Flatness for linear fractional systems with application to a thermal system

Abstract

This paper is devoted to the study of the flatness property of linear time-invariant fractional systems. In the framework of polynomial matrices of the fractional derivative operator, we give a characterization of fractionally flat outputs and a simple algorithm to compute them. We also obtain a characterization of the so-called fractionnally 0-flat outputs. We then present an application to a two dimensional heated metallic sheet, whose dynamics may be approximated by a fractional model of order 1/2. The trajectory planning of the temperature at a given point of the metallic sheet is obtained thanks to the fractional flatness property, without integrating the system equations. The pertinence of this approach is discussed on simulations.

Differential flatness‐based ADRC scheme for underactuated fractional‐order systems

“…In addition, there are other algorithms to find the flat output of the FOS, and an example will be given below.…”

“…For such underactuated systems, differential flatness features can be used for processing. In studying of fractional differential flatness, some calculation methods and applications of fractional flat output are available . The basic idea is to dynamically extend a class of nonlinear systems with incomplete feedback linearization in the sense of differential homeomorphism and avoid the complicated integration process, but not all systems are differentially flat systems, and its flat output is not unique and difficult to find even if it is a differentially flat system.…”

“…The basic idea is to dynamically extend a class of nonlinear systems with incomplete feedback linearization in the sense of differential homeomorphism and avoid the complicated integration process, but not all systems are differentially flat systems, and its flat output is not unique and difficult to find even if it is a differentially flat system. Victor et al give a characterization of fractionally flat outputs and a simple algorithm to compute them. They also obtained a characterization of the so‐called fractionally 0‐flat outputs in the framework of polynomial matrices of the fractional derivative operator.…”

Robust motion planning for a heat rod process

“…As a consequence of Propositions 15 of [37] (or e.g. [31, chap, 2, p. 57]), D γ a is an endomorphism from H a to itself and H a may be considered as the domain of D γ a .…”

“…In motion planning, a suitable approach consists in using the flatness-based trajectory design (see [14] [20] and the references therein) adapted to FOSs [37]. Recall that a system described by ordinary differential equations is said to be differentially flat if and only if there exists an output vector (flat output ) of the same dimension as the control vector, such that all system variables can be expressed as functions of this output and its derivatives.…”