On the Exact Linearization and Control of Flat Discrete-time Systems

The paper rigorously establishes (i) an iff criterion for the feasibility of using v = δ^A(ϕ) as a new input via linear independence of the differentials in (10) (Theorem 4.1), and (ii) a constructive dynamic/quasi-static feedback that yields the closed-loop input–output behavior y[A] = v with controller state dimension #A − n (Theorem 4.2). These are clearly documented and proved in the uploaded PDF, including the precise manifold/shift setting and minimal-κ results for outputs independent of future inputs, along with the corollary giving the form u = Fu ∘ φ(ζ[−q1,−1], x, v[0,R−κ]) and the #κ = n statement when there is no dependence on ζ-past. The relevant statements and proofs are in Theorem 4.1 (feasibility ⇔ independence of the differentials in (10)) and Theorem 4.2 (construction of Ψ, controller state z = ϕc of dimension #A − n, and the feedback achieving y[A] = v) , with the equivalence to a discrete-time Brunovsky form used in the proof (and the quasi-static interpretation when #A = n) . The minimal multi-index κ section and the associated corollary (including the form of the feedback and the #κ = n condition when the flat output does not depend on past ζ) are covered in Theorem 4.3 and Corollary 4.4 . The candidate solution reproduces the paper’s core results but frames (i) via a submersion argument: it defines an evaluation map T_A and appeals to the rank/inverse function theorem to obtain the equivalence between realizability of arbitrary v and linear independence of (10), and then builds (ii) using complementary coordinates z with dim z = #A − n and a reconstruction Ψ to implement u = Fu ∘ Ψ. This is mathematically consistent and essentially equivalent to the paper’s diffeomorphism-based construction and feedback law. One caveat: in the optional special case discussion, the model claims a non-anticipative, pointwise inversion u = K(ζ[−q1,−1], x, v) that achieves y[κ] = v without preview of future v. In discrete time, δ^κ(ϕ)(k) generally depends on future input coordinates u(k+α), so the feedback typically needs forward-shifts of v (quasi-static), as correctly stated in the paper (see the feedback form and Remark 1) . Aside from this overstatement, the core argument aligns with the paper’s results.