Sliding mode control for Markov jump linear uncertain systems with partly unknown transition rates

In this paper, based on sliding mode control approach, the robust stabilisation problem for a class of continuous-time Markovian jump linear uncertain systems with partly unknown transition rates is investigated. The transition rate matrix under consideration covers completely known, boundary known and completely unknown elements. By making use of linear matrix inequalities technique, sufficient conditions are presented to derive the linear switching surface and guarantee the stochastic stability of sliding mode dynamics. Then a sliding mode control law is designed to drive the state trajectory of the closed-loop system to the specified linear switching surface in finite time in spite of the existing uncertainties and unknown transition rates. Finally, an example is given to verify the validity of the theoretical results.     

转移概率一般不确知时滞Markov跳变神经网络的同步

研究了一类具有一般不确知转移概率的时滞Markov跳变神经网络的渐近同步问题.此类系统跳变过程的转移概率完全未知或者仅知道其估计值,因而更具有一般性.通过选择适当的Lyapunov-Krasovskii函数,利用线性矩阵不等式(LMIs)方法,得到了系统均方渐近同步的充分条件.最后,数值例子说明了所给结果的有效性.     

Filtering of discrete‐time Markov jump linear systems with uncertain transition probabilities

This article addresses the filtering design problem for discrete‐time Markov jump linear systems (MJLS) under the assumption that the transition probabilities are not completely known. We present the methods to determine ??₂‐ and ??_∞‐norm bounded filters for MJLS whose transition probability matrices have uncertainties in a convex polytope and establish an equivalence with the ones with partly unknown elements. The proposed design, based on linear matrix inequalities, allows different assumptions on Markov mode availability to the filter and on system parameter uncertainties to be taken into account. Under mode‐dependent assumption and internal model knowledge, observer‐based filters can be obtained and it is shown theoretically that our method outperforms some available ones in the literature to date. Numerical examples illustrate this claim. Copyright © 2010 John Wiley & Sons, Ltd.     

Fault estimation for a class of nonlinear Markov jump systems with general uncertain transition rates

This paper addresses the problem of fault estimation for a class of nonlinear Markov jump systems with Lipschitz-type nonlinearities and general transition rates allowed to be uncertain and unknown. First, by introducing a mode-dependent intermediate variable, an intermediate estimator is proposed to estimate faults and state simultaneously. Then, a vertex separator is exploited to develop a sufficient condition for the fault estimator design in terms of linear matrix inequalities. The design guarantees the boundedness in probability of the estimation errors if the derivations of the faults are bounded. Further, it is proved that the proposed approach is less conservative than the traditional methods. Finally, examples are given to show the advantages and effectiveness of the results.     

Anti-windup design of uncertain discrete-time Markovian jump systems with partially known transition probabilities and saturating actuator

This paper carries out a study on the design of anti-windup gains for uncertain discrete-time Markovian jump systems subject to both actuator saturation and partially known transition probabilities. The parameter uncertainties appearing in both the state and input matrices are assumed to be time-varying and norm-bounded. Under the assumption that a set of linear dynamic output feedback controllers have been designed to stabilise the Markovian jump system in the absence of actuator saturation, anti-windup compensation gains are designed for maximising the domain of attraction of the closed-loop system with actuator saturation. Then, by solving a convex optimisation problem with constraints of a set of linear matrix inequalities, the anti-windup compensation gains are obtained. A simulation example is provided to illustrate the effectiveness of the proposed technique.     

Finite-time dissipative control for semi-Markovian jump systems with time-varying delays and generally uncertain transition rates

In this paper, we mainly address the issue of finite-time stochastic boundedness (FTSB) and finite-time stochastic dissipation for semi-Markovian jump systems (SMJSs) with generally uncertain transition rates (TRs). Firstly, by resorting to linear matrix inequities (LMIs), some sufficient conditions on FTSB for SMJSs with uncertain TRs are introduced, which includes two cases. Then, dissipative control is developed and a state feedback controller is designed to ensure the FTSB of the closed-loop system by solving a series of LMIs. Finally, two examples are used to illustrate the results.     

