We address the problem of adaptive observer design for nonlinear time-varying systems which can be transformed in the so-called output feedback form (linear in the unmeasured variables). The observer design follows up previous work on adaptive observers for linear systems and has the form of the classical Luenberger observers for linear systems except that the observer gain is time-varying. A specific form of persistency of excitation is imposed to guarantee the convergence of the (state and parameter) estimation errors. As for the output feedback loop, we proceed using a cascade approach, i.e., we impose the appropriate conditions so that the closed loop system has a cascaded structure. Uniform global asymptotic stability may then be concluded based on cascaded systems theory.

%B Proc. 16th. IFAC World Congress %C Praha, Tcheck Republic %8 July %G eng %0 Conference Paper %B Proc. IEEE Int. Conf. on Decision and Control %D 2005 %T On the PE stabilization of time-varying systems: open questions and preliminary answers %A A. Lor %A A. Chaillet %A G. Besancon %A Y. Chitour %K Robotics %XWe address the following fundamental question: given a double integrator and a linear control that stabilizes it exponentially, is it possible to use the {\em same} control input in the case that the control input is multiplied by a time-varying term? Such question has many interesting motivations and generalizations: 1) we can pose the same problem for an input gain that depends on the state and time hence, a specific persistency of excitation property for nonlinear systems must be imposed; 2) the stabilization –with the same method– of chains of integrators of higher order than two is fundamentally more complex and has applications in the stabilization of driftless systems; 3) the popular backstepping method stabilization method for systems with non-invertible input terms. The purpose of this note is two-fold: we present some open questions that we believe are significant in time-varying stabilization and present some preliminary answers for simple, yet challenging case-studies.

%B Proc. IEEE Int. Conf. on Decision and Control %C Sevilla, Spain %P 6847–6852 %8 December %G eng