ℋ 2 filtering of discrete-time Markov jump linear systems through linear matrix inequalities

This paper addresses the ?₂ filtering design problem of discrete-time Markov jump linear systems. First, under the assumption that the Markov parameter is measured, the main contribution is on the LMI characterisation of all filters such that the estimation error remains bounded by a given ?₂ norm level, yielding the complete solution of the mode-dependent filtering design problem. Based on this result, a robust filter design to deal with convex bounded parameter uncertainty is considered. Second, from the same LMI characterisation, a design procedure for mode-independent filtering design is proposed. Some examples are solved for illustration and comparison.     

Robust H_∞ control for discrete-time polytopic uncertain systems with linear fractional vertices

The robust H∞ control problem for discrete-time uncertain systems is investigated in this paper. The uncertain systems are modelled as a polytopic type with linear fractional uncertainty in the vertices. A new linear matrix inequality (LMI) characterization of the H∞ performance for discrete systems is given by introducing a matrix slack variable which decouples the matrix of a Lyapunov function candidate and the parametric matrices of the system. This feature enables one to derive sufficient conditions for discrete uncertain systems by using parameter-dependent Lyapunov functions with less conservativeness. Based on the result, H∞ performance analysis and controller design are carried out. A numerical example is included to demonstrate the effectiveness of the proposed results.     

Kalman filtering for time-delayed linear systems

The problem of estimation has been one of the key research topics of control commu- nity since Wiener filtering[1]. However, only time-invariant single-variable stationary signal can be considered for Wiener filtering. In the 1960’s, Kalman filtering[2,3…     

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.     

Robust H∞ filtering for 2D continuous systems with finite frequency specifications

This paper investigates the design problem of robust H_∞ filtering for uncertain two-dimensional (2D) continuous systems described by Roesser model with polytopic uncertainties and frequency domain specifications. Our aim is to design a new filter guaranteeing an H_∞ performance level in specific finite frequency (FF) domains. Using the well-known generalised Kalman Yakubovich Popov lemma and homogeneous polynomially parameter-dependent matrices of arbitrary degrees, sufficient conditions for the existence of H_∞ filters for different FF ranges are proposed and then unified in terms of solving a set of linear matrix inequalities. Illustrative examples are provided to show the usefulness and potential of the proposed results.     

Robust ℓ 1 performance analysis for linear systems with parametric uncertainties

In this contribution, a computational approach for analysing the robust ?_∞-gain (or the robust ?₁ performance) of uncertain linear systems is developed. In particular, the system's state-space matrices may have a rational dependence on structured parametric time-invariant or time-varying uncertainties. The computation is based on robust semi-definite programming and provides a trade-off between accuracy and computational effort. A novel matrix inequality condition to determine the star-norm of discrete-time systems is derived as an auxiliary result.     

Asynchronous H∞ filtering for linear switched systems with average dwell time

This paper is concerned with the H_∞ filtering problem for a class of continuous-time linear switched systems with the asynchronous behaviours, where ‘asynchronous’ means that the switching of the filters to be designed has a lag to the switching of the system modes. By using the Lyapunov-like functions and the average dwell time technique, a sufficient condition is obtained to guarantee the asymptotic stability with a weighted H_∞ performance index for the filtering error system. Moreover, the results are formulated in the form of linear matrix inequalities that are numerical feasible. As a result, the filter design problem is solved. Finally, an illustrative numerical example is presented to show the effectiveness of the results.     

Robust H-infinity filtering for time-delay systems with missing measurements： a parameter-dependent approach

Robust H-infinity filtering for a class of uncertain discrete-time linear systems with time delays and missing measurements is studied in this paper. The uncertain parameters are supposed to reside in a convex polytope and the missing measurements are described by a binary switching sequence satisfying a Bernoulli distribution. Our attention is focused on the analysis and design of robust H-infinity filters such that, for all admissible parameter uncertainties and all possible missing measurements, the filtering error system is exponentially mean-square stable with a prescribed H-infinity disturbance attenuation level. A parameter-dependent approach is proposed to derive a less conservative result. Sufficient conditions are established for the existence of the desired filter in terms of certain linear matrix inequalities （LMIs）. When these LMIs are feasible, an explicit expression of the desired filter is also provided. Finally, a numerical example is presented to illustrate the effectiveness and applicability of the proposed method.     

