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Data-Driven Observability Analysis for Nonlinear Stochastic Systems

Pierre-François Massiani, Mona Buisson-Fenet, Friedrich Solowjow, Florent Di Meglio, Sebastian Trimpe

TL;DR

The paper addresses observability for nonlinear stochastic systems without relying on explicit dynamics by introducing distributional distinguishability, which compares output distributions under different initial conditions. It shows that, for linear or certain nonlinear systems, distributional and deterministic notions align, and proposes a practical, data-driven approach using kernels and maximum mean discrepancy (MMD) to quantify and test distinguishability from output data. A finite-sample MMD estimator and a two-sample test enable absolute and relative distinguishability assessments, with interpretations in terms of required data and observer design implications. Experiments on simulation and hardware (Duffing oscillator and Furuta pendulum) illustrate how noise influences indistinguishability classes, how to construct state-space distinguishability maps, and how sensor choices affect observability and observer performance. Overall, the framework provides a principled, model-free path to analyze and design observers and sensor configurations in uncertain, nonlinear settings.

Abstract

Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.

Data-Driven Observability Analysis for Nonlinear Stochastic Systems

TL;DR

The paper addresses observability for nonlinear stochastic systems without relying on explicit dynamics by introducing distributional distinguishability, which compares output distributions under different initial conditions. It shows that, for linear or certain nonlinear systems, distributional and deterministic notions align, and proposes a practical, data-driven approach using kernels and maximum mean discrepancy (MMD) to quantify and test distinguishability from output data. A finite-sample MMD estimator and a two-sample test enable absolute and relative distinguishability assessments, with interpretations in terms of required data and observer design implications. Experiments on simulation and hardware (Duffing oscillator and Furuta pendulum) illustrate how noise influences indistinguishability classes, how to construct state-space distinguishability maps, and how sensor choices affect observability and observer performance. Overall, the framework provides a principled, model-free path to analyze and design observers and sensor configurations in uncertain, nonlinear settings.

Abstract

Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from measurement data. The presence of noise further complicates this analysis, as standard notions of distinguishability are tailored to deterministic systems. We build on distributional distinguishability, which extends the deterministic notion by comparing distributions of outputs of stochastic systems. We first show that both concepts are equivalent for a class of systems that includes linear systems. We then present a method to assess and quantify distributional distinguishability from output data. Specifically, our quantification measures how much data is required to tell apart two initial states, inducing a continuous spectrum of distinguishability. We propose a statistical test to determine a threshold above which two states can be considered distinguishable with high confidence. We illustrate these tools by computing distinguishability maps over the state space in simulation, then leverage the test to compare sensor configurations on hardware.
Paper Structure (27 sections, 7 theorems, 19 equations, 3 figures, 1 table)

This paper contains 27 sections, 7 theorems, 19 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Stochastic Observability
  4. Metricizing Observability
  5. Kernel Mean Embeddings
  6. Notations
  7. Distinguishability
  8. Deterministic Distinguishability
  9. Distributional Distinguishability
  10. Noise and Distinguishability
  11. Measurement Noise
  12. Linear Systems
  13. Measuring Distributional Observability
  14. The MMD to Measure Relative Distinguishability
  15. Background
  16. ...and 12 more sections

Key Result

Theorem 1

Let $\mu_\mathrm{a}, \mu_\mathrm{b}\in\mathcal{M}^+_1(\mathbb{X})$ be two initial distributions, and let Assumption asmptn:measurement noise output map hold. If $\mu_\mathrm{a}$ and $\mu_\mathrm{b}$ are distributionally distinguishable, the distributions of $\hat{\Gamma}_\mathrm{a}\,\dot=\,(h(\Phi(t

Figures (3)

  • Figure 1: MMD over an $x_\mathrm{b}$-grid for linear system \ref{['def_syst_drift']} from reference point $x_\mathrm{a}=(1.5, 0.5)$ (white star). The empirical class of indistinguishability (red points) is computed using output data from the noisy system. It matches the analytical class of the nominal system.
  • Figure 2: MMD over an $x_\mathrm{b}$-grid for the Duffing oscillator \ref{['eq:duffing']} for different values of $x_\mathrm{a}$ (white stars). The trajectory without noise (orange) initialized in $x_\mathrm{a}$ is known to be a subset of the class of indistinguishability. The states where the test did not trigger (red points) constitute the empirical class of indistinguishability. In (a), this class contains not only the trajectory starting from $x_\mathrm{a}$, but also its symmetric w.r.t. the origin. In (c), the empirical class of distinguishability differs due to the significant process noise ($b_1=b_2=0.5$ rather than $0.05$). In all cases, vectors generating the null space of the nominal system's empirical Gramian (white arrow) are tangent to its class of indistinguishability.
  • Figure 3: Scheme of the Furuta pendulum (a), and behavior of the EKF over time (b and c): estimation error (median and quartiles) of the non-measured state (top), and distribution of the estimated $\sin(\theta_1)$ (bottom). In the distinguishable case (b), the EKF converges (top) and the distributions of the predictions differ during the transient (bottom). In the indistinguishable case (c), the estimation error remains high (top) and the distributions similar (bottom); the output of the observer is statistically the same for both initial states. Distributions (bottom) are interpolated by kernel density estimation.

Theorems & Definitions (15)

  • Definition 1: Son1998
  • Definition 2: Distributional Distinguishability
  • Example 1: Additive Noise
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1: A Kernel on Trajectories
  • Proposition 1
  • Theorem 3: GBR2012
  • Proposition 2
  • ...and 5 more