Exponential stability and L 1 -gain performance analysis for continuous-time delayed switched positive singular systems

Abstract

The problem of exponential stability and L₁-gain performance is solved for continuous-time delayed switched positive singular systems. First, a necessary and sufficient condition of positivity is established for the system via the singular value decomposition approach. Then, based on the co-positive Lyapunov function tool, we develop a sufficient condition for the switched positive singular system to be exponentially stable and meet a prescribed L₁-gain performance. Based on this condition, the optimal system performance level can be determined by solving a convex optimization problem. All the obtained conditions are in linear programming form, which suggests a good scalability and applicability to high-dimensional systems. Finally, a numerical example is given to illustrate the applicability of the obtained results.

Introduction

We study the stability as well as system performance for switched positive singular systems (SPSSs) in this article. Our motivations come from the following aspects. First, many systems in practical consist of non-negative variables like level of population, mass density, and absolute temperature, which are known as positive systems^1,2 and can be applied in many fields like, pharmacokinetics, ³ hydraulic systems,^4–7 chemical reaction,^8,9 and Internet congestion control. ¹⁰ Second, many practical systems such as power systems, economics systems, robotic systems, and biological systems^11–14 can be modeled as singular systems, which have much more advantages than the general state-space model, ¹⁵ and much attention have been received on analysis for singular systems.^16,17 Third, many systems often go through abrupt variations in their structures and parameters in practice; one effective method of characterizing the sudden changes is the switched system method.^18–21

For positive systems, L₁-norm is usually used as a performance measure because it accounts for the sum of the quantities. ²² Recently, there have been many important results for positive systems via the linear co-positive Lyapunov functional approach,^23–25 and the corresponding existence conditions are formulated in the linear programming (LP) form. When positive singular systems experience the switching-type changes in their parameters, SPSSs consisting of a family of linear positive singular systems and a switching signal can be modeled. For an SPSS, it is an area that is only beginning to be investigated, and there is further room for investigation. It should be pointed out that the analysis would be of theoretical challenges, since one needs to consider the coupling between the switching, the algebraic constraints in singular models, and the positivity limits in valuables.

Recently, some results on positive singular systems have been obtained.^26–28 As pointed out in Zhang et al., ²⁶ the internal positivity property of continuous descriptor systems is studied. In Phat and Sau, ²⁸ the problems of positivity and stability are addressed for discrete-time positive singular systems with time delay. It is widely recognized that few of them analyze the exponential stability for positive singular systems. The main difficulty is how to determine the exponential decay rate; the first problem to be solved is to determine the decay rate for positive singular systems via some effective approaches.

There are only a few recent publications investigating SPSSs.^29,30 Xia et al. ²⁹ dealt with the controller design problem for positive singular Markovian jump systems. In Qi and Gao, ³⁰ the stability problems for positive switched singular systems were investigated using the average dwell time approach. However, both the studies^29,30 only focused on the autonomous case without considering time delay and exogenous disturbance, which are frequently encountered in practice, and much attention has been received on analysis of systems with those two phenomena.^22,31,32 Thus, it becomes significant to investigate non-autonomous SPSSs with those two phenomena. When considering time delays and exogenous disturbances, the main difficulties are how to choose an appropriate Lyapunov function, and how to analyze the stability and system performance, since it becomes more challenging and complicated. To the best of the authors’ knowledge, this kind of system has not been studied up to date. Therefore, we are interested in investigating the exponential stability, the optimal L₁-gain performance for SPSSs with time-varying delay in this article.

The main contributions of this work are as follows: (1) provide a necessary and sufficient condition of positivity for delayed SPSSs via the singular value decomposition approach, (2) identify the switching signals via average dwell time approach to ensure the exponential stability of the system, (3) derive a sufficient delay-dependent condition for the first time for the system to be exponentially stable as well as satisfy a prescribed L₁-gain performance using the co-positive Lyapunov function tool, (4) propose a convex optimization problem for delayed SPSSs such that the optimal L₁-gain performance level can be determined, and (5) present all criteria obtained in LP form.

The remainder is as follows. The preliminaries and problem formulation are given in section “Problem formulation.” The positivity, exponential stability, and L₁-gain performance are analyzed in section “Exponential stability and L₁-gain performance analysis.” Section “Numerical results” provides a numerical example to illustrate the effectiveness of proposed method. Concluding remarks are given in section “Conclusion.”

Notations

$A⪰0(⪯,\succ ,\prec )$ denotes that all the entries of A are non-negative (non-positive, positive, and negative); $A\succ B(A⪰B)$ denotes that $A-B\succ 0(A-B⪰0)$ ; $A^T$ is the transpose of A; $\det (A)$ is the determinant of A; $\mathrm{deg}(·)$ represents the degree of a polynomial; $\mathrm{rank}(A)$ denotes the rank of A; $R(R_{+})$ denotes the collection consisting of all real (positive real) numbers; $R^n(R_{+}^n)$ is the space of all n-dimensional real (positive real) vector; $R^{m\times n}$ is the collection consisting of all m ×n -dimensional real matrices; $1_n\in R^n$ denotes a column vector in which each entry is 1; $[A{]}_{\mathrm{pq}}$ means the $(p,q)\mathrm{th}$ entry of a matrix A; and $[x{]}_p$ means the $p\mathrm{th}$ entry of a vector x. $||x|{|}_1=\sum _{p=1}^n|[x{]}_p|$ is the 1-norm of a vector $x\in R^n$ , $||A|{|}_1={\max }_{1\le q\le m}\sum _{p=1}^n|[A{]}_{\mathrm{pq}}|$ is the 1-norm of a matrix $A\in R^{n\times m}$ . $||v|{|}_{L_1}={\int }_{t=t_0}^{\infty }||\nu (t)|{|}_1\mathrm{dt}$ is the L₁-norm of a vector-valued function $\nu :R_{+}\to R^n$ , $L_1=\{\nu (t),t\ge t_0|||\nu |{|}_{L_1}<\infty \}$ . For any $\nu \in R^n$ , $\overline{\rho }(\nu )$ and $\underset{¯}{\rho }(\nu )$ stand for the maximal and minimal element of $\nu$ , respectively.

Problem formulation

Consider the following continuous-time delayed switched singular system

$$\{\begin{array}{c}E\stackrel{·}{x}(t)=A_{\sigma (t)}x(t)+A_{d\sigma (t)}x(t-d(t))+F_{\sigma (t)}w(t)\hfill \\ \multicolumn{1}{c}{z(t)=C_{\sigma (t)}x(t)+D_{\sigma (t)}w(t)}\\ \multicolumn{1}{c}{x(\theta )=\phi (\theta ),\theta =[-d_2,0]}\end{array}$$

(1)

where $x(t)\in R^n$ denotes the state, $w(t)\in R^u$ is the disturbance input which belongs to $L_1$ , and $z(t)\in R^m$ is the controlled output. $d(t)$ is a time-varying delay, which is assumed to satisfy $0\le d_1\le d(t)\le d_2$ and $\stackrel{·}{d}(t)\le d_d\le 1$ . $\phi (\theta )$ is a given continuous vector-valued initial condition of the system. $\{\sigma (t),t\ge 0\}$ is a switching signal taking values in a finite set $\underset{¯}{N}=\{1,2,\dots ,N\}$ with N being the number of subsystems; moreover, $\sigma (k)=i$ denotes that the $i\mathrm{th}$ subsystem is activated, $i\in \underset{¯}{N}$ . Meanwhile, for switching time sequence $t_0\le t_1\le t_2\le \cdots$ of the switching signal $\sigma (t)$ , the holding time between $[t_l,t_{l+1}]$ is called the dwell time of the currently engaged $l\mathrm{th}$ subsystem, where l is a non-negative integer. $A_i,A_{\mathrm{di}},F_i,C_i,\mathrm{and}{D}_i,i\in \underset{¯}{N}$ are known constant matrices with appropriate dimensions. $E\in R^{n\times n}$ is singular and $\mathrm{rank}E=r<n$ . Without losing generality, System (1) is assumed to be activated at $t_0=0$ .

