Abstract

This paper considers the guaranteed cost finite-time boundedness of discrete-time positive impulsive switched systems. Firstly, the definition of guaranteed cost finite-time boundedness is introduced. By using the multiple linear copositive Lyapunov function (MLCLF) and average dwell time (ADT) approach, a state feedback controller is designed and sufficient conditions are obtained to guarantee that the corresponding closed-loop system is guaranteed cost finite-time boundedness (GCFTB). Such conditions can be solved by linear programming. Finally, a numerical example is provided to show the effectiveness of the proposed method.

1. Introduction

As a special kind of positive systems [1–3], the positive switched systems whose output and state are nonnegative whenever the initial condition and input are nonnegative have been found in many applications such as communication networks [4], viral mutation [5], and formation flying [6]. There have been many available results about continuous-time positive switched systems [7–11] and discrete-time positive switched systems [12–14].

However, most results mentioned above focus on the classical Lyapunov stability, which guarantees the stability in an infinite-time interval. Different from the Lyapunov stability concept, the finite-time stability requires that the states do not exceed a certain bound during a fixed finite-time interval. The paper [15] firstly defined the definition of finite-time stability (FTS) for linear deterministic systems. Recently, [16] firstly extended the concept of FTS to positive switched systems and gave some FTS conditions of positive switched systems. So far, there have been a few meaningful results about FTS of positive switched systems; see [17–20]. In these results, to make the best of the nature of positivity, the MLCLF approach has been widely used and became a powerful tool for the analysis and synthesis of positive switched systems. Due to the wide application of digital controllers, some researches have been done on the FTS of discrete-time positive switched systems. The paper [21] investigated the problem of robust finite-time stability and stabilization of a class of discrete-time positive switched systems. The paper [22] studied the problem of finite-time control of a class of discrete impulsive switched positive time-delay systems under asynchronous switching, but the effect of disturbance was ignored.

Moreover, in most of practical applications, the researchers are more interested in designing the control system which is not only finite-time stable but also guarantees an adequate level of performance. One method to this problem is the so-called guaranteed cost finite-time control. Some remarkable results have been presented; see [23–27]. These results mainly focus on nonpositive systems. Very recently, in [28], guaranteed cost finite-time control was extended to fractional-order positive switched systems and a cost function for fractional-order positive systems (or fractional-order positive switched systems) was proposed. In [29], the problem of guaranteed cost finite-time control for positive switched linear systems with time-varying delays was considered and a cost function of positive systems (or positive switched systems) was also presented. Based on [29], [30] extended guaranteed cost finite-time control to positive switched nonlinear systems with -perturbation. It is worth noting that [28–30] are involved in continuous-time positive switched systems. However, the problem of guaranteed cost finite-time control for discrete-time positive impulsive switched systems is still open, which inspires us for this study.

In this paper, we consider the problem of GCFTB of discrete-time positive impulsive switched systems by constructing the MLCLF with average dwell time (ADT) technique. Firstly, the concept of guaranteed cost finite-time boundedness is extended to discrete-time positive impulsive switched systems. Secondly, a state feedback controller is designed and sufficient conditions are obtained to guarantee that the closed-loop system is GCFTB. Some sufficient conditions are obtained by linear programming.

The rest of the paper is organized as follows. Section 2 gives some necessary preliminaries and problem statements. In Section 3, the main results are given. In Section 4, a numerical example is provided. Section 5 concludes the paper.

Notations. The representation (0, ≺0, 0) means that (≥0, <0, ≤0), which is also applying to a vector. means that . is the -dimensional nonnegative (positive) vector space. denotes the space of matrices with real entries. represents the -dimensional vector . denotes the transpose of matrix . -norm is defined by . and are the sets of nonnegative and positive integers. denotes the set of positive integers. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.

2. Preliminaries and Problem Statements

Consider the following discrete-time positive impulsive switched systems:where , is the system state, and represents the control input. represents switching signal of system and takes values in a finite set , . In general, , , , and are the th subsystem if . is the initial time. denotes the th impulsive switching instant. Moreover, means that the th subsystem is active. and indicate that is a switching instant at which the system is switched from the th subsystem to the th subsystem. At switching instants, there exist impulsive jumps described by (1). , , , and are constant matrices with suitable dimensions, is the exogenous disturbance and defined as with a known scalar and a given finite-time threshold value .

Next, we will give some definitions and lemmas for system (1).

