On a New Optimization Based Lyapunov Methodology for the Estimation of Domains of Attraction

L. MATALLANA PÉREZ; A. M. BLANCO; J. A. BANDONI

Salt Lake City, EEUU

Congreso; AIChE Annual Meeting 2007; 2007

AIChE

The analysis of the eigenstructure of the Jacobean matrix of a nonlinear dynamic system at an equilibrium point provides local information on resonance, instability and growth or decay rates of the dynamic trajectories. In order to completely characterize an asymptotically stable equilibrium point it is also required some information about the size and shape of the region of the state space where asymptotically stable trajectories that converge to the point under study originate. Such a region constitutes the Domain of Attraction of the equilibrium point.  The Domain of Attraction is in general a complicated set, which does not admit analytical representation except in special cases with mild nonlinearities or particular structures. Several approaches, other than intensive dynamic simulation, have been proposed so far to address the estimation of Domains of Attraction. A particularly successful family of techniques was derived from the Lyapunov stability theory, which provides an “energetic” approach to dynamic stability analysis. Such an approach makes use the so-called Lyapunov functions. Roughly, the Domain of Attraction of a given stable equilibrium point can be approximated by a level set of a Lyapunov function of the equilibrium point in the case that such a function exist.  In this contribution, an optimization approach to estimate the Domains of Attraction of equilibrium points of autonomous dynamic systems is proposed. The Lyapunov-based methodology simultaneously provides an estimation of the Domain of Attraction and the parameters of the adopted Lyapunov candidate. A fermentation reactor is considered as case study.