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.     

New results on stability analysis and stabilization of time‐delay continuous Markovian jump systems with partially known rates matrix

In this note, the problems of stability analysis and controller synthesis of Markovian jump systems with time‐varying delay and partially known transition rates are investigated via an input–output approach. First, the system under consideration is transformed into an interconnected system, and new results on stochastic scaled small‐gain condition for stochastic interconnected systems are established, which are crucial for the problems considered in this paper. Based on the system transformation and the stochastic scaled small‐gain theorem, stochastic stability of the original system is examined via the stochastic version of the bounded realness of the transformed forward system. The merit of the proposed approach lies in its reduced conservatism, which is made possible by a precise approximation of the time‐varying delay and the new result on the stochastic scaled small‐gain theorem. The proposed stability condition is demonstrated to be much less conservative than most existing results. Moreover, the problem of stabilization is further solved with an admissible controller designed via convex optimizations, whose effectiveness is also illustrated via numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability and stabilization of discrete‐time singular Markov jump systems with time‐varying delay

The stochastic stability and stochastic stabilization of time‐varying delay discrete‐time singular Markov jump systems are discussed. For full and partial knowledge of transition probabilities cases, delay‐dependent linear matrix inequalities (LMIs) conditions for the systems to be regular, causal and stochastically stable are given. Sufficient conditions are proposed for the existence of state feedback controller in terms of LMIs. Finally, two numerical examples to illustrate the effectiveness of the method are given. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Delay‐dependent robust control for singular discrete‐time Markovian jump systems with time‐varying delay

The problem of delay‐dependent robust stabilization for uncertain singular discrete‐time systems with Markovian jumping parameters and time‐varying delay is investigated. In terms of free‐weighting‐matrix approach and linear matrix inequalities, a delay‐dependent condition is presented to ensure a singular discrete‐time system to be regular, causal and stochastically stable based on which the stability analysis and robust stabilization problem are studied. An explicit expression for the desired state‐feedback controller is also given. Some numerical examples are provided to demonstrate the effectiveness of the proposed approach. Copyright © 2009 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.     

Delay‐dependent passivity and passification for uncertain Markovian jump systems with time‐varying delays

This paper deals with the problems of passivity analysis and passivity‐based controller design for Markovian jump systems with both time‐varying delays and norm‐bounded parametric uncertainties. Firstly, new delay‐dependent conditions for the considered system to be passive are obtained by using a mode‐dependent Lyapunov functional and by introducing some slack variables. These conditions are expressed by means of LMIs that are easy to check. It is shown through a numerical example that the obtained passivity conditions are less conservative than the existing ones in the literature. Secondly, the passification problem is investigated. On the basis of the obtained passivity conditions, dynamic output‐feedback controllers are designed, which ensure that the resulting closed‐loop system is passive. The effectiveness of the proposed design method is demonstrated by a numerical example. Copyright © 2011 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     

Delay‐dependent stability criterion and H∞ analysis for Markovian jump systems with time‐varying delays

This paper deals with the problems of stochastic stability and H_∞ analysis for Markovian jump linear systems with time‐varying delays. In terms of linear matrix inequalities, a less conservative delay‐dependent stability criterion for Markovian jump systems is proposed by constructing a different Lyapunov‐Krasovskii functional and introducing improved integral‐equalities approach, and a sufficient condition is derived from the H_∞ performance. Numerical examples are provided to demonstrate the efficiency and reduced conservatism of the results in this paper. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

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.     

State and mode feedback control strategy for discrete‐time Markovian jump linear systems with time‐varying controllable mode transition probability matrix

In this article, the state and mode feedback control strategy is investigated for the discrete‐time Markovian jump linear system (MJLS) with time‐varying controllable mode transition probability matrix (MTPM). This strategy, consisting of a state feedback controller and a mode feedback controller, is proposed to ensure MJLS's stability and meanwhile improve system performance. First, a mode‐dependent state feedback controller is designed to stabilize the MJLS based on the time‐invariant part of the MTPM such that it can still keep valid even if the MTPM is adjusted by the mode feedback control. Second, a generalized quadratic stabilization cost is put forward for evaluating MJLS's performance, which contains system state, state feedback controller, and mode feedback controller. To reduce the stabilization cost, a mode feedback controller is introduced to adjust each mode's occurrence probability by changing the time‐varying controllable part of MTPM. The calculation of such mode feedback controller is given based on a value‐iteration algorithm with its convergence proof. Compared with traditional state feedback control strategy, this state and mode feedback control strategy offers a new perspective for the control problem of general nonhomogeneous MJLSs. Numerical examples are provided to illustrate the validity of the proposed strategy.     

Stochastic stability and stabilization of discrete‐time singular Markovian jump systems with partially unknown transition probabilities

This paper considers the stochastic stability and stabilization of discrete‐time singular Markovian jump systems with partially unknown transition probabilities. Firstly, a set of necessary and sufficient conditions for the stochastic stability is proposed in terms of LMIs, then a set of sufficient conditions is proposed for the design of a state feedback controller to guarantee that the corresponding closed‐loop systems are regular, causal, and stochastically stable by employing the LMI technique. Finally, some examples are provided to demonstrate the effectiveness of the proposed approaches. Copyright © 2014 John Wiley & Sons, Ltd.     

