Stabilization of Asynchronous Switched Systems with Constrained Control

Abstract

This paper investigates the method on stabilizing asynchronous switched linear systems with constrained inputs. Firstly, asynchronous systems mean the asynchronization between the system modes and state feedback controllers. Usually, it takes a period of time to identify which one of the state feedback controllers should be activated in practical application. Next, in consideration of the saturation effect of the controllers, this paper is aimed at stabilizing the systems with constrained inputs by mode-dependent average dwell time method. Besides, unstable subsystems are considered in this paper.

1 Introduction

The past decades have witnessed the fast growing interest in switched systems, which consist of many subsystems and a switching law (Zhao et al. 2015). Switched systems are used in practical applications widely, such as electrical networks (Cong 2016), sampled-data systems (Hetel et al. 2011) and sliding mode control systems (Ullah et al. 2016). Meanwhile, many theories of switched systems have been well established, such as reachability (Chao and Yu 2015), adaptive controller design (Niu et al. 2017) and output tracking control (Niu and Zhao 2013).

Till now, the stability analysis of switched systems is still a hot field. The mode-dependent average dwell time (MDADT) (Zhao et al. 2012) based on multi-Lyapunov functions has been verified to be a very valid and flexible method. The results get by MDADT are much less conservative than those get by average dwell time. While using MDADT to analyze the stability of switched systems, every subsystem must be Hurwitz stable. Otherwise, a state feedback controller should be designed for the unstable subsystem. Fiacchini and Jungers (2014) and Minh et al. (2011) have studied the method on stabilizing the unstable subsystems. Besides, their researches are based on the assumption that the switching of system modes and state feedback controllers is simultaneous. However, in practical application, it usually takes a period of time to identify which subsystem is activated. And then the certain state feedback controller can be chosen (Liu et al. 2016). This causes the inaccuracy while applying the established theories to practice. Next, when a state feedback controller is designed, the control input is supposed to be set arbitrarily or infinitely. However, the controllers with saturation effect in practice limit this assumption. Over the past two decays, the stabilization study of control system (not switched system) was very abundant. Many excellent theories were established, such as Hu and Lin (2001) and Hu et al. (2002). In the book Hu and Lin (2001), the authors had introduced the solutions to stabilization of control systems with constrained input in detail. After several years, these theories were generalized to switched systems. In Chen et al. (2012), the authors studied stability condition of switched systems based on minimum dwell time method. In Remark 1 of Chen et al. (2012), the authors presented that “we cannot employ average dwell time approach”. In fact, it is incorrect. We can use not only average dwell time approach but also MDADT approach by choosing appropriate initial states. Via this way, we can design a switching signal more flexibly.

Although there are some researches toward stabilization of switched systems with constrained input, to the best of the authors’ knowledge, no one considers asynchronous switching and constrained input jointly. For instance, authors in Ding et al. (2015) and Chen et al. (2012) studied the stability of switched systems with constrained input, but they all failed to analyze the asynchronous switching. Next, in Benzaouia et al. (2010), the arbitrary switching condition was studied. However, the results were restrictive to some extent because of the arbitrary switching law. Considering these, we aim at stabilizing the asynchronous switched linear systems with constrained inputs. It should be pointed out that all the subsystems may be unstable. Based on multi-Lyapunov functions, the mode-dependent average dwell time is obtained to guarantee that the system is exponentially stable. Furthermore, a Euclidean ball is found to limit the system states within it. By this method, the system can be stabilized with the constrained input. Moreover, most of the researchers assumed that each subsystem could be stabilized while analyzing the stability of switched systems with constrained input, such as Ding et al. (2015) and Wang and Zhao (2015). In fact, it is not impeccable. Because if certain subsystem state diverges quickly, the subsystem may not be stabilized by controllers with constrained input. In view of this, we present another two theorems (Theorems 2, 3). In the two theorems, the asynchronous switched systems with unstable subsystems can be exponentially stable with constrained control. Finally, while analyzing asynchronous switched systems, most of the researchers suppose that the delay time of state feedback controllers is a constant and equal. In this paper, the delay time of state feedback controllers can be different in different subsystems.

Notations Throughout this paper, the symbols used are quite standard. ${\mathbb {R}}^{n}$ and ${\mathbb {R}}^{n\times n}$ represent the n-dimensional Euclidean space and the space of $n\times n$ matrices with real entries, respectively. $\left\| \cdot \right\|$ denotes the Euclidean vector norm. For a matrix $P_i$, $\lambda _i^{\max } (P_i )$ and $\lambda _i^{\min } (P_i )$ are the maximal and minimal eigenvalue of $P_i$, respectively. $\lambda _{\min }$ is the minimum of $\lambda _i^{\min }$. For two vectors $x,y\in { \mathbb {R}}^{n}$, $x\;\underline{\prec }\;y$ denotes $x_i \le y_i$, $i=1,2,\ldots ,n$. $R(\cdot )$ denotes the range of a matrix. $\Delta t^{s}$ is the total running time of stable subsystems while $\Delta t^{u}$ denotes the total running time of unstable subsystems. Next, $\Delta t_i$ denotes the total running time of the ith subsystem. In this paper, max means the maximum. For example, $T_{\max }$, $\alpha _{\max }$, $\mu _{\max }$ and $\beta _{\max }$ denote the maximum of $T_i$, $\alpha _i$, $\mu _i$ and $\beta _i$, respectively.

