Shaping of system responses with minimax optimization in the time domain

Abstract

A minimax problem is introduced for the terminal control of a generic dynamical system without disturbances. The maximum magnitude of the weighted output of the system is minimized over a finite interval by the control input of a prescribed class. Such important characteristics of the controlled system appear explicitly in the proposed problem as the maximum magnitude and settling property of the output. Two numerical examples are shown to illustrate the problem. A slewing experiment is also presented to demonstrate the application of the minimax optimal control. ANY efforts have been made to develop optimal control problems with quadratic criteria, especially for linear systems, over the past many years. Linear control system design with quadratic criteria (the LQ method) has such an advantage that the optimal control is analytically obtained as linear state feedback or a dynamic compensator. On the other hand, there are difficulties in the selection of free design parameters in a control synthesis based on the LQ method. Although some guidelines exist for weight selection,1'2 and the exponential decay rate of the closed-loop state can be prescribed2'3 in the LQ method, the free parameters affect control performance in an indirect and complicated manner. Iterative adjustment of the free parameters is a laborious task for a control designer, and a computational algorithm such as the one presented in Refs. 4 and 5 may be required to satisfy design specifications. It is often the case that a criterion different from quadratic criteria is preferable for terminal control in which open-loop control is acceptable. A minimum-fuel or minimum-time problem provides the perfect criterion when the specific design objective is to minimize fuel or time. The number of free parameters is confined in such cases, and laborious adjustment may not be required. To achieve good control performance with optimal control, it is necessary to analyze the control objective and to define a meaningful performance index so that the optimization yields satisfactory control behavior for the objective. In this paper, the performance of terminal control is measured in terms of the maximum magnitude of a controlled output vector and decay rate of the output in order to evaluate the time responses of the output more directly than quadratic criteria. A minimax problem in the time domain is introduced to shape time responses by minimizing the maximum magnitude of the weighted output with the control input of a prescribed class. The weighting matrix of the output vector is dependent on time to prescribe the decay rate of the output. The minimax problem for terminal control is often called the Ghebyshev minimax optimal control problem and has been studied for many years. Johnson6 discussed the geometric properties of the minimax solution. Barry7 gave a Mayer-type formulation of the problem and proposed an approximation method to yield a suboptimal minimax control that is arbitrarily close and in many cases identical to optimal control.