Delay-Partitioning Approach to Stability of Linear Discrete-Time Systems with Interval-Like Time-Varying Delay

Abstract

This paper is concerned with the problem of global asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By utilizing the concept of delay partitioning, a new linear-matrix-inequality-(LMI-) based criterion for the global asymptotic stability of such systems is proposed. The proposed criterion does not involve any free weighting matrices but depends on both the size of delay and partition size. The developed approach is extended to address the problem of global asymptotic stability of state-delayed discrete-time systems with norm-bounded uncertainties. The proposed results are compared with several existing results.

1. Introduction

A source of instability for discrete-time systems is time delay which inevitably exists in various engineering systems. In many applications, time delays are unavoidable and must be taken into account in a realistic system design, for instance, chemical processes, echo cancellation, local loop equalization, multipath propagation in mobile communication, array signal processing, congestion analysis and control in high speed networks, neural networks, and long transmission line in pneumatic systems [1–6]. The stability analysis of time-delay systems has received considerable attention during the last two decades [3–7]. According to dependence of delay, the existing stability criteria are generally classified into two categories: delay-dependent criteria and delay-independent criteria. It is well known that delay-independent stability criteria are usually more conservative than the delay-dependent ones, especially if the size of time delay is small [6–10]. Therefore, much attention has been paid in recent years to the study of delay-dependent stability criteria.

A number of publications relating to the delay-dependent stability of continuous time-delay systems have appeared (see, e.g., [10–17] and the references cited therein). In contrast, less attention has been paid to studying the problem of stability of discrete time-delay systems. Several delay-dependent criteria for the stability of discrete-time systems have appeared [3, 6, 7, 18, 19]. Reference [7] (see [6] also) presents a novel delay-dependent linear-matrix-inequality-(LMI-) based condition for the global asymptotic stability of linear discrete-time systems with interval-like time-varying delay. The criteria proposed in [6, 7] are less conservative with smaller numerical complexity than [19–23]. The delay partitioning approach has been efficiently applied in [24–30] to the stability analysis of systems with time-varying delays. In the context of stability analysis of linear discrete systems with time-varying delay, the delay partitioning concept is first utilized in [30]. The approaches in [29, 30] divide the lower bound of the time-varying delay into a number of partitions. The criteria in [29, 30] are not only delay dependent but also dependent on the partitioning size. Though the approaches in [29, 30] provide less conservative stability results than [19, 20], these approaches would lead to heavier computational burden and more complicated synthesis procedure.

This paper studies the problem of stability analysis of linear discrete-time system with interval-like time-varying delay in the state. In this paper, inspired by the work of [6, 7, 30], an alternative to the approach presented in [29] for the stability analysis of linear discrete-time systems with interval-like time-varying delay in the state is brought out. The proposed method exploits the delay partitioning idea and does not introduce any free weighting matrices. The paper is organized as follows. Section 2 presents a description of the system under consideration. A novel LMI-based criterion for the global asymptotic stability of discrete-time state-delayed systems is proposed in Section 3. The proposed criterion depends on the size of delay as well as partition size. In Section 4, the approach is extended to derive global asymptotic stable conditions for delayed discrete-time systems with norm-bounded uncertainties. Finally, in Section 5, numerical examples are given to illustrate the effectiveness of the presented results.

2. System Description

The following notations are used throughout the paper: 𝐑^𝑝×𝑞: Set of 𝑝×𝑞 real matrices, 𝐑^𝑝: Set of 𝑝×1 real vectors, 𝐈: Identity matrix of appropriate dimension; the order is specified in subscript as the need arises, 𝟎: Null matrix or null vector of appropriate dimension; the orders are specified in subscripts as the need arises, 𝐁^𝑇: Transpose of the matrix (or vector) 𝐁, 𝐁>𝟎∶𝐁 is positive-definite symmetric matrix, 𝐁<𝟎∶𝐁 is negative-definite symmetric matrix.

