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Journal of Control Science and Engineering
Volume 2017 (2017), Article ID 7051658, 10 pages
https://doi.org/10.1155/2017/7051658
Research Article

Guaranteed Cost Finite-Time Control for Positive Switched Linear Systems with Time-Varying Delays

College of Information Engineering, Henan University of Science and Technology, Luoyang 471023, China

Correspondence should be addressed to Leipo Liu; moc.361@321opieluil

Received 21 September 2016; Accepted 16 January 2017; Published 19 February 2017

Academic Editor: Ai-Guo Wu

Copyright © 2017 Xiangyang Cao 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 guaranteed cost finite-time control for positive switched linear systems with time-varying delays. The definition of guaranteed cost finite-time boundedness is firstly given. Then, by using the mode-dependent average dwell time approach, a static output feedback law and a state feedback control law are constructed, respectively, and sufficient conditions are obtained to guarantee that the closed-loop system is guaranteed cost finite-time boundedness. Such conditions can be easily solved by linear programming. Finally, an example is given to illustrate the effectiveness of the proposed method.

1. Introduction

Positive switched systems are a class of hybrid systems consisting of a family of positive subsystems and a switching law that specifies which subsystem will be activated along the system trajectory at each instant of time. In recent years, many remarkable results about the problems of stability analysis and controller synthesis of positive switched systems have been presented [18]. Among these existing results, the average dwell time (ADT) approach and mode-dependent average dwell time (MDADT) technique are widely used; the latter allows that every subsystem has its own average dwell time to make the individual properties of each subsystem unneglected, which is more applicable and less conservative compared with ADT technique [9].

In fact, many studies focus on the asymptotic stability in the area of positive switched systems, which reflects the asymptotic behavior of the system in an infinite time interval. However, in many practical applications, one is more interested in what happens over a finite-time interval. Peter [10] has firstly defined finite-time stability (FTS) for linear deterministic systems, which means that, given a bound on the initial condition, the system state does not exceed a certain threshold during a specified time interval. Then Amato et al. [11] extended this definition to finite-time boundedness (FTB) when they dealt with the behavior of the state in the presence of external disturbances. Then many related results have been presented [1217]. Recently, [18] first extends the concept of FTS to positive switched systems and gives some FTS conditions of positive switched systems. In [19], the problem of finite-time control for a class of positive switched linear systems with time-varying delays is considered and the concept of finite-time boundedness is also proposed. In [20], the problem of output feedback finite-time control for switched positive systems with time-varying delays is investigated via MDADT approach. On the other hand, when controlling a real plant, it is also desirable to design a control system which not only is stable but also guarantees an adequate level of performance. One approach to this problem is the guaranteed cost control. It has the advantage of providing an upper bound on a given system performance index and thus the system performance degradation incurred by the uncertainties or time delays is guaranteed to be less than this bound [21]. So it is necessary to study the design problem of guaranteed cost finite-time controller. There are some results about this problem; see [2229] and references therein. These results mainly focus on nonpositive systems, involved in fuzzy systems [22, 27], continuous-time nonlinear systems [23], interconnected systems [24], Markov switching systems [25], stochastic systems [26, 28], neural networks [29], and so on. However, to the best of our knowledge, there are few results available on guaranteed cost finite-time control for positive switched linear systems with time-varying delays, which motivates our present study.

In this paper, the problem of the finite-time boundedness with guaranteed cost control for positive switched linear systems with time-varying delays is considered. Firstly, the definition of guaranteed cost finite-time boundedness is given, which is extended to positive switched linear systems with time-varying delays for the first time. Secondly, by using the MDADT approach and copositive Lyapunov-Krasovskii functional method, a static output feedback and a state feedback are designed, respectively, and sufficient conditions are obtained to guarantee that the closed-loop system is guaranteed cost finite-time boundedness (GCFTB). Such conditions can be easily solved by linear programming.

Notations. Throughout this paper, (0, 0, 0) means that all entries of matrix are nonnegative (positive, nonpositive, negative). is the -dimensional nonnegative (positive) vector space, the -dimensional Euclidean space is denoted by and represents the space of matrices with real entries, and is the transpose of a matrix . , where is the th element of ; if not explicitly stated, matrices are assumed to have compatible dimensions.

2. Problem Statements and Preliminaries

Consider the following positive switched linear systems with time-varying delays:where ,  , and represent the system state, control input, and control output, respectively. is the disturbance input, which satisfies is the switching signal, where is the number of subsystems; ,  ,  ,  ,  , and are constant matrices with appropriate dimensions, denotes th subsystems, and denotes the th switching instant; is the initial condition on ,  , and   denotes the time-varying delay satisfying ,   , where and are known positive constants.

Then we will present some definitions and lemmas for the following positive switched systems:

The following definitions and lemmas are necessary for our further study.

