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Sampled-data Output Regulation of Unstable Well-posed Infinite-dimensional Systems with Constant Reference and Disturbance Signals

Masashi Wakaiki, Hideki Sano

TL;DR

The paper tackles robust sampled-data output regulation for unstable well-posed infinite-dimensional systems with constant reference and disturbance inputs. It develops a two-tier design that first stabilizes a finite-dimensional unstable part and then embeds an internal model via a boundary interior Nevanlinna-Pick interpolation problem, leveraging coprime factorizations to synthesize finite-dimensional digital controllers. A key contribution is the explicit sufficient condition set ensuring the existence of such controllers, including a discretization-based reduction that preserves essential spectral and nonresonance properties, and an application to delay systems. Practically, the framework enables reliable regulation of PDE-like and delay-embedded plants using low-order digital controllers, with rigorous robustness to plant and exosystem perturbations.

Abstract

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna-Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays.

Sampled-data Output Regulation of Unstable Well-posed Infinite-dimensional Systems with Constant Reference and Disturbance Signals

TL;DR

The paper tackles robust sampled-data output regulation for unstable well-posed infinite-dimensional systems with constant reference and disturbance inputs. It develops a two-tier design that first stabilizes a finite-dimensional unstable part and then embeds an internal model via a boundary interior Nevanlinna-Pick interpolation problem, leveraging coprime factorizations to synthesize finite-dimensional digital controllers. A key contribution is the explicit sufficient condition set ensuring the existence of such controllers, including a discretization-based reduction that preserves essential spectral and nonresonance properties, and an application to delay systems. Practically, the framework enables reliable regulation of PDE-like and delay-embedded plants using low-order digital controllers, with rigorous robustness to plant and exosystem perturbations.

Abstract

We study the sample-data control problem of output tracking and disturbance rejection for unstable well-posed linear infinite-dimensional systems with constant reference and disturbance signals. We obtain a sufficient condition for the existence of finite-dimensional sampled-data controllers that are solutions of this control problem. To this end, we study the problem of output tracking and disturbance rejection for infinite-dimensional discrete-time systems and propose a design method of finite-dimensional controllers by using a solution of the Nevanlinna-Pick interpolation problem with both interior and boundary conditions. We apply our results to systems with state and output delays.

Paper Structure

This paper contains 20 sections, 23 theorems, 273 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Discrete-time output regulation
  3. Control objective
  4. Output regulation by a finite-dimensional controller
  5. Preliminary lemmas
  6. Proof of Theorem \ref{['thm:existence_servo_cont']}
  7. Sampled-data output regulation for constant reference and disturbance signals
  8. Preliminaries on well-posed systems
  9. Closed-loop system and control objective
  10. Assumptions on well-posed systems
  11. Properties of discretized systems
  12. Output regulation by a finite-dimensional digital controller
  13. Application to delay systems
  14. Decomposition of delay systems into stable and unstable parts
  15. Numerical simulation
  16. ...and 5 more sections

Key Result

theorem 2.1

Suppose that $A_{\rm e}$ is power stable. The controller eq:controller incorporates a $p$-copy internal model of the exosystem eq:exosystem if and only if it is a solution of Problem prob:ROR.

Figures (4)

  • Figure 1: Sampled-data system.
  • Figure 2: Sampled-data system with precompensator.
  • Figure 3: Time response of $y$ with $y_{\rm ref} = 1$.
  • Figure 4: Time response of $u$ with $y_{\rm ref} = 1$.

Theorems & Definitions (47)

  • definition 1: Set of perturbed plants and exosystems
  • definition 2: Definition 6.1 in Paunonen2010
  • theorem 2.1: Theorem IV.5 in Paunonen2017TAC
  • theorem 2.2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • ...and 37 more