The system under consideration is given by 𝐱(𝑘+1)=𝐀𝐱(𝑘)+𝐀₁𝐱((𝑘,𝐱(𝑘)(𝑘)𝑘−𝑑))=𝝓,𝑘=−ℎ₂,−ℎ₂+1,…,0, (1)(2) where 𝐱(𝑘)∈𝐑^𝑛 is the system state vector, 𝐀 and 𝐀₁ are constant matrices with appropriate dimensions, 𝑑(𝑘) is a positive integer representing interval-like time-varying delay satisfying 1≤ℎ₁≤𝑑(𝑘)≤ℎ₂, (3) where ℎ₁ and ℎ₂ are known positive integers representing the lower and upper delay bounds, respectively, and 𝝓(𝑘) is an initial value at time k. Let the lower bound of the delay ℎ₁ be divided into m number of partitions such that ℎ₁=𝜏𝑚, (4) where 𝜏 is an integer representing partition size.

3. Proposed Criterion

In this section, inspired by the work of [6, 7, 30], an LMI-based criterion for the global asymptotic stability of system (1)–(4) is established.

The main result may be stated as follows.

Theorem 1. For given positive integers 𝜏, 𝑚, and ℎ₂, the system in (1)–(4) is asymptotically stable if there exist real matrices 𝐏=𝐏^𝑇>𝟎, 𝐐_𝑖=𝐐_𝑖^𝑇>𝟎(𝑖=1,2,3), and 𝐙_𝑖=𝐙_𝑖^𝑇>𝟎(𝑖=1,2) such that Ψ₁=Ψ−𝟎_{𝑛×𝑚𝑛}𝐈_𝑛−𝐈_𝑛𝟎_𝑛^𝑇𝐙₂×𝟎_{𝑛×𝑚𝑛}𝐈_𝑛−𝐈_𝑛𝟎_𝑛Ψ<𝟎,₂=Ψ−𝟎_{𝑛×(𝑚+1)𝑛}𝐈_𝑛−𝐈_𝑛^𝑇𝐙₂×𝟎_{𝑛×(𝑚+1)𝑛}𝐈_𝑛−𝐈_𝑛<𝟎, (5) where Ψ=Λ^𝑇₁Φ₁Λ₁+Λ^𝑇₂Φ₂Λ₂+Λ^𝑇₃Φ₃Λ₃−Λ^𝑇₄Φ₄Λ₄+Λ^𝑇₁Φ₅Λ₃+Λ^𝑇₃Φ^𝑇₅Λ₁−Λ^𝑇₅𝐙₁Λ₅+Λ^𝑇₁𝐙₁Λ₅+Λ^𝑇₅𝐙₁Λ₁−Λ^𝑇₆𝐙₂Λ₆+Λ^𝑇₃𝐙₂Λ₄+Λ^𝑇₄𝐙₂Λ₃+Λ^𝑇₆𝐙₂Λ₃+Λ^𝑇₃𝐙₂Λ₆,Λ₁=𝐈_𝑛𝟎_{𝑛×(𝑚+2)𝑛},Λ₂=𝐈_𝑚𝑛𝟎_{𝑚𝑛×3𝑛}𝟎_{𝑚𝑛×𝑛}𝐈_𝑚𝑛𝟎_{𝑚𝑛×2𝑛}Λ₃,=𝟎_{𝑛×(𝑚+1)𝑛}𝐈_𝑛𝟎_𝑛,Λ₄=𝟎_{𝑛×(𝑚+2)𝑛}𝐈_𝑛Λ₅,=𝟎_𝑛𝐈_𝑛𝟎_{𝑛×(𝑚+1)𝑛},Λ₆=𝟎_{𝑛×𝑚𝑛}𝐈_𝑛𝟎_𝑛×2𝑛Φ₁,=𝐀^𝑇𝐏𝐀−𝐏+𝐐₂+ℎ₂−𝜏𝑚+1𝐐₃−𝐙₁+𝐀−𝐈_𝑛^𝑇𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂𝐀−𝐈_𝑛,Φ₂=𝐐₁𝟎𝟎−𝐐₁Φ₃,=𝐀^𝑇₁𝐏+𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂𝐀₁−𝐐₃−2𝐙₂,Φ₄=𝐐₂+𝐙₂,Φ₅=𝐀^𝑇𝐏𝐀₁+𝐀−𝐈_𝑛^𝑇𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂𝐀₁. (6)(7)(8)(9)(10)(11)(12)(13)(14)

