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.     

Delay-segment-dependent robust stability for uncertain discrete stochastic Markovian jumping systems with interval time delay

This article deals with the problem of robust stochastic stability for a class of uncertain discrete stochastic Markovian jumping systems with time-varying interval delay. By constructing a parameter-dependent Lyapunov–Krasovskii functional and checking its difference in two subintervals, respectively, some novel delay-segment-dependent stability criteria for the addressed system are derived. Two simulation examples are given to show effectiveness of the proposed method.     

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.     

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.     

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.     

Improved exponential stability for stochastic Markovian jump systems with nonlinearity and time‐varying delay

This paper is concerned with delay‐dependent exponential stability for stochastic Markovian jump systems with nonlinearity and time‐varying delay. An improved exponential stability criterion for stochastic Markovian jump systems with nonlinearity and time‐varying delay is proposed without ignoring any terms by considering the relationship among the time‐varying delay, its upper bound and their difference, and using both Itô's differential formula and Lyapunov stability theory. A numerical example is given to illustrate the effectiveness and the benefits of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.     

Reliable dissipative control for uncertain time‐delayed stochastic systems with Markovian jump switching and multiplicative noise

This paper considers the robust reliable dissipative control problem for a class of hybrid systems, which includes stochastics, Markovian jumping, state time delay, parameter uncertainty, possible actuator failure, multiplicative noises and impulsive effects. We propose a linear feedback memoryless controller and impulsive controller such that the hybrid system is stochastically stable and strictly (Q, S, R) dissipative, which include H_∞ performance as a special case, for all the admissible uncertainties and actuator failures occurring among a prescribed subset of actuators. Based on Itô's differential formula and Lyapunov stability theory, sufficient conditions are obtained in terms of linear matrix inequalities. A numerical example is constructed to show the effectiveness of the controller designed in this paper. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

Robust H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump

This paper deals with the problem of H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump. Based on the extended It^o stochastic differential formula, sufficient conditions for the solvability of these problems are obtained. Furthermore, It is shown that a desired filter can be constructed by solving a set of linear matrix inequalities. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.     

Robust H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump

This paper deals with the problem of H-infinity filter design for uncertain time-delay singular stochastic systems with Markovian jump. Based on the extended It^o stochastic differential formula, sufficient conditions for the solvability of these problems are obtained. Furthermore, It is shown that a desired filter can be constructed by solving a set of linear matrix inequalities. Finally, a simulation example is given to demonstrate the effectiveness of the proposed method.     

Robust exponential stabilization for Markovian jump systems with mode-dependent input delay

This paper is concerned with the problem of exponential stabilization for uncertain linear systems with Markovian jump parameters and mode-dependent input delays. Sufficient stabilization conditions are developed in terms of matrix inequalities, which can be solved by a proposed iterative algorithm based on the cone complementarity linearization (CCL) method. Memory controllers are also designed such that the closed-loop system is exponentially mean-square stable for all admissible uncertainties. Numerical examples are given to show that the developed method is efficient and less conservative.     

Moment exponential stability analysis of Markovian jump stochastic differential equations with uncertain transition jump rates

This paper is concerned with the moment exponential stability analysis of Markovian jump stochastic differential equations. The equations under consideration are more general, whose transition jump rates matrix Q is not precisely known. Sufficient conditions for testing the stability of such equations are established, and some numerical examples to illustrate the effectiveness of our results are presented.     

Robust sampled‐data control for Itô stochastic Markovian jump systems with state delay

In this paper, the problem of robust sampled‐data control for Itô stochastic Markovian jump systems (Itô SMJSs) with state delay is investigated. Using parameters‐dependent Lyapunov functionals and some stochastic equations, we give stochastic sufficient stability criteria for polytopic uncertain Itô SMJSs. As a corollary, stochastic sufficient stability criteria are given for nominal Itô SMJSs. For this two cases of Itô SMJSs, based on the obtained stochastic stability criteria, their time‐independent sampled‐data controllers are designed, respectively. Then, for designing a time‐dependent sampled‐data controller for Itô SMJSs, a parameters‐dependent time‐scheduled Lyapunov functional is developed. New stochastic sufficient stability criteria are obtained for polytopic uncertain Itô SMJSs and nominal Itô SMJSs. Furthermore, their time‐dependent sampled‐data controllers are designed, respectively. Lastly, a numerical example is provided to illustrate the effectiveness of the proposed method.     