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.     

随机非线性和部分转移概率未知时马尔科夫系统的H1控制

研究上述系统时: 1) 利用了非线性的概率分布信息; 2) 利用了转移概率中已知部分和未知部分的关系. 利用李雅普诺夫泛函方法和线性矩阵不等式方法, 本文得到了使得系统随机稳定的充分条件并得到了相应的反馈控制增益. 文中最后给出的例子表明了所建立模型和分析方法的有效性.     

Discrete‐time output feedback for Markov jump systems with uncertain transition probabilities

This article addresses the output feedback control for discrete‐time Markov jump linear systems. With fully known transition probability, sufficient conditions for an internal model based controller design are obtained. For the case where the transition probabilities are uncertain and belong to a convex polytope with known vertices, we provide a sufficient LMI condition that guarantees the norm of the closed‐loop system is below a prescribed level. That condition can be improved through an iterative procedure. Additionally, we are able to deal with the case of cluster availability of the Markov mode, provided that some system matrices do not vary within a given cluster, an assumption that is suitable to deal with packet dropout models for networked control systems. A numerical example shows the applicability of the design and compares it with previous results. Copyright © 2012 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.     

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.     

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.     

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.     

不确定状态滞后线性跳跃系统的时滞相关鲁棒非脆弱H∞控制

讨论了一类具有Markov跳跃参数的不确定混合线性时滞系统的鲁棒非脆弱控制问题.给出了使系统鲁棒随机稳定并具有给定的H∞性能的充分条件.并且通过参数变换和Schur补定理,将已得出的充分条件转化成一系列耦合的线性矩阵不等式形式以便于控制器参数的求解.仿真结果表明了本文提出的鲁棒非脆弱控制方法的有效性.     

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 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.     

Finite‐time guaranteed cost control for Itô Stochastic Markovian jump systems with incomplete transition rates

This paper is concerned with the finite‐time guaranteed cost control problem for stochastic Markovian jump systems with incomplete transition rates. By a mode‐dependent approach (MDA), several new sufficient conditions for the existence of state and output feedback finite‐time guaranteed cost controllers are provided, and the upper bound of cost function is more accurately expressed. Moreover, these results' superiorities are analyzed and shown. A new N‐mode optimization algorithm is given to minimize the upper bound of cost function. Finally, a detailed example is utilized to demonstrate the merit of the proposed results. Copyright © 2016 John Wiley & Sons, Ltd.     

Fuzzy-model-based robust control of Markov jump nonlinear systems with incomplete transition probabilities and uncertain packet dropouts

Interval type-2 fuzzy Markov jump systems (IT2FMJSs) have received much attention because they can well describe complex nonlinear systems with uncertainties and stochastic system mode switching. However, the transition probabilities of fuzzy MJSs (FMJSs) have been assumed to be completely known, limiting real-world applications of existing results. Different from the previous studies, transition probabilities between system modes switching are partially unknown, and packet dropouts of data transmission are uncertain in this study. Compared with the previous studies, the main advantages of this work are as follows: (1) To analyze stochastic stability and reduce conservatism of existing approaches, a novel Lyapunov function that depends on both system mode and fuzzy basis function is constructed; (2) the existence of a mode-dependent and fuzzy-basis-dependent state feedback controller is revealed; (3) stochastic stability of closed-loop system with a desired $H_{\infty }$ performance is established, and the problem of incomplete transition probabilities and uncertain packet dropouts has been completely addressed. An illustrative example of a robot arm is used to demonstrate the effectiveness and practicality of the proposed control strategy. By virtue of the proposed strategy, the effects of incomplete transition probabilities and uncertain packet dropouts on IT2FMJSs have been completely alleviated.