Definition 1

System (1) is said to be as follows:^33,34

1. Regular if every pair $(E,A_i)$ is regular, that is, $\det (\mathrm{sE}-A_i)\ne 0$ , $\forall i\in \underset{¯}{N}$ ;

2. Causal if every pair $(E,A_i)$ is causal, that is, $\mathrm{degree}(\det (\mathrm{sE}-A_i))=\mathrm{rank}E$ , $\forall i\in \underset{¯}{N}$ ;

3. Exponentially stable, if for any switching signal $\sigma (t)$ and any initial conditions $\phi (\theta )$ , $x(t)$ of System (1) with $w(t)=0$ satisfies $||x(t)|{|}_1\le \iota e^{-\chi t}||\phi |{|}_{1c}$ , $\forall t\ge 0$ , where $||\phi |{|}_{1c}={\sup }_{-d_2\le s\le 0}||\phi (s)|{|}_1$ , $\chi >0$ and $\iota >0$ are called the decay rate and decay efficient, respectively.

Remark 1

As in Zhang et al. ²⁶ and De La Sen, ³⁵ the time delay switched singular System (1) may have an impulsive solution. However, the regularity and causality of System (1) can guarantee the existence and uniqueness of the system’s solution for any given initial conditions.

Definition 2

System (1) is said to be positive if, for any initial conditions $\phi (\theta )⪰0$ , $w(t)⪰0$ and any switching signals $\sigma (t)$ , we have $x(t)⪰0$ and $z(t)⪰0$ for all $t\ge 0$ . ¹

Lemma 1

Let $A\in R^{n\times n}$ . Then, $e^{\mathrm{At}}\succ 0$ for $t\ge 0$ if and only if A is a Metzler matrix. ¹

For System (1), since $\mathrm{rank}E=r<n$ , then there exist non-singular matrices P, $Q\in R^{n\times n}$ such that $\mathrm{PEQ}=\left[\begin{array}{cc}{I}_r\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$ . In this article, for simplicity, let $E=\left[\begin{array}{cc}{I}_r\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$ and other system matrices have the following form, $\forall i\in \underset{¯}{N}$

$$\begin{array}{l}{A}_i=\left[\begin{array}{cc}{A}_{i1}& A_{i2}\\ A_{i3}& A_{i4}\end{array}\right],A_{di}=\left[\begin{array}{cc}{A}_{di1}& A_{di2}\\ A_{di3}& A_{di4}\end{array}\right],F_i=\left[\begin{array}{c}{F}_{i1}\\ F_{i2}\end{array}\right],C_i=\left[\begin{array}{cc}{C}_{i1}& C_{i2}\end{array}\right]\hfill \end{array}$$

(2)

Lemma 2

System (1) with (2) is causal if and only if $A_{i4}$ is non-singular, $\forall i\in \underset{¯}{N}$ . ¹²

In light of Lemma 2, when System (1) with (2) is regular and causal, under the coordinate transformation $x(t)=[x_1^T(t)x_2^T(t){]}^T$ , $x_1(t)\in R^r$ , $x_2(t)\in R^{n-r}$ , System (1) can be rewritten as

$$\{\begin{array}{c}{\stackrel{·}{x}}_1(t)={\overline{A}}_{\sigma (t)1}{x}_1(t)+{\overline{A}}_{d\sigma (t)1}{x}_1(t-d(t))+{\overline{A}}_{d\sigma (t)2}{x}_2(t-d(t))+{\overline{F}}_{\sigma (t)1}w(t)\hfill \\ \multicolumn{1}{c}{x_2(t)={\overline{A}}_{\sigma (t)3}{x}_1(t)+{\overline{A}}_{d\sigma (t)3}{x}_1(t-d(t))+{\overline{A}}_{d\sigma (t)4}{x}_2(t-d(t))+{\overline{F}}_{\sigma (t)2}w(t)}\\ \multicolumn{1}{c}{z(t)=C_{\sigma (t)1}{x}_1(t)+C_{\sigma (t)2}{x}_2(t)+D_{\sigma (t)}w(t)}\\ \multicolumn{1}{c}{x_1(\theta )={\phi }_1(\theta ),\theta =[-d_2,0]}\\ \multicolumn{1}{c}{x_2(\theta )={\phi }_2(\theta ),\theta =[-d_2,0]}\end{array}$$

(3)

where for each $i\in \underset{¯}{N}$ in equation (3)

$$\begin{array}{l}\begin{array}{l}{\overline{A}}_{i1}=A_{i1}-A_{i2}{A}_{i4}^{-1}{A}_{i3}\mathrm{,}{\overline{A}}_{di1}=A_{di1}-A_{i2}{A}_{i4}^{-1}{A}_{di3}\hfill \end{array}\\ \begin{array}{l}\begin{array}{l}{\overline{A}}_{di2}=A_{di2}-A_{i2}{A}_{i4}^{-1}{A}_{di4}\mathrm{,}{\overline{F}}_{i1}=F_{i1}-A_{i2}{A}_{i4}^{-1}{F}_{i2}\\ \begin{array}{l}{\overline{A}}_{i3}=-A_{i4}^{-1}{A}_{i3}\mathrm{,}{\overline{A}}_{di3}=-A_{i4}^{-1}{A}_{di3},{\overline{A}}_{di4}=-A_{i4}^{-1}{A}_{di4}\mathrm{,}{\overline{F}}_{i2}=-A_{i4}^{-1}{F}_{i2}\hfill \end{array}\end{array}\hfill \end{array}\end{array}$$

Lemma 3

Assume that $(E,A_i)$ , $\forall i\in \underset{¯}{N}$ , is regular and causal, then System (1) with (2) is positive if and only if System (3) is positive. ²⁸

Definition 3

For $\lambda >0$ and $\gamma >0$ , System (1) has a prescribed L₁-gain performance $\gamma$ if a switching signal $\sigma (t)$ exists, such that the following conditions occur:^24,36

1. When $w(t)=0$ , System (1) is regular, causal, and exponentially stable;

2. Under $\phi (\theta )=0$ , $\theta =[-d_2,0]$ , we have

$$\underset{t=0}{\overset{\infty }{\int }}{e}^{-\lambda t}{\Vert z(t)\Vert }_1<\gamma \underset{t=0}{\overset{\infty }{\int }}{\Vert w(t)\Vert }_1,\forall w(t)\in L_1$$

(4)

Definition 4

For the switching signal $\sigma (t)$ and any $T_2>T_1\ge 0$ , let $N_{\sigma }(T_1,T_2)$ denote the number of switches of $\sigma (t)$ over the interval $[T_1,T_2)$ . For given $T_a>0$ and $N_0\ge 0$ , if the inequality ¹⁹

$$N_{\sigma }(T_1,T_2)\le N_0+\frac{(T_2-T_1)}{T_a}$$

(5)

holds, then the positive constant $T_a$ is called the average dwell time, and $N_0$ is called the chattering bound. As commonly used in the literature, we choose $N_0=0$ .

The problem to be addressed in this article can be formulated as follows:

1. Provide a necessary and sufficient condition of the positivity for System (1);

2. Develop a sufficient delay-dependent LP condition and identify a class of switching signals $\sigma (k)$ , such that System (1) is exponentially stable and meets a prescribed L₁-gain performance level;

3. Propose a convex optimization problem for System (1), such that the optimal L₁-gain performance level can be determined.

Exponential stability and L₁-gain performance analysis

In this section, we aim at investigating the exponential stability and L₁-gain performance of the continuous-time SPSS (equation (1)). First, a necessary and sufficient condition for the positivity of System (1) is presented in the following theorem.

