Finite frequency H∞ filtering for discrete-time uncertain systems

The problem of finite frequency (FF) H∞ full-order filtering design for discrete-time linear systems, with polytopic uncertainties, is investigated in this paper. Based on the generalized Kalman-Yakubovich-Popov (GKYP) lemma and a parameter dependent Lyapunov function, a set of sufficient conditions for the existence of the H∞ filter are established in terms of matrix inequalities, ensuring that the filtering error system is stable and the H∞ attenuation level from disturbance to the estimation error is minimized in the FF domain of the external disturbances. Then, in order to linearize and relax the obtained matrix inequalities, we apply Finsler's lemma twice, which provides extra degrees of freedom in optimizing the guaranteed H∞ performance, and we show how we can calculate the filter gain matrices by using these conditions. Numerical example is given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods.

SUMMARY This paper deals with the problem of finite frequency H∞ full-order filter design for discrete-time and continuous-time linear systems, with polytopic uncertainties. Based on the generalized Kalman–Yakubovich–Popov lemma and a parameter-dependent Lyapunov function, a set of sufficient conditions are established in terms of matrix inequalities, ensuring that the filtering error system is stable and the H∞ attenuation level, from disturbance to the estimation error, is smaller than a given value over a prescribed finite frequency domain of the external disturbances. Then, in order to linearize and relax the obtained matrix inequalities, we introduce a large number of slack variables by applying Finsler's lemma twice, which provides extra degrees of freedom in optimizing the guaranteed H∞ performance. This leads to performance improvement and reduction of conservatism in the solution. It is shown later that the robust filter gains can be obtained by solving a set of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness and the less conservatism of the proposed approach in comparison with the existing methods. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, robustness and convergence properties of model reduction are investigated for discrete two-dimensional (2D) systems in the Fornasini-Marchesini (F-M) model with polytopic uncertainties. The goal is to design a reduced order model minimizing H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> performance in a known finite-frequency (FF) area of the noises able to reproduce the behavior of the uncertain 2D original system. Using Lyapunov function and generalized Kalman Yakubovich Popov (gKYP) lemma, sufficient conditions for the existence of the FF reduced order design approach are formulated as feasibility of a set of Linear Matrix Inequalities (LMIs). Numerical simulations are given to illustrate the validity and feasibility of the designed reduced-order model.

This paper studies the problem of stability analysis and controller design for discrete-time linear time invariant (LTI) systems with state saturation. Both full state saturation and partial state saturation are considered. Firstly, a new system model is constructed for solving the key problem. Then, a new method is presented for estimating the domain of attraction of the origin for a system with state saturation. Based on this method, an LMI-based ( linear matrix inequality) iterative algorithms is proposed for determining if a given ellipsoid is contractively invariant. Moreover, an LMI-based algorithm is developed for designing dynamic output-feedback controllers which guarantee that the domain of attraction of the origin for the closed-loop system is as large as possible. An example is given to illustrate the efficiency of the design method.

formula omitted] state feedback controller design for continuous-time T–S fuzzy systems in finite frequency domain

The paper is concerned with the problem of feedback control law design in the form of PS output controller for linear discrete-time systems. Since these tasks generally make towards bilinear optimization problems, using the standard quadratic form of Lyapunov function and a symmetric positive definite slack matrix, an enhanced form of algorithm is introduced. The solution, obtained through linear matrix inequalities and equalities formulation, also unifies the control law design procedure. The obtained results, offering the conditions of PS output controller existence, are illustrated with numerical examples to note effectiveness and applicability of the considered approach.

This paper deals with the sensor fault detection problem for linear time-invariant continuous-time systems with disturbances and sensor noises. Two finite frequency performance indexes are introduced to increase the fault sensitivity and attenuate the effects of disturbances and noises. Different from the existing techniques, the recently developed Generalized Kalman-Yakubovich-Popov (GKYP) lemma is applied in this paper to treat the finite frequency performance indexes directly, completely avoiding the complexity of weighting functions of the existing approaches. The design methods are presented in terms of solutions to a set of linear matrix inequalities (LMIs). A numerical example is studied to illustrate the effectiveness of the proposed method.

