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Robust Output Regulation of Uncertain Linear Time-Varying Systems

Jinmeng Zha, Zhen Zhang

TL;DR

The paper tackles robust output regulation for linear time-varying systems with parametric uncertainty by reframing the regulator equation via intrinsic system immersion, showing that robustness to plant uncertainty generally requires an infinite-dimensional internal model. It establishes a propagation condition linking robustness to how uncertainty propagates through the exosystem dynamics, and develops a comprehensive framework that includes minimum/internal-model reduction, time-varying canonical realizations, and both exact and approximate regulation results. Interaction uncertainty can be handled with finite-dimensional IMs, while plant uncertainty typically necessitates infinite-dimensional IMs, which can be truncated to achieve arbitrarily accurate approximate regulation. Theoretical results are complemented by illustrative examples demonstrating robustness under both interaction and plant uncertainties and by practical design procedures for implementing finite-dimensional approximations.

Abstract

Robust output regulation for linear time-varying systems has remained an open problem for decades. To address this, we propose intrinsic system immersion by reformulating the regulator equation in a more insightful form, indicating that finding an internal model is equivalent to reproducing the output trajectory of a forced system by constructing an unforced system. This perspective reveals the influence of parametric uncertainties, demonstrating that an infinite-dimensional controller is generally unavoidable for robustness against plant uncertainty. Consequently, a general robust design is proposed without explicitly solving the regulator equation. It ensures robustness against uncertainties in the exosystem interaction, and achieves approximate output regulation when an infinite-dimensional controller is necessary for regulation. Additionally, we study the regulator equation in a coordinate-free framework, extend the time-varying non-resonance condition, and provide a method to minimize the dimension of an internal model. Overall, these results provide a general systematic framework for constructing robust internal model-based controllers, and simplify the control implementation process.

Robust Output Regulation of Uncertain Linear Time-Varying Systems

TL;DR

The paper tackles robust output regulation for linear time-varying systems with parametric uncertainty by reframing the regulator equation via intrinsic system immersion, showing that robustness to plant uncertainty generally requires an infinite-dimensional internal model. It establishes a propagation condition linking robustness to how uncertainty propagates through the exosystem dynamics, and develops a comprehensive framework that includes minimum/internal-model reduction, time-varying canonical realizations, and both exact and approximate regulation results. Interaction uncertainty can be handled with finite-dimensional IMs, while plant uncertainty typically necessitates infinite-dimensional IMs, which can be truncated to achieve arbitrarily accurate approximate regulation. Theoretical results are complemented by illustrative examples demonstrating robustness under both interaction and plant uncertainties and by practical design procedures for implementing finite-dimensional approximations.

Abstract

Robust output regulation for linear time-varying systems has remained an open problem for decades. To address this, we propose intrinsic system immersion by reformulating the regulator equation in a more insightful form, indicating that finding an internal model is equivalent to reproducing the output trajectory of a forced system by constructing an unforced system. This perspective reveals the influence of parametric uncertainties, demonstrating that an infinite-dimensional controller is generally unavoidable for robustness against plant uncertainty. Consequently, a general robust design is proposed without explicitly solving the regulator equation. It ensures robustness against uncertainties in the exosystem interaction, and achieves approximate output regulation when an infinite-dimensional controller is necessary for regulation. Additionally, we study the regulator equation in a coordinate-free framework, extend the time-varying non-resonance condition, and provide a method to minimize the dimension of an internal model. Overall, these results provide a general systematic framework for constructing robust internal model-based controllers, and simplify the control implementation process.
Paper Structure (16 sections, 22 theorems, 110 equations, 2 figures)

This paper contains 16 sections, 22 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Solution to the regulator equation
  4. Solvability and the Solution
  5. LTV Non-Resonance Condition for Boundedness
  6. Asymptotic Uniqueness
  7. Intrinsic system immersion
  8. Minimum Internal Model
  9. Time-varying Canonical Realization
  10. Robust output regulation
  11. Robustness against Interaction Uncertainty
  12. Robustness against Plant Uncertainty
  13. Infinite-dimensional Internal Model
  14. Finite-dimensional Controller for Approximate Output Regulation
  15. Illustrative Example
  16. ...and 1 more sections

Key Result

Proposition 3.1

With Assumption ass:s, suppose the controller controller has stabilized the closed-loop system. Then, it solves Problem prob:1 if there exist UB mappings $\Pi\in C^{\infty}(\mathbb{R}\times \mathbb{R}^N, \mathbb{R}^{n\times\rho})$, $\Sigma\in C^{\infty}(\mathbb{R}\times \mathbb{R}^N , \mathbb{R}^{\n and Moreover, if the system is periodic, the above conditions become sufficient and necessary. Oth

Figures (2)

  • Figure 1: Simulation results of Example \ref{['eg:iu']} in the presence of interaction uncertainty. (a) Reference $r(t)$ and the outputs of the nominal controller $y_0(t)$ and the robust controller $y_1(t)$. (b) The errors of the nominal controller $e_0(t)$ and the robust controller $e_1(t)$.
  • Figure 2: Simulation results of Example \ref{['eg:pu']} in the presence of plant uncertainty. (a) Reference $r(t)$ and the outputs of the $0$th-order approximate controller $y_0(t)$ and the $1$st-order approximate controller $y_2(t)$. (b) The errors of the $0$th-order approximate controller $e_0(t)$ and the $1$st-order approximate controller $e_2(t)$.

Theorems & Definitions (63)

  • Proposition 3.1
  • Theorem 3.1
  • Proof
  • Remark 3.1
  • Lemma 3.1
  • Proof
  • Lemma 3.2: Solvability
  • Proof
  • Lemma 3.3
  • Proof
  • ...and 53 more