Abstract
The problem of input-output finite-time control of positive switched systems with time-varying and distributed delays is considered in this paper. Firstly, the definition of input-output finite-time stability is extended to positive switched systems with time-varying and distributed delays, and the proof of the positivity of such systems is also given. Then, by constructing multiple linear copositive Lyapunov functions and using the mode-dependent average dwell time (MDADT) approach, a state feedback controller is designed, and sufficient conditions are derived to guarantee that the corresponding closed-loop system is input-output finite-time stable (IO-FTS). Such conditions can be easily solved by linear programming. Finally, a numerical example is given to demonstrate the effectiveness of the proposed method.
1. Introduction
Positive switched systems are a class of dynamics whose state and output are nonnegative whenever the initial conditions and inputs are nonnegative. In the last decades, the research of positive switched systems is a hot topic due to their many practical applications in communication networks [1], the viral mutation dynamics under drug treatment [2], formation flying [3], systems theory [4, 5], and so forth. The stability analysis and controller synthesis of positive switched systems have been highlighted by many researchers [6–10]. However, most of the results mentioned above discussed the asymptotic stability (Lyapunov stability).
Contrary to asymptotic stability, some researchers focus on finite-time stability (FTS) of positive switched systems: that is, given a bound on the initial condition, the system state does not exceed a certain threshold during a specified time interval. Some works dealing with analysis and controller design of FTS control problems have been published [11–16]. Recently, the definition of IO-FTS is proposed by [17]; this definition is that a system is said to be IO-FTS if, given a class of norm bounded input signals over a specified time interval , the outputs of the system are also norm bounded over . IO-FTS involves signals defined over a finite-time interval and does not necessarily require the inputs and outputs to belong to the same class, and IO-FTS constraints permit specifying quantitative bounds on the controlled variables to be fulfilled during the transient response. This definition of IO-FTS is fully consistent with the definition of FTS. Some related results have been obtained in [18–25]. These results about IO-FTS are mainly involved in impulsive systems [18], linear systems [19–22], stochastic systems [23], Markovian jump systems [24], and impulsive switched linear systems [25]. It is worth noting that the results mentioned above are mainly concerned with nonpositive systems. Huang et al. [26] extended IO-FTS to a class of discrete-time positive switched systems with state delays via average dwell time (ADT), but the time-varying delay and distributed delay were not considered. As we know, multiple time delays, such as time-varying delays and distributed delays, usually occur in many practical systems and may result in system performance deterioration, even instability. Therefore, they must be taken into account in analyzing and implementing any controller scheme. Moreover, MDADT approach allows that every subsystem has its own ADT to make the individual properties of each subsystem unneglected, which is more applicable and less conservative compared with ADT. Then, it is necessary to investigate IO-FTS analysis of positive switched systems with time-varying and distributed delays via MDADT approach. However, this problem is still open.
Motivated by the aforementioned factors, the problem of IO-FTS of positive switched systems with time-varying and distributed delays is considered. The main contributions of this paper are as follows: () the positivity of positive switched systems with time-varying and distributed delays is proved; () the definition of IO-FTS is for the first time extended to positive switched systems with time-varying and distributed delays; () by using multiple copositive type Lyapunov function and MDADT approach, a state feedback controller is designed and sufficient conditions for IO-FTS of the corresponding closed-loop system are given. Such conditions can be easily solved by linear programming. The rest of this paper is organized as follows. In Section 2, problem statements and necessary lemmas are given. Main results are given in Section 3. A numerical example is provided in Section 4. Section 5 concludes this paper.
Notations. The representation (000) means that (000), which is applicable to a vector. means that . is the -dimensional nonnegative (positive) vector space. denotes the space of matrices with real entries. denotes the transpose of matrix . Let ; -norm is defined by . denotes the space of absolute integrable vector-valued functions on the interval ; that is, if holds. denotes the space of the uniformly bounded vector-valued functions on the interval ; that is, if holds. Matrices are assumed to have compatible dimensions for calculating if their dimensions are not explicitly stated.
2. Preliminaries and Problem Statements
Consider the following positive switched systems with time-varying and distributed delays:where is the system state and and represent the control input and output. is the system switching signal, where is the number of subsystems; , , , , , , and are constant matrices with appropriate dimensions, denotes time-varying delay, which satisfies , and is the distributed delay, where and are known positive constants. is the initial condition on , . is the exogenous disturbance and defined aswith a known scalar .
Next, we will present some definitions and lemmas for the positive switched system (1) to our further study.
Definition 1 (see [14]). System (1) is said to be positive if for any switching signals , any initial conditions , , and any disturbance input , the corresponding trajectory satisfies and for all .
Definition 2 (see [14]). is called a Metzler matrix if the off-diagonal entries of matrix are nonnegative.
Definition 3 (see [16]). For any switching signal and any , let denote the switching numbers where the th subsystem is activated over the interval and denote the total running time of the th subsystem over the interval . If there exist and such thatthen and are called MDADT and mode-dependent chattering bounds, respectively. Generally, we choose
Lemma 4 (see [16]). A matrix is a Metzler matrix if and only if there exists a positive constant such that .
Lemma 5 (see [9]). Let . Then , , if and only if is a Metzler matrix.
Lemma 6. System (1) is positive if and only if , , are Metzler matrices and , , , , , and .
Proof.
Sufficiency. For any , , let denote the switching instants on the interval . From system (1), , we haveIntegrating both sides of the first equation of the above equations from to , it yields It leads to By Lemma 5, if is a Metzler matrix, , , , , and , then it is easy to obtain that , , , and . Similar to the above process, , it follows that and . Recursively, if , , are Metzler matrices, , , , , and , then one has that , , .
