Robust Stabilization of Controlled Positive Switched Linear Systems by Output Feedback

Liu, Jinjin; Zhang, Ting

doi:https://doi.org/10.1155/2021/8783032

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Advances in Stability and Control of Dynamical Systems

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Research Article | Open Access

Volume 2021 | Article ID 8783032 | https://doi.org/10.1155/2021/8783032

Robust Stabilization of Controlled Positive Switched Linear Systems by Output Feedback

Jinjin Liu¹and Ting Zhang¹

Academic Editor: Abdul Qadeer Khan

Received07 May 2021

Accepted16 Aug 2021

Published27 Aug 2021

Abstract

This study focuses on the controller synthesis issues for constrained switched linear systems with uncertainties under mode-dependent average dwell time (MDADT) switching strategy. First, output feedback controllers ensure that the closed-loop systems are positive and asymptotically stable. Second, the bounded controllers are acquired based on system states with interval and polytopic uncertainties. Also, the proposed approach can be applied to the systems with the constrained output. Then, the presented conditions can be formulated in terms of linear programming. Finally, illustrative example is provided to show the effectiveness of the theoretical results.

1. Introduction

A switched system consists of a family of subsystems and a law determining which subsystem is active during a certain time interval [1, 2]. In real world, the associated states are always nonnegative, such as the population size of species, level of liquid, and concentration of substances. Such models are described by positive systems [3, 4]. As a special switched system, the positive switched system has not only the properties of the switched system but also the nature of the positive system. The positive switched linear system receives much attention in recent years due to the widely practical applications, such as congestion control in communication networks [5], medical treatment [6], general anesthesia [7, 8], multiagent system [9], and so on.

Many researchers have been concerned with the controller design problem, especially the constrained controller synthesis for the general system and positive system [10–12]. In fact, it is inevitable that the control system encounters limited situations, for example, maximum current or highest voltage in the circuits, the maximum temperature in thermal systems, and so on. Therefore, it is of great practical significance to design a controller which makes the control signal and state of the closed-loop system be bounded by the specified boundary. The study of bounded control for the positive system starts from [13, 14], where stability and stabilization conditions for the positive system by means of linear composite Lyapunov function are proposed under the constrained state feedback controller. Then, the results are extended to the positive system with delay in [15, 16], Takagi–Sugeno fuzzy system [17], fractional order system [18], and event-triggered system [19]. Meanwhile, stability and stabilization issues of positive switched systems are developing rapidly. Reference [20] considered the stability analysis problem for a class of positive switched systems with average dwell time switching; then, [21] continued this idea and further studied this issue under MDADT switching, that is, each subsystem has its own average dwell time. Reference [22] tried to adapt the constrained control problem to both continuous and discrete-time positive switched systems with delays, focusing on the nonnegative controller design under a prescribed upper bound with the arbitrary switching signal. [23, 24] investigated the controller synthesis for a class of discrete-time switched linear systems with bounds on the controls and the states, which guarantees positivity and stability of the closed-loop system. However, most of these results are based on the state feedback.

On the other hand, output variables are easy to directly measure and technically implement and also have a clear physical meaning in most cases, so the output feedback is a common form of the feedback mode. Based on linear programming, [25] solved the stabilization issue by the output feedback controller having one rank gains. On the basis of the singular value decomposition approach, [26] revisited the output feedback stabilization problem with and without interval uncertainties. By using the system augmentation approach, an iterative linear matrix inequality algorithm is presented to compute the feedback gain matrix [27]. Turning to positive switched systems, [28] continued the linear programming method and addressed feedback control under system input matrix. Reference [29] involved controller synthesis via a constrained output feedback, but the rank of the controller gain is one. Furthermore, it is well known that because of modeling errors and variations of plant parameters, there always exists uncertainty due to some unpredictable factors in many real systems, and some effective results have been achieved in this area [30–32].

