Stability Analysis of Linear Discrete-Time Systems with Interval Delay: A Delay-Partitioning Approach

Abstract

This paper considers the problem of asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By using a delay partitioning-based Lyapunov functional, a new criterion for the asymptotic stability of such systems is proposed in terms of linear matrix inequalities (LMIs). The proposed stability condition depends on both the size of delay and partition size. The presented approach is compared with previously reported approaches.

1. Introduction

Mathematical models with time delays are frequently encountered in various physical, industrial, and engineering systems due to measurement and computational delays, transmission and transport lags. During the last two decades, there has emerged a considerable interest in the system theoretic problems of time-delay systems. In many applications, time delays must be taken into account in a realistic system design, for instance, chemical processes, thermal 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 presence of time delays may result in instability of the designed systems. An excellent survey on the stability results of time-delay systems has been presented in [7]. According to dependence of delay, the available stability criteria are generally classified into two types: delay-dependent criteria and delay-independent criteria. Delay-dependent stability criteria generally lead to less conservative results than delay-independent ones, especially if the size of time delay is small [8–12].

Many publications relating to the delay-dependent stability analysis of continuous time-delay systems have appeared (see, e.g., [4, 12–21] and the references cited therein). In contrast, little effort has been made for studying the problem of stability of discrete time-delay systems. Utilizing Lyapunov functional method, several delay-dependent stability criteria for discrete-time systems have been reported in the literature [2, 22–27]. By employing novel techniques to estimate the forward difference of Lyapunov functional, stability criteria for linear discrete-time systems with interval-like time-varying delays have been proposed in [26]. The criteria proposed in [26] are less conservative with smaller numerical complexity than [22, 28–31]. The delay partitioning technique has been efficiently utilized in [32–36] to the stability analysis of systems with time-varying delays. Recently, by introducing free-weighting matrices and adopting the concept of delay partitioning, improved stability criteria have been established in [37]. The criteria reported in [37] are not only dependent on the delay but also dependent on the partitioning size. Though the approach in [37] provides less conservative results than [22, 29], it would lead to heavier computational burden and more complicated synthesis procedure.

Motivated by these developments, this paper studies the problem of stability analysis of linear discrete-time system with interval-like time-varying delay in the state. The paper is organized as follows. Section 2 introduces model description and preliminaries. By utilizing the delay partitioning idea of [37], a new linear-matrix-inequality- (LMI-) based criterion for the 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. An example highlighting the usefulness of the presented criterion is given in Section 4.

2. Model Description and Preliminaries

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,𝟎: null matrix or null vector of appropriate dimension; the orders are specified in subscripts as the need arises,𝐈_𝑝: 𝑝×𝑝 identity matrix,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.

In this paper, we consider a linear, autonomous, multivariable discrete-time system with interval-like time-varying delay in the state. Specifically, the system under consideration is represented by the difference equation: 𝐱(𝐱𝑘()𝑘=+𝝓1)(𝑘=),𝐀𝐱𝑘(𝑘=)+−ℎ𝐀₁₂𝐱(𝑘−𝑑(𝑘)),(2.1a),−ℎ₂+1,…,0,(2.1b)where 𝐱(𝑘)∈𝐑^𝑛 is the system state vector, 𝐀and𝐀₁ are constant matrices with appropriate dimensions, and 𝑑(𝑘) is a positive integer representing interval-like time-varying delay satisfying 1≤ℎ₁≤𝑑(𝑘)≤ℎ₂,(2.2) 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,(2.3) where 𝜏 is an integer representing partition size.

Definition 2.1. The equilibrium state 𝐱_𝑒=𝟎 of the system (2.1a) and (2.1b)–(2.3) is asymptotically stable if, for any 𝜀>0, there exists 𝛿>0 such that if ‖𝝓(𝑘)‖<𝛿, 𝑘=−ℎ₂,−ℎ₂+1,…,0, then ‖𝐱(𝑘)‖<𝜀, for every 𝑘≥0 and lim_𝑘→∞𝐱(𝑘)=𝟎.

Lemma 2.2 (see [30]). For any positive definite matrix 𝐖∈𝐑^𝑛×𝑛, two positive integers 𝑟 and 𝑟₀ satisfying 𝑟≥𝑟₀≥1, and vector function 𝐱(𝑖)∈𝐑^𝑛, one has ^𝑟_𝑖=𝑟0𝐱(𝑖)^𝑇𝐖^𝑟_𝑖=𝑟0𝐱(𝑖)≤𝑟−𝑟₀+1^𝑟_𝑖=𝑟0𝐱^𝑇(𝑖)𝐖𝐱(𝑖).(2.4)

3. Main Result

In this section, an LMI-based criterion for the asymptotic stability of system (2.1a) and (2.1b)–(2.3) is established. The main result may be stated as follows.

