# On the convergence of stochastic MPC to terminal modes of operation

## TL;DR

This paper establishes conditions under which stochastic MPC systems with disturbances converge in probability to terminal control laws, ensuring stability and performance bounds in nonlinear stochastic systems.

## Contribution

It introduces a Markov chain-based approach to prove convergence of stochastic MPC to terminal modes, filling a gap in existing stability analyses.

## Key findings

- Proves convergence of stochastic MPC to terminal control laws.
- Provides conditions for stability and performance bounds.
- Demonstrates convergence for two stochastic MPC formulations.

## Abstract

The stability of stochastic Model Predictive Control (MPC) subject to additive disturbances is often demonstrated in the literature by constructing Lyapunov-like inequalities that guarantee closed-loop performance bounds and boundedness of the state, but convergence to a terminal control law is typically not shown. In this work we use results on general state space Markov chains to find conditions that guarantee convergence of disturbed nonlinear systems to terminal modes of operation, so that they converge in probability to a priori known terminal linear feedback laws and achieve time-average performance equal to that of the terminal control law. We discuss implications for the convergence of control laws in stochastic MPC formulations, in particular we prove convergence for two formulations of stochastic MPC.