Table of Contents
ISRN Applied Mathematics
Volume 2011, Article ID 624127, 10 pages
http://dx.doi.org/10.5402/2011/624127
Research Article

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

Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology, Allahabad 211004, India

Received 7 September 2011; Accepted 16 October 2011

Academic Editors: C.-H. Lien, F. Tadeo, and Q. Zhang

Copyright © 2011 Priyanka Kokil et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [16]. 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 [812].

Many publications relating to the delay-dependent stability analysis of continuous time-delay systems have appeared (see, e.g., [4, 1221] 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, 2227]. 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, 2831]. The delay partitioning technique has been efficiently utilized in [3236] 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:𝐑𝑝×𝑞: set of 𝑝×𝑞 real matrices,𝐑𝑝: 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,𝐈𝑝: 𝑝×𝑝 identity matrix,𝐁𝑇: transpose of the matrix (or vector) 𝐁,𝐁>𝟎𝐁 is positive-definite symmetric matrix,𝐁<𝟎𝐁 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(𝑘+1)=𝐀𝐱(𝑘)+𝐀1𝐱(𝑘𝑑(𝑘)),(2.1a)𝐱(𝑘)=𝝓(𝑘),𝑘=2,2+1,,0,(2.1b)where 𝐱(𝑘)𝐑𝑛 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 11𝑑(𝑘)2,(2.2) where 1 and 2 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 1 be divided into 𝑚 number of partitions such that 1=𝜏𝑚,(2.3) where 𝜏 is an integer representing partition size.

Definition 2.1. The equilibrium state 𝐱𝑒=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=2,2+1,,0, then 𝐱(𝑘)<𝜀, for every 𝑘0 and lim𝑘𝐱(𝑘)=𝟎.

Lemma 2.2 (see [30]). For any positive definite matrix 𝐖𝐑𝑛×𝑛, two positive integers 𝑟 and 𝑟0 satisfying 𝑟𝑟01, and vector function 𝐱(𝑖)𝐑𝑛, one has 𝑟𝑖=𝑟0𝐱(𝑖)𝑇𝐖𝑟𝑖=𝑟0𝐱(𝑖)𝑟𝑟0+1𝑟𝑖=𝑟0𝐱𝑇(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 2, the system described by (2.1a) and (2.1b)–(2.3) is asymptotically stable if there exist real matrices 𝐏=𝐏𝑇>𝟎, 𝐐𝑖=𝐐𝑖𝑇>𝟎  (𝑖=1,2,3), 𝐙𝑖=𝐙𝑇𝑖>𝟎  (𝑖=1,2), 𝐗, 𝐘 and 𝐖 such that 𝐙2𝐘T𝐗𝐗𝐘𝚿𝐖T𝐖T𝐙2<𝟎,(3.1) where 𝚿=𝚲𝑇1𝚽1𝚲1+𝚲𝑇2𝚽2𝚲2+𝚲𝑇3𝚽3𝚲3𝚲𝑇4𝐐2𝚲4+𝚲𝑇1𝚽4𝚲3+𝚲𝑇3𝚽𝑇4𝚲1𝚲𝑇5𝐙1𝚲5+𝚲𝑇1𝐙1𝚲5+𝚲𝑇5𝐙1𝚲1++𝟎𝐘𝐘+𝐖𝐖𝟎𝐘𝐘+𝐖𝐖𝑇𝚲,where1=𝐈𝑛𝟎𝑛×(𝑚+2)𝑛,𝚲2=𝐈𝑚𝑛𝟎𝑚𝑛×3𝑛𝟎𝑚𝑛×𝑛𝐈𝑚𝑛𝟎𝑚𝑛×2𝑛,𝚲3=𝟎𝑛×(𝑚+1)𝑛𝐈𝑛𝟎𝑛,𝚲4=𝟎𝑛×(𝑚+2)𝑛𝐈𝑛,𝚲5=𝟎𝑛𝐈𝑛𝟎𝑛×(𝑚+1)𝑛,𝚽1=𝐀𝑇𝐏𝐀𝐏+𝐐2+2𝐐𝜏𝑚+13𝐙1+𝐀𝐈𝑛𝑇𝜏2𝐙1+2𝜏𝑚2𝐙2𝐀𝐈𝑛,𝚽2=𝐐1𝟎𝟎𝐐1,𝚽3=𝐀𝑇1𝐏+𝜏2𝐙1+2𝜏𝑚2𝐙2𝐀1𝐐3,𝚽4=𝐀𝑇𝐏𝐀1+𝐀𝐈𝑛𝑇𝜏2𝐙1+2𝜏𝑚2𝐙2𝐀1.(3.2)

