State Feedback |1-Optimal Controllers can be Dynamic

This paper considers discrete-time systems with full state feedback, scalar control and scalar disturbance. First, systems with a scalar regulated output are studied singular problems. It is shown that there is a large class of such systems characterized by the non-minimum phase zeros of the transfer function from the control to the regulated output, for which the 1-optimal controller is necessarily dynamic. Moreover, such controllers may have arbitrarily high order. Second, problems with two regulated outputs, one of them being the scalar control sequence, are considered non-singular problems. It is shown, by means of a fairly general example, that such problems may not have static controllers that are 1-optimal.