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

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.

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 by𝑥(𝑘+1)=𝐴_𝐼𝑥(𝑘)+𝐵_𝐼𝑥(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),…𝑥_𝑛(𝑘)}∈𝑅^𝑛, 𝐴_𝐼, 𝐵_𝐼 are the interval matrices described as 𝐴_𝐼≜[𝐴,𝐴]={𝐴=(𝑎_𝑖𝑗)∶𝑎_𝑖𝑗∈[𝑎_𝑖𝑗,𝑎_𝑖𝑗]}, 𝐵_𝐼≜[𝐵,𝐵]={𝐵=(𝑏_𝑖𝑗)∶𝑏_𝑖𝑗∈[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: 𝑅^𝑛(𝑅^𝑛₊): the space of 𝑛-dimensional (nonnegative) real column vectors; 𝑅^𝑛×𝑚(𝑅₊^𝑛×𝑚): the set of 𝑛×𝑚 (nonnegative) real matrices; 𝜌(𝐴): the spectral radius of matrix 𝐴∈𝑅^𝑛×𝑛; 𝐴≥𝐵(𝐴>𝐵): each pair of corresponding elements of 𝐴 and 𝐵 satisfies the inequality “≥(>)," where 𝐴,𝐵∈𝑅^𝑛×𝑚 or 𝐴,𝐵∈𝑅^𝑛; 𝑊_𝜌(𝐴): for 𝐴∈𝑅₊^𝑛×𝑛, 𝑊_𝜌(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]; 𝒜_𝑟(𝐵_𝑟): the set of matrices obtained by exchanging corresponding 𝑟 column (𝑠) of 𝐴 (𝐵) and 𝐴 (𝐵), so there are 2×(^𝑛_𝑟) matrices in each 𝒜_𝑟(𝐵_𝑟), where 𝑟∈𝑍^[0,𝑛], (^𝑛_𝑟)=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∈𝑅₊^𝑛×𝑛, then there is a nonnegative vector 𝑧≥0,𝑧≠0, such that 𝐴𝑧=𝜌(𝐴)𝑧.

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

Lemma 3.2. Let 𝑃,𝑄∈𝑅₊^𝑛×𝑛, 𝑢(𝑘)∈𝑅^𝑛₊ satisfy that 𝑢(𝑘+1)≤𝑃𝑢(𝑘)+𝑄𝑢(𝑘−𝑝),𝑘∈𝑍^[0]. (3.1) If𝜌(𝑃+𝑄)<1, (3.2) then there exists a constant 𝜆>0 such that𝑢(𝑘)≤𝑧𝑒^−𝜆𝑘,𝑘∈𝑍^[0], (3.3) for some 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛)^𝑇∈𝑊_𝜌(𝑃+𝑄𝑒^𝜆𝑝).

Proof. Since 𝜌(𝑃+𝑄)<1, using continuity, there must be a sufficiently small constant 𝜆>0 such that𝑒^𝜆(𝑝+1)𝜌(𝑃+𝑄𝑒^𝜆𝑝)≤1. (3.4) Let𝑦(𝑘)=𝑢(𝑘)𝑒^𝜆𝑘,𝑘∈𝑍^[0],𝑦(𝑘)=𝑢(𝑘),𝑘∈𝑍^[−𝑝,0], (3.5) so we have𝑢(𝑘)=𝑦(𝑘)𝑒^−𝜆𝑘≤𝑦(𝑘)𝑒^{−𝜆(𝑘−𝑝)},𝑘∈𝑍^[0]. (3.6) By (3.1), we have𝑦(𝑘+1)=𝑢(𝑘+1)𝑒^𝜆(𝑘+1)≤(𝑃𝑢(𝑘)+𝑄𝑢(𝑘−𝑝))𝑒^𝜆(𝑘+1),𝑘∈𝑍^[0]. (3.7) Since 𝑃,𝑄∈𝑅₊^𝑛×𝑛, we derive that𝑦(𝑘+1)≤(𝑃𝑦(𝑘)+𝑄𝑦(𝑘−𝑝)𝑒^𝜆𝑝)𝑒^𝜆(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 that𝑦_𝑚(𝑙+1)>𝑧_𝑚,𝑦(𝑘)≤𝑧for𝑘∈𝑍^{[−𝑝,𝑙]}. (3.10) By using (3.4) and (3.8), we obtain that𝑦(𝑙+1)≤𝑒^𝜆(𝑝+1)(𝑃+𝑄𝑒^𝜆𝑝)𝑧=𝑒^𝜆(𝑝+1)𝜌(𝑃+𝑄𝑒^𝜆𝑝)𝑧≤𝑧 (3.11) which contradicts the first inequality of (3.10). Thus (3.9) holds for all 𝑘∈𝑍^[0]. Therefore, we have𝑢(𝑘)≤𝑧𝑒^−𝜆𝑘,𝑘∈𝑍^[0], (3.12) and the proof is completed.

