On a Flat Triangular Form Based on the Extended Chained Form

Abstract

In this paper, we present a structurally flat triangular form which is based on the extended chained form. We provide necessary and sufficient conditions for an affine input system with two inputs to be static feedback equivalent to the proposed triangular form, and thus a sufficient condition for an affine input system to be flat.

A Structurally Flat Triangular Form Based on the Extended Chained Form

“…This system is not static feedback linearizable, but it is known to be flat. It is not static feedback equivalent to the triangular form proposed in Gstöttner et al (2020), but it is static feedback equivalent to the triangular form (8). In Section 5, we will systematically derive a state and input transformation, which brings (12) into the form…”

“…Among others, e. g. the planar VTOL aircraft, also considered in e. g. Fliess, Lévine, Martin, and Rouchon (1999) and Schöberl et al (2010) and the model of a gantry crane, considered in e. g. Fliess et al (1995). These systems cannot be handled with the triangular form presented in Gstöttner et al (2020). We again provide necessary and sufficient conditions for an AI-system to be static feedback equivalent to this triangular form.…”

“…In Gstöttner, Kolar, and Schöberl (2020), a triangular form which generalizes the extended chained form is considered and necessary and sufficient conditions for an two-input AI-system to be static feedback equivalent to this triangular form are provided. The proposed triangular form generalizes the extended chained form by augmenting it with two subsystems in Brunovsky normal form.…”

“…The structurally flat triangular form obtained this way contains the (extended) chained form as a special case. In this contribution, we further develop the ideas presented in Gstöttner et al (2020). We again augment the extended chained form with integrator chains, but here, the integrator chains attached to the inputs of the subsystem in (extended) chained form differ in length by one integrator.…”

“…This again provides a sufficient condition for an AI-system to be flat. In contrast to Gstöttner et al (2020), where proofs are only sketched, detailed proofs are provided in this contribution.…”