This paper studies the simultaneous fault detection and control problem for linear uncertain discrete-time systems. A design procedure dealing with polytopic uncertainties is proposed for residual generation, where both the fault detection objectives and some control objectives are considered simultaneously through certain performance indexes. To cope with the fault sensitivity performance index in a given frequency range directly, the generalised KYP lemma is introduced in this framework. A two-step procedure is adopted to obtain the solutions through satisfying a set of linear matrix inequalities. A numerical example is given to illustrate the effectiveness of the proposed methods.

This paper addresses the fault detection observer design problem for linear parameter-varying systems. Two finite frequency performance indexes are introduced to measure the fault sensitivity and the disturbance robustness. First, the H− index fault sensitivity condition in finite frequency domain is obtained by generalised Kalman–Yakubovich–Popov lemma and new linearisation techniques. Then, with the aid of Kalman–Yakubovich–Popov lemma and projection lemma, the stability and robustness conditions are derived. It turns out that the non-convexity problem which is caused by dealing with the above three conditions can be translated into a bilinear matrix inequality optimisation problem by increasing the dimensions of slack variable matrix. An iterative linear matrix inequality algorithm is proposed to get the solution. The effectiveness of the filter is shown via three numerical examples.

In this paper, the H filtering problem for a class of continuous-time polytopic uncertain systems is studied. Different from the traditional methods dealing with polytopic uncertainties in the full frequency domain, not only its values but also the finite frequency (FF) cases are considered. Our goal is to design the filters guaranteeing an H performance level in the finite frequency (FF) domains. By using the generalised Kalman-Yakubovich-Popov (gKYP) lemma, polynomially parameter-dependent Lyapunov function and some key matrices, we establish new sufficient conditions to characterise this problem in terms of linear matrix inequalities (LMIs). Finally, a two examples are used to show the superiority and effectiveness of the proposed methods.

This study is concerned with the fault detection problem for fractional‐order linear systems with disturbances. The and performance indices of fractional‐order systems in finite frequency domains are applied to measure the fault sensitivity and the disturbance robustness. A fractional‐order fault detection observer is designed to satisfy the two finite frequency performance indices simultaneously. Based on the generalised Kalman–Yakubovich–Popov lemma and the projection lemma, the design conditions are obtained in terms of linear matrix inequalities. Compared with the existing full frequency approaches, the proposed finite frequency one can get better results when the frequency ranges of faults and disturbances are known. The effectiveness of the proposed method is validated by numerical examples.

A contraction property for state feedback design of linear discrete-time systems

Interval observer design for unknown input estimation of linear time-invariant discrete-time systems

For continuous-time linear time-invariant (LTI) systems with polytopic uncertainties, we develop a robust sampled-data state-feedback control design scheme in terms of linear matrix inequalities (LMIs). Truncated power series expansions are used to approximate a discretized model of the original continuous-time system. The system matrices obtained by using the power series approximations are then expressed as homogeneous polynomial parameter-dependent (HPPD) matrices of finite degrees, and conditions for designing the controller are formulated as a HPPD matrix inequality, which can be solved by means of a recent LMI relaxation technique to test the positivity of HPPD matrices with variables in the simplex. To take care of the errors induced by the remainder terms of the truncated power series, the terms are considered as norm bounded uncertainties and then incorporated into the proposed LMI conditions. Finally, examples are used to illustrate the approach.

In this paper a basis is developed for the systematic design of linear multivariable discrete-time tracking systems for plants with inaccessible states, which thus extends the results for discrete-time plants with accessible states previously obtained by Bradshaw and Porter (1975). The design method is illustrated by the presentation of the results of computer simulation studies.