線形系の状態に対する観測と推定の協調化

概要

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The linear state estimation problem is revisited in a new formulation. The state to be estimated is that of a linear system driven by white noise. An estimator of Kalman-Luenberger type is used to estimate the unknown state on the basis of partial observation disturbed by white noise. By “partial observation” we mean that only a linear functional of the state is accessible with noise. The coefficient vector, which is reffered to as “observation gain”, is the object of suitable choice as well as the estimator gain while only the estimator gain is to be optimised in usual formulation. Performance criterion is a weighted quadratic mean of the state estimation error, in which the coeffficient is chosen as function of the observation gain so as to make the problem well-posed. The Kalman-Bucy filter applies for the purpose of minimising this criterion with respect to the estimator gain, while the LQ regulator applies for minimising the same criterion with respect to the observation gain. Alternate application of Kalman-Bucy filter and LQ regulator generates a point sequence in the product space of those of the two kind of gains. It is shown that the simultaneously optimal solution can be obtained as an accumulation point of the series. A simple example illustrates the results.