Robust stability and H-infinity control for uncertain discrete-time Markovian jump singular systems

The robust stability and stabilization, and H-infinity control problems for discrete-time Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a discrete-time Markovian jump standard linear system, and the linear matrix inequality (LMI) conditions for the discrete-time Markovian jump singular systems to be regular, causal, stochastically stable, and stochastically stable with °- disturbance attenuation are obtained, respectively. With these conditions, the robust state feedback stochastic stabilization problem and H-infinity control problem are solved, and the LMI conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper.     

Robust stability and H-infinity control for uncertain discrete-time Markovian jump singular systems

The robust stability and stabilization, and H-infinity control problems for discrete-time Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent （r.s.e.） transformation and by introducing new state vectors, the singular system is transformed into a discrete-time Markovian jump standard linear system, and the linear matrix inequality （LMI） conditions for the discrete-time Markovian jump singular systems to be regular, causal, stochastically stable, and stochastically stable with 7- disturbance attenuation are obtained, respectively. With these conditions, the robust state feedback stochastic stabilization problem and H-infinity control problem are solved, and the LMI conditions are obtained. A numerical example illustrates the effectiveness of the method given in the oaoer.     

Uniform stabilization of discrete-time switched and Markovian jump linear systems

The uniform stability of discrete-time switched linear systems, possibly with a strongly connected switching path constraint, and the existence of finite-path-dependent dynamic output feedback controllers uniformly stabilizing such a system are both shown to be characterized by the existence of a finite-dimensional feasible system of linear matrix inequalities. This characterization is based on the observation that a linear time-varying system is uniformly stable only if there exists a finite-path-dependent quadratic Lyapunov function. The synthesis of a uniformly stabilizing controller is done without conservatism by solving any feasible system of linear matrix inequalities among an increasing family of systems of linear matrix inequalities. The result carries over to the almost sure uniform stabilization of Markovian jump linear systems.     

Robust stochastic stability of uncertain discrete-time impulsive Markovian jump delay systems with multiplicative noises

In this paper, the robust stochastic stability is investigated for a class of uncertain discrete-time impulsive Markovian jump delay systems with multiplicative noises. Using the method of stochastic Lyapunov functionals construction, it is shown that impulses can stabilise the original impulse-free unstable systems. Moreover, the stability property of the impulse-free systems can be retained in the cases of appropriately large impulsive time interval. Some numerical examples are exploited to demonstrate the effectiveness and the superiority of the proposed results.     

Robust control of uncertain discrete-time Markovian jump systems with actuator saturation

In this paper, the robust stochastic stabilization problem for the class of discrete-time uncertain Markovian jump linear systems (MJLS) with actuator saturation is considered. The definition of domain of attraction in mean square sense (DoA-MSS) is introduced to analyze the stochastic stability of the closed-loop system. By using a class of stochastic Lyapunov function which is dependent on the jump mode and saturation function, design procedures for both the mode-dependent and mode-independent state feedback controllers are developed based on the Linear Matrix Inequality (LMI) approach. Finally, a numerical example is provided to show the usefulness of the proposed techniques.     

Robust one-step receding horizon control of discrete-time Markovian jump uncertain systems

This paper proposes a receding horizon control scheme for a set of uncertain discrete-time linear systems with randomly jumping parameters described by a finite-state Markov process whose jumping transition probabilities are assumed to belong to some convex sets. The control scheme for the underlying systems is based on the minimization of the worst-case one-step finite horizon cost with a finite terminal weighting matrix at each time instant. This robust receding horizon control scheme has a more general structure than the existing robust receding horizon control for the underlying systems under the same design parameters. The proposed controller is obtained using semidefinite programming.     

Robust H 2-control for discrete-time Markovian jump linear systems

This paper deals with the robust H₂-control of discrete-time Markovian jump linear systems. It is assumed that both the state and jump variables are available to the controller. Uncertainties satisfying some norm bounded conditions are considered on the parameters of the system. An upper bound for the H₂-control problem is derived in terms of a linear matrix inequality (LMI) optimization problem. For the case in which there are no uncertainties, we show that the convex formulation is equivalent to the existence of the mean square stabilizing solution for the set of coupled algebraic Riccati equations arising on the quadratic optimal control problem of discrete-time Markovian jump linear systems. Therefore, for the case with no uncertainties, the convex formulation considered in this paper imposes no extra conditions than those in the usual dynamic programming approach. Finally some numerical examples are presented to illustrate the technique.     

Robust H-infinity control for uncertain discrete-time Markovian jump systems with actuator saturation

The design of robust H-infinity controller for uncertain discrete-time Markovian jump systems with actuator saturation is addressed in this paper. The parameter uncertainties are assumed to be norm-bounded. Linear matrix inequality (LMI) conditions are proposed to design a set of controllers in order to satisfy the closed-loop local stability and closed-loop H-infinity performance. Using an LMI approach, a set of state feedback gains is constructed such that the set of admissible initial conditions is enlarged and formulated through solving an optimization problem. A numerical example is given to illustrate the effectiveness of the proposed methods.     

A stability and contractiveness analysis of discrete-time Markovian jump linear systems

We propose performance criteria for discrete-time linear systems with Markov jumping parameters. Exact convex conditions for internal stability and contractiveness of such systems are expressed in terms of linear matrix inequalities. These conditions are of the same form as that in the Kalman-Yacubovich-Popov lemma.     

