A Minimax Approach to the Design of Low Sensitivity State Estimators

Abstract

This paper proposes and explores a new approach to the design of state estimators for systems with large, but bounded uncertainties in plant and measurement noise covariances. A linear estimator with unspecified gain is chosen a priori. Useful sensitivity measures for this filter are its total mean square estimation error (S1), and the deviation of this error from the optimum, minimum estimation error in either an absolute (S2) or relative (S3) sense. These sensitivity measures are a function of the uncertain statistics and the unspecified filter gain. If a particular measure is first maximized over the set of uncertain covariances and then minimized with respect to the adjustable gains, a filter is obtained which yields a least upper bound on the actual measure regardless of the exact value of the statistics. Minimax filter design for plants with constant, but uncertain plant and measurement noise covariances is fully explored. First, for the S1 measure, it is shown that min-max equals max-min. Thus the minimax problem is replaced with a simple maximization of the optimal mean square error over the uncertain parameter set and the S1 filter is simply the Kalman filter for the maximizing noise statistics. Several properties of the required maximization for the infinite time case are then developed. Next the S2 and S3 filters are shown to be unique and optimal for at least one point in the set of uncertain parameters. Min-max does not equal max-min for the S2 and S3 measures. However, the convexity of S2 and S3 in the uncertain statistics is used to show that the maximum of these sensitivity measures is attained over a finite set of points. Three short examples are presented to illustrate the properties of minimax filters and the utility of the minimax design approach.