Robust design of dynamic observers

The two (identity) observer realizations z = Mz + Ky and z = transpose of Az + transpose of K(y - transpose of Cz), respectively called the open loop and closed loop realizations, for the linear system x = Ax, y = Cx are analyzed with respect to the requirement of robustness; i.e., the requirement that the observer continue to regulate the error x - z satisfactorily despite small variations in the observer parameters from the projected design values. The results show that the open loop realization is never robust, that robustness requires a closed loop implementation, and that the closed loop realization is robust with respect to small perturbations in the gains transpose of K if and only if the observer can be built to contain an exact replica of the unstable and underdamped dynamics of the system being observed. These results clarify the stringent accuracy requirements on both models and hardware that must be met before an observer can be considered for use in a control system.