In this context, we consider robust stabilization of enforced positive switched linear systems by the output feedback with and without boundary under the MDADT switching scheme. The so-called controlled positive systems mean that the resulting closed-loop systems we received is positive, even if the open-loop system is not positive at all. In order to overcome the incomplete controller gain, we start from the elements view of the system matrix. By introducing quantity matrices, sufficient conditions for the existence of the output feedback controller are formulated, such that the closed-loop system can not only guarantee robust asymptotic stability but also satisfy positive. Simultaneously, inputs and outputs are limited by the prescribed boundary provided that the initial condition is within the same boundary.

The rest of study is structured as follows. Section 2 deals with problem statement and presents some preliminaries. Robust stabilization of controlled positive switched linear systems interval uncertainties and polytopic uncertainties are solved in in Sections 3 and 4, respectively. Section 5 gives a numerical example to reveal our design. Finally, Section 6 concludes this study.

1.1. Notations

denotes the vector of -tuples of real numbers, and means the space of matrices with real entries. For a vector and a matrix , and imply that all its elements are positive (nonnegative), respectively. Furthermore, and stand for all its elements are negative (not positive), respectively. For two matrices , , and are the elements in the th row and th column of and for and . So, means that , and denotes for all and , respectively.

2. Problem Statement and Preliminaries

In this section, we present precise problem description for switched systems under consideration. Consider the following switched linear systems:where , , and denote the state vector, control input, and output, respectively. The switching signal is a piecewise constant and right continuous function of time , with being the number of modes of the overall switched system. implies that the th subsystem is active, and , , and are the system matrices.

The main problem we studied in this article is the following: under the premise of system matrices are not precisely known, the output feedback must be set up in such a way that the resulting closed-loop system is positive and asymptotically stable. In addition, the control law we designed should be not restricted in sign and limited within certain boundary. To be specific, consider the output feedback controller , leading to the corresponding closed-loop system.is positive and asymptotically stable under MDADT switching signal, while the open-loop system may not be positive. Here, .

Before presenting the main results of this study, the following definitions and lemmas need to list.

Definition 1 (see [22]). The continuous-time linear systemis said to be positive if for any switching signal and any initial condition , and the corresponding trajectory of system holds for all .

Definition 2 (see [4]). Matrix is called a Metzler matrix if the off-diagonal entries of are nonnegative.

Definition 3 (see [21]). For a switching signal and any , let be the switching number that the ^th subsystem is activated in time interval and denotes the total running of the ^th subsystem in time interval . If there exists a nonnegative constant , such thatthen is a MDADT of , and is the mode-dependent chattering bound.

Lemma 1 (see [22]). System (3) is positive if and only if is a Metzler matrix for each .
Here, it is worth mentioning that the closed-loop system (2) is positive only if system matrix is Metzler for each .

Lemma 2 (see [21]). If there exist vector and constant , such that , a SPLS is asymptotically stable under the MDADT switching signal , where for .

Lemma 3 (see [22]). Consider a SPLS ; if there exist vector and constant , such that sets up for , then the state trajectory of SPLS satisfies that for .

3. Robust Stabilization with Interval Uncertainties

In this section, we consider the controller synthesis issue if switched systems with interval uncertainties. Here, the subsystem matrices and are not precisely known but belong to the following interval uncertainty domain: and . In addition, we divide this section into three cases.

3.1. Sign-Restricted Controls

First of all, a fundamental result for robust stabilization of switched linear systems under MDADT switching strategy with nonnegative or negative controls is now put forward. Let us begin by the condition .

Theorem 1. Consider the switched linear system (1) with interval uncertainties. For prescribed constants , if there exist diagonal matrices with positive diagonal entries and matrices subjected tothen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under MDADT switching strategy , where , . Furthermore, holds for initial condition , where .

