Multi-input control-affine systems linearizable via one-fold prolongation and their flatness

Normal forms for multi‐input flat systems of minimal differential weight

“…We denote by  k the involutive closure of  k . According to our results 19,20 (see also other works of aforementioned authors 21,22 ), flatness and x-flatness of differential weight n + m + 1 are equivalent on an open and dense subset (that excludes the points at which some ranks drop) of either X or X × U, and, moreover, they are equivalent to linearizability via the one-fold prolongation of a suitably chosen control. Without the constant rank assumption, that equivalence no longer holds (see section 5 in the work of Nicolau and Respondek 20 devoted to that issue).…”

“…In the works of Nicolau and Respondek,19,20 see also other works of aforementioned authors, 21,22 we gave a complete geometric characterization of control-affine systems that become static feedback linearizable after a one-fold prolongation. The proposed normal forms apply to all systems described by the works of Nicolau and Respondek, 19,20 but we do not use results of the aforementioned works 19,20 to construct our normal forms.…”

Multi-Input Nonlinear Control Systems Linearizable via One-Fold Reduction

“…The conditions of the above theorem recall very much those for linearization via invertible one-fold prolongation, or, equivalently, for flatness of differential weight n + m + 1 (see [18] for the definition of the differential weight of a flat system, and [15] for a complete geometric characterization of flat systems of differential weight n + m + 1, where n is the state dimension and m is the number of controls). For those systems we have, as for the class described by Theorem 1, a sequence of inclusions of nested distributions.…”

“…That problem can be seen as the dual of the linearization via invertible one-fold prolongation: the simplest flat systems that are not static feedback linearizable are those that become linearizable via invertible one-fold prolongation of a suitably chosen control (which is the simplest dynamic feedback). That class of systems was completely characterized in [15] (see also [14]). The conditions for linearization via one-fold reduction reminds very much those for linearization via invertible one-fold prolongation.…”

On singularities of flat affine systems with n states and n−1 controls

“…. , n − 1, we note 20) and assume that Γ 0 (x 0 ) = Γ a 0 (x 0 ), thus meaning that the dimension of Γ 0 drops down from n − 1 to n − p at x 0 and that Γ b 0 (x 0 ) ⊂ Γ a 0 (x 0 ). For simplicity's sake, we note…”