Volume 14 Number 2
April 2017
Article Contents
Zhi Liu and Yu-Zhen Wang. Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points. International Journal of Automation and Computing, vol. 14, no. 2, pp. 213-220, 2017. doi: 10.1007/s11633-016-1003-5
Cite as: Zhi Liu and Yu-Zhen Wang. Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points. International Journal of Automation and Computing, vol. 14, no. 2, pp. 213-220, 2017. doi: 10.1007/s11633-016-1003-5

Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points

Author Biography:
  • Yu-Zhen Wang graduated from Tai'an Teachers College, China in 1986, received the M. Sc. degree from Shandong University of Science and Technology, China in 1995, and the Ph.D. degree from Institute of Systems Science, Chinese Academy of Sciences, China in 2001. Since 2003, he is a professor with School of Control Science and Engineering, Shandong University, China, and now the dean of the School of Control Science and Engineering, Shandong University. From 2001 to 2003, he worked as a postdoctoral fellow in Tsinghua University, China. From March 2004 to June 2004, from Februery 2006 to May 2006 and from November 2008 to January 2009, he visited City University of Hong Kong as a research fellow. From September 2004 to May 2005, he worked as a visiting research fellow at the National University of Singapore. He received the Prize of Guan Zhaozhi in 2002, the Prize of Huawei from the Chinese Academy of Sciences in 2001, the Prize of Natural Science from Chinese Education Ministry in 2005, and the National Prize of Natural Science of China in 2008. Currently, he is an associate editor IMA Journal of Mathematical Control and Information, and a Technical Committee member of IFAC. His research interests include nonlinear control systems, Hamiltonian systems and Boolean networks. E-mail:yzwang@sdu.edu.cn

  • Corresponding author: Zhi Liu received the B. Sc. degree from School of Mathematics and Information, Ludong University, China in 2009, and the M. Sc. degree from School of Mathematical Science, University of Jinan, China in 2013. Since 2013 she is a Ph.D. degree candidate at School of Control Science and Engineering, Shandong University, China. Her research interests include positive systems and switched systems. E-mail:liuzhishfd2008@163.com
  • Received: 2015-04-16
  • Accepted: 2015-09-08
  • Published Online: 2016-06-20
Fund Project:

National Natural Science Foundation of China 61374065

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Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points

  • Corresponding author: Zhi Liu received the B. Sc. degree from School of Mathematics and Information, Ludong University, China in 2009, and the M. Sc. degree from School of Mathematical Science, University of Jinan, China in 2013. Since 2013 she is a Ph.D. degree candidate at School of Control Science and Engineering, Shandong University, China. Her research interests include positive systems and switched systems. E-mail:liuzhishfd2008@163.com
Fund Project:

National Natural Science Foundation of China 61374065

Abstract: This paper studies the regional stability for positive switched linear systems with multi-equilibrium points (PSLS-MEP). First, a sufficient condition is presented for the regional stability of PSLS-MEP via a common linear Lyapunov function. Second, by establishing multiple Lyapunov functions, a dwell time based condition is proposed for the regional stability analysis. Third, a suprasphere which contains all equilibrium points is constructed as a stability region of the considered PSLS-MEP, which is less conservative than existing results. Finally, the study of an illustrative example shows that the obtained results are effective in the regional stability analysis of PSLS-MEP.

Zhi Liu and Yu-Zhen Wang. Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points. International Journal of Automation and Computing, vol. 14, no. 2, pp. 213-220, 2017. doi: 10.1007/s11633-016-1003-5
Citation: Zhi Liu and Yu-Zhen Wang. Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points. International Journal of Automation and Computing, vol. 14, no. 2, pp. 213-220, 2017. doi: 10.1007/s11633-016-1003-5
  • Switched systems whose subsystems operate according to a switching law have found wide applications in control systems, power systems, neural networks, process control, traffic control and many other fields[1-3]. The stability analysis of switched systems has attracted a great deal of attention in the last three decades[4-11]. Shorten and Narendra[8] proposed a necessary and sufficient condition for the existence of common Lyapunov function. In addition, multiple Lyapunov function method[10] and average dwell time technique[11] are viewed as two valid ways to study the stability for switched systems. However, most of the existing results[12-16] typically focused on the stability of switched systems with a common equilibrium point.

    During lots of switching events, the equilibrium point might change because of external disturbances. For example, in power systems, cutting machine and load cutting control could change the structure of the power network, meanwhile the operating point (i.e., equilibrium point) of the corresponding system may also change. In mechanical systems, mechanical vibration will lead to the position deviation of the equilibrium. In addition, the phenomenon of multi-equilibrium points (MEP) is found widely in the population dynamic system, network control system, missile launching system, unmanned aerial vehicle flight control, etc.

