+Qy(k−p)eλp)eλ(p+1),k∈Z[0].
We next show that for anyk∈Z[0],(3.9)y(k)≤z.  If this is not
true, then there must be a positive constant l>0 and some
integer m such that(3.10)ym(l+1)>zm,y(k)≤zfork∈Z[−p,l].  By using (3.4)
and (3.8), we obtain that(3.11)y(l+1)≤eλ(p+1)(P+Qeλp)z=eλ(p+1)ρ(P+Qeλp)z≤z which
contradicts the first inequality of (3.10). Thus (3.9) holds for all k∈Z[0]. Therefore, we have(3.12)u(k)≤ze−λk,k∈Z[0],  and the proof
is completed.Theorem 3.3.
For anyC∈𝒜≜⋃r=0n𝒜r, D∈B≜⋃r=0nBr, if the inequality(3.13)ρ([C]++[D]+)<1  holds, then the
discrete-time interval system (2.1) is asymptotically stable.Proof.
LetΓk={i∣xi(k)<0}≡{i1,i2,…,im}, Λk={j∣xj(k−p)<0}≡{j1,j2,…,jl} , where m,l satisfying 0≤m≤n,0≤l≤n,m+l=n are integers
and m=0 or l=0 is equivalent
to that Γk or Λk is empty, respectively.
Obviously, ⋃k=1∞Γk and ⋃k=1∞Λk are finite
sets.
By the definitions of𝒜 and B, we can obtain matrices A1k,A2k∈𝒜 by exchanging
the corresponding 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:(3.14)A1kx(k)≤Ax(k)≤A2kx(k),A∈AI,B1kx(k−p)≤Bx(k−p)≤B2kx(k−p),B∈BI.  So together with (2.1), we have(3.15)A1kx(k)+B1kx(k−p)≤x(k+1)≤A2kx(k)+B2kx(k−p).  From the above,
we see that A1k,A2k and B1k,B2k depend only on
the position of the negative components of x(k) and x(k−p), respectively.
Then, from (3.15), we derive(3.16)[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)]+}. So we have(3.17)[x(k+1)]+≤[A1k]+[x(k)]++[B1k]+[x(k−p)]+  or(3.18)[x(k+1)]+≤[A2k]+[x(k)]++[B2k]+[x(k−p)]+.
SinceA1k,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}, such that(3.19)[x(k+1)]+≤[Ai]+[x(k)]++[Bj]+[x(k−p)]+.  In view of
condition (3.13), we obtain that(3.20)ρ([Ai]++[Bj]+)<1,i,j∈{1,2,…,n}.  Thus, by
Lemma
3.2 and (3.19),
(3.20),  for any k∈Z[0], there exist constants λij>0 and some zij∈Wρ([Ai]++[Bj]+eλijp), i,j∈{1,2,…,n}, such that(3.21)x(k)≤zije−λijk.  Set λ=min⁡1≤i,j≤n{λij}, zij={zij(1),zij(2),…,zij(n)}T, z(h)=max⁡1≤i,j≤n{zij(h)}, 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), we derive that(3.22)x(k)≤ze−λk,k∈Z[0], which implies
that the conclusion of the  theorem holds.Remark 3.4.
By the meanings of𝒜r   and  Br, we know that 𝒜r=𝒜n−r,  Br=Bn−r, r∈Z[0,n]. So there are (∑r=0n2×(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=k is transformed
to be Ak=Ak (Bk=Bk) and then 𝒜k(Bk) contains only (2kk) different
matrices. Therefore, condition (3.13) can be verified easily and quickly by computer
software (such as MATLAB).Corollary 3.5 (see [11, Theorem IV]).
The discrete-time
interval system (2.1) is asymptotically stable if the following condition is
satisfied: (3.23)ρ(K+F)<1,  where matrices K and F are defined as K=(kij),⁢kij=max⁡{|a¯ij|,|a¯ij|}, F=(fij),⁢fij=max⁡{|b¯ij|,|b¯ij|}, i,j=1,2,…,n.Proof.
Clearly, for anyC∈𝒜=⋃i=1n𝒜i, D∈B=⋃i=1nBi, the inequality(3.24)[C]++[D]+≤K+F holds, then from [18] (i.e., for A,B∈R+n×n, if A≤B, then ρ(A)≤ρ(B)) and associated
with (3.23), we have(3.25)ρ([C]++[D]+)≤ρ(K+F)<1. 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(4.1)A¯=(−12−14014),A¯=(−1301412),B¯=(−14−16−14−18),B¯=(−1500110). For this case,  𝒜1={A1=(−1/3−1/41/41/4),A2=(−1/2001/2)},(4.2)𝒜2={A3=(−1301412),⁢A4=(−12−14014)},B1={B1=(−15−160−18),B2=(−140−14110)},B2={B3=(−1500110),⁢B4=(−14−16−14−18)}.    By simple calculation, we have ρ([Ai]++[Bj]+)≤ρ([A1]++[B4]+) for i,j=1,2,3,4,  and(4.3)[A1]++[B4]+=(13141414)+(14161418)=(7125121238). So we have(4.4)ρ([Ai]++[Bj]+)≤ρ([A1]++[B4]+)=0.9473<1,i,j=1,2,3,4.
Therefore, the system (2.1) is asymptotically stable by
means of Theorem 3.3.
In what follows, the simulation result is illustrated
in Figure1.Figure 1
Stability for discrete-time interval system.Remark 4.2.
If [11, Theorem IV] is applied to
Example 4.1, we obtain(4.5)K=(12141412),F=(14161418), where K, F are defined by
Corollary 3.5, that is,
[11, Theorem IV]. Then(4.6)ρ(K+F)=1.1482>1, that is,
[11, Theorem IV] cannot be applied. So the sufficient condition (3.13) proposed
in this paper 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 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 by [11]. So, the result of this
paper indeed allows us to have more freedom for checking 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.



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