New Method for the Estimation of Domains of Attraction of Fixed Points from Lyapunov Functions

The estimation of the domain of stability of xed points of nonlinear dierential systems constitutes a practical problem of much interest in engineering. The procedures based on Lyapunov’s second method congures an alternative worth consideration. It has the appeal of reducing calculation complexity and is time-saving with respect to the direct, computer crunching approach which requires a systematic numerical integration of the evolution equations from a gridlike pattern of initial conditions. However, it is not devoid of problems inasmuch as the Lyapunov function itself is problem-dependent and relies too much on presumptions. Additionally, the evaluation of its corresponding domain is produced in terms of a nonlinear programming problem with inequality constraints the resolution of which may sometimes require a large investment in computer time. These problems are in part avoided by restricting to quadratic Lyapunov functions, with the possible obvious consequence of limiting the estimation of the domain. In order to simplify the estimation of domains we exploit here a novel formulation of the issue of stability of invariant surfaces within Lyapunov’s direct method [D az-Sierra et al., 2001]. The resulting method addresses directly the optimization problem associated to the evaluation of the stability domain. The problem is recast in a new, simpler form by playing both on the Lyapunov function itself and on the constraints. The gains from the procedure permit to conceive increased returns in the application of Lyapunov’s direct method once it is realized that it is not prohibitive from a computational point of view to depart from the limited quadratic Lyapunov functions.