Definition 1 (see [19]). System (3) is said to be positive if, for any initial conditions ,  , disturbance input , and any switching signals , the corresponding trajectory holds for all .

Definition 2 (see [19]). is called a Metzler matrix if the off-diagonal entries of matrix are nonnegative.

Lemma 3 (see [6]). A matrix is a Metzler matrix if and only if there exists a positive constant such that .

Lemma 4 (see [19]). System (3) is positive if and only if ,  , are Metzler matrices and ,  ,  ,  , and .

Definition 5 (see [20]). For any switching signal and any , let denote the switching numbers that the th subsystem is activated over the interval and denote the total running time of the th subsystem over the interval . If there exist ,  ,  , such that Then and are called MDADT and mode-dependent chatter bounds, respectively. Generally, we choose .

Definition 6 (FTS). For a given time constant and vectors , positive switched system (3) with is said to be finite-time stable with respect to , ifIf the above condition is satisfied for any switching signals , then system (3) is said to be uniformly finite-time stable with respect to .

Definition 7 (FTB). For a given time constant and vectors , positive switched system (3) is said to be finite-time boundedness with respect to , where satisfies (2), if (5) applies.

Now we give some definitions about GCFTB for the positive switched systems (1).

Definition 8. Define the cost function of positive switched system (1) as follows: where and are two given vectors.

Remark 9. It should be noted that the proposed cost function is different from the existing literatures [2124]; it is for the first time introduced in positive switched systems. This definition provides a more useful description, because it takes full advantage of the characteristics of nonnegative states of positive switched systems.

Definition 10 (GCFTB). For a given time constant and vectors , consider positive switched system (1) and cost function (6); if there exist a feedback 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.

The aim of this paper is to design a static output feedback controller , a state feedback controller , and a class of switching signals for positive switched system (1) such that the corresponding closed-loop system is GCFTB.

3. Main Results

3.1. GCFTB Analysis

In this subsection, we will focus on the problem of GCFTB for positive switched system (3). The following theorem gives sufficient conditions of GCFTB for system (3) with the MDADT.

Theorem 11. Consider the positive switched system (3). For given constants ,  ,   and vectors and , if there exist a set of positive vectors ,  ,  ,  , and positive constants ,  ,  , and , the following inequalities hold: where represents the th column vector of the matrix ,  ,  , and , and  ,  , and represent the th elements of the vectors ,  , and , respectively; then under the following MDADT scheme system (3) is GCFTB with respect to , where    satisfies And the guaranteed cost value of system (3) is given by

Proof. Construct the following copositive Lyapunov-Krasovskii functional for system (3): where ,  , and ,  .
For the sake of simplicity, is written as in this paper.
Along the trajectory of system (3), we haveCombining (8), (9), (15), and (16) leads toSubstituting (7) into (17) yields It implies Integrating both sides of (19) during the period for leads to For any , let be the switching number of over , and denote as the switching instants over the interval . Then, for ,   is easily obtained from (13) and (15). Combining with (19), we have According to the definition of MDADT, one can obtain from (21)Choose . Therefore, for Noting the definition of and (8), we have Combining (21)–(25), we obtain Substituting (10) into (26), one has From Definition 7, we conclude that system (3) is FTB with respect to
Next, we will give the guaranteed cost value of system (3).
According to (17), denoting and integrating both sides of (18) from to for , it gives rise to Similar to the proof process of (21), for any , we can obtain where . From (29), we can get Multiplying both sides of (30) by leads to Noting that and , we obtain that ; that is, .
Then (31) can be turned intoLet ; then multiplying both sides of (32) by leads to Substituting (2) into (33) yields which can be rewritten as Substituting (25) into (35), the guaranteed cost value of system (3) is given by Therefore, according to Definition 10, we can conclude that system (3) is GCFTB with respect to . Thus, the proof is completed.

Remark 12. In Theorem 11, if in (12), then one can obtain , which means that the switching signal can be arbitrary.

When the exogenous disturbance , the result on guaranteed cost finite-time stability can be obtained as follows.

Corollary 13. Consider switched positive system (3) with . For given constants ,  , and vectors ,  , if there exist a set of positive vectors ,  ,  ,  , and positive constants ,  ,  , and , such that (7), (8), and the following inequality holdsthen under the following MDADT schemesystem (3) is guaranteed cost finite-time stable to , where satisfies (13); the corresponding guaranteed cost value is given by

3.2. GCFTB Controller Design

In this section, we are concerned with the GCFTB controller design of positive switched system (1). The static output feedback control law and state feedback control law will be designed to ensure the close-loop system (1) is GCFTB, respectively.

