# Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems

Authors: Yuangong Sun; Zhaorong Wu; Fanwei Meng
Journal: Complexity
 (2018)

---

## 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.


---

## Body

## 1. Introduction

Positive dynamical system is one for which nonnegative initial conditions give rise to nonnegative trajectories [1–3]. In recent years, stability issue for PSLS has been addressed for both practical applications in cooperative control of multiagent systems [4–8] and for theoretical reasons in [9–17] 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 [18–22].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 casesn=2 and n=3 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 Section2, 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,〈m〉 is the set of integers {1,2,…,m} for any positive integer m. If all entries of vector x are positive (nonpositive, negative), we denote x≻0(⪯0,≺0). For a matrix A, denote A⪯0 if all its entries are nonpositive. Denote the j-th column and the (i,j)-th component of matrix Ak by colj(Ak) and aij(k), respectively. In is an n-dimensional identity matrix. AMetzler matrix is a real square matrix, whose off-diagonal entries are nonnegative. A square matrix isHurwitz if the real part of each of its eigenvalues is negative.Consider the following continuous-time switched linear system:(1)x˙t=Aσtxt,t≥0,where x is the n-dimensional state vector, the piecewise continuous function σ:[0,+∞)→〈m〉 is the switching signal, and Ak is an n×n-matrix for each k∈〈m〉.As usual, system (1) is said to bepositive, if x(t)⪰0 for any t≥0, any x(0)⪰0, and arbitrary switching [12]. We know that system (1) is positive if and only if Ak is a Metzler matrix for each k∈〈m〉. A CLCLF method is usually used for asymptotic stability of PSLS (1) under arbitrary switching. Given an n-dimensional vector v≻0, V(x)=vTx (or briefly v) is said to be a CLCLF of PSLS (1) (or the family of Metzler matrices A={A1,A2,…,Am}) if(2)vTAk≺0,k∈m.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:(3)vTAk⪯0,k∈m.In order to guarantee asymptotic stability of PSLS (1) under appropriately chosen switching signals, CJCLFs were proposed in [27]. Given an n-dimensional vector v≻0, V(x)=vTx is said to be a CJCLF of PSLS (1) if (3) holds and(4)vT∑k=1mAk≺0.For the case m=2, 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 casesn=2 and n=3, 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 n-dimensional vector v≻0, V(x)=vTx (or briefly v) is said to be a CWCLF of PSLS (1) (or A) 