On Triangular Forms for x-Flat Control-Affine Systems With Two Inputs

The paper proves that for any x-flat two-input control-affine system with differential difference d, after at most d-fold prolonging each input, the system is static-feedback equivalent to the general triangular form (GTF) with the given flat output appearing as (z1, zk1+1). This is stated in Theorem 2 and proved in Appendix B by showing the codistributions Q_{R-1},...,Q_{R+d-1} are integrable via explicit span equalities (e.g., span{dφ[0,R−1]}=span{dx, dū1[0,d−1]}) . The candidate’s solution reaches the same conclusion using the same core idea: construct a flat chart where the codistributions become coordinate codistributions and hence integrable, invoking the GTF characterization based on integrability of codistributions . Differences: (i) the candidate adds an extra claim that any p with 2p≥d suffices; the paper only guarantees p=d (though it exhibits examples where fewer prolongations suffice), and it does not prove the general 2p≥d bound, so that strengthening is unsupported here . (ii) The candidate invokes a decoupling-matrix normalization step that is not needed in the paper’s proof. Overall, the main statement and the integrability mechanism are aligned, and the proofs are substantially the same at the level of core logic (integrability of the codistribution chain after d prolongations using the flat-output parameterization) .