A Global Optimization Approach for the Estimation of Domains of Attraction

LUIS GERONIMO MATALLANA PÉREZ; ANIBAL BLANCO; ALBERTO BANDONI

Salvador-Bahía-Brasil

Congreso; 10th International Symposium on Process Systems Engineering - PSE2009; 2009

Universidad de Sao Paulo

In order to completely characterize an asymptotically stable equilibrium point, some information about the size and shape of its region of asymptotic stability is required. Such a region is also referred as the Domain of Attraction (DOA) of the equilibrium, and represents the portion of the state space where asymptotically stable trajectories that converge to the point originate. The DOA is in general a complicated set, which does not admit analytical representation except in special cases. Many techniques have been proposed so far to address the estimation of DOAs including dynamic simulation based approaches. The Lyapunov stability theory, which provides an energetic approach to dynamic stability analysis, is the basis of a family of techniques whose rationale is to approximate the DOA by a level set of a Lyapunov function of the equilibrium point (Genesio et al., 1985). Several approaches have been proposed to identify the best level set of a given Lyapunov function by solving an optimization model. This problem is very challenging for the general case since the resulting optimization model is semi-infinite nonlinear. In order to overcome the inherent complexity the adoption of grids was suggested by several authors (Matallana et al., 2007; Tibken and Hachicho, 2000). Recently, formulations that make use of results on deterministic global optimization were proposed (Hachicho, 2007). The technique allows the identification of the best possible level set of a rational Lyapunov function that constitutes an estimation of the DOA of the equilibrium under study for nonlinear dynamic systems of polynomial type. In this contribution, an extension of the formulation presented in Hachicho (2007) is proposed. An optimization model is formulated, which includes additional constraints to avoid possible dummy solutions and makes use of global optimization software to address nonlinear systems of the general type. The technique is firstly illustrated by a two states system that presents a very rich nonlinear behavior and is then applied to typical chemical engineering reacting systems.