An exterior differential characterization of single-input local transverse feedback linearization

An Algorithm for Local Transverse Feedback Linearization

“…In this section we state the main result. The goal of this article is to demonstrate a constructive algorithm for producing a transverse output with respect to N at x 0 ∈ N. The computable necessary and sufficient conditions under which this algorithm succeeds are a slight variation of those proposed in [5]. The first of these is the controllability condition…”

“…Inspired by Gardner and Shadwick, the authors of this work sought an algorithmic procedure that could, at the minimum, produce the integrable distribution or, equivalently, the solvable system of partial differential equations. The algorithm for the single-input case was presented in [5,Remark 2]. This article generalizes that work and presents an algorithm that produces the state and feedback transformation in the multi-input case.…”

“…Unfortunately, their procedure relies on the existence of an integrable distribution adapted to the tangent space of N and to the specific system structure. Finding this distribution can be difficult even in the single-input case as demonstrated in [5]. Inspired by Gardner and Shadwick, the authors of this work sought an algorithmic procedure that could, at the minimum, produce the integrable distribution or, equivalently, the solvable system of partial differential equations.…”

“…After presenting our notation, we formally introduce the transverse feedback linearization problem in Section 2. Section 3 then presents a preliminary lifting of the problem into a setting that includes time and control in a way similar to both [6] and [5]. Section 4 then presents the conditions for multi-input transverse feedback linearization in the exterior differential setting.…”

“…Furthermore, system (2.1) with output h : R n → R m yields a vector relative degree at x 0 ∈ R n if, and only if, system (3.1) with output h • π : M → R m yields a vector relative degree at ι(x 0 ) ∈ M. We also consider the lift of the manifold N into M [5]. Define the closed, embedded submanifold…”