Cause versus Effect in Hybrid Systems: A Rigorous Non-standard Analysis Approach

A new approach for modeling nonlinear impulsive system is suggested based on nonstandard analysis. Basic properties of the hyperreals in nonstandard analysis are revisited, and extended to define generalized functions via a sequence approach. The extended generalized functions yield a non unique definition for a Heaviside function and the delta function, which are used to characterize a nonsmooth vector field. By using these extended generalized function, a causal way for characterizing jumps in discontinuous systems follows. An example for a simple affine system illustrates the usefulness of this theoretical development.