Finite-time stabilization for discrete-time switched stochastic linear systems under asynchronous switching

Abstract

This paper deals with the problem of state feedback stabilization with finite-time stochastic stability for a class of discrete-time switched stochastic linear systems under asynchronous switching. The attention is focused on designing the feedback controller that guarantees the finite-time stochastic stability of the dynamic system. The finite-time stochastic stability definition of discrete-time switched stochastic systems is introduced. The asynchronous switching idea originates from the fact that switching instants of the controllers lag behind or exceed those of subsystems. On the basis of the average dwell time method and multiple Lyapunov functions approach, a finite-time stochastic stability condition is established. Then, an asynchronously switched controller is designed and the corresponding switching law is derived to guarantee the considered system be finite-time stochastically stable. Two numerical examples are provided to show the effectiveness of the developed results.

Keywords

Asynchronous switching average dwell time discrete-time finite-time stochastic stability switched stochastic systems

Introduction

Switched systems have attracted the interest of several scientists over the last 20 years. Due to their strong engineering background in various areas, switched systems are often used as a unified modelling tool for a great number of real-world systems such as mechanical systems, automotive industry, aircraft and air traffic control, and many other domains (Tomlin et al., 1998; Varaiya, 1993; Wang and Brockett, 1997). A switched system usually consists of a family of systems of differential or difference equations and a rule that determines which system is activated at certain time interval. Different switching rules would produce different behaviours of the system and hence lead to different system performances. The issue of stability of switched systems, which has drawn considerable attention, is one of the basic research topics. Lyapunov stability theory and its variations or generalizations have played an important role in this research field. The common Lyapunov function method and multiple Lyapunov functions method for switched system have been presented by many researchers (Liberzon and Morse, 1999; Shorten et al., 2007; Sun et al., 2006a, 2006b; Zhao and Hill, 2009). For most switched systems, it is hard to find a common Lyapunov function; however, we can guarantee the switched system is still stable under some properly chosen for switching signals, which are found by using the multiple Lyapunov functions technique (Branicky, 1998). In addition, more researchers have paid attention to the average dwell time control of switched systems (Hespanha and Morse, 1999; Zhai et al., 2000). In particular, the average dwell time approach is employed to deal with the control, observation and filtering problems of switched delay systems or network control systems (Lian et al., 2011; Liu et al., 2009; Wang et al., 2009; Wang et al. 2012a). Considering that the subsystems are non-linear, the study of switched non-linear systems has been developed recently (Liu et al., 2012; Müller et al., 2012; Sun and Wang, 2013; Yang et al., 2012).

As an important class of switched systems, switched stochastic (SS) systems have found many practical applications in industry, especially in networked communication systems (Hespanha, 2005). One of the general formal models for stochastic hybrid systems is proposed by Hu et al. (2000). Discrete-time stochastic processes are found in a wide variety of applications, particularly in modelling of engineering, biological, medical and physical systems, which are subject to random perturbations (Chen, 1985; Costa et al., 2005; Davis, 1977; Kumar and Varaiya, 1986; Soderstrom, 2002). Discrete-time stochastic processes also arise in various numerical schemes for systems of stochastic differential equations (Kloeden and Platen, 1992). Compared with switched state-space models, the study of switched stochastic systems is more arduous, since not only stability, but also the qualitative properties of convergence and stochastic iterative processes should be considered simultaneously. Many results related to switched stochastic systems have been reported in the literature. For instance, the problem of state feedback stabilization of discrete-time stochastic processes under Markovian switching has been considered (Sathananthan et al., 2010), where the jump Markovian switching is modelled by a discrete-time Markov chain. Dwell time controllers for stochastic systems with a switching Markov chain are designed and it is proved that the system is stable under switching with guaranteed probability if the expectation of the time between two consecutive switching is sufficiently large (Battilotti and De Santis, 2005). The moment stability (M-S) and sample path stability (SP-S) are investigated for a class of switched stochastic systems (Feng et al., 2011). Based on the concept of the average dwell time, the mean square (MS) stability and exponential mean square (EMS) stability of multi-variable switched stochastic systems are investigated by Wei and Zhang (2006).

As we know, most of the results related to stability of switched stochastic systems are based on the concept of classical Lyapunov stability, which is defined over an infinite time interval. However, in some cases, we need to consider the stability or performance of the system in a finite time interval. In these cases, the concept of finite-time stability (FTS) could be used, which focuses its attention on the behaviour of a system response over a finite time interval. Therefore, to some degree, the study of finite-time stability of switched stochastic systems seems to be more important. Recently, only a few papers studied the finite-time stability for switched stochastic systems. Xiang et al. (2012) gave the definition of finite-time stochastic stability for these systems in the continuous-time case and presented a method to study the stabilization problem. However, as pointed out by Zhai et al. (2002), for continuous-time switched system it is easy to find a convex combination of subsystems, but for discrete-time switched systems we cannot derive such a combination. Therefore, it is non-trivial to discuss finite-time stochastic stability of discrete-time switched stochastic systems due to their some distinctive features.

Until now, most of the aforementioned results on finite-time stabilization for switched stochastic systems were under the synchronous switching assumption, i.e. the controller of each subsystem can match the practical system switching signal precisely. However, because the switching signal of the system is usually unknown, it is impractical to realize this point. In fact, it takes some time to identify the system mode and switching instant and then apply the matched controller; thus there inevitably exists asynchronous switching between the system mode and the controller. Sometimes, owing to the respond error of the controller, the switching instants of the controller may exceed those of the subsystem. So it is necessary to design an efficient controller that can tolerate asynchronous switching which will deteriorate the performance of systems, even make system out of control. Some typical examples can be found in mechanical and chemical systems (Hetel et al., 2007; Mhaskar et al., 2008). So far, some interesting results on stabilization of switched systems under asynchronous switching have been presented in the literature; see Xiang and Chen (2010), Xiang and Wang (2009), Xie et al. (2009), Zhang and Gao (2010) and Zhang and Shi (2009) for Lyapunov asymptotical or exponential stabilization design, and Wang et al. (2012b) and Zong et al. (2013) for finite-time stabilization design. However, to the best of our knowledge, the finite-time stabilization issue of discrete-time switched stochastic systems under asynchronous switching has not been fully investigated, which is quite an important issue for the switched system. Moreover, the procedures given in Wang et al. (2012b) and Zong et al. (2013) cannot be applied to the discrete-time case. This motivates the present study.

In this paper, we are concerned with the problem of finite-time stabilization for discrete-time switched stochastic systems under asynchronous switching. The main contribution of this paper lies in that the finite-time stochastic stability results for discrete-time switched stochastic systems are first proposed. Then the asynchronous switching control problem is studied in the sense of finite-time stochastic stability for the considered system. The result shows that it is unnecessary to guarantee that the closed-loop subsystem is finite-time stochastically stable by the designed asynchronously switched controller. During the finite-time interval, the switching frequency only needs to be limited in some value, then the switched system is finite-time stochastically stable by the designed controller despite of the asynchronous switching between the controllers and the practical subsystems. The remainder of the paper is organized as follows. In the next section, some preliminary definitions are provided, and the problem we deal with is precisely stated. Then we provide the main results of this paper: by applying the average dwell time method, a sufficient condition ensuring the finite-time stochastic stability of discrete-time switched stochastic system is first derived. Then a state feedback asynchronously switched controller is designed. Two numerical examples are presented by using the linear matrix inequality (LMI) toolbox to illustrate the efficiency of the proposed method, and finally, our conclusions are drawn.

Notation

Throughout this paper, $S > 0$ denotes $S$ is a positive definite symmetric matrix. $Z$ denotes the integer set and $Z^{+}$ denotes the non-negative integer set. $| | • | |$ denotes the usual 2-norm. $λ_{max} (P)$ and $λ_{min} (P)$ denote the maximum and minimum eigenvalues of matrix $P$ , respectively. $sup {•}$ and $inf {•}$ refer to the supremum and infimum. $diag {a_{i}}$ denotes the diagonal matrix with the diagonal elements $a_{i}$ , $i = 1, 2, \dots, n$ . $I$ is an identity matrix with appropriate dimension. $A^{T}$ denotes the transpose of matrix $A$ , $R^{n}$ is the $n$ -dimensional real Euclidean space. $R^{m \times n}$ is the set of all real $m \times n$ matrices.

