Almost Sure Exponential Stability of Recurrent Neural Networks With Markovian Switching

This paper presents new stability results for recurrent neural networks with Markovian switching. First, algebraic criteria for the almost sure exponential stability of recurrent neural networks with Markovian switching and without time delays are derived. The results show that the almost sure exponential stability of such a neural network does not require the stability of the neural network at every individual parametric configuration. Next, both delay-dependent and delay-independent criteria for the almost sure exponential stability of recurrent neural networks with time-varying delays and Markovian-switching parameters are derived by means of a generalized stochastic Halanay inequality. The results herein include existing ones for recurrent neural networks without Markovian switching as special cases. Finally, simulation results in three numerical examples are discussed to illustrate the theoretical results.     

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.     

Exponential stability of stochastic delay interval systems with Markovian switching

In the past few years, a lot of research has been dedicated to the stability of interval systems as well as the stability of systems with Markovian switching. However, little research has been on the stability of interval systems with Markovian switching, which is the topic of this paper. The system discussed is the stochastic delay interval system with Markovian switching. It is a very advanced system and takes all the features of interval systems, Ito equations, and Markovian switching, as well as time lag, into account. The theory developed is applicable in many different and complicated situations so the importance of the paper is clear.     

Stochastic stability analysis of fault-tolerant control systems inthe presence of noise

The stochastic stability of fault tolerant control systems (FTCSs) in the presence of noise are,analyzed using the Lyapunov function approach. In particular, the FTCS with a Markovian fault detection and isolation process having transition probabilities conditioned on the state of another Markovian process representing component failures, is considered. The almost sure asymptotic stability in probability and the exponential stability in the mean square are considered. Specifically, a testable necessary and sufficient condition for the exponential stability in the mean square is derived. A numerical example is presented to illustrate the theoretical analysis     

A survey on Markovian jump systems: Modeling and design

Markovian jump systems are a special class of hybrid and stochastic systems which can be used to describe many real world applications, such as manufacturing systems, power systems, chemical systems, economic systems, communication and control, etc. In this paper, a survey on recent developments of modeling, analysis and design of Markovian jump systems is presented. First, stability issues on Markovian jump systems are addressed. Then a variety of control and filter design methods are systematically recalled. Furthermore, the new trends of Markovian jump systems with uncertain transition rates as well as semi-Markovian jump systems are also discussed.     

RAZUMIKHIN-TYPE THEOREMS OF NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.     

Finite‐time stability and stabilization of semi‐Markovian jump systems with time delay

Semi‐Markovian jump systems are more general than Markovian jump systems in modeling practical systems. On the other hand, the finite‐time stochastic stability is also more effective than stochastic stability in practical systems. This paper focuses on the finite‐time stochastic stability, exponential stochastic stability, and stabilization of semi‐Markovian jump systems with time‐varying delay. First, a new stability condition is presented to guarantee the finite‐time stochastic stability of the system by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. Second, the stability criterion is further proved to guarantee the exponential stochastic stability of the system. Moreover, a controller design method is also presented according to the stability criterion. Finally, an example is provided to illustrate that the proposed stability condition is less conservative than other existing results. Additionally, we use the proposed method to design a controller for a load frequency control system to illustrate the effectiveness of the method in a practical system of the proposed method.     

Robust stability of Markovian jump discrete-time neural networks with partly unknown transition probabilities and mixed mode-dependent delays

This article discusses the robust stability problem for a class of uncertain Markovian jump discrete-time neural networks with partly unknown transition probabilities and mixed mode-dependent time delays. The transition probabilities of the mode jumps are considered to be partly unknown, which relax the traditional assumption in Markovian jump systems that all of them must be completely known a priori. The mixed time delays consist of both discrete and distributed delays that are dependent on the Markovian jump modes. By employing the Lyapunov functional and linear matrix inequality approach, some sufficient criteria are derived for the robust stability of the underlying systems. A numerical example is exploited to illustrate the developed theory.     

模型相关时变时滞马尔可夫跳变系统低保守稳定性

本文研究了时变时滞与模型相关的随机马尔可夫跳变系统的时滞相关稳定性问题. 通过建立时变时滞与模型相关的系统模型, 构造不同的Lyapunov-Krasovskii函数, 并通过引入改进的积分等式, 以线性矩阵不等式的形式提出了具有较小保守性的时滞依赖稳定性条件. 最后用几个数值算例说明本文结论的有效性及较低的保守性.     

p-Moment stability of stochastic differential equations with impulsive jump and Markovian switching

This paper introduces some new concepts of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching. Some stability criteria of p-moment stability for stochastic differential equations with impulsive jump and Markovian switching are obtained by using Liapunov function method. An example is also discussed to illustrate the efficiency of the obtained results.     

Delay-dependent stability analysis for Markovian jump systems with interval time-varying-delays

This paper proposes improved stochastic stability conditions for Markovian jump systems with interval time-varying delays. In terms of linear matrix inequalities (LMIs), less conservative delay-range-dependent stability conditions for Markovian jump systems are proposed by constructing a different Lyapunov-Krasovskii function. The resulting criteria have advantages over some previous ones in that they involve fewer matrix variables but have less conservatism. Numerical examples are provided to demonstrate the efficiency and reduced conservatism of the results in this paper.     

Finite-Time Stability of Stochastic Cohen–Grossberg Neural Networks with Markovian Jumping Parameters and Distributed Time-Varying Delays

In this paper, the finite-time stability problem is considered for a class of stochastic Cohen–Grossberg neural networks (CGNNs) with Markovian jumping parameters and distributed time-varying delays. Based on Lyapunov–Krasovskii functional and stability analysis theory, a linear matrix inequality approach is developed to derive sufficient conditions for guaranteeing the stability of the concerned system. It is shown that the addressed stochastic CGNNs with Markovian jumping and distributed time varying delays are finite-time stable. An illustrative example is provided to show the effectiveness of the developed results.     

