H ∞ model reduction for discrete-time Markov jump linear systems with partially known transition probabilities

In this article, the H _∞ model reduction problem for a class of discrete-time Markov jump linear systems (MJLS) with partially known transition probabilities is investigated. The proposed systems are more general, relaxing the traditional assumption in Markov jump systems that all the transition probabilities must be completely known. A reduced-order model is constructed and the LMI-based sufficient conditions of its existence are derived such that the corresponding model error system is internally stochastically stable and has a guaranteed H _∞ performance index. A numerical example is given to illustrate the effectiveness and potential of the developed theoretical results.     

Further results on H∞ control for discrete-time Markovian jump time-delay systems

This paper studies the problem of H_∞ control for a class of discrete-time Markovian jump systems with time delay. The purpose is to improve the existing results on H_∞ controller design for Markovian jump systems. A novel summation inequality is presented and an improved stability criterion for the system is derived by utilising the new inequality, which is proved to be less conservative than most results in the literature. Then the state feedback controller is designed, which guarantees the stochastic stability of the closed-loop system with a given disturbance attenuation. Numerical examples are provided to illustrate the effectiveness and advantages of the proposed techniques.     

H∞ control for conic non-linear jump systems with partially unknown transition probabilities

In this paper, the robust H_∞ control problem is investigated for conic non-linear stochastic jump systems with partially unknown transition probabilities and uncertain exogenous disturbance. By resorting to free-connection weighting matrix approach and matrix decomposition technique, two less conservative criteria are derived such that the nominal or uncertain Markovian jump non-linear systems are stochastically stable and have an H_∞ attention level. In addition, stochastic H_∞ control analysis and design are tackled for the class of stochastic jump systems by applying fixed weighting matrix method. Two numerical examples are presented to verify the effectiveness of theoretical results.     

H ∞ filtering for Markovian jump linear systems

The problem of H X filtering for continuous-time linear systems with Markovian jump is investigated. It was assumed that the jumping parameter was available. This paper develops necessary and sufficient conditions for designing a Markovian jump linear filter that ensures a prescribed bound on the L 2 -induced gain from the noise signals to the estimation error. The main result is tailored via linear matrix inequalities.     

Finite-time H∞ control for a class of discrete-time Markovian jump systems with partly unknown time-varying transition probabilities subject to average dwell time switching

An extension of a fixed transition probability (TP) Markovian switching model to combine time-varying TPs has offered another set of useful regime-switching models. This paper is concerned with the problem of finite-time H_∞ control for a class of discrete-time Markovian jump systems with partly unknown time-varying TPs subject to average dwell time switching. The so-called time-varying TPs mean that the TPs are varying but invariant within an interval. The variation of the TPs considered here is subject to a class of slow switching signal. Based on selecting the appropriate Lyapunov–Krasovskii functional, sufficient conditions of finite-time boundedness of Markovian jump systems are derived and the system trajectory stays within a prescribed bound. Finally, an example is given to illustrate the efficiency of the proposed method.     

Delay-dependent H ∞ performance analysis for Markovian jump systems with mode-dependent time varying delays and partially known transition rates

This paper deals with the H _∞ performance analysis problems for a class of Markovian jump systems with partially known transition rates and time-delays which are time varying and depend on system mode. Following the recent study on the class of systems, improved sufficient conditions for H _∞ performance of the underlying systems are derived in form of LMI by constructing a new Lyapunov-Krasovskii functional. Illustrative numerical examples are provided to demonstrate the effectiveness of the proposed approach.     

Robust ℋ∞ filtering of Markovian jump stochastic systems with uncertain transition probabilities

This article investigates the problem of robust ?_∞ filtering for a class of uncertain Markovian stochastic systems. The system under consideration not only contains Itô-type stochastic disturbances and time-varying delays, but also involves uncertainties both in the system matrices and in the mode transition rate matrix. Our aim is to design an ?_∞ filter such that, for all admissible parameter uncertainties and time-delays, the filtering error system can be guaranteed to be robustly stochastically stable, and achieve a prescribed ?_∞ disturbance rejection attenuation level. By constructing a proper stochastic Lyapunov–Krasovskii functional and employing the free-weighting matrix technique, sufficient conditions for the existence of the desired filters are established in terms of linear matrix inequalities, which can be readily solved by standard numerical software. Finally, a numerical example is provided to show the utility of the developed approaches.     

