A flat triangular form for nonlinear systems with two inputs: Necessary and sufficient conditions

Abstract

The present work establishes necessary and sufficient conditions for a nonlinear system with two inputs to be described by a specific triangular form. Except for some regularity conditions, such triangular form is flat. This may lead to the discovery of new flat systems. The proof relies on well-known results for driftless systems with two controls (the chained form) and on geometric tools from exterior differential systems. The paper also illustrates the application of its results on an academic example and on a reduced order model of an induction motor. where x ∈ U ⊂ R n is the state, U is open, n ≥ 2, u = (u 1 , u 2 ) ∈ R 2 is the control and f, g 1 , g 2 : U → R n are smooth (i.e. infinitely differentiable) mappings. Recall that (1) is driftless when f = 0 and that u = α(x)+β(x)v is a regular static state feedback defined on an open set V ⊂ U if the mappings α = (α 1 , α 2 ): V → R 2 and β = (β ij ): V → R 2×2 are smooth and the matrixIt was established in [1] necessary and sufficient conditions for (1) with f = 0 to be (locally) described around a given x 0 ∈ U by the chained formż

A structurally flat triangular form based on the extended chained form

“…referred to as extended chained form, was first considered in Li et al (2013). In Nicolau (2014) and Silveira et al (2015), geometric necessary and sufficient conditions for an AI-system with two inputs to be static feedback equivalent to (6) are provided. Those are summarised in the following theorem.…”

“…In Li et al (2013), an extension of the chained form for systems with drift, the so-called extended chained form, is considered. Geometric necessary and sufficient conditions for a two-input affine input system (AIsystem) to be static feedback equivalent to this extended chained form can be found in Silveira et al (2015). Conditions for the case with m ≥ 2 inputs are provided in Li et al (2016), Nicolau et al (2014) and Nicolau (2014).…”

A Structurally Flat Triangular Form Based on the Extended Chained Form

“…referred to as extended chained form, was first considered in Li et al (2013). In Nicolau (2014) and Silveira et al (2015) geometric necessary and sufficient conditions for an AI-system with two inputs to be static feedback equivalent to (6) are provided. Those are summarized in the following theorem.…”

“…In Li et al (2013) an extension of the chained form for systems with drift, the so called extended chained form, is considered. Geometric necessary and sufficient conditions for a two-input affine input system (AI-system) to be static feedback equivalent to this extended chained form can be found in Silveira, Pereira, and Rouchon (2015). Conditions for the case with m ≥ 2 inputs are provided in Nicolau (2014), Nicolau, Li, and Respondek (2014) and Li, Nicolau, and Respondek (2016).…”

Multi-input control-affine systems static feedback equivalent to a triangular form and their flatness

“…Our main results describing control-affine systems locally static feedback equivalent to the triangular form compatible to the chained form and to the m-chained form, are given by the two following theorems corresponding to two-input control-affine systems, i.e., m = 1 (Theorem 1), and to control-affine systems with m + 1 inputs, for m ≥ 2 (Theorem 2). Let us first consider the case m = 1, which has also been solved, using the formalism of differential forms and codistributions, by (Silveira, 2010) and by (Silveira et al, 2013).…”