Stability and stabilization of Markovian jump linear systems with partly unknown transition probabilities

In this paper, the stability and stabilization problems of a class of continuous-time and discrete-time Markovian jump linear system (MJLS) with partly unknown transition probabilities are investigated. The system under consideration is more general, which covers the systems with completely known and completely unknown transition probabilities as two special cases — the latter is hereby the switched linear systems under arbitrary switching. Moreover, in contrast with the uncertain transition probabilities studied recently, the concept of partly unknown transition probabilities proposed in this paper does not require any knowledge of the unknown elements. The sufficient conditions for stochastic stability and stabilization of the underlying systems are derived via LMIs formulation, and the relation between the stability criteria currently obtained for the usual MJLS and switched linear systems under arbitrary switching, are exposed by the proposed class of hybrid systems. Two numerical examples are given to show the validity and potential of the developed results.     

Robust stability,stabilization and ℋ∞ control of time‐delay systems with Markovian jump parameters

In this paper, the problems of stochastic stability and stabilization for a class of uncertain time‐delay systems with Markovian jump parameters are investigated. The jumping parameters are modelled as a continuous‐time, discrete‐state Markov process. The parametric uncertainties are assumed to be real, time‐varying and norm‐bounded that appear in the state, input and delayed‐state matrices. The time‐delay factor is constant and unknown with a known bound. Complete results for both delay‐independent and delay‐dependent stochastic stability criteria for the nominal and uncertain time‐delay jumping systems are developed. The control objective is to design a state feedback controller such that stochastic stability and a prescribed ?_∞‐performance are guaranteed. We establish that the control problem for the time‐delay Markovian jump systems with and without uncertain parameters can be essentially solved in terms of the solutions of a finite set of coupled algebraic Riccati inequalities or linear matrix inequalities. Extension of the developed results to the case of uncertain jumping rates is also provided. Copyright © 2003 John Wiley & Sons, Ltd.     

New results on stabilization of Markovian jump systems with time delay

This paper studies the problem of stochastic stabilization for a class of Markovian jump systems with time delay. A new delay-dependent stochastic stability criterion on the stochastic stability of the system is derived based on a novel Lyapunov-Krasovskii functional (LKF) approach. The equivalence and superiority to existing results are demonstrated. Then a state feedback controller, which guarantees the stochastic stability of the closed-loop system, is designed. Illustrative examples are provided to show the reduced conservatism and effectiveness of the proposed techniques.     

Robust H∞ deconvolution filtering for uncertain singular Markovian jump systems with time‐varying delays

The problem of H_∞ deconvolution filter design for a class of singular Markovian jump systems with time‐varying delays and parameter uncertainties is considered in this paper. By constructing a more comprehensive stochastic Lyapunov‐Krasovskii functional, novel delay‐dependent conditions are established to guarantee the filtering error system is not only stochastically admissible, but also satisfies a prescribed H_∞‐norm level for all admissible uncertainties. The desired filter parameters can be obtained by solving a set of strict linear matrix inequalities. Two examples and an electrical RLC circuit example are employed to verify the effectiveness and usefulness of the proposed methods in the paper. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Delay‐range‐dependent robust stability and stabilization for uncertain systems with time‐varying delay

This paper concerns delay‐range‐dependent robust stability and stabilization for time‐delay system with linear fractional form uncertainty. The time delay is assumed to be a time‐varying continuous function belonging to a given range. On the basis of a novel Lyapunov–Krasovskii functional, which includes the information of the range, delay‐range‐dependent stability criteria are established in terms of linear matrix inequality. It is shown that the new criteria can provide less conservative results than some existing ones. Moreover, the stability criteria are also used to design the stabilizing state‐feedback controllers. Numerical examples are given to demonstrate the applicability of the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Distributed delay‐dependent filtering for Markovian jump systems interconnected over an undirected graph with time‐delay

In this paper, the problems of delay‐dependent stochastic stability analysis and distributed filter synthesis are considered for Markovian jump systems interconnected over an undirected graph with state time‐invariant delay. A sufficient condition for the well‐posedness, delay‐dependent stochastic stability and contractiveness of the plant is developed in terms of linear matrix inequalities (LMIs). The distributed filter synthesis aims to design a distributed filter inheriting the structure of the plant such that the filtering error systems is well‐posed, delay‐dependent stochastically stable and contractive. Specifically, a corresponding sufficient condition to guarantee the filtering error system contractive is first presented by a set of nonlinear matrix inequalities. Next, for coupling these nonlinear matrix inequalities, a sufficient condition on the existence of such a distributed filter is proposed via a series of finite‐dimensional LMIs. Finally, a numerical simulation is presented to demonstrate the effectiveness of the proposed approach.