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On the controller form for linear hyperbolic MIMO systems with dynamic boundary conditions

Stefan Ecklebe, Frank Woittennek

TL;DR

This work addresses the design of a controller form for linear hyperbolic MIMO systems with dynamic boundary conditions, where a direct explicit state realization fails due to rank deficiencies in the input–output map for MIMO. It develops a generalized hyperbolic controller form and a flatness‑based, algebraic computation framework that uses generalised polynomials with real exponents to model predictions and delays, together with shift/interval reduction and generalised polynomial long division to obtain a realizable form. The main contributions are (i) a principled hyperbolic MIMO controller form, (ii) a concrete algorithm to compute it for a class of systems, and (iii) a benchmark reduction that yields a diagonal input map, illustrating feasibility and guiding future extensions. The approach provides a systematic method for controller design in PDE–ODE boundary‑coupled systems with MIMO inputs and flatness‑based parametrisations, with potential extensions to nonlinear boundary dynamics and plans to remove delays in future work.

Abstract

This contribution develops an algebraic approach to obtain a controller form for a class of linear hyperbolic MIMO systems, bidirectionally coupled with a linear ODE system at the unactuated boundary. After a short summary of established controller forms for SISO and MIMO ODE as well as SISO hyperbolic PDE systems, it is shown that the direct ap- proach to state a controller form fails already for a very simple MIMO example. Next, a generalised hyperbolic controller form with different variants is proposed and a new flatnesss-based scheme to compute said form is presented. Therein, the system is treated in an algebraic setting where generalised polynomials with real exponents are used to describe the predictions and delays in the system. The proposed algorithm is then applied to the motivating example.

On the controller form for linear hyperbolic MIMO systems with dynamic boundary conditions

TL;DR

This work addresses the design of a controller form for linear hyperbolic MIMO systems with dynamic boundary conditions, where a direct explicit state realization fails due to rank deficiencies in the input–output map for MIMO. It develops a generalized hyperbolic controller form and a flatness‑based, algebraic computation framework that uses generalised polynomials with real exponents to model predictions and delays, together with shift/interval reduction and generalised polynomial long division to obtain a realizable form. The main contributions are (i) a principled hyperbolic MIMO controller form, (ii) a concrete algorithm to compute it for a class of systems, and (iii) a benchmark reduction that yields a diagonal input map, illustrating feasibility and guiding future extensions. The approach provides a systematic method for controller design in PDE–ODE boundary‑coupled systems with MIMO inputs and flatness‑based parametrisations, with potential extensions to nonlinear boundary dynamics and plans to remove delays in future work.

Abstract

This contribution develops an algebraic approach to obtain a controller form for a class of linear hyperbolic MIMO systems, bidirectionally coupled with a linear ODE system at the unactuated boundary. After a short summary of established controller forms for SISO and MIMO ODE as well as SISO hyperbolic PDE systems, it is shown that the direct ap- proach to state a controller form fails already for a very simple MIMO example. Next, a generalised hyperbolic controller form with different variants is proposed and a new flatnesss-based scheme to compute said form is presented. Therein, the system is treated in an algebraic setting where generalised polynomials with real exponents are used to describe the predictions and delays in the system. The proposed algorithm is then applied to the motivating example.

Paper Structure

This paper contains 30 sections, 5 theorems, 41 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Established Controller Forms
  5. ODE SISO systems
  6. ODE MIMO systems
  7. hyperbolic SISO systems
  8. Benchmark System
  9. Parametrisation
  10. Stating a controller form
  11. A hyperbolic controller form
  12. Definition
  13. A classic controller form
  14. Generalised controller forms
  15. With input derivatives
  16. ...and 15 more sections

Key Result

Lemma 1

[lem]lem:sorting By permuting the input and/or flat output, $\mathbf{H}(\sigma,\partial_t)$ can be sorted, such that the degrees w.r.t. $\sigma$ in each column are ascending and the highest degrees w.r.t. $\partial_t$ appear on the main diagonal, thus with $j \in \{1,\dotsc,m\}$ hold for $i\in\{1,\dotsc,m-1\}$ and $l\in\{1,\dotsc,m\}$.

Figures (1)

  • Figure 1: Structure of the assumed system class

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof