New results on general observers for discrete-time nonlinear systems Original Research Article

In this paper, we establish that detectability is a necessary condition for the existence of general observers (asymptotic or exponential) for discrete-time nonlinear systems. Using this necessary condition, we show that there does not exist any general observer (asymptotic or exponential) for discrete-time 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 for maps, we derive necessary and sufficient conditions for the existence of general exponential observers for Lyapunov stable discrete-time nonlinear systems. As an application of this result, we show that for the existence of general exponential observers for Lyapunov stable discrete-time 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.