Proof. Consider a Lyapunov function [29, 30] 𝑉(𝑘)=𝐱^𝑇(𝑘)𝐏𝐱(𝑘)+^𝑘−1_{𝑖=𝑘−𝜏}𝚼^𝑇(𝑖)𝐐₁𝚼(𝑖)+^𝑘−1_{𝑖=𝑘−ℎ2}𝐱^𝑇(𝑖)𝐐₂𝐱(𝑖)+^−𝜏𝑚_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}𝐱^𝑇(𝑖)𝐐₃𝐱(𝑖)+⁻¹_𝑗=−𝜏^𝑘−1_{𝑖=𝑘+𝑗}𝜏Δ𝐱^𝑇(𝑖)𝐙₁(𝑖)+Δ𝐱_{−𝜏𝑚−1𝑗=−ℎ2𝑘−1𝑖=𝑘+𝑗}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖), (15) where 𝚼(𝑖)=𝐱^𝑇(𝑖)𝐱^𝑇(𝑖−𝜏)⋯𝐱^𝑇(𝑖−(𝑚−1)𝜏)^𝑇,Δ𝐱(𝑖)=𝐱(𝑖+1)−𝐱(𝑖). (16)(17) Taking the forward difference of (15) along the solution of the system (1)-(2), we have (𝑘)()(𝑘)=(𝑘)Δ𝑉=𝑉𝑘+1−𝑉𝐀𝐱+𝐀₁𝐱(𝑘−𝑑(𝑘))^𝑇𝐏𝐀𝐱(𝑘)+𝐀₁𝐱(𝑘−𝑑(𝑘))−𝐱^𝑇(𝑘)𝐏𝐱(𝑘)+𝚼^𝑇(𝑘)𝐐₁𝚼(𝑘)−𝚼^𝑇(𝑘−𝜏)𝐐₁𝚼(𝑘−𝜏)+𝐱^𝑇(𝑘)𝐐₂𝐱(𝑘)−𝐱^𝑇𝑘−ℎ₂𝐐₂𝐱𝑘−ℎ₂+ℎ₂−𝜏𝑚+1𝐱^𝑇(𝑘)𝐐₃𝐱(𝑘)+Δ𝐱^𝑇(𝑘)𝜏²𝐙₁Δ𝐱(𝑘)+Δ𝐱^𝑇(𝑘)ℎ₂−𝜏𝑚²𝐙₂Δ𝐱(𝑘)−^{𝑘−𝜏𝑚}_{𝑖=𝑘−ℎ2}𝐱^𝑇(𝑖)𝐐₃𝐱(𝑖)−^𝑘−1_{𝑖=𝑘−𝜏}𝜏Δ𝐱^𝑇(𝑖)𝐙₁(𝑖)−Δ𝐱_{𝑘−𝜏𝑚−1𝑖=𝑘−ℎ2}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖). (18) Using Lemma 1 in [22], we obtain [6, 7] −^𝑘−1_{𝑖=𝑘−𝜏}𝜏Δ𝐱^𝑇(𝑖)𝐙₁(𝑖)(𝐱(𝑘)(Δ𝐱≤−−𝐱𝑘−𝜏))^𝑇𝐙₁(𝐱(𝑘)−𝐱(𝑘−𝜏)). (19) Note that −^{𝑘−𝜏𝑚}_{𝑖=𝑘−ℎ2}𝐱^𝑇(𝑖)𝐐₃𝐱(𝑖)≤−𝐱^𝑇(𝑘−𝑑(𝑘))𝐐₃𝐱(𝑘−𝑑(𝑘)). (20) Define 𝜻(𝑘)=𝚼^𝑇(𝑘)𝐱^𝑇(𝑘−𝜏𝑚)𝐱^𝑇(𝑘−𝑑(𝑘))𝐱^𝑇𝑘−ℎ₂^𝑇. (21) It follows from (1) and (17) that Δ𝐱(𝑘)=𝐀−𝐈_𝑛𝐱(𝑘)+𝐀₁𝐱(𝑘−𝑑(𝑘)). (22) Now, we have the following relation [6, 7]: −^{𝑘−𝜏𝑚−1}_{𝑖=𝑘−ℎ2}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)=−^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−ℎ2}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂(𝑖)−Δ𝐱_{𝑘−𝜏𝑚−1(𝑘)𝑖=𝑘−𝑑}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