Analysis and design of model predictive control frameworks for dynamic operation -- An overview

TL;DR

The article surveys MPC approaches for dynamic operation in nonlinear constrained systems, focusing on how to achieve recursive feasibility, constraint satisfaction, and stability when the operational goal evolves online. It organizes the toolkit into four main families: (i) stabilizing MPC with terminal ingredients, (ii) tracking MPC with artificial references to handle infeasible or changing targets, (iii) economic MPC with periodic or artificial references for time-varying costs, and (iv) MPC without terminal constraints to analyze stability via horizon length. It details constructive methods for terminal ingredients (e.g., local CLFs, LPV-based parametrizations), artificial-reference strategies, and periodic reference handling, while discussing dissipativity-based stability, turnpike properties, and shifted-terminal-cost ideas to guarantee performance. The review also covers extensions to time-parametrization, distributed implementations, and robustness, and highlights open issues such as scalability, non-periodic operation, and design under model mismatch. Overall, the paper provides a comprehensive, unified view of how MPC can address dynamic operation in nonlinear constrained systems, offering a diverse toolbox for researchers and practitioners to tailor control strategies to online-changing objectives and constraints.

Abstract

This article provides an overview of model predictive control (MPC) frameworks for dynamic operation of nonlinear constrained systems. Dynamic operation is often an integral part of the control objective, ranging from tracking of reference signals to the general economic operation of a plant under online changing time-varying operating conditions. We focus on the particular challenges that arise when dealing with such more general control goals and present methods that have emerged in the literature to address these issues. The goal of this article is to present an overview of the state-of-the-art techniques, providing a diverse toolkit to apply and further develop MPC formulations that can handle the challenges intrinsic to dynamic operation. We also critically assess the applicability of the different research directions, discussing limitations and opportunities for further research.

Figures (5)

• Figure 1: Evasion manoeuvrer of a car with reference trajectory $r$ (red), (projected) terminal sets (blue ellipses) and state constraints (black), adapted from koehler2020nonlinearTAC. The terminal ingredients are optimized offline before knowing the exact reference trajectory $r$ (cf. Sec. \ref{['sec:terminal_4']}).
• Figure 2: Illustration how the constrained reference planning \ref{['problem:limon_partial']} ensures recursive feasibility of the tracking MPC at time $t_{i+1}$. Closed-loop state $x(t)$, $t\in\mathbb{I}_{[t_i,t_{i+1}]}$ (blue, solid), predicted state sequence $x_{\mathbf{u}}$ of the tracker (blue, dash-dotted); artificial reference $\mathbf{r}$ at time $t_i$ (red, dashed) with terminal set scaling $\alpha$ (red-dotted); artificial reference $\mathbf{r}$ at time $t_{i+1}$ (magenta, dashed) with terminal set scaling $\alpha$ (magenta, dotted). The predicted state sequence $x_{\mathbf{u}}$ of the tracker (blue, dash-dotted) satisfies the new terminal set constraint (magenta), since it contains the previous terminal set constraint (red), cf. \ref{['problem:limon_partial_3']}.
• Figure 3: Periodic tracking with a ball-and-plate system, adapted from koehler2020nonlinearAutomatica. Closed-loop state $x$ (blue, solid) resulting from periodic tracking MPC (Sec. \ref{['sec:artificial_3_1']}) with a time-varying target signal $y_{\mathrm{d}}$ (red) and state constraints $\mathbb{X}$ (black). Closed-loop state resulting from partially decoupled tracking and planning (Sec. \ref{['sec:artificial_4']}) with $M=2$ shown dashed in magenta.
• Figure 4: Temperature control in a building, adapted from Koehler2020Economic. Closed-loop state $x$ (blue, solid) resulting from periodic economic MPC (Thm. \ref{['thm:eco_periodic_artificial']}) with time-varying ambient temperature and price signal, which also changes unpredictably during online operation. Time-varying state constraints (black, solid). Optimal trajectory computed in hindsight with known price signal $y_{\mathrm{e}}$ shown in magenta, dashed.
• Figure 5: Chain of mass--spring--dampers, adapted from kohler2023stability. Sufficient horizon $\underline{N}$ for stability for varying $R$ based on Theorem \ref{['thm:UCON_detect']} in blue, solid. Bound $\underline{N}$ utilizing an additional approximate terminal cost (Sec. \ref{['sec:UCON_3_2']}) using a finite-horizon roll-out with $M=10$ shown in magenta, dotted. Theoretical bounds $\underline{N}$ using Theorem \ref{['thm:UCON']} shown in red, dashed.