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Inverse Unscented Kalman Filter

Himali Singh, Kumar Vijay Mishra, Arpan Chattopadhyay

TL;DR

This paper focuses on non-linear system dynamics and forms the inverse unscented KF (I-UKF) to estimate the defender's state based on the unscented transform, or equivalently, statistical linearization technique, and generalizes this framework to unknown systems by proposing reproducing kernel Hilbert space-based UKF (RKHS-UKF).

Abstract

Rapid advances in designing cognitive and counter-adversarial systems have motivated the development of inverse Bayesian filters. In this setting, a cognitive 'adversary' tracks its target of interest via a stochastic framework such as a Kalman filter (KF). The target or 'defender' then employs another inverse stochastic filter to infer the forward filter estimates of the defender computed by the adversary. For linear systems, the inverse Kalman filter (I-KF) has been recently shown to be effective in these counter-adversarial applications. In the paper, contrary to prior works, we focus on non-linear system dynamics and formulate the inverse unscented KF (I-UKF) to estimate the defender's state based on the unscented transform, or equivalently, statistical linearization technique. We then generalize this framework to unknown systems by proposing reproducing kernel Hilbert space-based UKF (RKHS-UKF) to learn the system dynamics and estimate the state based on its observations. Our theoretical analyses to guarantee the stochastic stability of I-UKF and RKHS-UKF in the mean-squared sense show that, provided the forward filters are stable, the inverse filters are also stable under mild system-level conditions. We show that, despite being a suboptimal filter, our proposed I-UKF is a conservative estimator, i.e., I-UKF's estimated error covariance upper-bounds its true value. Our numerical experiments for several different applications demonstrate the estimation performance of the proposed filters using recursive Cramér-Rao lower bound and non-credibility index (NCI).

Inverse Unscented Kalman Filter

TL;DR

This paper focuses on non-linear system dynamics and forms the inverse unscented KF (I-UKF) to estimate the defender's state based on the unscented transform, or equivalently, statistical linearization technique, and generalizes this framework to unknown systems by proposing reproducing kernel Hilbert space-based UKF (RKHS-UKF).

Abstract

Rapid advances in designing cognitive and counter-adversarial systems have motivated the development of inverse Bayesian filters. In this setting, a cognitive 'adversary' tracks its target of interest via a stochastic framework such as a Kalman filter (KF). The target or 'defender' then employs another inverse stochastic filter to infer the forward filter estimates of the defender computed by the adversary. For linear systems, the inverse Kalman filter (I-KF) has been recently shown to be effective in these counter-adversarial applications. In the paper, contrary to prior works, we focus on non-linear system dynamics and formulate the inverse unscented KF (I-UKF) to estimate the defender's state based on the unscented transform, or equivalently, statistical linearization technique. We then generalize this framework to unknown systems by proposing reproducing kernel Hilbert space-based UKF (RKHS-UKF) to learn the system dynamics and estimate the state based on its observations. Our theoretical analyses to guarantee the stochastic stability of I-UKF and RKHS-UKF in the mean-squared sense show that, provided the forward filters are stable, the inverse filters are also stable under mild system-level conditions. We show that, despite being a suboptimal filter, our proposed I-UKF is a conservative estimator, i.e., I-UKF's estimated error covariance upper-bounds its true value. Our numerical experiments for several different applications demonstrate the estimation performance of the proposed filters using recursive Cramér-Rao lower bound and non-credibility index (NCI).
Paper Structure (31 sections, 9 theorems, 53 equations, 5 figures, 2 algorithms)

This paper contains 31 sections, 9 theorems, 53 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Prior Art
  3. Our contributions
  4. System Model
  5. Inverse UKF
  6. Forward UKF
  7. I-UKF
  8. Continuous-time state evolution
  9. Complex-valued systems
  10. RKHS-UKF
  11. Performance Analyses
  12. Inverse UKF
  13. Forward UKF
  14. I-UKF
  15. Conservative estimator
  16. ...and 16 more sections

Key Result

Theorem 1

Consider the forward UKF with the non-linear stochastic system given by eqn:state transition x and eqn:observation y. The forward UKF's estimation error $\widetilde{\mathbf{x}}_{k}$ is exponentially bounded in mean-squared sense and bounded with probability one if the following conditions hold true. C2.$\mathbf{U}^{x}_{k}$ and $\mathbf{F}_{k}$ are non-singular for every $k\geq 0$. C3. The constant

Figures (5)

  • Figure 1: An illustration of I-UKF's recursion at $k$-th time step. The defender's true state at $k$-th time is $\mathbf{x}_{k}$, which the attacker observes as $\mathbf{y}_{k}$ through observation function $h(\cdot)$ with additive measurement noise $\mathbf{v}_{k}$. Forward UKF provides estimate $\hat{\mathbf{x}}_{k}$ of $\mathbf{x}_{k}$ using $\mathbf{y}_{k}$. The defender observes $\hat{\mathbf{x}}_{k}$ as $\mathbf{a}_{k}$ through observation function $g(\cdot)$ with additive measurement noise $\bm{\epsilon}_{k}$. Finally, with $\mathbf{a}_{k}$ and $\mathbf{x}_{k}$ as inputs, I-UKF computes estimate $\hat{\newline {\hat{\mathbf{x}}}}_{k}$ of $\hat{\mathbf{x}}_{k}$.
  • Figure 2: (a) Time-averaged RMSE and RCRLB, and (b) NCI for forward and inverse UKF for FM demodulator system.
  • Figure 3: (a) Time-averaged estimation error and RCRLB, and (b) NCI for forward and inverse UKF for vehicle reentry system averaged over $100$ runs.
  • Figure 4: (a) Time-averaged RMSE, and (b) NCI for forward and inverse RKHS-UKF for FM demodulator system, compared with forward and inverse EKF and RKHS-EKF.
  • Figure 5: (a) Time-averaged RMSE, and (b) Time-averaged NCI for forward and inverse RKHS-UKF for Lorenz system, compared with forward and inverse UKF, averaged over $50$ runs.

Theorems & Definitions (30)

  • Remark 1: Unknown $\kappa$
  • Remark 2: Differences from I-KF and I-EKF
  • Remark 3: Non-Gaussian noise
  • Remark 4: Difference from kernel KFs
  • Definition 1: Exponential mean-squared boundedness reif1999stochastic
  • Theorem 1: Stochastic stability of forward UKF
  • Theorem 2: Stochastic stability of I-UKF
  • proof
  • Definition 2: Conservative estimatebattistelli2014kullback
  • Theorem 3
  • ...and 20 more