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On real and observable rational realizations of input-output equations

Sebastian Falkensteiner, Dmitrii Pavlov, Rafael Sendra

TL;DR

The paper tackles the problem of reconstructing rational state-space realizations from a single IO-equation $F$ for single-output rational systems, with a focus on real coefficients and observability. It develops an algebraic-differential framework linking IO-equations to hypersurfaces $ abla(F)$ and leverages proper parametrizations to study observability, proving equivalence between realizability and observable realizability for first-order IO-equations. The authors provide algorithmic methods to compute observable and real realizations (ObservableRealization and RealRealization) and obtain partial results for higher-order IO-equations, illustrating the approach with examples. This work enables a systematic, algebraic route to derive real and observable rational dynamical systems from IO data, with implications for identifiability and observability in control theory.

Abstract

Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand sides. It has been shown that in the case where the input-output equation is of order one, rational realizations can be computed, if they exist. In this work, we focus first on the existence and actual computation of the so-called observable rational realizations, and secondly on rational realizations with real coefficients. The study of observable realizations allows to find every rational realization of a given first order input-output equation, and the necessary field extensions in this process. We show that for first order input-output equations the existence of a rational realization is equivalent to the existence of an observable rational realization. Moreover, we give a criterion to decide the existence of real rational realizations. The computation of observable and real realizations of first order input-output equations is fully algorithmic. We also present partial results for the case of higher order input-output equations.

On real and observable rational realizations of input-output equations

TL;DR

The paper tackles the problem of reconstructing rational state-space realizations from a single IO-equation for single-output rational systems, with a focus on real coefficients and observability. It develops an algebraic-differential framework linking IO-equations to hypersurfaces and leverages proper parametrizations to study observability, proving equivalence between realizability and observable realizability for first-order IO-equations. The authors provide algorithmic methods to compute observable and real realizations (ObservableRealization and RealRealization) and obtain partial results for higher-order IO-equations, illustrating the approach with examples. This work enables a systematic, algebraic route to derive real and observable rational dynamical systems from IO data, with implications for identifiability and observability in control theory.

Abstract

Given a single (differential-algebraic) input-output equation, we present a method for finding different representations of the associated system in the form of rational realizations; these are dynamical systems with rational right-hand sides. It has been shown that in the case where the input-output equation is of order one, rational realizations can be computed, if they exist. In this work, we focus first on the existence and actual computation of the so-called observable rational realizations, and secondly on rational realizations with real coefficients. The study of observable realizations allows to find every rational realization of a given first order input-output equation, and the necessary field extensions in this process. We show that for first order input-output equations the existence of a rational realization is equivalent to the existence of an observable rational realization. Moreover, we give a criterion to decide the existence of real rational realizations. The computation of observable and real realizations of first order input-output equations is fully algorithmic. We also present partial results for the case of higher order input-output equations.
Paper Structure (8 sections, 22 theorems, 49 equations, 2 algorithms)

This paper contains 8 sections, 22 theorems, 49 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics on Algebraic Geometry and Differential Algebra
  4. Input-Output Equations and Rational Realizations
  5. Observable realizations
  6. First order IO-equations
  7. Real Realizations
  8. First order IO-equations

Key Result

Lemma 2.11

Let $F \in \mathop{\mathrm{\mathbb{K}}}\nolimits[u^{(\infty)},y^{(\infty)}]$ be an irreducible polynomial of order $n$ w.r.t. $y$. Then, $F$ is realizable if and only if there exists a rational parametrization $\mathbf{P} =(P_0,\ldots,P_n) \in \overline{\mathop{\mathrm{\mathbb{K}}}\nolimits}(u^{(\in is in $\overline{\mathop{\mathrm{\mathbb{K}}}\nolimits}(u,\mathbf{x})^{n}$ where $\mathcal{J}$ deno

Theorems & Definitions (54)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 44 more