Quadratic and H∞ switching control for discrete-time linear systems with multiplicative noises

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H_∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H_∞ performance costs are bounded above by a constant value. These results are illustrated with some numerical examples.     

Conjugation and H ∞ control of discrete-time systems

We establish conjugation notion in discrete-time systems, first introduced into the H ^∞ control theory of continuous-time systems by Kimura (1989). In discrete-time systems, conjugation is a very elementary operation on rational transfer functions that replaces some of their poles by their reflections with respect to the unit circle. With the aid of J-lossless conjugation conjugation by a J.lossless system), it is shown that the parametrization of sub-optimal solutions of H ^∞ model-matching problems is reduced to a Lyapunov-type equation. The parametrization of all solutions is given in an extremely simple way. It is further proved that the J-lossless conjugation of the H ^∞ model-matching problem is a natural state-space representation of classical interpolation in discrete-time systems.     

H ∞ filtering for stochastic systems with time-varying delay

This article considers the problem of H _∞ filter design for stochastic systems with time-varying delay. The time delay is assumed to be of interval type. Attention is focused on the design of delay-dependent filters that guarantee the asymptotic stability in mean square and a prescribed noise attenuation level in an H _∞ sense for the filtering error dynamics. The delay-dependent H _∞ filter design scheme is proposed in terms of a linear matrix inequality. A numerical example is used to illustrate the effectiveness of the proposed approach.     

Decentralised H ∞ filtering of interconnected discrete-time fuzzy systems with time delays

This article considers the decentralised H _∞ filtering of interconnected discrete-time fuzzy systems with time delays based on piecewise Lyapunov–Krasovskii functionals. The fuzzy system consists of J interconnected time-delay discrete-time Takagi–Sugeno fuzzy subsystems and the decentralised H _∞ filter is designed for each subsystem. It is shown that the stability with H _∞ performance of overall filtering error system can be established if a piecewise Lyapunov–Kroasovskii functional can be constructed, and moreover, the functional can be obtained by solving a set of linear matrix inequalities that are numerically feasible. A simulation example is given to show the effectiveness of the presented approach.     

Finite-time boundness of H ∞ filtering for switching discrete-time systems

For switched discrete-time systems, switching behavior always affect the finite-time stability property, which was neglected by most previous research. This paper investigated the problem of finite-time boundness of H _?? filtering for switched discrete-time systems. Sufficient conditions which can ensure finite-time bounded and H _?? filtering finite-time boundness under arbitrary switching are derived. Based on the results of finite-time boundness and stochastic character, the closed-loop system trajectory stays within a prescribed bound. An example is given to illustrate the efficiency of the proposed method.     

Finite-time H∞ filtering for non-linear stochastic systems

This paper describes the robust H_∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H_∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton–Jacobi inequality. Based on this result, the existence of a finite-time H_∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.     

H ∞ filtering for discrete-time singular networked systems with communication delays and data missing

In this article, the problem of H _∞ filter design is investigated for discrete-time singular networked systems with both multiple stochastic time-varying communication delays and probabilistic missing measurements. Two kinds of stochastic time-varying communication delays, namely stochastic discrete delays and stochastic distributed delays, are simultaneously considered. The purpose of the addressed filtering problem is to design a filter such that, for the admissible random measurement missing and communication delays, the filtering error dynamics is asymptotically stable in the mean square with a prescribed H _∞ performance index. In terms of linear matrix inequality (LMI) method, a sufficient condition is established that ensures the asymptotical stability in the mean square with a prescribed H _∞ performance index of the filtering error dynamics and then the filter parameters are characterised by the solution to an LMI. A numerical example is introduced to demonstrate the effectiveness of the proposed design procedures.     

H2/H∞ control for stochastic systems with delay

This paper is concerned with the H₂/H_∞ control problem for stochastic linear systems with delay in state, control and external disturbance-dependent noise. A necessary and sufficient condition for the existence of a unique solution to the control problem is derived. The resulting solution is characterised by a kind of complex generalised forward–backward stochastic differential equations with stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the equivalent feedback solution via a new type of Riccati equations. To explain the theoretical results, we apply them to a population control problem.     

Passivity and H∞ control of switched discrete-time nonlinear systems using linearisation

Local passivity and H_∞ control of switched discrete-time nonlinear systems are studied using the linearisation technique in this paper. We first establish LMI-based sufficient conditions under which theconsidered system is locally strictly QSR-dissipative. Then, two special cases of QSR-dissipativity, local passivity and l₂ gain, are investigated. In view of the derived conditions being all convex in linearised system matrices, local feedback passification and H_∞ control problems of switched discrete-time nonlinear systems are solved. The efficiency of the proposed method is verified through numerical examples.     