Theorem 3.3. For any 𝐶∈𝒜≜⋃^𝑛_𝑟=0𝒜_𝑟, 𝐷∈𝐵≜⋃^𝑛_𝑟=0𝐵_𝑟, if the inequality𝜌([𝐶]⁺+[𝐷]⁺)<1 (3.13) holds, then the discrete-time interval system (2.1) is asymptotically stable.

Proof. Let Γ_𝑘={𝑖∣𝑥_𝑖(𝑘)<0}≡{𝑖₁,𝑖₂,…,𝑖_𝑚}, Λ_𝑘={𝑗∣𝑥_𝑗(𝑘−𝑝)<0}≡{𝑗₁,𝑗₂,…,𝑗_𝑙} , 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 Γ_𝑘 or Λ_𝑘 is empty, respectively. Obviously, ⋃^∞_𝑘=1Γ_𝑘 and ⋃^∞_𝑘=1Λ_𝑘 are finite sets.
By the definitions of 𝒜 and 𝐵, we can obtain matrices 𝐴_1𝑘,𝐴_2𝑘∈𝒜 by exchanging the corresponding 𝑖₁th,𝑖₂th,…,𝑖_𝑚th columns of 𝐴 and 𝐴 (if 𝑚=0, then 𝐴_1𝑘=𝐴,𝐴_2𝑘=𝐴), and matrices 𝐵_1𝑘,𝐵_2𝑘∈𝐵 by exchanging the corresponding 𝑗₁th,𝑗₂th,…,𝑗_𝑙th columns of 𝐵 and 𝐵 (if 𝑙=0, then 𝐵_1𝑘=𝐵,𝐵_2𝑘=𝐵) such that the following inequalities hold:𝐴_1𝑘𝑥(𝑘)≤𝐴𝑥(𝑘)≤𝐴_2𝑘𝑥(𝑘),𝐴∈𝐴_𝐼,𝐵_1𝑘𝑥(𝑘−𝑝)≤𝐵𝑥(𝑘−𝑝)≤𝐵_2𝑘𝑥(𝑘−𝑝),𝐵∈𝐵_𝐼. (3.14) So together with (2.1), we have𝐴_1𝑘𝑥(𝑘)+𝐵_1𝑘𝑥(𝑘−𝑝)≤𝑥(𝑘+1)≤𝐴_2𝑘𝑥(𝑘)+𝐵_2𝑘𝑥(𝑘−𝑝). (3.15) From the above, we see that 𝐴_1𝑘,𝐴_2𝑘 and 𝐵_1𝑘,𝐵_2𝑘 depend only on the position of the negative components of 𝑥(𝑘) and 𝑥(𝑘−𝑝), respectively.
Then, from (3.15), we derive[𝑥(𝑘+1)]⁺≤max{[𝐴_1𝑘𝑥(𝑘)+𝐵_1𝑘𝑥(𝑘−𝑝)]⁺,[𝐴_2𝑘𝑥(𝑘)+𝐵_2𝑘𝑥(𝑘−𝑝)]⁺}≤max{[𝐴_1𝑘]⁺[𝑥(𝑘)]⁺+[𝐵_1𝑘]⁺[𝑥(𝑘−𝑝)]⁺,[𝐴_2𝑘]⁺[𝑥(𝑘)]⁺+[𝐵_2𝑘]⁺[𝑥(𝑘−𝑝)]⁺}. (3.16) So we have[𝑥(𝑘+1)]⁺≤[𝐴_1𝑘]⁺[𝑥(𝑘)]⁺+[𝐵_1𝑘]⁺[𝑥(𝑘−𝑝)]⁺ (3.17) or[𝑥(𝑘+1)]⁺≤[𝐴_2𝑘]⁺[𝑥(𝑘)]⁺+[𝐵_2𝑘]⁺[𝑥(𝑘−𝑝)]⁺. (3.18)
Since 𝐴_1𝑘,𝐴_2𝑘∈𝒜, 𝐵_1𝑘,𝐵_2𝑘∈𝐵, by the definitions of 𝒜 and 𝐵 again, (3.17) and (3.18), for any 𝑘∈𝑍^[0], we can find corresponding matrices 𝐴_𝑖∈𝒜,𝐵_𝑗∈𝐵, 𝑖,𝑗∈{1,2,…,𝑛}, such that[𝑥(𝑘+1)]⁺≤[𝐴_𝑖]⁺[𝑥(𝑘)]⁺+[𝐵_𝑗]⁺[𝑥(𝑘−𝑝)]⁺. (3.19) In view of condition (3.13), we obtain that𝜌([𝐴_𝑖]⁺+[𝐵_𝑗]⁺)<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 𝜆_𝑖𝑗>0 and some 𝑧_𝑖𝑗∈𝑊_𝜌([𝐴_𝑖]⁺+[𝐵_𝑗]⁺𝑒^{𝜆𝑖𝑗𝑝}), 𝑖,𝑗∈{1,2,…,𝑛}, such that𝑥(𝑘)≤𝑧_𝑖𝑗𝑒^{−𝜆𝑖𝑗𝑘}. (3.21) Set 𝜆=min_{1≤𝑖,𝑗≤𝑛}{𝜆_𝑖𝑗}, 𝑧_𝑖𝑗={𝑧⁽¹⁾_𝑖𝑗,𝑧⁽²⁾_𝑖𝑗,…,𝑧^(𝑛)_𝑖𝑗}^𝑇, 𝑧^(ℎ)=max_{1≤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), we derive that𝑥(𝑘)≤𝑧𝑒^−𝜆𝑘,𝑘∈𝑍^[0], (3.22) which implies that the conclusion of the theorem holds.

