A Structurally Flat Triangular Form Based on the Extended Chained Form

The paper’s Theorem 4.1 states necessary and sufficient distributional conditions—(i) dim(Di)=2i up to i=n3+1 with C(Dn3+1) ≠ Dn3, and (ii) the existence of bp yielding ∆0, ∆1 satisfying items (a)–(e)—for static-feedback equivalence to the three-block triangular form (8) with x3 two Brunovsky chains of unequal lengths, an x2 extended chained part with last equation x3,1^1 + g(·)x3,2^1, and an x1 Brunovsky top block. The candidate solution reproduces this structure and follows the same distribution-flag strategy: using Di, ∆0/∆1 (via a privileged bp), and the upper flag Gi to establish necessity and sufficiency. Its proof sketch compresses the paper’s six constructive steps into three, but aligns in substance with the paper’s necessity (from the normal form) and sufficiency (constructive triangularization) arguments. Minor differences (e.g., picking bp=b2 in the necessity sketch versus the paper’s interpretation that bp corresponds to the longer x3 chain) do not affect correctness because the theorem requires existence of some bp. Overall, both arguments agree on the characterization and construction of the triangular form and its couplings, and the paper’s detailed proof fills in the technical steps the candidate sketches. See Theorem 4.1 and the surrounding discussion of (8)–(11) and items (a)–(e) for the precise statements and proof roadmap, as well as the six-step constructive sufficiency proof and examples that implement the bp-construction and triangularization .