Robust H ∞ filtering for uncertain discrete piecewise time-delay systems

This paper investigates the problem of robust H _∞ filter design for uncertain discrete piecewise time-delay systems based on a piecewise Lyapunov functional. The parametric uncertainties are assumed to be time-varying but norm bounded. The purpose is the design of a piecewise filter such that, for all admissible uncertainties, the resulting filtering error system is asymptotically stable and satisfies a prescribed H _∞ performance level. By introducing some different extra matrix variables, a sufficient condition for the solvability of this problem is obtained in terms of linear matrix inequalities (LMIs). When these LMIs are feasible, the explicit expression of a desired piecewise filter is given. Two numerical examples are provided to demonstrate the effectiveness of the proposed design method.     

Robust H ∞ filtering for discrete time-varying uncertain systems with a known deterministic input

In this paper, a new discrete-time dynamic game where two players seek to reinforce and suppress the output of a system under the influence of a known deterministic input is considered. A necessary and sufficient condition for the existence of a saddle-point solution to this game is derived in terms of a discrete-time Riccati difference equation. The result of the game is used to solve the robust H X filtering problem for discrete time-varying systems subject to a known deterministic input and norm-bounded parameter uncertainties occurring in both the state and the output matrices of the state-space model. There is no restriction on the form of the filter. This allows the effect of the known input on the estimation error due to system uncertainties to be optimally reduced without any prior assumption on the filter structure.     

Robust mixed H 2/H ∞ filtering for discrete-time delay fuzzy systems

A robust mixed H ₂/H∞ filtering problem for discrete-time fuzzy systems subject to parameter uncertainties and multiple time-varying delays in state variables is addressed in this paper. The uncertain systems are expressed as Takagi-Sugeno fuzzy models with linear nominal parts and norm-bounded uncertainties. The main objective is to design a stable filter which minimizes a guaranteed cost index and achieves a prescribed H∞ performance under worst-case disturbance. Based on Lyapunov theory, sufficient conditions guaranteeing stability and achieving prescribed performances are stated in terms of LMIs. Therefore, stable filters are obtained with existing convex algorithms. Lastly, two examples are given to illustrate the proposed design methodology.     

Event-triggered network-based ℓ1-gain filtering for positive linear systems

The ?₁-gain filtering problem of positive linear discrete-time systems based on networked communication is investigated in this paper. A filter system model in which the sampled signals are transmitted through the unreliable communication channels is constructed for a positive system. An event-triggered scheme in a linear form, which is different from the prior literatures, is designed to determine whether the signal packet should be transmitted to the filter or not. Network-induced delays are considered while handling the packet transmission. By using the linear Lyapunov function method, a sufficient condition to ensure the existence of the network-based positive filter satisfying ?₁-gain performance is proposed. The desired filter design method for the positive system is presented by using a linear programming approach. A numerical example with practical considerations is given to verify the proposed theoretical results.     

Using the Fredholm resolvent for computing the ℋ2-norm of linear periodic systems

This article presents a method for the ?₂-norm computation of single-loop finite-dimensional linear continuous periodic systems, which is based on the parametric transfer function concept and the calculus of Fredholm integral equations of the second kind. General formulae for the computation of the mean output variance are provided as relations of two integral functions of an auxiliary parameter. On the basis of the general solution, an approximate computation method is derived, and the deviation of the approximate solution is estimated. In principle, this approach allows one to perform the computation with any desired accuracy.     

Robust L2 − L∞ filtering for a class of dynamical systems with nonhomogeneous Markov jump process

This paper investigates the problem of robust L₂ ? L_∞ filtering for a class of dynamical systems with nonhomogeneous Markov jump process. The time-varying transition probabilities which evolve as a nonhomogeneous jump process are described by a polytope, and parameter-dependent and mode-dependent Lyapunov function is constructed for such system, and then a robust L₂ ? L_∞ filter is designed which guarantees that the resulting error dynamic system is robustly stochastically stable and satisfies a prescribed L₂ ? L_∞ performance index. A numerical example is given to illustrate the effectiveness of the developed techniques.     

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.