Abstract

This paper investigates the problem of reachable set bounding for discrete-time system with time-varying delay and bounded disturbance inputs. Together with a new Lyapunov-Krasovskii functional, discrete Wirtinger-based inequality, and reciprocally convex approach, sufficient conditions are derived to find an ellipsoid to bound the reachable sets of discrete-time delayed system. The main advantage of this paper lies in two aspects: first, the initial state vectors are not necessarily zero; second, the obtained criteria in this paper do not really require all the symmetric matrices involved in the employed Lyapunov-Krasovskii functional to be positive definite. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

1. Introduction

The reachable set of dynamic system is defined as a collection of system state vectors in presence of all admissible input disturbances. As is known to all, the phenomenon of disturbance inputs is usually unavoidable in engineering and practical systems because of data transformation, measurement errors, linearisation approximations, and some unknown disturbances. Therefore, the reachable set bounding for dynamic systems with input disturbances is an important and challenging research topic in robust control theory and practical engineering. Then, applications of reachable set bounding can be found in various fields such as peak-to-peak minimization, aircraft collision avoidance, parameter estimation, and constrained control design [1–3]. On the other hand, time delay often occurs in various real-world systems and the existence of it may lead to instability, oscillation, or bad system performance [4–6]. Thus, stability and control of time-delay systems plays an important role in the field of engineering. As a result, many scholars have devoted themselves to the study of time-delay systems [7–18] and varieties of methods have been introduced to solve the stability analysis and control synthesis of delayed systems, such as delay-partitioning method [10, 11], reciprocally convex approach [13], and the free-weighting matrices technique [14].

In many practical applications, we often require to find a bound of a set of all the states that are reachable from a given set. Two definitions on forward and backward reachable sets are introduced in [18–20]. Those notions have been widely applied to state bounding observers, safety verification, and model checking. Forward reachable set with regard to a given initial set of a dynamic system is the set of all the states starting from the given initial set. It should be pointed out that reachable set defined in [21–25] is a special case of forward reachable set when the given initial set contains only the origin point. Most of existing results about forward reachable set bounding were reported for dynamic systems without time delay. As for the reachable set bounding problem of time-delay systems, it was firstly solved by Fridman and Shaked in [2] based on Lyapunov-Razumikhin approach. Then, some excellent achievements on reachable set bounding for dynamical systems have been proposed in the literature [21–31]. By using an enlarging Lyapunov-Krasovskii functional and delay-partitioning method, an improved condition was provided in [21]. Zuo et al. introduced the maximal Lyapunov functional method to deal with the problem of reachable set bounding for linear systems in [22]. Subsequently, the maximal Lyapunov functional approach has been adopted to research the reachable set estimation for delayed neural network [23]. In [25], the authors studied the problem of reachable set estimation and synthesis for delayed systems by using delay-decomposition technique and reciprocally convex approach. Recently, the problem of reachable set estimation for delayed singular systems was considered in [31] based on Lyapunov method. In [27], new explicit delay-independent conditions are obtained by utilizing a novel method which does not involve the Lyapunov-Krasovskii functional way. In [32], the authors presented a new approach to obtain the smallest box which bounds all reachable sets of a class of nonlinear system. The problem of state bounding for homogeneous positive nonlinear systems was studied in [33]. In this paper, we will further study the forward reachable set bounding problem for delayed system.