H∞ control for discrete-time Takagi-Sugeno fuzzy systems

We consider the design problem of H X controllers for discrete-time fuzzy systems. We introduce two Riccati inequalities from the standard H X theory, one for the state feedback controller, the other for the filter gain. We rewrite the second inequality and obtain an equivalent inequality for the inverse. We then rewrite this using LMI to obtain the final form of the inequality and show that the proposed output feedback controller is n -suboptimal. We give two examples and construct n -suboptimal controllers for them.     

Delay-dependent H ∞ control of linear discrete-time systems with an interval-like time-varying delay

The free-weighting-matrix approach is developed to study the H _∞ control of linear discrete-time systems with an interval-like time-varying delay. First, a delay- and range-dependent criterion for a given H _∞ performance is derived. Second, a memoryless H _∞ state-feedback controller is designed based on a performance analysis. Finally, two numerical examples demonstrate the effectiveness of the proposed method and show that both the upper bound and range of an interval-like time-varying delay affect the stability and/or H _∞ performance of a system.     

Delay-dependent H ∞ and generalized H 2 filtering for stochastic neural networks with time-varying delay and noise disturbance

This paper presents the delay-dependent \(H_\infty\) and generalized H ₂ filters design for stochastic neural networks with time-varying delay and noise disturbance. The stochastic neural networks under consideration are subject to time-varying delay in both the state and measurement equations. The aim is to design a stable full-order linear filter assuring asymptotical mean-square stability and a prescribed \(H_\infty\) or generalized H ₂ performance indexes for the filtering error systems. Delay-dependent sufficient conditions for the existence of \(H_\infty\) and generalized H ₂ filters are both proposed in terms of linear matrix inequalities. Finally, numerical example demonstrates that the proposed approaches are effective.     

Robust H 2 and H ∞ control of discrete-time systems with polytopic uncertainties via dynamic output feedback

This paper addresses the problem of robust H ₂ and H _∞ control of discrete linear time-invariant (LTI) systems with polytopic uncertainties via dynamic output feedback. The problem has been known to be difficult when a parameter dependent Lyapunov function is to be applied for a less conservative design due to non-convexity. Our approach is based on a novel bounding technique that converts the non-convex optimization into a convex one together with a line search, which is simple but may be conservative. To further reduce the design conservatism, an algorithm based on the sequentially linear programming method (SLPMM) is proposed. A numerical example is given which demonstrates the feasibility of the proposed design methods.     

H∞ filtering for discrete-time systems subject to stochastic missing measurements: a decomposition approach

This paper deals with the problem of H_∞ filtering for discrete-time systems with stochastic missing measurements. A new missing measurement model is developed by decomposing the interval of the missing rate into several segments. The probability of the missing rate in each subsegment is governed by its corresponding random variables. We aim to design a linear full-order filter such that the estimation error converges to zero exponentially in the mean square with a less conservatism while the disturbance rejection attenuation is constrained to a given level by means of an H_∞ performance index. Based on Lyapunov theory, the reliable filter parameters are characterised in terms of the feasibility of a set of linear matrix inequalities. Finally, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed design approach.     

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.     

Gain-scheduled H ∞ filtering of parameter-varying discrete-time systems via parameter-dependent Lyapunov functions

This paper is concerned with the problem of gain-scheduled H _∞ filter design for a class of parameter-varying discrete-time systems. A new LMI-based design approach is proposed by using parameter-dependent Lyapunov functions. Recommended by Editorial Board member Huanshui Zhang under the direction of Editor Jae Weon Choi. This work was supported in part by the National Natural Science Foundation of P. R. China under Grants 60874058, by 973 program No 2009CB320600, but also the National Natural Science Foundation of Province of Zhejiang under Grants Y107056, and in part by a Research Grant from the Australian Research Council. Shaosheng Zhou received the B.S. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1992, July 1996 and October 2001, from Qufu Normal University and Southeast University. His research interests include nonlinear control and stochastic systems. Baoyong Zhang received the B.S. and M.Sc. degrees in Applied Mathematics, in July 2003 and July 2006, all from Qufu Normal University. His research interests include and nonlinear systems, robust control and filtering. Wei Xing Zheng received the B.Sc. degree in Applied Mathematics and the M.Sc. and Ph.D. degrees in Electrical Engineering, in January 1982, July 1984 and February 1989, respectively, all from the Southeast University, Nanjing, China. His research interests include signal processing and system identification.