The Stochastic Robustness of Nominal and Stochastic Model Predictive Control

In this work, we establish and compare the stochastic and deterministic robustness properties achieved by nominal model predictive control (MPC), stochastic MPC (SMPC), and a proposed constraint-tightened MPC (CMPC) formulation, which represents an idealized version of tube-based MPC. We consider three definitions of robustness for nonlinear systems and bounded disturbances: robustly asymptotically stable (RAS), robustly asymptotically stable in expectation (RASiE), and RASiE with respect to the stage cost $\ell (\cdot)$ used in these MPC formulations ($\ell$-RASiE). Via input-to-state stability (ISS) and stochastic ISS (SISS) Lyapunov functions, we establish that MPC, subject to sufficiently small disturbances, and CMPC ensure all three definitions of robustness without a stochastic objective function. While SMPC is RASiE and $\ell$-RASiE, SMPC is not neccesarily RAS for nonlinear systems. Through a few simple examples, we illustrate the implications of these results and demonstrate that, depending on the definition of robustness considered, SMPC is not necessarily more robust than nominal MPC even if the disturbance model is exact.

I. Introduction

In Practice, a control algorithm must ensure some margin of robustness to disturbances. Nominal model predictive control (MPC) is known to be robustly asymptotically stable (RAS) with respect to (w.r.t.) sufficiently small disturbances [1], [11], [29], [39]. We use the term inherent robustness to describe this property of nominal MPC as this robustness is achieved through feedback, and disturbances are not explicitly considered in the problem formulation. This inherent robustness is often sufficient for successful implementation of MPC.