Optimum Design of Control System Compensators By Geometric Programming

Abstract

A process of condensation is used to develop a geometric programming approach to the necessary conditions for a constrained optimum in an elementary way. Special features of geometric programming not found in other optimization procedures and which offer significant advantages in the design of optimum compensators are developed. Compensator design is formulated as an optimum pole placement problem subject to a time delay response constraint. The geometric programming approach results in simple explicit equations for the optimum compensator parameters. IntroductionThe modern design approach to dynamic compensation is based on optimization and simulation using digital computers. There is, however, no general agreement on which of many optimization procedures is best suited for a given set of circumstances and a given design problem. By extension of well-known nonlinear programming methods in conjunction with optimal control theory it is relatively easy to develop algorithms for the numerical solution of problems in the optimization of dynamic systems [1][2][3][4]. Such algorithms, however, are computationally complex and involve repetitive solution of vector differential equations during each step of gradient-based search procedures. A more fundamental difficulty is that an optimum system is dependent on the choice of constraints and performance index weighting factors. Many of these choices are arbitrary. As a consequence, it is often necessary to obtain numerical solutions to a class of optimum systems and then use simulation to decide which system is "best." Such procedures can be expensive in design time and/or can be expensive in terms of computational and data processing facilities. The above reasons undoubtedly contribute to the continuing popularity of classical frequency response methods for the design of singleinput single-output systems [5]. The classical approach with its emphasis on pole-zero locations offers a designer considerable insight into how compensator dynamics effect system response. The main disadvantages of the classical approach are the lack of systematic procedures for choosing compensators which result in satisfaction of constraints and in determining optimum compensators. Classical and modern design procedures tend to merge when design problems are formulated with emphasis on optimum pole placement. However, deriving useful optimum relations between system and compensator parameters remains a basic problem. In [4] such relations are derived by use of polynomial approximations. Extensive simulation and repetitive solution of