Numerical algorithms and issues concerning the discrete-time optimal projection equations for systems with white parameters

The discrete-time optimal projection equations for systems with white parameters are strengthened. For the class of minimal mean square stabilizing compensators the strengthened discrete-time optimal projection equations are proved to be equivalent to first-order necessary optimality conditions for optimal reduced-order dynamic compensation of systems with white parameters. The conventional discrete-time optimal projection equations are proved to be weaker. As a result solutions of the conventional discrete-time optimal projection equations may not correspond to optimal reduced-order compensators. To compute optimal reduced-order compensators two numerical algorithms are proposed. One is a homotopy algorithm and one is based on iteration of the strengthened discrete-time optimal projection equations. The latter algorithm is a generalization of the algorithm that solves the full-order problem, which in turn is a generalization of the algorithm that solves the two Riccati equations of full-order LQG control through iteration. Therefore the efficiency of these three types of algorithms is comparable. It is demonstrated that, despite the strengthening of the optimal projection equations, the optimal reduced-order compensation problem, in general, may possess multiple extrema