Salted Fisher Information for Hybrid Systems

TL;DR

This paper introduces the salted Fisher information matrix (SFIM), a new formulation for hybrid systems that accounts for both continuous dynamics and discrete event-induced sensitivity updates.

Contribution

It derives a Fisher information matrix framework compatible with hybrid systems using the saltation matrix, unifying continuous and discrete sensitivity analysis.

Findings

SFIM effectively captures sensitivities in hybrid systems.

Hybrid persistence of excitation ensures SFIM's positive definiteness.

Demonstrated on power system example.

Abstract

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