2 Preliminary

Consider the switched linear system

$$\begin{aligned} \dot{x}=A_{\sigma (t)} x+B_{\sigma (t)} u_{\sigma (t)} \end{aligned}$$

(1)

where $x\in {\mathbb {R}}^{n}$ is the system state, $\sigma (t):\;[0,\infty )\rightarrow {\mathbb {Z}}=\{1,2,\ldots ,N\}$ is the switching law, N is the number of subsystems. For a switching sequence $t_0<t_1<\cdots<t_k <\cdots$, $\sigma (t)$ is continuous from right everywhere. Throughout this paper, $\sigma (t_k )=i$, $\sigma (t_k^- )=\sigma (t_{k-1} )=j$, $i\ne j$ and $\sigma (t)\in {\mathbb {Z}}$. So when $t\in [t_k ,t_{k+1} )$, we say the ith subsystem is activated. $A_i \in {\mathbb {R}}^{n\times n}$, $B_i \in {\mathbb {R}}^{n\times m_k }$.

Definition 1

(Liberzon 2003) The equilibrium $x=0$ of system (1) is globally uniformly exponentially stable (GUES) under certain switching signal if for initial conditions $x(t_0 )$, there exist constants $\eta _1 >0$, $\eta _2 >0$ such that the solution of the system satisfies

$$\begin{aligned} \left\| {x(t)} \right\| \le \eta _1 e^{-\eta _2 (t-t_0 )}\left\| {x(t_0 )} \right\| , \quad \forall t\ge t_0 . \end{aligned}$$

Definition 2

(Zhao et al. 2012) For a switching signal $\sigma$ and any $t_2>t_1 >t_0$, let $N_{\sigma i} (t_1 ,t_2 )$ be the switching numbers of the ith subsystem over the interval $[t_1 ,t_2 )$. If $N_{\sigma i} (t_1 ,t_2 )\le N_{0i} +(t_2 -t_1 )/\tau _{ai}$ holds, then $\tau _{ai}$ is mode-dependent average dwell time and $N_{0i}$ is mode-dependent chatter bound.

Definition 3

(Xie et al. 2013) For a switching signal $\sigma$ and any $t_2>t_1 >t_0$, let $N_\sigma ^u (t_1 ,t_2 )$ be the switching numbers of unstable subsystems over the interval $[t_1 ,t_2 )$. If $N_\sigma ^u (t_1 ,t_2 )\ge N_0^u +\Delta t^{u}/\tau _a^u$ holds, then $\tau _a^u$ is average dwell time of unstable subsystems and $N_0^u$ is called chatter bound.

Remark 1

To introduce the three definitions and the following lemma, the asynchronization between system modes and state feedback controllers is not considered. Besides, Definition 3 is used to analyze the stability of fast switched systems, which means the average dwell time cannot exceed an upper bound value.

Next, a method to stabilize linear systems (not switched linear system) is introduced.

Consider a linear system

$$\begin{aligned} \dot{x}=Ax+Bu \end{aligned}$$

(2)

where $A\in {\mathbb {R}}^{n\times n}$   ,   $B\in {\mathbb {R}}^{n\times m}$   ,    $rank(B)=m\le n$, $u\in U\subset {\mathbb {R}}^{m}$, U is the admissible control set which is defined by $U=\{u\in { \mathbb {R}}^{m}\left| \right. -u_{\min } \underline{\prec }\;u\;\underline{\prec }\;u_{\max } \}$   , besides, $u_{\min } ,u_{\max } \in {\mathbb {R}}^{m}$ are two vectors only with positive components. Set $K\in {\mathbb {R}}^{m\times n}$ such that $A-BK$ is Hurwitz matrix. Then the closed-loop system becomes

$$\begin{aligned} \dot{x}=(A-BK)x \end{aligned}$$

(3)

Define

$$\begin{aligned} D=\{x\in {\mathbb {R}}^{n}\left| \right. -u_{\min } \underline{\prec }\;Kx\;\underline{\prec }\;u_{\max } \} \end{aligned}$$

(4)

Lemma 1

(Ni and Cheng 2012) For the system matrix A, using Schur unitary triangularization Theorem (Bhatia 1991), there exists an orthogonal matrix $Q\in {\mathbb {R}}^{n\times n}$ such that

$$\begin{aligned} Q^{T}AQ=\left( {{\begin{array}{ll} {A_0}&{} \quad {A_2 } \\ 0&{} \quad {A_1 } \\ \end{array} }} \right) \end{aligned}$$

(5)

where $A_0$ has all negative real part eigenvalues and $A_1$ has all nonnegative real part eigenvalues.

Consider the linear system (2). Suppose A has r nonnegative real part eigenvalues. Q is an orthogonal matrix satisfying (5), besides,

$$\begin{aligned} Q^{T}B=\left( {{\begin{array}{ll} {(B_0 )_{(n-r)\times r} }&{} {*} \\ {(B_1 )_{r\times r} }&{} {*} \\ \end{array} }} \right) \end{aligned}$$

where ${*}$ is the element we do not concern. Let $H=(h_{ij} )_{n\times n}$ be a Hurwitz matrix such that

$$\begin{aligned} \left( {{\begin{array}{ll} {H^{+}}&{} \quad {H^{-}} \\ {H^{-}}&{}\quad {H^{+}} \\ \end{array} }} \right) ({\begin{array}{ll} {u_{\max }^T }&{} {u_{\min }^T } \\ \end{array} })^{T}\underline{\prec }\;0 \end{aligned}$$

where

$$\begin{aligned} H^{+}(i,j)= & {} \left\{ {{\begin{array}{l} {{\begin{array}{ll} {h_{ij} }&{} {\;\qquad \qquad \; \quad \mathrm{if}\;i=j} \\ \end{array} }} \\ {{\begin{array}{ll} {\max (h_{ij} ,0)}&{} \quad {\mathrm{if}\;i\ne j} \\ \end{array} }} \\ \end{array} }} \right. \\ H^{-}(i,j)= & {} \left\{ {{\begin{array}{ll} {{\begin{array}{ll} 0&{} {\;\quad \qquad \qquad \;\qquad \mathrm{if}\;i=j} \\ \end{array} }} \\ {{\begin{array}{ll} {\max (-h_{ij} ,0)}&{} \quad {\mathrm{if}\;i\ne j} \\ \end{array} }} \\ \end{array} }} \right. \end{aligned}$$

