Directional sensitivity of continuous 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 the optimal solution to a least-squares problem in some vector or functional space. For instance, the conventional L2L2-optimal finite-memory observer is shown to be a special case of the continuous least-squares state estimator. Sensitivity of these estimators to structured uncertainty in the system matrix of the plant is studied using the Fréchet derivative. It is shown that the state estimation error caused by the plant model’s mismatch is proportional to the Fréchet derivative of the symbol of the parameterization 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 parameterization operator choice on the state estimator sensitivity to the plant model’s uncertainty is investigated in detail.