A riccati equation approach to the singular LQG problem

Abstract

The problem of optimal fixed-order dynamic compensation for the singular LQG problem is considered. Necessary conditions characterizing the optimal compensator for the case involving both singular measurement noise and singular control weighting are given. The solution consists of a set of two algebraic Riccati equations and two Lyapunov equations coupled by three projection matrices. One projection is the standard order reduction projection while the other two projections reflect the two types of singularity that exist in the system. The three projections are shown to satisfy disjointness conditions. In addition to order reduction, an advantage of the fixed-structure approach is that differentiation, which is often undersirable from a practical point of view and which may exist in the unconstrained optimal control, can be avoided. It is shown that the fixed-order compensator agrees with the unconstrained solution when the latter possesses the same number of differentiations as are included in the prespecified controller structure and when the order is selected appropriately.