Salted Fisher Information for Hybrid Systems
This work addresses a theoretical gap in parameter estimation for hybrid systems, which is incremental as it extends existing Fisher information methods to include discrete events.
The paper tackled the problem of Fisher information formulations neglecting sensitivity updates from discrete events in hybrid systems, resulting in a new salted Fisher information matrix (SFIM) that unifies continuous accumulation with discrete updates and establishes conditions for its positive definiteness.
Discrete events alter how parameter influence propagates in hybrid systems. Prevailing Fisher information formulations assume that sensitivities evolve smoothly according to continuous-time variational equations and therefore neglect the sensitivity updates induced by discrete events. This paper derives a Fisher information matrix formulation compatible with hybrid systems. To do so, we use the saltation matrix, which encodes the first order transformation of sensitivities induced by discrete events. The resulting formulation is referred to as the salted Fisher information matrix (SFIM). The proposed framework unifies continuous information accumulation during flows with discrete updates at event times. We further establish that hybrid persistence of excitation provides a sufficient condition for positive definiteness of the SFIM. Examples are provided to demonstrate the merit of the proposed approach, including a three bus generator wind turbine differential algebraic power system