A FREQUENCY-DOMAIN DIFFERENTIAL CORRECTOR FOR QUASI-PERIODIC TRAJECTORY DESIGN AND ANALYSIS

The candidate solution differentiates the tapered DFT with respect to the initial state, builds the L-NAFF and GMS-C sensitivity formulas via the implicit function theorem, and assembles a Gauss–Newton update. These steps match the paper’s derivations: the DFT and Cq/Sq definitions (Eqs. (17), (22)-(23)) and continuous-frequency version (Eqs. (31)-(35)) underpin ∂Dq/∂X0 and the identities ∂Cq = 2 Re(∂Dq), ∂Sq = −2 Im(∂Dq) (Eq. (119)) . The X0-derivatives of dCq/df and dSq/df agree exactly with Eqs. (120)-(121) . For L-NAFF, the implicit-form sensitivities in Eq. (72) match the model’s expression and include the quotient-rule term that the paper writes compactly via L1–L4 (Eqs. (112)-(116)) . For GMS-C, the assembled residual and sensitivity mapping dξ/dX0 = (∂FG/∂ξ)^{-1}[∂Cq; ∂Sq; ∂(CSq)] coincide with Eqs. (64), (73)-(74); the conditioning-driven choice of C/S at the adjacent bin is also mirrored by the model’s comment . Finally, the paper notes a minimum-norm step for underdetermined targeting, consistent with the model’s Moore–Penrose/LM update suggestion .