Proof. Choose a Lyapunov functional candidate as 𝑉(𝑘)=𝐱𝑇(𝑘)𝐏𝐱(𝑘)+𝑘1𝑖=𝑘𝜏𝚼𝑇(𝑖)𝐐1𝚼(𝑖)+𝑘1𝑖=𝑘2𝐱𝑇(𝑖)𝐐2𝐱(𝑖)+𝜏𝑚𝑗=2𝑘1𝑖=𝑘+𝑗𝐱𝑇(𝑖)𝐐3+𝐱(𝑖)1𝑗=𝜏𝑘1𝑖=𝑘+𝑗𝜏Δ𝐱𝑇(𝑖)𝐙1Δ𝐱(𝑖)+𝜏𝑚1𝑗=2𝑘1𝑖=𝑘+𝑗2𝜏𝑚Δ𝐱𝑇(𝑖)𝐙2Δ𝐱(𝑖),(3.3) where 𝐱𝚼(𝑖)=𝑇(𝑖)𝐱𝑇(𝑖𝜏)𝐱𝑇(𝑖(𝑚1)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(𝑘+1)𝑉(𝑘)𝐀𝐱(𝑘)+𝐀1𝐱(𝑘𝑑(𝑘))𝑇𝐏𝐀𝐱(𝑘)+𝐀1𝐱(𝑘𝑑(𝑘))𝐱𝑇(𝑘)𝐏𝐱(𝑘)+𝚼𝑇(𝑘)𝐐1𝚼(𝑘)𝚼𝑇(𝑘𝜏)𝐐1𝚼(𝑘𝜏)+𝐱𝑇(𝑘)𝐐2𝐱(𝑘)𝐱𝑇𝑘2𝐐2𝐱𝑘2+2𝐱𝜏𝑚+1𝑇(𝑘)𝐐3𝐱(𝑘)𝑘𝜏𝑚𝑖=𝑘2𝐱𝑇(𝑖)𝐐3𝐱(𝑖)+Δ𝐱𝑇𝜏(𝑘)2𝐙1+2𝜏𝑚2𝐙2Δ𝐱(𝑘)𝑘1𝑖=𝑘𝜏𝜏Δ𝐱𝑇(𝑖)𝐙1Δ𝐱(𝑖)𝑘𝜏𝑚1𝑖=𝑘22𝜏𝑚Δ𝐱𝑇(𝑖)𝐙2Δ𝐱(𝑖).(3.6) Using Lemma 2.2, we obtain 𝑘1𝑖=𝑘𝜏𝜏Δ𝐱𝑇(𝑖)𝐙1Δ𝐱(𝑖)(𝐱(𝑘)𝐱(𝑘𝜏))𝑇𝐙1(𝐱(𝑘)𝐱(𝑘𝜏)).(3.7) Note that 𝑘𝜏𝑚𝑖=𝑘2𝐱𝑇(𝑖)𝐐3𝐱(𝑖)𝐱𝑇(𝑘𝑑(𝑘))𝐐3𝐱(𝑘𝑑(𝑘)).(3.8) Now, we have the following relations: 0=2𝜁𝑇(𝑘)𝐘𝐱(𝑘𝜏𝑚)𝐱(𝑘𝑑(𝑘))𝑘𝜏𝑚1𝑖=𝑘𝑑(𝑘),Δ𝐱(𝑖)0=2𝜁𝑇(𝑘)𝐖𝐱(𝑘𝑑(𝑘))𝐱𝑘2𝑘𝑑(𝑘)1𝑖=𝑘2,Δ𝐱(𝑖)(3.9) where 𝐘 and 𝐖 are constant matrices of appropriate dimensions and 𝚼𝜁(𝑘)=𝑇(𝑘)𝐱𝑇(𝑘𝜏𝑚)𝐱𝑇(𝑘𝑑(𝑘))𝐱𝑇𝑘2𝑇.(3.10) It follows from (2.1a) and (3.5) that Δ𝐱(𝑘)=𝐀𝐈𝑛𝐱(𝑘)+𝐀1𝐱(𝑘𝑑(𝑘)).(3.11) Employing (3.6)–(3.11), we have the following inequality: Δ𝑉(𝑘)𝜁𝑇(𝑘)𝚿𝜁(𝑘)2𝜁𝑇(𝑘)𝐘𝑘𝜏𝑚1𝑖=𝑘𝑑(𝑘)Δ𝐱(𝑖)2𝜁𝑇(𝑘)𝐖𝑘𝑑(𝑘)1𝑖=𝑘2Δ𝐱(𝑖)𝑘𝜏𝑚1𝑖=𝑘𝑑(𝑘)2𝜏𝑚Δ𝐱𝑇(𝑖)𝐙2Δ𝐱(𝑖)𝑘𝑑(𝑘)1𝑖=𝑘22𝜏𝑚Δ𝐱𝑇(𝑖)𝐙2Δ𝐱(𝑖),(3.12) where Ψ is given by (3.2). Equation (3.12) can also be rearranged as 1Δ𝑉(𝑘)2𝜏𝑚𝑘𝜏𝑚1𝑖=𝑘𝑑(𝑘)𝜁(𝑘)2𝜏𝑚Δ𝐱(𝑖)𝑇𝐘𝚿𝐘𝑇𝐙2𝜁(𝑘)2+1𝜏𝑚Δ𝐱(𝑖)2𝜏𝑚𝑘𝑑(𝑘)1𝑖=𝑘2𝜁(𝑘)(2𝜏𝑚)Δ𝐱(𝑖)𝑇𝐖𝚿𝐖𝑇𝐙2𝜁(𝑘)2.𝜏𝑚Δ𝐱(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 𝐘𝚿𝐘𝑇𝐙2𝐖<𝟎,𝚿𝐖𝑇𝐙2<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].

