Using Genetic Algorithms to Optimize Control Lyapunov Functions

Abstract

The problem of simultaneously tuning the control law and the control Lyapunov function (CLF) for nonlinear systems is considered, and a genetic algorithm optimization method is proposed as a general solution. It is shown that direct sampling of the nonlinear effects on the CLF offers an advantage over robust linear control methods that assume a linearized system with the nonlinearities treated as uncertainties. While the genetic algorithm approach is not guaranteed to find a solution, experimentation suggests that it does possess a high likelihood of finding "good" solutions to some nontrivial problems. The control law and the local CLF are tuned simultaneously to maximize the rate of convergence and minimize the control effort of nonlinear systems. Two control laws are tested: 1. an LQR full-state feedback controller, and 2. a Sontag-like nonlinear full-state feedback controller. It is assumed that a quadratic Lyapunov function is a local CLF for the nonlinear systems considered. The proposed optimization method does not offer a strict guarantee on controller performance. However, it is suggested that with enough randomized performance sampling, the controller will achieve the estimated performance level with sufficiently high confidence, making the proposed method a practical solution for real-world controller design.