Theorem 3.1. For given positive integers 𝜏, 𝑚, and ℎ₂, the system described by (2.1a) and (2.1b)–(2.3) is asymptotically stable if there exist real matrices 𝐏=𝐏^𝑇>𝟎, 𝐐_𝑖=𝐐_𝑖𝑇>𝟎  (𝑖=1,2,3), 𝐙_𝑖=𝐙^𝑇𝑖>𝟎  (𝑖=1,2), 𝐗, 𝐘 and 𝐖 such that ⎡⎢⎢⎢⎢⎣−𝐗𝐘𝐙₂𝐘𝚿^T𝐖_T𝐗𝐖^T−𝐙₂⎤⎥⎥⎥⎥⎦<𝟎,(3.1) where 𝚿=𝚲^𝑇1𝚽₁𝚲₁+𝚲^𝑇2𝚽₂𝚲₂+𝚲^𝑇3𝚽₃𝚲₃−𝚲^𝑇4𝐐₂𝚲₄+𝚲^𝑇1𝚽₄𝚲₃+𝚲^𝑇3𝚽^𝑇4𝚲₁−𝚲^𝑇5𝐙₁𝚲₅𝚲+𝚲^𝑇1𝐙₁𝚲₅+𝚲^𝑇5𝐙₁𝚲₁+𝟎𝐘−𝐘+𝐖−𝐖+𝟎𝐘−𝐘+𝐖−𝐖^𝑇,where₁𝚲₂=𝐈_𝑛𝟎_{𝑛×(𝑚+2)𝑛},=⎡⎢⎢⎢⎣𝐈^𝑚𝑛𝟎_{𝑚𝑛×3𝑛}𝟎_{𝑚𝑛×𝑛}𝐈_𝑚𝑛𝟎_{𝑚𝑛×2𝑛}𝚲₃⎤⎥⎥⎥⎦,𝚲₄=𝟎_{𝑛×(𝑚+1)𝑛}𝐈_𝑛𝟎_𝑛,𝚲₅=𝟎_{𝑛×(𝑚+2)𝑛}𝐈_𝑛,𝚽₁=𝟎_𝑛𝐈_𝑛𝟎_{𝑛×(𝑚+1)𝑛},𝚽₂=𝐀^𝑇𝐏𝐀−𝐏+𝐐₂+ℎ₂−𝜏𝑚+1𝐐₃−𝐙₁+𝐀−𝐈_𝑛^𝑇𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂𝐀−𝐈^𝑛,=⎡⎢⎢⎣𝐐¹𝚽₃𝟎−𝟎𝐐₁⎤⎥⎥⎦,𝚽₄=𝐀^𝑇1𝐏+𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂𝐀¹−𝐐₃,=𝐀^𝑇𝐏𝐀₁+𝐀−𝐈_𝑛^𝑇𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂𝐀¹.(3.2)

