Stabilization for switched stochastic neutral systems under asynchronous switching
Introduction
Switched systems are an important class of hybrid systems which are composed of several dynamical subsystems and a switching rule that orchestrates the switching among them to ensure stability and satisfied performance. Many physical and engineering systems can be modeled as switched systems, such as chemical processes, power electronic, automatic highway systems [3], [15], [20]. In the past few decades, switched systems have attracted considerable attention. Many remarkable achievements have been made on the issues such as controllability, observability, stability, stabilization, observer design, filtering, and fault detection [1], [4], [7], [14], [16], [17], [18]. Among them, stability analysis and stabilization control problems are the main concerns, and large numbers of excellent results have been published; see, e.g., [4], [12], [13], [17], [21] and the references therein.
As is well known, time delay as a source of instability and poor performance often appears in many dynamical systems, such as neural networks, nuclear reactors, manual control and microwave oscillator systems. Actually, there are many physical processes which can be described by differential equations of neutral type, i.e. neutral systems, where time delay appears both in state and state derivative. In the literature, various analysis techniques have been utilized to study the neutral systems, see for example [2], [11], [27]. However, due to the complicated behavior of switched neutral systems, only a few results on such systems have appeared [8], [26], [28]. On the other hand, stochastic phenomena exist widely in engineering applications and social systems. Stochastic systems have received much attention and many results have been reported [2], [6], [30]. It should be noted that so far the synthesis issue for switched stochastic neutral systems has not been fully investigated.
In the ideal case, the switching of the controllers coincides exactly with that of corresponding subsystems. In engineering application, however, since it inevitably takes some time to identify the active subsystem and apply for the matched controller, the switching time of controllers may lag behind that of practical subsystems, which results in asynchronous switching between the controllers and system modes. The necessities of considering asynchronous switching for efficient controller design have been shown in a class of chemical systems [10]. Some primary studies on the asynchronous switching problems for switched systems have been proposed [5], [23], [24], [25]. Recently, Lyapunov stability theory is used to investigate the problem [9], [22], [29]. In [9], asynchronous H∞ filtering is considered for discrete-time switched systems. [22] analyzes the robust reliable control for a class of uncertain switched neutral systems under asynchronous switching. Sufficient conditions for the stability of such systems are given by two inequalities. In [29], stability, L2-gain and asynchronous H∞ control are considered for discrete-time switched systems.
Based on the above discussion, switched stochastic neutral time-delay systems under asynchronous switching are worth studying. To the best of our knowledge, few results on the stabilization for such systems have been reported, which motivates this study for us. Applying Itô’s differential formula and the Lyapunov stability theory, a sufficient condition with respect to mean-square exponential stability of the given system is obtained and the controllers with asynchronous switching are designed. The presented condition is applicable to the case where the subsystems are unstable during mismatched periods resulted from asynchronous switching between subsystems and the corresponding controllers. Furthermore, the switching signal of the Lyapunov function constructed in the paper is dependent on the controllers’ switching signal, which is convenient for the analysis of the proposed issue.
The rest of this paper is organized as follows. In Section 2, the problem to be studied is formulated and some definitions and a lemma are introduced. In Section 3, based on the average dwell time approach, the controllers are developed to ensure the mean-square exponential stability of the switched stochastic neutral system under asynchronous switching. A numerical example is provided to illustrate the method in Section 4 and we conclude this paper in Section 5.
Notations: Throughout this paper, the notations are standard. The superscript “T” denotes matrix transposition, “∗” the symmetric term in a symmetric matrix, diag{⋯} a block-diagonal matrix and I the identity matrix. stands for the n-dimensional Euclidean space, the expectation operator and ∥ · ∥ the Euclidean norm. P > 0 means that P is real symmetric and positive definite, and λmin(P) (λmax(P)) is the minimum (maximum) eigenvalue of P.
Section snippets
Problem formulation and preliminaries
Consider the following switched stochastic neutral system described by the Itô’s form:where is the state vector, is the control input, ϖ(t) is a one-dimensional Brownian motion satisfying and , the switching signal σ(t):[t0, ∞) → M = {1, 2, … , l} is a piecewise continuous (from the right) function, where l is the number of subsystems. Specifically, denote
Main results
The objective of the paper is to design the controllers such that the closed-loop system is stabilizable under asynchronous switching. In the sequel, applying the average dwell time approach, we give sufficient conditions for the mean-square exponential stability of system (1), and the controllers are designed.