Reliable finite-time H∞ control of uncertain singular nonlinear Markovian jump systems with bounded transition probabilities and time-varying delay

This paper investigates the problem of reliable finite-time H_∞ control for one class of uncertainsingular nonlinear Markovian jump systems with time-varying delay subject to partial information on the transition probabilities. Continuous fault model is more general and practical to serve as the actuator fault. Time delay is a kind of positive time-varying differentiable bounded delays. First, based on a state estimator the resulting closed-loop error system is constructed and sufficient criteria are provided to guarantee that the augmented system is singular stochastic finite-time boundedness and singular stochastic H_∞ finite-time boundedness in both normal and fault cases via constructing a delay-dependent Lyapunov–Krasonskii function. Then, the gain matrices of state-feedback controller and state estimator are fixed by solving a feasibility problem in terms of linear matrix inequalities through decoupling technique, respectively. Finally, numerical examples are given to show the validity of the proposed design approach.     

H∞ filtering for piecewise homogeneous Markovian jump nonlinear systems

This paper concerns the problem of H_∞ filtering for piecewise homogeneous Markovian jump nonlinear systems. Different from the existing studies in the literatures, the existence of variations in transition rates for Markovian jump nonlinear systems is considered. The purpose of the paper is to design mode-dependent and mode-independent filters, such that the dynamics of the filtering errors are stochastic integral input-to-state stable with H_∞ performance index. Using the linear matrix inequality method and the Lyapunov functional method, sufficient conditions for the solution to the H_∞ filtering problem are derived. Finally, three examples are proposed to illustrate the effectiveness of the given theoretical results.     

Delay-dependent robust H ∞ control for uncertain fuzzy Markovian jump systems

This paper is concerned with the problem of delay-dependent robust H _∞ control for uncertain fuzzy Markovian jump systems with time delays. The purpose is to design a mode-dependent state-feedback fuzzy controller such that the closed-loop system is robustly stochastically stable and satisfies an H _∞ performance level. By introducing slack matrix variables, a delay-dependent sufficient condition for the solvability of the problem is proposed in terms of linear matrix inequalities. An illustrative example is finally given to show the applicability and effectiveness of the proposed method. Recommended by Editorial Board member Young Soo Suh under the direction of Editor Jae Weon Choi. This work is supported by the National Science Foundation for Distinguished Young Scholars of P. R. China under Grant 60625303, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20060288021, and the Natural Science Foundation of Jiangsu Province under Grant BK2008047. Yashun Zhang received the B.S. and M.S. degrees in Control Science and Control Engineering from Hefei University of Science and Technology in 2003 and 2006. He is currently a Ph.D. student in Control Science and Control Engineering, Nanjing University of Science and Technology. His research interests include fuzzy control, sliding mode control and nonlinear control. Shengyuan Xu received the Ph.D. degree in Control Science and Control Engineering from Nanjing University of Science and Technology in 1999. His research interests include robust filtering and control, singular systems, time-delay systems and nonlinear systems. Jihui Zhang is a Professor in the School of Automation Engineering of Qingdao University, China. His main areas of interest are discrete event dynamic systems, production planning and control, and operations research.     

H∞ model reduction for negative imaginary systems

This paper is concerned with the H_∞ model reduction for negative imaginary (NI) systems. For a given linear time-invariant system that is stable and NI, our goal is to find a stable reduced-order NI system satisfying a pre-specified H_∞ approximation error bound. Sufficient conditions in terms of matrix inequalities are derived for the existence and construction of an H_∞ reduced-order NI system. Iterative algorithms are provided to solve the matrix inequalities and to minimise the H_∞ approximation error. Finally, a numerical example is used to demonstrate the effectiveness of the proposed model reduction method.     

Improved H ∞ analysis of Markovian jumping stochastic systems with time-varying delays

This article investigates the problems of H _∞ analysis for Markovian jump stochastic systems with both nonlinear disturbance and time-varying delays. By virtue of the delay partition approach, the improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian jump stochastic systems are obtained in terms of linear matrix inequalities (LMIs). The proposed approach involves neither free weighting matrices nor any model transformation, and it is shown that the new criteria have the capability of providing less conservative results than the state-of-the-art. Two numerical simulations are conducted to demonstrate the effectiveness of the proposed method in comparison with existing methods.     