Then there exists a unique solution to the equation

$$\begin{aligned} A_1 X-XH=-B_1 \end{aligned}$$

(6)

Moreover, if $R(B_1 )\subset R(X)$, system (2) with

$$\begin{aligned} K=-\left( {{\begin{array}{ll} 0&{} \quad {X^{-1}} \\ 0&{}\quad 0 \\ \end{array} }} \right) Q^{T} \end{aligned}$$

is asymptotically stable for all $x_0 \in D$. The control input $u=Kx$ is admissible and domain D in (4) is positively invariant.

Remark 2

Lemma 1 is used to stabilize linear systems with constrained inputs. We can stabilize all the subsystems according to Lemma 1, firstly, switched system may not be stable with all subsystem stable (the mode-dependent average dwell time must exceed a constant), secondly, how to choose an appropriate initial state? Thirdly, what if the switched system is with asynchronous switching? Finally, what if some of the subsystem cannot be stabilized? So this paper is aimed at solving the four problems.

In what follows, the method of stabilizing switched linear systems with constrained inputs will be presented.

3 Main Results

Consider system (1). $u_i =[u_i^1 ,u_i^2 ,\ldots ,u_i^{m_i } ]^{T}\in {\mathbb {R}}^{m_i }$ is the control inputs, besides,

$$\begin{aligned} -u_i^{\min } \underline{\prec }\;u_i \underline{\prec }\;u_i^{\max } , \quad i\in {\mathbb {Z}} \end{aligned}$$

(7)

where $u_i^{\min }$ and $u_i^{\max }$ are only with positive elements. Suppose that system (1) is controllable. According to Lemma 1, feedback matrices $K_i$ and admissible region $D_i =\{x\in {\mathbb {R}}^{n}\left| \right. -u_i^{\min } \underline{\prec }\;K_i x\;\underline{\prec }\;u_i^{\max } \}$ can be obtained. Assume that the switched controllers lag behind system modes for $T_i$, besides, $T_i <(t_{k+1} -t_k )$ . Then system (1) becomes

$$\begin{aligned} \dot{x}=(A_{\sigma (t)} -B_{\sigma (t)} K_{\sigma (t-T_i )} )x \end{aligned}$$

(8)

Because of asynchronous switching, time interval $[t_k ,t_{k+1} )$ is divided into matched period (denoted by $T_i^+ )$ and mismatched period (denoted by $T_i )$. It should be pointed out that during all the mismatched periods, the system is supposed to be divergent. Let $\Omega$ be the biggest Euclidean ball which is centered at the origin and inside the intersection of $D_i$,   $i\in {\mathbb {Z}}$. For any $z\in \Omega$,

$$\begin{aligned} d=\max (\left\| z \right\| ). \end{aligned}$$

In what follows, the main results of this paper will be presented.

Theorem 1

Consider system (8), let $K_i$ be obtained in Lemma 1, if there exist constants $\alpha _i >0$, $\beta _i >0$, $\mu _i \ge 1$ and two class $\kappa { }_\infty$ functions $\kappa { }_1$, $\kappa { }_2$ such that (9), (10), (11), (12) and (13) hold, then system (8) is exponentially stable by admissible control input $u_i =K_i x$, $\forall i\in { \mathbb {Z}}$.

$$\begin{aligned} \left\| {x(t_0 )} \right\| ^{2}\le & {} d^{2}/\psi _1 \end{aligned}$$

(9)

$$\begin{aligned} \kappa _1(\left\| {x(t)} \right\| )\le & {} V_i (x(t))\le \kappa { }_2(\left\| {x(t)} \right\| ) \end{aligned}$$

(10)

$$\begin{aligned} \dot{V}_i \left( {x(t)} \right)\le & {} \left\{ {{\begin{array}{l} {-\alpha _i V_i \left( {x(t)} \right) \quad t\in [t_k +T_i ,t_{k+1} )} \\ {\beta _i V_i \left( {x(t)} \right) \quad t\in [t_k ,t_k +T_i )} \\ \end{array} }} \right. \end{aligned}$$

(11)

$$\begin{aligned} V_i \left( {x(t)} \right)\le & {} \mu _i V_j \left( {x(t)} \right) \end{aligned}$$

(12)

$$\begin{aligned} \tau _{ai}> & {} \frac{T_i (\alpha _i +\beta _i )+\ln \mu _i }{\alpha _i } \end{aligned}$$

(13)

where

$$\begin{aligned} \psi _1= & {} \frac{\lambda _{\sigma (t_0 )}^{\max } }{\lambda _{\min } }\cdot \exp \{\beta _{\max } T_{\max } \} \\&\quad \cdot \prod _{i=1}^N {\mu _i^{N_{0i} } } \cdot \exp \left\{ {T_i (\alpha _i +\beta _i )N_{0i} } \right\} . \end{aligned}$$

Proof

Denote $g_i (t)=e^{\alpha _i t}V_i (t)$. Then

$$\begin{aligned} \frac{\mathrm{d}g_i (t)}{\mathrm{d}t}=e^{\alpha _i t}\left[ {\dot{V}_i (t)+\alpha _i V_i (t)} \right] \end{aligned}$$

