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Complexity
Volume 2018, Article ID 1365960, 7 pages
https://doi.org/10.1155/2018/1365960
Research Article

Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems

1School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China
3School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Yuangong Sun; moc.361@gnognauynus

Received 7 November 2017; Accepted 31 January 2018; Published 26 February 2018

Academic Editor: Arturo Buscarino

Copyright © 2018 Yuangong Sun 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

Lyapunov functions play a key role in the stability analysis of complex systems. In this paper, we study the existence of a class of common weak linear copositive Lyapunov functions (CWCLFs) for positive switched linear systems (PSLSs) which generalize the conventional common linear copositive Lyapunov functions (CLCLFs) and can be used as handy tool to deal with the stability of PSLSs not covered by CLCLFs. We not only establish necessary and sufficient conditions for the existence of CWCLFs but also clearly describe the algebraic structure of all CWCLFs. Numerical examples are also given to demonstrate the effectiveness of the obtained results.

1. Introduction

Positive dynamical system is one for which nonnegative initial conditions give rise to nonnegative trajectories [13]. In recent years, stability issue for PSLS has been addressed for both practical applications in cooperative control of multiagent systems [48] and for theoretical reasons in [917] to name a few. For PSLSs, linear copositive Lyapunov functions play an important role in the stability analysis. It is well known that the existence of CLCLFs implies asymptotic stability of PSLSs under arbitrary switching. Moreover, necessary and sufficient conditions for the existence of CLCLFs have been extensively investigated in [1822].

Since the existence of CLCLFs is only a sufficient condition for asymptotic stability of PSLSs under arbitrary switching [18], it is necessary and significant to study asymptotic stability of the PSLS when it does not have a CLCLF. Motivated by the idea in [23, 24], where common joint quadratic Lyapunov functions were introduced for the first time, a class of common joint linear copositive Lyapunov functions (CJCLFs) were proposed to design time-dependent switching signals under which the PSLS is asymptotically stable [25, 26]. Moreover, such a method in [26] has been successfully applied to consensus of multiagent systems.

Notice that CJCLFs play an important role in the stability analysis of the PSLS. It is necessary to make it clear whether the PSLS has a CJCLF. So far, the existence of CJCLFs is still untouched except for the simpler cases and in [27]. Unlike CLCLFs, CJCLFs are determined by a series of nonstrict inequalities on each individual system combined with a strict inequality satisfied jointly, which leads to some difficulty in the analysis of the existence of CJCLFs.

In order to better solve the existence of CJCLFs, we will first introduce a class of common weak linear copositive Lyapunov functions (CWCLFs) determined only by a series of nonstrict inequalities on each individual system. By using matrix theory, necessary and sufficient conditions for the existence of CWCLFs have been established. What is more, the algebraic structure of all CWCLFs for PSLSs has been portrayed clearly. Consequently, the existence of CJCLFs becomes easily verifiable based on the algebraic structure of CWCLFs.

The paper is organized as follows. In Section 2, we will present the notations used throughout this paper as well as some preliminary results that are used later. Section 3 then focuses on deriving necessary and sufficient conditions for the existence of CWCLFs for PSLSs. In Section 4, we give two examples to demonstrate the effectiveness of the obtained theoretical results. Finally, conclusions are drawn in Section 5.

2. Problem Statement and Preliminaries

Throughout this paper, is the set of integers for any positive integer . If all entries of vector are positive (nonpositive, negative), we denote . For a matrix , denote if all its entries are nonpositive. Denote the -th column and the -th component of matrix by and , respectively. is an -dimensional identity matrix. A Metzler matrix is a real square matrix, whose off-diagonal entries are nonnegative. A square matrix is Hurwitz if the real part of each of its eigenvalues is negative.

Consider the following continuous-time switched linear system:where is the -dimensional state vector, the piecewise continuous function is the switching signal, and is an -matrix for each .

As usual, system (1) is said to be positive, if for any , any , and arbitrary switching [12]. We know that system (1) is positive if and only if is a Metzler matrix for each . A CLCLF method is usually used for asymptotic stability of PSLS (1) under arbitrary switching. Given an -dimensional vector , (or briefly ) is said to be a CLCLF of PSLS (1) (or the family of Metzler matrices ) if

Note that (2) is only a sufficient condition for asymptotic stability of PSLS (1) under arbitrary switching. There are obviously many examples where such a sufficient condition does not hold even if PSLS (1) is asymptotically stable under arbitrary switching. Therefore, we consider the following weaker condition:

In order to guarantee asymptotic stability of PSLS (1) under appropriately chosen switching signals, CJCLFs were proposed in [27]. Given an -dimensional vector , is said to be a CJCLF of PSLS (1) if (3) holds andFor the case , it was shown in [25] that PSLS (1) is asymptotically stable under arbitrary switching if it has a CJCLF. Therefore, CJCLFs play an important role in the analysis for asymptotic stability of PSLS (1).

