Robust H2 filtering for uncertain systems withmeasurable inputs

This paper deals with the robust minimum variance filtering problem for linear time-varying systems subject to a measurable input and to norm bounded parameter uncertainty in the state and/or the output matrices of the state-space model. The problem addressed is the design of linear filters having an error variance with a guaranteed upper bound for any allowed uncertainty and any input of bounded energy. Three types of input signals are considered: a signal that is a priori known for the whole time interval, an unknown signal of very large bandwidth that is perfectly measured on-line, and a large bandwidth signal that is measured ahead of time in a fixed preview time interval. Both the time-varying finite-horizon and stationary infinite-horizon cases are treated     

Optimal linear filtering under parameter uncertainty

This paper addresses the problem of designing a guaranteed minimum error variance robust filter for convex bounded parameter uncertainty in the state, output, and input matrices. The design procedure is valid for linear filters that are obtained from the minimization of an upper bound of the error variance holding for all admissible parameter uncertainty. The results provided generalize the ones available in the literature to date in several directions. First, all system matrices can be corrupted by parameter uncertainty, and the admissible uncertainty may be structured. Assuming the order of the uncertain system is known, the optimal robust linear filter is proved to be of the same order as the order of the system. In the present case of convex bounded parameter uncertainty, the basic numerical design tools are linear matrix inequality (LMI) solvers instead of the Riccati equation solvers used for the design of robust filters available in the literature to date. The paper that contains the most important and very recent results on robust filtering is used for comparison purposes. In particular, it is shown that under the same assumptions, our results are generally better as far as the minimization of a guaranteed error variance is considered. Some numerical examples illustrate the theoretical results     

Robust filtering under randomly varying sensor delay with variance constraints

This paper deals with a new filtering problem for linear uncertain discrete-time stochastic systems with randomly varying sensor delay. The norm-bounded parameter uncertainties enter into the system matrix of the state space model. The system measurements are subject to randomly varying sensor delays, which often occur in information transmissions through networks. The problem addressed is the design of a linear filter such that, for all admissible parameter uncertainties and all probabilistic sensor delays, the error state of the filtering process is mean square bounded, and the steady-state variance of the estimation error for each state is not more than the individual prescribed upper bound. We show that the filtering problem under consideration can effectively be solved if there are positive definite solutions to a couple of algebraic Riccati-like inequalities or linear matrix inequalities. We also characterize the set of desired robust filters in terms of some free parameters. An illustrative numerical example is used to demonstrate the usefulness and flexibility of the proposed design approach.     

Robust filtering for bilinear uncertain stochastic discrete-timesystems

This paper deals with the robust filtering problem for uncertain bilinear stochastic discrete-time systems with estimation error variance constraints. The uncertainties are allowed to be norm-bounded and enter into both the state and measurement matrices. We focus on the design of linear filters, such that for all admissible parameter uncertainties, the error state of the bilinear stochastic system is mean square bounded, and the steady-state variance of the estimation error of each state is not more than the individual prespecified value. It is shown that the design of the robust filters can be carried out by solving some algebraic quadratic matrix inequalities. In particular, we establish both the existence conditions and the explicit expression of desired robust filters. A numerical example is included to show the applicability of the present method     

Delay-Dependent H∞ Control and Filtering for Uncertain Markovian Jump Systems With Time-Varying Delays

This paper deals with the problems of delay-dependent robust H_infin control and filtering for Markovian jump linear systems with norm-bounded parameter uncertainties and time-varying delays. In terms of linear matrix inequalities, improved delay-dependent stochastic stability and bounded real lemma (BRL) for Markovian delay systems are obtained by introducing some slack matrix variables. The conservatism caused by either model transformation or bounding techniques is reduced. Based on the proposed BRL, sufficient conditions for the solvability of the robust H_infin control and H_infin filtering problems are proposed, respectively. Dynamic output feedback controllers and full-order filters, which guarantee the resulting closed-loop system and the error system, respectively, to be stochastically stable and satisfy a prescribed H_infin performance level for all delays no larger than a given upper bound, are constructed. Numerical examples are provided to demonstrate the reduced conservatism of the proposed results in this paper.     