Problem formulation and preliminary

Description of problem

Consider a class of discrete-time switched stochastic systems of the form as follows

\begin{matrix} x (k + 1) = A_{σ (k)} x (k) + B_{σ (k)} u (k) \\ + (C_{σ (k)} x (k) + D_{σ (k)} u (k)) ω (k) \end{matrix}

(1)

x (0) = x_{0}

(2)

where $x (k) \in R^{n}$ is the system state. $u (k) \in R^{p}$ is the control input, $x (0) = x_{0}$ is a finite deterministic value and represents the initial state of the system. $ω (k)$ is a zero-mean real scalar process on a probability space $(Ω, F, P)$ relative to an increasing family ${(F_{k})}_{k \in L}$ of $σ$ -algebras, $F_{k} \subset F$ generated by ${(ω (k))}_{k \in L}$ , where $L$ is the set of natural numbers. It is supposed that

E {ω (k)} = 0, E {ω {(k)}^{2}} = α

(3)

where the scalar $α > 0$ , and the stochastic process $ω (0), ω (1), \dots$ are independent, i.e. $E {ω (r) ω (h)} = 0$ , $r \neq h$ . $σ (k) : Z^{+} \to \underline{N} = {1, 2, \dots, N}$ is a switching signal, which is a piecewise constant function depending on time $k$ , $N$ is a finite positive integer and denotes the number of subsystems. Moreover, corresponding to the switching signal $σ (k)$ , we denote the switching sequence by $S = {(i_{0}, k_{0}), (i_{1}, k_{1}), \dots, (i_{l}, k_{l}), \dots | i_{l} \in \underline{N}, l \in Z^{+}}$ with $k_{0} = 0$ , which means that the $i_{l}$ th subsystem is activated when $k_{l} \leq k < k_{l + 1}$ or equivalently, $σ (k) = i_{l}$ when $k_{l} \leq k < k_{l + 1}$ . $(A_{i_{l}}, B_{i_{l}}, C_{i_{l}}, D_{i_{l}})$ represents the $i_{l}$ th subsystem or $i_{l}$ th mode of system (1). $A_{i} \in R^{n \times n}, B_{i} \in R^{n \times p}$ , $C_{i} \in R^{n \times n}$ and $D_{i} \in R^{n \times p}$ , $\forall i \in \underline{N}$ are real-valued matrices with appropriate dimensions.

Owing to asynchronous switching, the practical switching instant of controller is different from that of system. For convenience, $σ' (k)$ is used to denote the practical switching signal of controller, $σ' (k)$ can be written as $σ' (k) : {(k_{0} + Δ_{k_{0}}, σ (k_{0})), (k_{1} + Δ_{k_{1}}, σ (k_{1})), \dots, (k_{l} + Δ_{k_{l}}, σ (k_{l}), \dots)}$ , where $Δ_{k_{l}} < inf_{l \geq 0} (k_{l + 1} - k_{l})$ and $Δ_{k_{l}} > 0$ , here $Δ_{k_{l}}$ represents the period when the switching instant of the controller lags behind one of the system. Or $| Δ_{k_{l}} | < inf_{l \geq 1} (k_{l} - k_{l - 1})$ and $Δ_{k_{l}} < 0$ , here $Δ_{k_{l}}$ represents the period when the switching instant of the controller exceeds one of the system. In these cases, $Δ_{k_{l}}$ is said to be the mismatched period between the controller and the system.

Remark 1. Generally speaking, the proposed condition that $Δ_{k_{l}} < inf_{l \geq 0} (k_{l + 1} - k_{l})$ or $| Δ_{k_{l}} | < inf_{l \geq 1} (k_{l} - k_{l - 1})$ is reasonable. In fact, $Δ_{k_{l}}$ is the switching time error or delayed switching time between the controller and the system, however, in practice this period is often less than the dwell time of the system. The condition guarantees that there always exists a synchronous period (or matched period) to work together between the controller and the system, which makes it possible to design the stabilizing controller for the system.

Definition 1. Given switching the sequence $S$ and the switching signal $σ' (k)$ , let $Δ_{[0, k)} = Δ_{k_{0}} + Δ_{k_{1}} + \dots + Δ_{k_{l}}$ denote the total mismatched control time during the interval $[0, k)$ , and call the ratio $Δ_{[0, k)} / k$ the mismatched control rate of the switched controllers in the system.

Let us review the definition of average dwell time, which will be useful in designing the stabilizing controller to guarantee the finite-time stochastic stability of the system.

Definition 2. (Zhang et al., 2008). For the switching signal $σ (k)$ and $k_{0} \leq k \leq K$ , let $N_{σ} (k, K)$ denotes the number of switching over $[k, K)$ . If

N_{σ} (k, K) \leq N_{0} + \frac{K - k}{τ_{a}}

(4)

holds for $τ_{a} > 0$ and an integer $N_{0} \geq 0$ , then $τ_{a}$ is called an average dwell time and $N_{0}$ is called the chatter bound.

Without loss of generality, in this paper we choose $N_{0} = 0$ , as in Sun et al. (2006a) and Lin et al. (2011).

Remark 2. The concept of average dwell time was originally proposed for continuous time switched systems in Hespanha and Morse (1999), and it has been modified to fit the discrete-time case in some literatures, for instance Zhai et al. (2002) and Zhang et al. (2008). Definition 2 is borrowed from these existing results.

Next, we shall introduce some concepts on finite-time stochastic stability and finite-time asynchronous switching stabilization for discrete-time switched stochastic systems.

Definition 3. (Finite-time stochastic stability, FTSS). Given positive constants $c_{1}$ , $c_{2}$ and $M$ with $c_{1} < c_{2}$ and a switching signal $σ (k)$ , system (1) with $u (k) \equiv 0$ is said to be finite-time stochastically stable with respect to $(c_{1}, c_{2}, M, σ (k))$ , if

‖ x (0) ‖ \leq c_{1} \Rightarrow E {‖ x (k) ‖} \leq c_{2}, \forall k \in [1, M]

(5)

Remark 3. In Amato et al. (2001), it is pointed out that finite-time stability (FTS) and asymptotic stability are two independent concepts for switched system. From the definition above, we can see that FTSS and asymptotic mean square stability (Wei and Zhang, 2006) are also independent for discrete-time switched stochastic system.

Definition 4. (Finite-time stochastic stabilization under asynchronous switching). System (1) is said to be finite-time stochastically stabilizable via asynchronous switching, if there is an asynchronous switching state feedback controller

u (k) = K_{σ' (k)} x (k)

(6)

such that the resulting closed-loop system is finite-time stochastically stable with respect to $(c_{1}, c_{2}, M, σ (k))$ , where $c_{1} < c_{2}$ , $σ (t)$ is a switching signal of the system and $σ' (t)$ is a switching signal of the controller.

The main issue in this paper is given as follows. Given switched system (1), find a state feedback asynchronously switched controller in the form of (6) such that the resulting closed-loop system is finite-time stochastically stable with respect to $(c_{1}, c_{2}, M, σ (k))$ .

Preliminary results

In this subsection, several lemmas are offered, which will be useful to present our main results.

