Directional sensitivity of least-squares state estimators

Least-squares state estimators present an alternative to Luenberger observers and yield an exact (deadbeat) estimate of the state vector of a dynamic system as an optimal solution to a least-squares problem in some vector or functional space. Sensitivity of these estimators to structured uncertainty in the system matrix of the plant is studied in a common for continuous and discrete case framework using the Frechet derivative. It is shown that the state estimation error caused by the plant model mismatch is proportional to the Frechet derivative of the symbol of the parametrization operator used for the estimator implementation, evaluated for the nominal value of the system matrix. For the special case of state estimation in a single-tone continuous oscillator, the crucial impact of the parametrization operator choice on the observer sensitivity to plant model uncertainty is investigated in detail.