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

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.


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 highspeed 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: delaydependent 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 timedelay 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 freeweighting 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.

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: where x k ∈ R n is the system state vector, A and A 1 are constant matrices with appropriate dimensions, and d k is a positive integer representing interval-like time-varying delay satisfying where h 1 and h 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 h 1 be divided into m number of partitions such that 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 Lemma 2.2 see 30 .For any positive definite matrix W ∈ R n×n , two positive integers r and r 0 satisfying r ≥ r 0 ≥ 1, and vector function

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 h 2 , the system described by 2.1a and 2.1b -2.3 is asymptotically stable if there exist real matrices where where

3.2
Proof.Choose a Lyapunov functional candidate as where Taking the forward difference of 3.3 along the solution of 2.1a and 2.1b , we have

3.6
Using Lemma 2.2, we obtain Now, we have the following relations: where Y and W are constant matrices of appropriate dimensions and

3.10
It follows from 2.1a and 3.5 that Employing 3.6 -3.11 , we have the following inequality: where Ψ is given by 3.2 .Equation 3.12 can also be rearranged as

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 n/2 n 3m 2 18m 41 3m 11 , the total row size of the LMIs is L 2 n 7m 31 , and the numerical complexity is proportional to L 2 D 3  2 .Therefore, Theorem 3.1 has much smaller numerical complexity than Theorem 2 of 37 .
Remark 3.3.For a given h 1 , the allowable maximum value of h 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 3.5.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 h 1 τm ≥ 1.For the situation where h 1 0, we have the following result. where where

3.16
Proof.Choosing the Lyapunov functional as 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.

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, it is clear that Theorem 3.1 can provide a larger upper bound h 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 h 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 .

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 statedelayed 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.

Remark 3 . 4 .
With m 1, 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.

Table 1 :
Comparison of the number of decision variables involved in various methods.The system 2.1a and 2.1b with 0 ≤ d k ≤ h 2 is asymptotically stable if there exist real matrices P P

Table 2 :
Admissible upper bound of h 2 for various h 1 .
≤ h 2 .This example has been considered in22, 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 h 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 h 1 , the admissible upper bound h 2 is listed in Table2.From Table