Lemma 1. Consider the following discrete stochastic system

x (k + 1) = Ax (k) + Cx (k) ω (k)

(7)

For a given constant scalar $β \geq 1$ , if there exists positive definite symmetric matrices $P > 0$ such that

[\begin{matrix} - P & A^{T} P & C^{T} P \\ PA & - β P & 0 \\ PC & 0 & - (β / α) P \end{matrix}] < 0

(8)

holds, then for the Lyapunov function candidate

V (x (k)) = x {(k)}^{T} Px (k)

(9)

along the trajectory of system (7) there holds the inequality

E {V (x (k))} < β^{k} V (x (0))

(10)

Proof. Consider the Lyapunov function candidate as follows

V (x (k)) = x {(k)}^{T} Px (k)

(11)

Then, by some calculations, we have

\begin{matrix} E {V (x (k + 1)) | F_{k}} - V (x (k)) \\ = x {(k + 1)}^{T} Px (k + 1) - x {(k)}^{T} Px (k) \\ = {[Ax (k) + Cx (k) ω (k)]}^{T} P {[Ax (k) + Cx (k) ω (k)] - x (k)}^{T} Px (k) \\ = x {(k)}^{T} A^{T} PAx (k) + 2 x {(k)}^{T} A^{T} PCx (k) ω (k) \\ + ω {(k)}^{T} x {(k)}^{T} C^{T} PCx (k) ω (k) - x {(k)}^{T} Px (k) \end{matrix}

(12)

for all $x (k) \neq 0$ . Now, from (3), taking expectation of both sides of (12) gives

\begin{matrix} E {V (x (k + 1))} - E {V (x (k))} \\ = E {x {(k)}^{T} A^{T} PAx (k) - x {(k)}^{T} Px (k) + ω {(k)}^{T} x {(k)}^{T} C^{T} PCx (k) ω (k)} \\ = E {x {(k)}^{T} A^{T} PAx (k) - x {(k)}^{T} Px (k) + {‖ P^{1 / 2} Cx (k) ‖}^{2} ω {(k)}^{2}} \\ = E {x {(k)}^{T} A^{T} PAx (k) - x {(k)}^{T} Px (k) + α x {(k)}^{T} C^{T} PCx (k)} \end{matrix}

(13)

From (8), it is easy to deduce

E {V (x (k + 1))} < β E {V (x (k))}

(14)

which implies (10). This completes the proof.

Remark 4. Lemmas 1 provides the methods for expected value estimation of the Lyapunov function candidate that will be used to design the asynchronously switched controller.

Applying the control (6) to system (1), the resulting closed-loop system is given by $\forall σ (k - Δ_{k_{l}}) = i_{l - 1} \in \underline{N}$ , $σ (k) = i_{l} \in \underline{N}$ , $i_{l - 1} \neq i_{l}$

{\begin{matrix} x (k + 1) = {\bar{A}}_{i_{l} i_{l - 1}} x (k) + {\bar{C}}_{i_{l} i_{l - 1}} x (k) ω (k), \forall k \in [k_{l}, k_{l} + Δ_{k_{l}}) \\ x (k + 1) = {\bar{A}}_{i_{l}} x (k) + {\bar{C}}_{i_{l}} x (k) ω (k), \forall k \in [k_{l} + Δ_{k_{l}}, k_{l + 1}) \end{matrix}

(15)

where ${\bar{A}}_{i_{l} i_{l - 1}} = A_{i_{l}} + B_{i_{l}} K_{i_{l - 1}}$ , ${\bar{C}}_{i_{l} i_{l - 1}} = C_{i_{l}} + D_{i_{l}} K_{i_{l - 1}}$ , ${\bar{A}}_{i_{l}} = A_{i_{l}} + B_{i_{l}} K_{i_{l}}$ and ${\bar{C}}_{i_{l}} = C_{i_{l}} + D_{i_{l}} K_{i_{l}}$ .

From Lemma 1, the finite-time stochastic stability result for discrete-time switched stochastic system (15) can be obtained. Based on the analysis of finite-time stochastic stability, finite-time stabilizing controller under asynchronous switching will be designed for system (1).

Main results

In this section, we will first give the finite-time stochastic stability condition of discrete-time switched stochastic system, and further identify and design the stabilizing switching law and the corresponding controllers for the system (1).

Finite-time stochastic stability analysis

Consider the switched stochastic system without the control input as follows

x (k + 1) = A_{σ (k)} x (k) + C_{σ (k)} x (k) ω (k)

(16)

In this subsection, a sufficient condition under which system (16) is finite-time stochastically stable will first be developed.

Theorem 1.For any $i, j \in \underline{N}$ , $i \neq j$ , suppose that there exist positive definite matrices $P_{i} > 0$ , and constant scalars $β \geq 1$ , $μ \geq 1$ , such that

P_{i} \leq μ P_{j}

(17)

[\begin{matrix} - P_{i} & A_{i}^{T} P_{i} & C_{i}^{T} P_{i} \\ P_{i} A_{i} & - β P_{i} & 0 \\ P_{i} C_{i} & 0 & - (β / α) P_{i} \end{matrix}] < 0

(18)

If the average dwell time of the switching signal $σ (k)$ satisfies

τ_{a} > τ_{a}^{*} = \frac{M \ln μ}{\ln (\frac{c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i \in \underline{N}} {λ_{min} (P_{i})}}{sup_{i \in \underline{N}} {λ_{max} (P_{i})}}) - M \ln β}

(19)

then system (16) is finite-time stochastically stable with respect to $(c_{1}, c_{2}, M, σ (k))$ .

Proof. Construct a Lyapunov-like function as follows

V (x (k)) = V_{σ (k)} (x (k)) = x {(k)}^{T} P_{σ (k)} x (k)

(20)

According to the switching sequence $S$ of the system, the $i_{l}$ th subsystem is activated when $k \in [k_{l}, k_{l + 1})$ . Without loss of generality, assume $i_{l} = i$ . By virtue of system (16), Lemma 1 and (18), we have

E {V_{i} (x (k))} < β E {V_{i} (x (k - 1))}

(21)

Note that the switching occurs at the time $k_{l}$ , Without loss of generality, assume $σ (k_{l}) = i$ , $σ (k_{l} - 1) = j$ . Then,

V (x (k_{l})) = V_{σ (k_{l})} (x (k_{l})) = V_{i} (x (k_{l})) = x {(k_{l})}^{T} P_{i} x (k_{l})

(22)

By (17), we obtain

V_{σ (k_{l})} (x (k_{l})) = x {(k_{l})}^{T} P_{i} x (k_{l}) \leq μ x {(k_{l})}^{T} P_{j} x (k_{l}) = μ V_{σ (k_{l} - 1)} (x (k_{l}))

(23)

Let $0 = k_{0} < k_{1} < k_{2} < \dots < k_{N_{σ} (0, M)} < M$ denote the switching instants of the switching signal $σ (k)$ over the interval $[0, M)$ . For $\forall k \in [0, M)$ , there must exist $0 \leq r \leq N_{σ} (0, M)$ such that $k \in [k_{r}, k_{r + 1})$ . Owing to $k_{r} - k_{r - 1} \geq 1$ , we have

V_{σ (k_{r} - 1)} (x (k_{r})) = V_{σ (k_{r - 1})} (x (k_{r}))

(24)

From (23) and (24), using the iterative formulas (20) and (21), it can be concluded that

\begin{matrix} E {V (x (k))} < β^{k - k_{r}} E {V_{σ (k_{r})} (x (k_{r}))} \\ < β^{k - k_{r}} μ E {V_{σ (k_{r} - 1)} (x (k_{r}))} \\ = β^{k - k_{r}} μ E {V_{σ (k_{r - 1})} (x (k_{r}))} \\ < β^{k - k_{r}} μ β^{k_{r} - k_{r - 1}} E {V_{σ (k_{r - 1})} (x (k_{r - 1}))} \\ = μ β^{k - k_{r - 1}} E {V_{σ (k_{r - 1})} (x (k_{r - 1}))} \\ < \dots \\ < μ^{r} β^{k} E {V_{σ (0)} (x (0))} \\ = μ^{r} β^{k} E {V (x (0))} \end{matrix}

(25)

Noticing $r \leq N_{σ} (0, M) \leq \frac{M}{τ_{a}}$ , one has

E {V (x (k))} < μ^{\frac{M}{τ_{a}}} β^{M} E {V (x (0))}

(26)

