Observer for Lipschitz Nonlinear Systems

Prof. Gauthier Sallet, Ph.D.

   Zeit: Dienstag, 6. 11. 2001, 16:00
   Ort: Hörsaal V 9.31 Pfaffenwaldring 9, Universitätsbereich Stuttgart-Vaihingen

Abstract:

In an interesting paper R.Rajamani and Y.M.Cho have proposed a systematic methodology to design observers. The problem is to design observers for the following class of nonlinear systems: dx/dt = Ax+Phi(x,u), y=Cx, where Phi(x,u) is global Lipschitz with respect to the state x, uniformly in the control u, i.e. there exists a constant L such that norm(Phi(x,u)-Phi(y,u))<= L norm(x-y) and the pair (C,A) is observable. They have introduced a new problem: relation between distance to unobservability and convergence of "Luenberger-like" observers. A result for the convergence of the observer has also been given. They have presented a quantity denoted by delta, claimed as the distance to unobservability. We show that this number is not actually the distance from the system to the set of unobservable systems. Moreover the result for the observer is incorrect. We provide a counterexample for the result of convergence. In this paper results for the convergence are obtained, with additional strengthened hypothesis, to correct Rajamani and Cho's result. Moreover we propose a practical algorithm to compute effectively the different quantity, so that the design can become practical. The results are used to design an observer for a, now classical, single-link flexible robot joint. The behavior of the observer to noisy output is quite satisfactory. A a by product of the proceeding results we obtain a stability robustness result for systems dx/dt = Ax+f(x,t), norm(f(t,x)) <= L norm(x).

Biographical Sketch:

Sallet Gauthier is currently full professor in the CNRS Laboratory MMAS (Mathematical Methods for Analyzing Systems) of the University of Metz. Before this G. Sallet spend two years in INRIA (the French National Institute for Research in Computer Sciences and Control) as a research director and member of the scientific evaluation committee of the Institute. G. Sallet is currently the scientific leader of the INRIA research project GONGE (Geometric control of Nonlinear Systems, www.inria.fr/recherche/equipes/conge.en.html) The research project has 9 permanent resarchers, 7 PHD students, 3 post doc students and one assistant secretary. G. Sallet has been elected chairman of the scientific committee of the University of Metz.
G. Sallet published 40 papers in numerous International Journal (SIAM J. Control, IEEE Trans. Automatic Control, MCSS, System and Control Letters, Automatica, International Journal of Control, Kybernetica, Transection AMS, Journal of Differential Equations, Math System Theory...) and International conferences (IEEE-CDC, IFAC, Nolcos, ECC).

Scientific interests: Nonlinear Control
(a) controllability and observability
(b) Stabilization par Jurdjevic-Quinn methods, Lyapunov and semidefinite Lyapunov functions
(c) Nonlinear observers. Stabilization by means of an observer, dynamic stabilization
(d) Industrial applications: fluid power systems, Waste Water Treatment, Plant, Biochemical reactors, telecommunications
(e) Application of Nonlinear Control to Epidemiological systems. Especially Malaria and Tuberculosis