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Salted Fisher Information for Hybrid Systems

Bukunmi G. Odunlami, Marcos Netto, Hai Lin

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

Salted Fisher Information for Hybrid Systems

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

Paper Structure

This paper contains 18 sections, 3 theorems, 56 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximum likelihood estimation
  4. Fisher information
  5. Hybrid systems
  6. Saltation matrix
  7. Motivational example
  8. Salted Fisher Information
  9. Fisher information change at hybrid events
  10. Parameter sensitivity of event times
  11. Hybrid persistence of excitation and SFIM full rank
  12. Numerical Experiment
  13. Differential-algebraic equations
  14. Hybrid dynamics
  15. Rotor speed guard
  16. ...and 3 more sections

Key Result

Proposition C.2

The matrix containing the sensitivities of the state variables with respect to parameters, $Z(t)\coloneqq \partial x(t,\theta)/\partial\theta$, evolves as follows. For a mode $q \in Q$, while the dynamics are flowing, At event time $\tau_j$, the state undergoes a reset where $\eta_j(\theta)$ denotes the reset noise that may depend on the parameter. Differentiating eq:reset_with_noise with respec

Figures (6)

  • Figure 3: One-line diagram of a buck converter.
  • Figure 4: Left: State trajectories obtained using the hybrid and continuous (averaged) models. Right: Corresponding FIM eigenvalues.
  • Figure 5: The pre-event perturbation set, $\delta x^-$, mapped across a discrete event via the reset Jacobian, $D_tR$, and the saltation matrix, $\Xi$. Representative perturbations (unfilled black circles) and their images under $D_tR$ (red dots) and $\Xi$ (blue squares) highlight the induced geometric deformation.
  • Figure 6: Three-bus system with a wind turbine generator.
  • Figure 7: Top: Hybrid mode sequence showing switching between $q_1$ and $q_2$ over time. Middle: Evolution of $\log \det(F+\varepsilon I)$ for the smooth FIM and the SFIM. Bottom: Zoomed-in view of the middle plot highlighting the stepwise increases in the SFIM at switching instants.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition C.2: Hybrid sensitivity evolution
  • Definition C.3: Salted Fisher information
  • Proposition C.4: Event-time sensitivity
  • proof
  • Definition C.5: Hybrid persistence of excitation
  • Theorem C.6: HPE implies SFIM positive definite
  • proof