For particular cases and , the existence of CJCLFs of PSLS (1) has been studied in [27]. For the general case, it remains unexplored so far. In this paper, we will introduce the definition of CWCLFs. Given an -dimensional vector , (or briefly ) is said to be a CWCLF of PSLS (1) (or ) if (3) holds. If the algebraic structure of all CWCLFs can be clearly described, condition (4) becomes easily verifiable, and hence the existence of CJCLFs can be solved accordingly.

Under the assumption that there exists a CWCLF of , we have (H1): for any and ; for all if for some and . In the following, it is always assumed that (H1) holds. Otherwise, does not have a CWCLF.

Note that has a CWCLF if and only if the family of Metzler matrices has a CWCLF, where is a diagonal matrix with For the sake of convenience, assume throughout this paper that for and .

In the sequel, we define a sequence of positive integers (SPI) for such that . Denote the -matrix: where for and the nonempty index set for . Let where is a -matrix obtained from by deleting all its rows except those labelled by .

Remark 1. It follows from assumption (H1) that has a CWCLF if and only if has a CWCLF. Moreover, for any SPI and any , has a CWCLF if has a CWCLF.
For , decompose the matrix as follows:where and are the corresponding -dimensional column vectors; for . If the matrix is invertible, the equationhas a unique solution, where is a -dimensional column vector. We denote the solution of (9) by when it has a unique solution. Let

We now introduce several lemmas required in the proof of the main results. Since new notations are introduced in this paper, the following lemmas in Wu and Sun (2013) are rewritten appropriately.

Lemma 2 (see [22]). Given an SPI and , if has a CLCLF, then there exists a -tuple , , such thatwhere , .

For , noting that for , for any ; it is obvious that has a CLCLF. By Lemma 2, let That is, and are always well defined.

Generally speaking, given an SPI and , if has a CLCLF, by Lemma 2, we can define

Lemma 3 (see [22]). Given an SPI and , has a CLCLF if and only if .

Lemma 4 (see [28]). For an -Metzler matrix , if is Hurwitz, then .

3. Main Results

We first present the following lemma which plays a key role in the proof of the main results.

Lemma 5. Given (), assume that has a CLCLF for any SPI . Suppose also that has a CWCLF for some SPI ; then , and all CWCLFs of have the form when , where is a constant.

Proof. Since has a CLCLF, by using Lemma 2, we see that and are well defined by (14) and (15) for the given SPI . Suppose that is a CWCLF of . Set , where is a -dimensional column vector and is an appropriate constant. It is obvious that is also a CWCLF of Consequently, we get from (8) thatOn the other hand, by the definition of , we have that where , , are defined as in Lemma 2. This together with (16) yields thatNoting that is Hurwitz since has a CLCLF, it follows from Lemma 4 and (19) thatSubstituting (20) into (17) givesIt implies that .
Next, we show that if . By the definition of , there exists an index such thatFrom (21) and (22), we haveIf , we can directly conclude that from (20) and (23). Otherwise, has at least one zero entry. For the sake of convenience, assume that the last component of is zero, and all the others are positive. That is,where is a -dimensional column vector. Setwhere and are the corresponding -dimensional column vectors; and are appropriate constants. From (22)–(25), we obtainSince , we can get from (20) and (27) thatNow, it is sufficient to show that . Otherwise, from (20). We now decompose the -matrix into the following form: From (19), (25), and (28), we have Since , the above inequality implies that is a zero vector. Based on (9) and (25), a straightforward computation yields that It implies that . From (12), (14), (15), and (26), we get which is a contradiction with the fact that has a CLCLF. Consequently, when ; that is, all CWCLFs of have the form . This completes the proof of Lemma 5.

Remark 6. Assume that the family of Metzler matrices has a CWCLF. Noting that there is always a CLCLF of for any , we get from Lemma 5 that for any SPI . If for any SPI , Based on Lemma 3, has a CLCLF for any SPI . By using Lemma 5 again, we further get for any SPI . Therefore, the existence of CWCLFs of implies that there is at least one such that for any SPI and any . For the trivial case , we denote .