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 ofA, we have (H1): ajj(k)≤0 for any j∈〈n〉 and k∈〈m〉; aij(k)=0 for all i∈〈n〉 if ajj(k)=0 for some j∈〈n〉 and k∈〈m〉. In the following, it is always assumed that (H1) holds. Otherwise, A does not have a CWCLF.Note thatA has a CWCLF if and only if the family of Metzler matrices {AkDk-1:k∈〈m〉} has a CWCLF, where Dk=diag⁡{d1(k),d2(k),…,dn(k)} is a diagonal matrix with (5)dik=-aiik,ifaiik<0,1,ifaiik=0. For the sake of convenience, assume throughout this paper that aii(k)=-1(0) for i∈〈n〉 and k∈〈m〉.In the sequel, we define a sequence of positive integers (SPI){j1,j2,…,jp} for p∈〈n〉 such that 1≤j1<j2<⋯<jp≤n. Denote the n×p-matrix: (6)Akj1kj2⋯kjp=colj1Akj1,colj2Akj2,…,coljpAkjp, where kji∈Λji for i∈〈p〉 and the nonempty index set Λi={k∈〈m〉:aii(k)<0} for i∈〈n〉. Let (7)Lj1j2⋯jp=A~kj1kj2⋯kjp:kji∈Λji,i∈p, where A~kj1kj2⋯kjp is a p×p-matrix obtained from Akj1kj2⋯kjp by deleting all its rows except those labelled by j1,j2,…,jp.Remark 1.
It follows from assumption (H1) thatA has a CWCLF if and only if L12⋯n has a CWCLF. Moreover, for any SPI {j1,j2,…,jp} and any p∈〈n〉, Lj1j2⋯jp has a CWCLF if L12⋯n has a CWCLF.
Forp∈〈n-1〉(n≥2), decompose the matrix A~kj1kj2⋯kjp+1 as follows:(8)A~kj1kj2⋯kjp+1=A~kj1kj2⋯kjpξkjp+1ηkjp+1T-1,where ξkjp+1 and ηkjp+1 are the corresponding p-dimensional column vectors; kji∈Λji for i∈〈p+1〉. If the matrix A~kj1kj2⋯kjp is invertible, the equation(9)yTA~kj1kj2⋯kjp=-ηkjp+1Thas a unique solution, where y is a p-dimensional column vector. We denote the solution of (9) by θkj1kj2⋯kjp+1 when it has a unique solution. Let (10)Θj1j2⋯jp+1=θkj1kj2⋯kjp+1:kji∈Λji,i∈p+1.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{j1,j2,…,jp+1} and p∈〈n-1〉(n≥2), if Lj1j2⋯jp has a CLCLF, then there exists a p+1-tuple (k¯j1,k¯j2,…,k¯jp+1), k¯ji∈Λji,i∈〈p+1〉, such that(11)θk¯j1k¯j2⋯k¯jp+1TA~kj1kj2⋯kjp⪯-ηkjp+1T,(12)θkj1kj2⋯kjp+1⪯θk¯j1k¯j2⋯k¯jp+1,where kji∈Λji, i∈〈p+1〉.Forp=1, noting that A~kj1=-1 for kj1∈Λj1, Lj1={-1} for any j1∈〈n〉; it is obvious that Lj1 has a CLCLF. By Lemma 2, let (13)θj1j2=θk¯j1k¯j2,λj1j2=max⁡θj1j2Tξkj2:kj2∈Λj2.That is, θj1j2 and λj1j2 are always well defined.Generally speaking, given an SPI{j1,j2,…,jp+1} and p∈〈n-1〉, if Lj1j2⋯jp has a CLCLF, by Lemma 2, we can define(14)θj1j2⋯jp+1=θk¯j1k¯j2⋯k¯jp+1,(15)λj1j2⋯jp+1=max⁡θj1j2⋯jp+1Tξkjp+1:kjp+1∈Λjp+1.Lemma 3 (see [22]).
Given an SPI{j1,j2,…,jp+1} and p∈〈n-1〉(n≥2), Lj1j2⋯jp+1 has a CLCLF if and only if λj1j2<1,λj1j2j3<1,…,λj1j2⋯jp+1<1.Lemma 4 (see [28]).
For ann×n-Metzler matrix A, if A is Hurwitz, then A-1⪯0.


