New methods for suboptimal feedback control of parabolic systems

Abstract

A new class of algorithms for the suboptimal feedback control of parabolic systems is presented. Three specific techniques within this class are developed and tested numerically using a linear example with three different types of boundary conditions. Performance of two of these techniques is found to compare favorably with that obtained by use of a rigorous open-loop optimal control computed by the gradient method. It is also shown for linear systems having quadratic performance criteria that the new algorithms are superior to the technique presented by Vermeychuk and Lapidus (1973) in cases where state deviation has a weighting coefficient much greater than that for control energy. J. GREGORY VERMEYCHUK Department of Chemical EngineeringState University of New York Buffalo, New York 14214 SCOPEFor purposes of control implementation, many engineering processes are best modeled by a system of partial differential equations. The packed bed tubular reactor is one such example.The difficulties associated with the implementation of classical optimal control techniques on systems governed by partial differential equations are often so great as to preclude the use of optimal control for such systems, even when the results would be economically advantageous. The specific nature of these difficulties, discussed by Vermeychuk and Lapidus (1973), can be traced to two central causes.1. The classical optimal control is entirely precomputed using ah idealized system model, then applied in feedforward mode. Insensitivity of the resulting control to change in initial conditions, stochastic parameter variations, and modeling errors is a serious problem.2. The synthesis of any distributed optimal control, either in open loop form or in feedback form, as is possible in linear-quadratic situation, requires a large, general purpose digital computer. This requirement may conceivably be met in a practical situation, but only in very special cases. More realistically, a small processcontrol type digital computer would be employed in practice.The previously mentioned work presented examples of control algorithms which produced a near-optimal feedback control for certain classes of distributed systems at the expense of very little computational effort. One of these algorithms is based upon the successive minimization of the kernel of the performance functional. The particular technique considered was only one of a class of such techniques, in which various functions of the performance functional could be used. This paper further develops the subject of suboptimal feedback control via use of the performance functional kernel by presenting new algorithms in the general class, and by addressing two questions left unanswered in the initial treatment, relating to the stability of the controlled system and the proper choice of the best value for the free parameter appearing in the algorithms. CONCLUSIONS AND SIGNIFICANCETwo versions of a suboptimal feedback control algorithm based on the successive instantaneous minimization of the time deriva...