Numerical example
In this section, an example is given to illustrate the effectiveness of the proposed approach. Consider system (1) composed of two subsystems with the following parameters:
Choose α = 0.3, β = 0.3, υ = 0.35, μ = 1.1, h = 0.3, then the average dwell time is . By solving (29)
Conclusions
The stabilization problem for switched stochastic neutral systems under asynchronous switching has been studied. By using the average dwell time approach, delay-dependent sufficient conditions have been given to guarantee the mean-square exponential stability of such system in terms of LMIs. The controllers have been designed. The Lyapunov function constructed in the paper is dependent on the controllers’ switching signal and has been allowed to increase during mismatched periods. Finally,
Acknowledgement
This work is supported in part by the NSF of China under Grants 61004040, 61104114 and 61074096.
References (30)
- et al.
A new result on stability analysis for stochastic neutral systems
Automatica
(2010) - et al.
Sliding mode control for stochastic systems subject to packet losses
Information Sciences
(2012) - et al.
Stability analysis for uncertain switched neutral systems with discrete time-varying delay: a delay-dependent method
Mathematics and Computers in Simulation
(2009) - et al.
Asynchronous H∞ filtering of discrete-time switched systems
Signal Processing
(2012) - et al.
On robust stabilization for neutral delay-differential systems with parametric uncertainties and its application
Applied Mathematics and Computation
(2005) - et al.
Stability and L2-gain analysis for switched delay systems: a delay-dependent method
Automatica
(2006) - et al.
Analysis and synthesis of switched linear control systems
Automatica
(2005) - et al.
Robust control of a class of uncertain nonlinear systems
Systems & Control Letters
(1992) - et al.
Robust reliable stabilization of uncertain switched neutral systems with delayed switching
Applied Mathematics and Computation
(2011) - et al.
Stabilization of switched linear systems with time-delay in detection of switching signal
Journal of Mathematical Analysis and Applications
(2005)
New stability and stabilization for switched neutral control systems
Chaos, Solitons & Fractals
H∞ output tracking control for neutral systems with time-varying delay and nonlinear perturbations
Communications in Nonlinear Science and Numerical Simulation
Exponential stability analysis for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations
Nonlinear Analysis: Hybrid Systems
Stabilization for T–S model based uncertain stochastic systems
Information Sciences
L2 − L∞ filtering for time-delayed switched Hopfield neural networks
International Journal of Innovative Computing, Information and Control
Cited by (41)
Stability analysis of cyclic switched linear systems: An average cycle dwell time approach
2021, Information SciencesAsynchronous stabilization of switched neutral systems: A cooperative stabilizing approach
2019, Nonlinear Analysis: Hybrid SystemsPermutation matrix based robust stability and stabilization for uncertain discrete-time switched TS fuzzy systems with time-varying delays
2016, NeurocomputingCitation Excerpt :A switched system is a hybrid system composed of continuous-time or discrete-time subsystems and a rule that orchestrates the switching between them [1–9]. This class of systems can be found in many areas, such as computer science, mechanical systems, electrical engineering and technology, automotive industry, air traffic control and many other fields [10,11]. Takagi–Sugeno (TS) fuzzy model-based method is widely adopted to investigate the stability analysis and stabilization problems of nonlinear systems [12,13].
Robust stability for switched positive systems with D-perturbation and time-varying delay
2016, Information SciencesSampled-data-based stabilization of switched linear neutral systems
2016, AutomaticaCitation Excerpt :Lots of works are devoted to switched systems in the past two decades, see, for example Branicky (1998), Chen and Zheng (2010), Fu, Ma, and Chai (2015), Lian, Ge, and Han (2013), Liberzon (2003), Lin and Antsaklis (2009), Sun, Du, Shi, Wang, and Wang (2014), Sun and Ge (2005), Sun, Liu, David, and Wang (2008), Sun, Zhao, and Hill (2006), Wu and Dong (2006), Xiang, Sun, and Chen (2012), Zhai, Hu, Yasuda, and Michel (2001), Zhang and Gao (2010), Zhang and Yu (2009), Zhang, Zhuang, and Shi (2015), Zhang, Zhuang, Shi, and Zhu (2015), Zhao and Hill (2008) and Zhao, Shi, and Zhang (2012). Among all problems studied for switched systems, asynchronous switching stemming from the delay between the active subsystem and its matched controller is one important issue (Lian et al., 2013; Wang, Zhao, & Jiang, 2013; Xiang et al., 2012; Zhang & Gao, 2010; Zhao et al., 2012). A majority of existing literature on asynchronous switching focus on either for continuous-time switched systems with continuous controllers or for discrete-time switched systems with discrete-time controllers.