Abstraction by projection and application to multi-affine systems

Abstract

In this paper we present an abstraction method for nonlinear continuous systems. The main idea of our method is to project out some continuous variables, say z, and treat them in the dynamics of the remaining variables x as uncertain input. Therefore, the dynamics of x is then described by a differential inclusion. In addition, in order to avoid excessively conservative abstractions, the domains of the projected variables are divided into smaller regions corresponding to different differential inclusions. The final result of our abstraction procedure is a hybrid system of lower dimension with some important properties that guarantee convergence results. The applicability of this abstraction approach depends on the ability to deal with differential inclusions. We then focus on uncertain bilinear systems, a simple yet useful class of nonlinear differential inclusions, and develop a reachability technique using optimal control. The combination of the abstraction method and the reachability analysis technique for bilinear systems allows to treat multi-affine systems, which is illustrated with a biological system. © Springer-Verlag 2004.