## 3. Main Results

We first present the following lemma which plays a key role in the proof of the main results.Lemma 5.
Givenp∈〈n-1〉 (n≥2), assume that Lj1j2⋯jp has a CLCLF for any SPI {j1,j2,…,jp}. Suppose also that Lj1j2⋯jp+1 has a CWCLF for some SPI {j1,j2,…,jp+1}; then λj1j2⋯jp+1≤1, and all CWCLFs of Lj1j2⋯jp+1 have the form μ(θj1j2⋯jp+1T,1)T when λj1j2⋯jp+1=1, where μ>0 is a constant.Proof.
SinceLj1j2⋯jp has a CLCLF, by using Lemma 2, we see that θj1j2⋯jp+1 and λj1j2⋯jp+1 are well defined by (14) and (15) for the given SPI {j1,j2,…,jp+1}. Suppose that v≻0 is a CWCLF of Lj1j2⋯jp+1. Set v=μ(uT,1)T, where u≻0 is a p-dimensional column vector and μ>0 is an appropriate constant. It is obvious that (uT,1)T is also a CWCLF of Lj1j2⋯jp+1. Consequently, we get from (8) that(16)uTA~kj1kj2⋯kjp⪯-ηkjp+1T,∀kji∈Λji,i∈p+1,(17)uTξkjp+1≤1,∀kjp+1∈Λjp+1.On the other hand, by the definition of θj1j2⋯jp+1, we have that (18)θj1j2⋯jp+1TA~k¯j1k¯j2⋯k¯jp=-ηk¯jp+1T,where k¯ji∈Λji, i∈〈p+1〉, are defined as in Lemma 2. This together with (16) yields that(19)u-θj1j2⋯jp+1TA~k¯j1k¯j2⋯k¯jp⪯0.Noting that A~k¯j1k¯j2⋯k¯jp is Hurwitz since Lj1j2⋯jp has a CLCLF, it follows from Lemma 4 and (19) that(20)u⪰θj1j2⋯jp+1.Substituting (20) into (17) gives(21)θj1j2⋯jp+1Tξkjp+1≤uTξkjp+1≤1,∀kjp+1∈Λjp+1.It implies that λj1j2⋯jp+1≤1.
Next, we show thatu=θj1j2⋯jp+1 if λj1j2⋯jp+1=1. By the definition of λj1j2⋯jp+1, there exists an index k¯jp+1∈Λjp+1 such that(22)θj1j2⋯jp+1Tξk¯jp+1=1.From (21) and (22), we have(23)u-θj1j2⋯jp+1Tξk¯jp+1=0.If ξk¯jp+1≻0, we can directly conclude that u=θj1j2⋯jp+1 from (20) and (23). Otherwise, ξk¯jp+1 has at least one zero entry. For the sake of convenience, assume that the last component of ξk¯jp+1 is zero, and all the others are positive. That is,(24)ξk¯jp+1=ξ~k¯jp+1T,0T,where ξ~k¯jp+1≻0 is a (p-1)-dimensional column vector. Set(25)u=u~T,u^T,θj1j2⋯jp+1=θ~j1j2⋯jp+1T,θ^j1j2⋯jp+1T,where u~ and θ~j1j2⋯jp+1 are the corresponding (p-1)-dimensional column vectors; u^ and θ^j1j2⋯jp+1 are appropriate constants. From (22)–(25), we obtain(26)θ~j1j2⋯jp+1Tξ~k¯jp+1=1,(27)u~-θ~j1j2⋯jp+1Tξ~k¯jp+1=0.Since ξ~k¯jp+1≻0, we can get from (20) and (27) that(28)u~=θ~j1j2⋯jp+1.Now, it is sufficient to show that u^=θ^j1j2⋯jp+1. Otherwise, u^>θ^j1j2⋯jp+1 from (20). We now decompose the (p+1)×(p+1)-matrix A~k¯j1k¯j2⋯k¯jp+1 into the following form: (29)A~k¯j1k¯j2⋯k¯jp-1ξk¯jpξ~k¯jp+1ηk¯jpT-10η~k¯jp+1Tη^k¯jp+1-1.From (19), (25), and (28), we have (30)u^-θ^j1j2⋯jp+1ηk¯jpT⪯0.Since u^>θ^j1j2⋯jp+1, the above inequality implies that ηk¯jp is a zero vector. Based on (9) and (25), a straightforward computation yields that (31)θ~j1j2⋯jp+1TA~k¯j1k¯j2⋯k¯jp-1=-η~k¯jp+1T.It implies that θ~j1j2⋯jp+1∈Θj1j2⋯jp-1jp+1. From (12), (14), (15), and (26), we get (32)λj1j2⋯jp-1jp+1≥θ~j1j2⋯jp+1Tξ~k¯jp+1=1,which is a contradiction with the fact that Lj1j2⋯jp-1jp+1 has a CLCLF. Consequently, u=θj1j2⋯jp+1 when λj1j2⋯jp+1=1; that is, all CWCLFs of Lj1j2⋯jp+1 have the form μ(θj1j2⋯jp+1,1)T. This completes the proof of Lemma 5.Remark 6.