    Therefore, the model of switched systems with MEP can better describe many natural phenomena[17-19]. In references [17, 18], by using the method of quadratic Lyapunov-like function, the authors provided a rough estimation of stability region. The stability properties for a kind of nonlinear switched systems with MEP were studied in [19], and the trajectory is demonstrated to globally converge to a connected superset. However, it should be pointed out that the boundaries of stability region given by all mentioned methods are relatively conservative.

    On the other hand, a positive system whose states, inputs and outputs only take nonnegative values has been studied widely in the last two decades[20-28]. A finite number of positive linear systems running under some switching law is called a positive switched linear system (PSLS), which has numerous applications in engineering, economics, social science, management science, biology, medicine, etc. During the past two or three decades, the stability theory of PSLSs with a common equilibrium point has been well studied, and a number of effective methods have been proposed. Among them, the copositive Lyapunov function[24, 26] is specifically constructed for the PSLSs and is proved to have more excellent characteristics.

    In this paper we study the regional stability of PSLS-MEP by using the Lyapunov function method, and present several new results. The main contributions of this paper are as follows. 1) Both common linear Lyapunov function method and multiple Lyapunov function method are proposed for the regional stability of PSLS-MEP. It is noted that one can hardly construct a linear Lyapunov function for the regional stability of general switched systems with MEP. This is the main difference between our results and the existing ones[17-19]. 2) Using our methods, it is very convenient to obtain that the trajectory of PSLS-MEP enters and remains in a suprasphere which contains all equilibrium points. Compared with [17-19], the stability region given in our work is much less conservative (please see Remark 3.5 and practical example below).

    The remainder of the paper is organized as follows. Some necessary preliminaries are introduced in Section 2. The regional stability analysis for PSLS-MEP are carried out in Section 3. An illustrative example is given in Section 4 to show the effectiveness of our main results, which is followed by conclusions in Section 5.

    Notations. The notations of this paper are fairly standard. ${\bf R}$ ( ${\bf R}_+$ ), ${\bf R}^n$ ( ${\bf R}^n_+$ ) and ${\bf R}^{n\times n}$ denote the set of all real (nonnegative) numbers, the set of $n$ -dimensional (nonnegative) vectors, and the space of $n\times n$ -dimensional real matrices, respectively. $A\succeq0$ $(\preceq0, \succ0, \prec0)$ denotes that all elements of matrix $A$ are nonnegative (non-positive, positive, negative). $A^{\rm T}$ stands for the transpose of matrix $A$ . ${M_n}$ denotes the set of $n\times n$ Metzler matrix (A real matrix is called a Metzler matrix if its off-diagonal elements are nonnegative). $\|\cdot\|$ means the Euclidean norm of vector in ${\bf R}^n$ , i.e., $\|x\|=\sqrt{| x_1|^2+|x_2|^2+\cdots+|x_n|^2}$ . For any two points $x_1\in {\bf R}^n$ and $x_2\in {\bf R}^n$ , the distance between $x_1$ and $x_2$ is denoted by $d(x_1, x_2)=\displaystyle\parallel x_1-x_2\parallel.$ " $\mathop{\rm sgn}$ " denotes the sign function, i.e., ${\rm{sgn}}(x) = \left\{ {\begin{array}{*{20}{l}} {1,}&{if{\rm{ }}x > 0}\\ {0,}&{if{\rm{ }}x = 0}\\ { - 1,}&{if{\rm{ }}x < 0.} \end{array}} \right.$

  • Consider the following PSLS-MEP:

    $ \begin{align} \label{2.1}\left\{ \begin{array}{ll} \dot{x}(t)=A_{\sigma(t)}\Big(x(t)-x_{\sigma(t)}^e\Big)\\ x(t_0)=x_0 \end{array} \right.\end{align} $

    (1)

    where $x(t): {\bf R}_+\rightarrow{\bf R}^n_+$ is the state vector, the switching signal $\sigma(t): [t_0, +\infty)\rightarrow\underline{l}=\{1, 2, \cdots, l\}$ is a piecewise constant function, $l$ is the number of subsystems, $A_\sigma$ $(\sigma\in\underline{l})$ are the system matrices, $x_\sigma^e=(x_{\sigma1}^e, \cdots, x_{\sigma n}^e)^{\rm T}$ $(\sigma\in \underline{l})$ are the equilibrium points, and the initial state $x_0\succeq0$ . The switching moment sequence is denoted by $\{t_k\}_{k=0}^{+\infty}$ , which satisfies $t_0 < t_1 < t_2 < \cdots < t_k < \cdots$ , where $t_0$ ( $\geq0$ ) denotes the initial moment, $t_k$ denotes the $k$ -th switching instant, $\tau_k=t_{k}-t_{k-1}$ $(k=1, 2, \cdots)$ are the dwell time of the $k$ -th activated subsystem, and $\tau_k>0$ . Throughout this paper, we suppose that: 1) There are finite switching times in any bounded time interval. 2) Every subsystem can be activated in our considered operating intervals. 3) All the subsystems are asymptotically stable.