From (20), we have

V (x (k)) \geq inf_{i \in \underline{N}} {λ_{min} (P_{i})} ‖ x (k) ‖^{2}

(27)

On the other hand, for $i \in \underline{N}$ we have

V (x (0)) \leq sup_{i \in \underline{N}} {λ_{max} (P_{i})} ‖ x (0) ‖^{2}

(28)

Using the fact

‖ x (0) ‖ \leq c_{1}

(29)

we obtain

V (x (0)) \leq sup_{i \in \underline{N}} {λ_{max} (P_{i})} c_{1}^{2}

(30)

Combining (25)–(29), the following inequality can be derived

E {{‖ x (k) ‖}^{2}} < \frac{sup_{i \in \underline{N}} {λ_{max} (P_{i})}}{inf_{i \in \underline{N}} {λ_{min} (P_{i})}} c_{1}^{2} μ^{\frac{M}{τ_{a}}} β^{M}

(31)

From (19), we obtain

\frac{sup_{i \in \underline{N}} {λ_{max} (P_{i})}}{inf_{i \in \underline{N}} {λ_{min} (P_{i})}} c_{1}^{2} μ^{\frac{M}{τ_{a}}} β^{M} < c_{2}^{2}

(32)

According to (31) and (32), we have

E {‖ x (k) ‖} < c_{2}

(33)

The proof is completed.

Remark 5. Unlike the classical Lyapunov function for switched systems in the case of asymptotic stability, there is no requirement of negative definiteness or negative semi-definiteness on $Δ E {V (x (k))}$ in the proof procedure of Theorem 1.

Remark 6. From above proof procedure, we can see that for a given bound (29) on the initial condition, the expected estimation of the system state remains within prescribed bounds (33) in the fixed time interval. It should be noted that a system could be finite-time stable but not Lyapunov asymptotically stable, and vice versa. While Lyapunov asymptotic stability deals with the behaviour of a system within a sufficiently long time interval, finite-time stability, which is more practical, is used to study the behaviour of the system within a finite interval.

Remark 7. By (21), we know that if $0 < β < 1$ , the subsystem is exponentially mean-square stable, which can be obtained from Feng and Xu (2009). So it clearly shows that a finite-time stable switched stochastic system may not require the subsystems are mean-square stable. Moreover, exponentially mean-square stable subsystems cannot ensure the system is finite-time stable, as the transient of a system response may exceed the bound. In fact, we can see that there is no any strict requirement on the stability of subsystems and it is very different from mean-square stability of switched stochastic systems.

Remark 8. If we preset $μ = sup_{i, j \in \underline{N}} {\frac{λ_{max} (P_{i})}{λ_{min} (P_{j})}, i \neq j}$ , (17) may be unnecessary. In this case, the number of matrix inequalities to be solved is $i$ . Otherwise, the number of matrix inequalities we need to solve will be $i^{2}$ . When the number of subsystems is not very large, the solution of matrix inequalities may be not complex, but the number of matrix inequalities to be solved will increase more with the increasing number of subsystems.

Remark 9. By (19), $μ = 1$ means that $τ_{a} > 0$ . In fact, from (31), the supremum of $E {{‖ x (k) ‖}^{2}}$ is independent of $τ_{a}$ when $μ = 1$ . So we can choose $τ_{a}$ without any restriction. Besides, (19) implies that

c_{2}^{2} inf_{i \in \underline{N}} {λ_{min} (P_{i})} > c_{1}^{2} sup_{i \in \underline{N}} {λ_{max} (P_{i})} β^{M}

which guarantees that $τ_{a}$ is non-negative.

The function $V (x (k))$ in the proof procedure of Theorem 1 belongs to multiple Lyapunov-like functions. The advantages of multiple Lyapunov-like functions lie in their flexibility. However, there are some constraints on the switching signals such as condition (19). Hence, Theorem 1 may not be suitable for the case of fast switching or unknown switching sequence. The following corollary will give some conditions, which can guarantee discrete-time switched stochastic systems (16) finite-time stochastically stable under an arbitrarily switching signal.

Corollary 1. For any $i \in \underline{N}$ , $i \neq j$ , suppose that there exists positive definite matrix $P > 0$ , and a constant scalar $β \geq 1$ , such that

[\begin{matrix} - P & A_{i}^{T} P & C_{i}^{T} P \\ P A_{i} & - β P & 0 \\ P C_{i} & 0 & - (β / α) P \end{matrix}] < 0

(34)

λ_{max} (P) β^{M} c_{1}^{2} < λ_{min} (P) c_{2}^{2}

(35)

then system (16) is finite-time stochastically stable with respect to $(c_{1}, c_{2}, M)$ , which is independent of $σ (k)$ .

Finite-time stabilization under asynchronous switching

In this subsection, based on Theorem 1 finite-time stabilizing controller under asynchronous switching will be designed for system (1). The finite-time stabilization problem addressed here is to find an asynchronously switched controller.

Consider the closed-loop system (15) running in a mismatched control period $[k_{l}, k_{l} + Δ_{k_{l}})$ , i.e.

x (k + 1) = {\bar{A}}_{ij} x (k) + {\bar{C}}_{ij} x (k) ω (k)

(36)

where ${\bar{A}}_{ij} = A_{i} + B_{i} K_{j}$ , ${\bar{C}}_{ij} = C_{i} + D_{i} K_{j}$ . For the sake of simplicity, in (36), we use the subscripts $i$ and $j$ to substitute for $i_{l}$ and $i_{l - 1}$ , respectively. Choose the following Lyapunov-like function for system (36)

V_{ij} (x (k)) = x {(k)}^{T} P_{ij} x (k)

(37)

where $P_{ij} > 0$ . By Theorem 1, it is easy to obtain an expected increase estimation of $V_{ij} (x (k))$ in (37) along the state trajectory of system (36).

Lemma 2. For any $i, j \in \underline{N}$ , $i \neq j$ , if there exist positive definite matrices $P_{ij} > 0$ , and a constant scalar $ξ_{1} \geq 1$ , such that

[\begin{matrix} - P_{ij} & {\bar{A}}_{ij}^{T} P_{ij} & {\bar{C}}_{ij}^{T} P_{ij} \\ P_{ij} {\bar{A}}_{ij} & - ξ_{1} P_{ij} & 0 \\ P_{ij} {\bar{C}}_{ij} & 0 & - (ξ_{1} / α) P_{ij} \end{matrix}] < 0

(38)

then along any state trajectory of system (36), the function $V_{ij} (x (k))$ in (37) ensures the following expected increase estimation

E {V_{ij} (x (k))} < ξ_{1} E {V_{ij} (x (k - 1))}

(39)

Consider the closed-loop system (15) running in a matched control period $[k_{l} + Δ_{k_{l}}, k_{l + 1})$ , i.e.

x (k + 1) = {\bar{A}}_{i} x (k) + {\bar{C}}_{i} x (k) ω (k)

(40)

where ${\bar{A}}_{i} = A_{i} + B_{i} K_{i}$ , ${\bar{C}}_{i} = C_{i} + D_{i} K_{i}$ . For conciseness, in (40), we also use the subscript $i$ instead of $i_{l}$ . Choose the following Lyapunov function for system (40)

V_{i} (x (k)) = x {(k)}^{T} P_{i} x (k)

(41)

where $P_{i} > 0$ . From Theorem 1, we can obtain an expected increase estimation of $V_{i} (x (k))$ in (41) along the state trajectory of system (40).