Theorem 1

Assume that $(E,A_i)$ , $\forall i\in \underset{¯}{N}$ , is regular and causal. System (1) is positive if and only if ${\overline{A}}_{i1}$ is a Metzler matrix, ${\overline{A}}_{\mathrm{di}1}⪰0$ , ${\overline{A}}_{\mathrm{di}2}⪰0$ , ${\overline{F}}_{i1}⪰0$ , ${\overline{A}}_{i3}⪰0$ , ${\overline{A}}_{\mathrm{di}3}⪰0$ , ${\overline{A}}_{\mathrm{di}4}⪰0$ , ${\overline{F}}_{i2}⪰0$ , $C_{i1}⪰0$ , $C_{i2}⪰0$ , $D_i⪰0$ , $\forall i\in \underset{¯}{N}$ , where ${\overline{A}}_{i1}$ , ${\overline{A}}_{\mathrm{di}1}$ , ${\overline{A}}_{\mathrm{di}2}$ , ${\overline{F}}_{i1}$ , ${\overline{A}}_{i3}$ , ${\overline{A}}_{\mathrm{di}3}$ , ${\overline{A}}_{\mathrm{di}4}$ , ${\overline{F}}_{i2}$ , $C_{i1}$ , and $C_{i2}$ are defined as in equation (3).

Proof: Sufficiency

We will show that System (1) is positive. For $T=0$ and $t_0=0$ , we obviously have from the initial condition $\phi (\theta )⪰0$ , $\theta =[-d_2,0]$ , such that $x(0)⪰0$ .

For any $T>0$ and $t_0=0$ , we denote $t_1,t_2,\dots ,t_l,t_{l+1},\dots ,t_{N_{\sigma }(t_0,T)}$ the switching instances on the interval $[t_0,T)$ . First, we prove that the solution $x(t)=[x_1^T(t)x_2^T(t){]}^T$ is positive on $[t_0,t_1)$ . It follows from Lagrange’s formula and System (3) that $\forall t\in [t_0,t_1)$

$$\begin{array}{c}\exp \{-{\overline{A}}_{\sigma (t_0)1}t\}({\stackrel{·}{x}}_1(t)-{\overline{A}}_{\sigma (t_0)1}{x}_1(t))=\exp \{-{\overline{A}}_{\sigma (t_0)1}t\}\hfill \\ [{\overline{A}}_{\sigma (t_0)1}{x}_1(t-d(t))+{\overline{A}}_{\sigma (t_0)2}{x}_2(t-d(t))+{\overline{F}}_{\sigma (t_0)1}w(t)]\hfill \end{array}$$

(6)

$$\begin{array}{c}\multicolumn{1}{c}{x_2(t)={\overline{A}}_{\sigma (t_0)3}{x}_1(t)+{\overline{A}}_{d\sigma (t_0)3}{x}_1(t-d(t))}\\ +{\overline{A}}_{d\sigma (t_0)4}{x}_2(t-d(t))+{\overline{F}}_{\sigma (t_0)2}w(t)\hfill \end{array}$$

(7)

$$z(t)=C_{\sigma (t_0)1}{x}_1(t)+C_{\sigma (t_0)2}{x}_2(t)+D_{\sigma }(t_0)w(t)$$

(8)

By integrating both sides of equation (6) from $t=t_0\mathrm{to}{t}_1$ , it holds that

$$\begin{array}{c}\exp \{-{\overline{A}}_{\sigma (t_0)1}t\}{x}_1(t)=\underset{t_0}{\overset{t}{\int }}\exp \{-{\overline{A}}_{\sigma (t_0)1}s\}[{\overline{A}}_{d\sigma (t_0)1}{x}_1(s-d(s))\hfill \\ +{\overline{A}}_{d\sigma (t_0)2}{x}_2(s-d(s))+{\overline{F}}_{\sigma (t_0)1}w(s)]\mathrm{ds}+x_1(t_0)\hfill \end{array}$$

(9)

It follows from equation (9) that

$$\begin{array}{c}\multicolumn{1}{c}{x_1(t)=\underset{t_0}{\overset{t}{\int }}\exp \{{\overline{A}}_{\sigma (t_0)1}(t-s)\}[{\overline{A}}_{d\sigma (t_0)1}{x}_1(s-d(s)+{\overline{A}}_{d\sigma (t_0)2}{x}_2(s-d(s))+{\overline{F}}_{\sigma (t_0)1}w(s)]\mathrm{ds}+\exp \left\{{\overline{A}}_{\sigma (t_0)1}t\right\}{x}_1(t_0)}\\ =\underset{t_0}{\overset{t}{\int }}\exp \{{\overline{A}}_{\sigma (t_0)1}(t-s)\}{\overline{A}}_{d\sigma (t_0)1}{x}_1(s-d(s)\mathrm{ds}\hfill \\ +\underset{t_0}{\overset{t}{\int }}\exp \{{\overline{A}}_{\sigma (t_0)1}(t-s)\}{\overline{A}}_{d\sigma (t_0)2}{x}_2(s-d(s))\mathrm{ds}\hfill \\ +\underset{t_0}{\overset{t}{\int }}\exp \{{\overline{A}}_{\sigma (t_0)1}(t-s)\}{\overline{F}}_{\sigma (t_0)1}w(s)\mathrm{ds}+\exp \left\{{\overline{A}}_{\sigma (t_0)1}t\right\}{x}_1(t_0)\hfill \end{array}$$

(10)

By Lemma 2 and Theorem 1, if ${\overline{A}}_{\sigma (t_0)1}$ is a Metzler matrix, ${\overline{A}}_{d\sigma (t_0)1}⪰0$ , ${\overline{A}}_{d\sigma (t_0)2}⪰0$ , ${\overline{F}}_{\sigma (t_0)1}⪰0$ , ${\overline{A}}_{\sigma (t_0)3}⪰0$ , ${\overline{A}}_{d\sigma (t_0)3}⪰0$ , ${\overline{A}}_{d\sigma (t_0)4}⪰0$ , ${\overline{F}}_{\sigma (t_0)2}⪰0$ , $C_{\sigma (t_0)1}⪰0$ , $C_{\sigma (t_0)2}⪰0$ , and $D_{\sigma (t_0)}⪰0$ , it is straightforward to obtain $x_1(t)⪰0$ , $x_2(t)⪰0$ , $z(t)⪰0$ , $\forall t\in [t_0,t_1)$ , and $x_1(t_1)=x_1(t_1^{-})⪰0$ from equation (10). Now, similar to the aforementioned processes, one can have that $\forall t\in [t_1,t_2)$

$$\begin{array}{c}\multicolumn{1}{c}{x_1(t)=\underset{t_1}{\overset{t}{\int }}\exp \{{\overline{A}}_{\sigma (t_1)1}(t-s)\}[{\overline{A}}_{d\sigma (t_1)1}{x}_1(s-d(s))+{\overline{A}}_{d\sigma (t_1)2}{x}_2(s-d(s))+{\overline{F}}_{\sigma (t_1)1}w(s)]\mathrm{ds}+\exp \left\{{\overline{A}}_{\sigma (t_1)1}t\right\}{x}_1(t_1)⪰0}\\ x_2(t)={\overline{A}}_{\sigma (t_1)3}{x}_1(t)+{\overline{A}}_{d\sigma (t_1)3}{x}_1(t-d(t))+{\overline{A}}_{d\sigma (t_1)4}{x}_2(t-d(t))+{\overline{F}}_{\sigma (t_1)2}w(t)⪰0\hfill \\ z(t)=C_{\sigma (t_1)1}{x}_1(t)+C_{\sigma (t_1)2}{x}_2(t)+D_{\sigma (t_1)}w(t)⪰0\hfill \end{array}$$

and $x_1(t_2)=x_1(t_2^{-})⪰0$ . Recursively, if ${\overline{A}}_{i1}$ is a Metzler matrix, ${\overline{A}}_{\mathrm{di}1}⪰0$ , ${\overline{A}}_{\mathrm{di}2}⪰0$ , ${\overline{F}}_{i1}⪰0$ , ${\overline{A}}_{i3}⪰0$ , ${\overline{A}}_{\mathrm{di}3}⪰0$ , ${\overline{A}}_{\mathrm{di}4}⪰0$ , ${\overline{F}}_{i2}⪰0$ , $C_{i1}⪰0$ , $C_{i2}⪰0$ , and $D_i⪰0$ , $\forall i\in \underset{¯}{N}$ , we can obtain that $x_1(T)⪰0$ , $x_2(T)⪰0$ , $z(T)⪰0$ , $\forall T\ge 0$ .

