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Risk averse deterministic Kalman filters for uncertain dynamical systems

Karl Kunisch, Jesper Schröder

TL;DR

This paper develops risk-averse state estimation for linear deterministic systems with parametric uncertainty in A_sigma by extending Kalman filtering through a deterministic energy-minimization framework. It introduces three estimators: x0 (expected energy), x_infty (worst-case energy), and x_theta (entropic risk) and proves existence, regularity, and convergence properties, including x_theta -> x0 as theta -> 0 and x_theta -> x_infty as theta -> infinity. It derives error bounds relative to the true parameter estimator and demonstrates on two numerical examples showing improved protection against large reconstruction errors under model uncertainty. The results provide a robust filtering approach for uncertain linear systems and point to future work on time-varying, nonlinear, and PDE-based models.

Abstract

Taking a deterministic viewpoint this work investigates extensions of the Kalman-Bucy filter for state reconstruction to systems containing parametric uncertainty in the state operator. The emphasis lies on risk averse designs reducing the probability of large reconstruction errors. In a theoretical analysis error bounds in terms of the variance of the uncertainties are derived. The article concludes with a numerical implementation of two examples allowing for a comparison of risk neutral and risk averse estimators.

Risk averse deterministic Kalman filters for uncertain dynamical systems

TL;DR

This paper develops risk-averse state estimation for linear deterministic systems with parametric uncertainty in A_sigma by extending Kalman filtering through a deterministic energy-minimization framework. It introduces three estimators: x0 (expected energy), x_infty (worst-case energy), and x_theta (entropic risk) and proves existence, regularity, and convergence properties, including x_theta -> x0 as theta -> 0 and x_theta -> x_infty as theta -> infinity. It derives error bounds relative to the true parameter estimator and demonstrates on two numerical examples showing improved protection against large reconstruction errors under model uncertainty. The results provide a robust filtering approach for uncertain linear systems and point to future work on time-varying, nonlinear, and PDE-based models.

Abstract

Taking a deterministic viewpoint this work investigates extensions of the Kalman-Bucy filter for state reconstruction to systems containing parametric uncertainty in the state operator. The emphasis lies on risk averse designs reducing the probability of large reconstruction errors. In a theoretical analysis error bounds in terms of the variance of the uncertainties are derived. The article concludes with a numerical implementation of two examples allowing for a comparison of risk neutral and risk averse estimators.

Paper Structure

This paper contains 19 sections, 21 theorems, 117 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Kalman filter
  3. The stochastic formulation
  4. The deterministic formulation
  5. Kalman filtering for uncertain systems
  6. Modeling uncertain parameters
  7. Minimizing the expected energy
  8. Minimizing the worst case energy
  9. Minimizing the entropic risk of the energy
  10. Continuity of the state estimator in the risk aversion parameter
  11. Error analysis
  12. Numerical experiments
  13. General setup
  14. Harmonic oscillator with uncertain damping
  15. Connected amplidynes with uncertain inductances
  16. ...and 4 more sections

Key Result

Proposition 3.1

[proposition]prop: charEstMin The state estimator $\widehat{x}_0$ is well-defined and given in terms of the Kalman filter trajectories and precision matrices associated with the individual Kalman filters. For $t \in [0,T]$ it holds where $\mathcal{P}(t) = \sum_{k=1}^N P_k(t)$.

Figures (2)

  • Figure 1: Harmonic oscillator with uncertain damping parameter
  • Figure 2: Amplidyne with uncertain inductance

Theorems & Definitions (45)

  • Proposition 3.1
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Definition 3.5
  • Theorem 3.6
  • proof
  • Definition 3.8
  • ...and 35 more