(14)

From the first inequality in (11), it can be obtained that $g_i (t)$ is decreasing when $t\in [t_k +T_i , t_{k+1} )$. Thus,

$$\begin{aligned} V_{\sigma (t_k )} \left( {x(t_{k+1} )} \right) \le e^{-\alpha _i (t_{k+1} -t_k -T_i )}V_{\sigma (t_k )} \left( {x(t_k +T_i )} \right) \end{aligned}$$

(15)

From the second inequality in (11), when $t\in [t_k ,t_k +T_i )$, by using the same method, the following can be got

$$\begin{aligned} V_{\sigma (t_k )} \left( {x(t_k +T_i )} \right) \le e^{\beta _i T_i }V_{\sigma (t_k )} \left( {x(t_k )} \right) \end{aligned}$$

(16)

Combine (15) and (16), it follows that

$$\begin{aligned} V_{\sigma (t_k )} \left( {x(t_{k+1} )} \right) \le e^{-\alpha _i T_i^+ +\beta T_i }V_{\sigma (t_k )} \left( {x(t_k )} \right) \end{aligned}$$

(17)

Multiply both sides of (17) by $\mu _{\sigma (t_{k+1} )}$ and then apply (12) to it, we can get

$$\begin{aligned} V_{\sigma (t_{k+1} )} \left( {x(t_{k+1} )} \right) \le e^{-\alpha _i T_i^+ +\beta _i T_i }\mu _{\sigma (t_{k+1} )} V_{\sigma (t_k )} \left( {x(t_k )} \right) \end{aligned}$$

(18)

It follows from (18) that,

$$\begin{aligned}&V_{\sigma (t_{k+1} )} \left( {x(t_{k+1} )} \right) \le e^{-\alpha _i (T_i^+ +T_i )} \frac{e^{\beta _i T_i^- }}{e^{-\alpha _i T_i^- }}\mu _{\sigma (t_{k+1} )} V_{\sigma (t_k )} \left( {x(t_k )} \right) \nonumber \\&\quad =e^{-\alpha _i (t_{k+1} -t_k )}e^{T_i (\alpha _i +\beta _i )}\mu _{\sigma (t_{k+1} )} V_{\sigma (t_k )} \left( {x(t_k )} \right) \nonumber \\&\quad \le e^{-\alpha _i (t_{k+1} -t_k )}e^{T_i (\alpha _i +\beta _i )}\mu _{\sigma (t_{k+1} )} \mu _{\sigma (t_k )} V_{\sigma (t_{k-1} )} \left( {x(t_k )} \right) \nonumber \\&\quad \vdots \nonumber \\&\quad \le \prod _{i=1}^N {\mu _i^{N_{\sigma i} } e^{-\alpha _i \Delta t_i }e^{T_i (\alpha _i +\beta _i )N_{\sigma _i } }} V_{\sigma (t_0 )} \left( {x(t_0 )} \right) \end{aligned}$$

(19)

$$\begin{aligned}&\quad <\exp \{\beta _{\max } T_{\max } \} \nonumber \\&\quad \prod _{i=1}^N {\mu _i^{N_{\sigma i} } e^{-\alpha _i \Delta t_i }e^{T_i (\alpha _i +\beta _i )N_{\sigma i} }} V_{\sigma (t_0 )} \left( {x(t_0 )} \right) \end{aligned}$$

(20)

Then Lyapunov functions are set as

$$\begin{aligned} V_i \left( {x(t)} \right) =x^{T}P_i x \end{aligned}$$

(21)

According to the Rayleigh theorem,

$$\begin{aligned} \lambda _{\min } (P)\le \frac{x^{T}Px}{x^{T}x}\le \lambda _{\max } (P) \end{aligned}$$

(22)

the following can be got from (19) and (22)

$$\begin{aligned} \left\| {x(t_{k+1} )} \right\| ^{2}\le & {} \frac{\lambda _{\sigma (t_0 )}^{\max } }{\lambda _{\sigma (t_{k+1} )}^{\min } }\cdot \exp \{\beta _{\max } T_{\max } \} \nonumber \\&\cdot \prod \limits _{i=1}^N {\mu _i^{N_{\sigma i} } e^{-\alpha _i \Delta t_i }e^{T_i (\alpha _i +\beta _i )N_{\sigma _i } }} \left\| {x(t_0 )} \right\| ^{2} \end{aligned}$$

(23)

Because $N_{\sigma i} (t_1 ,t_2 )\le N_{0i} +\frac{\Delta t_i }{\tau _{ai} }$ in Definitions 2 and (13) hold, (23) can be simplified to

$$\begin{aligned}&\left\| {x(t_{k+1} )} \right\| ^{2}\le \psi _1\nonumber \\&\quad \exp \left\{ {\sum _{i=1}^N {\Delta t_i \cdot \left[ {\frac{\ln \mu _i }{\tau _{ai} }} \right. } } \right. -\alpha _i \left. {\left. {+\frac{T_i (\alpha _i +\beta _i )}{\tau _{ai} }} \right] } \right\} \left\| {x(t_0 )} \right\| ^{2}\nonumber \\ \end{aligned}$$

(24)

Denote $\xi _1 =\mathop {\max }\limits _{i\in {\mathbb {Z}}} \left( {\frac{\ln \mu _i }{\tau _{ai} }-\alpha _i +\frac{T_i (\alpha _i +\beta _i )}{\tau _{ai} }} \right)$, then (24) becomes

$$\begin{aligned} \left\| {x(t_{k+1} )} \right\| ^{2}\le \psi _1 \exp \{\xi _1 (t-t_0 )\}\left\| {x(t_0 )} \right\| ^{2} \end{aligned}$$

