Finite-time stability and asynchronous resilient control for Itô stochastic semi-Markovian jump systems

https://doi.org/10.1016/j.jfranklin.2022.01.004Get rights and content

Abstract

In this paper, the finite-time stability and asynchronous resilient control for a class of Itô stochastic semi-Markov jump systems are studied. Firstly, the sufficient conditions of the finite-time stability for stochastic semi-Markovian jump systems are given. Secondly, the state feedback and observer-based finite-time asynchronous resilient controllers are designed. By multiple Lyapunov functions approach, the sufficient conditions for the existence of these two types of controllers which make the system stochastically stabilizable in finite time are given. Compared with nonresilient case, the existence of the resilient controller can eliminate the influence of the uncertainties and get better results. Finally, a numerical example is given to verify the effectiveness of our results.

Introduction

In recent years, due to the needs of practical problems, the research of stochastic systems has developed rapidly and a series of excellent results have been achieved [1], [2]. With the development of stochastic theory, stochastic switching systems have also been extensively used in actual engineering, such as fault detection [3]. This kind of system can be described as a hybrid system, which consists of a series of discrete or continuous subsystems and the rules to coordinate the switching between these subsystems. Markovian jump systems (MJSs), as a special kind of stochastic switching systems, have been widely concerned by scholars. At present, some achievements have been obtained in stability analysis [4], [5], [6], controller design [7], [8], [9], [10] and so on. However, from the perspective of practical application, semi-Markovian jump systems (S-MJSs) can be used to simulate more complex systems than MJSs [11], [12], [13], [14], [15]. Unlike MJSs, the sojourn-time (ST) of S-MJSs is not limited to the exponential distribution, it can also satisfy other random distributions such as Weibull distribution. Therefore, the transition probability of S-MJSs can be time-varying, which means that S-MJSs have a wider practical application space. For example, in multi-bus systems [16] and electromechanical systems [17], S-MJSs are more suitable to simulate such systems than MJSs.

As we all know, stability is one of the prerequisites for the normal operation of the systems. Currently, a lot of results have been made on Lyapunov asymptotic stability [18], [19], [20], which studied the asymptotic behavior of the systems when time tends to infinity. However, in many practical systems, the behavior of the system in a short finite time interval is an important research objective, which is not considered by Lyapunov asymptotic stability. For example, reference [21] shows that the current of power system must be limited within a certain range to prevent system damage. Therefore, for the sake of describing the phenomenon accurately, the definition of finite-time stability (FTS) was proposed in Kamenkov [22], Lebedev [23,24] to guarantee that E[x(t)Rx(t)] of the system does not exceed a predetermined domain in a finite time under the given initial conditions E[x(0)Rx(0)]. So far, a series of achievements have been made in FTS [25], [26], [27], [28], [29], but there is still a lot of research space in the field of FTS for S-MJSs.

In the controller design of switched systems, due to the delay of switching signals detection, there will inevitably be a mismatch between the controller and the systems, i.e., the switching of the controller is ahead of or lagging behind the switching of the systems. Moreover, in MJSs or S-MJSs, there even exists the problem of system mode uncertainty, which makes it more difficult to design suitable controller. Therefore, asynchronous control is an indispensable control method. So far, some results on asynchronous control have been studied [30], [31], [32], [33]. On the other hand, problems such as parameter uncertainties and nonlinearity often occur in engineering. These problems will result in the instability of the practical systems, and may also degrade the performance of the systems. Hence, the researches on the above factors have also attracted the attention of many scholars [34], [35]. In the control process, in order to avoid these problems caused by accidents, resilient controller is considered. The advantage of resilient control is that the designed controller has robust response to unexpected errors and uncertainties. However, as far as we know, there are few studies on asynchronous resilient control for Itô stochastic S-MJSs.

The main purpose of this paper is to study the stochastic FTS and design state feedback and observer-based asynchronous resilient controllers to make S-MJSs stochastically stabilizable in finite time. ST of S-MJSs obeys general distribution, which enables S-MJSs to simulate more complex systems, such as DNA analysis. In addition, the existence of asynchronous resilient controller can solve the asynchronous phenomenon and eliminate the error caused by system uncertainties, simultaneously. Based on the above discussion, the problems of FTS and asynchronous resilient control for Itô stochastic S-MJSs are proposed. The contributions are listed as follows:

  • 1)

    There exists asynchronous phenomenon between Itô stochastic S-MJSs and the controllers. In order to solve this problem, the stochastic systems and controllers are connected by a conditional probability, which is applied to the design schemes of the state feedback and observer-based asynchronous controllers.

  • 2)

    On the basis of these two kinds of controllers, the resilient control is considered simultaneously to eliminate the influence of system uncertainties and reduce some unexpected errors. Compared with nonresilient case, the control effect of resilient controller is better.

  • 3)

    The sufficient conditions of FTS for S-MJSs are given by choosing a semi-Markov-based Lyapunov function. Moreover, the sufficient conditions for the existence of two kinds of controllers which make S-MJSs stochastically stabilizable in finite time are given.