Assume that the family of Metzler matricesA has a CWCLF. Noting that there is always a CLCLF of Lj1 for any j1∈〈n〉, we get from Lemma 5 that λj1j2≤1 for any SPI {j1,j2}. If λj1j2<1 for any SPI {j1,j2}, Based on Lemma 3, Lj1j2 has a CLCLF for any SPI {j1,j2}. By using Lemma 5 again, we further get λj1j2j3≤1 for any SPI {j1,j2,j3}. Therefore, the existence of CWCLFs of A implies that there is at least one p∈{2,3,…,n}(n≥2) such that λj1j2⋯jq<1 for any SPI {j1,j2,…,jq} and any q∈〈p-1〉. For the trivial case q=1, we denote λj1=0.We first establish a result for the case whenp=n in Remark 6. That means (H2) λj1j2⋯jq<1 for any SPI {j1,j2,…,jq} and any q∈〈n-1〉(n≥2).Theorem 7.
Assume that (H2) holds. There exists a CWCLF ofA if and only if λ12⋯n≤1 and θ12⋯n≻0 when λ12⋯n=1. Moreover, all CWCLFs of A are the same as CLCLFs if λ12⋯n<1, and all CWCLFs of A have the form μ(θ12⋯nT,1)T if λ12⋯n=1, where μ>0 is a constant.Proof.
Necessity. We first get from (H2) and Lemma 3 that Lj1j2⋯jq has a CLCLF for any SPI {j1,j2,…,jq} and any q∈〈n-1〉. By using Lemma 5, we have that λ12⋯n≤1. If λ12⋯n<1, we conclude from Lemma 3 that A has a CLCLF. Therefore, all CWCLFs of A are the same as CLCLFs. If λ12⋯n=1, from Lemma 5, we see that all CWCLFs of A have the form μ(θ12⋯nT,1)T, and hence θ12⋯n≻0.
Sufficiency. If λ12⋯n<1, we get from (H2) and Lemma 3 that there is a CLCLF of A, which is also a CWCLF of A. If λ12⋯n=1, we get from (11), (14), and (15) that (θ12⋯nT,1)T is a CWCLF of A since θ12⋯n≻0. This completes the proof of Theorem 7.For the particular case whenani(k)>0 for i∈〈n-1〉 and k∈〈m〉, if there is a CWCLF of A, then there is a CLCLF of L12⋯n-1, and hence λ12<1, λ123<1,…, λ12⋯n-1<1 by Lemma 3. Suppose also that ain(k)>0 for i∈〈n-1〉 and k∈〈m〉. Following the proof of Lemma 5, we see that Lemma 5 holds true under the assumption that L12⋯n-1 has a CLCLF. Similar to the proof of Theorem 7, we have the following corollary.Corollary 8.
Assume thatani(k)>0 and ain(k)>0 for i∈〈n-1〉 and k∈〈m〉. There exists a CWCLF of A if and only if λ12<1, λ123<1,…, λ12⋯n-1<1, λ12⋯n≤1, and θ12⋯n≻0 when λ12⋯n=1. Moreover, all CWCLFs of A are the same as CLCLFs if λ12⋯n<1, and all CWCLFs of A have the form μ(θ12⋯nT,1)T if λ12⋯n=1, where μ>0 is a constant.Next, we consider the case when the assumption in Theorem7 does not hold. By Remark 6, we have that (H3) there exists an integer p∈〈n-2〉 with n≥3 such that λj1j2⋯jq<1 for any SPI {j1,j2,…,jq} and any q∈〈p〉. In addition, there exists an SPI {j¯1,j¯2,…,j¯p+1} such that λj¯1j¯2⋯j¯p+1=1.For the sake of convenience, we assume in the sequel thatj¯i=i for i∈〈p+1〉. Otherwise, we can adjust the corresponding columns and rows of all matrices in A 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 A.If assumption (H3) holds, by using Lemmas2 and 3, we know that the p-dimensional vector θ12⋯p+1 is well defined by (14). Construct the (n-p)×n-matrix of the form(33)Bk=diag⁡θ12⋯p+1T,1,In-p-1Ak.Let (34)Mq=Bqk:k∈m, where q∈〈p+1〉 and the (n-p)×(n-p)-matrix Bqk has the form(35)Bqk=colqBk,colp+2Bk,colp+3Bk,…,colnBk.Theorem 9.