    Next, we give some preliminary results on positive systems.

    Definition 1[12]. The system (1) is said to be positive if for any initial condition $x(t_0)\succeq 0$ and any switching signal $\sigma(t)$ , the corresponding trajectory $x(t)$ satisfies $x(t)\succeq 0, \ \forall\ t\geq t_0$ .

    Lemma 1[12]. Let $A\in {\bf R}^{n\times n}$ . Then, $e^{At}\succeq0$ , $\forall$ $ t\geq0$ , if and only if $A\in {M_n}$ .

    We present a necessary and sufficient condition for the positivity of the system (1).

    Lemma 2. For any $x(t_0)\succeq0$ and any switching signal $\sigma(t)$ , system (1) is positive if and only if $A_{\sigma}\in {M_n}$ and $A_\sigma x_\sigma^e\preceq 0$ , $\forall\ \sigma\in\underline{l}$ .

    Proof. We first consider the following system:

    $ \begin{align}\label{2.2}\left\{ \begin{array}{ll} \dot{x}(t)=A(x(t)-x^e)\\ x(t_0)=x_0. \end{array} \right.\end{align} $

    (2)

    Apparently, the trajectory of system (2) can be described by

    $ \begin{align*} x(t)={\rm e}^{A(t-t_0)}x_0+\int_{t_0}^{t}\Big(-Ax^e{\rm e}^{A(t-\tau)}\Big){\rm d}\tau. \end{align*} $

    From Lemma 1, we can conclude that for any initial state $x(t_0)\succeq0$ , the corresponding trajectory $x(t)\succeq0$ is equivalent to $A\in {M_n}$ , and $Ax^e\preceq 0$ . This together with Lemma 4 in [13] shows that the conclusion holds.

    Remark 1. It should be point out that the system matrices satisfying $A_{\sigma}\in {M_n}$ and $A_\sigma x_\sigma^e\preceq 0$ $(x_\sigma^e\succeq0)$ are widespread. For example, there is a 2-dimensional matrix $A=\left( \begin{array}{cc} a & b\\ c & d\\ \end{array} \right) $ and an equilibrium point $x^e=\left( \begin{array}{cc} e \\ f \end{array} \right) $ , where $a, b, c, d, e, f$ are constants. A simple calculation shows that when $a, b, c, d, e, f$ satisfy

    $ \begin{align*}\left\{ \begin{array}{ll} b\geq 0\\ c\geq 0\\ e\geq 0\\ f\geq 0\\ ae+bf\leq 0 \\ ce+df\leq 0 \end{array} \right. \end{align*} $

    $x^e\succeq0$ , $A\in {M_n}$ and $A x^e\preceq 0$ hold.

    Finally, we give the concept of regional stability for system (1).

    Definition 2[17]. System (1) is said to be regionally stable under arbitrary switching signal $\sigma(t)$ , if there exists a nonempty bounded set ${N}\subseteq {\bf R}^n_+$ , such that for any $x(t_0)\succeq0$ , $\lim\limits_{t\rightarrow+\infty} x(t)\in {N}$ . The set ${N}$ is called a stable region of system (1).

    Remark 2. Although all the subsystems are asymptotically stable, system (1) cannot converge to a point under arbitrary switching signals because all the subsystems share different equilibrium points. In fact, the trajectory of (1) will enter into a region which contains all the equilibrium points and some trajectories.

  • In this section, we study the regional stability of the system (1), and present the main results of this paper. To this end, we give the following two propositions, which are crucial in the proof of our main results.

    Given $a_i\in {\bf R}^n$ $(i\in \underline{n}=\{1, 2, \cdots, n\})$ , denote by

    $ \begin{align*} \widehat{a}=\displaystyle\frac1n\sum\limits_{i=1}^{n}a_i \end{align*} $

    and

    $ \begin{align*} {M}=\{x\in{\bf R}^n\ |\ d(x, \widehat{a}) \leq\max\limits_{i\in\underline{n}}d(\widehat{a}, a_i)\}.\end{align*} $

    Proposition 1. If there exists a point $\widetilde{a}\in {\bf R}^n$ such that

    $ \begin{align*} \sum\limits_{i=1}^{n}d(\widetilde{a}, a_i)= \min\limits_{x\in{\bf R}^n}\sum\limits_{i=1}^{n}d(x, a_i) \end{align*} $

    then $\widetilde{a}\in {M}$ .

