ℋ︁∞ filtering for discrete‐time linear systems with Markovian jumping parameters†

This paper investigates the problem of ??_∞ filtering for discrete‐time linear systems with Markovian jumping parameters. It is assumed that the jumping parameter is available. This paper develops necessary and sufficient conditions for designing a discrete‐time Markovian jump linear filter which ensures a prescribed bound on the ?₂‐induced gain from the noise signals to the estimation error. The proposed filter design is given in terms of linear matrix inequalities. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

ℋ︁∞ and l2–l∞ filtering for two‐dimensional linear parameter‐varying systems

In this paper, the ??_∞ and l₂–l_∞ filtering problem is investigated for two‐dimensional (2‐D) discrete‐time linear parameter‐varying (LPV) systems. Based on the well‐known Fornasini–Marchesini local state‐space (FMLSS) model, the mathematical model of 2‐D systems under consideration is established by incorporating the parameter‐varying phenomenon. The purpose of the problem addressed is to design full‐order ??_∞ and l₂–l_∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in ??_∞ and l₂–l_∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method. Copyright © 2007 John Wiley & Sons, Ltd.     

Exponential H ∞ filtering for singular systems with Markovian jump parameters

This paper is concerned with the H _∞ filter design for continuous‐time singular systems with Markovian jump parameters, whose system mode is transmitted through an unreliable network. In contrast to the traditionally mode‐dependent or mode‐independent filtering method, a new partially mode‐dependent filter is established via using a mode‐dependent Lyapunov function, where the stochastic property of mode available to a filter is considered. Sufficient conditions for the existence of H _∞ filter are obtained as strict linear matrix inequalities. Finally, numerical examples are used to show the effectiveness of the given theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

Robust H ∞ filtering for singular time‐delayed systems with uncertain Markovian switching probabilities

This paper is concerned with the robust H _∞ filter design for a class of uncertain singular time‐delayed Markovian jump systems, whose transition rate matrix has elementwise bounded uncertainties. By the LMI approach, a novel bounded real lemma is proposed such that the singular Markovian jump system is robustly exponentially mean‐square admissible with a prescribed H _∞ performance index. Based on this, a sufficient condition for the existence of a robust H _∞ filter is developed in terms of LMIs. Finally, a numerical example is provided to show the effectiveness of the theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust H 2 Filtering for Discrete‐Time Markovian Jump Linear Systems

This paper studies the problems of H₂ filtering for discrete‐time Markovian jump linear systems with non‐accessible mode information. The objective is to design a less‐conservative mode‐independent H₂ filter such that the filtering error system is stochastically stable. To this end, sufficient conditions for the existence of an upper bound of H₂ norm are given in terms of linear matrix inequalities. With the introduction of a slack variable, a less‐conservative filter is derived. The proposed robust H₂ filter design method is also applicable to cover the cases where the system matrices are subject to polytopic uncertainty, as well as the case where the transition probability matrix is subject to a given polytopic uncertainty. Application of the proposed method to three examples from the literature demonstrates the favorable performance of the proposed solution to existing approaches.     

Exponential H ∞ filtering for stochastic Markovian jump systems with time delays

This paper investigates the problem of exponential H _∞ filtering for stochastic systems with time delays and Markovian jumping parameters. On the basis of Lyapunov–Krasovskii functional theory and generalized Finsler lemma, a delay‐dependent bounded real lemma is established without using any model transformations, bounding techniques for cross terms, or additional free matrix variables. The obtained bounded real lemma guarantees that the filtering error system is both mean‐square exponentially stable and almost surely exponentially stable with a prescribed H _∞ noise attenuation level. Then an exponential H _∞ filter is designed for stochastic retarded Markovian jump systems in terms of a set of LMIs. Meanwhile, the mathematical equivalence of the proposed method to one recent method is presented, but our proposed method is more computationally efficient with fewer matrix variables than that recent method. The validity of the method is verified by a numerical example.Copyright © 2012 John Wiley & Sons, Ltd.     

