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A Unified Estimation--Guidance Framework Based on Bayesian Decision Theory

Liraz Mudrik, Yaakov Oshman

TL;DR

The paper tackles the degradation of perfect-information DGL1 guidance in stochastic environments by embedding Bayesian decision theory within a GST framework. It couples an IMMPF-based estimator to produce a posterior state distribution and uses a multi-hypothesis Bayesian decision rule to select optimal guidance commands, with trajectory shaping exploited when decisions are ambiguous. Two GST-compliant guidance designs are proposed: IETS, which actively shapes trajectories to improve estimation, and EADGL1, which uses estimation awareness without shaping. Large-scale Monte Carlo simulations show that the stochastic guidance laws significantly reduce miss distance and required kill radius compared to the classical DGL1, while maintaining real-time implementability on standard hardware.

Abstract

Using Bayesian decision theory, we modify the perfect-information, differential game-based guidance law (DGL1) to address the inevitable estimation error occurring when driving this guidance law with a separately-designed state estimator. This yields a stochastic guidance law complying with the generalized separation theorem, as opposed to the common approach, that implicitly, but unjustifiably, assumes the validity of the regular separation theorem. The required posterior probability density function of the game's state is derived from the available noisy measurements using an interacting multiple model particle filter. When the resulting optimal decision turns out to be nonunique, this feature is harnessed to appropriately shape the trajectory of the pursuer so as to enhance its estimator's performance. In addition, certain properties of the particle-based computation of the Bayesian cost are exploited to render the algorithm amenable to real-time implementation. The performance of the entire estimation-decision-guidance scheme is demonstrated using an extensive Monte Carlo simulation study.

A Unified Estimation--Guidance Framework Based on Bayesian Decision Theory

TL;DR

The paper tackles the degradation of perfect-information DGL1 guidance in stochastic environments by embedding Bayesian decision theory within a GST framework. It couples an IMMPF-based estimator to produce a posterior state distribution and uses a multi-hypothesis Bayesian decision rule to select optimal guidance commands, with trajectory shaping exploited when decisions are ambiguous. Two GST-compliant guidance designs are proposed: IETS, which actively shapes trajectories to improve estimation, and EADGL1, which uses estimation awareness without shaping. Large-scale Monte Carlo simulations show that the stochastic guidance laws significantly reduce miss distance and required kill radius compared to the classical DGL1, while maintaining real-time implementability on standard hardware.

Abstract

Using Bayesian decision theory, we modify the perfect-information, differential game-based guidance law (DGL1) to address the inevitable estimation error occurring when driving this guidance law with a separately-designed state estimator. This yields a stochastic guidance law complying with the generalized separation theorem, as opposed to the common approach, that implicitly, but unjustifiably, assumes the validity of the regular separation theorem. The required posterior probability density function of the game's state is derived from the available noisy measurements using an interacting multiple model particle filter. When the resulting optimal decision turns out to be nonunique, this feature is harnessed to appropriately shape the trajectory of the pursuer so as to enhance its estimator's performance. In addition, certain properties of the particle-based computation of the Bayesian cost are exploited to render the algorithm amenable to real-time implementation. The performance of the entire estimation-decision-guidance scheme is demonstrated using an extensive Monte Carlo simulation study.
Paper Structure (22 sections, 3 theorems, 50 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 22 sections, 3 theorems, 50 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. The Perfect Information DGL1 Guidance Law
  4. Target Tracking IMMPF
  5. The IMMPF Algorithm
  6. Target Tracking IMMPF
  7. Mode Estimation
  8. Filter Initialization
  9. Estimation-Aware Guidance
  10. Bayesian Decision Theory
  11. Stochastic Guidance as a Bayesian Decision Problem
  12. Computational Effort Reduction
  13. Information-Enhancement Trajectory Shaping
  14. GST-Compliant Guidance Law
  15. Numerical Simulation Study
  16. ...and 7 more sections

Key Result

Lemma V.1

At time $t_{k}$ where $z_{ij'}(t_{go;j'}-\tau)$ is the ZEM of particle $j'$, which is propagated ahead over the interval $[t_{go;j'},t_{go;j'}-\tau]$ by applying the acceleration command as hypothesis $H_{i}$ is true.

Figures (11)

  • Figure 1: Planar engagement geometry
  • Figure 2: DGL1 game space and its singular region (grey). Equicost level lines (dashed) and boundary lines (corresponding to zero miss distance) of the singular region, $\partial \mathcal{Z}^{*}$ (solid).
  • Figure 3: Initializing radar geometry
  • Figure 4: IMMPF particle cloud distribution, color-coded by hypothesis, relative to the singular region boundaries at a particular moment during the interception.
  • Figure 5: Time-varying transition probability from mode 1 to mode 2.
  • ...and 6 more figures

Theorems & Definitions (4)

  • Lemma V.1
  • Lemma V.2
  • Theorem V.3
  • proof