Computational Schemes for Optimizing Domains of Attraction in Dynamical Systems

In this paper, several computational schemes are presented for the optimal tuning of the global behavior of nonlinear dynamical systems. Specifically, the maximization of the size of domains of attraction associated with invariants in parametrized dynamical systems is addressed. Cell Mapping (CM) techniques are used to estimate the size of the domains, and such size is then maximized via different optimization tools. First, a genetic algorithm is tested whose performance shows to be good for determining global maxima at the expense of high computational cost. Secondly, an iterative scheme based on a Stochastic Approximation procedure (the Kiefer-Wolfowitz algorithm) is evaluated showing acceptable performance at low cost. Finally, several schemes combining neural network based estimations and optimization procedures are addressed with promising results. The performance of the methods is illustrated with two applications: first on the well-known van der Pol equation with standard parametrization, and second the tuning of a controller for saturated systems.