3.2.1. Static Output Feedback Controller Design

Consider system (1); under the controller , the corresponding closed-loop system is given by

By Lemma 3, to guarantee the positivity of system (40), should be Metzler matrices, Theorem 14 gives some sufficient conditions to guarantee that the closed-loop system (1) is GCFTB.

Theorem 14. Consider the positive switched system (1). For given constants ,  ,   and vectors ,  , and , if there exist a set of positive vectors ,  ,  ,  , and positive constants ,  ,  , and , (8)–(10) and the following conditions hold: where, where represents the th elements of vector ,   represents the th column vector of the matrix , and ,  , and ,  ,  , and represent the th elements of the vectors ,  , and , respectively, and    satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (40) is GCFTB with respect to and the guaranteed cost value of system (40) is given by

Proof. By Lemma 3, we know that is a Metzler matrix for each . According to Lemma 4, system (40) is positive if ,  ,  , and ,   are all nonnegative. Replacing in (7) with and letting , similar to Theorem 11, we easily obtain that the resulting closed-loop system (40) is GCFTB with respect to and the guaranteed cost value is given by (44).
The proof is completed.

3.2.2. State Feedback Controller Design

Consider system (1); under the controller , the corresponding closed-loop system is given by

By Lemma 3, to guarantee the positivity of system (45), should be Metzler matrices, The following Theorem 15 gives some sufficient conditions to guarantee that closed-loop system (1) is GCFTB.

Theorem 15. Consider the positive switched system (1). For given constants ,  ,   and vectors ,  , and , if there exist a set of positive vectors ,  ,  ,  , and positive constants ,  ,  , and , (8)–(10) and the following conditions hold:where, where   represents the th elements of vector ,   represents the th column vector of the matrix , and ,  , and ,  ,  , and represent the th elements of the vectors ,  , and , respectively, and    satisfies (13); then under the MDADT scheme (12), the resulting closed-loop system (45) is GCFTB with respect to and the guaranteed cost value of system (45) is given by

Proof. By Lemma 3, we know that is a Metzler matrix for each . According to Lemma 4, system (45) is positive if ,  ,  , and are all nonnegative. Replacing in (7) with and letting , similar to Theorem 11, we easily obtain that the resulting closed-loop system (45) is GCFTB with respect to and the guaranteed cost value is given by (49).
The proof is completed.

Next, a procedure is presented to obtain the feedback gain matrices ,  .

Step 1. By adjusting the parameters and solving (8)–(10), (13), (42), and (47) via linear programming, positive vectors ,  ,  , and can be obtained.

Step 2. Substituting and into   (or ),   can be obtained.

Step 3. The gain is substituted into   (or ). If   (or ) are Metzler matrices, then are admissible. Otherwise, return to Step 1.

4. Numerical Example

In this section, a numerical example in [6] is provided to show the effectiveness of the proposed approach. Consider positive switched system (1) with the parameters as follows: Assuming that , we can get and .

Let Choosing the parameters ,  ,  ,  , and   and solving the inequalities in Theorem 14 by linear programming, we get (I) By , we obtain the static output feedback controller: (II) By , we obtain the state feedback controller:

It is easy to verify that (41) and (46) are satisfied. Then, according to (12), we get and  .

The simulation results are shown in Figures 15; the initial conditions of system (1) are and ,  , which satisfy the condition . Figure 1 depicts the switching signal with MDADT. Under the static output feedback control law, Figures 2 and 3 are obtained to show the state trajectories and the evolution of of the closed-loop system (1). Under the state feedback control law, Figures 4 and 5 depict the state trajectories and the evolution of of the closed-loop system (1). The cost value is , which can be obtained by (44) or (49).

Figure 1: Switching signal of system (1).
Figure 2: State trajectories of the closed-loop system (1) under static output feedback law.
Figure 3: The evolution of of system (1) under static output feedback law.
Figure 4: State trajectories of the closed-loop system (1) under state feedback law.
Figure 5: The evolution of of system (1) under state feedback law.

Remark 16. Reference [6] considered the problem of exponential output tracking control for positive switched linear systems with ADT, which focuses on the asymptotic stability and the exponential performance index. There are some differences between [6] and this paper. Firstly, the finite-time stability in this paper and the asymptotic stability in [6] are two absolutely different concepts. Secondly, the performance indexes are also different. Finally, this paper uses MDADT approach instead of ADT method in [6]. So we do not compare the simulation results.

5. Conclusions

This paper has dealt with the problem of guaranteed cost finite-time control for positive switched systems with tine-varying delays. A novel guaranteed cost performance index is introduced. Based on the MDADT approach, some sufficient conditions and two kinds of controllers are obtained to guarantee that the closed-loop system is GCFTB. Finally, a numerical example is provided to show the effectiveness of the proposed method.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful for the supports of the National Natural Science Foundation of China under Grants U1404610, 61473115, 61374077, and 61203047.

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