The Stochastic Robustness of Nominal and Stochastic Model Predictive Control

• Published in: IEEE transactions on automatic control Vol. 68; no. 10; pp. 1 - 13
• Main Authors: McAllister, Robert D., Rawlings, James B.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.10.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic properties; Asymptotic stability; Closed loop systems; Costs; Disturbances; Formulations; Liapunov functions; Model predictive control; Nonlinear systems; Optimization; Predictive control; Robustness; stability of nonlinear systems; Stochastic models; stochastic optimal control; Stochastic processes; stochastic systems; Trajectory
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2022.3226712

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: robust asymptotic stability (RAS), robust asymptotic stability in expectation (RASiE), and RASiE w.r.t. the stage cost <inline-formula><tex-math notation="LaTeX">\ell (\cdot)</tex-math></inline-formula> used in these MPC formulations (<inline-formula><tex-math notation="LaTeX">\ell</tex-math></inline-formula>-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 also ensures RASiE and <inline-formula><tex-math notation="LaTeX">\ell</tex-math></inline-formula>-RASiE, SMPC does not guarantee 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.