    Proof. For any $a^{*}\notin{M}$ , let $a'$ be the intersection of the line segment $\widehat{a}a^{*}$ and ${\partial M}$ (the boundary of the suprasphere ${M}$ ). For a special case, the three points $a_i$ $(i\in\underline{n})$ , $a'$ , $a^{*}$ are collinear, and thus one immediately finds that the length of the line segments satisfy $a^{*}a_i>a'a_i$ , namely

    $ \begin{align*} d(a^{*}, a_i)>d(a', a_i). \end{align*} $

    If the three points $a_i$ $(i\in\underline{n})$ , $a'$ , $a^{*}$ are not collinear, we know that the three points $a^{*}$ , $a'$ , $a_i$ $(i\in\underline{n})$ can be used as three vertices in a triangle. Next, we will explain all the triangles are obtuse triangles, and the line segments $a^{*}a_i$ are the edges with respect to obtuse angles in the triangle $\triangle a^{*}a'a_i$ . First, one can construct a tangent plane of ${M}$ through the point $a'$ . Let $a_i^*$ be the intersection of the tangent plane and the line segment $a_ia^{*}$ . It is easy to see that the points $a_i$ , $a'$ , $a_i^*$ and $a^*$ are in the same plane, and the angle $\angle a^*a'a_i^*$ is a right angle. Then, the angle $\angle a_ia'a^*$ is an obtuse angle in the triangle $\triangle a^*a'a_i$ . From the relationship between side and angle, one can conclude that the size of the line segment $a^{*}a_i$ is greater than the line segment $a'a_i$ , i.e.,

    $ \begin{align*} d(a^{*}, a_i)>d(a', a_i)\end{align*} $

    holds for any $i\in\{1, 2, \cdots, n\}$ . That is to say

    $ \begin{align*} \sum\limits_{i=1}^{n}d(a^{*}, a_i)>\sum\limits_{i=1}^{n}d(a', a_i).\end{align*} $

    Hence, for any $a^{*}\notin {M}$ , we are always able to find a point $a'\in{\partial M}\subseteq{M}$ such that

    $ \begin{align*}\sum\limits_{i=1}^{n}d(a^{*}, a_i)>\min\limits_{x\in {\bf R}^n} \sum\limits_{i=1}^{n}d(x, a_i)=\sum\limits_{i=1}^{n}d(\widetilde{a}, a_i).\end{align*} $

    Therefore, $\widetilde{a}\in{M}$ .

    Remark 3. From Proposition 1, we know that for any $x\in{\bf R}^n$ , $\sum\limits_{i=1}^{n}d(x, a_i)$ reaches the minimum value if and only if $x\in{M}$ . For any $a^{*}\notin{M}$ , we are always able to find a point $a'\in\mathcal{\partial M}\subseteq{M}$ such that

    $ \begin{align*} \sum\limits_{i=1}^{n}d(a', a_i) < \sum\limits_{i=1}^{n}d(a^{*}, a_i). \end{align*} $

    Proposition 2. Assume that $x(t)$ is a solution of system (1). Then,

    $ \begin{align} D_+|x_i(t)-x_{\sigma i}^e|\leq \sum_{j=1}^{n}a_{ij}^{(\sigma)}|x_j(t)-x_{\sigma j}^e| \end{align} $

    (3)

    where

    $ \begin{align*} D_+|x_i(t)|=\lim\limits_{h\rightarrow0+}\frac{|x_i(t+h)|-|x_i(t)|}{h} \end{align*} $

    denotes the right derivative of $|x_i(t)|$ , $i\in\underline{n}$ , and $A_\sigma=(a_{ij})^{(\sigma)}$ , $\sigma\in\underline{l}$ .

    Proof. 1) When $x_i(t)=x_{\sigma i}^e$ , i.e., $x_i(t)-x_{\sigma i}^e=0$ , we get

    $ \begin{align*} &D_+|x_i(t)-x_{\sigma i}^e|=|D_+x_i(t)|=\\ &\qquad \displaystyle |\sum\limits_{j=1}^n a_{ij}^{(\sigma) }(x_j(t)-x_{\sigma j}^e)|\leq\\ &\qquad \displaystyle \sum\limits_{j=1, j\neq i}^n a_{ij}^{(\sigma) }|x_j(t)-x_{\sigma j}^e| +\\ &\qquad \displaystyle a_{ii}^{(\sigma)}|(x_i(t)-x_{\sigma i}^e)|=\\ &\qquad \displaystyle\sum\limits_{j=1}^n a_{ij}^{(\sigma)}|x_j(t)-x_{\sigma j}^e|. \end{align*} $

