Gildas Besançon

Remarks on nonlinear adaptive observer design

Abstract A solution to the problem of state estimation for systems with unknown parameters is to use some on-line adaptation of the observer parameters. On the basis of various existing results for such adaptive observer designs, a unifying “adaptive observer form” is proposed in this paper. This form indeed emphasizes properties allowing some asymptotic state estimation in spite of unknown parameters, as well as additional properties which further allow parameter estimation. As an example, it is shown how an adaptive observer can be designed for a class of state affine systems.

On adaptive observers for state affine systems

In this paper, the problem of adaptive observer design for the class of state affine systems is discussed. The discussion is based on recent results on adaptive observer with exponential rate of convergence obtained for multi-input–multi-output linear time varying systems on the one hand, and the well-known Kalman-like design for state affine systems on the other. In particular the relationship between both designs is emphasized, showing how they can even be equivalent. The interest of such an adaptive observer for state affine systems is illustrated by the example of state and parameter estimation for the Lorenz chaotic system. The observer performances are illustrated via simulation.

Observer Synthesis for a Class of Nonlinear Control Systems

This paper provides an observer for the block-stateaffine cascade systems, up to cascade non-linear injections. This observer can obviously be extended to all systems diffeomorphic to one of this form, but characterizing this whole class is not easy. Nevertheless, necessary and sufficient conditions characterizing a subset of this class are given, together with a symbolically computable test. The proposed methods are illustrated on the example of a blending process.

Technical Communique: Global output feedback tracking control for a class of Lagrangian systems

The problem of global tracking control without velocity measurement of the so-called Euler-Lagrange systems has been paid a lot of attention for the past several years. In this note, a nice property of one-degree-of-freedom Euler-Lagrange systems is highlighted, which in particular allows us to obtain a new solution to the problem of tracking control for this class of systems. The solution is under the form of a linear-like observer-based controller which gives fairly good results, as illustrated in the simulation. The method can be generalized to any system having the same property.

On uniform observation of nonuniformly observable systems

Abstract In [4,7,9,12], classes of nonlinear systems are considered for which observers can be designed. Although observability of nonlinear systems is known to be dependent on the input, the proposed observers have the property that the estimation error decays to zero irrespective of the input. In the first part of this paper, it is shown that this phenomenon follows from a common property of these systems: for all of them, the “unobservable states” with respect to some input, are in some sense “stable” (in the linear case, these systems are called detectable), and for this reason, a reduced order observer can be designed. In the second part is given a more general class of nonlinear systems for which such an observer can be designed.

An Immersion-Based Observer Design for Rank-Observable Nonlinear Systems

In this note, an observer design for a wide class of nonlinear systems is proposed. It is based on some high-gain design for the particular case of state affine systems up to triangular nonlinearity on the one hand, and immersion of nonlinear systems into such a form on the other hand. Conditions for the observer convergence are given in terms of appropriate excitation, and a constructive procedure for the immersion is presented. The methodology is illustrated by examples of application

On output transformations for state linearization up to output injection

Steering a nonlinear system into a so-called state-affine one up to output injection, by change of coordinates, is of much interest for observer design. This interest is preserved by also considering changes of output. The purpose of the paper is then to provide computable conditions for existence of output transformations making a system which is not transformable into a state affine one up to output injection by only a change of coordinates to become so after a change of output. Such a characterization enlarges the class of systems for which an observer is available.

State-Affine Systems and Observer-Based Control

Abstract Several tools are now available to handle systems which are affine in the unmeasured states, and various results have recently been presented towards a general characterization of such systems. However, most of them remain not fully constructIve. The present paper proposes a new method to recognize systems which can be transformed into some triangular state-affine form, and as an illustration of its Interest, an observer-based control is presented when all state coefficients are bounded.

Observer Scheme for State and Parameter Estimation in Asynchronous Motors with Application to Speed Control

The paper presents a method for simultaneous state and parameter estimation in induction motors, based on an exact Kalman-like observer. The use of such an observer is made possible by an immersion of the nonlinear model of the machine into a model which is affine with respect to the variables to be estimated. This method can be used in various cases of unknown variables to be estimated, from the most simple case when only the stator flux components are unknown, to some “worst” case when all the electrical parameters, along with the stator flux angular velocity and load torque are to be estimated. In any case, global exponential estimation is guaranteed under appropriate excitation, following Kalman theory. In practice, the method is evaluated through simulations within an estimate-based speed control scheme.

Brief Analysis of a two-relay system configuration with application to Coulomb friction identification

This paper presents an analysis of a class of two-relay feedback systems, motivated by the idea of estimating a relay-type nonlinearity on the feedback path of a plant by adding another relay. The time response of the resulting system is first analyzed, and stability conditions for possible limit cycles are then proposed using Poincares method. On this basis, a simple procedure for estimating the unknown value (e.g. Coulomb friction) of the relay on a plant feedback, by a closed-loop experiment, is finally presented and illustrated on both simulated and real-time examples.