Necessity

Conversely, we first suppose that there exists an element $[{\overline{A}}_{i1}{]}_{\mathrm{pq}}<0$ , $\forall i\in \underset{¯}{N}$ , $p\ne q$ , where $[{\overline{A}}_{i1}{]}_{\mathrm{pq}}$ is the $(p,q)\mathrm{th}$ scalar entry of ${\overline{A}}_{i1}$ . From System (3), we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_{1p}(t)=\sum _{g=1,g\ne p,g\ne q}^r{[{\overline{A}}_{i1}]}_{\mathrm{pg}}{x}_{1g}(t)+{[{\overline{A}}_{i1}]}_{\mathrm{pp}}{x}_{1p}(t)}\\ +{[{\overline{A}}_{i1}]}_{\mathrm{pq}}{x}_{1q}(t)+\sum _{g=1}^r{[{\overline{A}}_{\mathrm{di}1}]}_{\mathrm{pg}}{x}_{1g}(t-d(t))\hfill \\ +\sum _{g=r+1}^n{[{\overline{A}}_{\mathrm{di}2}]}_{\mathrm{pg}}{x}_{2g}(t-d(t))+\sum _{g=1}^u{[{\overline{F}}_{i1}]}_{\mathrm{pg}}{w}_g(t)\hfill \end{array}$$

where $x_{1g}(t)$ , $x_{2g}(t)$ , and $w_g(t)$ represent the $g\mathrm{th}$ element of $x_1(t)$ , $x_2(t)$ , and $w(t)$ , respectively. Then, if $x_{1q}(t)\ne 0$ , we can see that ${\stackrel{·}{x}}_{1p}(t)<0$ is possible whenever $x_{1p}(t)=0$ , $x_{2p}(t)=0$ , and $w_p(t)=0$ , which means that $x_{1p}(t^{+})<0$ . It follows that System (1) is not positive. Second, further assume that ${\overline{A}}_{\mathrm{di}1}$ has an element $[{\overline{A}}_{\mathrm{di}1}{]}_{\mathrm{pq}}<0$ , $\forall i\in \underset{¯}{N}$ , we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{x}}_{1p}(t)=\sum _{g=1,g\ne p}^r{[{\overline{A}}_{i1}]}_{\mathrm{pg}}{x}_{1g}(t)+{[{\overline{A}}_{i1}]}_{\mathrm{pp}}{x}_{1p}(t)}\\ +\sum _{g=1,g\ne q}^r{[{\overline{A}}_{\mathrm{di}1}]}_{\mathrm{pg}}{x}_{1g}(t-d(t))\hfill \\ +[{\overline{A}}_{\mathrm{di}1}{]}_{\mathrm{pq}}{x}_{1q}(t-d(t))+\sum _{g=1}^u{[{\overline{F}}_{i1}]}_{\mathrm{pg}}{w}_g(t)\hfill \\ +\sum _{g=r+1}^n{[{\overline{A}}_{\mathrm{di}2}]}_{\mathrm{pg}}{x}_{2g}(t-d(t))\hfill \end{array}$$

Then, it can be obtained that $x_{1p}(t^{+})<0$ is possible whenever $x_{1p}(t)=0$ , $x_{2p}(t)=0$ , and $w_p(t)=0$ ; that is, System (1) is not positive. Similarly, if ${\overline{A}}_{\mathrm{di}2}$ has an element $[{\overline{A}}_{\mathrm{di}2}{]}_{\mathrm{pq}}<0$ , or ${\overline{F}}_{i1}$ has an element $[{\overline{F}}_{i1}{]}_{\mathrm{pq}}<0$ , one can get that System (1) is not positive.

Continuously, suppose that there exists an element $[{\overline{A}}_{i3}{]}_{\mathrm{pq}}<0$ , $\forall i\in \underset{¯}{N}$ , $p\ne q$ , where $[{\overline{A}}_{i3}{]}_{\mathrm{pq}}$ is the $(p,q)\mathrm{th}$ scalar entry of ${\overline{A}}_{i3}$ . From System (3), we obtain

$$\begin{array}{c}\multicolumn{1}{c}{x_{2p}(t)=\sum _{g=1,g\ne p,g\ne q}^r{[{\overline{A}}_{i3}]}_{\mathrm{pg}}{x}_{1g}(t)+{[{\overline{A}}_{i3}]}_{\mathrm{pp}}{x}_{1p}(t)}\\ +{[{\overline{A}}_{i3}]}_{\mathrm{pq}}{x}_{1q}(t)\hfill \\ +\sum _{g=1}^r{[{\overline{A}}_{\mathrm{di}3}]}_{\mathrm{pg}}{x}_{1g}(t-d(t))\hfill \\ +\sum _{g=r+1}^n{[{\overline{A}}_{\mathrm{di}4}]}_{\mathrm{pg}}{x}_{2g}(t-d(t))+\sum _{g=1}^u{[{\overline{F}}_{i2}]}_{\mathrm{pg}}{w}_g(t)\hfill \end{array}$$

Then, if $x_{1q}(t)\ne 0$ , we can see that $x_{2p}(t)<0$ is possible whenever $x_{1p}(t)=0$ , $x_{2p}(t)=0$ and $w_p(t)=0$ . It follows that System (1) is not positive. Similarly, if ${\overline{A}}_{\mathrm{di}3}$ has an element $[{\overline{A}}_{\mathrm{di}3}{]}_{\mathrm{pq}}<0$ , or ${\overline{A}}_{\mathrm{di}4}$ has an element $[{\overline{A}}_{\mathrm{di}4}{]}_{\mathrm{pq}}<0$ , or ${\overline{F}}_{i2}$ has an element $[{\overline{F}}_{i2}{]}_{\mathrm{pq}}<0$ , $\forall i\in \underset{¯}{N}$ , one can get that System (1) is not positive.

Finally, suppose that there exists an element $[C_{i1}{]}_{\mathrm{pq}}<0$ , $\forall i\in \underset{¯}{N}$ , $p\ne q$ , where $[C_{i1}{]}_{\mathrm{pq}}$ is the $(p,q)\mathrm{th}$ scalar entry of $[C_{i1}]$ . From System (3), we obtain

$$\begin{array}{c}\multicolumn{1}{c}{z_p(t)=\sum _{g=1,g\ne p,g\ne q}^r{[C_{i1}]}_{\mathrm{pg}}{x}_{1g}(t)+{[C_{i1}]}_{\mathrm{pp}}{x}_{1p}(t)+{[C_{i1}]}_{\mathrm{pq}}{x}_{1q}(t)}\\ +\sum _{g=r+1}^n{[C_{i2}]}_{\mathrm{pg}}{x}_{2g}(t)+\sum _{g=1}^u{[D_i]}_{\mathrm{pg}}{w}_g(t)\hfill \end{array}$$

Then, if $x_{1q}(t)\ne 0$ , we can see that $z_p(t)<0$ is possible whenever $x_{1p}(t)=0$ , $x_{2p}(t)=0$ , and $w_p(t)=0$ . It follows that System (1) is not positive. Similarly, if $C_{i2}$ has an element $[C_{i2}{]}_{\mathrm{pq}}<0$ , or $D_i$ has an element $[D_i{]}_{\mathrm{pq}}<0$ , $\forall i\in \underset{¯}{N}$ , one can get that System (1) is not positive.