Proof. Using the output feedback gain matrix, it follows thatNote the fact that , and it is easy to giveUsing the fact (5) and recalling the condition , one can derive for . In consequence, is the Metzler matrix and the closed-loop (2) system is positive by Lemma 1.
With the vector and matrix operation rule in mind, we haveBy (6), we can get , that is, the closed-loop system (2) is asymptotically stable by Lemma 2.
Finally, it derives that from (7). Since , it yields . Thus, under the output feedback controller and (7), sets up.

Remark 1. We do not restrict the state of the original system because our goal is to design the output feedback controller, so that the closed-loop system is positive. This is so-called the controlled positive system.
Next, we turn our attention on negative control, that is, .

Theorem 2. Consider the switched linear system (1) with interval uncertainties. For predetermined constants , if there exist diagonal matrices with positive diagonal entries and matrices , such thatthen the closed-loop system (2) is positive and asymptotically stable under MDADT switching strategy , where , . Moreover, the output feedback gain is given by . Also, sets up, and assume that the initial condition is for .

Proof. By (11) and (12), it verifies that the closed-loop system (2) is positive and asymptotically stable using a similar proof to Theorem 1. Taking and (13) into account, we can obtain . So, it follows that

Remark 2. For the model of a positive system, system control is also required to be nonnegative. On the contrary, negative controllers are also very common in real life. So, it is important to talk about sign-constrained controllers.

3.2. Bounded Controls

In this subsection, the bounded control problems are investigated for the switched system with interval uncertainties under MDADT switching. That is, the designed output feedback controllers allow positivity and asymptotic stability of the closed-loop system with the initial condition , and the controllers should be limited to boundaries decided in advance. In this sequel, we consider the condition of for fixed .

Theorem 3. Consider the switched linear system (1) with interval uncertainties. For fixed constants , if there exist diagonal matrices with positive diagonal entries and matrices , such thatthen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under MDADT switching strategy , where and . Furthermore, the restricted condition meets if initial condition is , where .

Proof. By Theorem 1, the closed-loop system is positive and asymptotically stable, and also sets up. Furthermore, it is clear that holds for the initial condition meets by Lemma 3. Then, we haveLet . We have . Recalling the output feedback gain matrix, we can obtainThen, the inequality holds by (17).
In the following, we aim to the condition of controlled positive switched systems for prescribed .

Theorem 4. Consider the switched linear system (1) with interval uncertainties. For given constants , if there exist diagonal matrices with positive diagonal entries and matrices , such thatthen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under MDADT switching strategy , where , . Moreover, fulfills suppose that initial condition satisfies , where .

Proof. The proof is similar to Theorem 3 and therefore omitted here.
At the end of this subsection, asymmetrically bounded control is addressed to stabilize switched systems with interval uncertainties for given and .

Theorem 5. Consider the switched linear system (1) with interval uncertainties. For given constants , if there exist diagonal matrices with positive diagonal entries and matrices , , such thatthen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under MDADT switching strategy , where , . Moreover, holds if the initial condition is , where .

Proof. According to the proof of Theorem 1, the closed-loop system is positive and asymptotically stable for the output feedback gain matrix . Since, inequalities (21) and (22) are together with Lemma 3, , sets up for the initial condition . Note that vector and combining with (23)–(26), we can getUsing the properties of the inequality, (27) can be rewritten asSo, one hasThis completes the proof.

3.3. Constrained Output

The matrix is nonnegative. Under this general premise, this subsection handles the constrained output issue for controlled positive switched systems. The objective is to design the output feedback controller, such that the corresponding closed-loop system is positive and asymptotically stable via MDADT switching signals. Here, the output satisfies the bound for fixed .

Theorem 6. Consider the switched linear system (1) with interval uncertainties. Assume that . For given constants , if there exist diagonal matrices with positive diagonal entries and matrices satisfyingthen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under MDADT switching strategy , where , . In addition, satisfies if initial condition is , where .

Proof. It is similar to the previous proof of Theorem 1, and we know that the closed-loop system (2) is positive and asymptotically stable from (31) and (32). By using and Lemma 3, it follows . Together with (33), we immediately arrive at constraints .

