Regulator problems with a cost for applying or changing control

Abstract

An examination is made of linear regulator problems with an additional cost for applying a nonzero control at each instant. The dual of this problem is the least-squares filtering problem with a cost for taking measurements. The regulator problem with a cost for performing a non-zero change in control is also examined by reducing it to the problem with a cost for applying control. Because a solution to the general problem involves solving a two-point boundary value problem in nonlinear matrix differential or difference equations, two easily computed suboptimal control schemes are proposed and their properties discussed. Bounds on the total length of time the optimum control is nonzero are derived, as well as upper bounds on the difference in performance between the optimum control and either of the proposed suboptimum schemes. For a class of continuous-time problems, a bound is derived on the number of times the control switches from zero to nonzero or vice versa. A class of problems is also delineated for which the optimum solution has some special properties which enable it to be computed simply and directly.