Design of nonlinear observers with approximately linear error dynamics using multivariable Legendre polynomials

This paper presents a numerical approach to the design of nonlinear observers by approximate error linearization. By using a Galerkin approach on the basis of multivariable Legendre polynomials an approximate solution to the singular PDE of the observer design technique proposed by Kazantzis and Krener (see (Syst. Control Lett. 1998; 34 :241–247; SIAM J. Control Optim. 2002; 41 :932–953)) is determined. It is shown that the L₂‐norm of the remaining nonlinearity in the resulting error dynamics can be made small on a specified multivariable interval in the state space. Furthermore, a linear matrix equation is derived for determining the corresponding change of co‐ordinates and output injection such that the proposed design procedure can easily be implemented in a numerical software package. A simple example demonstrates the properties of the new numerical observer design. Copyright © 2006 John Wiley & Sons, Ltd.