Lemma 3.For any $i \in \underline{N}$ , if there exist positive definite matrices $P_{i} > 0$ , and a constant scalar $ξ_{2} \geq 1$ , such that

[\begin{matrix} - P_{i} & {\bar{A}}_{i}^{T} P_{i} & {\bar{C}}_{i}^{T} P_{i} \\ P_{i} {\bar{A}}_{i} & - ξ_{2} P_{i} & 0 \\ P_{i} {\bar{C}}_{i} & 0 & - (ξ_{2} / α) P_{i} \end{matrix}] < 0

(42)

then along any state trajectory of system (40), the function $V_{i} (x (k))$ in (41) ensures the following expected increase estimation

E {V_{i} (x (k))} < ξ_{2} E {V_{i} (x (k - 1))}

(43)

Remark 10. Lemmas 2 and 3 provide sufficient conditions of the growth expected estimation of Lyapunov functions (37) and (41), respectively. It is worth pointing out that owing to $ξ_{1}, ξ_{2} \geq 1$ , the supremums of $E {V_{ij} (x (k))}$ and $E {V_{i} (x (k))}$ are growing with increasing of $ξ_{1}, ξ_{2}$ , and they are called the growth expected estimation of functions. From the following proof line, we shall find that conditions (39) and (43) can be used to design the stabilizing controller and switching law of the system, and it is unnecessary to require $0 < ξ_{1}, ξ_{2} < 1$ , which may lead to some conservatism.

Next, we will give the design method of asynchronous switching law of system (15). In order to illustrate the proof line more clearly, it is assumed that the $i_{l - 1}$ th subsystem switched to the $i_{l}$ th subsystem at the switching instant $k_{l}$ . Owing to asynchronous switching, the switching instant of the $i_{l - 1}$ th controller is $k_{l} + Δ_{k_{l}}$ , then there exists mismatched period at time interval $[k_{l}, k_{l} + Δ_{k_{l}})$ , $Δ_{k_{l}} > 0$ (or $[k_{l} + Δ_{k_{l}}, k_{l})$ , $Δ_{k_{l}} < 0$ ). In this period, the controller $K_{i_{l - 1}}$ affects the $i_{l}$ th subsystem (or the controller $K_{i_{l}}$ affects the $i_{l - 1}$ th subsystem).

Remark 11. We consider the case of $Δ_{k_{l}} > 0$ , that is to say, the switching time of the controller is lag of the switching time of the system. Figure 1 illustrates the asynchronous switching mode between the controller and the subsystems. From Figure 1, we can see that the controller $K_{i_{l - 1}}$ of the $i_{l - 1}$ th subsystem affects the $i_{l - 1}$ th subsystem in the matched period $[k_{l - 1} + Δ_{k_{l - 1}}, k_{l})$ and affects the $i_{l}$ th subsystem in the mismatched period $[k_{l}, k_{l} + Δ_{k_{l}})$ .

Based on Lemmas 2 and 3, the following theorem presents a sufficient condition for the existence of a stabilizing switching law for the closed-loop system (15) under asynchronous switching.

Theorem 2.For any $i, j \in \underline{N}$ , $i \neq j$ , if there exist positive definite matrices $P_{i} > 0$ , $P_{ij} > 0$ , and constant scalars $ξ_{1} \geq 1$ , $ξ_{2} \geq 1$ , such that (38) and (42) hold, then the closed-loop system (15) is finite-time stochastically stable with respect to $(c_{1}, c_{2}, M, σ (k))$ under the switching signal $σ (k)$ with the following condition

\begin{matrix} τ_{a} > τ_{a}^{*} = \\ \frac{M \ln μ_{1} μ_{2}}{\ln (\frac{μ_{2} c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i, j \in} {λ_{min} (P_{i}), λ_{min} (P_{ij})}}{sup_{i, j \in} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}) - (M - Δ_{[0, M)}) \ln ξ_{2} - Δ_{[0, M)} \ln ξ_{1}} \end{matrix}

(44)

where $μ_{1}, μ_{2} \geq 1$ , $P_{i} \leq μ_{1} P_{ij}, P_{ij} \leq μ_{2} P_{i}$ , $M$ is finite control interval of the system and $Δ_{[0, M)}$ denotes the total mismatched control time during the interval $[0, M)$ .

Proof. Here, we only discuss the situation of $Δ_{k_{l}} > 0$ . For $Δ_{k_{l}} < 0$ , the proof method is similar, and we can reach the same conclusion.

For any given $M \geq 1$ , we let $0 = k_{0} < k_{1} < k_{2} < \dots < k_{N_{σ} (0, M)} = K < M$ denote the switching instants of the switching signal $σ (k)$ over the interval $[0, M)$ . As mentioned earlier, $σ (k) = i_{l}$ when $k \in [k_{l}, k_{l + 1})$ . Choose the following piecewise Lyapunov-like function for system (15)

\begin{matrix} V (x (k)) = \\ {\begin{matrix} V_{i_{l}} (x (k)), \forall k \in [k_{l} + Δ_{k_{l}}, k_{l + 1}), l = 0, 1, \dots, N_{σ} (0, M) - 1 \\ V_{i_{l} i_{l - 1}} (x (k)), \forall k \in [k_{l}, k_{l} + Δ_{k_{l}}), l = 0, 1, \dots, N_{σ} (0, M) - 1 \end{matrix} \end{matrix}

(45)

where $V_{i_{l}} (x (k))$ and $V_{i_{l} i_{l - 1}} (x (k))$ follow the same definition as those in (41) and (37), respectively, except the subscript $i$ , $j$ are instead of $i_{l}$ , $i_{l - 1}$ . Then, it holds from (39), (43) and (45) that

\begin{matrix} E {V (x (k))} \leq \\ {\begin{matrix} {ξ_{2}}^{k - k_{l} - Δ_{k_{l}}} E {V_{i_{l}} (x (k_{l} + Δ_{k_{l}}))}, \forall k \in [k_{l} + Δ_{k_{l}}, k_{l + 1}), l = 0, 1, \dots, N_{σ} (0, M) - 1 \\ {ξ_{1}}^{k - k_{l}} E {V_{i_{l} i_{l - 1}} (x (k_{l}))}, \forall k \in [k_{l}, k_{l} + Δ_{k_{l}}), l = 0, 1, \dots, N_{σ} (0, M) - 1 \end{matrix} \end{matrix}

(46)

Moreover, it can be easily deduced that $\forall (i_{l - 1}, i_{l}) \in \underline{N} \times \underline{N}$ , $i_{l - 1} \neq i_{l}$

\begin{matrix} V_{i_{l}} (x (k_{l} + Δ_{k_{l}})) \leq μ_{1} V_{i_{l} i_{l - 1}} (x (k_{l} + Δ_{k_{l}})), \\ V_{i_{l} i_{l - 1}} (x (k_{l})) \leq μ_{2} V_{i_{l}} (x (k_{l})) \end{matrix}

(47)

When $k \in [k_{l} + Δ_{k_{l}}, k_{l + 1})$ , it follows from (46), (47) and Definition 1 that