That is to say, System (1) is positive under arbitrary switching signals if and only if ${\overline{A}}_{i1}$ is a Metzler matrix, ${\overline{A}}_{\mathrm{di}1}⪰0$ , ${\overline{A}}_{\mathrm{di}2}⪰0$ , ${\overline{F}}_{i1}⪰0$ , ${\overline{A}}_{i3}⪰0$ , ${\overline{A}}_{\mathrm{di}3}⪰0$ , ${\overline{A}}_{\mathrm{di}4}⪰0$ , ${\overline{F}}_{i2}⪰0$ , $C_{i1}⪰0$ , $C_{i2}⪰0$ , $D_i⪰0$ , $\forall i\in \underset{¯}{N}$ .

This completes the proof.

Remark 2

When $E=I$ , positivity condition for positive switched non-singular systems has been proposed in Shen and Lam; ²² that is, System (1) with $E=I$ is positive if and only if $A_i$ is a Metzler matrix, $A_{\mathrm{di}}⪰0$ , $F_i⪰0$ , $C_i⪰0$ , $D_i⪰0$ , $\forall i\in \underset{¯}{N}$ .

In the sequence, by applying the average dwell time approach and the co-positive Lyapunov function technique, a sufficient condition for exponential stability with an L₁-gain performance of System (1) is proposed.

Theorem 2

Suppose that $(E,A_i)$ , $\forall i\in \underset{¯}{N}$ , is regular and causal. Let $\alpha >0$ and $\gamma >0$ be prescribed scalars, ${\overline{A}}_{i1}$ is a Metzler matrix, ${\overline{A}}_{\mathrm{di}1}⪰0$ , ${\overline{A}}_{\mathrm{di}2}⪰0$ , ${\overline{F}}_{i1}⪰0$ , ${\overline{A}}_{i3}⪰0$ , ${\overline{A}}_{\mathrm{di}3}⪰0$ , ${\overline{A}}_{\mathrm{di}4}⪰0$ , ${\overline{F}}_{i2}⪰0$ , $C_{i1}⪰0$ , $C_{i2}⪰0$ , $D_i⪰0$ , if $0<||{\overline{A}}_{\mathrm{di}4}|{|}_1<1$ , and there exist vectors ${\nu }_i$ , ${\upsilon }_{1i}$ , ${\upsilon }_{2i}$ , ${\upsilon }_{3i}\in R_{+}^n$ , such that $\forall i\in \underset{¯}{N}$

$$A_i^T{\nu }_i+\alpha E^T{\nu }_i+{\upsilon }_{1i}+{\upsilon }_{2i}+{\upsilon }_{3i}+C_i^T{1}_m⪯0$$

(11)

$$A_{\mathrm{di}}^T{\nu }_i-(1-d_d)e^{-\alpha d_2}{\upsilon }_{3i}⪯0$$

(12)

$$F_i^T{\nu }_i+D_i^T{1}_m-\gamma 1_u⪯0$$

(13)

then, System (1) is positive and exponentially stable with its decay rate estimate $\beta =\alpha -(\ln \mu )/T_a$ and achieves a prescribed L₁-gain performance level $\gamma$ for any switching signal $\sigma (t)$ with average dwell time

$$T_a>T_a^{*}=\frac{\ln \mu }{\alpha }$$

(14)

where $\mu \ge 1$ satisfies

$$\begin{array}{c}{\nu }_i⪯\mu {\nu }_j,{\upsilon }_{1i}⪯\mu {\upsilon }_{1j},{\upsilon }_{2i}⪯\mu {\upsilon }_{2j},\hfill \\ {\upsilon }_{3i}⪯\mu {\upsilon }_{3j},\forall i,j\in \underset{¯}{N}\hfill \end{array}$$

(15)

Proof

First, we prove that System (1) with $w(t)=0$ is exponentially stable. Choose the following co-positive Lyapunov function

$$V_i(t)=V_{1i}(t)+V_{2i}(t),\forall t\in [t_l,t_{l+1})$$

(16)

where

$$\begin{array}{c}{V}_{1i}(t)=x^T(t)E^T{\nu }_i\hfill \\ V_{2i}(k)=\underset{s=t-d_1}{\overset{t}{\int }}{e}^{\alpha (s-t)}{x}^T(s){\upsilon }_{1i}\mathrm{ds}\hfill \\ +\underset{s=t-d_2}{\overset{t}{\int }}{e}^{\alpha (s-t)}{x}^T(s){\upsilon }_{2i}\mathrm{ds}+\underset{t-d(t)}{\overset{t}{\int }}{e}^{\alpha (s-t)}{x}^T(s){\upsilon }_{3i}\mathrm{ds}\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}{\stackrel{·}{V}}_{1i}(t)+\alpha V_{1i}(t)={\stackrel{·}{x}}^T(t)E^T{\nu }_i+\alpha x^T(t)E^T{\nu }_i\hfill \\ =x^T(t)A_i^T{\nu }_i+x^T(t-d(t))A_{\mathrm{di}}^T{\nu }_i+\alpha x^T(t)E^T{\nu }_i\hfill \end{array}$$

(17)

$$\begin{array}{c}{\stackrel{·}{V}}_{2i}(t)+\alpha V_{2i}(t)=x^T(t){\upsilon }_{1i}-e^{-\alpha d_1}{x}^T(t-d_1){\upsilon }_{1i}\hfill \\ +x^T(t){\upsilon }_{2i}-e^{-\alpha d_2}{x}^T(t-d_2){\upsilon }_{2i}\hfill \\ +x^T(t){\upsilon }_{3i}-(1-\stackrel{·}{d}(t))e^{-\alpha d(t)}{x}^T(t-d(t)){\upsilon }_{3i}\hfill \\ \le x^T(t)[{\upsilon }_{1i}+{\upsilon }_{2i}+{\upsilon }_{3i}]-(1-d_d)e^{-\alpha d_2}{x}^T(t-d(t)){\upsilon }_{3i}\hfill \\ -e^{-\alpha d_1}{x}^T(t-d_1){\upsilon }_{1i}-e^{-\alpha d_2}{x}^T(t-d_2){\upsilon }_{2i}\hfill \end{array}$$

(18)

Adding the terms from equations (17)–(18) into ${\stackrel{·}{V}}_i(t)+\alpha V_i(t)$ , and from equations (11)–(12), we have

$$\begin{array}{c}{\stackrel{·}{V}}_i(t)+\alpha V_i(t)\le x^T(t)[A_i^T{\nu }_i+\alpha E^T{\nu }_i+{\upsilon }_{1i}+{\upsilon }_{2i}+{\upsilon }_{3i}]\hfill \\ +x^T(t-d(t))[A_{\mathrm{di}}^T{\nu }_i-(1-d_d)e^{-\alpha d_2}{\upsilon }_{3i}]\hfill \\ -e^{-\alpha d_1}{x}^T(t-d_1){\upsilon }_{1i}-e^{-\alpha d_2}{x}^T(t-d_2){\upsilon }_{2i}\le 0\hfill \end{array}$$

(19)

When $t\in [t_l,t_{l+1})$ , from equation (19), we have

$$V_{\sigma (t)}(t)\le e^{-\alpha (t-t_l)}{V}_{\sigma (t_l)}(t_l)$$

(20)

and from equations (15) and (16), at the switching instant $t_l$ , we can easily obtain

$$V_{\sigma (t_l)}(t_l)\le \mu V_{\sigma (t_l^{-})}(t_l^{-}),\forall l=1,2,\dots$$

(21)

where $t_l^{-}$ denotes the left limitation of $t_l$ . Then, it follows from equations (14), (20), and (21) and the relation $t=N_{\sigma }(t_0,t)\le t-t_0/T_a$ that

$$V_{\sigma (t)}(t)\le e^{-(\alpha -\frac{\ln \mu }{T_a})(t-t_0)}{V}_{\sigma (t_0)}(t_0)$$

(22)

On the other hand, let ${\nu }_i=[{\nu }_{i1}^T{\nu }_{i2}^T{]}^T$ , with ${\nu }_{i1}\in R^r$ , ${\nu }_{i2}\in R^{n-r}$ , then it follows from the Lyapunov function (equation (16)) that $V_{1i}(t)=x_1^T(t){\nu }_{i1}$ .