We first establish a result for the case when in Remark 6. That means (H2) for any SPI and any .

Theorem 7. Assume that (H2) holds. There exists a CWCLF of if and only if and when . Moreover, all CWCLFs of are the same as CLCLFs if , and all CWCLFs of have the form if , where is a constant.

Proof.
Necessity. We first get from (H2) and Lemma 3 that has a CLCLF for any SPI and any . By using Lemma 5, we have that . If , we conclude from Lemma 3 that has a CLCLF. Therefore, all CWCLFs of are the same as CLCLFs. If , from Lemma 5, we see that all CWCLFs of have the form , and hence .
Sufficiency. If , we get from (H2) and Lemma 3 that there is a CLCLF of , which is also a CWCLF of . If , we get from (11), (14), and (15) that is a CWCLF of since . This completes the proof of Theorem 7.

For the particular case when for and , if there is a CWCLF of , then there is a CLCLF of , and hence , , by Lemma 3. Suppose also that for and . Following the proof of Lemma 5, we see that Lemma 5 holds true under the assumption that has a CLCLF. Similar to the proof of Theorem 7, we have the following corollary.

Corollary 8. Assume that and for and . There exists a CWCLF of if and only if , , , , and when . Moreover, all CWCLFs of are the same as CLCLFs if , and all CWCLFs of have the form if , where is a constant.

Next, we consider the case when the assumption in Theorem 7 does not hold. By Remark 6, we have that (H3) there exists an integer with such that for any SPI and any . In addition, there exists an SPI such that .

For the sake of convenience, we assume in the sequel that for Otherwise, we can adjust the corresponding columns and rows of all matrices in by permutation matrices such that the above assumption holds. It is not difficult to see that such a transformation does not change the existence of CWCLFs of .

If assumption (H3) holds, by using Lemmas 2 and 3, we know that the -dimensional vector is well defined by (14). Construct the -matrix of the formLet where and the -matrix has the form

Theorem 9. Assume that (H3) holds. There exists a CWCLF of if and only if , and there exists a CWCLF of . In addition, all CWCLFs of have the form , where is a constant, is a -dimensional vector, and is a CWCLF of .

Proof.
Necessity. We first get from (H3) and Lemma 3 that has a CLCLF for any SPI and any . By using (H3) and Lemma 5, all CWCLFs of have the form with , and hence . Assume that is a CWCLF of . Then, there exists appropriate such that , where is the corresponding -dimensional vector. Based on a straightforward computation, it is not difficult to conclude from (33) and (35) that is a CWCLF of .
Sufficiency. Since , we first have that is a CWCLF of from (11), (14), and (15). If is a CWCLF of , we can get from (33) and (35) that is a CWCLF of according to a direct computation. This completes the proof of Theorem 9.

Remark 10. By virtue of Theorem 9, the existence of CWCLFs of reduces to the existence of CWCLFs of lower dimensional Metzler matrices.

4. Numerical Examples

In this section, we present two examples to illustrate the main results.

Example 1. Consider the family of Metzler matrices with Since the combination matrix has a zero eigenvalue, there is not any CLCLF of . We now verify whether has CWCLF.
Step 1. For the SPI , we have that and .
Step 2. From (33) and (35), a straightforward computation yields that Step 3. It is not difficult to see that all CWCLFs of have the form for .
Therefore, we get from Theorem 9 that all CWCLFs of have the form for . Moreover, it is easy to see that there is not a CJCLF of .

Example 2. Consider the family of Metzler matrices with Since the combination matrix has a zero eigenvalue, there is not any CLCLF of . We now verify whether has CWCLF.
Step 1. Note that and . That is, for any SPI .
Step 2. For an SPI , a straightforward computation yields that and . We now adjust the corresponding columns and rows of all matrices in by permutation matrices such that they take the following form: Step 3. According to a direct computation, we get from (33) and (35) that Step 4. It can be seen that all CWCLFs of have the form for and .
Therefore, we get from Theorem 9 that all CWCLFs of have the form with and . In addition, it is easy to verify that is a CJCLF of .

5. Conclusion

The existence of a class of CWCLFs has been investigated in this paper, which generalize the usual CLCLFs and can be applied to stability analysis of positive switched linear systems. By using matrix theory, necessary and sufficient conditions for the existence of CWCLFs have been established. Moreover, the algebraic structure of all CWCLFs is described clearly. Two numerical examples are also given to illustrate the effectiveness of the obtained results.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61473133, 11671227, and 61374074) and the Natural Science Foundation of Shandong Province (Grant no. JQ201119).

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