Sampled-data H∞ filtering for Markovian jump singularly perturbed systems with time-varying delay and missing measurements

In this paper, sampled-data H_∞ filtering problem is considered for Markovian jump singularly perturbed systems with time-varying delay and missing measurements. The sampled-data system is represented by a time-delay system, and the missing measurement phenomenon is described by an independent Bernoulli random process. By constructing an ?-dependent stochastic Lyapunov–Krasovskii functional, delay-dependent sufficient conditions are derived such that the filter error system satisfies the prescribed H_∞ performance for all possible missing measurements. Then, an H_∞ filter design method is proposed in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the feasibility and advantages of the obtained results.     

Robust H∞ control of uncertain stochastic Markovian jump systems with mixed time-varying delays

In this paper, robust H_∞ control for a class of uncertain stochastic Markovian jump systems (SMJSs) with interval and distributed time-varying delays is investigated. The jumping parameters are modelled as a continuous-time, finite-state Markov chain. By employing the Lyapunov-Krasovskii functional and stochastic analysis theory, some novel sufficient conditions in terms of linear matrix inequalities are derived to guarantee the mean-square asymptotic stability of the equilibrium point. Numerical simulations are given to demonstrate the effectiveness and superiority of the proposed method comparing with some existing results.     

Mixed H2/H∞ pitch control of wind turbine with a Markovian jump model

This paper proposes a Markovian jump model and the corresponding H₂/H_∞ control strategy for the wind turbine driven by the stochastic switching wind speed, which can be used to regulate the generator speed in order to harvest the rated power while reducing the fatigue loads on the mechanical side of wind turbine. Through sampling the low-frequency wind speed data into separate intervals, the stochastic characteristic of the steady wind speed can be represented as a Markov process, while the high-frequency wind speed in the each interval is regarded as the disturbance input. Then, the traditional operating points of wind turbine can be divided into separate subregions correspondingly, where the model parameters and the control mode can be fixed in each mode. Then, the mixed H₂/H_∞ control problem is discussed for such a class of Markovian jump wind turbine working above the rated wind speed to guarantee both the disturbance rejection and the mechanical loads objectives, which can reduce the power volatility and the generator torque fluctuation of the whole transmission mechanism efficiently. Simulation results for a 2 MW wind turbine show the effectiveness of the proposed method.     

Mode-dependent quantised H∞ filtering for Markovian jump singular system

This paper considers the H_∞ filtering problem for Markovian jump singular systems. In addition, when taking into account the effect of the quantisation, the designed filter not only guarantee admissibility but also satisfy a prescribed H_∞ filtering performance for the filter error system. By using Lyapunov method, a new bounded real lemma is proposed in terms of linear matrix inequalities (LMIs) to ensure that the filtering error system has the mentioned properties. Furthermore, a sufficient condition for the existence of filter is obtained in terms of LMIs. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed approach.     

ℋ ∞ model reduction for linear parameter-varying systems with distributed delay

This paper is concerned with the ?_∞ model reduction for linear parameter-varying (LPV) systems with both discrete and distributed delays. For a given stable system, our attention is focused on the construction of reduced-order models, which approximate the original system well in an ?_∞ norm sense. First, a sufficient condition is proposed for the asymptotic stability with an ?_∞ performance of the error system by using the parameter-dependent Lyapunov functional method. Then, the decoupling technique is applied, such that there does not exist any product term between the Lyapunov matrices and the system matrices in the parametrised linear matrix inequality (PLMI) constraints; thus a new sufficient condition is obtained. Based on the new condition, two different approaches are developed to solve the model reduction problem. One is the convex linearisation approach and the other is the projection approach. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method.     

Stabilisation and H∞ control of neutral stochastic delay Markovian jump systems

This paper investigates the exponential stabilisation and H_∞ control problem of neutral stochastic delay Markovian jump systems. First, a delay feedback controller is designed to stabilise the neutral stochastic delay Markovian jump system in the drift part. Second, sufficient conditions for the existence of feedback controller are proposed to ensure that the resulting closed-loop system is exponentially stable in mean square and satisfies a prescribed H_∞ performance level. Finally, numerical examples are provided to show the effectiveness of the proposed design methods.