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Data-driven harmonic output regulation of a class of nonlinear systems

Zhongjie Hu, Claudio De Persis, John W. Simpson-Porco, Pietro Tesi

TL;DR

This work presents a data-driven framework for designing harmonic output regulators for unknown nonlinear systems by augmenting the plant with an internal model and enforcing exponential contractivity through a data-dependent semidefinite program. A key contribution is a disturbance-filtering data representation that eliminates dependence on unmeasured exogenous signals, enabling direct controller synthesis from data without requiring a normal form. The developed SDP yields a stabilizing gain that drives the regulation error to a $D$-periodic signal with vanishing first $\ell+1$ Fourier coefficients, with a specialized linear-system result delivering exact regulation without perturbation measurements. The approach advances data-driven control of nonlinear uncertain plants and suggests practical extensions to discrete-time settings and output-feedback regulators. Overall, the paper provides a principled, data-centric route to harmonic regulation with provable contraction properties and disturbance-insensitive design features.

Abstract

The paper deals with the data-based design of state-feedback controllers that solve the output regulation problem for a class of nonlinear systems. Inspired by recent developments in model-based output regulation design techniques and in data-driven control design for nonlinear systems, we derive a data-dependent semidefinite program that, when solved, directly returns a controller that steers the regulation error to a periodic signal whose Fourier series has identically zero coefficients up to a certain order set by the controller. When specialized to the case of linear systems, the result appears to improve upon existing work. Numerical results illustrate the findings

Data-driven harmonic output regulation of a class of nonlinear systems

TL;DR

This work presents a data-driven framework for designing harmonic output regulators for unknown nonlinear systems by augmenting the plant with an internal model and enforcing exponential contractivity through a data-dependent semidefinite program. A key contribution is a disturbance-filtering data representation that eliminates dependence on unmeasured exogenous signals, enabling direct controller synthesis from data without requiring a normal form. The developed SDP yields a stabilizing gain that drives the regulation error to a -periodic signal with vanishing first Fourier coefficients, with a specialized linear-system result delivering exact regulation without perturbation measurements. The approach advances data-driven control of nonlinear uncertain plants and suggests practical extensions to discrete-time settings and output-feedback regulators. Overall, the paper provides a principled, data-centric route to harmonic regulation with provable contraction properties and disturbance-insensitive design features.

Abstract

The paper deals with the data-based design of state-feedback controllers that solve the output regulation problem for a class of nonlinear systems. Inspired by recent developments in model-based output regulation design techniques and in data-driven control design for nonlinear systems, we derive a data-dependent semidefinite program that, when solved, directly returns a controller that steers the regulation error to a periodic signal whose Fourier series has identically zero coefficients up to a certain order set by the controller. When specialized to the case of linear systems, the result appears to improve upon existing work. Numerical results illustrate the findings
Paper Structure (10 sections, 9 theorems, 94 equations, 2 figures, 1 table)

This paper contains 10 sections, 9 theorems, 94 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Systems of interest and the output regulation problem
  3. The dataset and a data-dependent representation
  4. Data-based design of a harmonic regulator
  5. The regulator design under a relaxed Assumption \ref{['assum:w']}
  6. The case of linear systems
  7. Conclusions and future work
  8. Proof of Lemma \ref{['lem:forma-nuova-risposta-esosistema']}
  9. Periodic exponentially stable solution of the system \ref{['system-(9b)-(16)']}
  10. A few facts about the Fourier series

Key Result

Lemma 1

Consider the system with initial condition $w(0)$ and $w\in \mathbb{R}^{n_w}$. Let $\mathcal{T}\in \mathbb{R}^{n_w\times n_w}$ be the similarity matrix that transforms $S$ into its real Jordan form $J$, namely $J= \mathcal{T}^{-1}S\mathcal{T}$. Then there exist positive integers $k_1, \ldots, k_{r+s}$ satisfying $k_1+\ Furthermore, the matrices $M_{k_i}(\lambda_i, t)$, $M_{k_{r+i}}(\lambda_{r+i},

Figures (2)

  • Figure 1: Evolution of the error $e$ with an internal model composed of 4 oscillators.
  • Figure 2: Evolution of $x$ (left), $u$ (center) and $h$ (right) with initial state uniformly distributed in $[-1,1]$.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Definition 1
  • Proposition 1
  • Theorem 1
  • Example 1
  • Theorem 2
  • Example 2
  • Definition 2
  • ...and 3 more