Stochastic stability of semi‐Markovian jump systems with mode‐dependent delays

Semi‐Markovian jump systems, due to the relaxed conditions on the stochastic process, and its transition rates are time varying, can be used to describe a larger class of dynamical systems than conventional full Markovian jump systems. In this paper, the problem of stochastic stability for a class of semi‐Markovian systems with mode‐dependent time‐variant delays is investigated. By Lyapunov function approach, together with a piecewise analysis method, a sufficient condition is proposed to guarantee the stochastic stability of the underlying systems. As more time‐delay information is used, our results are much less conservative than some existing ones in literature. Finally, two examples are given to show the effectiveness and advantages of the proposed techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices

This paper focuses on stability and stabilization for a class of continuous-time Markovian jump systems with partial information on transition probability. The free-connection weighting matrix method is proposed to obtain a less conservative stability criterion of Markovian jump systems with partly unknown transition probability or completely unknown transition probability. As a result, a sufficient condition for the state feedback controller design is derived in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the effectiveness and the merits of the proposed method.     

Stability analysis and saturation control for nonlinear positive Markovian jump systems with randomly occurring actuator faults

This article investigates the stability analysis and control design of a class of nonlinear positive Markovian jump systems with randomly occurring actuator faults and saturation. It is assumed that the actuator faults of each subsystem are varying and governed by a Markovian process. The nonlinear term is located in a sector. First, sufficient conditions for stochastic stability of the underlying systems are established using a stochastic copositive Lyapunov function. Then, a family of reliable L₁‐gain controller is proposed for nonlinear positive Markovian jump systems with actuator faults and saturation in terms of a matrix decomposition technique. Under the designed controllers, the closed‐loop systems are positive and stochastically stable with an L₁‐gain performance. An optimization method is presented to estimate the maximum domain of attraction. Furthermore, the obtained results are developed for general Markovian jump systems. Finally, numerical examples are given to illustrate the effectiveness of the proposed techniques.     

Robust Stability for Uncertain Delayed Fuzzy Hopfield Neural Networks With Markovian Jumping Parameters

This paper is concerned with the problem of the robust stability of nonlinear delayed Hopfield neural networks (HNNs) with Markovian jumping parameters by Takagi-Sugeno (T-S) fuzzy model. The nonlinear delayed HNNs are first established as a modified T-S fuzzy model in which the consequent parts are composed of a set of Markovian jumping HNNs with interval delays. Time delays here are assumed to be time-varying and belong to the given intervals. Based on Lyapunov-Krasovskii stability theory and linear matrix inequality approach, stability conditions are proposed in terms of the upper and lower bounds of the delays. Finally, numerical examples are used to illustrate the effectiveness of the proposed method.     

Stochastic model predictive control for constrained discrete-time Markovian switching systems

In this paper we study constrained stochastic optimal control problems for Markovian switching systems, an extension of Markovian jump linear systems (MJLS), where the subsystems are allowed to be nonlinear. We develop appropriate notions of invariance and stability for such systems and provide terminal conditions for stochastic model predictive control (SMPC) that guarantee mean-square stability and robust constraint fulfillment of the Markovian switching system in closed-loop with the SMPC law under very weak assumptions. In the special but important case of constrained MJLS we present an algorithm for computing explicitly the SMPC control law off-line, that combines dynamic programming with parametric piecewise quadratic optimization.     

Stochastic stability of uncertain fuzzy recurrent neural networks with Markovian jumping parameters

In this paper, the global robust stability of uncertain recurrent neural networks with Markovian jumping parameters which are represented by the Takagi–Sugeno fuzzy model is considered. A novel linear matrix inequality-based stability criterion is obtained by using Lyapunov functional theory to guarantee the asymptotic stability of uncertain fuzzy recurrent neural networks with Markovian jumping parameters. Finally, numerical examples are given to demonstrate the correctness of the theoretical results. Our results are also compared with results discussed in Arik [On the global asymptotic stability of delayed cellular neural networks, IEEE Trans. Circ. Syst. I 47 (2000), pp. 571–574], Cao [Global stability conditions for delayed CNNs, IEEE Trans. Circ. Syst. I 48 (2001), pp. 1330–1333] and Lou and Cui [Delay-dependent stochastic stability of delayed Hopfield neural networks with Markovian jump parameters, J. Math. Anal. Appl. 328 (2007), pp. 316–326] to show the effectiveness and conservativeness.     

Stabilization of linear systems over networks with bounded packet loss

This paper is concerned with the stabilization problem of networked control systems where the main focus is the packet-loss issue. Two types of packet-loss processes are considered. One is the arbitrary packet-loss process, the other is the Markovian packet-loss process. The stability conditions of networked control systems with both arbitrary and Markovian packet losses are established via a packet-loss dependent Lyapunov approach. The corresponding stabilizing controller design techniques are also given based upon the stability conditions. These results are also extended to the unit time delay case. Finally, the numerical example and simulations have demonstrated the usefulness of the developed theory.     

转移概率部分未知的时滞不确定Markov跳跃系统稳定性分析

本文研究转移概率部分未知的时滞不确定的Markov 跳跃系统随机稳定性问题,基于Lyapunov稳定理论,构造合适的Lyapunov泛函,使用自由权矩阵技术和凸结合技术来估计积分项的上界,同时也充分考虑时滞下界和上界的关系,得到保证Markov 跳跃系统随机稳定性的充分性条件,该条件以线性矩阵不等式的形式表出。最后,一个数值例子和其仿真验证了我们所提方法的有效性和优越性。