General observer design for continuous-time and discrete-time nonlinear systems

Abstract

In this work, it is established that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for nonlinear systems. Using this necessary condition, it is shown that there does not exist any general observer (asymptotic or exponential) for nonlinear systems with real parametric uncertainty if the state equilibrium does not change with the parameter values and if the plant output function is purely a function of the state. Next, using center manifold theory, necessary and sufficient conditions are derived for the existence of general exponential observers for Lyapunov stable nonlinear systems. As an application of this general result, it is shown that for the existence of general exponential observers for Lyapunov stable nonlinear systems, the dimension of the state of the general exponential observer should not be less than the number of critical eigenvalues of the linearization matrix of the state dynamics of the plant. Results have been derived for both continuous-time and discrete-time nonlinear systems.