(25)

From Definition 1, it can be obtained that system (8) is exponentially stable under condition (13). Next, since (9) holds and $\Omega$ is an Euclidean ball, besides, $D_i$, $i\in {\mathbb {Z}}$, are positively invariant, then for any $\sigma (t_k )\in {\mathbb {Z}}$, state response $x(t)\in \Omega$. Thus, the control input $u_i =K_i x$ is admissible. In conclusion, the system is exponentially stable with constrained control. $\square$

Remark 3

To solve $\tau _{ai}$, the Lyapunov function can be chosen as $V_i \left( {x(t)} \right) =x^{T}P_i x$, then, (11) and (12) become (26) and (27), respectively.

$$\begin{aligned}&\left\{ {{ \begin{array}{l} {(A_i -B_i K_i )^{T}P_i +P_i (A_i -B_i K_i )\le -\alpha _i P_i ,\;t\in [t_k +T_i ,t_{k+1} )} \\ {(A_i -B_i K_j )^{T}P_i +P_i (A_i -B_i K_j )\le \beta _i P_i ,\;t\in [t_k ,t_k +T_i )}\\ \end{array} }} \right. \nonumber \\ \end{aligned}$$

(26)

$$\begin{aligned}&\quad P_i \le \mu _i P_j \end{aligned}$$

(27)

By using LMI toolbox in Matlab, $\alpha _i$, $\beta _i$ and $\mu _i$ can be got. Then $\tau _{{a}_{i}}$ can be solved.

Remark 4

As has been supposed that the system is divergent during all the mismatched periods (the divergence speed can be described by $\beta _i )$ and the delay time ($T_i )$ of state feedback controllers can be different and variable. With $\beta _i$ and $T_i$ getting larger, $\psi _1$ is getting larger simultaneously. This means the feasible initial states that can be chosen are shrunken.

As has been assumed that all the subsystems can be stabilized by Lemma 1, but what if some of the subsystems cannot be stabilized? Besides, in practice, it is usually difficult to design some state feedback controllers. Considering this, we suppose that only the subsystems $i\in [1,M]$ can be stabilized, $\Omega \subset \bigcap \nolimits _{i=1}^M {D_i }$is the biggest Euclidean ball centered at the origin and the rest of subsystems $(i\in [M+1,N])$ are without state feedback controllers. For any $z\in \Omega$, $d=\max (\left\| z \right\| )$. Besides, $V_i^s \left( {x(t)} \right)$ and $V_i^u \left( {x(t)} \right)$ are Lyapunov functions for stable subsystems and unstable subsystems, respectively. $\tau _{ai}^s$ and $\tau _{ai}^u$ are MDADT of stable subsystems and unstable subsystems, respectively. Moreover,

$$\begin{aligned} \Delta _{\max } =\mathop {\max }\limits _{M+1\le \sigma (t_k )\le N} (t_{k+1} -t_k ). \end{aligned}$$

Now we will give another two theorems where there are unstable subsystems without state feedback controllers.

Theorem 2

Consider system (8), let $K_i$, $i\in [1,M]$ be obtained in Lemma 1 and $u_i =0$, $\forall i\in [M+1,N]$. If there exist constants $\alpha _i >0$, $\alpha ^{u}>0$, $\beta _i >0$, $\beta ^{u}>0$, $\mu _i \ge 1$, $0<\mu ^{u}<1$ and two class $\kappa { }_\infty$ functions $\kappa { }_1$, $\kappa { }_2$ such that (28), (29), (30), (31) and (32) hold, then system (8) is exponentially stable by admissible control input $u_i =K_i x$, $i\in [1,M]$.

$$\begin{aligned}&\left\| {x(t_0 )} \right\| ^{2}\le d^{2}/\psi _2 \end{aligned}$$

(28)

$$\begin{aligned}&\quad \left\{ \begin{array}{l} \kappa _1(\left\| {x(t)} \right\| )\le V_i^s (x(t))\le \kappa { }_2(\left\| {x(t)} \right\| ) \\ \kappa _1(\left\| {x(t)} \right\| )\le V_i^u (x(t))\le \kappa { }_2(\left\| {x(t)} \right\| ) \\ \end{array} \right. \end{aligned}$$

(29)

$$\begin{aligned}&\quad \left\{ \begin{array}{l} \dot{V}_i^s \left( {x(t)} \right) \le \left\{ \begin{array}{l} -\alpha _i V_i^s \left( {x(t)} \right) ,\quad t\in [t_k +T_i ,t_{k+1}) \\ \beta _i V_i^s \left( {x(t)} \right) ,\quad t\in [t_k ,t_k +T_i) \\ \end{array} \right. \\ \dot{V}_i^u \left( x(t) \right) \le \left\{ \begin{array}{l} \alpha ^{u}V_i^u \left( x(t) \right) ,\quad t\in [t_k +T_i ,t_{k+1}) \\ \beta ^{u}V_i^u \left( x(t) \right) ,\quad t\in [t_k ,t_k +T_i) \\ \end{array} \right. \end{array} \right. \end{aligned}$$

(30)

$$\begin{aligned}&\quad \left\{ \begin{array}{l} V_i^s \left( x(t) \right) \le \mu _i V_j^s \left( {x(t)} \right) \\ V_i^s \left( x(t) \right) \le \mu _i V_j^u \left( {x(t)} \right) \\ V_i^u \left( x(t) \right) \le \mu ^{u}V_j^s \left( {x(t)} \right) \\ \end{array} \right. \end{aligned}$$