Proof. Choose a Lyapunov functional candidate as 𝑉(𝑘)=𝐱^𝑇(𝑘)𝐏𝐱(𝑘)+^𝑘−1_𝑖=_𝑘−𝜏𝚼^𝑇(𝑖)𝐐₁𝚼(𝑖)+^𝑘−1_𝑖=_𝑘−ℎ2𝐱^𝑇(𝑖)𝐐₂𝐱(𝑖)+^−𝜏𝑚_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}+𝐱^𝑇(𝑖)𝐐₃𝐱(𝑖)₋₁_{𝑗=−𝜏𝑘−1𝑖=𝑘+𝑗}𝜏Δ𝐱^𝑇(𝑖)𝐙₁Δ𝐱(𝑖)+^{−𝜏𝑚−1}_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖),(3.3) where 𝚼(𝑖)=𝐱^𝑇(𝑖)𝐱^𝑇(𝑖−𝜏)⋯𝐱^𝑇(𝑖−(𝑚−1)𝜏)^𝑇,(3.4)Δ𝐱(𝑖)=𝐱(𝑖+1)−𝐱(𝑖).(3.5) Taking the forward difference of (3.3) along the solution of (2.1a) and (2.1b), we have Δ𝑉(𝑘)==𝑉𝐀(𝐱𝑘(+𝑘)1+)−𝐀₁𝑉(𝑘)+𝚼_𝑇𝐱(𝑘−𝑑(𝑘))^𝑇𝐏𝐀𝐱(𝑘)+𝐀₁𝐱(𝑘−𝑑(𝑘))−𝐱^𝑇(𝑘)𝐏𝐱(𝑘)−𝐱_𝑇(𝑘)𝐐₁𝚼(𝑘)−𝚼^𝑇(𝑘−𝜏)𝐐₁𝚼(𝑘−𝜏)+𝐱^𝑇(𝑘)𝐐₂𝐱(𝑘)−𝑘−ℎ₂𝐐₂𝐱𝑘−ℎ₂+ℎ₂−𝜏𝑚+1𝐱^𝑇(𝑘)𝐐₃𝐱(𝑘)_{𝑘−𝜏𝑚𝑖=}_𝑘−ℎ2−𝐱^𝑇(𝑖)𝐐₃𝐱(𝑖)+Δ𝐱^𝑇(𝑘)𝜏²𝐙₁+ℎ₂−𝜏𝑚²𝐙₂Δ𝐱(𝑘)_{𝑘−1𝑖=}_𝑘−𝜏𝜏Δ𝐱^𝑇(𝑖)𝐙₁Δ𝐱(𝑖)−^{𝑘−𝜏𝑚−1}_𝑖=_𝑘−ℎ2ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖).(3.6) Using Lemma 2.2, we obtain −^𝑘−1_𝑖=_𝑘−𝜏𝜏Δ𝐱^𝑇(𝑖)𝐙₁Δ𝐱(𝑖)≤−(𝐱(𝑘)−𝐱(𝑘−𝜏))^𝑇𝐙₁(𝐱(𝑘)−𝐱(𝑘−𝜏)).(3.7) Note that −^{𝑘−𝜏𝑚}_𝑖=_𝑘−ℎ2𝐱^𝑇(𝑖)𝐐₃𝐱(𝑖)≤−𝐱^𝑇(𝑘−𝑑(𝑘))𝐐₃𝐱(𝑘−𝑑(𝑘)).(3.8) Now, we have the following relations: 0=2𝜁^𝑇(𝑘)𝐘𝐱(𝑘−𝜏𝑚)−𝐱(𝑘−𝑑(𝑘))−^{𝑘−𝜏𝑚−1}_𝑖=𝑘_{−𝑑(𝑘)}0=2𝜁_𝑇Δ𝐱(𝑖),(𝑘)𝐖⎡⎢⎢⎣𝐱(𝑘−𝑑(𝑘))−𝐱𝑘−ℎ₂−^{𝑘−𝑑(𝑘)−1}_𝑖=_𝑘−ℎ2Δ𝐱(𝑖)⎤⎥⎥⎦,(3.9) where 𝐘 and 𝐖 are constant matrices of appropriate dimensions and 𝜁(𝑘)=𝚼^𝑇(𝑘)𝐱^𝑇(𝑘−𝜏𝑚)𝐱^𝑇(𝑘−𝑑(𝑘))𝐱^𝑇𝑘−ℎ₂^𝑇.(3.10) It follows from (2.1a) and (3.5) that Δ𝐱(𝑘)=𝐀−𝐈_𝑛𝐱(𝑘)+𝐀₁𝐱(𝑘−𝑑(𝑘)).(3.11) Employing (3.6)–(3.11), we have the following inequality: Δ𝑉(𝑘)≤𝜁^𝑇(𝑘)𝚿𝜁(𝑘)−2𝜁^𝑇(𝑘)𝐘^{𝑘−𝜏𝑚−1}_𝑖=𝑘_{−𝑑(𝑘)}Δ𝐱(𝑖)−2𝜁^𝑇(𝑘)𝐖^{𝑘−𝑑(𝑘)−1}_𝑖=_𝑘−ℎ2−Δ𝐱(𝑖)_{𝑘−𝜏𝑚−1𝑖=𝑘}_{−𝑑(𝑘)}ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖)−^{𝑘−𝑑(𝑘)−1}_𝑖=_𝑘−ℎ2ℎ₂−𝜏𝑚Δ𝐱^𝑇(𝑖)𝐙₂Δ𝐱(𝑖),(3.12) where Ψ is given by (3.2). Equation (3.12) can also be rearranged as Δ𝑉(𝑘)≤1ℎ₂−𝜏𝑚^{𝑘−𝜏𝑚−1}_𝑖=𝑘_{−𝑑(𝑘)}⎡⎢⎢⎣−ℎ𝜁(𝑘)₂−𝜏𝑚Δ𝐱(𝑖)⎤⎥⎥⎦^𝑇⎡⎢⎢⎣𝐘𝚿𝐘_𝑇−𝐙₂⎤⎥⎥⎦⎡⎢⎢⎣−ℎ𝜁(𝑘)₂+1−𝜏𝑚Δ𝐱(𝑖)⎤⎥⎥⎦ℎ₂−𝜏𝑚^{𝑘−𝑑(𝑘)−1}_𝑖=_𝑘−ℎ2⎡⎢⎢⎣−(ℎ₂𝜁(𝑘)−𝜏𝑚)Δ𝐱(𝑖)⎤⎥⎥⎦^𝑇⎡⎢⎢⎣𝐖𝚿𝐖_𝑇−𝐙₂⎤⎥⎥⎦⎡⎢⎢⎣−ℎ𝜁(𝑘)₂−𝜏𝑚Δ𝐱(𝑖)⎤⎥⎥⎦.(3.13) Using [17, Lemma 4.1], it can be shown that there exists a matrix 𝐗 of appropriate dimensions satisfying (3.1) if and only if ⎡⎢⎢⎣𝐘𝚿𝐘_𝑇−𝐙₂⎤⎥⎥⎦<𝟎,⎡⎢⎢⎣𝐖𝚿𝐖_𝑇−𝐙₂⎤⎥⎥⎦<𝟎,(3.14) which together with (3.13) implies Δ𝑉(𝑘)<0 for all nonzero 𝜁(𝑘). This completes the proof.

