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Data-enabled Predictive Repetitive Control

Rogier Dinkla, Tom Oomen, Sebastiaan Mulders, Jan-Willem van Wingerden

TL;DR

The paper introduces DeePRC, a data-driven predictive control framework that attenuates periodic disturbances in $P$-periodic LPTV systems by lifting to an $LTI$ representation and extending Willems' fundamental lemma to systems with exogenous disturbances. It combines the internal model principle with a noise-robust closed-loop DeePC (CL-DeePC) approach in a lifted domain, relaxing controllability requirements and enabling effective disturbance rejection even under noise. Theoretical extensions (including a generalized fundamental lemma for disturbances) are complemented by simulations on a $P{=}20$ periodic system, where DeePRC outperforms a non-lifted CL-DeePC in constraint satisfaction and convergence. The work provides a practical data-driven pathway to handle known-period dynamics and disturbances, with future directions targeting unknown periods and periodic data differencing.

Abstract

Many systems are subject to periodic disturbances and exhibit repetitive behaviour. Model-based repetitive control employs knowledge of such periodicity to attenuate periodic disturbances and has seen a wide range of successful industrial implementations. The aim of this paper is to develop a data-driven repetitive control method. In the developed framework, linear periodically time-varying (LPTV) behaviour is lifted to linear time-invariant (LTI) behaviour. Periodic disturbance mitigation is enabled by developing an extension of Willems' fundamental lemma for systems with exogenous disturbances. The resulting Data-enabled Predictive Repetitive Control (DeePRC) technique accounts for periodic system behaviour to perform attenuation of a periodic disturbance. Simulations demonstrate the ability of DeePRC to effectively mitigate periodic disturbances in the presence of noise.

Data-enabled Predictive Repetitive Control

TL;DR

The paper introduces DeePRC, a data-driven predictive control framework that attenuates periodic disturbances in -periodic LPTV systems by lifting to an representation and extending Willems' fundamental lemma to systems with exogenous disturbances. It combines the internal model principle with a noise-robust closed-loop DeePC (CL-DeePC) approach in a lifted domain, relaxing controllability requirements and enabling effective disturbance rejection even under noise. Theoretical extensions (including a generalized fundamental lemma for disturbances) are complemented by simulations on a periodic system, where DeePRC outperforms a non-lifted CL-DeePC in constraint satisfaction and convergence. The work provides a practical data-driven pathway to handle known-period dynamics and disturbances, with future directions targeting unknown periods and periodic data differencing.

Abstract

Many systems are subject to periodic disturbances and exhibit repetitive behaviour. Model-based repetitive control employs knowledge of such periodicity to attenuate periodic disturbances and has seen a wide range of successful industrial implementations. The aim of this paper is to develop a data-driven repetitive control method. In the developed framework, linear periodically time-varying (LPTV) behaviour is lifted to linear time-invariant (LTI) behaviour. Periodic disturbance mitigation is enabled by developing an extension of Willems' fundamental lemma for systems with exogenous disturbances. The resulting Data-enabled Predictive Repetitive Control (DeePRC) technique accounts for periodic system behaviour to perform attenuation of a periodic disturbance. Simulations demonstrate the ability of DeePRC to effectively mitigate periodic disturbances in the presence of noise.
Paper Structure (20 sections, 4 theorems, 39 equations, 2 figures)

This paper contains 20 sections, 4 theorems, 39 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Periodic System Model
  4. Notation and Definitions
  5. DeePRC
  6. Lifting the LPTV to an LTI system
  7. Applying the Internal Model Principle
  8. Disturbances with Willems' Fundamental Lemma
  9. Imposing full row rank on the state-input data matrix
  10. Relaxing the assumption of controllability
  11. Implications for a constant disturbance
  12. DeePRC Formulation and Noise Mitigation
  13. Closed-loop DeePC applied to a lifted system
  14. Optimal Control Problem Formulation
  15. Receding horizon implementation
  16. ...and 5 more sections

Key Result

Lemma 1

Consider the deterministic LTI system $\mathcal{P}$ from eq:SS_LTI and assume it to be controllableNote that Willems2005 employs a behavioural definition of controllability (see, e.g., Markovsky2021) that is implied by classical state controllability. This latter notion of state controllability is u with $g\in\mathbb{R}^{N+{n_\mathrm{x}}}$.

Figures (2)

  • Figure 1: Performance of DeePRC and CL-DeePC controllers using respectively data from the lifted LTI and the non-lifted periodic domain under the influence of a periodic disturbance and noise. DeePRC effectively compensates the input disturbance, whilst CL-DeePC performs worse here w.r.t. the case without control. Dashed grey lines indicate contraints.
  • Figure 2: Obtained iteration cost as specified by \ref{['eq:iter_cost']} of the DeePRC and CL-DeePC controllers under conditions with and without noise. The iteration cost of DeePRC is lower than that of CL-DeePC and illustrates faster convergence.

Theorems & Definitions (6)

  • Lemma 1: Willems' fundamental lemma Willems2005
  • Theorem 1
  • proof
  • Theorem 2: Fundamental lemma for systems with an exogenous disturbance
  • proof
  • Corollary 1