4. Robust Stabilization with Polytopic Uncertainties

This section is focused on robust stabilization for controlled positive switched systems with polytopic uncertainties. Suppose that the subsystem matrices and are not precisely known but belong to the following polytopic uncertainty set:where and are the certain matrices and stand for extreme points of the ^th subsystem, and denotes the total number of extreme points. Except, we are talking about asymmetrically constrained control and constrained output cases. Others can be discussed similarly.

Theorem 7. Consider the switched linear system (1) with polytopic uncertainties. For given constants , if there exist diagonal matrices with positive diagonal entries and matrices , , such thatthen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under MDADT switching strategy , where , . Moreover, can be achieved if the initial condition , where .

Proof. First, from the closed-loop system matrix, one hasThen, it is not hard to see thatNotice the matrix element of the output feedback gain, and it follows that . Hence,By virtue of (35), holds for . Thus, is the Metzler matrix due to . This means that is also the Metzler matrix by a convexity of argument, so the closed-loop (2) system is positive by Lemma 1.
Next, consider vector , and we haveThe above together with (36) means that , and it follows . Thus, the closed-loop system is asymptotically stable by Lemma 2.
The rest of the proof is similar to Theorem 5.

Remark 3. In fact, the processes of interval uncertainties and polytopic uncertainties can convert each other. If and , Theorem 7 can be treated in the same way as Theorem 5.

Theorem 8. Consider the switched linear system (1) with polytopic uncertainties. Assume that . For given constants , if there exist diagonal matrices with positive diagonal entries and matrices subjected tothen the closed-loop system (2) with the output feedback gain is positive and asymptotically stable under the MDADT switching strategy , where , . And then, holds if , where .

Proof. It is similar to the previous proof of Theorem 7, and we know that the closed-loop system (2) is positive and asymptotically stable from (45) and (46). The rest section can be deduced by Theorem 6.

Remark 4. It should be pointed out that we discuss the general case of switching signals in this study. When we choose and for , MDADT switching strategy degenerates into the simple case of average dwell time. Furthermore, if we make , , and , the presentation of our conclusion becomes that under the arbitrary switching signal.

5. Numerical Examples

Consider system (1) with the interval uncertainties, where the bounds of the system matrices are given by

In this part, we take the constrained output feedback controller , for example, take . Address inequalities (14)–(17) in Theorem 3 by utilizing the linear programming algorithm in MATLAB, and it can readily obtain that , , and matrices

Then, the output feedback gain matrix can be calculated by

It is easy to see that the gain matrices and are nonsingular, and the rank of the matrix is full. Furthermore, the extreme closed-loop system matrices are

We can observe that and are the Metzler matrices. Select and for ; hence, the close-loop system is positive. Let , and , ; we get the minimum MDADT and . The state trajectories of the closed-loop system together with the extreme plants under initial are given in Figures 1 and 2. Figures 3 and 4 plot the control input, which illustrates that the controls meet the fixed bound when the initial value is limited to . The switching signal with MDADT is shown in Figure 5.

6. Conclusion

The problem of robust stabilization of controlled positive switched linear systems by the output feedback is addressed in this article. First, we pay attention to find constrained output feedback controllers rendering the positiveness and asymptotic stability of the closed-loop systems, eventhough the open-loop system is not positive. Second, by utilizing the MDADT switching scheme, the robust stabilization of the controlled positive switched system is solved under interval and polytopic uncertainties. It is worth pointing out that all the controllers we designed satisfy certain limiting conditions, including bounded control inputs and constrained outputs. All the conditions can be solvable in terms of linear programming, instead of LMI. Along the approach of this study, further works may refer to delay problems and nonfragile reliable control.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of the People Republic of China (61803144, 11872175, and 62073122) and the Key Scientific Research Projects for Colleges and Universities of Henan Province (19A120001).

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Copyright © 2021 Jinjin Liu and Ting Zhang. 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.

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