    2) When $x_i(t)\neq x_{\sigma i}^e$ , by the fact that

    $ \begin{align*} a_{ii}^{(\sigma)}\leq0, \qquad a_{ij}^{(\sigma)}\geq0, \quad i\neq j \end{align*} $

    we have

    $ \begin{align*} &D_+|x_i(t)-x_{\sigma i}^e|=\\ &\qquad D_+(x_i(t)-x_{\sigma i}^e)\mathop{\rm sgn} (x_i(t)-x_{\sigma i}^e)=\\ &\qquad \displaystyle\sum\limits_{j=1}^n a_{ij}^{(\sigma)}(x_j(t)-x_{\sigma j}^e)\mathop{\rm sgn} (x_i(t)-x_{\sigma i}^e)=\\ &\qquad \displaystyle\sum\limits_{j=1, j\neq i}^n a_{ij}^{(\sigma)}(x_j(t)-x_{\sigma j}^e)\mathop{\rm sgn} (x_i(t)-x_{\sigma i}^e)+\\ &\qquad a_{ii}^{(\sigma)}(x_i(t)-x_{\sigma i}^e)\mathop{\rm sgn} (x_i(t)-x_{\sigma i}^e)=\\ &\qquad \displaystyle\sum\limits_{j=1, j\neq i}^n a_{ij}^{(\sigma)}(x_j(t)-x_{\sigma j}^e)\mathop{\rm sgn} (x_i(t)-x_{\sigma i}^e)+\\ &\qquad a_{ii}^{(\sigma)}|(x_i(t)-x_{\sigma i}^e)|\leq\\ &\qquad \displaystyle\sum\limits_{j=1}^n a_{ij}^{(\sigma)}|x_j(t)-x_{\sigma j}^e|. \end{align*} $

    For the sake of convenience, we introduce symbols

    $ \begin{align*} \bar{x}\doteq\displaystyle\frac1l\sum\limits_{\sigma\in\underline{l}}x_\sigma^e\succeq 0\end{align*} $

    and

    $ \begin{align} r\doteq\max\limits_{\sigma\in\underline{l}}\{d(\bar{x}, x^e_\sigma)\}\geq 0. \end{align} $

    (4)

    The suprasphere is defined as

    $ \begin{align} {N}=\{x\in {\bf R}^n_+\ |\ d(x, \bar{x})\leq r\}. \end{align} $

    (5)

    In the following, we establish sufficient conditions for the regional stability of system (1) by two different methods.

    Theorem 1. System (1) is regionally stable if there exists a vector $\xi=(\xi_1, \xi_2, \cdots, \xi_n)^{\rm T}\in{\bf R}^n_+$ satisfying

    $ \begin{align} \xi^{\rm T}A_{\sigma}\preceq0, \ \forall\ \sigma\in\underline{l}. \end{align} $

    (6)

    Proof. For any $t>t_0$ , assume that $t\in[t_k, t_{k+1})$ holds for some $k\geq0$ . We establish the candidate common linear Lyapunov function as

    $ \begin{align} V(t)=\displaystyle\sum\limits_{\sigma=1}^{l}\sum\limits_{i=1}^{n}\xi_i|x_i(t)-x_{{\sigma}i}^e|, \ t\geq t_0. \end{align} $

    (7)

    Calculate the right derivative

    $ \begin{align*} D_+V=\lim\limits_{h\rightarrow0+}\frac{V(t+h)-V(t)}{h} \end{align*} $

    along the trajectories of system (1) under an arbitrary switching signal $\sigma(t)$ . It is easy to see from Proposition 2 that

    $ \begin{align*}\begin{array}{rcl} &D_+V\leq\displaystyle\sum\limits_{\sigma=1}^{l}\sum\limits_{i=1}^{n}\xi_i \sum\limits_{j=1}^{n}a_{ij}^{(\sigma)}|x_j(t)-x_{{\sigma}j}^e|=\\ &\qquad \displaystyle\sum\limits_{\sigma=1}^{l}\xi^{\rm T}A_{\sigma}{G}_{\sigma}(t) \end{array} \end{align*} $

    where

    $ \begin{align*} \begin{array}{l} {G}_{\sigma}(t)=\\ \qquad (|x_1(t)-x_{\sigma1}^e|, |x_2(t)-x_{\sigma2}^e|, \cdots, |x_n(t)-x_{\sigma n}^e|)^{\rm T}\succeq0. \end{array} \end{align*} $

    This together with (6) implies that

    $ \begin{align} D_+V\leq0. \end{align} $

    (8)

    In addition, $V(t)$ is a positive function and has the minimum value $V_{\min}$ if $\sum\limits_{\sigma=1}^{l} d(x, x_{\sigma}^e)$ reaches the minimum value. From the monotone bounded theorem, we know that

    $ \begin{align*}\lim\limits_{t\rightarrow+\infty}V(t)=V_{\min}.\end{align*} $

    In other words, $\sum\limits_{\sigma=1}^{l} d(x, x_{\sigma}^e)$ will reach the minimum value as the time $t\rightarrow+\infty$ . By Proposition 1, it yields that $x(t)$ enters ${N}$ after some moment.

