Approximate Observer Error Linearization by Dissipativity Methods

Abstract

In this paper a method to design observers for nonlinear systems is proposed. The basic idea is to decompose the system in a nonlinear part, that can be transformed into a nonlinear observer form, and a perturbation term connected in the feedback loop. In transformed coordinates the observer error becomes linear with a feedback perturbation. By using the dissipativity theory it is possible to design the observer gains so, that the closed loop is internally stable if some LMIlike conditions are satisfied. This method is very flexible and allows the design of nonlinear observers using exact linearization methods to a much bigger class of systems. The method is shown to be very general, since it includes (and generalizes) as special cases several observer design methods, as for example the exact error linearization method, the high-gain method, the circle criterion design method, and the design for Lipschitz nonlinear systems. The design is computationally simple in many cases, since it reduces to the solution of a feasibility LMI problem, for which highly efficient numerical methods are available. The method offers great flexibility in the design, since the particular properties of the nonlinearities can be characterized by means of one or several quadratic forms, i.e. supply rates. This feature can be used to design observers for systems with special properties in the nonlinearities.

Keywords: Dissipativity, absolute stability, nonlinear observer design, approximate error linearization, LMI.