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.     

Further criterion for stochastic stability analysis of semi‐Markovian jump linear systems

This article is devoted to provide further criterion for stochastic stability analysis of semi‐Markovian jump linear systems (S‐MJLSs), in which more generic transition rates (TRs) will be studied. As is known, the time‐varying TR is one of the key issues to be considered in the analysis of S‐MJLS. Therefore, this article is to investigate general cases for the TRs that covered almost all types, especially for the type that the jumping information from one mode to another is fully unknown, which is merely investigated before. By virtue of stochastic functional theory, sufficient conditions are developed to check stochastic stability of the underlying systems via linear matrix inequalities formulation combined with a maximum optimization algorithm. Finally, a numerical example is given to verify the validity and effectiveness of the obtained results.     

Exponential stability of switched Markovian jumping neutral‐type systems with generally incomplete transition rates

In this paper, the exponential mean‐square stability of neutral switching Markovian jump systems with generally incomplete transition probabilities is investigated. The model discussed in this paper concludes both deterministic switching signals and Markovian jumping signals. The transition rates of the jumping process are assumed to be partly available, that is, some elements have been exactly known, some have been merely known with lower and upper bounds, and others may have no information to use. Based on the Lyapunov‐Krasovskii functional method, sufficient conditions on the exponential mean‐square stability of the considered system are derived in terms of liner matrix inequalities. A numerical example is provided to show the feasibility and effectiveness of the proposed results.     

Stability and dissipativity analysis for uncertain Markovian jump systems with random delays via new approach

This paper studies the problem of stability and dissipativity analysis for uncertain Markovian jump systems (UMJSs) with random time-varying delays. Based on the auxiliary function-based integral inequality (AFBII) and with the help of some mathematical tools, a new double integral inequality (NDII) is developed. Then, to show the efficiency of the proposed inequality, a suitable Lyapunov-Krasovskii functional (LKF) is constructed with augmented delay-dependent terms. By employing integral inequalities, new delay-dependent sufficient conditions are derived in terms of linear matrix inequalities (LMIs). Finally, illustrative examples are given to show the effectiveness and less conservatism of the results.     

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.     

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.     

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.     

Improved robust stability criteria for uncertain linear neutral‐type systems via novel Lyapunov‐Krasovskii functional

The stability problem for the uncertain time‐varying delayed neutral‐type system is concerned in this paper. By introducing a novel Lyapunov‐Krasovskii functional (LKF) related to a delay‐product‐type function and two delay‐dependent matrices, some new delay‐dependent robust stability sufficient conditions are derived, which are based on convex linear matrix inequality (LMI) framework. The sufficient conditions in this paper can reduce the conservativeness of some recent previous ones. In the end, some numerical examples, including a linear neutral‐type system, the partial element equivalent circuit and a general linear system, show the effectiveness of the proposed method.     

Stochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters

In this paper, we first study the problems of robust quadratic mean-square stability and stabilization for a class of uncertain discrete-time linear systems with both Markovian jumping parameters and Frobenius norm-bounded parametric uncertainities. Necessary and sufficient conditions for the above problems are proposed, which are in terms of positive-definite solutions of a set of coupled algebraic Riccati inequalities. Then, the problem of robust quadratic guaranteed cost control for the underlying systems is investigated. A guaranteed cost control is designed to ensure the cost function is within a certain bound, irrespective of all admissible uncertainities. © 1998 John Wiley & Sons, Ltd.     

Exponential stabilization of uncertain time‐delay linear systems with Markovian jumping parameters

This paper studies the exponential stabilization problem of uncertain time‐delay linear systems with Markovian jumping parameters. A novel delay decomposition approach is developed to derive delay‐dependent conditions under which the closed‐loop control system is mean square exponentially stable for all admissible uncertainties. It is shown that the feedback gain matrices and the decay rate can be obtained by solving coupled linear matrix inequalities. Moreover, the difficulties arising from searching for tuning parameters in the existing methods are overcome. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Stability analysis and stabilization of Markovian jump systems with time‐varying delay and uncertain transition information

The paper investigates the problems of stability and stabilization of Markovian jump systems with time‐varying delays and uncertain transition rates matrix. First, the stochastic scaled small‐gain theorem is introduced to analyze the stability of the Markovian jump system. Then, a new stability criterion is proposed by using a new Lyapunov‐Krasovskii functional combined with Wirtinger‐based integral inequality. The proposed stability condition is demonstrated to be less conservative than other existing results. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a new precise triangle inequality and a new Lyapunov‐Krasovskii functional. Moreover, a controller design criterion is presented according to the stability criterion. Furthermore, the transition rate matrix is treated as partially known and with uncertainty, and the relevant stability and stabilization criteria are proposed. Finally, 3 numerical examples are provided to illustrate the superior result of the stability criteria and the effectiveness of the proposed controller design method.     

Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

Observer‐based consensus control of nonlinear multiagent systems under semi‐Markovian switching topologies and cyber attacks

This article investigates the leader‐following consensus of nonlinear multiagent systems under semi‐Markovian switching topologies and cyber attacks. Unlike the related works, the communication channels considered herein are subjected to successful but recoverable attacks, and when the channels work well, the network topology is time‐varying and described by a semi‐Markovian switching topology. Due to the effect of attacks, the communication network is intermittently paralyzed. For the cases that the transition rates of semi‐Markovian switching topologies are completely known and partially unknown, observer‐based control protocols and sufficient conditions are proposed, respectively, to ensure the consensus of the systems in the mean square sense. Finally, simulation examples are given to illustrate the validity of the theoretical results.     

Further results on delay‐dependent robust stability conditions of uncertain neutral systems

This paper deals with the problem of robust stability analysis for uncertain neutral systems. In terms of a linear matrix inequality (LMI), an improved delay‐dependent asymptotic stability criterion is developed without using bounding techniques on the related cross product terms. Based on this, a new delay‐dependent LMI condition for robust stability is obtained. Numerical examples are provided to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

Stochastic robust finite‐time boundedness for semi‐Markov jump uncertain neutral‐type neural networks with mixed time‐varying delays via a generalized reciprocally convex combination inequality

This article investigates the stochastic robust finite‐time boundedness problem for semi‐Markov jump uncertain (SMJU) neutral‐type neural networks with distributed and additive time‐varying delays (TDs). To derive less conservative stability criteria, a generalized reciprocally convex combination inequality (RCCI) is first proposed, which includes the existing RCCIs as its special cases. By taking full advantage of the characteristics of various TDs and SMJU parameters, a novel suitable Lyapunov‐Krasovskii functional is provided. Then, with the virtue of the new RCCI and other analysis approaches, some new criteria guaranteeing the underlying systems are stochastically robustly finite‐time bounded or stable and are derived in the form of linear matrix inequalities. Finally, three numerical examples are given to show the validity of the approaches presented in this article.     

Robust Stochastic Stability for a Class of Singular Systems with Uncertain Markovian Jump and Time‐Varying Delay

This paper deals with the problem of the robust stochastic stability for a class of singular systems with uncertain Markovian jump and time‐varying delay. Sufficient conditions on the stochastic stability are presented. The obtained results are formulated in terms of strict linear matrix inequalities. A numerical example is provided to show the effectiveness of the proposed approaches.     

Augmented Lyapunov functional and delay‐dependent stability criteria for neutral systems

In this paper, an augmented Lyapunov functional is proposed to investigate the asymptotic stability of neutral systems. Two methods with or without decoupling the Lyapunov matrices and system matrices are developed and shown to be equivalent to each other. The resulting delay‐dependent stability criteria are less conservative than the existing ones owing to the augmented Lyapunov functional and the introduction of free‐weighting matrices. The delay‐independent criteria are obtained as an easy corollary. Numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability of discrete‐time systems with time‐varying delay via a novel Lyapunov‐Krasovskii functional

This article is concerned with the problem of stability analysis of discrete‐time systems with time‐varying delay. Unlike previous articles concentrating on the improvement of summation inequalities, this article constructs a novel augmented Lyapunov‐Krasovskii (L‐K) functional by fully considering the coupling information among state‐related vectors. Based on the newly developed L‐K functional, a more relaxed stability criterion is derived by exploring the interest of existing inequalities. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

Stability analysis for stochastic BAM neural networks with Markovian jumping parameters

This paper is concerned with the stability analysis issue for stochastic delayed bidirectional associative memory (BAM) neural network with Markovian jumping parameters. Assume that the jumping parameters are generated from continue-time discrete-state homogeneous Markov process and the delays are time-invariant. By employing the Lyapunov stability theory, some inequality techniques and the stochastic analysis, sufficient conditions are derived to achieve the global exponential stability in the mean square of the stochastic BAM neural network. One example is also provided in the end of this paper to illustrate the effectiveness of our results.     

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。