    In the following part, we will explain the trajectory $x(t)$ not only enters the suprasphere ${N}$ but also remains in it. Might as well it can be supposed that the moment $T$ is the first time that trajectory $x(t)$ enters ${N}$ , i.e.,

    $ \begin{align*} d(x(T), \bar{x})=r, \qquad T>0. \end{align*} $

    Assume, for the sake of contradiction, that for any $\epsilon>0$ , there exists $t^*=T+\epsilon$ such that $x(t^*)\notin{N}$ , i.e.,

    $ \begin{align*} d(x(t^*), \bar{x})>r, \quad t^*>T. \end{align*} $

    Due to the arbitrariness of $\epsilon$ , the three points $x(t^*)$ , $x(T)$ , $\bar{x}$ are approximated to be on the same line. From the Proposition 1 and Remark 2, for any $x(t^*)\notin {N}$ , one can claim that

    $ \begin{align} \sum\limits_{\sigma=1}^{l}d(x(t^*), x_\sigma^e)>\sum\limits_{\sigma=1}^{l}d(x(T), x_\sigma^e). \end{align} $

    (9)

    On the other hand, together with (8), we have

    $ \begin{align*} V(t^*)\leq V(T), \quad\text{for}\quad t^*>T \end{align*} $

    i.e.,

    $ \begin{align} \sum\limits_{\sigma=1}^{l}d(x(t^*), x_\sigma^e)\leq\sum\limits_{\sigma=1}^{l}d(x(T), x_\sigma^e). \end{align} $

    (10)

    Apparently, (9) and (10) are contradictory. That is to say, the trajectory $x(t)$ will not escape from ${N}$ , once it enters ${N}$ . Hence, for any initial condition $x(t_0)\succeq0$ and any switching laws, system (1) is regionally stable, and ${N}$ is the stability region.

    Remark 4. Based on the method of quadratic Lyapunov-like function, Guo and Wang[17, 18] provided the estimation of stability region as

    $ \begin{align*} \Omega=\{x\in {\bf R}^n:(x-\bar{x})^{\rm T}P(x-\bar{x})\leq C_p\} \end{align*} $

    where $C_p=4\lambda\mu^2$ , $\lambda=\lambda_{\max}(P)$ , $\mu=\max\limits_{\sigma\in\mathcal{L}}\|A_\sigma(\bar{x}-x_\sigma^e)\|$ . By a simple calculation, we have

    $ \begin{align} \|x-\bar{x}\|\leq \displaystyle2\sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}}\max\limits_{\sigma\in\mathcal{L}}\|A_\sigma(\bar{x}-x_\sigma^e)\|. \end{align} $

    (11)

    Comparing (11) with (5), it is obvious to see that

    $ \begin{align*} r\leq \displaystyle2\sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}}\max\limits_{\sigma\in\mathcal{L}}\{\|A_\sigma\|\times\|\bar{x}-x_\sigma^e\|\}. \end{align*} $

    Therefore, our estimation is better than [17, 18].

    Remark 5. From the above analysis, one can see that the suprasphere ${N}$ is closely connected to the Lyapunov function. It is noted that how to find the accurate boundary of stability region for a PSLS-MEP is a very challenging problem.

    When the number of subsystems is large, it is difficult to find a common $\xi$ satisfying $ \xi^{\rm T} A_\sigma\preceq0$ . In view of this kind of situation, we can weaken the conditions in Theorem 1 and obtain the following theorem.

    Theorem 2. Suppose that there exist $l$ constant vectors $\xi^{(\sigma)}=(\xi_1^{(\sigma)}, \xi_2^{(\sigma)}, $ $ \cdots, \xi_n^{(\sigma)})^{\rm T}\in {\bf R}^n_+$ $(\sigma\in\underline{l})$ and constants $\mu>1$ , $\alpha>0$ such that the following conditions hold:

    1) $(\xi^{(i_k)})^{\rm T}A_k+\alpha(\xi^{(i_k)})^{\rm T}\preceq0$ , $k\in\underline{l}$ , $t\in[t_k, t_{k+1})$ , $\sigma=i_k\in\underline{l}$ .

    2) $(\xi^{(i)})^{\rm T}\leq\mu(\xi^{(j)})^{\rm T}$ , $i, j\in\underline{l}$ .

    3) The average dwell time $\displaystyle\tau_a\geq\frac{\ln\mu}{\alpha}$ .

    Then, system (1) is regionally stable under switching signals $\sigma(t)$ , and ${N}$ defined in (4) is an estimation of stability region.

    Proof. For any $t>0$ , assume that $t\in[t_k, t_{k+1})$ for some $k\geq0$ and $\sigma(t)=i_k\in\underline{l}$ . Denote $N=N(t, t_0)$ by the number of switching times on the interval $[t_0, t]$ , and for each subsystem, we establish the multiple Lyapunov function as

    $ \begin{align} V_{\sigma}(t)=V_{i_k}(t)=\sum\limits_{i_k=1}^{l}\sum\limits_{i=1}^{n}\xi_i^{(i_k)}|x_i(t)-x_{{i_k}i}^e| \end{align} $

    (12)

    for $t\geq t_0$ and $\sigma(t)=i_k$ .

