Salted Fisher Information for Hybrid Systems

1. Problem Statement

Fisher Information (FI) is a fundamental metric used to assess parameter identifiability and estimator variance bounds in dynamical systems. Traditionally, FI formulations assume that system states and their sensitivities evolve smoothly over time according to continuous differential equations.

However, many engineered systems (e.g., power systems with protection mechanisms, power electronics) are hybrid systems, characterized by continuous flows interspersed with discrete events (jumps/switches).

• The Gap: Prevailing FI methods neglect the sensitivity updates induced by discrete events. They often treat mode transitions as simple reparameterizations or ignore the fact that parameter perturbations affect not only the state trajectory but also the timing of the events (guard crossings) and the geometry of the state reset.
• The Consequence: Ignoring these discrete effects leads to an underestimation of information content, potentially resulting in incorrect conclusions about parameter identifiability (e.g., a system appearing unidentifiable when it is actually identifiable due to discrete jumps).

2. Methodology

The authors propose a unified framework to derive the Salted Fisher Information Matrix (SFIM) for hybrid systems. The methodology relies on three core components:

A. Saltation Matrix for Sensitivity Propagation

To accurately propagate parameter sensitivities ($Z(t) = \partial x / \partial \theta$) across discrete events, the paper utilizes the Saltation Matrix ($\Xi$).

• Unlike standard reset maps which only update the state, the saltation matrix provides a first-order approximation of how sensitivities change when a trajectory crosses a guard condition.
• It accounts for two simultaneous effects:
  1. The change in vector fields (from pre-event mode $f_q$ to post-event mode $f_{q'}$).
  2. The shift in event timing ($\delta t$) caused by parameter perturbations altering when the guard condition is met.
• The sensitivity update at an event time $\tau_j$ is given by:
  $$Z(\tau_j^+) = \Xi_j Z(\tau_j^-) + D_\theta R_{qq'}(x(\tau_j^-), \theta)$$
  where $\Xi_j$ is the saltation matrix derived from the guard condition, reset map, and vector fields.

B. Derivation of the SFIM

The SFIM unifies continuous information accumulation with discrete jumps.

• Continuous Flow: Information accumulates via the standard integral of output sensitivities ($J(t)$) over time intervals.
• Discrete Events: At each guard crossing $\tau_j$, an instantaneous information increment ($\Delta I_j$) is added. This increment is derived from the quadratic form of the jump in output sensitivity, which is driven by the saltation matrix.
• Event-Time Sensitivity: The paper explicitly derives the sensitivity of the switching time $\tau$ with respect to parameters ($\partial \tau / \partial \theta$), showing that parameter changes alter when the switch occurs, further influencing the accumulated information.

C. Theoretical Conditions for Identifiability

The authors establish a sufficient condition for the SFIM to be positive definite (ensuring parameter identifiability):

• Hybrid Persistence of Excitation (HPE): This condition requires that the regressors (sensitivities) remain informative across both continuous flows and discrete jumps.
• Theorem: If the system satisfies HPE, the resulting SFIM is guaranteed to be positive definite ($I_S \succ 0$).

3. Key Contributions

1. Salted Fisher Information Matrix (SFIM): A novel formulation that rigorously incorporates discrete event effects (saltation) into Fisher Information calculations, correcting the deficiencies of smooth or reset-only approximations.
2. Unified Framework: The derivation seamlessly combines continuous-time variational equations with discrete-time sensitivity updates, capturing the full hybrid arc dynamics.
3. Theoretical Guarantee: Proof that Hybrid Persistence of Excitation (HPE) is a sufficient condition for the SFIM to be positive definite, providing a theoretical basis for identifiability in hybrid systems.
4. Event-Time Sensitivity Analysis: Explicit derivation of how parameter perturbations affect the timing of discrete events, a factor often overlooked in previous hybrid system analyses.

4. Results and Numerical Experiments

The proposed framework was validated using two examples:

A. DC-DC Buck Converter (Motivational Example)

• Comparison: The SFIM was compared against a standard "averaged" continuous model.
• Outcome: The averaged model showed slow, gradual growth in Fisher Information eigenvalues, indicating limited information in certain parameter directions. In contrast, the SFIM showed significantly larger eigenvalues.
• Insight: The saltation matrix captures the geometric deformation of the sensitivity set at jumps, revealing information that the averaged model completely misses.

B. Three-Bus Power System with Wind Turbine Generator (WTG)

• System: A differential-algebraic power system where the WTG switches between pitch-inactive and pitch-active modes based on rotor speed or power thresholds.
• Parameters: Identified a subset of parameters including mechanical inertia, gain coefficients, and time constants.
• Performance Metrics:
  • Rank: The standard "Smooth FIM" was rank-deficient (rank 6), implying unidentifiable parameters. The SFIM achieved full rank (7).
  • Eigenvalues: The smallest eigenvalue ($\lambda_{min}$) increased from $1.07 \times 10^{-1}$ (Smooth) to $1.56 \times 10^{-1}$ (SFIM).
  • Information Scale: The log-determinant (a measure of total information volume) increased dramatically from 8.14 (Smooth) to 62.56 (SFIM).
• Event-Time Impact: When using a power-triggered guard (where switching time depends explicitly on parameters), the SFIM showed distinct eigenvalue distributions compared to reset-only approximations, proving that parameter-dependent event timing is a critical source of information.

5. Significance

• Improved Parameter Estimation: By accounting for discrete jumps, the SFIM provides a more accurate lower bound on estimator variance (Cramér-Rao bound), leading to better experimental design and more reliable parameter identification in hybrid systems.
• Power System Applications: The results demonstrate that ignoring hybrid dynamics in power systems (e.g., protection relays, turbine pitch control) can lead to false conclusions about system observability. The SFIM reveals that discrete events often provide the "excitation" necessary to identify parameters that are otherwise unobservable in continuous flow.
• Theoretical Advancement: The work bridges the gap between hybrid systems theory (saltation matrices) and statistical estimation theory (Fisher Information), offering a rigorous tool for analyzing complex, switching dynamical systems.

Limitations & Future Work: The current formulation assumes deterministic hybrid dynamics with known guard/reset maps and relies on first-order approximations. Future directions include extending the framework to stochastic hybrid systems and developing trajectory design algorithms to maximize SFIM.