Assume that (H3) holds. There exists a CWCLF ofA if and only if θ12⋯p+1≻0, and there exists a CWCLF of ⋃q∈〈p+1〉Mq. In addition, all CWCLFs of A have the form μ(θ12⋯p+1T,1,ϕT)T, where μ>0 is a constant, ϕ≻0 is a (n-p-1)-dimensional vector, and (1,ϕT)T is a CWCLF of ⋃q∈〈p+1〉Mq.Proof.
Necessity. We first get from (H3) and Lemma 3 that Lj1j2⋯jq has a CLCLF for any SPI {j1,j2,…,jq} and any q∈〈p〉. By using (H3) and Lemma 5, all CWCLFs of L12⋯p+1 have the form μ(θ12⋯p+1T,1)T with μ>0, and hence θ12⋯p+1≻0. Assume that v is a CWCLF of A. Then, there exists appropriate μ>0 such that v=μ(θ12⋯p+1T,1,ϕT)T, where ϕ is the corresponding (n-p-1)-dimensional vector. Based on a straightforward computation, it is not difficult to conclude from (33) and (35) that (1,ϕT)T is a CWCLF of ⋃q∈〈p+1〉Mq.
Sufficiency. Since θ12⋯p+1≻0, we first have that (θ12⋯p+1T,1)T is a CWCLF of L12⋯p+1 from (11), (14), and (15). If (1,ϕT)T is a CWCLF of ⋃q∈〈p+1〉Mq, we can get from (33) and (35) that (θ12⋯p+1T,1,ϕT)T is a CWCLF of A according to a direct computation. This completes the proof of Theorem 9.Remark 10.
By virtue of Theorem9, the existence of CWCLFs of A 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 of3×3 Metzler matrices A={A1,A2} with (36)A1=-113451-14500-1,A2=-11015-112120-1.Since the combination matrix (37)A122=-1101-11200-1has a zero eigenvalue, there is not any CLCLF of A. We now verify whether A has CWCLF.
Step 1. For the SPI {1,2}, we have that θ12=1 and λ12=1.
Step 2. From (33) and (35), a straightforward computation yields that (38)M1=0850-1,-451212-1,M2=-23850-1,0120-1.
Step 3. It is not difficult to see that all CWCLFs of M1∪M2 have the form μ(1,8/5) for μ>0.
Therefore, we get from Theorem9 that all CWCLFs of A have the form μ(1,1,8/5) for μ>0. Moreover, it is easy to see that there is not a CJCLF of A.Example 2.
Consider the family of4×4 Metzler matrices A={A1,A2} with (39)A1=-101300-113000-114110-1A2=-1013120-1131200-1013130-1.Since the combination matrix (40)A1112=-1013120-1131200-10110-1has a zero eigenvalue, there is not any CLCLF of A. We now verify whether A has CWCLF.
Step 1. Note that λ12=λ13=λ23=λ34=0 and λ14=λ24=1/2. That is, λj1j2<1 for any SPI {j1,j2}.
Step 2. For an SPI {1,2,4}, a straightforward computation yields that θ124=(1,1)T and λ124=1. We now adjust the corresponding columns and rows of all matrices in A by permutation matrices such that they take the following form: (41)A1=-100130-101311-100014-1A2=-1012130-112131313-10000-1.
Step 3. According to a direct computation, we get from (33) and (35) that (42)M1=M2=0230-1,-23230-1,M3=-12314-1,0230-1.
Step 4. It can be seen that all CWCLFs of M1∪M2∪M3 have the form a(1,b)T for a>0 and 2/3≤b≤4.
Therefore, we get from Theorem9 that all CWCLFs of A have the form a(1,1,b,1)T with a>0 and 2/3≤b≤4. In addition, it is easy to verify that (1,1,1,1)T is a CJCLF of A.


## 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.



---

*Source: 1365960-2018-02-26.xml*