STATE ESTIMATION WITH OPTIMAL SELECTION OF THE OUTPUT MATRIX FOR DISCRETE-TIME LINEAR-SYSTEMS

This paper deals with the minimization of a scalar measure of the matr ix which solves the Lyapunov equation P = [A - HC] P[A - HC]' + BWB' HVH' with respect to the pair (H, C). This problem arises in the pred iction of the state of a discrete-time, stochastic linear system, when one is concerned with minimizing the prediction error covariance both with respect to the predictor gain and with respect to the output mat rix, which accounts for the physical device by which the state of the system is observed. By duality, the problem can be interpreted as the optimization of a suitable performance index for a linear regulator wi th respect both to the input matrix and to the feedback regulator gain . First, the index to be minimized is carefully chosen, in order to ob tain a meaningful optimization problem, and the optimization with resp ect to the predictor gain or with respect to the output matrix, when t he other matrix is fixed, is considered. The related results suggest a n algorithm which generates a sequence of pairs (output matrix, predic tor gain) which ensure stability of the matrix [A - HC], and to which a decreasing sequence of values of the index there corresponds. Result s about boundedness and convergence of the sequence thus obtained are proved. The problem and the related solution algorithm is also extende d to deal with the singular case of no measurement noise. Numerical ex periments, in which comparison is made with the behaviour of a standar d gradient-based method, confirm the robustness of the proposed algori thm, anticipated on the basis of its theoretical properties.