Remark 3.2. 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 [22, 26, 31] is only related to state dimension 𝑛, whereas the complexity of [37] and Theorem 3.1 depends on both 𝑛 and 𝑚.
The total number of scalar decision variables of Theorem 3.1 is 𝐷₁=(𝑛/2)[𝑛(𝑚²+4𝑚+19)+(𝑚+5)], and the total row size of the LMIs is 𝐿₁=2𝑛(𝑚+5). The numerical complexity of Theorem 3.1 is proportional to 𝐿₁𝐷³¹ [16]. In [37], the total number of scalar decision variables of Theorem 2 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 3.1 has much smaller numerical complexity than Theorem 2 of [37].

Table 1 

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

Remark 3.3. For a given ℎ₁, the allowable maximum value of ℎ₂ for guaranteeing the asymptotic stability of system (2.1a) and (2.1b)–(2.3) can be obtained by iteratively solving the LMI (3.1).

Remark 3.4. With 𝑚=1, 𝐘=[𝐘^𝑇𝟏𝐘^𝑇2𝟎_𝑛×2𝑛]^𝑇, and 𝐖=[𝐖^𝑇1𝐖^𝑇2𝟎_𝑛×2𝑛]^𝑇, Theorem 3.1 reduces to an equivalent form of [26, Proposition 1]. Thus, as compared to [26, Proposition 1], Theorem 3.1 provides additional degrees of freedom in the selection of 𝑚, 𝐘, and 𝐖 which would result in an enhanced stability region in the parameter space.

Remark 3.5. Using similar steps as in the proof of [37, Proposition 8], it is easy to establish that the conservatism of the stability result obtained via Theorem 3.1 is nonincreasing as the number of partitions increases.
It may be noted that, in the derivation of Theorem 3.1, the lower bound of the delay is assumed to be ℎ₁=𝜏𝑚≥1. For the situation where ℎ₁=0, we have the following result.