Definition 1. System (1) is said to be positive if for any switching signals , any disturbance input , and control input , the corresponding trajectory satisfies for all .

Lemma 2 (see [25]). System (1) is positive if and only if , , , and , where .

Definition 3. For any switching signal and any , let denote the switching numbers over the interval . For given and , if the inequalityholds, then is called an average dwell time, and is called a chattering bound. Generally, we choose

Definition 4 (finite-time stability (FTS)). For a given time and two vectors , discrete-time positive impulsive switched system (1) with is said to be FTS with respect to , if where is an any time point on the time interval .

Definition 5 (finite-time boundedness (FTB)). For a given constant , and two vectors , discrete-time positive impulsive switched system (1) is said to be FTB with respect to , where ) satisfies (2), if where is an any time point on the time interval .

Now we give some new definitions for our further study.

Definition 6. Define the cost function of discrete-time positive impulsive switched system (1) as follows:where and are two given vectors.

Remark 7. It should be noted that the proposed cost function is different from the general one, such as [26–28]; this definition provides a more useful description, because it takes full advantage of the characteristics of nonnegative states of discrete-time positive impulsive switched systems.

Definition 8 (GCFTB). For a given time constant and two vectors , consider discrete-time positive impulsive switched system (1) and cost function (6); if there exist a control law and a positive scalar such that the closed-loop system is FTB with respect to and the cost function satisfies , then the closed-loop system is called GCFTB, where is a guaranteed cost value and is a guaranteed cost finite-time controller.

3. Main Results

3.1. Guaranteed Cost Finite-Time Boundedness Analysis

In this subsection, we will focus on the problem of GCFTB for discrete-time positive impulsive switched system (1) with . The following theorem gives sufficient conditions of GCFTB for system (1) with .

Theorem 9. Consider the discrete-time positive impulsive switched system (1) with , for a given time constant , vectors and ; if there exist a set of positive vectors ,, , and positive constants , , , , such that the following inequalities hold:where and represents the th elements of the vectors , respectively, then under the following ADT scheme:system (1) with is GCFTB with respect to and the guaranteed cost value of system (1) with is given by

Proof. Construct the multiple linear copositive Lyapunov function (MLCLF) for system (1) with as follows:where .
Suppose a switching sequence . Without loss of generality, we assume that subsystem is activated at the switching instant and the subsystem is activated at the switching instant .
When , , along the trajectory of system (1) with , the difference of the MLCLF isFrom (7), (8) and , we haveIt implies thatWhen , , , . Along the trajectory of system (1) with , we haveFrom (9) and , , we haveSo, when , from (19), we getRepeating the procedure of (20) and noting , we obtainBy iterative operation, we get According to (2), and , we also have From (10) and (14), we have From (23) to (24) and , we obtainSubstituting (12) into (25), one hasAccording to Definition 5, we conclude that system (1) with is FTB with respect to .
Next, we will give the guaranteed cost value of system (1) with .
When , , according to (16), we knowSimilar to the proof process of (17)–(22), for any and , we can obtainNoting that , (28) can be rewritten as Substituting (2) into (29) and letting , we getThen we can obtain Therefore, according to Definition 8, we can conclude that system (1) with is GCGTB. Thus, the proof is completed.

3.2. Guaranteed Cost Finite-Time Controller Design

In this subsection, we are concerned with the guaranteed cost finite-time controller design of discrete-time positive impulsive switched system (1). Under the controller , the corresponding closed-loop system is given byBy Lemma 2, to guarantee the positivity of system (32), should be satisfied, . The following Theorem 10 gives some sufficient conditions to guarantee that the closed-loop system (32) is GCFTB.

Theorem 10. Consider the discrete-time positive impulsive switched system (32), for a given time constant , vectors , , and ; if there exist positive vectors , , and positive constants , , , , , such that ((8)–(11)) and the following inequalities hold:where , , and represents the th elements of the vectors , then under the following ADT scheme (12), the resulting closed-loop system (32) is GCFTB with respect to and the guaranteed cost value of system (32) is given by

Proof. From (33), we know that . According to Lemma 2, system (32) is positive. Next, we prove the guaranteed cost finite-time stability of system (32).
Replacing in (27) with , we haveSimilar to the proof process of (17)–(22), for any and , we can obtainNoting that , (37) can be rewritten asSubstituting (2) into (38) and letting , we getFrom (12), we can obtainThe proof is completed.