The rest of this paper is organized as follows: Section 2 proposes the research system and some preparatory work. In Section 3, FTS of stochastic S-MJSs is considered. Section 4 and Section 5 discuss the state feedback asynchronous resilient control and observer-based asynchronous resilient control, respectively. In Section 6, an example is utilized to illustrate and compare the results. The conclusion is drawn in Section 7.

Notations: G: the transpose of a matrix G. G0(G>0): G is a positive semi-definite(positive definite) matrix. λmax(G)(λmin(G)): the maximal(minimal) eigenvalue of a matrix G. E[·]: the mathematical operator. sym(A): A+A. “*”: the term for symmetrical part in a matrix. diag{}: a block-diagonal matrix. “w.r.t.”: the abbreviation of “with respect to”.

Section snippets

System description and preliminaries

Consider the following stochastic S-MJS{dx(t)=[A˜1(υt)x(t)+B1(υt)u(t)]dt+[A˜2(υt)x(t)+B2(υt)u(t)]dw(t),y(t)=F(υt)x(t),x(0)=x0Rn,where A˜1(υt)=A1(υt)+ΔA1(υt),A˜2(υt)=A2(υt)+ΔA2(υt) and A˜1(υt), A˜2(υt), A1(υt), A2(υt), B1(υt), B2(υt) and F(υt) are constant matrices, which are abbreviated as A˜1p, A˜2p, A1p, A2p, B1p, B2p and Fp when υt=p. ΔA1(υt) and ΔA2(υt) are uncertain matrices. x(t)Rn is the state, u(t)Rl is the control input and y(t)Rm is the measured output. x0 is the initial state. w(t

Stochastic finite-time stability

Consider the following stochastic S-MJS{dx(t)=A˜1(υt)x(t)dt+A˜2(υt)x(t)dw(t),x(0)=x0Rn,Now, the definition of stochastic FTS is given.

Definition 1

The autonomous system Eq. (2) is said to be stochastic FTS w.r.t. (d1,d2,T,R) if there are positive scalars d1, d2, T and matrix R>0 such thatx0Rx0d1E[x(t)Rx(t)]<d2,t[0,T],where d1<d2.

In the following lemma, the sufficient conditions of FTS for stochastic S-MJS Eq. (2) are given.

Lemma 3

Given a positive constant φ, the system Eq. (2) is stochastic FTS w.r.t. (d1,d2

State feedback asynchronous resilient control

In this section, we will study the state feedback asynchronous resilient control. In view of the fact that it is difficult or impossible to access the system modes accurately in real time, the concept of asynchronous control is proposed. Moreover, resilient control is used to eliminate the impact of the system uncertainties. In order to represent the asynchronous phenomenon of the controller, σt is used to denote a stochastic process taking values in M={1,2,,S}, and it is related to υt in the

Observer-based asynchronous resilient control

Because some states are not available, we attempt to design an observer-based asynchronous resilient controller as follows{dx^(t)=[A1(σt)x^(t)+B1(σt)u(t)+L(σt)(y(t)y^(t))]dt,y^(t)=F(σt)x^(t),u(t)=K(σt)x^(t),x^(0)=x^0,where L(σt)=L(σt)+ΔL(σt) is the resilient estimator gain, and ΔL(σt)=C3(σt)Γ(σt,t)D4(σt). Combining Eq. (34) and system Eq. (1), we can getdx˜(t)=Mpπx˜(t)dt+Npπx˜(t)dw(t),where Mpπ=A1(υt)+B1(υt)K1(σt)F(υt), Npπ=B2(υt)K2(σt), andx˜(t)=[x(t)x^(t)],A1(υt)=[A˜1(υt)B1(υt)F(υt)000],B1(υ

Example

In this section, an example is given to illustrate the effectiveness of the results and the superiority of the method.

Given d1 = 2, d2 = 6, ε = 3, ε¯ = 10, T = 0.3s, R = I2×2, Γ1(t) = Γ2(t) = sin(t), x(0) = [0.950.95], x^(0) = [0.90.9], φ = ψ = 3. The coefficients of the two-mode S-MJS Eq. (1) are shown in Table 1. In addition, the deterministic parameters used in dealing with uncertain parameters are shown in Table 2.

Given the following transition rate matrices[ρ̲pq]2×2=[0.70.70.80.8],[ρ¯

Conclusion

In this paper, FTS and the asynchronous resilient control for Itô stochastic S-MJS have been studied. The sufficient conditions of stochastic FTS for S-MJS have been given, and the state feedback and observer-based asynchronous resilient controllers have been designed to make the system stochastically stabilizable in finite time. Through the comparison of the resilient controller and the nonresilient controller, we can find that the existence of the resilient controller makes the system have a

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61973198 and SDUST Research Fund (No.2015TDJH105) in part by the Research Fund for the Taishan Scholar Project of Shandong Province of China.

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