Robust stability of uncertain Markovian jump discrete-time recurrent neural networks with time delays

This article is concerned with robust stochastic stability for a class of uncertain Markovian jump discrete-time recurrent neural networks (MJDRNNs) with time delays. The uncertainty is assumed to be of the norm-bounded form. By employing the Lyapunov functional and linear matrix inequality (LMI) approach, some sufficient criteria are proposed for the robust stochastic stability in the mean square of the MJDRNNs with constant or mode-dependent time delays. The proposed LMI-based results are computationally efficient as they can be solved numerically using standard commercial software. The validity of the obtained results are further illustrated by two simulation examples.     

Robust fuzzy control for uncertain discrete-time nonlinear Markovian jump systems without mode observations

This paper studies the robust fuzzy control problem of uncertain discrete-time nonlinear Markovian jump systems without mode observations. The Takagi and Sugeno (T-S) fuzzy model is employed to represent a discrete-time nonlinear system with norm-bounded parameter uncertainties and Markovian jump parameters. As a result, an uncertain Markovian jump fuzzy system (MJFS) is obtained. A stochastic fuzzy Lyapunov function (FLF) is employed to analyze the robust stability of the uncertain MJFS, which not only is dependent on the operation modes of the system, but also directly includes the membership functions. Then, based on this stochastic FLF and a non-parallel distributed compensation (non-PDC) scheme, a mode-independent state-feedback control design is developed to guarantee that the closed-loop MJFS is stochastically stable for all admissible parameter uncertainties. The proposed sufficient conditions for the robust stability and mode-independent robust stabilization are formulated as a set of coupled linear matrix inequalities (LMIs), which can be solved efficiently by using existing LMI optimization techniques. Finally, it is also demonstrated, via a simulation example, that the proposed design method is effective.     

时变时滞离散广义Markov 跳变系统的鲁棒稳定性

研究一类具有区间时变时滞的离散不确定广义Markov跳变系统的时滞相关鲁棒稳定性问题.通过将Jensen不等式与一个新的定界不等式相结合,得到了一个新的稳定性判据,该判据中仅含有Lyapunov变量,具有较小的计算负担.进而,基于凸组合方法得到了另一个新的稳定性判据,该判据引入了一些自由矩阵变量,具有较小的保守性.数值算例表明了所提出方法的有效性.     

Static output-feedback stabilization of discrete-time Markovian jump linear systems: A system augmentation approach

This paper studies the static output-feedback (SOF) stabilization problem for discrete-time Markovian jump systems from a novel perspective. The closed-loop system is represented in a system augmentation form, in which input and gain-output matrices are separated. By virtue of the system augmentation, a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities, which possess a monotonic structure for a linearized computation, and a convergent iteration algorithm is given to solve such inequalities. In addition, a special property of the feasible solutions enables one to further improve the solvability via a simple D-K type optimization on the initial values. An extension to mode-independent SOF stabilization is provided as well. Compared with some existing approaches to SOF synthesis, the proposed one has several advantages that make it specific for Markovian jump systems. The effectiveness and merit of the theoretical results are shown through some numerical examples.     

Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay

This paper deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete-time systems with state delay. The system under consideration involves state time delay and time-varying norm-bounded parameter uncertainties appearing in all the matrices of the state-space model. Necessary and sufficient conditions for quadratic stability and quadratic stabilization are presented in terms of certain matrix inequalities, respectively. A robustly stabilizing state feedback controller can be constructed by using the corresponding feasible solution of the matrix inequalities. Two examples are presented to demonstrate the effectiveness of the proposed approach.     

Robust decentralized stabilization of uncertain large-scale discrete-time systems

This article describes the synthesis of robust decentralized controllers for large-scale discrete-time systems with uncertainties. Based on the Lyapunov method, a sufficient condition for robust stability is derived in terms of a linear matrix inequality (LMI). The solutions of the LMI can be easily obtained using efficient convex optimization techniques. A numerical example is given to illustrate the proposed method.     

Robust stability and control of uncertain linear discrete-time periodic systems with time-delay

This paper deals with the problems of robust stability analysis and robust control of linear discrete-time periodic systems with a delayed state and subject to polytopic-type parameter uncertainty in the state-space matrices. A robust stability criterion independent of the time-delay length as well as a delay-dependent criterion is proposed, where the former applies to the case of a constant time-delay and the latter allows for a time-varying delay lying in a given interval. The developed robust stability criteria are based on affinely uncertainty-dependent Lyapunov–Krasovskii functionals and are given in terms of linear matrix inequalities. These stability conditions are then applied to solve the problems of robust stabilization and robust _H∞$H_{\infty }$  control via static periodic state feedback. Numerical examples illustrate the potentials of the proposed robust stability and control methods.     

Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

This paper proposes new sufficient conditions for the exponential stability and stabilization of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities (LMIs) and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponetial stability rate of the solution. Numerical examples illustrating the conditions are given.     

Robust exponential stability and stabilization of linear uncertain polytopic time-delay systems

This paper proposes new sufficient conditions for the exponential stability and stabilization.of linear uncertain polytopic time-delay systems. The conditions for exponential stability are expressed in terms of Kharitonov-type linear matrix inequalities （LMIs） and we develop control design methods based on LMIs for solving stabilization problem. Our method consists of a combination of the LMI approach and the use of parameter-dependent Lyapunov functionals, which allows to compute simultaneously the two bounds that characterize the exponetial stability rate of the solution. Numerical examples illustrating the conditions are given.