# Convergence of stochastic nonlinear systems and implications for Stochastic Model Predictive Control

arXiv: https://arxiv.org/abs/1908.00483 · HTML: https://mlsanity.com/paper/1908.00483/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2019.

## Built on by (most cited first, 2 of 2)

- [Constrained Reinforcement Learning using Distributional Representation for Trustworthy Quadrotor UAV Tracking Control](https://mlsanity.com/paper/2302.11694/) (2023)
- [Interpretable Stochastic Model Predictive Control using Distributional Reinforced Estimation for Quadrotor Tracking Systems](https://mlsanity.com/paper/2205.07150/) (2022)

## Abstract (arXiv)

The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that ensure closed-loop performance bounds and boundedness of the state, but tight ultimate bounds for the state and non-conservative performance bounds are typically not determined. In this work we use an input-to-state stability property to find conditions that imply convergence with probability 1 of a disturbed nonlinear system to a minimal robust positively invariant set. We discuss implications for the convergence of the state and control laws of stochastic MPC formulations, and we prove convergence results for several existing stochastic MPC formulations for linear and nonlinear systems.