Event-Triggered Finite-Time H ∞ Control for Switched Stochastic Systems

This paper investigates the problem of event-triggered finite-time H ∞ control for a class of switched stochastic systems. The main objective of this study is to design an event-triggered state feedback H ∞ controller such that the resulting closed-loop system is finite-time bounded and satisfies a prescribed H ∞ level in some given finite-time interval. Based on stochastic differential equations theory and average dwell time approach, sufficient conditions are derived to ensure the finite-time stochastic stability with the prescribed H ∞ performance for the relevant closed-loop system by employing the linear matrix inequality technique. Finally, the desired state feedback H ∞ controller gain matrices can be expressed in an explicit form.


Introduction
In the last few decades, switched systems have attracted much attention in the field of control systems [1] [2]. This is mainly due to the fact that switched system is an important subclass of hybrid systems and has found many practical and broad applications [3] [4] [5] [6]. A switched system is composed of a family of interconnected subsystems, featured with continuous and discrete-time dynamics, appropriately described by differential or difference equations, respectively, along with a switching law governing the switching among the subsystems.
Many practical systems exist that can be well modeled as switched systems, which motivated a large number of researchers to investigate it widely. Quantities of important conclusions have been developed in the literature [7]- [12]. Sta-switched stochastic systems. The main contributions can be summarized as follows. The coupling between the switching signals and triggered signals is analyzed. A novel framework of finite-time stability for augmented closed-loop switched stochastic system is established. The sufficient condition for eventtriggered finite-time H ∞ controller of switched stochastic systems is obtained by adopting the average dwell time technique and multiple Lyapunov-Krasovskii functional method with LMIs. The design of controller parameters are presented which can guarantee the mentioned system is finite-time bounded and satisfies a weighted H ∞ disturbance attenuation performance, which can avoid some unnecessary data transmission.
The rest of this paper is arranged as follows. In Section 2, the problem formulation and necessary preliminaries are presented. We give a sufficient condition for finite-time ETC of the mentioned augmented system in terms of LMIs in Section 3. Moreover, a designing approach of an event-triggered finite-time H ∞ controller is presented. Finally, some conclusions are summarized in Section 4.
Notation: The notations used in this paper are quite standard. n R stands for the n-dimensional Euclidean space. The notation X Y where X and Y are real symmetric matrices) means that the matrix X Y The symbol * in a matrix denotes a term that is defined by symmetry of the matrix.

Problem Formulation and Preliminaries
Consider the following continuous-time switched stochastic system: are state vector, control input vector, external disturbance, and controlled output respectively, where ( ) v t satisfies the constraint condition with respect to the finite-time interval are the sub-controller gains, ( ) k x t is the current sampled system state, 1 k t + is the next sampled instant, which can be determined by the event-trigger, is chosen as the initial sampled state. In this paper, the event-triggering schemes are described by µ is a constant and Ω is a symmetric and positive definite matrix with appropriate dimension to be determined.

t e x t x t
µ Ω = Ω is satisfied, the sampler will be triggered to sample the system state immediately. Then the sampled data is transmitted to the subcontroller for calculating the control input which will be further used by the subsystem.
Remark 2.3 It should be pointed out that the parameter µ has great influence on the event-trigger instants, i.e. different values of µ correspond to different event-trigger frequencies. The less µ is selected, the shorter the event-trigger period is. Hence, µ should be selected in accordance with the specific control requirement and control capacity.
There is a number 0 s τ > such that any two switches are separated by at least s τ to evade zeno phenomena, which means 1 q q s r r τ + − ≥ for any 0 q > [32].
Substituting the state-feedback controllers (4) into (1) (2), the event-triggered switched stochastic closed-loop system is obtained for We introduce the following definitions and lemmas, which will be useful in the succeeding discussion.
Definition 2.1 (Average dwell time [33]) Given time instants t and T such is called the chatter bound), then a τ is called an average dwell time.
Definition 2.2 (Finite-time stochastic stabilizable [34]) The system (7) (8) with event-triggered control input (4) is said to be finite-time stochastic stable if for a given time-constant 0 T > , the following relation holds: 2) Under the zero initial condition, there is where the prescribed value γ is the attenuation level.
Lemma 2.1 [35] For any real matrices , X Y with appropriate dimensions and a positive scalar 0 ε > , one has , the following three conditions are equivalent to each other: 2) 11 0 φ < , and

Main Results
In this section, we focus on the finite-time stabilization of the switched stochas-Journal of Applied Mathematics and Physics tic system (7) (8) with event-triggered control input (4), and some sufficient conditions which can ensure the switched stochastic system (7) (8) such that the following LMIs hold 0 Θ < Then, under the event-triggering strategy (5), the event-triggered state-feedback controllers (4) and any switching signal with the average dwell time satisfying the switched closed-loop stochastic system (7) (8) By Itô formula, define a weak infinitesimal operator L, then, the stochastic derivative of ( ) We have the weak infinitesimal operator of ( ) ( ) i LV x t as follows:

t P A x t B K e t D v t A x t B K e t D v t Px t A x t B K e t D v t P A x t B K e t D v t
The relationship between the switching instants and event-triggered instants will be discussed as the following two cases.  (15) and (21), there is Integrate both sides of the inequality (23) from q r to t, and obtain   (24) can also be similarly received respectively. Then, the following inequalities can be established.
On the other hand, it can be derived from (14) Taking (27) (28) (29) and (17) Therefore, the H ∞ control performance is obtained from Definition 2.3.
This completes the proof.