H∞ Stabilization of Discrete-Time Nonlinear Semi-Markov Jump Singularly Perturbed Systems With Partially Known Semi-Markov Kernel Information

Abstract

In this paper, the H ∞ stabilization problem is studied for discrete-time semi-Markov jump singularly perturbed systems (SMJSPSs) with repeated scalar nonlinearities. As the exact statistical information of the sojourn time or the mode transition is difficult to obtain, the case with only partial semi-Markov kernel information available is considered. Furthermore, introducing an external disturbance or nonlinearity into the analysis of discrete-time semi-Markov jump systems (DTSMJSs) meets critical obstacles, since the relation between the system state vectors at two nonadjacent instants is difficult to determine. To address this issue, the variation trend of the Lyapunov function for a semi-Markov jump sequence is analyzed in detail. Subsequently, criteria of mean-square exponential stability (MSES) for DTSMJSs are established for the first time based on the Lyapunov stability theory. By virtue of the criteria obtained and the cone complementary linearization algorithm, a controller ensuring MSES and H ∞ performance for discrete-time nonlinear SMJSPSs is constructed. Finally, the effectiveness and applicability of the proposed method are validated by simulation examples including an inverted pendulum model.

Asynchronous Stabilization for Two Classes of Stochastic Switching Systems with Applications on Servo Motors

“…We have only considered the stabilization problem for semi-Markov jump systems. However, the proposed method can be extended to solve the controller design problem under performance constraints, such as mixed and passivity [ 3 , 4 ]. Nevertheless, note that the state equation of the semi-Markov jump system needs to be iterated from time instant k to time to describe the relationship between Lyapunov functions at time instant k to time .…”

“…Switched systems, as a special class of hybrid systems, have received considerable attention in the past few years [ 1 , 2 ]. It usually contains a family of subsystems and a switching law that coordinates between them [ 3 ]. A variety of physical systems, such as mechanical systems [ 4 , 5 ] and network control systems [ 6 , 7 , 8 , 9 ], can be well modeled by the switched systems.…”

SMC for Discrete-Time Networked Semi-Markovian Switching Systems With Random DoS Attacks and Applications

“…According to [55], the discrete-time semi-Markovian kernel (SMK) approach has been proposed to overcome the constraint of single distribution for each mode and the switching among different modes is mutually governed by the transition probability and the ST distribution function. On this basis, the developed theories have been investigated for discrete-time S-MSSs; see, for example, [56], [57], [58], [59], [60], [61]. On the premise that both the upper and lower bounds of ST are restricted, an active fault-tolerant control scheme has been proposed for discretetime S-MSSs [59].…”

“…On the premise that both the upper and lower bounds of ST are restricted, an active fault-tolerant control scheme has been proposed for discretetime S-MSSs [59]. Furthermore, by using partially available SMK information, the H ∞ controller design problem has been studied for discrete-time singularly perturbed S-MSSs with repetitive scalar nonlinearity and external disturbances [61].…”

“…Almost all the existing results consider that each system mode possesses a single distribution of ST [49], [50], [51], [52], [53], [54], [66], while there is a strong possibility that the types of distributions and/or the parameters in the same type of distribution of ST are different. Moreover, most of the existing results about S-MSSs are obtained under an ideal data transmission environment [46], [47], [49], [50], [51], [52], [53], [55], [56], [57], [58], [59], [60], [61], that is, there are no malicious cyberattacks during signal transmission. Furthermore, many factors, such as SMK, DTSMC mechanism, uncertainty, and DoS attacks, play an important part in describing practical stochastic switching systems.…”