A generalized robust H ∞ filtering for singular Markovian jump systems and its applications

In this paper, a generalized robust H _∞ filtering method is proposed for a class of singular Markovian jump systems, whose generality is mainly embodied that the desired filter could bear perturbances in terms of uncertainties on its parameter matrices. Firstly, an LMI condition of robust mode‐dependent filter is developed. Based on the given result, a new approach to mode‐independent H _∞ filter is presented, which establishes a direct connection between mode‐dependent and mode‐independent filters. Secondly, when the transition rate matrix is with elementwise bounded uncertainties or partially unknown, sufficient conditions of such robust mode‐dependent and mode‐independent filters are all developed within LMI frameworks. Finally, a numerical example is used to demonstrate the effectiveness of the proposed methods. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Gain‐scheduled ℋ︁2 controller synthesis for linear parameter varying systems via parameter‐dependent Lyapunov functions

This paper deals with the problem of gain‐scheduled ??₂ control for linear parameter‐varying systems. The system state–space model matrices are affinely parameterized and the admissible values of the parameters and their rate of variation are supposed to belong to a given convex bounded polyhedral domain. Based on a parameter‐dependent Lyapunov function, a linear matrix inequality methodology is proposed for designing a gain‐scheduled state feedback ??₂ controller, where the feedback gain is a matrix fraction of polynomial matrices with quadratic dependence on the scheduling parameters. Copyright © 2005 John Wiley & Sons, Ltd.     

Gain‐scheduled dynamic output feedback control for discrete‐time LPV systems

This paper presents synthesis conditions for the design of gain‐scheduled dynamic output feedback controllers for discrete‐time linear parameter‐varying systems. The state‐space matrix representation of the plant and of the controller can have a homogeneous polynomial dependency of arbitrary degree on the scheduling parameter. As an immediate extension, conditions for the synthesis of a multiobjective ??_∞ and ??₂ gain‐scheduled dynamic feedback controller are also provided. The scheduling parameters vary inside a polytope and are assumed to be a priori unknown, but measured in real‐time. If bounds on the rate of parameter variation are known, they can be taken into account, providing less conservative results. The geometric properties of the uncertainty domain are exploited to derive finite sets of linear matrix inequalities based on the existence of a homogeneous polynomially parameter‐dependent Lyapunov function. An application of the control design to a realistic engineering problem illustrates the benefits of the proposed approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Simultaneous ℋ︁2/ℋ︁∞ control of uncertain jump systems with functional time‐delays

This paper presents new results pertaining to the control design of a class of linear uncertain systems with Markovian jump parameters. An integral part of the system dynamics is a delayed state in which the time‐delays are mode dependent. The jumping parameters are modelled as a continuous‐time, discrete‐state Markov process and the uncertainties are norm‐bounded. We construct an appropriate Lyapunov–Krasovskii functional and design a simultaneous ℋ︁₂/ℋ︁_∞ controller which minimizes a quadratic ℋ︁₂ performance measure while satisfying a prescribed ℋ︁_∞‐norm bound on the closed‐loop system. It is established that sufficient conditions for the existence of the simultaneous ℋ︁₂/ℋ︁_∞ controller and the associated performance upper bound are cast in the form of linear matrix inequalities. Simulation results are provided and extension to the case where the jumping rates are subject to uncertainties is presented. Copyright © 2007 John Wiley & Sons, Ltd.     

H∞ filtering for singular Markovian jump systems with time delay

The problem of H_∞ filtering is considered for singular Markovian jump systems with time delay. In terms of linear matrix inequality (LMI) approach, a delay‐dependent bounded real lemma (BRL) is proposed for the considered system to be stochastically admissible while achieving the prescribed H_∞ performance condition. Based on the BRL and under partial knowledge of the jump rates of the Markov process, both delay‐dependent and delay‐independent sufficient conditions that guarantee the existence of the desired filter are presented. The explicit expression of the desired filter gains is also characterized by solving a set of strict LMIs. Some numerical examples are given to demonstrate the effectiveness of the proposed methods. Copyright © 2009 John Wiley & Sons, Ltd.     

ℒ︁2‐Gain analysis and control of uncertain nonlinear systems with bounded disturbance inputs

This paper proposes a convex approach to regional stability and ℒ︁₂‐gain analysis and control synthesis for a class of nonlinear systems subject to bounded disturbance signals, where the system matrices are allowed to be rational functions of the state and uncertain parameters. To derive sufficient conditions for analysing input‐to‐output properties, we consider polynomial Lyapunov functions of the state and uncertain parameters (assumed to be bounded) and a differential‐algebraic representation of the nonlinear system. The analysis conditions are written in terms of linear matrix inequalities determining a bound on the ℒ︁₂‐gain of the input‐to‐output operator for a class of (bounded) admissible disturbance signals. Through a suitable parametrization involving the Lyapunov and control matrices, we also propose a linear (full‐order) output feedback controller with a guaranteed bound on the ℒ︁₂‐gain. Numerical examples are used to illustrate the proposed approach. Copyright © 2007 John Wiley & Sons, Ltd.     

