Márcio J. Lacerda

Robust H 2 and H ∞ filter design for uncertain linear systems via LMIs and polynomial matrices

This paper presents new convex optimization procedures for full order robust H 2 and H ∞ filter design for continuous and discrete-time uncertain linear systems. The time-invariant uncertain parameters are supposed to belong to a polytope with known vertices. Thanks to the use of a larger number of slack variables, linear matrix inequalities for the design of robust filters can be derived from the proposed conditions, outperforming the existing methods. The superiority and efficiency of the proposed method for filter design are illustrated by means of numerical comparisons in benchmark examples from the literature.

A new approach to handle additive and multiplicative uncertainties in the measurement for LPV filtering

This paper presents a general framework to cope with full-order linear parameter-varying (LPV) filter design subject to inexactly measured parameters. The main novelty is the ability of handling additive and multiplicative uncertainties in the measurements, for both continuous and discrete-time LPV systems, in a unified approach. By conveniently modelling scheduling parameters and uncertainties affecting the measurements, the filter design problem can be expressed in terms of robust matrix inequalities that become linear when two scalar parameters are fixed. Therefore, the proposed conditions can be efficiently solved through linear matrix inequality relaxations based on polynomial solutions. Numerical examples are presented to illustrate the improved efficiency of the proposed approach when compared to other methods and, more important, its capability to deal with scenarios where the available strategies in the literature cannot be used.

Robust H 2 and H ∞ memory filter design for linear uncertain discrete-time delay systems

This paper is concerned with the problems of robust full-order H 2 and H ∞ filter design for linear uncertain discrete-time systems with multiple state delays. The uncertain parameters affecting the matrices of the system are supposed to be time-invariant and to belong to a polytopic domain. The main novelty is the fact that the filter contains an arbitrary number of past states and past output measures of the system, yielding a filtering system with memory. Linear matrix inequality relaxations based on polynomially parameter-dependent Lyapunov matrices and slack variables are proposed for the H 2 and H ∞ filter design. Due to the extra dynamics introduced through the delayed states, the robust memory filter is able to provide less conservative results in terms of the H ∞ and the H 2 performance when compared to the memoryless case. Throughout the paper, the multiple delays are considered to be fixed and time-invariant, but an extension of the conditions to cope with unknown delays belonging to a given interval is also presented for both time-varying and time-invariant delay cases. Numerical examples are given to demonstrate the improvements of the proposed approach with respect to other methods from the literature. HighlightsFilter with arbitrary number of past states and output measures of the system.LMI relaxations based on polynomially parameter-dependent variables for filter design.Unknown (possibly time-varying) delays in an interval can affect the uncertain system.

ℋ 2 filter design through multi-simplex modeling for discrete-time Markov jump linear systems with partly unknown transition probability matrix

This paper is concerned with the ℋ2 robust filtering problem for discrete-time Markov jump linear systems (MJLS) with transition probability matrix affected by uncertainties. Differently from previous approaches in the literature, the proposed strategy presents a systematic way to handle, simultaneously, different types of uncertainties commonly appearing in the transition probability matrix of MJLS. Full-order filters with partial, complete or null Markov mode observation are synthesized via a linear matrix inequality (LMI) based formulation. The main novelty of the proposed filter design procedure is the use of parameter-dependent Lyapunov matrices of arbitrary degree to certify the stochastic stability and to guarantee an upper bound to the ℋ2 norm of the filtering error system. Moreover, the proposed conditions also include slack variables and scalars. For fixed values of the scalar parameters, the conditions become LMIs. Numerical examples borrowed from the literature illustrate that the proposed filter can provide better ℋ2 guaranteed costs when compared to other existing methods.

ℋ∞ filter design for nonlinear polynomial systems☆

The problem of ℋ∞ filter design for continuous-time nonlinear polynomial systems is addressed in this paper. The aim is to design a full order dynamic filter that depends polynomially on the filter states. The strategy relies on the use of a quadratic Lyapunov function and an inequality condition that assures an ℋ∞ performance bound for the augmented polynomial system, composed by the original system and the filter to be designed, in a regional (local) context. Then, by using Finsler’s lemma, an enlarged parameter space is created, where the Lyapunov matrix appears separated from the system matrices in the conditions. Imposing structural constraints to the decision variables and fixing some values for a scalar parameter, design conditions for the ℋ∞ filter can be obtained in terms of linear matrix inequalities. As illustrated by numerical experiments, the proposed conditions can improve the ℋ∞ performance provided by standard linear filtering by including the polynomial terms in the filter dynamics.

Robust ℋ 2 filter design for polytopic linear systems via LMIs and polynomial matrices

This paper presents new linear matrix inequality conditions for full order robust ℋ2 filter design for continuous and discrete-time polytopic linear systems with time-invariant uncertainty. Thanks to the use of a larger number of slack variables, the proposed conditions are less conservative than the existing conditions in the literature, containing recently published results as particular cases. Examples illustrate the better performance of the proposed filters when compared to other approaches for robust filter design.

Robust ℋ ℞ filter design for polytopic linear discrete-time delay systems via LMIs and polynomial matrices

This paper presents new robust linear matrix inequality conditions for full order robust ℋ℞ filter design of discrete-time polytopic linear systems affected by a time-varying delay. Thanks to the use of a larger number of slack variables, the proposed conditions are less conservative than the existing methods. Numerical experiments illustrate the better performance of the proposed filter design procedure when compared to other approaches available in the literature.

On the Use of Interval Extensions to Estimate the Largest Lyapunov Exponent from Chaotic Data

A method to estimate the (positive) largest Lyapunov exponent (LLE) from data using interval extensions is proposed. The method differs from the ones available in the literature in its simplicity since it is only based on three rather simple steps. Firstly, a polynomial NARMAX is used to identify a model from the data under investigation. Secondly, interval extensions, which can be easily extracted from the identified model, are used to calculate the lower bound error. Finally, a simple linear fit to the logarithm of lower bound error is obtained and then the LLE is retrieved from it as the third step. To illustrate the proposed method, the LLE is calculated for the following well-known benchmarks: sine map, Rossler Equations, and Mackey-Glass Equations from identified models given in the literature and also from two identified NARMAX models: a chaotic jerk circuit and the tent map. In the latter, a Gaussian noise has been added to show the robustness of the proposed method.

Linear filter design for continuous-time polynomial systems with ℒ2-gain guaranteed bound

This paper proposes a technique for the design of full and reduced-order linear filters for continuous-time polynomial systems satisfying a prescribed induced ℒ2-gain. The filter variables appear affinely in a polynomial inequality and are computed by solving a semidefinite program (SDP). Bounds on the filter performance, established in terms of the induced ℒ2 norms between the input noise and the filtering error are then assessed via polynomial storage functions. Results from numerical experiments are presented.

Robust ℋ ∞ memory filters for uncertain discrete-time linear systems

This paper is concerned with the problem of robust full-order ℋ∞ filter design for uncertain discrete-time linear systems. The uncertainties are supposed to be time-invariant and to belong to a polytopic domain. The main novelty is the fact that the filter contains an arbitrary number of past states and past system output measures, yielding a filtering system with memory. Linear matrix inequality relaxations based on polynomially parameter-dependent Lyapunov matrices and slack variables are proposed for the ℋ∞ filter design. Due to the extra dynamics, the robust memory filter is able to provide less conservative results in terms of the ℋ∞ performance when compared to the memoryless case. Numerical examples are given to demonstrate the improvements of the proposed method.