Mohamed Boutayeb

Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems

In this paper, convergence analysis of the extended Kalman filter (EKF), when used as an observer for nonlinear deterministic discrete-time systems, is presented. Based on a new formulation of the first-order linearization technique, sufficient conditions to ensure local asymptotic convergence are established. Furthermore, it is shown that the design of the arbitrary matrix plays an important role in enlarging the domain of attraction and then improving the convergence of the modified EKF significantly. The efficiency of this approach, compared to the classical version of the EKF, is shown through a nonlinear identification problem as well as a state and parameter estimation of nonlinear discrete-time systems.

Observers for a class of Lipschitz systems with extension to H∞ performance analysis

Abstract In this paper, observer design for a class of Lipschitz nonlinear dynamical systems is investigated. One of the main contributions lies in the use of the differential mean value theorem (DMVT) which allows transforming the nonlinear error dynamics into a linear parameter varying (LPV) system. This has the advantage of introducing a general Lipschitz-like condition on the Jacobian matrix for differentiable systems. To ensure asymptotic convergence, in both continuous and discrete time systems, such sufficient conditions expressed in terms of linear matrix inequalities (LMIs) are established. An extension to H ∞ filtering design is obtained also for systems with nonlinear outputs. A comparison with respect to the observer method of Gauthier et al. [A simple observer for nonlinear systems. Applications to bioreactors, IEEE Trans. Automat. Control 37(6) (1992) 875–880] is presented to show that the proposed approach avoids high gain for a class of triangular globally Lipschitz systems. In the last section, academic examples are given to show the performances and some limits of the proposed approach. The last example is introduced with the goal to illustrate good performances on robustness to measurement errors by avoiding high gain.

A strong tracking extended Kalman observer for nonlinear discrete-time systems

The authors show how the extended Kalman filter, used as an observer for nonlinear discrete-time systems or extended Kalman observer (EKO), becomes a useful state estimator when the arbitrary matrices, namely R/sub k/ and Q/sub k/, are adequately chosen. As a first step, we use the linearization technique given by Boutayed et al. (1997), which consists of introducing unknown diagonal matrices to take the approximation errors into account. It is shown that the decreasing Lyapunov function condition leads to a linear matrix inequality (LMI) problem, which points out the connection between a good convergence behavior of the EKO and the instrumental matrices R/sub k/ and Q/sub k/. In order to satisfy the obtained LMI, a particular design of Q/sub k/ is given. High performances of the proposed technique are shown through numerical examples under the worst conditions.

On LMI conditions to design observers for Lipschitz nonlinear systems

This paper addresses the problem of observer design for Lipschitz nonlinear systems via LMI. The goal of this note is to clarify some recent results and to propose a new design methodology. This is based on the reformulation of the Lipschitz property using some mathematical tools. This reformulation is a relevant and useful Lipschitz condition, which leads to less restrictive LMI conditions. To show the superiority of this latter, two numerical examples are presented.

Brief Extension of minimum variance estimation for systems with unknown inputs

In this paper, we address the problem of minimum variance estimation for discrete-time time-varying stochastic systems with unknown inputs. The objective is to construct an optimal filter in the general case where the unknown inputs affect both the stochastic model and the outputs. It extends the results of Darouach and Zasadzinski (Automatica 33 (1997) 717) where the unknown inputs are only present in the model. The main difficulty in treating this problem lies in the fact that the estimation error is correlated with the systems noises, this fact leads generally to suboptimal filters. Necessary and sufficient conditions for the unbiasedness of this filter are established. Then conditions under which the estimation error and the system noises are uncorrelated are presented, and an optimal estimator and a predictor filters are derived. Sufficient conditions for the existence of these filters are given and sufficient conditions for their stability are obtained for the time-invariant case. A numerical example is given in order to illustrate the proposed method.

Static output feedback stabilization with H/sub /spl infin// performance for linear discrete-time systems

This note presents a new sufficient condition for the static output feedback stabilization of linear discrete-time systems. This new condition is expressed as a linear matrix inequality feasibility problem and hence easily tractable numerically. An extension of this condition is given in order to incorporate H/sub /spl infin// performance objectives. The applicability of the proposed approach is shown through numerical examples and compared to some recent methods.

Observer Design for Lipschitz Nonlinear Systems: The Discrete-Time Case

This brief deals with observer design for a class of discrete-time nonlinear systems, namely, linear systems with Lipschitz nonlinearities. Perhaps one of the main features, with respect to the existing results, is the use of new particular Lyapunov functions to deduce nonconservative conditions for asymptotic convergence of the state estimation errors. The established sufficient conditions are expressed in terms of linear matrix inequalities, which are easily and numerically tractable by standard software algorithms. By means of simple transformations, a reduced-order version is established where the observer gain is computed in an optimal manner. Performances of the proposed approach are illustrated through simulation and experimental results; one of them concerns synchronization of chaotic nonlinear models

A unified H∞ adaptive observer synthesis method for a class of systems with both Lipschitz and monotone nonlinearities

Abstract This paper investigates the problem of the H ∞ adaptive observer design for a class of nonlinear dynamical systems. The main contribution consists in providing a unified synthesis method for systems with both Lipschitz and monotone nonlinearities (not necessarily Lipschitz). Thanks to the innovation terms into the nonlinear functions [M. Arcak, P. Kokotovic, Observer-based control of systems with slope-restricted nonlinearities, IEEE Transactions on Automatic Control 46 (7) (2001) 1146–1150] and to the differential mean value theorem [A. Zemouche, M. Boutayeb, G.I. Bara, Observers for a class of Lipschitz systems with extension to H ∞ performance analysis, Systems and Control Letters 57 (1) (2008) 18–27], the stability analysis leads to the solvability of a Linear Matrix Inequality (LMI) with several degrees of freedom. For simplicity, we start by presenting the result in an H ∞ adaptive-free context. Furthermore, we propose an H ∞ adaptive estimator that extends easily the obtained results to systems with unknown parameters in the presence of disturbances. We show, in particular, that the matching condition in terms of an equality constraint required in several works is not necessary and therefore allows reducing the conservatism of the existing conditions. Performances of the proposed approach are shown through a numerical example with a polynomial nonlinearity.

Observer design for one-sided Lipschitz discrete-time systems

Abstract This note focuses on state observer design for a general class of nonlinear discrete-time systems that satisfies the one-sided Lipschitz condition. It has been shown that this condition may encompass a large class of nonlinearities. However, challenging problems arise such as relevant choice of the Lyapunov function or non convexity of the obtained stability conditions. Both full-order and reduced-order observer designs are considered. In this work, the main contribution is to provide first some mathematical artifacts on the Lyapunov function to obtain simple and workable stability conditions, furthermore we show how to obtain LMI conditions to ensure asymptotic convergence. On the other hand, we extend the obtained results to n − p reduced order observer design. High performances are shown through simulation results.

Observers design for linear time-delay systems

Abstract In this contribution we propose a simple and useful approach to design observers for discrete-time systems with delays in the state and output variables. The main feature is that the necessary and sufficient conditions for the existence of such observer are derived. The stability analysis is performed by the Lyapunov approach, where the obtained conditions are expressed in terms of a modified Riccati equation. Numerical examples are provided to show efficiency of the proposed observer.