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Model Predictive Control with High-Probability Safety Guarantee for Nonlinear Stochastic Systems

Zishun Liu, Liqian Ma, Yongxin Chen

TL;DR

The paper tackles safety guarantees for stochastic nonlinear systems by transforming probabilistic trajectory safety constraints into deterministic constraints via set erosion, enabling use of standard deterministic MPC. A tight probabilistic-tube radius r_{δ,t} is derived and shown to depend on the open-loop Lipschitz constant, ensuring trajectory-level safety with probability at least 1−δ. The authors prove recursive feasibility and trajectory-wide safety, and demonstrate substantial feasibility advantages at high safety levels through unicycle and 2D quadrotor experiments. The framework provides a scalable, theory-backed approach for safety-critical stochastic control with broad applicability to nonlinear dynamics.

Abstract

We present a model predictive control (MPC) framework for nonlinear stochastic systems that ensures safety guarantee with high probability. Unlike most existing stochastic MPC schemes, our method adopts a set-erosion that converts the probabilistic safety constraint into a tractable deterministic safety constraint on a smaller safe set over deterministic dynamics. As a result, our method is compatible with any off-the-shelf deterministic MPC algorithm. The key to the effectiveness of our method is a tight bound on the stochastic fluctuation of a stochastic trajectory around its nominal version. Our method is scalable and can guarantee safety with high probability level (e.g., 99.99%), making it particularly suitable for safety-critical applications involving complex nonlinear dynamics. Rigorous analysis is conducted to establish a theoretical safety guarantee, and numerical experiments are provided to validate the effectiveness of the proposed MPC method.

Model Predictive Control with High-Probability Safety Guarantee for Nonlinear Stochastic Systems

TL;DR

The paper tackles safety guarantees for stochastic nonlinear systems by transforming probabilistic trajectory safety constraints into deterministic constraints via set erosion, enabling use of standard deterministic MPC. A tight probabilistic-tube radius r_{δ,t} is derived and shown to depend on the open-loop Lipschitz constant, ensuring trajectory-level safety with probability at least 1−δ. The authors prove recursive feasibility and trajectory-wide safety, and demonstrate substantial feasibility advantages at high safety levels through unicycle and 2D quadrotor experiments. The framework provides a scalable, theory-backed approach for safety-critical stochastic control with broad applicability to nonlinear dynamics.

Abstract

We present a model predictive control (MPC) framework for nonlinear stochastic systems that ensures safety guarantee with high probability. Unlike most existing stochastic MPC schemes, our method adopts a set-erosion that converts the probabilistic safety constraint into a tractable deterministic safety constraint on a smaller safe set over deterministic dynamics. As a result, our method is compatible with any off-the-shelf deterministic MPC algorithm. The key to the effectiveness of our method is a tight bound on the stochastic fluctuation of a stochastic trajectory around its nominal version. Our method is scalable and can guarantee safety with high probability level (e.g., 99.99%), making it particularly suitable for safety-critical applications involving complex nonlinear dynamics. Rigorous analysis is conducted to establish a theoretical safety guarantee, and numerical experiments are provided to validate the effectiveness of the proposed MPC method.

Paper Structure

This paper contains 16 sections, 2 theorems, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Safety of Stochastic Systems
  4. Stochastic MPC with Chance Constraint
  5. Stochastic MPC with High-Probability Safety Guarantee
  6. Tractable Set Erosion-Based MPC
  7. Recursive Feasibility and Safety Guarantee
  8. Proof of Theorem \ref{['thm 1']}-a)
  9. Proof of Theorem \ref{['thm 1']}-b)
  10. A Tight Radius of PT
  11. Justification on Lipschitz Constants
  12. Performance Analysis for Lipschitz Cost
  13. Numerical Examples
  14. Unicycle
  15. 2D Quadrotor
  16. ...and 1 more sections

Key Result

Theorem 1

kohler2024predictive Given stochastic system sys: d-t ss, safe set $\mathcal{C}$, cost functions $\mathcal{L}_t(\cdot,\cdot)$, terminal cost $\Phi(\cdot)$, initial state set $\mathcal{X}_0\in\mathcal{C}$, safe probability level $1-\delta$, and terminal state set $\mathcal{X}_f$, under Assumptions as

Figures (3)

  • Figure 1: An illustration of the presented stochastic MPC scheme. Blue curve: current nominal trajectory $\{x_t\}$. Green curve: prediction of $\{x_t\}$. Red curve: Current actual trajectory $\{X_t\}$. Purple curve: Prediction of $\{X_t\}$. The gray areas are the eroded part of the safe set $\mathcal{C}$.
  • Figure 2: Experiments on a unicycle modeled by \ref{['sys: uni']}. (a): 1000 sampled stochastic trajectories of \ref{['sys: uni']} under the control of the proposed MPC. (b): Instantaneous cost $\mathcal{L}_t(X_t,u_t)$ (in red) of the 1000 trajectories in (a), compared with $\mathbb{E}(\mathcal{L}_t(X_t,u_t))$ (in blue) and $\mathcal{L}_t(\bar{x}_t,\bar{u}_t)$ defined in Section III.E. (c) The nominal trajectory $\{x_t\}$ controlled by the proposed MPC (in blue), which is compared with the eroded safe set (outside the pink areas), the nominal trajectory $\{x_t^{st}\}$ with stabilizers only (in purple), and the associated stochastic trajectories of $x_t^{st}$.
  • Figure 3: Experiments on the quadrotor. Left: 1000 independent stochastic trajectories of the quadrotor. Right: Comparison between the nominal trajectory $\{x_t\}$ (in red) and that without safety MPC control input (in blue, its associated stochastic trajectory is in green).

Theorems & Definitions (6)

  • Remark II.1
  • Definition II.1
  • Theorem 1
  • Example 1
  • Theorem 2
  • proof