Minimum Effort Control Of Several Terminal Components

Abstract

The stochastic control problem of minimizing the total average velocity correction with several prescribed terminal variances in the presence of random injection and measurement errors is considered. It is shown that, for the case of linear feedback, this can be formulated as an optimization problem for an equivalent deterministic system whose states are the covariances of the predicted miss. However, the deterministic optimization problem is “degenerate” causing some difficulty in the computation of the feedback gain. It is shown that the optimum linear corrective strategy is, in general, discontinuous and consists of an initial period of no control, followed by a period of continuous control and finally a period of no control and possibly an impulse at the end. Equations are derived from which the variable feedback gain and the various time intervals can be computed. Two simple examples involving (1) the control of two terminal position components, and (2) the control of both the terminal position and the terminal velocity are considered in detail. Numerical results are given showing the comparison between this solution and that obtained by using the well known theory for the quadratic loss criterion. In particular, the computation includes, for the two position case, a gap in the information.