Ali Zemouche

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.

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.

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 Nonlinear Systems: An Approach Based on the Differential Mean Value Theorem.

In this note, observers design for a class of non linear dynamical systems has been investigated. The main contribution lies in the use of the differential mean value theorem (DMVT) to transform the nonlinear error dynamics into a LPV system. The stability analysis is, therefore, performed using a standard Lyapunov function that leads to the solvability of a set of Linear Matrix Inequalities (LMIs), easily tractable. Numerical examples are provided to show high performances of the proposed approach and the large class of nonlinear dynamical systems that are concerned.

Technical communique: On LMI conditions to design observer-based controllers for linear systems with parameter uncertainties

This paper deals with the problem of observer-based stabilization for linear systems with parameter inequality. A new design methodology is proposed thanks to a judicious use of the famous Young relation. This latter leads to a less restrictive synthesis condition, expressed in term of Linear Matrix Inequality (LMI), than those available in the literature. Numerical comparisons are provided in order to show the validity and superiority of the proposed method.

Observer synthesis method for Lipschitz nonlinear discrete-time systems with time-delay: An LMI approach

Abstract In this paper, we address the problem of observer design for a class of Lipschitz nonlinear discrete-time systems with time-delay. The main contribution lies in the use of a new structure of the proposed observer with a novel Lyapunov–Krasovskii functional. Thanks to these designs, new nonrestrictive synthesis conditions, expressed in terms of linear matrix inequalities (LMIs), are obtained. Indeed, the obtained LMIs contain more degree of freedom than those established by the approaches available in the literature which consider a simple Luenberger observer with a simple Lyapunov function for the stability analysis. An extension of the presented result to H ∞ performance analysis is given in order to take into account the noise (if it exists) affecting the considered system.

ℋ −/ℋ ∞ fault detection filter for a class of nonlinear descriptor systems

This article addresses the problem of designing a robust fault detection filter (RFDF) for a class of nonlinear descriptor systems whose nonlinear term is globally Lipschitz and linear term is described by a linear parameter varying form. The effect of perturbations on the residuals is minimised using the ℋ∞ norm while the sensitivity performance is guaranteed using the ℋ− index. Thanks to a particular Lyapunov function: which depends on the faults and the filter design problem is formulated as a convex optimisation problem solved via linear matrix inequality techniques. Performances of the proposed filter are shown through the application of an activated sludge process model.

Nonlinear-Observer-Based

The aim of this work is to provide a unified observer design method for nonlinear Lipschitz discrete-time systems with extension to H infin synchronization and input recovery used for communication systems. The stability analysis is performed using a suitable Lyapunov function that leads to the solvability of linear matrix inequalities. One of the main challenges is to establish nonconservative stability conditions, particularly with respect to large Lipschitz constants. Simulations and experimental results have been provided in order to show the performances of the proposed method.

{\cal H}_{\infty}

We present a new synthesis solution for the state estimation problem of a class of discrete-time systems, namely linear systems up to Lipschitz nonlinearities. Our solution is obtained by using the Lipschitz condition jointly with the Lyapunov stability condition in order to guarantee the asymptotic stability of the estimation error. This result uses the linear matrix inequality (LMI) formulation.