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

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.

💡 Research Summary

The paper addresses the problem of safely controlling nonlinear stochastic systems of the form Xₜ₊₁ = f(Xₜ, uₜ) + wₜ, where the disturbance wₜ is sub‑Gaussian and may be unbounded. The safety requirement is expressed as a chance constraint P(Xₜ ∈ C, ∀ t ≤ T) ≥ 1 − δ, i.e., the entire trajectory must stay inside a prescribed safe set C with high probability. Existing stochastic MPC approaches either rely on Monte‑Carlo sampling, which is computationally heavy, or convert chance constraints into deterministic ones only for individual time steps, which does not guarantee trajectory‑level safety.

To overcome these limitations, the authors adopt a set‑erosion strategy. They define a nominal (noise‑free) trajectory {xₜ} by xₜ₊₁ = f(xₜ, uₜ) with the same control inputs as the stochastic system. If one can guarantee that the stochastic trajectory stays within a tube of radius r_{δ,t} around the nominal trajectory for all t with probability at least 1 − δ, then ensuring that the nominal trajectory remains inside the eroded safe set \tilde C_t = C ⊖ B(r_{δ,t}) implies the original chance constraint. Hence the problem reduces to a deterministic MPC that enforces \tilde x_{t+k|t} ∈ \tilde C_{t+k} for all prediction steps k.

The key technical contribution is a tight bound on the tube radius r_{δ,t}. Building on recent concentration results, the authors derive a closed‑form expression (Equation 8) that scales as O(p log (1/δ)) with the confidence level δ, and, when the system’s Lipschitz constant L < 1, grows only as ˜O(√log T) with the horizon length T. This bound is substantially less conservative than previous works, which often lead to overly large erosion depths that render the deterministic MPC infeasible. Moreover, the bound depends only on the open‑loop Lipschitz constant, not on the closed‑loop one, simplifying practical implementation.

The deterministic MPC problem (Equation 5) is formulated with standard cost functions, dynamics constraints, input constraints, a terminal set X_f, and the eroded safety constraints. Because the erosion is performed offline (or updated infrequently), the optimization remains a standard nonlinear program that can be solved with any off‑the‑shelf solver. The authors prove recursive feasibility (Theorem 1‑a) and the overall safety guarantee (Theorem 1‑b) under three assumptions: (1) the system is Lipschitz, (2) the disturbance is sub‑Gaussian, and (3) the terminal set is invariant under some admissible control. The proofs use a sliding‑window argument and the definition of the tube radius.

Numerical experiments on high‑dimensional nonlinear robotic models with complex obstacle configurations demonstrate that the proposed method achieves the desired safety level (e.g., 99.99 %) while maintaining real‑time computational performance. Compared to prior stochastic MPC schemes, the new approach yields higher feasibility rates and smaller erosion depths, confirming the practical advantage of the tighter tube bound.

In summary, the paper presents a theoretically sound and computationally efficient stochastic MPC framework that converts probabilistic safety constraints into deterministic ones via a rigorously derived set‑erosion depth. The method leverages existing deterministic MPC tools, making it attractive for safety‑critical applications such as autonomous driving, aerial robotics, and industrial automation where high‑probability safety guarantees are essential.