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Harmonic model predictive control for tracking sinusoidal references and its application to trajectory tracking

Pablo Krupa, Daniel Limon, Alberto Bemporad, Teodoro Alamo

TL;DR

This work extends Harmonic Model Predictive Control (HMPC) to track harmonic and arbitrary periodic references by embedding a parameterized harmonic reference within the MPC optimization. The resulting controller preserves recursive feasibility and asymptotic stability, with the key advantage that problem complexity does not grow with the reference period, thanks to second-order cone constraints that enforce admissibility of the harmonic reference. The method also enables tracking of non-harmonic or complex periodic trajectories by using local harmonic approximations, maintaining feasibility and offering a large domain of attraction. Numerical case studies on a ball-and-plate system demonstrate competitive computational performance and effective tracking for both harmonic and generic periodic references, supported by a solver (SPCIES/SCS) whose runtime is largely independent of the reference period. The approach has potential applications in trajectory tracking, obstacle avoidance, and aerospace rendezvous where long-period references or partial knowledge of future references are common.”

Abstract

Harmonic model predictive control (HMPC) is a recent model predictive control (MPC) formulation for tracking piece-wise constant references that includes a parameterized artificial harmonic reference as a decision variable, resulting in an increased performance and domain of attraction with respect to other MPC formulations. This article presents an extension of the HMPC formulation to track periodic harmonic/sinusoidal references and discusses its use for tracking arbitrary trajectories. The proposed formulation inherits the benefits of its predecessor, namely its good performance and large domain of attraction when using small prediction horizons, and that the complexity of its optimization problem does not depend on the period of the reference. We show closed-loop results discussing its performance and comparing it to other MPC formulations.

Harmonic model predictive control for tracking sinusoidal references and its application to trajectory tracking

TL;DR

This work extends Harmonic Model Predictive Control (HMPC) to track harmonic and arbitrary periodic references by embedding a parameterized harmonic reference within the MPC optimization. The resulting controller preserves recursive feasibility and asymptotic stability, with the key advantage that problem complexity does not grow with the reference period, thanks to second-order cone constraints that enforce admissibility of the harmonic reference. The method also enables tracking of non-harmonic or complex periodic trajectories by using local harmonic approximations, maintaining feasibility and offering a large domain of attraction. Numerical case studies on a ball-and-plate system demonstrate competitive computational performance and effective tracking for both harmonic and generic periodic references, supported by a solver (SPCIES/SCS) whose runtime is largely independent of the reference period. The approach has potential applications in trajectory tracking, obstacle avoidance, and aerospace rendezvous where long-period references or partial knowledge of future references are common.”

Abstract

Harmonic model predictive control (HMPC) is a recent model predictive control (MPC) formulation for tracking piece-wise constant references that includes a parameterized artificial harmonic reference as a decision variable, resulting in an increased performance and domain of attraction with respect to other MPC formulations. This article presents an extension of the HMPC formulation to track periodic harmonic/sinusoidal references and discusses its use for tracking arbitrary trajectories. The proposed formulation inherits the benefits of its predecessor, namely its good performance and large domain of attraction when using small prediction horizons, and that the complexity of its optimization problem does not depend on the period of the reference. We show closed-loop results discussing its performance and comparing it to other MPC formulations.
Paper Structure (11 sections, 7 theorems, 23 equations, 5 figures, 1 table)

This paper contains 11 sections, 7 theorems, 23 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Admissible harmonic signals
  3. HMPC for harmonic reference tracking
  4. Properties of the HMPC formulation \ref{['eq:HMPC']}
  5. Numerically solving the HMPC formulation \ref{['eq:HMPC']}
  6. Tracking arbitrary references
  7. Case study
  8. Tracking a harmonic reference
  9. Tracking an arbitrary periodic reference
  10. Computational results and performance
  11. Conclusions

Key Result

Proposition 1

Let $\hat{x}(\cdot)$ and $\hat{u}(\cdot)$ be harmonic signals with the same frequency $w$ parametrized by ${ \rm \bf{\hat{x}}} \doteq (\hat{x}_e, \hat{x}_s, \hat{x}_c)$ and ${ \rm \bf{\hat{u}}} \doteq (\hat{u}_e, \hat{u}_s, \hat{u}_c)$, and Then, $({ \rm \bf{\hat{x}}}, { \rm \bf{\hat{u}}}) \in {\mathcal{D}}$ implies $\hat{x}(t+1) = A \hat{x}(t) + B \hat{u}(t)$, $\forall t$.

Figures (5)

  • Figure 1: Harmonic signal expressed in the relative time $k$.
  • Figure 2: Tracking an admissible harmonic reference.
  • Figure 3: Tracking a non-admissible harmonic reference.
  • Figure 4: Tracking an admissible arbitrary reference.
  • Figure 5: Tracking a non-admissible arbitrary reference.

Theorems & Definitions (17)

  • Definition 1: Harmonic signal
  • Definition 2: Admissible harmonic signals
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • Definition 3: Optimal reachable harmonic reference
  • Lemma 1
  • Corollary 2
  • ...and 7 more