Robust H∞ filtering for a class of Markovian jump systems with time‐varying delays based on delta operator approach

In this paper, the robust H_∞ filtering problem for a class of discrete Markovian jump systems with time‐varying delays and linear fractional uncertainties is investigated based on delta operator approach. Based on Lyapunov‐Krasovskii functional in delta domain, new delay‐dependent sufficient conditions for the solvability of this problem are presented in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, an explicit expression of a desired jump H_∞ filter is given. The proposed method can unify some previous related continuous and discrete systems into the delta operator systems framework. Numerical examples are given to illustrate the effectiveness of the developed techniques. © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Exponential H∞ filter design for stochastic time‐varying delay systems with Markovian jumping parameters

In this paper, the exponential H_∞ filter design problem is investigated for a general class of stochastic time‐varying delay system with Markovian jumping parameters. The stochastic uncertainties appear in both the dynamic and the measurement equations and the state delay is assumed to be time‐varying. Attention is focused on the design of mean‐square exponentially stable and Markovian jump filter such that the filtering error systems are mean‐square exponentially stable and the estimation error satisfies a given H_∞ performance. By introducing some slack matrix variables, delay‐dependent sufficient conditions for the solvability of the above problem are presented in terms of linear matrix inequalities (LMIs). In addition, the decay rate can be a given positive value without any other constraints. When the proposed LMIs are feasible, an explicit expression of the desired H_∞ filter can be given. A numerical example is provided to illustrate the effectiveness of the proposed design approach. Copyright © 2009 John Wiley & Sons, Ltd.     

An augmented system approach to static output‐feedback stabilization with ℋ︁∞ performance for continuous‐time plants

This paper revisits the static output‐feedback stabilization problem of continuous‐time linear systems from a novel perspective. The closed‐loop system is represented in an augmented form, which facilitates the parametrization of the controller matrix. Then, new equivalent characterizations on stability and ??_∞ performance of the closed‐loop system are established in terms of matrix inequalities. On the basis of these characterizations, a necessary and sufficient condition with slack matrices for output‐feedback stabilizability is proposed, and an iteration algorithm is given to solve the condition. An extension to output‐feedback ??_∞ control is provided as well. The effectiveness and merits of the proposed approach are shown through several examples. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust delay‐dependent ℋ︁∞ control of time‐delay systems with state and input delays

This paper studies the design problem of robust delay‐dependent ??_∞ controller for a class of time‐delay control systems with time‐varying state and input delays, which are assumed to be noncoincident. The system is subject to norm‐bounded uncertainties and ??₂ disturbances. Based on the selection of an augmented form of Lyapunov–Krasovskii (L‐K) functional, first a Bounded Real Lemma (BRL) is obtained in terms of linear matrix inequalities (LMIs) such that the nominal, unforced time‐delay system is guaranteed to be globally asymptotically stable with minimum allowable disturbance attenuation level. Extending BRL, sufficient delay‐dependent criteria are developed for a stabilizing ??_∞ controller synthesis involving a matrix inequality for which a nonlinear optimization algorithm with LMIs is proposed to get feasible solution to the problem. Moreover, for the case of existence of norm‐bounded uncertainties, both the BRL and ??_∞ stabilization criteria are easily extended by employing a well‐known bounding technique. A plenty of numerical examples are given to illustrate the application of the proposed methodology of this note. The achieved numerical results on the maximum allowable delay bound and minimum allowable disturbance attenuation level are exhibited to be less conservative in comparison to those of existing methods in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

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 filter design for piecewise discrete‐time systems with time‐varying delays

A novel delay‐dependent filtering design approach is developed for a class of linear piecewise discrete‐time systems with convex‐bounded parametric uncertainties and time‐varying delays. The time‐delays appear in the state as well as the output and measurement channels. The filter has a linear full‐order structure and guarantees the desired estimation accuracy over the entire uncertainty polytope. The desired accuracy is assessed in terms of either ??_∞‐performance or ??₂–??_∞ criteria. A new parametrization procedure based on a combined Finsler's Lemma and piecewise Lyapunov–Krasovskii functional is established to yield sufficient conditions for delay‐dependent filter feasibility. The filter gains are determined by solving a convex optimization problem over linear matrix inequalities. In comparison to the existing design methods, the developed methodology yields the least conservative measures since all previous overdesign limitations are almost eliminated. By means of simulation examples, the advantages of the developed technique are readily demonstrated. Copyright © 2009 John Wiley & Sons, Ltd.