Discrete-time fast regulator with fast observer

A discrete-time fast regulator with fast observer is considered in this paper. The system is assumed as a linear time-invariant nth order r-input, m-output system. Complete controllability and complete observability are assumed. No assumptions are made on the non-singularity of the system matrix and integrality of n/r and n/m. It is shown that the fast regulator with fast observer transfers any initial state to the origin within the number of steps equal to the sum of observability and controllability indices. Simple design methods for fast regulator and fast observer are presented. Also, it is illustrated geometrically how the state point is transferred to a subspace with successively decreased dimension for each step until it arrives at the origin.