\begin{matrix} E {V (x (k))} \leq {ξ_{2}}^{k - (k_{N_{σ} (0, K) - 1} + Δ_{k_{N_{σ} (0, K) - 1}})} E {V (k_{N_{σ} (0, K) - 1} + Δ_{k_{N_{σ} (0, K) - 1}})} \\ \leq {ξ_{2}}^{k - (k_{N_{σ} (0, K) - 1} + Δ_{k_{N_{σ} (0, K) - 1}})} μ_{1} {ξ_{1}}^{Δ_{k_{N_{σ} (0, K) - 1}}} E {V (k_{N_{σ} (0, K) - 1})} \\ \leq μ_{1} {ξ_{2}}^{k - (k_{N_{σ} (0, K) - 1} + Δ_{k_{N_{σ} (0, K) - 1}})} {ξ_{1}}^{Δ_{k_{N_{σ} (0, K) - 1}}} μ_{2} {ξ_{2}}^{k_{N_{σ} (0, K) - 1} - (k_{N_{σ} (0, K) - 2} + Δ_{k_{N_{σ} (0, K) - 2}})} E {V (k_{N_{σ} (0, K) - 2} + Δ_{k_{N_{σ} (0, K) - 2}})} \\ \leq μ_{1} μ_{2} {ξ_{2}}^{k - (k_{N_{σ} (0, K) - 2} + Δ_{k_{N_{σ} (0, K) - 1}} + Δ_{k_{N_{σ} (0, K) - 2}})} {ξ_{1}}^{Δ_{k_{N_{σ} (0, K) - 1}}} μ_{1} {ξ_{1}}^{Δ_{k_{N_{σ} (0, K) - 2}}} E {V (k_{N_{σ} (0, K) - 2})} \\ \leq {(μ_{1} μ_{2})}^{2} {ξ_{2}}^{k - (k_{N_{σ} (0, K) - 3} + Δ_{k_{N_{σ} (0, K) - 1}} + Δ_{k_{N_{σ} (0, K) - 2}} + Δ_{k_{N_{σ} (0, K) - 3}})} {ξ_{1}}^{Δ_{k_{N_{σ} (0, K) - 1}} + Δ_{k_{N_{σ} (0, K) - 2}}} E {V (k_{N_{σ} (0, K) - 3} + Δ_{k_{N_{σ} (0, K) - 3}})} \\ \leq \dots \\ \leq μ_{1} {(μ_{1} μ_{2})}^{N_{σ} (0, K) - 1} {ξ_{2}}^{k - (Δ_{k_{N_{σ} (0, K) - 1}} + Δ_{k_{N_{σ} (0, K) - 2}} + \dots + Δ_{0})} {ξ_{1}}^{Δ_{k_{N_{σ} (0, K) - 1}} + Δ_{k_{N_{σ} (0, K) - 2}} + \dots + Δ_{0}} E {V (0)} \\ \leq {μ_{2}}^{- 1} {(μ_{1} μ_{2})}^{N_{σ} (0, M)} {ξ_{2}}^{M - Δ_{[0, M)}} {ξ_{1}}^{Δ_{[0, M)}} E {V (0)} \end{matrix} (48)

(48)

From (45), for $\forall i, j \in \underline{N}$ , $i \neq j$ , we have

V (x (k)) \geq inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})} ‖ x (k) ‖^{2}

(49)

On the other hand

V (x (0)) \leq sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})} ‖ x (0) ‖^{2}

(50)

using the fact

‖ x (0) ‖ \leq c_{1}

(51)

we obtain

V (x (0)) \leq sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})} c_{1}^{2}

(52)

Combining (48)–(52), the following inequality can be derived

\begin{matrix} E {{‖ x (k) ‖}^{2}} \leq {μ_{2}}^{- 1} {(μ_{1} μ_{2})}^{N_{σ} (0, M)} {ξ_{2}}^{M - Δ_{[0, M)}} {ξ_{1}}^{Δ_{[0, M)}} \\ \frac{sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}{inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})}} c_{1}^{2} \end{matrix}

(53)

From Definition 2, we have

N_{σ} (0, M) \leq \frac{M}{τ_{a}}

(54)

From (44) and (54), we obtain

\begin{matrix} {μ_{2}}^{- 1} {(μ_{1} μ_{2})}^{N_{σ} (0, M)} {ξ_{2}}^{M - Δ_{[0, M)}} {ξ_{1}}^{Δ_{[0, M)}} \\ \frac{sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}{inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})}} c_{1}^{2} < c_{2}^{2} \end{matrix}

(55)

According to (53) and (55), we have

E {‖ x (k) ‖} < c_{2}

(56)

The proof is completed.

Remark 12. As mentioned in Theorem 1, finite-time stochastic stability of the whole system has no strict requirement on stability of subsystems, finite-time stochastic stability of switched stochastic system under the asynchronous switching also does not request the closed-loop subsystem is stable in the matched control period or mismatched control period.

Remark 13. Theorem 2 gives the design method of a stabilizing switching law for the closed-loop system (15), and the total mismatched control period $Δ_{[0, M)}$ needs to be pre-specified in order to obtain the average dwell time $τ_{a}$ . However, in practical engineering, it is difficult to obtain the mismatched control period $Δ_{[0, M)}$ before designing the controller. In order to make the problem be computable, the switching signal $σ (k)$ with the following conditions can guarantee the closed-loop system (15) is finite-time stochastically stable.

Con d_{1} : \frac{Δ_{[0, M)}}{M} \leq \frac{1}{1 + \log_{(\bar{ξ} / ξ_{1})} (ξ_{2} / \bar{ξ})}

(57)

Con d_{2} : τ_{a} > τ_{a}^{*} = \frac{M \ln μ_{1} μ_{2}}{\ln (\frac{μ_{2} c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i, j \in} {λ_{min} (P_{i}), λ_{min} (P_{ij})}}{sup_{i, j \in} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}) - M \ln \bar{ξ}}

(58)

where $ξ_{1} < \bar{ξ} < ξ_{2}$ . $Con d_{1}$ restricts the mismatched control rate of the switched controllers in the system. Then by restricting the upper bound of the ratio between the total mismatched control period and the overall time, we can easily compute the average dwell time $τ_{a}$ . The problem that $Δ_{[0, M)}$ needs to be pre-specified can be avoided although $Con d_{1}$ may lead to some conservatism.

On the other hand, when $ξ_{1} = ξ_{2} = ξ$ , (44) becomes

τ_{a} > τ_{a}^{*} = \frac{M \ln μ_{1} μ_{2}}{\ln (\frac{μ_{2} c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})}}{sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}) - M \ln ξ}

(59)

Here the average dwell time $τ_{a}$ is computable via imposing restriction on parameters $ξ_{1}, ξ_{2}$ . It is worth pointing out that the two methods make the average dwell time $τ_{a}$ be computable at the cost of increased conservatism.

Remark 14. In fact, (44) in Theorem 2 implies that if switching sequence $σ (k) : {(k_{0}, σ (k_{0})), (k_{1}, σ (k_{1})), \dots, (k_{l}, σ (k_{l}), \dots)}$ of the system can be pre-specified, i.e. $τ_{a}$ is a known constant, the matched period $M - Δ_{[0, M)}$ and the mismatched period $Δ_{[0, M)}$ should satisfy the following relation

ξ_{1}^{Δ_{[0, M)}} ξ_{2}^{M - Δ_{[0, M)}} < \frac{μ_{2} c_{2}^{2}}{{(μ_{1} μ_{2})}^{M / τ_{a}} c_{1}^{2}} • \frac{inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})}}{sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}

(60)

Moreover, if $ξ_{1} > ξ_{2}$ , we can obtain the constraint condition of mismatched control rate from (60)

\begin{matrix} \frac{Δ_{[0, M)}}{M} < \log_{ξ_{1} / ξ_{1} ξ_{2} ξ_{2}} \\ {{[\frac{μ_{2} c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})}}{sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}]}^{1 / M} • \frac{1}{ξ_{2} {(μ_{1} μ_{2})}^{1 / τ_{a}}}} \end{matrix}

(61)

From (44) and (61), we know that when the switching sequence is unknown, the mismatched control rate can be designed freely to guarantee the finite-time stochastic stability of the system by the asynchronously switched controller. But if switching sequence of the system is pre-specified, the mismatched control rate may need to be limited. On the other hand, for the average dwell time scheme with finite-time stochastic stability, we can pre-determine one value among two parameters of $τ_{a}$ and $\frac{Δ_{[0, M)}}{M}$ , then the other value can be determined by (44).

Remark 15. Theorem 2 shows that within a relatively small mismatched control rate and a relatively large average dwell time, the closed-loop system (15) can be finite-time stochastically stable despite the fact that all the closed-loop subsystems may be unstable.

