New results and examples on observer designof nonlinear systems around equilibria

In this paper, we present some new results and examples on nonlinear observer designfor continuous-time and discrete-time nonlinear systems defined in the neighborhood of equilibria. We also show that detectability is a necessary condition for the existence of nonlinear observers for nonlinear systems around equilibria. Using this necessary condition, we establish that, for the classical bifurcation case when the state equilibrium point does not change with the bifurcation parameter, and when the plant output function is purely a function of the state vector, there does not exist even a local asymptotic observer for the plant. Finally, we illustrate our results with classical examples from bifurcation theory.