Exponential Observers for Nonlinear Dynamic Systems 1

There is a basic assumption in using state feedback control, i.e., we assume that all the state variables are available for direct measurements. However, in many situations the complete state is not available; so it is necessary to obtain an estimate of the current state of the given system from measurements of its output. For this purpose, Luenberger (1964, 1966) showed how to construct an observer, which is a state estimator given as a dynamic system. T h e observer theory for linear systems was then extended by several researchers ( IEEE, 1971). I n this paper, we present an observer theory for nonlinear dynamic systems. Only identi ty observers having the same dimension as that of the given system are considered. In Section 2, we define the exponential observer, which is an asymptotic state est imator with exponentially decaying error. We show that in Section 3 the existence of a certain Lyapunov-l ike function can be used as a sufficient condition for the existence of an exponential observer. Conditions on the system structure such that such a Lyapunov-l ike function exists are given in Section 4. In Section 5, we discuss the exponential observers for a class of nonlinear systems including those of Thau (1973). T h e conclusions are in Section 6.