Optimal hold functions for digital control systems

Abstract

This chapter presents the solutions to two classes of optimal design problems in GSHF control: regulation and tracking in a standard 4-block configuration. The basic premise is that the hold function itself is the design variable. The distinguishing feature of the design procedure is that the performance index explicitly penalizes the intersampling behavior of the closed loop system by using penalty indices that are time integrals of a quadratic function of the state vector and control input of the analog plant. For regulation, it presented solutions to the optimal LQG and LQ regulation problems. It was found that for fixed monodromy problems, existence and uniqueness of an optimal hold function could be guaranteed and computed directly by solving a linear two point boundary value problem of Hamiltonian structure. The chapter provides a sufficient condition under which the optimal GSHF is exactly a ZOH. Several examples are presented to illustrate the point that, under certain conditions, regulation by optimal GSHF control that results in significant improvement of performance, as measured by the performance index; however, in other cases the optimal GSHF is far from a ZOH with corresponding little improvement in performance. A limitation of this result is that it is not known a priori the circumstances under which the optimal GSHF gives a significant improvement in performance. It also presents a solution to the deterministic tracking problem using GSHF control. A necessary and sufficient condition is given for suppressing the dynamics of the reference model from the error signal at the sample times. An optimal hold function for minimizing a criterion that reflects the intersample output tracking error and the control energy is proved to exist and be unique.