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Model predictive control for tracking using artificial references: Fundamentals, recent results and practical implementation

Pablo Krupa, Johannes Köhler, Antonio Ferramosca, Ignacio Alvarado, Melanie N. Zeilinger, Teodoro Alamo, Daniel Limon

TL;DR

The paper addresses the challenge that online reference changes can destabilize or render classical MPC infeasible, by introducing an artificial reference as a decision variable and an offset cost to guarantee recursive feasibility and stability.Starting from the original linear formulation, it surveys extensions to robust, periodic, harmonic, and economic MPC, as well as nonlinear variants, including setpoint and periodic tracking and learning-based applications, with clear theoretical guarantees under standard assumptions.It provides practical guidance on optimization, invariant-set computation, and solver choices, highlighting the trade-offs between complexity and real-time performance, and discusses applications in learning-based MPC and multi-agent scenarios.Overall, MPC for tracking with artificial references increases the domain of attraction and enables robust operation under online reference changes, while offering scalable implementations through tailored solvers and decomposition strategies.

Abstract

This paper provides a comprehensive tutorial on a family of Model Predictive Control (MPC) formulations, known as MPC for tracking, which are characterized by including an artificial reference as part of the decision variables in the optimization problem. These formulations have several benefits with respect to the classical MPC formulations, including guaranteed recursive feasibility under online reference changes, as well as asymptotic stability and an increased domain of attraction. This tutorial paper introduces the concept of using an artificial reference in MPC, presenting the benefits and theoretical guarantees obtained by its use. We then provide a survey of the main advances and extensions of the original linear MPC for tracking, including its non-linear extension. Additionally, we discuss its application to learning-based MPC, and discuss optimization aspects related to its implementation.

Model predictive control for tracking using artificial references: Fundamentals, recent results and practical implementation

TL;DR

The paper addresses the challenge that online reference changes can destabilize or render classical MPC infeasible, by introducing an artificial reference as a decision variable and an offset cost to guarantee recursive feasibility and stability.Starting from the original linear formulation, it surveys extensions to robust, periodic, harmonic, and economic MPC, as well as nonlinear variants, including setpoint and periodic tracking and learning-based applications, with clear theoretical guarantees under standard assumptions.It provides practical guidance on optimization, invariant-set computation, and solver choices, highlighting the trade-offs between complexity and real-time performance, and discusses applications in learning-based MPC and multi-agent scenarios.Overall, MPC for tracking with artificial references increases the domain of attraction and enables robust operation under online reference changes, while offering scalable implementations through tailored solvers and decomposition strategies.

Abstract

This paper provides a comprehensive tutorial on a family of Model Predictive Control (MPC) formulations, known as MPC for tracking, which are characterized by including an artificial reference as part of the decision variables in the optimization problem. These formulations have several benefits with respect to the classical MPC formulations, including guaranteed recursive feasibility under online reference changes, as well as asymptotic stability and an increased domain of attraction. This tutorial paper introduces the concept of using an artificial reference in MPC, presenting the benefits and theoretical guarantees obtained by its use. We then provide a survey of the main advances and extensions of the original linear MPC for tracking, including its non-linear extension. Additionally, we discuss its application to learning-based MPC, and discuss optimization aspects related to its implementation.
Paper Structure (19 sections, 7 theorems, 48 equations, 1 figure)

This paper contains 19 sections, 7 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fundamentals of MPC for tracking using an artificial reference: the original formulation
  3. Extensions of MPC for tracking
  4. Robust MPC
  5. Tracking periodic references
  6. Harmonic MPC for tracking
  7. Economic MPC for tracking
  8. Other extensions of MPC for tracking
  9. MPC for tracking with non-linear systems
  10. Setpoint tracking
  11. Tracking periodic references
  12. Robust tracking MPC
  13. Other extensions and experimental results
  14. Application to Learning-based MPC
  15. Optimization and aspects and practical implementation of MPC for tracking
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\hat{{ \rm \bf{x}}} \doteq (\hat{x}_0, \dots, \hat{x}_N)$, $\hat{{ \rm \bf{u}}} \doteq (\hat{u}_0, \dots, \hat{u}_{N-1})$ be any feasible solution of eq:linMPCT for the current state $x(t)$. Then, problem eq:linMPCT is feasible for the successor state $A x(t) + B \hat{u}_0$ for any value of $y_

Figures (1)

  • Figure 1: Domain of attraction of standard MPC \ref{['eq:stanMPC']} and MPC for tracking \ref{['eq:linMPCT']}. Legend: ${\mathcal{Z}}$ are the state constraints; ${\mathcal{Z}}_{s}$ the states that belong to the manifold of steady states; ${\mathcal{X}}_a$ the invariant set for tracking of \ref{['eq:linMPCT']}; ${\mathcal{X}}_f$ the terminal invariant set of \ref{['eq:stanMPC']}; ${\mathcal{D}}_{(4)}$ and ${\mathcal{D}}_{(6)}$ the domains of attraction of \ref{['eq:stanMPC']} and \ref{['eq:linMPCT']}, respectively.

Theorems & Definitions (12)

  • Definition 1: Reachable setpoints
  • Definition 2: Optimal reachable reference
  • Theorem 1: Recursive feasibility
  • Theorem 2: Asymptotic stability
  • Example 1: Domain of attraction
  • Theorem 3: Limon_JPC_2010, Theorem 1
  • Definition 3: Reachable periodic trajectory
  • Definition 4: Optimal economic setpoint
  • Theorem 4: ferramosca2014economic
  • Theorem 5: Recursive feasibility Limon_TAC_2018
  • ...and 2 more