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

Handling model mismatch is a common challenge in model predictive control (MPC). While robust MPC is effective, its conservatism often makes it less desirable. Certainty-equivalence MPC (CE-MPC), which uses a nominal model, offers an appealing alternative due to its design simplicity and low computational costs. This paper investigates CE-MPC for uncertain nonlinear systems with multiplicative parametric uncertainty and input constraints that are inactive at the steady state. The primary contributions are two-fold. First, a novel perturbation analysis of the MPC value function is provided, without assuming the Lipschitz continuity of the stage cost, better tailoring the widely used quadratic cost and having broader applicability in value function approximation, learning-based MPC, and performance-driven MPC design. Second, the stability and performance analysis of CE-MPC are provided, quantifying the suboptimality of CE-MPC compared to the infinite-horizon optimal controller with perfect model knowledge. The results provide insights in how the prediction horizon and model mismatch jointly affect stability and the worst-case 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.

💡 Research Summary

This paper investigates certainty‑equivalence model predictive control (CE‑MPC) for discrete‑time nonlinear systems subject to multiplicative parametric uncertainty and input constraints that are inactive at the equilibrium. The true system is described by (x_{t+1}=f(x_t,u_t;\theta^{*})) while the controller only knows a nominal parameter (\hat\theta). The mismatch is bounded by (\varepsilon_\theta) and the set (\Theta(\hat\theta,\varepsilon_\theta)) contains all possible true parameters.

The authors first formulate the infinite‑horizon optimal control problem (IHOPC) and define the oracle controller that knows (\theta^{*}). They then introduce the finite‑horizon MPC problem without terminal ingredients, using a separable stage cost (\ell(x,u)=\ell_x(x)+\ell_u(u)) that is strongly smooth and strongly convex. A key technical contribution is a perturbation analysis of the MPC value function that does not rely on Lipschitz continuity of the cost. By exploiting only the (C^2) smoothness and convexity of (\ell), they derive explicit first‑ and second‑order bounds on the deviation of the value function caused by the model error (\Delta f(x,u;\theta)=f(x,u;\theta)-f(x,u;\hat\theta)). The bound is expressed through a scalar (\beta(N,\varepsilon_\theta)) that decreases with the prediction horizon (N) and increases with the uncertainty level (\varepsilon_\theta).

Using this perturbation result, the paper establishes two main theoretical guarantees:

1. Stability – If the mismatch is sufficiently small and the horizon is long enough, the closed‑loop system under CE‑MPC remains inside the region of attraction (X_{\text{ROA}}) of the true system and converges to the equilibrium. The authors provide an explicit contraction factor (\gamma(N,\varepsilon_\theta)<1) that quantifies how the Lyapunov function (\ell_x) decays along the CE‑MPC closed‑loop trajectory.

2. Infinite‑horizon performance (competitive ratio) – They compare the cost incurred by CE‑MPC with the optimal infinite‑horizon cost of the oracle controller. The ratio
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