Using genetic algorithms for controller design - Simultaneous stabilization and eigenvalue placement in a region

Abstract

In this paper we investigate the possibility of using genetic algorithms to solve several optimization problems arising in the design of controllers. First we consider the problem of stabilizing a finite number of plants via dynamic output feedback (the simultaneous stabilization problem). For a single plant, one could design a controller by using a low order Luenberger observer, Ref. 1. Alternatively, the methods in Ref. 2, 3 give sufficient conditions on the order of the compensator for arbitrary eigenvalue assignment. However, it may be possible to achieve stability by a compensator of lower order than that required by these theories. Ref. 4-6 are examples of such approaches. For the problem of simultaneously stabilizing a finite number of single input plants, Ref. 7 shows that if the plants are minimum phase, stability can be achieved with a compensator of order k = n m 1, where m is the order of the plant transfer function numerator and n the order of the denominator. Here we do not impose a minimum phase assumption and present a numerical technique for obtaining simultaneous stabilization via a low order compensator. A numerical optimization technique, called the genetic algorithm (GA), has been successfully applied to a variety of problems for many years. It is based on the principles of survival-of-the-fittest and basic genetic operators in biological systems. The application of the GA in control systems has G n e d interest in recent years. We have applied th$ technique to solve the problem of stabilizing a set of plants using full state feedback, Ref. 8. This paper extends this application to the problem of stabilizing a set of plants using dynamic output feedback. Two forms of output feedback controllers are used to stabilize the system. One of the forms allows the elements of the matrices defining the compensator to be completely arbitrary, while the other imposes a canonical structure on these matrices and thereby reduces the number of free parameters. The benchmark example from Ref. 9 is studied to show the utility of the GA. The performance of three variations of the GA algorithm are also compared. We also consider the problem of choosing the controller to place the eigenvalues in a wedge-shaped portion of the left half plane to achieve performance criteria as well as stabilization. An example demonstrates that the GA can be used to solve this problem.