According to equations (16) and (22), we obtain

$$V_{\sigma (t)}(t)\ge {\beta }_1\Vert x_1(t){\Vert }_1$$

(23)

$$V_{\sigma (t_0)}(t_0)\le {\beta }_2\Vert \phi {\Vert }_{1c}$$

(24)

where ${\beta }_1=\underset{\forall i\in \underset{¯}{N}}{min}\underset{¯}{\rho }({\nu }_{i1})$ , ${\beta }_2=\underset{\forall i\in \underset{¯}{N}}{max}\overline{\rho }({\nu }_{i1})+1/\alpha (1-e^{-\alpha d_1})\underset{\forall i\in \underset{¯}{N}}{max}\overline{\rho }({\upsilon }_{1i})+1/\alpha (1-e^{-\alpha d_2})\underset{\forall i\in \underset{¯}{N}}{max}\overline{\rho }({\upsilon }_{2i})+1/\alpha (1-e^{-\alpha d_2})\underset{\forall i\in \underset{¯}{N}}{max}\overline{\rho }({\upsilon }_{3i})$ . Then, combining equations (22)–(24) yields

$$\Vert x_1(t){\Vert }_1\le \frac{{\beta }_2}{{\beta }_1}{e}^{-(\alpha -\frac{\ln \mu }{T_a})(t-t_0)}\Vert \phi {\Vert }_{1c}=\tau e^{-\beta t}\Vert \phi {\Vert }_{1c},\forall t\ge 0$$

(25)

where $\tau ={\beta }_2/{\beta }_1$ and $\beta =\alpha -(\ln \mu /T_a)$ . Therefore, from the average dwell time condition $T_a>T_a^{*}=\ln \mu /\alpha$ , one can readily obtain $\beta >0$ , which means that $x_1(t)$ is exponentially stable with decay rate $\beta$ . Moreover, it is easy to obtain from equation (25) that

$$\Vert x_1(t){\Vert }_1\le \tau e^{\beta d_2}\Vert \phi {\Vert }_{1c}{e}^{-\beta t},\forall t\ge 0$$

(26)

In the next equation, the second component solution $x_2(t)$ will be proven to be stable with the same exponential decay rate $\beta$ . Denote that $p(t)={\overline{A}}_{i3}{x}_1(t)+{\overline{A}}_{\mathrm{di}3}{x}_1(t-d(t))$ . Observe that if $t>d(t)$ , then

$$\begin{array}{c}\Vert x_1(t-d(t)){\Vert }_1\le \frac{{\beta }_2}{{\beta }_1}{e}^{-\beta (t-d(t))}\Vert \phi {\Vert }_{1c}\hfill \\ \le \frac{{\beta }_2}{{\beta }_1}{e}^{-\beta t}{e}^{\beta d_2}\Vert \phi {\Vert }_{1c}=\tau e^{\beta d_2}\Vert \phi {\Vert }_{1c}^{-\beta t}\hfill \end{array}$$

(27)

For $t\in [0,d(t)]$ , we have

$$\begin{array}{c}\Vert x_1(t-d(t)){\Vert }_1=\Vert {\phi }_1{\Vert }_{1c}\le \Vert \phi {\Vert }_{1c}\le \Vert \phi {\Vert }_{1c}{e}^{-\beta (t-d(t))}\hfill \\ \le e^{\beta d_2}\Vert \phi {\Vert }_{1c}{e}^{-\beta t}\le \tau e^{\beta d_2}\Vert \phi {\Vert }_{1c}{e}^{-\beta k}\hfill \end{array}$$

(28)

From equations (27) and (28), we have

$$\Vert x_1(t-d(t)){\Vert }_1\le \tau e^{\beta d_2}\Vert \phi {\Vert }_{1c}{e}^{-\beta t},\forall t\ge 0$$

(29)

From equations (26) and (29), we have

$$\begin{array}{c}\multicolumn{1}{c}{\Vert p(t){\Vert }_1\le \Vert {\overline{A}}_{i3}{\Vert }_1\Vert x_1(t){\Vert }_1+\Vert {\overline{A}}_{\mathrm{di}3}{\Vert }_1\Vert x_1(t-d(t)){\Vert }_1}\\ \multicolumn{1}{c}{={\tau }_1\Vert \phi {\Vert }_{1c}{e}^{-\beta t},\forall t\ge 0}\end{array}$$

(30)

where ${\tau }_1=\tau e^{\beta d_2}(||{\overline{A}}_{i3}|{|}_1+||{\overline{A}}_{\mathrm{di}3}|{|}_1)$ . Moreover, equation (3) yields

$$x_2(t)={\overline{A}}_{\mathrm{di}4}{x}_2(t-d(t))+p(t)$$

(31)

Therefore, $\forall t\ge 0$

$$\Vert x_2(t){\Vert }_1\le \Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1\Vert x_2(t-d(t)){\Vert }_1+\Vert p(t){\Vert }_1$$

(32)

Set $\delta =\max \{{\tau }_1;e^{\beta d_2}\}$ . If $t\in [0,d(t)]$ , then $t-d(t)\in [-d(t),0]$ .

So, from equation (32), we have

$$\Vert x_2(t){\Vert }_1\le \Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1\Vert \phi {\Vert }_{1c}+\Vert p(t){\Vert }_1\le ({\Vert {\overline{A}}_{\mathrm{di}4}\Vert }_1\delta +\delta )\Vert \phi {\Vert }_{1c}{e}^{\beta k}$$

(33)

If $t\in [d(t),2d(t)]$ , then $t-d(t)\in [0,d(t)]$ . From equations (32) and (33), we have

$$\Vert x_2(t){\Vert }_1\le (\Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1^2\delta +{\Vert {\overline{A}}_{\mathrm{di}4}\Vert }_1\delta +\delta )\Vert \phi {\Vert }_{1c}{e}^{-\beta k}$$

Suppose that $\forall t\in [(l-1)d(t),\mathrm{ld}(t)]$ , then

$${\Vert x_2(t)\Vert }_1\le \left({\Vert {\overline{A}}_{di4}\Vert }_1^l\delta +{\Vert {\overline{A}}_{di4}\Vert }_1^{l-1}\delta +\cdots +{\Vert {\overline{A}}_{di4}\Vert }_1\delta +\delta \right){\Vert \phi \Vert }_{1c}{e}^{-\beta k}$$

Thus, when $t\in [\mathrm{ld}(t),(l+1)d(t)]$ , $t-d(t)\in [(l-1)d(t),\mathrm{ld}(t)]$ , through the inductive supposition method, and we have from equations (32) and (33) that

$$\begin{array}{c}\Vert x_2(t){\Vert }_1\le \Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1(\delta +\delta {\Vert {\overline{A}}_{\mathrm{di}4}\Vert }_1+\cdots +\delta \Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1^l)\hfill \\ \Vert \phi {\Vert }_{1c}{e}^{-\beta t}+\Vert p(t){\Vert }_1\hfill \\ \le (\delta +\delta {\Vert {\overline{A}}_{\mathrm{di}4}\Vert }_1+\cdots +\delta \Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1^{l+1})\Vert \phi {\Vert }_{1c}{e}^{-\beta t}\hfill \end{array}$$

If $0<||{\overline{A}}_{\mathrm{di}4}|{|}_1<1$ , by induction, we obtain

$$\begin{array}{c}\Vert x_2(t){\Vert }_1\le (\delta +\delta {\Vert {\overline{A}}_{\mathrm{di}4}\Vert }_1+\cdots +\delta \Vert {\overline{A}}_{\mathrm{di}4}{\Vert }_1^l+\cdots )\hfill \\ \Vert \phi {\Vert }_{1c}{e}^{-\beta t}\hfill \\ \le \frac{\delta }{1-{\Vert {\overline{A}}_{\mathrm{di}4}\Vert }_1}\Vert \phi {\Vert }_{1c}{e}^{-\beta t}\hfill \end{array}$$

(34)

From equations (25) and (34), we finally have $||x(t)|{|}_1\le M||\phi |{|}_{1c}{e}^{-\beta t}$ , $M=\max \{\tau ;\delta /1-||{\overline{A}}_{\mathrm{di}4}|{|}_1\}$ , $\forall t\ge 0$ ; thus, we conclude that System (3) with $w(t)=0$ is exponentially stable with decay rate $\beta$ , which infers that System (1) with $w(t)=0$ is exponentially stable with decay rate $\beta$ .