    By calculating the right derivative, we get

    $ \begin{align*} D_+V_{i_k}\leq\displaystyle\sum\limits_{i_k=1}^{l}\sum\limits_{i=1}^{n}\xi_i^{(i_k)} \sum\limits_{j=1}^{n}a_{ij}^{(i_k)}|x_j(t)-x_{{i_k}j}^e| \end{align*} $

    and

    $ \begin{align*} &D_+V_{i_k}+\alpha V_{i_k}\leq\\ &\quad \displaystyle\sum\limits_{i_k=1}^{l}\sum\limits_{i=1}^{n}\xi_i^{(i_k)} \sum\limits_{j=1}^{n}a_{ij}^{(i_k)}|x_j(t)-x_{{i_k}j}^e|+\\ &\quad \alpha \displaystyle\sum\limits_{i_k=1}^{l}\sum\limits_{i=1}^{n}\xi_i^{(i_k)}|x_i(t)-x_{{i_k}i}^e| =\\ &\quad \displaystyle \sum\limits_{\sigma=1}^{l}(\xi^{(\sigma)})^{\rm T}A_{\sigma}{G}_{\sigma}(t)+\\ &\quad \displaystyle \alpha\sum\limits_{\sigma=1}^{l}(\xi^{(\sigma)})^{\rm T}{G}_{\sigma}(t)=\\ &\quad \displaystyle \sum\limits_{\sigma=1}^{l}[(\xi^{(\sigma)})^{\rm T}A_{\sigma}+\alpha(\xi^{(\sigma)})^{\rm T}]{G}_{\sigma}(t). \end{align*} $

    According to the condition 1), we derive that

    $ \begin{align*} D_+V_\sigma+\alpha V_\sigma\leq0 \end{align*} $

    for $t\in [t_k, t_{k+1})$ . Hence,

    $ \begin{align*} V_{\sigma}(t)\leq {\rm e}^{-\alpha(t-t_k)}V_{\sigma}(t_k^+). \end{align*} $

    By the conditions 2) and 3), we have

    $ \begin{align*} &V_{i_N}(t)\leq {\rm e}^{-\alpha(t-t_N)}V_{i_N}(t_N^+)\leq\\ &\qquad \displaystyle \mu {\rm e}^{-\alpha(t-t_{N})} V_{i_{N-1}}(t_{N}^-)\leq\cdots\leq\\ &\qquad \displaystyle \mu^N {\rm e}^{-\alpha(t-t_0)}V_{i_0}(t_0). \end{align*} $

    Note that the switching times and the average dwell time satisfy

    $ \begin{align*} \displaystyle N\leq N_0+\frac{t-t_0}{\tau_a} \end{align*} $

    which can be reduced to

    $ \begin{align*} \displaystyle V_{i_N}(t)\leq \mu^{N_0}{\rm e}^{-(\alpha-\frac{\ln\mu}{\tau_a})(t-t_0)}V_{i_0}(t_0). \end{align*} $

    For a small positive real number $\epsilon$ , the Lyapunov function $V(t)\rightarrow \epsilon$ as the time $t\rightarrow+\infty$ . That is to say, $\sum\limits_{\sigma=1}^{l}d(x, x_{\sigma}^e)$ will reach the minimum value as the time $t\rightarrow+\infty$ . By the similar analysis method as Theorem 1, it yields that $x(t)$ enters and remains in ${N}$ .

    Remark 6. It is noted that the common linear Lyapunov function method and the multiple Lyapunov function method have their own advantages, and cannot contain each other. On one hand, the condition $(\xi^{(i_k)})^{\rm T}A_k+\alpha(\xi^{(i_k)})^{\rm T}\preceq0$ in Theorem 2 is weaker than the condition in Theorem 1. On the other hand, Theorem 1 is applicable to arbitrary switching signals, while Theorem 2 is based on the average dwell time switching signals.

  • In this section, we apply the obtained results to the analysis of data communication networks.

    Generally speaking, a data communication network consists of four parts: control center, communication terminal, communication links and data-interchange nodes. Starting from the control center, different data packets are sent to the communication terminal through the corresponding communication links and data-interchange nodes. If the amount of data transmitted over the network is too large, then the network might occur, the phenomenon of congestion and the data packets might get lost in the process of transmission. On the contrary, if the amount of data transmitted over the network is too small, it is a huge waste of network resources. In order to achieve a win-win situation, every subnetwork should have a relatively appropriate amount of data in practice.

