Universal regulators for optimal damping and tracking in discrete-time systems

We consider the problem of controlling a discrete-time linear system by output feedback so as to have a second output z_t track an observed reference signal r_t. First, as a preliminary, we consider the problem of asymptotic tracking. This problem has been studied intensely in the literature, mainly in the continuous-time case. We show that only under special conditions does there exist a linear regulator which achieves this design goal and which is universal in the sense that it works for all reference signals and does not depend on them. However, if r_t is an harmonic signal with known frequencies but with unknown amplitudes and phases, a situation frequently studied in the literature, there exist such regulators under mild conditions, provided the dimension of r_t is no larger than the number of controls. However, if this dimensionality condition is not satisfied, or the plant itself is corrupted by an additive harmonic disturbance w_t of the same type as r_t, asymptotic tracking is not possible, but a steady state tracking error remains. Therefore we turn to another approach to the tracking problem, and this is our main result. The measure of performance is given by a natural quadratic cost function. The object is to design an optimal regulator which is universal in the sense that it does not depend on the unknown amplitudes and phases of r_t and w_t and is optimal for all choices of r_t and w _t. We prove that an optimal universal regulator exists in a wide class of stabilizing and possibly nonlinear regulators under natural technical conditions and that this regulator is in fact linear, provided that the dimension of the output used for feedback is no smaller than the dimension of the disturbance w_t. On the other hand, it is shown that if the dimensionality condition is not satisfied, the existence of an optimal universal regulator is not a generic property, so as a rule no optimal universal regulator exists