Observer for Lipschitz Nonlinear Systems
Prof. Gauthier Sallet, Ph.D.
Zeit: Dienstag, 6. 11. 2001, 16:00
Ort: Hörsaal V 9.31 Pfaffenwaldring 9,
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).
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,
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
(c) Nonlinear observers. Stabilization by means of an observer, dynamic
(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