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Efficient implementation of MPC for tracking using ADMM by decoupling its semi-banded structure

Victor Gracia, Pablo Krupa, Daniel Limon, Teodoro Alamo

TL;DR

This work tackles the computational burden of MPCT by exploiting its semi-banded optimization problem through an ADMM-based solver. By reformulating MPCT into a split problem with a z-update solved via a Woodbury-based semi-banded decomposition and a simple v-update, the method recovers the banded structure that enables efficient solves. Compared to the existing EADMM approach, the proposed ADMM offers unconditional convergence guarantees for any $\rho>0$ and often demonstrates superior practical performance, particularly for reachable references, as shown on a ball-and-plate example. The results support the viability of fast, embedded MPCT implementations by leveraging the semi-banded ADMM decomposition.

Abstract

Model Predictive Control (MPC) for tracking formulation presents numerous advantages compared to standard MPC, such as a larger domain of attraction and recursive feasibility even when abrupt changes in the reference are produced. As a drawback, it includes some extra decision variables in its related optimization problem, leading to a semi-banded structure that differs from the banded structure encountered in standard MPC. This semi-banded structure prevents the direct use of the efficient algorithms available for banded problems. To address this issue, we present an algorithm based on the alternating direction method of multipliers that explicitly takes advantage of the underlying semi-banded structure of the MPC for tracking.

Efficient implementation of MPC for tracking using ADMM by decoupling its semi-banded structure

TL;DR

This work tackles the computational burden of MPCT by exploiting its semi-banded optimization problem through an ADMM-based solver. By reformulating MPCT into a split problem with a z-update solved via a Woodbury-based semi-banded decomposition and a simple v-update, the method recovers the banded structure that enables efficient solves. Compared to the existing EADMM approach, the proposed ADMM offers unconditional convergence guarantees for any and often demonstrates superior practical performance, particularly for reachable references, as shown on a ball-and-plate example. The results support the viability of fast, embedded MPCT implementations by leveraging the semi-banded ADMM decomposition.

Abstract

Model Predictive Control (MPC) for tracking formulation presents numerous advantages compared to standard MPC, such as a larger domain of attraction and recursive feasibility even when abrupt changes in the reference are produced. As a drawback, it includes some extra decision variables in its related optimization problem, leading to a semi-banded structure that differs from the banded structure encountered in standard MPC. This semi-banded structure prevents the direct use of the efficient algorithms available for banded problems. To address this issue, we present an algorithm based on the alternating direction method of multipliers that explicitly takes advantage of the underlying semi-banded structure of the MPC for tracking.
Paper Structure (10 sections, 5 theorems, 24 equations, 2 figures, 1 table, 3 algorithms)

This paper contains 10 sections, 5 theorems, 24 equations, 2 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. MPC for tracking formulation
  3. Efficiently solving MPCT using ADMM
  4. Alternating Direction Method of Multipliers
  5. Applying ADMM to MPCT
  6. Efficient computation of $z^{k+1}$
  7. Computation of $v^{k+1}$
  8. Comparison with the EADMM-based solver
  9. Numerical results
  10. Conclusion

Key Result

Proposition 1

Consider the OP z_problem, where $P \in \mathbb{R}^{n_z \times n_z}$ is positive semi-definite, $p \in \mathbb{R}^{n_z}$, $G \in \mathbb{R}^{m_z \times n_z}$ and $b \in \mathbb{R}^{m_z}$. A vector $z^* \in \mathbb{R}^{n_z}$ is an optimal solution of this problem if and only if there exists a vector

Figures (2)

  • Figure 1: Closed-loop trajectory of the ball and plate system with \ref{['MPCT_formulation']} for an admissible reference.
  • Figure 2: Closed-loop trajectory of the ball and plate system with \ref{['MPCT_formulation']} for a non-admissible reference.

Theorems & Definitions (10)

  • Proposition 1: boyd2004convex
  • Definition 1
  • Lemma 1: Woodbury matrix identity
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 1
  • Remark 1
  • Remark 2