Local Observer Design for Nonlinear Control Systems around Equilibria

In this paper, new results have been derived for the local observer design for nonlinear control systems with real parametric uncertainty around equilibia. In this paper, it is first shown that equilibrium-state detectability is a necessary condition for the existence of local asymptotic observers for any nonlinear system and using this result, it is shown that for the classical case, when the state equilibrium does not change with the real parametric uncertainty, and when the plant output is purely a function of the state, there is no local asymptotic observer for the plant. Next, it is shown that in sharp contrast to this case, for the general case of problems where we allow the state equilibrium to change with the real parametric uncertainty, there generically exist local exponential observers even when the plant output is purely a function of the state. In this paper, a characterization and construction procedure for local exponential observers for a general class of nonlinear systems with real parametric uncertainty has also been derived under some stability assumptions. It is also shown that for the general class of nonlinear systems considered, the existence of local exponential observers in the presence of inputs implies, and is implied by the existence of local exponential observers in the absence of inputs.