Robust stability and H∞ control for uncertain discrete Markovian jump singular systems with mode‐dependent time‐delay

The robust stochastic stability, stabilization and H_∞ control for mode‐dependent time‐delay discrete 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 standard linear system, and delay‐dependent linear matrix inequalities (LMIs) conditions for the mode‐dependent time‐delay discrete Markovian jump singular systems to be regular, causal and stochastically stable, and stochastically stable with γ‐disturbance attenuation are obtained, respectively. With these conditions, robust stabilization problem and robust H_∞ control problem are solved, and the LMIs sufficient conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Transition probability bounds for the stochastic stability robustness of continuous- and discrete-time Markovian jump linear systems

This paper considers the robustness of stochastic stability of Markovian jump linear systems in continuous- and discrete-time with respect to their transition rates and probabilities, respectively. The continuous-time (discrete-time) system is described via a continuous-valued state vector and a discrete-valued mode which varies according to a Markov process (chain). By using stochastic Lyapunov function approach and Kronecker product transformation techniques, sufficient conditions are obtained for the robust stochastic stability of the underlying systems, which are in terms of upper bounds on the perturbed transition rates and probabilities. Analytical expressions are derived for scalar systems, which are straightforward to use. Numerical examples are presented to show the potential of the proposed techniques.     

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.     

Robust stabilisation of uncertain delayed Markovian jump systems and its applications

This paper is concerned with the robust stabilsation of uncertain delayed Markovian jump systems. Given a Markovian jump system with time delay and Brownian motion simultaneously, we allow the uncertainty added in the form of additive perturbations and existing in the drift and diffusion sections at the same time. A sufficient condition on the mean square stability of system in the face of such disturbances is obtained, which is similar to small-gain theorem. A kind of partially delay-dependent controller stabilising the resulting closed-loop system is firstly designed to relate to the probability distribution of delay, whose key idea is applied to construct a delayed controller with disordering phenomenon. It is seen that the existence conditions established here could be solved easily. Based on the proposed results, some applications on robust synchronisation of uncertain delayed multi-agent systems with Markovian switching are considered. It is shown that the robust synchronisation of such an uncertain multi-agent network could be achieved by a protocol that each controller being partially delay-dependent or disordering could robustly stabilise a given single Markovian jump system. As for these cases, the proposed protocols could be obtained by solving certain algebraic Riccati equations and inequalities, which also involve weighting factors and depend on the eigenvalues of the Laplacian graph.     

Further results on finite-time stabilisation for stochastic Markovian jump systems with time-varying delay

In this paper, we study the issue of finite-time stabilisation for stochastic Markovian jump systems with time-varying delay by considering a new criterion on finite-time stability. By constructing more appropriate Lyapunov–Krasovskii functional, some new conditions for verifying the finite-time stability of the plant as well as controller synthesis are established in standard linear matrix inequalities. The practical example about a single-link robot arm model demonstrates the validity of the main results.     

Exponential stability of stochastic singular delay systems with general Markovian switchings

In recent years, Markovian jump systems have received much attention. However, there are very few results on the stability of stochastic singular systems with Markovian switching. In this paper, the discussed system is the stochastic singular delay system with general transition rate matrix in terms of uncertain and partially unknown transition rate matrix. The aim is to answer the question whether there are conditions guaranteeing the underlying system having a unique solution and being exponentially admissible simultaneously. The proposed results show that all the features of the underlying system such as time delay, diffusion, and general Markovian switchings play important roles in the system analysis of exponential admissibility. A numerical example is used to demonstrate the effectiveness of the proposed methods. Copyright © 2014 John Wiley & Sons, Ltd.