M12 PARIS
An overview on observer design methods for nonlinear systems
The purpose of this course is to give an overview of the main synthesis techniques of state observers for nonlinear dynamical systems. The lecture will start by addressing some general comments on the ``estimation problem'', that is, reconstructing the full information of a dynamical process on the basis of partial observed data. We will then introduce a particular type of algorithm: the asymptotic observer. Some necessary conditions that ensure convergence of the estimate toward the state of the system will be introduced: the detectability and its infinitesimal characterization.
Then, based on a characterization of detectability, a first class of observers will be presented. Some methods to design such observers will be introduced based on numerical methods.
The next part of the course will consist in presenting the main three families of observers based on stronger observability properties:
● Kalman and Kalman-like observers for state-affine systems, based on a persistence of excitation of the
gramian of
observability;
● High-gain observers and differentiators, based on differential observability assumptions;
● Kazantzis/Kravaris-Luenberger observers, based on backward distinguishability conditions.
We will show that each class of observer relies on transforming the plant's dynamics in a particular normal form which allows the design of an observer. We will explain how each observability condition guarantees the invertibility of its associated transformation and the convergence of the observer. The most important and informative proofs will be detailed, and the advantages/drawbacks of each design discussed.
At the end of the course, we will study some implementation issues and open problems. For instance, the case of time discretization of the output will be considered. Another issue related to the left inversion problem in observers will also be discussed. Finally, we show how an estimate given by the observer may be used in combination with a stabilizing state feedback in order to guarantee asymptotic stabilization of the origin by means of output feedback.
Throughout the course, the various concepts encountered will be illustrated with examples and followed by homework assignments designed to enhance their understanding.