Theorem 3.6. The system (2.1a) and (2.1b) with 0≤𝑑(𝑘)≤ℎ₂ is asymptotically stable if there exist real matrices 𝐏=𝐏^𝑇>𝟎, 𝐐_𝑖=𝐐^𝑇𝑖>𝟎(𝑖=1,2), 𝐙=𝐙^𝑇>𝟎, 𝐗, 𝐘, and 𝐖 such that ⎡⎢⎢⎢⎢⎣−𝐗𝐘𝐙_𝑇𝐘𝚿^𝑇𝐖𝐗𝐖^𝑇−𝐙⎤⎥⎥⎥⎥⎦<𝟎,(3.15) where 𝚿=𝚲^𝑇1𝚽₁𝚲₁+𝚲^𝑇1𝚽₂𝚲₂+𝚲^𝑇2𝚽_2𝑇𝚲₁+𝚲^𝑇2𝚽₃𝚲₂−𝚲^𝑇3𝐐₁𝚲₃𝚲+𝐘−𝐘+𝐖−𝐖+𝐘−𝐘+𝐖−𝐖^𝑇,where₁𝚲₂=𝐈_𝑛𝟎_𝑛×2𝑛,𝚲₃=𝟎_𝑛𝐈_𝑛𝟎_𝑛,𝚽₁=𝟎_𝑛×2𝑛𝐈_𝑛,𝚽₂=𝐀^𝑇𝐏𝐀−𝐏+ℎ₂₂𝐀−𝐈_𝑛^𝑇𝐙𝐀−𝐈_𝑛+𝐐₁+ℎ₂+1𝐐₂,𝚽₃=𝐀^𝑇𝐏𝐀₁+ℎ₂₂𝐀−𝐈_𝑛^𝑇𝐙𝐀₁,=𝐀_1𝑇𝐏+ℎ₂₂𝐙𝐀₁−𝐐₂.(3.16)

Proof. Choosing the Lyapunov functional as 𝑉(𝑘)=𝐱^𝑇(𝑘)𝐏𝐱(𝑘)+^𝑘−1_𝑖=_𝑘−ℎ2+𝐱^𝑇(𝑖)𝐐₁𝐱(𝑖)₀_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}𝐱^𝑇(𝑖)𝐐₂𝐱(𝑖)+⁻¹_𝑗=−ℎ2^𝑘−1_{𝑖=𝑘+𝑗}ℎ₂Δ𝐱^𝑇(𝑖)𝐙Δ𝐱(𝑖)(3.17) and employing the steps of the proof of Theorem 3.1 discussed earlier, one can easily arrive at the above theorem. The details of the proof of Theorem 3.6 are, therefore, omitted.

Remark 3.7. With 𝐿₁=9𝑛 rows and 𝐷₁=9𝑛²+2𝑛 scalar decision variables, LMI (3.15) in Theorem 3.6 has the numerical complexity proportional to 𝐿₁𝐷³¹ [16], while the condition in [37, Proposition 10] has the numerical complexity proportional to 𝐿₂𝐷³², where 𝐿₂=21𝑛 and 𝐷₂=(𝑛/2)(25𝑛+7). So, Theorem 3.6 has smaller numerical complexity than [37, Proposition 10].

4. A Numerical Example

To demonstrate the applicability of the presented results and compare them with previous results, we now consider a specific example of system (2.1a), (2.1b) with 𝐀=⎡⎢⎢⎣00..08500.9⎤⎥⎥⎦,𝐀₁=⎡⎢⎢⎣−−00..12−00.1⎤⎥⎥⎦(4.1) and 0≤𝑑(𝑘)≤ℎ₂. This example has been considered in [22, 29, 37]. Using Matlab LMI toolbox [38, 39], it is found from Theorem 3.6 that the present system is stable for 0≤𝑑(𝑘)≤17. In this case, the number of decision variables involved in Theorem 3.6 is 40. On the other hand, to obtain the upper bound ℎ₂=17, [37, Proposition 10] requires 57 decision variables. This demonstrates the numerical efficiency of the proposed method.

Next, consider the system described by (2.1a), (2.1b)–(2.3), and (4.1). For different values of ℎ₁, the admissible upper bound ℎ₂ is listed in Table 2. From Table 2, it is clear that Theorem 3.1 can provide a larger upper bound ℎ₂ than the previously reported stability results [22, 23, 26, 29]. It may also be observed that, for the example under consideration, Theorem 3.1 leads to upper bound ℎ₂ which is identical to that arrived at via Theorem 2 in [37]. However, as discussed in Remark 3.2, Theorem 3.1 has much smaller numerical complexity than Theorem 2 in [37].

Table 2 

Admissible upper bound of ℎ₂ for various ℎ₁.

5. Conclusion

In this paper, the problem concerning the asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state has been considered. Using the concept of delay partitioning, an LMI-based criterion for the asymptotic stability of such systems has been established. The criterion depends on the size of delay as well as partition size. The presented approach may imply asymptotic stability for a broader class of time-varying state-delayed systems, as compared to previous approaches [22, 23, 26, 29]. The presented criterion is numerically less complex than [37]. The stability results discussed in this paper can easily be extended to delayed discrete-time systems with parameter uncertainties.