(31)

$$\begin{aligned}&\quad \left\{ \begin{array}{l} \tau _{ai}^s >\frac{T_i (\alpha _i +\beta _i )+\ln \mu _i }{\alpha _i } \\ \tau _a^u <\frac{-\ln \mu ^{u}}{\gamma } \\ \end{array} \right. \end{aligned}$$

(32)

where $\psi _2 =\frac{\lambda _{\sigma (t_0 )}^{\max } }{\lambda _{\min } }\cdot \mu _{\max } \exp \{\gamma \Delta _{\max } +\beta _{\max } T_{\max } \}(\mu ^{u})^{N_0^u }\cdot \prod _{i=1}^M {\mu _i^{N_{0i} } } \cdot \exp \{T_i (\alpha _i +\beta _i )N_{0i} \}$ is a constant, $\tau _a^u$ is the average dwell time (except MDADT) of unstable subsystems and $\gamma =\max (\alpha ^{u},\beta ^{u})$.

Proof

The proof of Theorem 2 is similar to that of Theorem 1. So some part of the proof which has been appeared in Theorem 1 is omitted. It follows from (18) that

$$\begin{aligned}&V_{\sigma (t_{k+1} )} \left( {x(t_{k+1} )} \right) \le e^{-\alpha _i T_i^+ +\beta _i T_i }\mu _{\sigma (t_{k+1} )} V_{\sigma (t_k )} \left( {x(t_k )} \right) \nonumber \\&\vdots \nonumber \\&\le e^{\gamma \Delta t^{u}}\mu _u^{N_{\sigma }^{u}}\cdot \prod _{i=1}^M e^{-\alpha _i \Delta t_i}e^{T_i (\alpha _i +\beta _i )N_{{\sigma }_{i}}}\mu _{i}^{N_{\sigma i}} V_{\sigma (t_0 )} \left( {x(t_0 )} \right) \nonumber \\ \end{aligned}$$

(33)

The Lyapunov functions are set as $V_i \left( {x(t)} \right) =x^{T}P_i x$ then the following can be got.

$$\begin{aligned}&\left\| {x(t_{k+1} )} \right\| ^{2}\le \frac{\lambda _{\sigma (t_0 )}^{\max } }{\lambda _{\min } }e^{\gamma \Delta t^{u}}(\mu ^{u})^{N_\sigma ^u } \nonumber \\&\qquad \cdot \prod _{i=1}^M {e^{-\alpha _i \Delta t_i }e^{T_i (\alpha _i +\beta _i )N_{{\sigma } _{i}} }\mu _i ^{N_{{\sigma } _{i}} }} \left\| {x(t_0 )} \right\| ^{2}\end{aligned}$$

(34)

$$\begin{aligned}&<\frac{\lambda _{\sigma (t_0 )}^{\max } }{\lambda _{\min } }\mu _{\max } \exp \{\gamma \Delta _{\max } +\beta _{\max } T_{\max } \}e^{\gamma \Delta t^{u}}(\mu ^{u})^{N_{{\sigma } ^{u}} } \nonumber \\&\qquad \cdot \prod _{i=1}^M {e^{-\alpha _i \Delta t_i }e^{T_i (\alpha _i +\beta _i )N_{{\sigma } _{i}} }\mu _i ^{N_{{\sigma } _{i}} }} \left\| {x(t_0 )} \right\| ^{2} \end{aligned}$$

(35)

From Definitions 2, 3 and inequality (32), (34) can be simplified to

$$\begin{aligned} \begin{array}{l} \left\| {x(t_{k+1} )} \right\| ^{2}\le \psi _2 \exp \left\{ {\sum _{i=1}^M {\left[ {\frac{\ln \mu _i +T_i (\alpha _i +\beta _i )}{\tau _{ai}^s }-\alpha _i } \right] } } \right. \cdot \Delta t_i \\ \quad \quad +\,\left. {\left( {\frac{\ln \mu ^{u}}{\tau _a^u }+\gamma } \right) \cdot \Delta t^{u}} \right\} \left\| {x(t_0 )} \right\| ^{2} \\ \end{array} \end{aligned}$$

(36)

Denote $\xi _2 =\max \left( {\frac{\ln \mu _i +T_i (\alpha _i +\beta _i )}{\tau _{ai}^s }-\alpha _i ,\frac{\ln \mu ^{u}}{\tau _a^u }+\gamma } \right)$ Then (36) becomes

$$\begin{aligned} \left\| {x(t_{k+1} )} \right\| ^{2}\le \psi _2 \exp \{\xi _2 (t-t_0 )\}\left\| {x(t_0 )} \right\| ^{2} \end{aligned}$$

(37)

Compared with Definition 1, it can be concluded that system (8) is exponentially stable under condition (32). Next, since (28) holds, and $D_i$ is positively invariant, then from (36), for any $\sigma (t_k )\in {\mathbb {Z}}$, state response $x(t)\in \Omega$. Thus the control input $u_i =K_i x$, $i\in [1,M]$ is admissible. In conclusion, the system is exponentially stable with constrained control. $\square$

Remark 5

As has been stated, both $\tau _{ai}$ and $\tau _a^u$ can be calculated by using the LMI toolbox in Matlab. Next, from (31), it should be noticed that once the unstable subsystem is activated, then the following subsystem must be a stable subsystem. In what follows, another theorem is presented where the unstable subsystems can be activated one by one. But its total running time must be limited.