Next the L₁-gain performance will be considered. Noticing equations (11)–(13), we have

$$\begin{array}{c}{\stackrel{·}{V}}_i(t)+\alpha V_i(t)+\Gamma (t)\le x^T(t)[A_i^T{\nu }_i\hfill \\ +\alpha E^T{\nu }_i+{\upsilon }_{1i}+{\upsilon }_{2i}+{\upsilon }_{3i}+C_i^T{1}_m]\hfill \\ +x^T(t-d(t))[A_{\mathrm{di}}^T{\nu }_i-(1-d_d)e^{-\alpha d_2}{\upsilon }_{3i}]\hfill \\ -e^{-\alpha d_1}{x}^T(t-d_1){\upsilon }_{1i}-e^{-\alpha d_2}{x}^T(t-d_2){\upsilon }_{2i}\hfill \\ +w^T(t)[F_i^T{\nu }_i+D_i^T{1}_m-\gamma 1_u]\le 0\hfill \end{array}$$

(35)

where $\Gamma (t)=||z(t)|{|}_1-\gamma ||w(t)|{|}_1$ .

Then, it follows $\forall t\in [t_l,t_{l+1})$

$$V_{\sigma (t)}(t)\le e^{-\alpha (t-t_l)}{V}_{\sigma (t_l)}(t_l)-\underset{t_l}{\overset{t}{\int }}{e}^{-\alpha (t-s)}\Gamma (s)$$

(36)

From equations (15) and (16), at the switching instant $t_l$ , we can easily obtain

$$V_{\sigma (t_l)}(t_l)\le \mu V_{\sigma (t_l^{-})}(t_l^{-}),\forall l=1,2,\dots$$

(37)

Combining equations (36) and (37) leads to

$$\begin{array}{c}\multicolumn{1}{c}{V_{\sigma (t)}(t)\le e^{-\alpha (t-t_0)+N_{\sigma }(t_0,t)\ln \mu }{V}_{\sigma (t_0)}(t_0)}\\ -\underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)+N_{\sigma }(s,t)\ln \mu }\Gamma (s)\mathrm{ds}\hfill \end{array}$$

(38)

Under the zero initial condition, from equation (38), we have

$$0\le -\underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)+N_{\sigma }(s,t)\ln \mu }\Gamma (s)\mathrm{ds}$$

namely

$$\begin{array}{c}\underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)+N_{\sigma }(s,t)\ln \mu }{\Vert z(s)\Vert }_1\mathrm{ds}\hfill \\ \le \gamma \underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)+N_{\sigma }(s,t)\ln \mu }{\Vert w(s)\Vert }_1\mathrm{ds}\hfill \end{array}$$

(39)

Multiplying both sides of equation (39) by $e^{-N_{\sigma }(t_0,t)\ln \mu }$ yields

$$\begin{array}{c}\underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)-N_{\sigma }(t_0,s)\ln \mu }{\Vert z(s)\Vert }_1\mathrm{ds}\hfill \\ \le \gamma \underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)-N_{\sigma }(t_0,s)\ln \mu }{\Vert w(s)\Vert }_1\mathrm{ds}\hfill \end{array}$$

(40)

Notice that $N_{\sigma }(t_0,s)<(s-t_0)/T_a$ and $T_a>T_a^{*}=\ln \mu /\alpha$ ; then, we have

$$N_{\sigma }(t_0,s)\ln \mu \le \alpha (s-t_0)$$

(41)

Combining equations (40) and (41) leads to

$$\underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-t_0)}{\Vert z(s)\Vert }_1\mathrm{ds}\le \gamma \underset{t_0}{\overset{t}{\int }}{e}^{-\alpha (t-s)}{\Vert w(s)\Vert }_1\mathrm{ds}$$

(42)

Integrating both sides of equation (41) from $t=0$ to $\infty$ leads to inequality

$$\underset{0}{\overset{\infty }{\int }}{\Vert z(t)\Vert }_1\mathrm{dt}\le \gamma \underset{0}{\overset{\infty }{\int }}{\Vert w(t)\Vert }_1\mathrm{dt}$$

This completes the proof.

Remark 3

When no parameters switches or jump occurs in System (1), one can obtain the corresponding sufficient condition that guarantees the exponential stability of the system by removing equation (15). In this case, the decay rate is only related to $\alpha$ , which can be tuned according to practical applications.

Remark 4

From Theorem 2, one can see that the minimal average dwell time $T_a^{*}$ is determined by the scalars $\alpha$ and $\mu$ . One can choose different $\alpha$ and $\mu$ to have different $T_a^{*}$ if equations (11)–(15) hold. In addition, when $\mu =1$ in equation (14), which leads to ${\nu }_i=\mu {\nu }_j$ , ${\upsilon }_{1i}=\mu {\upsilon }_{1j}$ , ${\upsilon }_{2i}=\mu {\upsilon }_{2j}$ , ${\upsilon }_{3i}=\mu {\upsilon }_{3j}$ , $\forall i,j\in \underset{¯}{N}$ , and $T_a^{*}\to 0$ , then, System (1) possesses a common Lyapunov function, and switching signals can be arbitrary.

When $E=I$ , the results on exponential stability and L₁-gain performance analysis for positive switched non-singular system can be obtained easily from Theorem 2.

Corollary 1

Let $\alpha >0$ and $\gamma >0$ be prescribed scalars, $A_i$ is a Metzler matrix, $A_{\mathrm{di}}⪰0$ , $F_i⪰0$ , $C_i⪰0$ , $D_i⪰0$ , if there exist vectors ${\nu }_i$ , ${\upsilon }_{1i}$ , ${\upsilon }_{2i}$ , ${\upsilon }_{3i}\in R_{+}^n$ , such that $\forall i\in \underset{¯}{N}$

$$\begin{array}{c}{A}_i^T{\nu }_i+\alpha {\nu }_i+{\upsilon }_{1i}+{\upsilon }_{2i}+{\upsilon }_{3i}+C_i^T{1}_m⪯0\hfill \\ A_{\mathrm{di}}^T{\nu }_i-(1-d_d)e^{-\alpha d_2}{\upsilon }_{3i}⪯0\hfill \\ F_i^T{\nu }_i+D_i^T{1}_m-\gamma 1_u⪯0\hfill \end{array}$$

then, System (1) with $E=I$ is positive and exponentially stable with its decay rate estimate $\beta =\alpha -(\ln \mu )/T_a$ and achieves a prescribed L₁-gain performance level $\gamma$ for any switching signal $\sigma (t)$ with average dwell time (equation (14)).

Problem 1

The exponential stability and L₁-gain performance can be analyzed by solving feasibility problem of LPs (equations (11)–(15)) in Theorem 2. Similarly, for given $\alpha$ , $d_1$ , and $d_2$ , the optimal L₁-gain performance level can be determined by solving the following convex optimization problem

$$\begin{array}{c}\underset{{\nu }_i,{\upsilon }_{1i},{\upsilon }_{2i},{\upsilon }_{3i}}{min}\gamma \hfill \\ s.t.(11)-(15),\forall i\in \underset{¯}{N}\hfill \end{array}$$

Numerical results

In this section, we give an example to show the validity of our theoretical results.

