Symbolic Computation of Lyapunov Functions using Evolutionary Algorithms

J.S. McGough, A.W. Christianson, and R.C. Hoover (USA)


Lyapunov Functions, Grammatical Evolution, Evolutionary Computing, Symbolic Computation, Dynamical Systems.


This paper is concerned with the question of stability in dynamical systems, specifically the issue of computing symbolic forms of Lyapunov functions for given dynami cal systems. Due to the non-constructive form of the the Lyapunov constraints, we employ a type of evolutionary algorithm to construct candidate Lyapunov functions. Evo lutionary Algorithms have demonstrated results in a vast array of optimization problems and are regularly employed in engineering design. We study the application of a variant of Genetic Pro gramming known as Grammatical Evolution (GE). GE dis tinguishes itself from more traditional forms of genetic pro grams in that it separates the internal representation of a potential solution from the actual target expression. Strings of integers are evolved, with the candidate expressions be ing generated by performing a mapping using a problem specific grammar. Traditional approaches using Genetic Programming have been plagued by unrestrained expres sion growth, stagnation and lack of convergence. These are addressed by the more biologically realistic gene represen tation and variations in the genetic operators. Illustrative examples are presented to validate the proposed technique.

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