Partially Singular Control of Linear Stochastic System

The partially singular stochastic regulator problem is solved in this paper for a time-invariant linear system with noisy observation. The performance index is taken here as J=E{x'Qx+u'Ru}, in which the control weighting matrix R is not non-singular. The explicit form of the optimal dynamic controller is first derived. It is then shown that the dimension of the optimal dynamic controller is lower than that of a non-singular case. The singular optimal filtering problem in which the covariance matrix of the observation noise is singular is also briefly discussed, and the duality between the both problems is demonstrated.