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Risk-Aware Fixed-Time Stabilization of Stochastic Systems under Measurement Uncertainty

Mitchell Black, Georgios Fainekos, Bardh Hoxha, Dimitra Panagou

TL;DR

This work tackles risk-aware finite-time stabilization for stochastic nonlinear systems with output feedback under measurement noise. It introduces two Lyapunov-based certificates, the RA-FxT-CLF and the RA-PI-CLF, that guarantee probabilistic fixed-time convergence to a goal set with bounds $T_g$ and probabilities $p_g$ or $p_g^*$, even with observer error. The theoretical results are supported by stochastic analysis (level-crossing and It\^o isometry) and validated through a 2-D nonlinear system and a simulated fixed-wing UAV reach-avoid scenario, showing improved convergence characteristics as risk aversion increases. This framework enables robust, provably safe reach-avoid behavior in noisy, uncertain environments, with practical reformulations as QP-based controllers for real-time implementation.

Abstract

This paper addresses the problem of risk-aware fixed-time stabilization of a class of uncertain, output-feedback nonlinear systems modeled via stochastic differential equations. First, novel classes of certificate functions, namely risk-aware fixed-time- and risk-aware path-integral-control Lyapunov functions, are introduced. Then, it is shown how the use of either for control design certifies that a system is both stable in probability and probabilistically fixed-time convergent (for a given probability) to a goal set. That is, the system trajectories probabilistically reach the set within a finite time, independent of the initial condition, despite the additional presence of measurement noise. These methods represent an improvement over the state-of-the-art in stochastic fixed-time stabilization, which presently offers bounds on the settling-time function in expectation only. The theoretical results are verified by an empirical study on an illustrative, stochastic, nonlinear system and the proposed controllers are evaluated against an existing method. Finally, the methods are demonstrated via a simulated fixed-wing aerial robot on a reach-avoid scenario to highlight their ability to certify the probability that a system safely reaches its goal.

Risk-Aware Fixed-Time Stabilization of Stochastic Systems under Measurement Uncertainty

TL;DR

This work tackles risk-aware finite-time stabilization for stochastic nonlinear systems with output feedback under measurement noise. It introduces two Lyapunov-based certificates, the RA-FxT-CLF and the RA-PI-CLF, that guarantee probabilistic fixed-time convergence to a goal set with bounds and probabilities or , even with observer error. The theoretical results are supported by stochastic analysis (level-crossing and It\^o isometry) and validated through a 2-D nonlinear system and a simulated fixed-wing UAV reach-avoid scenario, showing improved convergence characteristics as risk aversion increases. This framework enables robust, provably safe reach-avoid behavior in noisy, uncertain environments, with practical reformulations as QP-based controllers for real-time implementation.

Abstract

This paper addresses the problem of risk-aware fixed-time stabilization of a class of uncertain, output-feedback nonlinear systems modeled via stochastic differential equations. First, novel classes of certificate functions, namely risk-aware fixed-time- and risk-aware path-integral-control Lyapunov functions, are introduced. Then, it is shown how the use of either for control design certifies that a system is both stable in probability and probabilistically fixed-time convergent (for a given probability) to a goal set. That is, the system trajectories probabilistically reach the set within a finite time, independent of the initial condition, despite the additional presence of measurement noise. These methods represent an improvement over the state-of-the-art in stochastic fixed-time stabilization, which presently offers bounds on the settling-time function in expectation only. The theoretical results are verified by an empirical study on an illustrative, stochastic, nonlinear system and the proposed controllers are evaluated against an existing method. Finally, the methods are demonstrated via a simulated fixed-wing aerial robot on a reach-avoid scenario to highlight their ability to certify the probability that a system safely reaches its goal.
Paper Structure (14 sections, 5 theorems, 44 equations, 6 figures, 3 tables)

This paper contains 14 sections, 5 theorems, 44 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Mathematical Preliminaries
  4. Problem Formulation
  5. Risk-Aware Fixed-Time Stabilization
  6. Risk-Aware Fixed-Time CLF
  7. Risk-Aware Path Integral CLF
  8. Control Design for Risk-Aware FxT-Stabilization
  9. Numerical Case Studies
  10. Empirical Study: 2D Nonlinear System
  11. Fixed-Wing UAV Example
  12. Conclusion
  13. Proof of Lemma \ref{['lem.wiener_crossing_probability']}
  14. Proof of Lemma \ref{['lem.multidim_ito_isometry']}

Key Result

Lemma 1

Given $a>0$, the probability that $w_t < a$, $\forall t \in \mathcal{T}$, is given by

Figures (6)

  • Figure 1: State paths ($x$) starting from initial condition $x_0$ (marked by X) for the RA-FxT-CLF controlled system with EKF state estimation ($\hat{x}$) and $p_g=0.55$. All paths displayed (translucent), with a selection highlighted (sims. 1, 25, 50, 90).
  • Figure 2: Control trajectories for the RA-FxT-CLF controlled system with EKF state estimation and $p_g=0.55$. All controls displayed (translucent), with a selection highlighted (sims. 1, 25, 50, 90).
  • Figure 3: Lyapunov function values of the true state ($V(\boldsymbol{x})$) versus time for the RA-FxT-CLF controlled system with EKF state estimation and $p_g=0.55$. All runs displayed (translucent), with a selection highlighted (sims. 1, 25, 50, 90).
  • Figure 4: Fixed-wing UAV paths in the XY plane starting from initial condition (marked by X).
  • Figure 5: Fixed-wing UAV paths in the YZ plane starting from initial condition (marked by X). Note from Figure \ref{['fig.fixed_wing_xy_paths']} that the vehicle actually avoids the shown obstacles.
  • ...and 1 more figures

Theorems & Definitions (16)

  • Lemma 1: Level Crossing
  • proof
  • Lemma 2: Multivariate It$\hat{\mathrm{o}}$ Isometry
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 3
  • proof
  • Definition 4
  • ...and 6 more