New approaches to robust minimum variance filter design

This paper is concerned with the design of robust filters that ensure minimum filtering error variance bounds for discrete-time systems with parametric uncertainty residing in a polytope. Two efficient methods for robust Kalman filter design are introduced. The first utilizes a recently introduced relaxation of the quadratic stability requirement of the stationary filter design. The second applies the new method of recursively solving a semidefinite program (SDP) subject to linear matrix inequalities (LMIs) constraints to obtain a robust finite horizon time-varying filter. The proposed design techniques are compared with other existing methods. It is shown, via two examples, that the results obtained by the new methods outperform all of the other designs     

Robust Kalman filters for linear time-varying systems withstochastic parametric uncertainties

We present a robust recursive Kalman filtering algorithm that addresses estimation problems that arise in linear time-varying systems with stochastic parametric uncertainties. The filter has a one-step predictor-corrector structure and minimizes an upper bound of the mean square estimation error at each step, with the minimization reduced to a convex optimization problem based on linear matrix inequalities. The algorithm is shown to converge when the system is mean square stable and the state space matrices are time invariant. A numerical example consisting of equalizer design for a communication channel demonstrates that our algorithm offers considerable improvement in performance when compared with conventional Kalman filtering techniques     

Robust Filtering With Randomly Varying Sensor Delay: The Finite-Horizon Case

In this paper, we consider the robust filtering problem for discrete time-varying systems with delayed sensor measurement subject to norm-bounded parameter uncertainties. The delayed sensor measurement is assumed to be a linear function of a stochastic variable that satisfies the Bernoulli random binary distribution law. An upper bound for the actual covariance of the uncertain stochastic parameter system is derived and used for estimation variance constraints. Such an upper bound is then minimized over the filter parameters for all stochastic sensor delays and admissible deterministic uncertainties. It is shown that the desired filter can be obtained in terms of solutions to two discrete Riccati difference equations of a form suitable for recursive computation in online applications. An illustrative example is presented to show the applicability of the proposed method.      

Robust H∞ Filtering for Uncertain Discrete Stochastic Systems with Time Delays

This paper considers the problem of robust H_∞ filtering for uncertain discrete-time stochastic systems with time-varying delays. The parameter uncertainties are assumed to be real time-varying norm-bounded in both the state and measurement equations. The problem to be addressed is the design of a stable filter that guarantees stochastic stability and a prescribed H_∞ performance level of the filtering error system for all admissible uncertainties and time delays. A sufficient condition for the existence of such filters is obtained in terms of a linear matrix inequality (LMI). For the case when this LMI is feasible, an explicit expression for a desired filter is given. An illustrative example is also provided to demonstrate the effectiveness and applicability of the proposed method.     

Robust discrete-time minimum-variance filtering

The bounded-variance filtered estimation of the state of an uncertain, linear, discrete-time system, with an unknown norm-bounded parameter matrix, is considered. An upper bound on the variance of the estimation error is found for all admissible systems, and estimators are derived that minimize the latter bound. We treat the finite-horizon, time-varying case and the infinite-time case, where the nominal system model is time invariant. In the special stationary case, where it is known that the uncertain system is time invariant, we provide a robust filter for all uncertainties that still keep the system asymptotically stable     

Improved Delay-Dependent H∞ Filtering Design for Discrete-Time Polytopic Linear Delay Systems

This brief revisits the problem of delay-dependent robust H_infin filtering design for discrete-time polytopic linear systems with interval-like time-varying delay. Under the condition whether the unknown parameters can be measured online or not, a parameter-dependent or a parameter-independent filter is respectively developed which guarantees the asymptotic stability of the resulting filtering error system with robust H_infin performance gamma. It is shown that by using a new linearization technique incorporating a bounding technique, a unified framework can be developed such that the full-order and reduced-order, the parameter-dependent and parameter-independent filters can be obtained by solving a set of linear matrix inequalities. Finally, a numerical example is provided to illustrate the effectiveness and merits of the proposed approach.     

Further Results on Robust Variance-Constrained Filtering for Uncertain Stochastic Systems with Missing Measurements

This paper revisits the problem of robust filtering for uncertain discrete-time stochastic systems with missing measurements. The measurements of the system may be unavailable at any sample time. Our aim is to design a new filter such that the error state of the filtering process is mean-square bounded. Furthermore, the steady-state variance of the estimation error of each state does not exceed the individual prescribed upper bound subject to all admissible uncertainties and all possible incomplete observations. It is shown that the design of a robust filter can be carried out by directly solving a set of linear matrix inequalities. The nonsingular assumption on the system matrix A and the inequality which is used to handle the uncertainties are not necessary in the derivation process of our results. Thus, it is expected that a less conservative condition can be obtained. The advantage of the new method is demonstrated via an illustrative example.     