Example 1

Consider System (1) with two modes, that is, $\underset{¯}{N}=\{1,2\}$ , and $E=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right]$

$$\begin{array}{c}{A}_1=\left[\begin{array}{ccc}-6\hfill & 0\hfill & 0.5\hfill \\ \multicolumn{1}{c}{0}& -8\hfill & 0.9\hfill \\ \multicolumn{1}{c}{1}& 1\hfill & -1\hfill \end{array}\right],A_{d1}=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0.2\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0.1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0.1\hfill \end{array}\right],\hfill \\ F_1=\left[\begin{array}{c}0.2\hfill \\ \multicolumn{1}{c}{0.2}\\ \multicolumn{1}{c}{0.1}\end{array}\right]C_1=\left[\begin{array}{ccc}0.1\hfill & 0.1\hfill & 0.2\hfill \end{array}\right],D_1=0.01,\hfill \\ A_2=\left[\begin{array}{ccc}-5\hfill & 0\hfill & 0.3\hfill \\ \multicolumn{1}{c}{0}& -9\hfill & 0.6\hfill \\ \multicolumn{1}{c}{1}& 1\hfill & -1\hfill \end{array}\right],A_{d2}=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0.1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0.2\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0.2\hfill \end{array}\right],\hfill \\ F_2=\left[\begin{array}{c}0.1\hfill \\ \multicolumn{1}{c}{0.1}\\ \multicolumn{1}{c}{0.2}\end{array}\right]C_2=\left[\begin{array}{ccc}0.2\hfill & 0.1\hfill & 0.1\hfill \end{array}\right],D_2=0.02\hfill \end{array}$$

Direct calculation shows that

$$\begin{array}{c}\det (\mathrm{sE}-A_1)=s^2+12.6s+38.6\ne 0,\mathrm{for}s=0\hfill \\ \mathrm{deg}(\det (\mathrm{sE}-A_1))=\mathrm{deg}(s^2+12.6s+38.6)=\mathrm{rank}E=2\hfill \\ \det (\mathrm{sE}-A_2)=s^2+13.1s+39.3\ne 0,\mathrm{for}s=0\hfill \\ \mathrm{deg}(\det (\mathrm{sE}-A_2))=\mathrm{deg}(s^2+13.1s+39.3)=\mathrm{rank}E=2\hfill \end{array}$$

Thus, the switched singular System (1) is said to be regular and casual. Furthermore, by simple computation, we obtain

$$\begin{array}{c}{\overline{A}}_{11}=\left[\begin{array}{cc}-5.5\hfill & 0.5\hfill \\ \multicolumn{1}{c}{0.9}& -7.1\hfill \end{array}\right],{\overline{A}}_{d11}=\left[\begin{array}{cc}1\hfill & 0.5\hfill \\ \multicolumn{1}{c}{0}& 1.9\hfill \end{array}\right],\hfill \\ {\overline{A}}_{d12}=\left[\begin{array}{c}0.25\hfill \\ \multicolumn{1}{c}{0.19}\end{array}\right],{\overline{F}}_{11}=\left[\begin{array}{c}0.25\hfill \\ \multicolumn{1}{c}{0.29}\end{array}\right]\hfill \\ {\overline{A}}_{13}=\left[\begin{array}{cc}1\hfill & 1\hfill \end{array}\right],{\overline{A}}_{d13}=\left[01\right],{\overline{A}}_{d14}=0.1,{\overline{F}}_{12}=0.1\hfill \\ {\overline{A}}_{21}=\left[\begin{array}{cc}-4.7\hfill & 0.3\hfill \\ \multicolumn{1}{c}{0.6}& -8.4\hfill \end{array}\right],{\overline{A}}_{d21}=\left[\begin{array}{cc}1\hfill & 0.3\hfill \\ \multicolumn{1}{c}{0}& 1.6\hfill \end{array}\right],\hfill \\ {\overline{A}}_{d22}=\left[\begin{array}{c}0.16\hfill \\ \multicolumn{1}{c}{0.32}\end{array}\right],{\overline{F}}_{21}=\left[\begin{array}{c}0.16\hfill \\ \multicolumn{1}{c}{0.22}\end{array}\right]\hfill \\ {\overline{A}}_{23}=\left[11\right],{\overline{A}}_{d23}=\left[01\right],{\overline{A}}_{d24}=0.20,{\overline{F}}_{22}=0.2\hfill \end{array}$$

Therefore, by Theorem 1, System (1) is positive.

Let $d(t)=0.2+0.1\sin t$ , $\alpha =0.5$ , solve the optimization Problem 1, we have ${\gamma }^{*}=0.4167$ , the corresponding solutions are as follows

$$\begin{array}{c}{\nu }_1=\left[\begin{array}{c}0.5059\hfill \\ \multicolumn{1}{c}{0.6807}\\ \multicolumn{1}{c}{1.6061}\end{array}\right],{\nu }_2=\left[\begin{array}{c}0.6277\hfill \\ \multicolumn{1}{c}{0.5011}\\ \multicolumn{1}{c}{1.2560}\end{array}\right],\hfill \\ {\upsilon }_{11}=\left[\begin{array}{c}0.1069\hfill \\ \multicolumn{1}{c}{0.1156}\\ \multicolumn{1}{c}{0.0286}\end{array}\right],{\upsilon }_{12}=\left[\begin{array}{c}0.1426\hfill \\ \multicolumn{1}{c}{0.1661}\\ \multicolumn{1}{c}{0.0330}\end{array}\right]\hfill \\ {\upsilon }_{21}=\left[\begin{array}{c}0.1069\hfill \\ \multicolumn{1}{c}{0.1156}\\ \multicolumn{1}{c}{0.0286}\end{array}\right],{\upsilon }_{22}=\left[\begin{array}{c}0.1426\hfill \\ \multicolumn{1}{c}{0.1661}\\ \multicolumn{1}{c}{0.0330}\end{array}\right],\hfill \\ {\upsilon }_{31}=\left[\begin{array}{c}0.7555\hfill \\ \multicolumn{1}{c}{3.0525}\\ \multicolumn{1}{c}{0.4548}\end{array}\right],{\upsilon }_{32}=\left[\begin{array}{c}0.9408\hfill \\ \multicolumn{1}{c}{2.4053}\\ \multicolumn{1}{c}{0.5679}\end{array}\right]\hfill \end{array}$$

By analysis, it can be found that the allowable minimum of $\mu$ is $\mu =1.4487$ when $\alpha =0.5$ is fixed; in this case, $T_a^{*}=0.7414$ . One can also get the decay rate as $\beta =\alpha -\ln (\mu )/T_a=0.0945$ with $\mu =1.5$ and $T_a=1$ . Thus, according to Theorem 2, we can conclude that System (1) is exponentially stable with an optimal L₁-gain performance level $\gamma =0.4167$ for any switching signal satisfying $T_a>0.7414$ .

The simulation results are shown in Figures 1–3, where the initial condition is $x(0)=[23{]}^T$ , $x(t)=[00{]}^T$ , $t=[-0.3,-0.1)$ , and the disturbance is $w(t)=0.5e^{-0.5t}$ . The switching signal $\sigma (t)$ is showed in Figure 1; Figures 2–3 plot x and z of the system over 0–25 s under the switching signal, respectively. It is easy to see that System (1) is positive and exponentially stable. This demonstrates the proposed results.

Figure 1.

Switching signal.

Figure 2.

The state $x(t)$ of the given system.

Figure 3.

The output $z(t)$ of the given system.

Conclusion

The problem of exponential stability and L₁-gain performance analysis for continuous-time delayed SPSSs has been solved in this article. First, a necessary and sufficient condition for the positivity of this system has been derived via the singular value decomposition method. Then, a condition has been derived to solve the problem of exponential stability and L₁-gain performance analysis. The decay rate estimation has been derived, and the optimal system performance level can be determined by solving a convex optimization problem. Finally, a numerical example is provided to demonstrate the effectiveness of the obtained results.