Remark 3.4. By the meanings of 𝒜_𝑟 and 𝐵_𝑟, we know that 𝒜_𝑟=𝒜_𝑛−𝑟, 𝐵_𝑟=𝐵_𝑛−𝑟, 𝑟∈𝑍^[0,𝑛]. So there are (∑^𝑛_𝑟=02×(^𝑛_𝑟))/2=2^𝑛 matrices in the sets 𝒜 and 𝐵, respectively. Furthermore, if number 𝑛 is even, that is, 𝑛=2𝑘,𝑘=1,2,…, then the equality 𝐴_𝑟=𝐴_𝑛−𝑟 (𝐵_𝑟=𝐵_𝑛−𝑟) for 𝑟=𝑘 is transformed to be 𝐴_𝑘=𝐴_𝑘 (𝐵_𝑘=𝐵_𝑘) and then 𝒜_𝑘(𝐵_𝑘) contains only (_𝑘^2𝑘) 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: 𝜌(𝐾+𝐹)<1, (3.23) where matrices 𝐾 and 𝐹 are defined as 𝐾=(𝑘_𝑖𝑗),𝑘_𝑖𝑗=max{|𝑎_𝑖𝑗|,|𝑎_𝑖𝑗|}, 𝐹=(𝑓_𝑖𝑗),𝑓_𝑖𝑗=max{|𝑏_𝑖𝑗|,|𝑏_𝑖𝑗|}, 𝑖,𝑗=1,2,…,𝑛.

Proof. Clearly, for any 𝐶∈𝒜=⋃^𝑛_𝑖=1𝒜_𝑖, 𝐷∈𝐵=⋃^𝑛_𝑖=1𝐵_𝑖, the inequality[𝐶]⁺+[𝐷]⁺≤𝐾+𝐹 (3.24) 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 have𝜌([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 and𝐴=−12−14014,−1𝐴=301412,𝐵=−14−16−14−18,−1𝐵=500110. (4.1) For this case, 𝒜₁={𝐴₁=(^−1/3−1/4_1/41/4),𝐴₂=(^−1/20_01/2)},𝒜₂={𝐴₃=−1301412,𝐴₄=−12−14014𝐵},₁={𝐵₁=−15−1610−8,𝐵₂=−140−141𝐵10},₂={𝐵₃=−1500110,𝐵₄=−14−16−14−18}. (4.2) By simple calculation, we have 𝜌([𝐴_𝑖]⁺+[𝐵_𝑗]⁺)≤𝜌([𝐴₁]⁺+[𝐵₄]⁺) for 𝑖,𝑗=1,2,3,4, and[𝐴₁]⁺+[𝐵₄]⁺=13141414+14161418=7512112238. (4.3) So we have𝜌([𝐴_𝑖]⁺+[𝐵_𝑗]⁺)≤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.

Figure 1 

Stability for discrete-time interval system.

Remark 4.2. If [11, Theorem IV] is applied to Example 4.1, we obtain1𝐾=21414121,𝐹=4161418, (4.5) where 𝐾, 𝐹 are defined by Corollary 3.5, that is, [11, Theorem IV]. Then𝜌(𝐾+𝐹)=1.1482>1, (4.6) 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.

Acknowledgments

The author wishes to thank the editor and the referees for their helpful and interesting comments. 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 Telecommunications A2007-41.