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Risk-Aware Control of Discrete-Time Stochastic Systems: Integrating Kalman Filter and Worst-case CVaR in Control Barrier Functions

Masako Kishida

TL;DR

This work addresses risk-aware control for discrete-time linear stochastic systems with partial observability by integrating a Kalman filter-based state estimator with worst-case CVaR optimization within a Control Barrier Function framework. The approach yields risk-aware CBF constraints that are tractable for half-space and ellipsoidal safe sets, enabling two controller synthesis methods: (i) modifying a nominal controller to satisfy safety constraints, and (ii) a CLF-CBF-based optimization that balances stabilization with safety. Key contributions include a Kalman-filter-based risk quantification of safety constraints, semidefinite-program-compatible worst-case CVaR handling for quadratic losses, and practical controller designs demonstrated on a vehicle-navigation example. The results highlight improved safety under stochastic disturbances and demonstrate how tail-risk considerations enhance the reliability of safety-critical systems in real-time control contexts.

Abstract

This paper proposes control approaches for discrete-time linear systems subject to stochastic disturbances. It employs Kalman filter to estimate the mean and covariance of the state propagation, and the worst-case conditional value-at-risk (CVaR) to quantify the tail risk using the estimated mean and covariance. The quantified risk is then integrated into a control barrier function (CBF) to derive constraints for controller synthesis, addressing tail risks near safe set boundaries. Two optimization-based control methods are presented using the obtained constraints for half-space and ellipsoidal safe sets, respectively. The effectiveness of the obtained results is demonstrated using numerical simulations.

Risk-Aware Control of Discrete-Time Stochastic Systems: Integrating Kalman Filter and Worst-case CVaR in Control Barrier Functions

TL;DR

This work addresses risk-aware control for discrete-time linear stochastic systems with partial observability by integrating a Kalman filter-based state estimator with worst-case CVaR optimization within a Control Barrier Function framework. The approach yields risk-aware CBF constraints that are tractable for half-space and ellipsoidal safe sets, enabling two controller synthesis methods: (i) modifying a nominal controller to satisfy safety constraints, and (ii) a CLF-CBF-based optimization that balances stabilization with safety. Key contributions include a Kalman-filter-based risk quantification of safety constraints, semidefinite-program-compatible worst-case CVaR handling for quadratic losses, and practical controller designs demonstrated on a vehicle-navigation example. The results highlight improved safety under stochastic disturbances and demonstrate how tail-risk considerations enhance the reliability of safety-critical systems in real-time control contexts.

Abstract

This paper proposes control approaches for discrete-time linear systems subject to stochastic disturbances. It employs Kalman filter to estimate the mean and covariance of the state propagation, and the worst-case conditional value-at-risk (CVaR) to quantify the tail risk using the estimated mean and covariance. The quantified risk is then integrated into a control barrier function (CBF) to derive constraints for controller synthesis, addressing tail risks near safe set boundaries. Two optimization-based control methods are presented using the obtained constraints for half-space and ellipsoidal safe sets, respectively. The effectiveness of the obtained results is demonstrated using numerical simulations.
Paper Structure (18 sections, 6 theorems, 40 equations, 2 figures)

This paper contains 18 sections, 6 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Notation
  4. Conditional Value-at-Risk
  5. Control Barrier Function
  6. Preparation: Integrating Kalman Filter and Worst-case CVaR in CBF
  7. System Model
  8. Kalman Filter
  9. Risk-Aware CBF
  10. Charactering the Set of Admissible Control Input Using Kalman Filter
  11. Half-space safe set
  12. Ellipsoidal safe set
  13. Controller Synthesis
  14. The gain $K_{t+1}$
  15. Method 1: Modifying a nominal controller
  16. ...and 3 more sections

Key Result

Proposition II.1

The worst-case CVaR is a coherent risk measure, i.e., it satisfies the following properties: Let $L_1 = L_1(\xi)$ and $L_2 = L_2(\xi)$ be two measurable loss functions, then the followings hold.

Figures (2)

  • Figure 1: Method 1
  • Figure 2: Method 2

Theorems & Definitions (13)

  • Definition II.1: Conditional Value-at-Risk (CVaR) RocU00ZymKR13-b
  • Definition II.2: Worst-case CVaR ZhuF09ZymKR13-b
  • Proposition II.1: Coherence properties ZhuF09Art99
  • Lemma II.1: Quadratic Loss Function ZymKR13-b, ZymKR13
  • Lemma II.2: A bound on linear $L(\xi)$Kis23c
  • Definition II.3: Control Barrier Function (CBF) ZenZS21
  • Definition III.1: Risk-Aware Control Barrier Function
  • Theorem IV.1
  • proof
  • Proposition IV.1
  • ...and 3 more