A partially delay‐dependent and disordered controller design for discrete‐time delayed systems

This paper considers the stabilization problem for a class of discrete‐time delayed systems by exploiting a partially delay‐dependent controller whose gains suffer a disordering phenomenon simultaneously. Two stochastic variables are used to describe the partially delay‐dependent and disordering properties, which are not independent, and referred to the original operation modes here. By introducing an augmented Markov chain, the corresponding closed‐loop system is transformed into a Markovian jump system with four new operation modes (NOMs). Based on the proposed model, a kind of controller depending on NOMs is firstly proposed with linear matrix inequalities forms. Moreover, without designing a controller containing NOMs directly, another kind of stabilizing controller referring to one depending on original operation modes is developed, which is composed of a series of NOM‐dependent controllers and satisfies a minimum variance approximation. Finally, two numerical examples are used to demonstrate the utility and superiority of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

不确定广义跳变时滞系统的鲁棒指数稳定性

讨论不确定广义跳变时滞系统的鲁棒指数稳定性问题. 利用线性矩阵不等式(LMI)技术给出一个时滞区间依赖条件保证标称系统正则、无脉冲且均方指数稳定. 同时该准则也被推广至不确定系统. 数值例子说明本文的结果改进了已有的结论.     

Delay-dependent stability analysis for discrete-time singular Markovian jump systems with time-varying delay

This article considers the problem of delay-dependent stability analysis for a class of discrete-time singular systems with Markovian jump and time-varying delay. The transition probabilities in Markov chain are assumed to be partially unknown. In terms of linear matrix inequality approach, the delay-dependent criteria are proposed to ensure the underlying system to be regular, causal and stochastically stable. Several numerical examples are given to demonstrate the effectiveness and less conservatism of the obtained results.     

Real-time guaranteed cost control of MIMO networked control systems with packet disordering

This paper discusses guaranteed cost control for multi-input and multi-output (MIMO) networked control systems (NCSs) with multi-channel packet disordering. Considering the time-varying and bounded network transmission delay, packet dropout and packet disordering, a novel model of NCSs is proposed by introducing the concept of packet displacement. It is worthwhile mentioned that this model can fully describe the dynamic characteristic of network and always guarantee the newest control input executed by the plant, which makes that the plant can be controlled in real time. The resulting closed-loop systems are jump linear systems due to the newest signals executed subject to Markovian chains. A real-time controller is designed for uncertain and certain NCSs based on Markovian theory combined with linear matrix inequality (LMI) techniques such that the closed-loop cost function value is not more than a specified upper bound that varies according to Quality of Services (QOS). Finally, numerical examples are given to illustrate the effectiveness of the proposed method.     

Passivity analysis and passification of uncertain Markovian jump systems with partially known transition rates and mode-dependent interval time-varying delays

In this paper, the problem of delay-dependent robust passivity analysis and robust passification of uncertain Markovian jump linear systems (MJLSs) with partially known transition rates and mode-dependent time-varying delays are investigated. In the deterministic model, the time-varying delay is in a given range and the uncertainties are assumed to be norm bounded. By constructing an appropriate Lyapunov–Krapunov functional (LKF) combining with Jensen’s inequality and the free-weighting matrix method, delay-dependent passification conditions are obtained in term of linear matrix inequalities(LMIs). For the robust passification problem, desired passification controllers are designed, which guarantee that the closed-loop MJLS is passive. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach.     

A Markovian jump system approach to consensus of heterogeneous multiagent systems with partially unknown and uncertain attack strategies

The problem of robust leader‐following consensus of heterogeneous multiagent systems subject to deny‐of‐service attacks is investigated, where attack strategies are partially unknown and uncertain to defender. A Markovian jump system approach is proposed, that is, capable of describing the occurrence of different attack strategies, and the occurring probability of each attack strategy is represented by the transition probability of the Markovian jump model. Then, sufficient conditions are derived such that the output tracking performance can be guaranteed. In order to design the controller gains, some slack matrices are introduced, which can provide some design freedom. Finally, it is shown that the controller design results can be applied to the multivehicle position‐tracking system. The simulation results reveal that the consensus performance is much better if one has more statistics information on attacks.     

具有时变输入与状态时滞的Markov跳变系统鲁棒H∞控制

研究一类同时具有时变输入和状态时滞的不确定Markov跳变系统的时滞相关鲁棒H∞控制问题. 采用新的Lyapunov-Krasovskii泛函, 基于积分不等式方法, 使所得结果具有较小的保守性. 避免对时变时滞变化率等的限制. 控制器可以通过锥补线性化方法求解. 数值算例显示所提出方法的有效性.     

Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays

This paper focuses on the problem of delay-dependent robust stochastic stability analysis and controller synthesis for Markovian jump systems with state and input delays. It is assumed that the delays are constant and unknown, but their upper bounds are known. By constructing a new Lyapunov-Krasovskii functional and introducing some appropriate slack matrices, new delay-dependent stochastic stability and stabilization conditions are proposed by means of linear matrix inequalities (LMIs). An important feature of the results proposed here is that all the robust stability and stabilization conditions are dependent on the upper bound of the delays. Memoryless state feedback controllers are designed such that the closed-loop system is robustly stochastically stable. Some numerical examples are provided to illustrate the effectiveness of the proposed method.     

On reachable set synthesis of delay Markovian jump systems under partially delay-dependent asynchronous control

This paper pays close attention to the reachable set synthesis problem for discrete-time delayed Markovian jump systems (DDMJSs) under partially asynchronous control. Different from the traditional controller, a class of partially asynchronous controller is proposed, where delay-free and delay states are comprised and cannot take place at the same time. In addition, Bernoulli variable is used to describe the probability distribution of the above two kinds of states, and the design of feedback controller is considered. On this basis, the sufficient conditions which the state feedback controller is existent are solved effectively by using linear matrix inequalities (LMIs). Furthermore, the results of reachable set synthesis are extended to the DDMJs with uncertain transition probabilities. Finally, several numerical simulations are given to show the effectiveness of the presented methods.     

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.     

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.     

Delay-dependent stabilization of singular Markovian jump systems with state delay

This paper deals with the delay-dependent stabilization problem for singular systems with Markovian jump parameters and time delays. A delay-dependent condition is established for the considered system to be regular, impulse free and stochastically stable. Based on the condition, a design algorithm of the desired state feedback controller which guarantees the resultant closed-loop system to be regular, impulse free and stochastically stable is proposed in terms of a set of strict linear matrix inequalities （LMIs）. Numerical examples show the effectiveness of the proposed methods.     

Robust delayed-state-feedback stabilization of uncertain stochastic systems

Due to time spent in computation and transfer, control input is usually subject to delays. Problems of deterministic systems with input delay have received considerable attention. However, relatively few works are concerned with problems of stochastic system with input delay. This paper studies delayed-feedback stabilization of uncertain stochastic systems. Based on a new delay-dependent stability criterion established in this paper, a robust delayed-state-feedback controller that exponentially stabilizes the uncertain stochastic systems is proposed. Numerical examples are given to verify the effectiveness and less conservativeness of the proposed method.     

Delay-range-dependent robust stabilisation and H ∞ control for nonlinear uncertain stochastic fuzzy systems with mode-dependent time delays and Markovian jump parameters

This article focuses on the problems of robust stabilisation and H _∞ control for nonlinear uncertain stochastic systems with mode-dependent time delay and Markovian jump parameters represented by the Takagi–Sugeno (T-S) fuzzy model approach. The system under consideration involves parameter uncertainties, Itô-type stochastic disturbances, Markovian jump parameters and unknown nonlinear disturbances. The purpose is to design a state feedback controller such that the closed-loop system is robustly exponentially stable in the mean square and satisfies a prescribed H _∞ performance level. Novel delay-range-dependent conditions in the form of linear matrix inequalities (LMIs) are derived for the solvability of robust stabilisation and H _∞ control problem. A desired fuzzy controller can be constructed by solving a set solutions of LMIs and can be easily calculated by Matlab LMI control toolbox. Finally, a numerical example is presented to illustrate the proposed method.     

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.     

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 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.     

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.     

Stabilization of discrete-time Markovian jump linear systems via time-delayed controllers

This paper is concerned with the stabilization problem for a class of discrete-time Markovian jump linear systems with time-delays both in the system state and in the mode signal. The delay in the system state may be time-varying. The delay in the mode signal is manifested as a constant mismatch of the modes between the controller and the system. We first show that the resulting closed-loop system is a time-varying delayed Markovian jump linear system with extended state space. Then a sufficient condition is proposed for the design of a controller such that the closed-loop system is stochastically stable. Finally, numerical simulation is used to illustrate the developed theory.     

On robust stabilization of Markovian jump systems with uncertain switching probabilities

This brief paper is concerned with the robust stabilization problem for a class of Markovian jump linear systems with uncertain switching probabilities. The uncertain Markovian jump system under consideration involves parameter uncertainties both in the system matrices and in the mode transition rate matrix. First, a new criterion for testing the robust stability of such systems is established in terms of linear matrix inequalities. Then, a sufficient condition is proposed for the design of robust state-feedback controllers. A globally convergent algorithm involving convex optimization is also presented to help construct such controllers effectively. Finally, a numerical simulation is used to illustrate the developed theory.