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Probabilistic reachable sets of stochastic nonlinear systems with contextual uncertainties

Xun Shen, Ye Wang, Kazumune Hashimoto, Yuhu Wu, Sebastien Gros

TL;DR

The paper tackles probabilistic reachable sets for stochastic nonlinear systems with contextual uncertainties, where disturbance distributions depend on the current context. It formulates minimal-volume polynomial sublevel sets under contextual chance constraints and develops a resampling-based SAA using least-squares conditional density estimation to model conditional disturbances. The authors prove almost-sure convergence of the approximate solution to the original problem and provide finite-sample feasibility bounds, then validate the method on a quadrotor example, showing reduced bias relative to methods that ignore conditioning. The approach offers a principled, data-driven way to compute safer reachability sets applicable to context-aware safety-critical control.

Abstract

Validating and controlling safety-critical systems in uncertain environments necessitates probabilistic reachable sets of future state evolutions. The existing methods of computing probabilistic reachable sets normally assume that stochastic uncertainties are independent of system states, inputs, and other environment variables. However, this assumption falls short in many real-world applications, where the probability distribution governing uncertainties depends on these variables, referred to as contextual uncertainties. This paper addresses the challenge of computing probabilistic reachable sets of stochastic nonlinear states with contextual uncertainties by seeking minimum-volume polynomial sublevel sets with contextual chance constraints. The formulated problem cannot be solved by the existing sample-based approximation method since the existing methods do not consider conditional probability densities. To address this, we propose a consistent sample approximation of the original problem by leveraging conditional density estimation and resampling. The obtained approximate problem is a tractable optimization problem. Additionally, we prove the proposed sample-based approximation's almost uniform convergence, showing that it gives the optimal solution almost consistently with the original ones. Through a numerical example, we evaluate the effectiveness of the proposed method against existing approaches, highlighting its capability to significantly reduce the bias inherent in sample-based approximation without considering a conditional probability density.

Probabilistic reachable sets of stochastic nonlinear systems with contextual uncertainties

TL;DR

The paper tackles probabilistic reachable sets for stochastic nonlinear systems with contextual uncertainties, where disturbance distributions depend on the current context. It formulates minimal-volume polynomial sublevel sets under contextual chance constraints and develops a resampling-based SAA using least-squares conditional density estimation to model conditional disturbances. The authors prove almost-sure convergence of the approximate solution to the original problem and provide finite-sample feasibility bounds, then validate the method on a quadrotor example, showing reduced bias relative to methods that ignore conditioning. The approach offers a principled, data-driven way to compute safer reachability sets applicable to context-aware safety-critical control.

Abstract

Validating and controlling safety-critical systems in uncertain environments necessitates probabilistic reachable sets of future state evolutions. The existing methods of computing probabilistic reachable sets normally assume that stochastic uncertainties are independent of system states, inputs, and other environment variables. However, this assumption falls short in many real-world applications, where the probability distribution governing uncertainties depends on these variables, referred to as contextual uncertainties. This paper addresses the challenge of computing probabilistic reachable sets of stochastic nonlinear states with contextual uncertainties by seeking minimum-volume polynomial sublevel sets with contextual chance constraints. The formulated problem cannot be solved by the existing sample-based approximation method since the existing methods do not consider conditional probability densities. To address this, we propose a consistent sample approximation of the original problem by leveraging conditional density estimation and resampling. The obtained approximate problem is a tractable optimization problem. Additionally, we prove the proposed sample-based approximation's almost uniform convergence, showing that it gives the optimal solution almost consistently with the original ones. Through a numerical example, we evaluate the effectiveness of the proposed method against existing approaches, highlighting its capability to significantly reduce the bias inherent in sample-based approximation without considering a conditional probability density.
Paper Structure (19 sections, 5 theorems, 88 equations, 3 figures, 1 algorithm)

This paper contains 19 sections, 5 theorems, 88 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. System description
  4. Problem formulation
  5. Challenging issue and the goals of the paper
  6. The proposed algorithm
  7. Targeted approximation of the original problem
  8. Conditional density estimation
  9. Resampling-based probabilistic reachable sets
  10. Almost uniform convergence
  11. Probabilistic feasibility with finite samples
  12. Example
  13. Description
  14. Results and discussions
  15. Conclusions
  16. ...and 4 more sections

Key Result

Lemma 1

As $N\rightarrow\infty,$ if $\lambda_N\rightarrow 0$ with $\lambda_N^{-1}=o(N^{2/(2+\gamma)}),\ \gamma\in(0,2),$ then where $\|\cdot\|_{2}$ is the $L_2\left(\mu_{\bm{\xi}} \times \mu_{\bm{\mathrm{W}}}\right)$-norm, and $\mathcal{O}_p$ is the asymptotic order in probability.

Figures (3)

  • Figure 1: The overall structure of the proposed algorithm compared to traditional sample-based methods (PRS: probabilistic reachable set).
  • Figure 2: The confidence regions obtained by Proposed with $N=1000,\ N_{\mathsf{r}}=500,\ d=2,\ \alpha_{\mathsf{s}}=0.235$. Here, the points are the data. The colored regions are the obtained confidence regions.
  • Figure 3: The confidence regions obtained by Proposed with $N=1000,\ N_{\mathsf{r}}=500,\ d=2,\ \alpha_{\mathsf{s}}=0.235$. Here, the points are the data. The colored regions are the obtained confidence regions.

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Theorem 3