tab1
Table 1: Comparison of the number of decision variables involved in various methods.

Remark 3.3. For a given 1, the allowable maximum value of 2 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=[𝐘𝑇𝟏𝐘𝑇2𝟎𝑛×2𝑛]𝑇, and 𝐖=[𝐖𝑇1𝐖𝑇2𝟎𝑛×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 1=𝜏𝑚1. For the situation where 1=0, we have the following result.

Theorem 3.6. The system (2.1a) and (2.1b) with 0𝑑(𝑘)2 is asymptotically stable if there exist real matrices 𝐏=𝐏𝑇>𝟎, 𝐐𝑖=𝐐𝑇𝑖>𝟎(𝑖=1,2), 𝐙=𝐙𝑇>𝟎, 𝐗, 𝐘, and 𝐖 such that 𝐘𝐙𝑇𝐗𝐘𝚿𝐖𝐗𝑇𝐖𝑇𝐙<𝟎,(3.15) where 𝚲𝚿=𝑇1𝚽1𝚲1+𝚲𝑇1𝚽2𝚲2+𝚲𝑇2𝚽2𝑇𝚲1+𝚲𝑇2𝚽3𝚲2𝚲𝑇3𝐐1𝚲3+𝐖+𝐖𝐘𝐘+𝐖𝐘𝐘+𝐖𝑇𝚲,where1=𝐈𝑛𝟎𝑛×2𝑛,𝚲2=𝟎𝑛𝐈𝑛𝟎𝑛,𝚲3=𝟎𝑛×2𝑛𝐈𝑛,𝚽1=𝐀𝑇𝐏𝐀𝐏+22𝐀𝐈𝑛𝑇𝐙𝐀𝐈𝑛+𝐐1+2𝐐+12,𝚽2=𝐀𝑇𝐏𝐀1+22𝐀𝐈𝑛𝑇𝐙𝐀1,𝚽3=𝐀1𝑇𝐏+22𝐙𝐀1𝐐2.(3.16)

Proof. Choosing the Lyapunov functional as 𝑉(𝑘)=𝐱𝑇(𝑘)𝐏𝐱(𝑘)+𝑘1𝑖=𝑘2𝐱𝑇(𝑖)𝐐1+𝐱(𝑖)0𝑗=2𝑘1𝑖=𝑘+𝑗𝐱𝑇(𝑖)𝐐2𝐱(𝑖)+1𝑗=2𝑘1𝑖=𝑘+𝑗2Δ𝐱𝑇(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 𝐿1=9𝑛 rows and 𝐷1=9𝑛2+2𝑛 scalar decision variables, LMI (3.15) in Theorem 3.6 has the numerical complexity proportional to 𝐿1𝐷31 [16], while the condition in [37, Proposition 10] has the numerical complexity proportional to 𝐿2𝐷32, where 𝐿2=21𝑛 and 𝐷2=(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=0.800.050.9,𝐀1=0.100.20.1(4.1) and 0𝑑(𝑘)2. 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 0𝑑(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 2=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 1, the admissible upper bound 2 is listed in Table 2. From Table 2, it is clear that Theorem 3.1 can provide a larger upper bound 2 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 2 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].

tab2
Table 2: Admissible upper bound of 2 for various 1.

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.

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