Necessary. Conversely, suppose that there exists an element , , , where is in the th row and th column of . From system (1), we can obtain where , , and represent the th elements of , , and , respectively. Then, if , it is obvious that is possible whenever , which means that . It follows that system (1) is not positive.
Then, suppose that has an element , ; we obtain It can be obtained that is possible whenever . It shows that system (1) is not positive.
In the same way, if has an element , has an element , or has an element , , then system (1) is exactly not positive.
Finally, assume that there exists an element , , , where is in the th row and th column of . According to system (1), It is easy to obtain that is possible whenever ; it means that system (1) is not positive.
From the above, system (1) is positive under any switching signals if and only if are Metzler matrices, , , , , and , .
The proof is completed.
Next, we will give the definitions of input-output finite-time stability for the positive switched system (1).
Definition 7 (IO-FTS). Consider zero initial condition , for a given time constant , disturbances signals defined by (2), and a vector ; system (1) is said to be IO-FTS with respect to , ifIf the disturbance satisfies , is defined as then we give Definition 8.
Definition 8 (IO-FTS). Consider zero initial condition , for a given time constant , disturbances signals , and a vector ; system (1) is said to be IO-FTS with respect to , if
Remark 9. [26] gives two definitions of discrete positive switched systems for two classes of exogenous disturbances, respectively. Similarly, for continuous positive switched systems, two classes of exogenous disturbances for IO-FTS problem can be also proposed, that is, norm bounded integrable signals and the uniformly bounded signals . Because of the similar process, in this paper, we only focus on the former.
The aim of this paper is to design a state feedback controller and find a class of switching signals for the positive switched system (1) such that the corresponding closed-loop system is IO-FTS.
3. Main Results
3.1. IO-FTS Analysis
In this subsection, we will focus on the problem of IO-FTS of positive switched system (1) with . The following theorem gives sufficient conditions of IO-FTS of system (1) via the MDADT approach.
Theorem 10. Consider system (1) with . Given positive constants , , , and and a vector , if there exist positive vectors , , , , the following inequalities hold:where , , , represents the th column vector of the matrix , , , , and , , and represent the th elements of the vectors , , and , respectively. satisfies (18); then system (1) is IO-FTS for any switching signal with the MDADT
Proof. Construct the multiple copositive type Lyapunov-Krasovskii functional for system (1) as follows:where , , and , .
Along the trajectory of system (1) with , we haveCombining (20) and (21) yieldsNoting that , we haveFrom (22) and (23), we obtainFrom (14), (15), and (24), we haveIntegrating both sides of (25) during the period for leads toOn the other hand, from (18) and (20), one can easily obtainLet be the switching number of over , and denote as the switching instants over the interval . Combining with (2), (26), and (27), we haveAccording to (3) and choosing , for , one can obtain from (28)Under the zero initial condition , noting the definition of and (29), we haveFrom (3) and (30), one hasFrom (17) and (31), we haveFrom (19) and (32), we obtainThus, the proof is completed.
3.2. IO-FTS Controller Design
In this subsection, the state feedback controller will be designed. For system (1), under the controller , the corresponding closed-loop system is given byAccording to Lemma 4, to guarantee the positivity of system (34), should be Metzler matrices, . Theorem 11 gives some sufficient conditions to guarantee that the closed-loop system (34) is IO-FTS.
Theorem 11. Consider the positive switched system (34). Given positive constants , , and and a vector , if there exists a set of positive vectors , , , , , (15)–(17) and the following conditions hold:where , , , , represents the th column vector of the matrix , represents the th elements of vector , , , , and , , and represent the th elements of the vectors , , and , respectively. satisfies (18); then the closed-loop system (34) is IO-FTS for any switching signal with MDADT (19).
Proof. According to Lemma 4, we get that is a Metzler matrix for each . Replacing in (14) with , then letting , similar to Theorem 10, we can get (36) and the resulting closed-loop system (34) is IO-FTS with the MDADT (19).
The proof is completed.
Next, an algorithm is presented to obtain the feedback gain matrices , .
Step 1. By adjusting the parameters and solving (15)–(18) and (36) via linear programming, positive vectors , , , and can be obtained.
Step 2. Substituting and into , can be obtained.
Step 3. The gain is substituted into . If are Metzler matrices, then are admissible. Otherwise, return to Step 1.
4. Numerical Example
Consider the positive switched system (1) with the parameters as follows: Let , then we get . Choose . Solving the inequalities in Theorem 11 by linear programming, we haveBy , we get It is easy to verify that are Metzler matrices. Then, from (19), we can obtain , . Choose and . The simulation results are shown in Figures 1–3, where . The state trajectories of the closed-loop system with MDADT are shown in Figure 1. The switching signal with MDADT is depicted in Figure 2. Figure 3 plots the evolution of of system (1). From Figure 3, we easily know that , which implies that the resulting closed-loop system (1) is IO-FTS.
5. Conclusions
In this paper, the problem of IO-FTS of positive switched systems with time-varying and distributed delays is investigated. The definition of IO-FTS of continuous positive switched systems is proposed. Then, by constructing multiple linear copositive Lyapunov functions, a state feedback controller is designed. Based on the MDADT approach, some sufficient conditions are obtained to guarantee that the closed-loop system is IO-FTS. Such sufficient conditions can be solved by linear programming. Our further work will focus on the IO-FTS of positive switched nonlinear systems with multiple delays.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are grateful for the support of the National Natural Science Foundation of China under Grants U1404610, 61473115, 61374077, and 61203047.