)=−^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−ℎ2}ℎ₂−𝑑(𝑘)Δ𝐱^𝑇(𝑖)𝐙₂(𝑖)−Δ𝐱_{𝑘−𝑑(𝑘)−1𝑖=𝑘−ℎ2}(𝑑(𝑘)−𝜏𝑚)Δ𝐱^𝑇(𝑖)𝐙₂(𝑖)−Δ𝐱_{𝑘−𝜏𝑚−1(𝑘)𝑖=𝑘−𝑑}ℎ₂−𝑑(𝑘)Δ𝐱^𝑇(𝑖)𝐙₂(𝑖)−Δ𝐱_{𝑘−𝜏𝑚−1(𝑘)𝑖=𝑘−𝑑}(𝑑(𝑘)−𝜏𝑚)Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖). (23) Letting 𝛽=(𝑑(𝑘)−𝜏𝑚)/(ℎ₂−𝜏𝑚), one obtains [6] −^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−ℎ2}(𝑑(𝑘)−𝜏𝑚)Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)=−𝛽^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−ℎ2}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)≤−𝛽^{𝑘−𝑑(𝑘)−1}_{𝑖=𝑘−ℎ2}ℎ₂−𝑑(𝑘)Δ𝐱^𝑇(𝑖)𝐙₂(𝑖),−Δ𝐱_{𝑘−𝜏𝑚−1(𝑘)𝑖=𝑘−𝑑}ℎ₂−𝑑(𝑘)Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)=−1−𝛽^{𝑘−𝜏𝑚−1}_{(𝑘)𝑖=𝑘−𝑑}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)≤−1−𝛽^{𝑘−𝜏𝑚−1}_{(𝑘)𝑖=𝑘−𝑑}(𝑑(𝑘)−𝜏𝑚)Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖). (24) Applying Lemma 1 in [22] and using (23)-(24), we have [6, 7] −^{𝑘−𝜏𝑚−1}_{𝑖=𝑘−ℎ2}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂(𝑖)𝐱((𝑘Δ𝐱≤−𝑘−𝑑))−𝐱𝑘−ℎ₂^𝑇×𝐙₂𝐱(𝑘−𝑑(𝑘))−𝐱𝑘−ℎ₂−(𝐱(𝑘−𝜏𝑚)−𝐱(𝑘−𝑑(𝑘)))^𝑇×𝐙₂(𝐱()((𝑘𝐱((𝑘𝑘−𝜏𝑚−𝐱𝑘−𝑑)))−𝛽𝑘−𝑑))−𝐱𝑘−ℎ₂^𝑇×𝐙₂𝐱(𝑘−𝑑(𝑘))−𝐱𝑘−ℎ₂−1−𝛽(𝐱(𝑘−𝜏𝑚)−𝐱(𝑘−𝑑(𝑘)))^𝑇×𝐙₂(𝐱(𝑘−𝜏𝑚)−𝐱(𝑘−𝑑(𝑘))). (25) Employing (18)–(22) and (25), we have the following inequality:Δ𝑉(𝑘)≤𝜻^𝑇(𝑘)1−𝛽Ψ₁+𝛽Ψ₂𝜻(𝑘). (26) Since 0≤𝛽≤1, Δ𝑉(𝑘)<0 for all nonzero 𝜻(𝑘) if (5) hold true. This completes the proof of Theorem 1.