Theorem 3

Consider system (8), let $K_i$, $i\in [1,M]$ be obtained in Lemma 1. If there exist constants $\mu _i \ge 1$, $\beta _i >0$, $\alpha _i$ and two class $\kappa _\infty$ functions $\kappa { }_1$, $\kappa { }_2$ such that (38), (39), (40), (41) and (42) hold, then system (8) is exponentially stable by admissible control input $u_i =K_i x$, $i\in [1,M]$.

$$\begin{aligned} \left\| {x(t_0 )} \right\| ^{2}\le & {} d^{2}/\psi _3\end{aligned}$$

(38)

$$\begin{aligned} \kappa _1(\left\| {x(t)} \right\| )\le & {} V_i (x(t))\le \kappa { }_2(\left\| {x(t)} \right\| ) \end{aligned}$$

(39)

$$\begin{aligned} \dot{V}_i \left( {x(t)} \right)\le & {} \left\{ {{\begin{array}{l} {-\alpha _i V_i \left( {x(t)} \right) ,\;t\in [t_k +T_i ,t_{k+1} )\;} \\ {\beta _i V_i \left( {x(t)} \right) ,t\in [t_k ,t_k +T_i )} \\ \end{array} }} \right. \end{aligned}$$

(40)

$$\begin{aligned} V_i \left( {x(t)} \right)\le & {} \mu _i V_j \left( {x(t)} \right) \end{aligned}$$

(41)

$$\begin{aligned}&\left\{ {{\begin{array}{l} {\tau _{ai}^s>\frac{T_i (\alpha _i +\beta _i )+\ln \mu _i }{\alpha _i }} \\ {\tau _{ai}^u \ge 0} \\ {\frac{\Delta t^{s}}{\Delta t^{u}}>\frac{\zeta ^{{*}}-\zeta _2 }{\zeta _1 -\zeta ^{{*}}},\;(\zeta _1<\zeta ^{{*}}<0)} \\ \end{array} }} \right. \end{aligned}$$

(42)

where $\zeta _1 =\mathop {\max }\limits _{1\le i\le M} \left( {\frac{T_i (\alpha _i +\beta _i )}{\tau _{ai}^s }-\alpha _i +\frac{\ln \mu _i }{\tau _{ai}^s }} \right)$ is a negative constant, $\zeta _2 =\mathop {\max }\limits _{M+1\le i\le N} \left( {\frac{T_i (\alpha _i +\beta _i )}{\tau _{ai}^u }-\alpha _i +\frac{\ln \mu _i }{\tau _{ai}^u }} \right)$ is a positive constant and $\psi _3 =\frac{\lambda _{\sigma (t_0 )}^{\max } }{\lambda _{\min } }\mu _{\max } \exp \{\gamma \Delta _{\max } +\beta _{\max } T_{\max } \}\prod \nolimits _{i=1}^N {\exp \{T_i (\alpha _i } +\beta _i )N_{0i} +N_{0i} \ln \mu _i \}$ is a positive constant.

Proof

The proof of Theorem 3 is similar to that of Theorem 1. So some part of the proof which has been appeared in Theorem 1 is omitted. It follows from (18) that

$$\begin{aligned}&V_{\sigma (t_{k+1} )} \left( {x(t_{k+1} )} \right) \le e^{-\alpha _i T_i^+ +\beta _i T_i }\mu _{\sigma (t_{k+1} )} V_{\sigma (t_k )} \left( {x(t_k )} \right) \nonumber \\&\quad \le e^{-\alpha _i T_i^+ +\beta _i T_i }\mu _{\sigma (t_{k+1} )} \mu _{\sigma (t_k )} V_{\sigma (t_k^- )} \left( {x(t_k )} \right) \end{aligned}$$

(43)

$$\begin{aligned}&\quad \vdots \nonumber \\&\quad \le \prod _{i=1}^M {\mu _i^{N_{\sigma i} } \exp \left\{ {-\alpha _i \Delta t_i +T_i (\alpha _i +\beta _i )N_{\sigma i} } \right\} } \nonumber \\&\quad \cdot \prod _{i=M+1}^N {\mu _i^{N_{\sigma i} } \exp \left\{ {-\alpha _i \Delta t_i +T_i (\alpha _i +\beta _i )N_{\sigma i} } \right\} } \end{aligned}$$

(44)

$$\begin{aligned}&\quad <\mu _{\max } \exp \{\gamma \Delta _{\max } +\beta _{\max } T_{\max } \} \nonumber \\&\quad \prod _{i=1}^M {\mu _i^{N_{\sigma i}} \exp \{-\alpha _i \Delta t_i +T_i (\alpha _i} +\beta _i )N_{\sigma i} \}\nonumber \\&\quad \cdot \prod _{i=M+1}^N \mu _i^{N_{\sigma i} } \exp \left\{ {-\alpha _i \Delta t_i +T_i (\alpha _i +\beta _i )N_{\sigma i} } \right\} \end{aligned}$$

(45)

The Lyapunov functions are set as $V_i \left( {x(t)} \right) =x^{T}P_i x$ then the following can be got,

$$\begin{aligned} \left\| {x(t_{k+1} )} \right\| ^{2}\le \psi _3 \exp \{\Delta t^{s}\zeta _1 +\Delta t^{u}\zeta _2 \}\left\| {x(t_0 )} \right\| ^{2} \end{aligned}$$

(46)

Because $\zeta _1<\zeta ^{{*}}<0$ holds, (46) becomes,

$$\begin{aligned} \left\| {x(t_{k+1} )} \right\| ^{2}\le \psi _3 \exp \{\zeta ^{{*}}(t-t_0 )\}\left\| {x(t_0 )} \right\| ^{2} \end{aligned}$$

(47)

So (42) can verify exponential stability of the system. Besides, (38) makes the control input admissible. This completes the proof of the theorem. $\square$

Remark 6

In Theorem 1, $\psi _1$ will get larger with $T_i$ and $\beta _i$ getting larger. In Theorems 2 and 3, $\psi _2$ and $\psi _3$ not only have a relationship with $T_i$ and $\beta _i$, but also have a relationship with $\Delta _{\max }$ and $\gamma$. In this way, the trajectory of state response with unstable subsystems can remain in the feasible region all the time.

