Generalized Derivatives for Hybrid Systems

Established sensitivity results for hybrid discrete/continuous dynamic systems are generalized by relaxing smoothness assumptions on the functions governing the systems' continuous evolution and discrete event handling. The new results only require L-smoothness of these functions in the sense of Nesterov, instead of continuous differentiability. Parametric lexicographic derivatives for such a hybrid system provide useful local first-order sensitivity information, and are described as the unique solutions of auxiliary hybrid systems. This sensitivity analysis framework permits generalized derivative evaluation even for certain hybrid systems in which small changes in parameters can change the sequence of discrete modes visited. To handle parametric sensitivities of event times that are not known explicitly, conditions are provided under which a local inverse function or implicit function is L-smooth, with lexicographic derivatives that are described as the unique solutions of certain equation systems. These equation systems are readily solved when the functions involved are piecewise differentiable.