Remark 2. To prove Theorem 1, a methodology similar to [6, 7] has been adopted.

Remark 3. It may be noted that the inequalities (5) are LMIs and can be effectively solved by using MATLAB LMI toolbox [31, 32].

Remark 4. For a given 𝜏 and 𝑚, the allowable maximum value of ℎ₂ for guaranteeing the global asymptotic stability of system (1)–(4) can be obtained by iteratively solving (5).

Remark 5. For 𝑚=1, after some algebraic manipulations, it can be shown that Theorem 1 is equivalent to Theorem 1 in [7]. The parameters 𝑚 and 𝜏 (subject to (4)) in Theorem 1 represent some additional degrees of freedom, that is, in comparison with Theorem 1 in [7] which is free of these parameters.

Remark 6. A comparison of the number of the decision variables involved in several recent stability results is summarized in Table 1. It may be observed that the size of complexity in [6, 7, 18, 19, 23] is only related to state dimension 𝑛, whereas the complexity of [29, Theorem 3.1], [30, Theorem 2], and Theorem 1 depends on both 𝑛 and 𝑚.

Table 1 

Comparison of the number of decision variables involved in various methods.

The total number of scalar decision variables of Theorem 1 is 𝐷₁=(𝑛/2)[𝑛(𝑚²+5)+(𝑚+5)], and the total row size of the LMIs is 𝐿₁=𝑛(3𝑚+11). The numerical complexity of Theorem 1 is proportional to 𝐿₁𝐷³₁ [14]. The total number of scalar decision variables of Theorem 2 in [30] is 𝐷₂=(𝑛/2)[𝑛(3𝑚²+18𝑚+41)+(3𝑚+11)], the total row size of the LMIs is 𝐿₂=𝑛(7𝑚+31), and the numerical complexity is proportional to 𝐿₂𝐷³₂. Therefore, Theorem 1 has much smaller numerical complexity than Theorem 2 in [30]. Since no free weighting matrix has been introduced in the presented method, Theorem 1 involves less number of decision variables as compared to Theorem 3.1 in [29] (see Table 1). Thus, in general, Theorem 1 is numerically less complex than Theorem 3.1 in [29].

Remark 7. Using similar steps as in the proof of Proposition 8 in [30], it is easy to establish that the conservatism of the stability result obtained via Theorem 1 is nonincreasing as the number of partitions increases.

4. Extensions to Uncertain State-Delayed Discrete-Time Systems

In this section, we extend the previously discussed approach to derive global asymptotic stable conditions for delayed discrete-time systems with norm-bounded uncertainties.

Consider the system given by 𝐱(𝑘+1)=(𝐀+Δ𝐀)𝐱(𝑘)+𝐀₁+Δ𝐀₁𝐱(𝑘−𝑑(𝑘)). (27) The parameter uncertainties Δ𝐀 and Δ𝐀₁ are assumed to be norm-bounded and of the following form: Δ𝐀Δ𝐀₁=𝐇𝐅𝐄₀𝐄₁, (28) where 𝐇∈𝐑^𝑛×𝑞 and 𝐄_𝑖∈𝐑^𝑝×𝑛(𝑖=0,1) are known constant matrices and 𝐅∈𝐑^𝑞×𝑝 is an unknown matrix which satisfies 𝐅^𝑇𝐅≤𝐈. (29) The parameter uncertainties Δ𝐀 and Δ𝐀₁ are said be admissible if both (28) and (29) are satisfied.

The following lemma is needed in the proof of our next result.

Lemma 8 (see [5, 25]). Let 𝚺,Γ,𝐅, and 𝐌 be the real matrices of appropriate dimensions with 𝐌 satisfying 𝐌=𝐌^𝑇; then 𝐌+𝚺𝐅Γ+Γ^𝑇𝐅^𝑇𝚺^𝑇<𝟎, (30) for all 𝐅^𝑇𝐅≤𝐈, if and only if there exists a scalar 𝜀>0 such that 𝐌+𝜀⁻¹𝚺𝚺^𝑇+𝜀Γ^𝑇Γ<𝟎. (31)

