A toolbox for the analysis of linear systems with delays

Abstract

This paper considers linear time varying control systems with delays. Roughly speaking, a so called π-flat output of such a system, if it exists, allows to parameterize the states and inputs of the system using derivatives and delays of the π-flat output. Additionally, also predictions of the π-flat output are allowed, which are characterized by a prediction operator π. We present a toolbox for the computer algebra system Maple, which implements the algorithm recently been proposed in [1] for the computation of π-flat outputputs. We also give an equivalent description of hyper-regularity of polynomial matrices using one-sided inverses. This allowed us to replace the transformation to Smith-Jacobson form by the efficient method of row-/column-reduction for checking hyper-regularity. The implementation of the toolbox is illustrated and its application is explained by means of examples.

On Symbolic Computation of Flat Outputs for Differentially Flat Systems

“…To illustrate the results of the preceding sections and the usefulness of the concept for the feedforward controller design for linear time-delay differential systems, we demonstrate all steps for the system of Example 1. Note that all the necessary computations can be done with a package for the computer algebra system Maple, which has been presented in [25]. The package can be obtained from the first author upon request.…”

“…Constructive algorithms, relying on standard computer algebra environments, may be found e.g. in [23,24] for nonlinear finite-dimensional systems, or [25,26] for linear systems over Ore algebras.…”

Flatness for linear fractional systems with application to a thermal system

“…Note, again according to [2], that the technique of row-or column-reduction described in this reference requires less computations for left-or right-inverses than the above mentioned Smith decomposition and might indeed be preferred.…”

“…The role played by flat outputs for motion planning is thus clear: all system trajectories being parametrized by such an output, which, in addition, does not need to satisfy any differential equation, it suffices to construct an output curve by interpolation and deduce the desired state trajectory which, moreover, may be obtained without integrating the system equations, thus making this solution particularly simple and elegant. Our aim in this paper is therefore (1) to extend this property to fractional systems, a notion that will be called fractional flatness, (2) to characterize fractionally flat systems and fractionally flat outputs, (3) to provide an algorithm to compute such flat outputs, and (4) to show the usefulness of our approach on the motion planning example of a thermal system. Preliminary results on fractional flatness have been presented by some of the authors [45,44] without algebraic foundations and in the particular case where the matrix B of System (5) (see Section 3.1) is a 0-degree polynomial matrix, leading to a less precise characterization of the so-called defining matrices.…”

On Hyper–Regularity and Unimodularity of Ore Polynomial Matrices

“…However, the computation is rather costly and thus not very practical. Instead, methods based on row and column reduction have been developed by Beckermann et al (2006) as well as Antritter et al (2014), Antritter and Middeke (2011) or Verhoeven (2016), with the last referencing a Maple toolbox. These methods also describe how to compute hyper-regular and unimodular inverses.…”