It is worth noting that all the above-mentioned research has been focused on continuous-time systems. However, with the progress of computer-based computational techniques, discrete-time systems play a considerable role in many practical applications. Therefore, the stability and control of discrete-time delayed systems is an important issue and has been studied by many researchers [34–40]. Up to now, there are few papers considering the reachable set estimation of discrete-time systems with time-varying delay. In [38], an ellipsoidal-like bounding of the reachable set for discrete-time system with time-varying delay is shown by using the delay-decomposition and the free-weighting matrix techniques. In [39], the problem of reachable set bounding for discrete-time polytopic systems with multiple constant delays was studied; delay-dependent conditions were derived in the form of relaxed matrix inequalities. Compared with the results in [38, 39], some less conservative conditions are obtained in [40] by employing a novel Lyapunov-Krasovskii functional combining with delay-decomposition technique and reciprocally convex method. In [41], the reachable set estimation of delayed discrete-time T-S fuzzy systems was considered. The reachable set estimation problem for discrete-time Markovian jump neural networks was firstly investigated in [42]. However, there is still much room to investigate the problem of forward reachable set bounding for discrete-time system with time-varying delay, which is our motivation.

In this paper, we concentrate on the forward reachable set bounding problem for discrete-time system with time-varying delay. The ellipsoidal reachable set estimation conditions of discrete-time linear systems are obtained by using discrete Wirtinger-based inequality and reciprocally convex approach. The novelty of this paper is three aspects: first, a relaxed Lyapunov-Krasovskii functional, which does not require all the involved symmetric matrices to be positive definite, is employed to solve the addressed problem; second, the initial state vectors are not necessarily zero, which is more general than the existing results [38, 39]; third, discrete Wirtinger-based inequality is taken into account in this paper to deal with the problem of reachable set bounding for delayed discrete-time systems. Finally, two numerical examples are provided to validate the effectiveness of the proposed method.

Notations. Throughout this paper, means that the matrix is positive definite (positive semidefinite, negative definite, and negative semidefinite); is the set of real matrices; the superscripts and denote the inverse and transpose of a matrix, respectively; denotes the symmetric block in symmetric matrix; denotes the identity matrix with compatible dimensions; denotes the set of natural number; and denotes the set of integers.

2. Problem Statement and Preliminaries

We consider the following discrete-time system with time-varying delay:where is the state vector of the system, , , and are known constant matrices with appropriate dimensions, is the time-varying delay and satisfies , and represents a disturbance which satisfies is the initial condition function satisfyingDefine that , which satisfiesIn this paper, we denote reachable set of system (1) asOur aim is to find an ellipsoid to bound the reachable set of system (1).

In addition, we give some lemmas which will be used in deriving our results.

Lemma 1. Let be a positive-definite function with satisfying ; if there exists a scalar , such thatthen, we have .

Proof. From (6), we can get then, we have then, we can get This completes the proof.

Lemma 2 (see [18]). For a given matrix , and three given nonnegative integers satisfying , vector function , , we denote then, we havewhere .

Lemma 3 (see [13]). Let have positive values in an open subset and ; then, the reciprocally convex combination of over satisfies

3. Main Results

Before main results, we define block entry matrices as (e.g., ). The other vectors are defined as

Theorem 4. For given scalar , if there exist positive scalars , , , , , symmetric matrices , , , , , , and matrices , , , , such thatthen, the reachable set of system (1) is contained in the following ellipsoid: where

Proof. We construct the following Lyapunov-Krasovskii functional for system (1):where First of all, the functional is required to be positive definite. By using Jensen inequality [37], the following can be obtained:Noticing thatwe haveIt follows from (23)–(25) thatwhereFrom condition (15), we haveTogether with (26)–(28), we can getThen, taking the forward difference of with respect to (1), we can getBased on Lemma 2, we haveWhen , by using Lemma 2, we obtainwhere , , , and .
Noting and and using Lemma 3, we have where and .
If , we have , ; if , we also have , . Accordingly, when or , (33) still holds.
Thus, we obtainFrom (30)–(34), we havewhere .
For any matrices with appropriate dimensions, we haveFrom (35)-(36), we can getNoting that , we obtainFurthermore, we describe the initial value of Lyapunov functional as By condition (17) and Lemma 1, we have . Then, together with (15), it is implied that . Thus, this means that the reachable set of system (1) is contained in the ellipsoid defined by (19). This completes the proof.