Theorem 9. For given scalars ℎ₁ and ℎ₂, system described by (27)–(29) and (2)–(4) is globally asymptotically stable for all the admissible uncertainties, if there exist real matrices 𝐏=𝐏^𝑇>𝟎, 𝐐_𝑖=𝐐^𝑇_𝑖>𝟎(𝑖=1,2,3), and  𝐙_𝑖=𝐙^𝑇_𝑖>𝟎(𝑖=1,2) and a scalar 𝜀>0 such that the following LMIs hold: Ψ₁+𝜀𝐄^𝑇𝐄Δ^𝑇₁𝐏Δ^𝑇₂𝐙𝟎_{(𝑚+3)𝑛×𝑞}𝐏Δ₁−𝐏𝟎_𝑛𝐏𝐇𝐙Δ₂𝟎_𝑛−𝐙𝟎𝐙𝐇_{𝑞×(𝑚+3)𝑛}𝐇^𝑇𝐏𝐇^𝑇𝐙−𝜀𝐈_𝑞<𝟎,Ψ₂+𝜀𝐄^𝑇𝐄Δ^𝑇₁𝐏Δ^𝑇₂𝐙𝟎_{(𝑚+3)𝑛×𝑞}𝐏Δ₁−𝐏𝟎_𝑛𝐏𝐇𝐙Δ₂𝟎_𝑛−𝐙𝟎𝐙𝐇_{𝑞×(𝑚+3)𝑛}𝐇^𝑇𝐏𝐇^𝑇𝐙−𝜀𝐈_𝑞<𝟎, (32)(33) whereΨ₁=Ψ−Λ^𝑇₇𝐙₂Λ₇,Ψ₂=Ψ−Λ^𝑇₈𝐙₂Λ₈,Ψ=Λ^𝑇₁Φ₁Λ₁+Λ^𝑇₂Φ₂Λ₂+Λ^𝑇₃Φ₃Λ₃−Λ^𝑇₄Φ₄Λ₄−Λ^𝑇₅𝐙₁Λ₅+Λ^𝑇₁𝐙₁Λ₅+Λ^𝑇₅𝐙₁Λ₁−Λ^𝑇₆𝐙₂Λ₆+Λ^𝑇₃𝐙₂Λ₄+Λ^𝑇₄𝐙₂Λ₃+Λ^𝑇₆𝐙₂Λ₃+Λ^𝑇₃𝐙₂Λ₆,Λ₇=𝟎_{𝑛×𝑚𝑛}𝐈_𝑛−𝐈_𝑛𝟎_𝑛,Λ₈=𝟎_{𝑛×(𝑚+1)𝑛}𝐈_𝑛−𝐈_𝑛,Φ₁=−𝐏+𝐐₂+ℎ₂−𝜏𝑚+1𝐐₃−𝐙₁,Φ₂=𝐐₁𝟎𝟎−𝐐₁Φ₃,=−𝐐₃−2𝐙₂,Φ₄=𝐐₂+𝐙₂,𝐙=𝜏²𝐙₁+ℎ₂−𝜏𝑚𝐙₂,Δ₁=𝐀𝟎_{𝑛×𝑚𝑛}𝐀₁𝟎_𝑛,Δ₂=𝐀−𝐈_𝑛𝟎_{𝑛×𝑚𝑛}𝐀₁𝟎_𝑛,𝐄=𝐄₀𝟎_{𝑝×𝑚𝑛}𝐄₁𝟎_𝑝×𝑛, (34)(35)(36)(37)(38)(39)(40) and Λ_𝑖(𝑖=1,…,6) in (35) are given by (7)–(9).

Proof. Applying Theorem 1, the sufficient conditions for the global asymptotic stability of system described by (27)–(29) and (2)–(4) are obtained as follows: 𝐀^𝑇𝐏𝐀−𝐈_𝑛^𝑇𝐙Ψ₁𝟎_{𝑚𝑛×𝑛}𝟎_{𝑚𝑛×𝑛}𝐀^𝑇₁𝐏𝐀^𝑇₁𝐙𝟎_𝑛𝟎_𝑛𝐏𝐀𝟎_{𝑛×𝑚𝑛}𝐏𝐀₁𝟎_𝑛−𝐏𝟎_𝑛𝐙𝐀−𝐈_𝑛𝟎_{𝑛×𝑚𝑛}𝐙𝐀₁𝟎_𝑛𝟎_𝑛−𝐙+𝟎_{(𝑚+3)𝑛×𝑞}𝐏𝐇𝐅𝐄𝐙𝐇⁰𝟎_{𝑝×𝑚𝑛}𝐄₁𝟎_𝑝×3𝑛+𝐄^𝑇₀𝟎_{𝑚𝑛×𝑝}𝐄^𝑇₁𝟎_3𝑛×𝑝𝐅^𝑇𝟎_{𝑞×(𝑚+3)𝑛}𝐇^𝑇𝐏𝐇^𝑇𝐙<𝟎,𝐀^𝑇𝐏𝐀−𝐈_𝑛^𝑇𝐙Ψ₂𝟎_{𝑚𝑛×𝑛}𝟎_{𝑚𝑛×𝑛}𝐀^𝑇₁𝐏𝐀^𝑇₁𝐙𝟎_𝑛𝟎_𝑛𝐏𝐀𝟎_{𝑛×𝑚𝑛}𝐏𝐀₁𝟎_𝑛−𝐏𝟎_𝑛𝐙𝐀−𝐈_𝑛𝟎_{𝑛×𝑚𝑛}𝐙𝐀₁𝟎_𝑛𝟎_𝑛−𝐙+𝟎_{(𝑚+3)𝑛×𝑞}𝐏𝐇𝐅𝐄𝐙𝐇⁰𝟎_{𝑝×𝑚𝑛}𝐄₁𝟎_𝑝×3𝑛+𝐄^𝑇₀𝟎_{𝑚𝑛×𝑝}𝐄^𝑇₁𝟎_3𝑛×𝑝𝐅^𝑇𝟎_{𝑞×(𝑚+3)𝑛}𝐇^𝑇𝐏𝐇^𝑇𝐙<𝟎. (41)(42) By Lemma 8 and Schur complement, it can be shown that (41) and (42) are equivalent to (32) and (33), respectively. This completes the proof of Theorem 9.