4 Numerical Examples

Example 1

Consider system (8) with

$$\begin{aligned} A_1= & {} \left[ {{\begin{array}{ll} {-1.4402}&{} \quad {-3.36} \\ {-3.52}&{} \quad {-1.28} \\ \end{array} }} \right] ,B_1 =\left[ {{\begin{array}{l} {-3.36} \\ {3.52} \\ \end{array} }} \right] \\ A_2= & {} \left[ {{\begin{array}{ll} {-2.928}&{} \quad {2.88} \\ {2.24}&{} \quad {-1.488} \\ \end{array} }} \right] ,B_2 =\left[ {{\begin{array}{l} {1.92} \\ {2.24} \\ \end{array} }} \right] \end{aligned}$$

$T_1 =T_2 =0.1\,\mathrm{s}. \quad N_{01} =N_{02} =0.$ According to Lemma 1, feedback matrices can be chosen as: $K_1 =\left[ {{\begin{array}{ll} {-1}&{} 1 \\ \end{array} }} \right]$, $K_2 =\left[ {{\begin{array}{ll} 1&{} {1.5} \\ \end{array} }} \right]$.

By solving the LMIs in (26) and (27), we can get $\alpha _1 =9.5999$, $\beta _1 =0.224$, $\mu _1 =1.4455$, $\alpha _2 =9.6938$, $\beta _2 =0.3198$, $\mu _2 =1$ ($\mu _2$ is smaller than one so it is enlarged to one.), $\psi _1 =1.7211$. According to (13), $\tau _{a1} >0.1023$, $\tau _{a2} >0.1033$. The control input is constrained with $-50\le u_1 \le 50$ and $-40\le u_2 \le 40$. Because $u_i =K_i x$, we can get the biggest Euclidean ball, which is plotted in a dotted line in Fig. 1. The circle plotted in a solid line illustrates the feasible region of initial states. We choose $x(t_0 )=[{\begin{array}{ll} {-20}&{} {17.5601} \\ \end{array} }]$, $\tau _{a1} =0.12$ s, $\tau _{a2} =0.13$ s. Figure 2 illustrates the state responses, while Fig. 3 is the switching signal. Next, we choose $\tau _{a1} =0.19$ s, $\tau _{a2} =0.2$ s, Fig. 4 illustrates the state responses, while Fig. 5 is the switching signal.

Fig. 1

The feasible region of initial states

Fig. 2

State responses of Example 1 with shorter MDADT

Fig. 3

Switching signal of Example 1 with shorter MDADT

Fig. 4

State responses of Example 1 with longer MDADT

Fig. 5

Switching signal of Example 1 with longer MDADT

Example 2

Consider system (8) in Example 1 with the same subsystems and another subsystem:

$$\begin{aligned} A_3 =\left[ {{\begin{array}{ll} {0.048}&{} \quad 0 \\ 0&{} \quad {0.048} \\ \end{array} }} \right] ,\;B_3 =\left[ {{\begin{array}{l} {0.16} \\ {0.32} \\ \end{array} }} \right] \end{aligned}$$

Suppose subsystem 3 cannot be stabilized and $N_{03} =0$, $T_3 =0.1$ s. By solving the LMIs in Theorem 2 we can get $\alpha _1 =9.5999$, $\beta _1 =0.224$, $\mu _1 =1$, $\alpha _2=9.6838$, $\beta _2 =0.96$, $\mu _2 =1$, $\gamma =3.8398$. $\mu ^{u}=0.207$. According to (32), we can get $\tau _{a1} >0.102$ s, $\tau _{a2} >0.110$ s, $\tau _{a3} <0.400$ s. Then we choose $\tau _{a1} =0.12$ s, $\tau _{a2} =0.13$ s, $\tau _{a3} =0.3$ s, then $\psi _2 =5.446$, we choose $x(t_0 )=[{\begin{array}{ll} {-14}&{} {2.7184} \\ \end{array} }]$. Figures 6 and 7 illustrate the state responses and switching signal, respectively. Next, we choose $\tau _{a1} =0.22$ s, $\tau _{a2} =0.23$ s, $\tau _{a3} =0.2$ s, then $\psi _2 =2.386$, we choose $x(t_0 )=[{\begin{array}{ll} {-19}&{} {10.1601} \\ \end{array} }]$. Figures 8 and 9 illustrate the state responses and switching signal, respectively.

Fig. 6

State responses of Example 2 with shorter MDADT

Fig. 7

Switching signal of Example 2 with shorter MDADT

Fig. 8

State responses of Example 2 with longer MDADT

Fig. 9

Switching signal of Example 2 with longer MDADT

From the two examples, it can be noticed that by choosing the appropriate initial states and MDADT, the asynchronous switched system is exponentially stable. Besides, although there may be unstable subsystem, the trajectory of state responses can also remain in the feasible region all the time. The example for Theorem 3 is omitted. The advantage of Theorem 3 is that unstable subsystems can be activated one by one. But the total running time of stable and unstable subsystems must be calculated.

5 Conclusion

This paper provides three theorems which can be used to stabilize asynchronous switched systems with constrained inputs and unstable subsystems. The mode-dependent average dwell time can guarantee that the system is exponentially stable. Even if there are unstable subsystems, the restriction of the running time of unstable subsystems can also guarantee the system is exponentially stable. Furthermore, choosing the initial states within a specific Euclidean ball makes the control inputs admissible all the time.