Remark 16. In fact, (53) shows that the longer the finite time period is, the larger the upper bound of $E {{‖ x (k) ‖}^{2}}$ will be. It should be pointed out that if $M$ becomes large, other relevant parameters, such as $ξ_{1}$ , $ξ_{2}$ , $μ_{1}$ , $μ_{2}$ , etc., have to vary to guarantee that ${μ_{2}}^{- 1} {(μ_{1} μ_{2})}^{N_{σ} (0, M)} {ξ_{2}}^{M - Δ_{[0, M)}} {ξ_{1}}^{Δ_{[0, M)}} \frac{sup_{i, j \in \underline{N}} {λ_{max} (P_{i}), λ_{max} (P_{ij})}}{inf_{i, j \in \underline{N}} {λ_{min} (P_{i}), λ_{min} (P_{ij})}} c_{1}^{2}$ is sufficiently small such that $E {{‖ x (k) ‖}^{2}}$ is no larger than $c_{2}^{2}$ .

Now, we are in a position to present the result on the existence of state-feedback controllers for the discrete-time switched stochastic system (1) under asynchronous switching via Theorem 2.

Theorem 3.For any $i, j \in \underline{N}$ , $i \neq j$ , if there exist positive definite matrices $X_{i} > 0$ , $X_{ij} > 0$ , matrices $Y_{i}$ constant scalars $ξ_{1}, ξ_{2}, μ_{1}, μ_{2} \geq 1$ , such that

X_{ij} \leq μ_{1} X_{i}, X_{i} \leq μ_{2} X_{ij}

(62)

[\begin{matrix} - X_{i} & {(A_{i} X_{i} + B_{i} Y_{i})}^{T} & {(C_{i} X_{i} + D_{i} Y_{i})}^{T} \\ A_{i} X_{i} + B_{i} Y_{i} & - ξ_{2} X_{i} & 0 \\ C_{i} X_{i} + D_{i} Y_{i} & 0 & - (ξ_{2} / α) X_{i} \end{matrix}] < 0

(63)

[\begin{matrix} - X_{ij} & X_{ij} {(A_{i} + B_{i} Y_{j} X_{j}^{- 1})}^{T} & X_{ij} {(C_{i} + D_{i} Y_{j} X_{j}^{- 1})}^{T} \\ (A_{i} + B_{i} Y_{j} X_{j}^{- 1}) X_{ij} & - ξ_{1} X_{ij} & 0 \\ (C_{i} + D_{i} Y_{j} X_{j}^{- 1}) X_{ij} & 0 & - (ξ_{1} / α) X_{ij} \end{matrix}] < 0

(64)

hold, then under asynchronously switched controller $u (k) = K_{σ' (k)} x (k)$ , $K_{i} = Y_{i} X_{i}^{- 1}$ , the system (1) is finite-time stochastically stabilizable with respect to $(c_{1}, c_{2}, M, σ (k))$ for any switching signal $σ (k)$ satisfying

\begin{matrix} τ_{a} > τ_{a}^{*} = \\ \frac{M \ln μ_{1} μ_{2}}{\ln (\frac{μ_{2} c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i, j \in \underline{N}} {λ_{min} (X_{i}^{- 1}), λ_{min} (X_{ij}^{- 1})}}{sup_{i, j \in \underline{N}} {λ_{max} (X_{i}^{- 1}), λ_{max} (X_{ij}^{- 1})}}) - (M - Δ_{[0, M)}) \ln ξ_{2} - Δ_{[0, M)} \ln ξ_{1}} \end{matrix}

(65)

Proof. Setting $X_{i} = P_{i}^{- 1}$ and $Y_{i} = K_{i} P_{i}^{- 1}$ , and substituting them to (63), we have

[\begin{matrix} - P_{i}^{- 1} & {(A_{i} P_{i}^{- 1} + B_{i} K_{i} P_{i}^{- 1})}^{T} & {(C_{i} P_{i}^{- 1} + D_{i} K_{i} P_{i}^{- 1})}^{T} \\ A_{i} P_{i}^{- 1} + B_{i} K_{i} P_{i}^{- 1} & - ξ_{2} P_{i}^{- 1} & 0 \\ C_{i} P_{i}^{- 1} + D_{i} K_{i} P_{i}^{- 1} & 0 & - (ξ_{2} / α) P_{i}^{- 1} \end{matrix}] < 0

(66)

Using $diag {P_{i}, P_{i}, P_{i}}$ to pre- and post-multiply (66), respectively, it follows that (42). Setting $X_{ij} = P_{ij}^{- 1}$ and $Y_{j} = K_{j} P_{j}^{- 1}$ , similarly, we can also obtain that (64) is equivalent to (38). In addition, $X_{ij} \leq μ_{1} X_{i}, X_{i} \leq μ_{2} X_{ij}$ ensures $P_{i} \leq μ_{1} P_{ij}, P_{ij} \leq μ_{2} P_{i}$ . Therefore, by Theorem 2, the desired result follows immediately. The proof is thus completed.

Remark 17. It is noticed that the matrix inequalities (62), (63) and (64) are coupled. Therefore, we can firstly solve the LMI (63) to obtain the solution of matrices $X_{i}$ and $Y_{i}$ . Then we solve the matrix inequality (62), (64) by substituting $X_{i}$ and $Y_{i}$ into them. By adjusting the parameters $μ_{1}$ , $μ_{2}$ , $ξ_{1}$ and $ξ_{2}$ appropriately, we seek the feasible solution $X_{ij}$ such that the matrix inequalities (62) and (64) hold. If the chosen parameters $μ_{1}$ , $μ_{2}$ , $ξ_{1}$ and $ξ_{2}$ make (62) and (64) have no feasible solution, we can adjust $μ_{1}$ , $μ_{2}$ , $ξ_{1}$ and $ξ_{2}$ to be larger. Following this guideline, the solution to the matrix inequalities (62) to (64) will be found. By this method, one can avoid to solve bilinear matrix in equalities (BMIs) of (64) which will bring some complexity on computation.

If there is no asynchronous switching in system (1), i.e. $Δ_{k_{l}} = 0$ , then Theorem 3 reduces to the following corollary.

Corollary 2. For any $i, j \in \underline{N}$ , $i \neq j$ , if there exist positive definite matrices $X_{i} > 0$ , and a constant scalar $μ \geq 1$ , such that (63) holds, then under asynchronously switched controller $u (k) = K_{σ' (k)} x (k)$ , $K_{i} = Y_{i} X_{i}^{- 1}$ , the system (1) is finite-time stochastically stabilizable with respect to $(c_{1}, c_{2}, M, σ (k))$ for any switching signal $σ (k)$ satisfying with the average dwell time

τ_{a} > τ_{a}^{*} = \frac{M \ln μ}{\ln (\frac{c_{2}^{2}}{c_{1}^{2}} • \frac{inf_{i \in \underline{N}} {λ_{min} (X_{i}^{- 1})}}{sup_{i \in \underline{N}} {λ_{max} (X_{i}^{- 1})}}) - M \ln ξ_{2}}

(67)

where $μ \geq 1$ , $X_{j} \leq μ X_{i}$ .

Numerical examples

In this section, two examples are given to show the effectiveness of the proposed method.

Example 1 (Stability)

Consider the discrete-time switched stochastic system (16) with $N = 2$ (i.e. two subsystems) and the following parameters:

A_{1} = [\begin{matrix} 1 & - 0.5 \\ 2 & 0 \end{matrix}], A_{2} = [\begin{matrix} 0 & - 2 \\ 0.5 & 1 \end{matrix}], C_{1} = C_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

Suppose that $ω (k)$ is white Gaussian noise with $E {ω (k)} = 0$ , $E {ω {(k)}^{2}} = 1$ , and $x (0) = {(\begin{matrix} 0.5 & - 0.2 \end{matrix})}^{T}$ . Note that $λ_{1, 2} (A_{1}) = λ_{1, 2} (A_{2}) = 0.5 \pm 0.866 i$ , so no subsystem is asymptotically stable. Now we consider the finite-time stochastic stability of the systems. The parameters are given as follows:

c_{1} = 1, c_{2} = 100, M = 6

Then, we apply Theorem 1 and solve corresponding matrix inequalities. Solving (17) and (18) for $μ = 2$ , $β = 3$ leads to feasible solutions

P_{1} = [\begin{matrix} 1.1831 & - 0.2332 \\ - 0.2332 & 0.5136 \end{matrix}], P_{2} = [\begin{matrix} 0.5136 & 0.2332 \\ 0.2332 & 1.1831 \end{matrix}]

According to (19), one obtains $τ_{a}^{*} = 2.6$ . Then, by virtue of Theorem 1, the system (16) is finite-time stochastically stable with respect to $(1, 100, 6, σ (k))$ . Figure 2 shows the state response of the system over 0-6 under a periodic switching signal $σ (k)$ with interval time $τ_{a} = 3$ . From Figure 3, it is easy to see that the system (16) is finite-time stochastically stable.

