Observability analysis of discontinuous dynamical systems via algebraic geometry

The main objective of this paper is to provide computational methods allowing one to characterize the observability of a class of systems having discontinuous righthand side. In order to pursue this objective, it is first shown how tools borrowed from algebraic geometry can be used to characterize the observability of polynomial systems. Thus, by considering that elementary systems can be recast into polynomial form and that several systems having discontinuous right-hand side can be approximated by a 1-parameter family of elementary systems, such tools are adapted to deal with a class of elementary systems having discontinuous right-hand side. The key advantage of the proposed method is that it allows one to use classical tools, such as high-gain observers, to design state observers for systems having discontinuous right-hand side.