Remark 10. With𝑚=1, Theorem 9 reduces to an equivalent form of Theorem 4 in [7]. The parameters 𝑚 and 𝜏 (subject to (4)) in Theorem 9 may be treated as additional degrees of freedom, that is, in comparison with Theorem 4 in [7] which is free of such parameters.

5. Numerical Examples

To demonstrate the applicability of the presented results and compare them with previous results, we now consider the following examples.

Example 1 (see [6, 7, 19, 20, 29]). Consider the delayed discrete-time system (1) with the following parameters: 𝐀=0.800.050.9,𝐀₁=−0.10−0.2−0.1. (43)

For given 𝜏 and 𝑚, the admissible maximum value of ℎ₂ guaranteeing the asymptotically stability of the delayed discrete-time system can be obtained by solving the LMIs (5). For a given ℎ₁, a comparison of the maximum values of ℎ₂ calculated by different methods is summarized in Table 2, from which one can easily see that the stability conditions obtained in this paper are less conservative than those in [7, 19, 20].

Table 2 

Comparison of the maximum values of ℎ₂ for given ℎ₁.

A comparison of [29, Table 2] and Table 2 reveals that Theorem 3.1 in [29] may provide a larger upper delay bound ℎ₂ than Theorem 1. However, as discussed in Remark 6, Theorem 1 involves less number of decision variables as compared to Theorem 3.1 in [29].

Example 2 (see [7, 19]). Consider the following delayed discrete-time system with parameter uncertainty: 𝐱()=(𝑘)0𝐱(𝑘)+𝐱((𝑘,𝑘+10.8+𝛼00.9−0.10−0.1−0.1𝑘−𝑑)) (44) where 𝐱(𝑘)=𝑥₁(𝑘)𝑥₂(𝑘)^𝑇 and |𝛼(𝑘)|≤𝛼. System (44) can be represented in the form of (27) with 𝐀=0.8000.9,𝐀₁=−0.10−0.1−0.1,𝐇=𝛼0,𝐄₀=01,𝐄₁=𝛼(𝑘)00,𝐅=𝛼. (45) For given ℎ₁ and ℎ₂, we wish to find 𝛼 such that the uncertain system (44) is asymptotically stable for any |𝛼(𝑘)|≤𝛼. Table 3 presents a comparison of the values of 𝛼 obtained by various methods. From Table 3, it is clear that Theorem 9 can provide less conservative results than [7, 19, 20]. In the situation where 20≤𝑑(𝑘)≤26, the stability criteria in [7, 19, 20] are invalid whereas the value of 𝛼 obtained by Theorem 9 is 0.0837 for 𝑚=5,𝜏=4.

Table 3 

Comparison of upper bound 𝛼 for given ℎ₁ and ℎ₂.

6. Conclusion

In this paper, we have considered the problem of global asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By utilizing the concept of delay partitioning, an LMI-based criterion for the global asymptotic stability of such systems has been established. The proposed criterion depends on both the size of delay and partition size. With the help of numerical examples, it has been illustrated that the proposed stability condition may provide less conservative result than most of the existing results [6, 7, 19, 20] due to the delay partitioning, and it may become even less conservative when the partitioning goes finer. Since no free weighting matrix has been introduced in the presented method, Theorem 1 involves less number of decision variables as compared to Theorem 3.1 in [29]. However, a comparison of [29, Table 2] and Table 2 reveals that Theorem 3.1 in [29] may provide a larger upper delay bound ℎ₂ than Theorem 1. The proposed criterion is numerically less complex than [30]. The method has been extended to deal with the problem of global asymptotic stability of state-delayed discrete-time systems with norm-bounded uncertainties.