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Recent advancements on MPC for tracking: periodic and harmonic formulations

Pablo Krupa, Daniel Limon, Teodoro Alamo

TL;DR

This chapter summarizes MPC formulations recently proposed that have been designed to address asymptotic stability and recursive feasibility guarantees regardless of the reference provided by the user, even if it is changed online or if it violates the system constraints.

Abstract

The main benefit of model predictive control (MPC) is its ability to steer the system to a given reference without violating the constraints while minimizing some objective. Furthermore, a suitably designed MPC controller guarantees asymptotic stability of the closed-loop system to the given reference as long as its optimization problem is feasible at the initial state of the system. Therefore, one of the limitations of classical MPC is that changing the reference may lead to an unfeasible MPC problem. Furthermore, due to a lack of deep knowledge of the system, it is possible for the user to provide a desired reference that is unfeasible or non-attainable for the MPC controller, leading to the same problem. This chapter summarizes MPC formulations recently proposed that have been designed to address these issues. In particular, thanks to the addition of an artificial reference as decision variable, the formulations achieve asymptotic stability and recursive feasibility guarantees regardless of the reference provided by the user, even if it is changed online or if it violates the system constraints. We show a recent formulation which extends this idea, achieving better performance and larger domains of attraction when working with small prediction horizons. Additional benefits of these formulations, when compared to classical MPC, are also discussed and highlighted with illustrative examples.

Recent advancements on MPC for tracking: periodic and harmonic formulations

TL;DR

This chapter summarizes MPC formulations recently proposed that have been designed to address asymptotic stability and recursive feasibility guarantees regardless of the reference provided by the user, even if it is changed online or if it violates the system constraints.

Abstract

The main benefit of model predictive control (MPC) is its ability to steer the system to a given reference without violating the constraints while minimizing some objective. Furthermore, a suitably designed MPC controller guarantees asymptotic stability of the closed-loop system to the given reference as long as its optimization problem is feasible at the initial state of the system. Therefore, one of the limitations of classical MPC is that changing the reference may lead to an unfeasible MPC problem. Furthermore, due to a lack of deep knowledge of the system, it is possible for the user to provide a desired reference that is unfeasible or non-attainable for the MPC controller, leading to the same problem. This chapter summarizes MPC formulations recently proposed that have been designed to address these issues. In particular, thanks to the addition of an artificial reference as decision variable, the formulations achieve asymptotic stability and recursive feasibility guarantees regardless of the reference provided by the user, even if it is changed online or if it violates the system constraints. We show a recent formulation which extends this idea, achieving better performance and larger domains of attraction when working with small prediction horizons. Additional benefits of these formulations, when compared to classical MPC, are also discussed and highlighted with illustrative examples.
Paper Structure (9 sections, 10 theorems, 21 equations, 8 figures)

This paper contains 9 sections, 10 theorems, 21 equations, 8 figures.

Table of Contents

  1. Introduction
  2. MPC for tracking piecewise-constant references
  3. Case study: ball and plate system
  4. MPC for tracking periodic references
  5. Applying periodic MPCT to the ball and plate system
  6. Harmonic MPC for tracking
  7. The harmonic MPC formulation
  8. Performance benefits of the HMPC formulation: case study on the ball and plate system
  9. Concluding remarks

Key Result

theorem 1

Let $x(t)$ belong to the feasibility region of the MPCT formulation eq:MPCT. Let $\tilde{{ \rm \bf{x}}}$, $\tilde{{ \rm \bf{u}}}$, $\tilde{x}_s$ and $\tilde{u}_s$ be any feasible solution of eq:MPCT for a given reference $(x_r(t), u_r(t))$. Then, the successor state $A x(t) + B \tilde{u}(0|t)$ belon

Figures (8)

  • Figure 1: Intuitive representation of the MPCT artificial reference. The blue line represents the space of steady states of the system and the red dot the desired reference $x_r$.
  • Figure 2: Ball and plate system.
  • Figure 3: Closed-loop results of the ball and plate system using MPCT \ref{['eq:MPCT']}.
  • Figure 4: Closed-loop results of the ball and plate system using periodic MPCT \ref{['eq:perMPCT']}.
  • Figure 5: Example \ref{['example:HMPC']}: performance issue of MPCT with small prediction horizons.
  • ...and 3 more figures

Theorems & Definitions (24)

  • definition 1: Admissible steady state
  • theorem 1: Recursive feasibility of MPCT
  • definition 2: Optimal reachable reference
  • theorem 2: Asymptotic stability of MPCT
  • remark 1
  • remark 2
  • definition 3: Admissible trajectory
  • theorem 3: Recursive feasibility of periodic MPCT
  • definition 4: Optimal reachable periodic reference
  • theorem 4: Asymptotic stability of periodic MPCT
  • ...and 14 more