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Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals

Zirui Niu, Daniele Astolfi, Giordano Scarciotti

TL;DR

The paper tackles output regulation for linear systems driven by non-periodic, non-smooth exogenous signals generated by an explicit exosystem. It first develops a full-information regulator using regulator equations under a new non-resonance condition, then extends to an error-feedback regulator with a canonical internal model and high-gain stabilization under minimum-phase assumptions. To address model uncertainties, two robust internal-model designs are proposed: an augmentation-based approach and an integral-immersion method, each enabling regulation across uncertain parameters with differing regulator dimensions. A circuit regulation example demonstrates bounded regulator solutions and showcases the dimension advantages of the immersion-based method. Overall, the work broadens the applicability of output regulation to non-smooth, non-periodic disturbances and provides practically implementable, robust regulator designs.

Abstract

We address the output regulation problem of linear systems with non-smooth and non-periodic exogenous signals. Specifically, we first formulate and solve the full-information problem by designing a state-feedback controller. We study the solvability of the regulator equations, providing a new non-resonance condition. We then focus on the error-feedback problem, for which we design a (non-robust) internal model leveraging the concept of canonical realisation and applying a high-gain method for the stabilisation of the closed-loop system under the minimum-phase assumption. Finally, we study the regulation problem involving model parameter uncertainties. Drawing ideas from both hybrid and time-varying (smooth) output regulation, we propose two methods to establish an internal model that is robust to uncertainties. The first method is an extension of the hybrid internal model, while the second relies on a new concept of immersion. In this non-smooth case, the immersion is established based on integrals rather than derivatives. The effectiveness of the proposed solutions is illustrated by a circuit regulation example.

Output Regulation of Linear Systems with Non-periodic Non-smooth Exogenous Signals

TL;DR

The paper tackles output regulation for linear systems driven by non-periodic, non-smooth exogenous signals generated by an explicit exosystem. It first develops a full-information regulator using regulator equations under a new non-resonance condition, then extends to an error-feedback regulator with a canonical internal model and high-gain stabilization under minimum-phase assumptions. To address model uncertainties, two robust internal-model designs are proposed: an augmentation-based approach and an integral-immersion method, each enabling regulation across uncertain parameters with differing regulator dimensions. A circuit regulation example demonstrates bounded regulator solutions and showcases the dimension advantages of the immersion-based method. Overall, the work broadens the applicability of output regulation to non-smooth, non-periodic disturbances and provides practically implementable, robust regulator designs.

Abstract

We address the output regulation problem of linear systems with non-smooth and non-periodic exogenous signals. Specifically, we first formulate and solve the full-information problem by designing a state-feedback controller. We study the solvability of the regulator equations, providing a new non-resonance condition. We then focus on the error-feedback problem, for which we design a (non-robust) internal model leveraging the concept of canonical realisation and applying a high-gain method for the stabilisation of the closed-loop system under the minimum-phase assumption. Finally, we study the regulation problem involving model parameter uncertainties. Drawing ideas from both hybrid and time-varying (smooth) output regulation, we propose two methods to establish an internal model that is robust to uncertainties. The first method is an extension of the hybrid internal model, while the second relies on a new concept of immersion. In this non-smooth case, the immersion is established based on integrals rather than derivatives. The effectiveness of the proposed solutions is illustrated by a circuit regulation example.

Paper Structure

This paper contains 17 sections, 70 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Full-Information Problem
  4. Solution of the Full-Information Problem
  5. Solvability of the Regulator Equations
  6. Error-Feedback Problem
  7. Internal Model Property
  8. Achieving Internal Model Property
  9. Robust stabilisation
  10. Internal Model Principle
  11. Problem Formulation
  12. Augmentation-based Internal Model
  13. Immersion-Based Robust Internal Model
  14. Illustrative Example
  15. Conclusion
  16. ...and 2 more sections

Figures (6)

  • Figure 1: Time histories of the functions $a_1(t)$, $a_2(t)$, $a_3(t)$, and $a_4(t)$.
  • Figure 2: Schematics of the RLC circuit.
  • Figure 3: Top: time history of solution $\Pi_x(t)$ of the regulator equations. Bottom: time history of solution $\Delta(t)$ of the regulator equations.
  • Figure 4: (a) Time history of system state $x(t)$. (b) Time history of $H_{\mathrm{im}}(t)$ for the canonical realization. (c) Time history of the input $u(t)$ generated by the robust augmentation-based regulator. (d) Time history of the regulation error $e(t)$.
  • Figure 5: Top: Time histories of functions $a_1(t)$, $a_2(t)$, $a_3(t)$, $a_4(t)$. Middle: Time history of $H_{\mathrm{im}}(t)$ for the canonical realization. Bottom: Time history of the input $u(t)$ generated by the robust immersion-based regulator.
  • ...and 1 more figures

Theorems & Definitions (13)

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  • ...and 3 more