    According to [14-16], the data communication network showed in Fig. 1 can be described by the following PSLS-MEP

    Figure 1.  Data communication networks with three nodes

    $ \begin{align}\left\{ \begin{array}{ll} \dot{x}(t)=A_{\sigma}\Big(x(t)-x_{\sigma}^e\Big)\\ x(t_0)=x_0 \end{array} \right.\end{align} $

    (13)

    where $x(t)\in {\bf R}^3_+$ denotes the amount of data transmitted over the network, $x_\sigma^e$ is the known appropriate amount of data, $\sigma=\{1, 2\}$ , and

    $ \begin{align*} A_1=\left( \begin{array}{ccc} -0.6 & 0.02 & 0.03\\ 0.03 & -0.4 & 0.04\\ 0.02 & 0.03 & -0.5\\ \end{array} \right)\\ A_2=\left( \begin{array}{ccc} -0.3 & 0.04 & 0.04\\ 0.02 & -0.5 & 0.03\\ 0.05 & 0.06 & -0.4\\ \end{array} \right). \end{align*} $

    Note that $\sigma=1$ denotes the idle-time network, while $\sigma=2$ denotes the busy-time network. Without loss of generality, we suppose that

    1) The maximum amount of data that the idle-time (busy-time) network can bear is 2Gb (4Gb). That is to say, $x_1(t)+x_2(t)+x_3(t)\leq2$ ( $x_1(t)+x_2(t)+x_3(t)\leq4$ ).

    2) The appropriate amount of data for idle-time model is $x_1^e=(0.6, 0.6, 0.7)^{\rm T}$ , and for busy-time model is $x_2^e=(1.4, 1.2, 1.3)^{\rm T}$ .

    3) Take the initial moment $t_0=0$ , and the initial amount of data that needs to be sent across the network as $x(t_0)=(0.45, 0.9, 1.5)^{\rm T}$ .

    It is easy to verify that $A_\sigma x_\sigma^e\preceq0$ ( $\sigma\in \{1, 2\}$ ) and both subsystems are asymptotically stable (see Fig. 2).

    Figure 2.  Stability of each subsystem

    At the same time, by a direct calculation, we obtain

    $ \begin{align*} &\bar{x}=(1, 0.9, 1)^{\rm T}\\ &d(\bar{x}, x_1^e)=d(\bar{x}, x_2^e)= 0.5831 \end{align*} $

    and the stable region

    $ \begin{align*} {N}=\{x\in{\bf R}^3_+|d(x, \bar{x})\leq 0.5831\} \end{align*} $

    is a sphere in ${\bf R}^3_+$ (see Fig. 3). Choose $\xi=(1, 1, 1)^{\rm T}$ such that the condition in Theorem 1, i.e., $\xi^{\rm T}A_{\sigma}\preceq0$ holds. In our simulation, $\sigma$ is a periodic switching signal and the dwell time is randomly generated.

    Figure 3.  Stability region and trajectory of the system (13)

    Fig. 3 tells us that for a given initial amount of data $x_0\succeq 0$ , the trajectory of system (13) enters and remains in ${N}$ at last. Our simulation result is consistent with the theoretical result.

    In order to better illustrate the effectiveness of the proposed method, we list the stability regions given in [17-19], respectively.

    1) By a simple calculation, we have $\|A_1\|=0.6073$ , $\|A_2\|=0.5184$ . Choose $P=I_3$ . Together with (11), we get the stability region (provided in [17, 18]) $\Omega=\{x:\|x(t)-\bar{x}\|\leq0.7082\}$ . Obviously, ${N}$ is smaller than $\Omega$ ;

    2) From reference [19], $N^{(i)}(k)=\{x\in{\bf R}^3_+:\|x-x_i^e\|\leq k\}, $ $N(k)=\bigcup\limits_{i=1, 2}N^{(i)}(k), $ $\alpha^{(i)}(k)=\max\limits_{x_1, x_2\in N(k)}\|x_1-x_2\|, $ $M^{(i)}(k)=\{x\in{\bf R}^3_+:\|x-x_i^e\|\leq\alpha^{(i)}(k)\}, $ and the stability region is

    $ \begin{align*} L(k)=\bigcup\limits_{i=1, 2}M^{(i)}(k). \end{align*} $

    Choose $k=0.02$ .

    In order to make a more intuitive comparison between ${N}$ and $L(k)$ , we provide the figures of stability regions in different coordinate planes, please see Figs. 4-6, where the region bounded by solid line is ${N}$ , and the region bounded by dashed line is $L(k)$ . Clearly, our result is less conservative than [19].

    Figure 4.  Stability regions in $ x_1-x_2$ plan

    Figure 5.  Stability regions in $x_1-x_3 $ plane

    Figure 6.  Stability regions in $ x_2-x_3$ plane

  • In this paper, we have studied the regional stability of positive switched linear systems with multiple equilibrium points. The innovation of our work is that both common linear Lyapunov function and multiple Lyapunov functions are constructed to solve the regional stability of PSLS-MEP. Moreover, an effective estimation of stable region has been given by using our method. A numerical example has been given to illustrate our results.

    Compared with [17-19], the stability region given in our work is much less conservative. Future works will study how to obtain the accurate boundary of stability region for PSLS-MEP.

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