H/sub 2/ optimal linear robust sampled-data filtering design using polynomial approach

A new frequency domain approach to robust multi-input-multi-output (MIMO) linear filter design for sampled-data systems is presented. The system and noise models are assumed to be represented by polynomial forms that are not perfectly known except that they belong to a certain set. The optimal design guarantees that the error variance is kept below an upper bound that is minimized for all admissible uncertainties. The design problem is cast in the context of H/sub 2/ via the polynomial matrix representation of systems with norm bounded unstructured uncertainties. The sampled-data mix of continuous and discrete time systems is handled by means of a lifting technique; however, it does not increase the dimensionality or alter the computational cost of the solution. The setup adopted allows dealing with several filtering problems. A simple deconvolution example illustrates the procedure.     

filtering for discrete-time linear systems with bounded time-varying parameters

In this paper, the problem of linear parameter varying (LPV) filter design for time-varying discrete-time polytopic systems with bounded rates of variation is investigated. The design conditions are obtained by means of a parameter-dependent Lyapunov function and extra variables for the filter design, expressed as bilinear matrix inequalities. An LPV filter, which minimizes an upper bound to the performance of the estimation error, is obtained as the solution of an optimization problem. A convex model to represent the parameters and their variations as a polytope is proposed in order to provide less conservative design conditions. Robust filters for time-varying polytopic systems can be obtained as a particular case of the proposed method. Numerical examples illustrate the results.     

Robust steady-state filtering for systems with deterministic and stochastic uncertainties

For uncertain systems containing both deterministic and stochastic uncertainties, we consider two problems of optimal filtering. The first is the design of a linear time-invariant filter that minimizes an upper bound on the mean energy gain between the noise affecting the system and the estimation error. The second is the design of a linear time-invariant filter that minimizes an upper bound on the asymptotic mean square estimation error when the plant is driven by a white noise. We present filtering algorithms that solve each of these problems, with the filter parameters determined via convex optimization based on linear matrix inequalities. We demonstrate the performance of these robust algorithms on a numerical example consisting of the design of equalizers for a communication channel.     

An LMI approach to the H/sub /spl infin// filter design for uncertain systems with distributed delays

This paper investigates the problem of robust H/sub /spl infin// filter design for linear distributed delay systems with norm-bounded time-varying parameter uncertainties. The distributed delays are assumed to appear in both the state and measurement equations. The problem we address is the design of a filter, such that for all admissible uncertainties, the resulting error system is asymptotically stable and satisfies a prescribed H/sub /spl infin// performance level. A sufficient condition is obtained to guarantee the existence of desired H/sub /spl infin// filters, which can be constructed by solving certain linear matrix inequalities. The effectiveness of the proposed design method is demonstrated by a numerical example.     

Robust ℋ∞ filtering for uncertaindiscrete-time state-delayed systems

This paper addresses the problem of robust ℋ_∞ filtering for linear discrete-time systems subject to parameter uncertainties in the system state-space model and with multiple time delays in the state variables. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. A methodology is developed to design a stable linear filter that assures asymptotic stability and a prescribed ℋ_∞ performance for the filtering error, irrespective of the uncertainty and the time delays. The proposed design is given in terms of linear matrix inequalities, which has the advantage in that it can be implemented numerically very efficiently     

A New Parameter-Dependent Approach to Robust Energy-to-Peak Filter Design

This paper presents a new parameter-dependent approach to the design of robust energy-to-peak filters for linear uncertain systems. Given a system containing polytopic parameter uncertainties, our purpose is to design a robust filter such that the filtering error system is asymptotically stable with a guaranteed L₂-L_∞ disturbance attenuation level γ. This problem is solved by introducing new energy-to-peak performance characterizations, and by utilizing an idea of structured parameter-dependent matrices. New sufficient conditions are obtained for the existence of desired filters in terms of linear matrix inequalities, which can be easily tested by using standard numerical software. If these conditions are satisfied, a desired filter can be readily constructed. Both continuous- and discrete-time systems are considered, and the effectiveness and advantages of the proposed filter design methods are shown via two numerical examples.     

Robust L<Subscript>2</Subscript> - L<Subscript>∞</Subscript> Filtering for Uncertain Nonlinearly Parameterized Stochastic Systems with Time-Varying Delays

This paper is concerned with the problem of robust L₂ - L_∞ filtering for uncertain stochastic time-delay systems in the nonlinear fractional transformation (NFT) form. Attention is focused on the design of both full-order and reduced-order filters, guaranteeing robust stochastic stability and a prescribed level of L₂ - L_∞ performance for the filtering error system. A parameter-dependent Lyapunov functional is employed to solve the problem. The concerned filters are not only parameter dependent but also have the NFT form. Sufficient conditions for the existence of such filters are presented in terms of certain linear matrix inequalities (LMIs). Desired filters are explicitly expressed based on the solutions to the proposed LMIs. A numerical example is provided to demonstrate the effectiveness of the proposed method.