Semi-global finite-time observers for nonlinear systems

Abstract

It is well known that high gain observers exist for single output nonlinear systems that are uniformly observable and globally Lipschitzian. Under the same conditions, we show that these systems admit semi-global and finite-time converging observers. This is achieved with a derivation of a new sufficient condition for local finite-time stability, in conjunction with applications of geometric homogeneity and Lyapunov theories.

Introduction

Research on nonlinear observers has achieved remarkable progress since the formal introduction of the concept and the Lyapunov approach based results of existence and design in Thau (1973). With the advance of the nonlinear observability theory (Hermann & Krener, 1997) in the differential geometric framework (Isidori, 1995), quite a number of early works have been devoted to establishing the link between nonlinear observer and nonlinear observability. The existence of exponential observers is closely related to the observability of the linearized system (Kou et al., 1975, Xia and Gao, 1988). Uniform observability of a single output nonlinear system results in a triangular structure useful for observer design (see Gauthier et al., 1992, Gauthier and Kupka, 1994, Hammouri et al., 2002 and their other works). These findings are employed in all three major classes of nonlinear observer design methods that abound in the literature. Linearized observability is a standing assumption for both the Lyapunov based approach (Raghavan and Hedrick, 1994, Thau, 1973) and the observer canonical form approach (Bestle and Zeitz, 1983, Krener and Isidori, 1983). High-gain observers are very much associated with the triangular structure derived from the uniform observability of nonlinear systems (Gauthier et al., 1992, Gauthier and Kupka, 1994). New developments of all three design methods have been carried out in various directions (Kazantzis and Kravaris, 1998, Krener and Respondek, 1985, Rajamani and Cho, 1998, Shim et al., 2001, Xia and Gao, 1989).

Observers with finite-time convergence have certain advantages and are therefore desirable in some situations of control and supervision (Menold, Findeisen, & Allgöwer, 2003a). There is a series of methods that achieve finite-time convergence (Engel and Kreisselmeier, 2002, Haskara et al., 1998, Hong et al., 2001, Michalska and Mayne, 1995). Some of these observers, such as the sliding mode observers, are not continuous. The continuity property and its importance in finite-time stability are realized in Bhat and Bernstein, 2000, Bhat and Bernstein, 2005. It is also interesting to point out that continuous observers are realized to be different and unique in the nonlinear context (Krener, 1986, Xia and Zeitz, 1997). For instance, linearized observability is no longer necessary for the existence of a continuous observer (Xia & Zeitz, 1997). A first approach to design such an observer is a dedicated introduction of time-delay in the observers (Engel & Kreisselmeier, 2002). This approach was extended to linear time-varying systems in Menold et al. (2003a) and to nonlinear systems that can be transformed into the observer canonical form Menold, Findeisen, and Allgöwer (2003b). Sauvage, Guay, and Dochain (2007) also proposed nonlinear finite-time observers for a class of nonlinear systems, with a time-delay in the observers. A finite-time observer for a class of observer error linearizable systems has recently been constructed in Perruquetti, Floquet, and Moulay (2008). The major technique used is homogeneity (Qian & Lin, 2001).

The aim of this paper is to prove a general result: a uniformly observable and globally Lipschitzian single output nonlinear system admits semi-global finite-time observers. This paper is organized as follows. The definition of finite-time stability and its criteria are reviewed in Section 2. In Section 3, we present the semi-globally finite-time stable observers for single output nonlinear systems. Finally, the paper is concluded in Section 4.