Example 2 (Stabilizability)

Consider the discrete-time switched stochastic system (1) with $N = 2$ and the following parameters:

Subsystems 1( $S_{1}$ ): $A_{1} = [\begin{matrix} 1 & - 0.5 \\ 2 & 0 \end{matrix}]$ , $B_{1} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}]$ , $C_{1} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , $D_{1} = [\begin{matrix} 1 & 0 \\ 0 & 0.1 \end{matrix}]$

Subsystems 2( $S_{2}$ ): $A_{2} = [\begin{matrix} 0 & - 2 \\ 0.5 & 1 \end{matrix}]$ , $B_{2} = [\begin{matrix} 0.1 & 0 \\ 0 & 0.1 \end{matrix}]$ , $C_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ , $D_{2} = [\begin{matrix} 0.2 & 0 \\ 0 & 1 \end{matrix}]$

Based on Corollary 2, setting $c_{1} = 1$ , $c_{2} = 100$ , $M = 40$ , $μ = 10$ , $ξ_{2} = 1$ , we can obtain the controller gains as follows:

K_{1} = [\begin{matrix} - 1.3835 & 0.1876 \\ - 7.8034 & - 6.2857 \end{matrix}], K_{2} = [\begin{matrix} - 4.7578 & 3.2202 \\ - 0.0526 & - 1.2539 \end{matrix}]

and the average dwell time $τ_{a} > τ_{a}^{*} = 8.49$ . Notice that the results obtained above are for the case of no asynchronous switching (i.e. the synchronous control). However, in fact, asynchronous switching cannot be avoided. Here, we choose the initial condition as $x (0) = {(\begin{matrix} 0.3 & - 0.4 \end{matrix})}^{T}$ , $τ_{a} = 10$ and mismatched period $Δ_{k_{l}} = 1$ , $l = 0, 1, \dots$ . Then the above result still be used to stabilize the switched system, and the simulation result of the state responses of closed-loop switched system is shown in Figure 4 and the corresponding value of $‖ x (k) ‖^{2}$ is presented in Figure 5.

Obviously, it can be seen that $E {{‖ x (k) ‖}^{2}} > c_{2}^{2} = 10^{4}$ , $\forall k \in [20, 40]$ and the system is not finite-time stochastically stable with respect to $(1, 100, 40, σ (k))$ due to ignoring the asynchronous switching during design process.

Now, we consider the asynchronous switching control problem by Theorem 3. For convenient comparison, the parameters of subsystems are still chosen as above. The values of $μ_{1}$ , $μ_{2}$ , $ξ_{1}$ , $ξ_{2}$ , $c_{1}$ , $c_{2}$ and $M$ are given by

c_{1} = 1, c_{2} = 100, M = 40, μ_{1} = μ_{2} = 10, ξ_{1} = ξ_{2} = 1

Based on Theorem 3, we can obtain a set of feasible solutions as follows

X_{1} = [\begin{matrix} 69.4939 & 63.9256 \\ 63.9256 & 142.9840 \end{matrix}], X_{2} = [\begin{matrix} 154.2010 & - 52.2203 \\ - 52.2203 & 50.2685 \end{matrix}]

X_{12} = [\begin{matrix} 90.7283 & 114.4665 \\ 114.4665 & 196.4122 \end{matrix}], X_{21} = [\begin{matrix} 55.5097 & - 45.4227 \\ - 45.4227 & 57.1621 \end{matrix}]

K_{1} = [\begin{matrix} - 1.3865 & 0.1883 \\ - 7.6356 & - 6.4085 \end{matrix}], K_{2} = [\begin{matrix} - 4.8689 & 3.1527 \\ - 0.0459 & - 1.2520 \end{matrix}]

By computation, we can obtain that

\begin{matrix} λ_{1, 2} (A_{1} + B_{1} K_{1}) = 0.1102 \pm 0.1755 i, \\ λ_{1, 2} (A_{2} + B_{2} K_{2}) = 0.1940 \pm 0.6092 i, \end{matrix}

\begin{matrix} λ_{1, 2} (A_{1} + B_{1} K_{2}) = 0.1940 \pm 0.5165 i, \\ λ_{1} (A_{2} + B_{2} K_{1}) = - 0.6540, λ_{2} (A_{2} + B_{2} K_{1}) = 0.8745 \end{matrix}

So it can be checked that not all the closed-loop subsystems in the matched or mismatched period are asymptotically stable. According to (65), the average dwell time satisfies that $τ_{a} > τ_{a}^{*} = 22.2$ . We still set the initial condition as $x (0) = {(\begin{matrix} 0.3 & - 0.4 \end{matrix})}^{T}$ and $Δ_{k_{l}} = 1$ , $l = 0, 1, \dots$ . From Theorem 3, if the switching signal $σ (k)$ is designed as (65), the closed-loop system can be finite-time stochastically stable under asynchronous switching. Fix $τ_{a} = 23$ and denote the closed-loop subsystems $(A_{1} + B_{1} K_{1}, C_{1} + D_{1} K_{1})$ , $(A_{2} + B_{2} K_{2}, C_{2} + D_{2} K_{2})$ , $(A_{1} + B_{1} K_{2}, C_{1} + D_{1} K_{2})$ and $(A_{2} + B_{2} K_{1}, C_{2} + D_{2} K_{1})$ by $S_{1}^{sub}$ , $S_{2}^{sub}$ , $S_{12}^{sub}$ , $S_{21}^{sub}$ , respectively. The switching sequence of the opened-loop system is $S_{1} S_{2} S_{1} S_{2} \dots$ . Thus, based on the aforementioned switching signal $σ' (k)$ , the closed-loop subsystems are activated in the following sequence

S_{1}^{sub} S_{12}^{sub} S_{2}^{sub} S_{21}^{sub} S_{1}^{sub} S_{12}^{sub} \dots

The state response of the resulting closed-loop system is shown in Figure 6 under the given condition. It can be clearly seen from Figure 7 that the designed controllers and the switching law are effective despite asynchronous switching. Thus, it is necessary for practical applications to take asynchronous switching into account.

Conclusions

Motivated by the recent works (Lin et al., 2011; Xiang and Wang, 2009; Wang et al., 2012b; Zong et al., 2013) showing the asynchronous switching phenomenon widely existing in switched systems and the importance of finite-time stochastic stability for switched systems, sufficient conditions for finite-time stochastic stability and stabilization of a class of discrete-time switched stochastic system under asynchronous switching have been developed. In contrast to the conception of the asymptotical mean square stability and the exponential mean square stability for switched stochastic systems, we give the definition of finite-time stochastic stability and finite-time stochastic stabilizability via asynchronous switching for the system in a discrete-time case. All conditions are given in a matrix inequality format and depend on the switching mode of the system. The results are used to deal with systems whose subsystems subject to stochastic property and asynchronous switching control. The usefulness of the obtained results has been demonstrated by two numerical examples. Our future work will focus on extending the proposed finite-time stabilization method to solve finite-time bounded stabilization problem for discrete-time switched stochastic systems with time-varying delays, which may be a more general problem.

Footnotes

Acknowledgements

The authors would like to thank the Associate Editor and all the reviewers for their helpful comments.

Funding

This work is supported by the National Natural Science Foundation of China under grant number 61273120.

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