Common Weak Linear Copositive Lyapunov Functions for Positive Switched Linear Systems

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.

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  = 2 and  = 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 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 2 Complexity obtained theoretical results.Finally, conclusions are drawn in Section 5.

Problem Statement and Preliminaries
Throughout this paper, ⟨⟩ is the set of integers {1, 2, . . ., } for any positive integer .If all entries of vector  are positive (nonpositive, negative), we denote  ≻ 0 (⪯ 0, ≺ 0).For a matrix , denote  ⪯ 0 if all its entries are nonpositive.Denote the -th column and the (, )-th component of matrix   by col  (  ) 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  : [0, +∞) → ⟨⟩ is the switching signal, and   is an  × -matrix for each  ∈ ⟨⟩.As usual, system (1) is said to be positive, if () ⪰ 0 for any  ≥ 0, any (0) ⪰ 0, 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 V ≻ 0, () = V   (or briefly V) is said to be a CLCLF of PSLS (1) (or the family of Metzler matrices A = { 1 ,  2 , . . .,   }) 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 V ≻ 0, () = V   is said to be a CJCLF of PSLS (1) if (3) holds and For the case  = 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 cases  = 2 and  = 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 -dimensional vector V ≻ 0, () = V   (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.
Note that A has a CWCLF if and only if the family of Metzler matrices {   −1  :  ∈ ⟨⟩} has a CWCLF, where For the sake of convenience, assume throughout this paper that  ()  = −1(0) for  ∈ ⟨⟩ and  ∈ ⟨⟩.In the sequel, we define a sequence of positive integers where    ∈ Λ   for  ∈ ⟨⟩ and the nonempty index set Λ  = { ∈ ⟨⟩ :  ()  < 0} for  ∈ ⟨⟩.Let where as follows: where    +1 and    +1 are the corresponding -dimensional column vectors;    ∈ Λ   for  ∈ ⟨ + 1⟩.If the matrix Ã  1   2 ⋅⋅⋅   is invertible, the equation has 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.

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

Theorem 7. Assume that (H2) holds. There exists a CWCLF of A if and only if 𝜆
Moreover, all CWCLFs of A are the same as CLCLFs if  12⋅⋅⋅ < 1, and all CWCLFs of A have the form (  12⋅⋅⋅ , 1)  if  12⋅⋅⋅ = 1, where  > 0 is a constant.
In addition, there exists an SPI { 1 ,  2 , . . .,  +1 } such that For the sake of convenience, we assume in the sequel that   =  for  ∈ ⟨ + 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.
Remark 10.By virtue of Theorem 9, the existence of CWCLFs of A reduces to the existence of CWCLFs of lower dimensional Metzler matrices.

Numerical Examples
In this section, we present two examples to illustrate the main results.(36) Since the combination matrix has a zero eigenvalue, there is not any CLCLF of A. We now verify whether A has CWCLF.
Step 2. From (33) and (35), a straightforward computation yields that Step 3. It is not difficult to see that all CWCLFs of M 1 ∪ M 2 have the form (1, 8/5) for  > 0.
Therefore, we get from Theorem 9 that all CWCLFs of A have the form (1, 1, 8/5) for  > 0.Moreover, it is easy to see that there is not a CJCLF of A. has a zero eigenvalue, there is not any CLCLF of A. We now verify whether A has CWCLF.
Step 2. For an SPI {1, 2, 4}, a straightforward computation yields that  124 = (1, 1)  and  124 = 1.We now adjust the corresponding columns and rows of all matrices in A