Certainty-Equivalence Model Predictive Control: Stability, Performance, and Beyond

TL;DR

This paper analyzes Certainty-Equivalence Model Predictive Control (CE-MPC) for uncertain nonlinear systems with input constraints, focusing on stability and infinite-horizon performance when the controller uses a nominal model. It develops a perturbation analysis of the MPC value function that does not require a Lipschitz assumption on the stage cost, enabling broad applicability and connections to learning-based MPC. The authors derive a sufficient stability condition tying together the prediction horizon and model mismatch, along with suboptimality and competitive-ratio bounds that quantify how CE-MPC approximates the infinite-horizon oracle controller. The results are specialized to uncertain linear systems with quadratic costs, yielding the first competitive-ratio bound for constrained MPC under multiplicative uncertainty. These insights provide practical horizon-selection guidelines and deepen the theoretical understanding of CE-MPC under parametric uncertainty, with future directions including terminal-cost extensions and online model learning integration.

Abstract

Handling model mismatch is a common challenge in model-based controller design, particularly in model predictive control (MPC). While robust MPC is effective in managing uncertainties, its conservatism often makes it less desirable in practice. Certainty-equivalence MPC (CE-MPC), which relies on a nominal model, offers an appealing alternative due to its design simplicity and low computational requirements. Contrary to the existing analyses where MPC has access to the true model, this paper investigates CE-MPC for uncertain nonlinear systems with input constraints and parametric uncertainty. The primary contributions of the paper are two-fold. First, a novel perturbation analysis of the MPC value function is provided, without relying on the common assumption of Lipschitz continuity of the stage cost, better tailoring the popular quadratic cost and having broader applicability to value function approximation, online model learning in MPC, and performance-driven MPC design. Second, the stability and performance analysis of CE-MPC are provided, with a quantification of the suboptimality of CE-MPC compared to the infinite-horizon optimal controller with perfect model knowledge. The results provide valuable insights in how the prediction horizon and model mismatch jointly affect stability and performance. Furthermore, the general results are specialized to linear quadratic control, and a competitive ratio bound is derived, serving as the first competitive-ratio bound for MPC of uncertain linear systems with input constraints and multiplicative uncertainty.

Figures (7)

• Figure 1: Analysis pipeline for the stability and performance of certainty-equivalence model predictive control (CE-MPC).
• Figure 2: Input perturbation of $10$ scenarios for a given $\varepsilon_{\theta}$.
• Figure 3: Input perturbation of $100$ scenarios for a given $\varepsilon_{\theta}$. For the considered quantity $||\mathbf{u}^\star_N(\theta^\ast) - \mathbf{u}^\star_N(\hat{\theta})||$, the solid line is the mean value, the dashed lines are the mean plus (minus) variance, and the dotted lines are the max (min).
• Figure 4: Scalable input perturbation simulation of $100$ scenarios for a given $\varepsilon_{\theta}$, and the worst-case value of $||\mathbf{u}^\star_N(\theta^\ast) - \mathbf{u}^\star_N(\hat{\theta})||$ is plotted.
• Figure 5: Worst-case compatitive ratio simulation of $100$ scenarios for a given $\varepsilon_{\theta}$. $10$ different initial states are simulated for a given norm of the initial state.

Theorems & Definitions (42)

• Definition 1: Region of Attraction
• Corollary 1
• Corollary 2
• Remark 1: State Constraints
• Corollary 3
• Remark 2: Terminal Cost
• Lemma 1: Exponential Decay of Sensitivity lin2022boundedshin2022exponential
• Lemma 2: Model Difference Bound
• Corollary 4: Cost Bounds
• Proposition 1: Consistent Error Matching