Caputo fractional operational matrices for pseudospectral nonlinear model predictive control: Construction, convergence, and comparative analysis

We construct and systematically compare Caputo fractional differentiation operational matrices in shifted Chebyshev and shifted Legendre polynomial bases, developed for constrained Nonlinear Model Predictive Control (NMPC) over a finite prediction horizon [0, T_p ]. Three contributions are established. First, the matrices D_α^C (Chebyshev) and D_α^L (Legendre) are derived constructively: the Legendre projection coefficients p_k,j^L are expressed in closed form via the Beta function, while the Chebyshev coefficients p_k,j^C are evaluated by a numerically exact Gauss–Chebyshev scheme. Second, convergence theorems are proved establishing the rate O(N^−(s−α)) in appropriate weighted Sobolev spaces—with explicit constants and the sharp regularity conditions s>α+12 (Chebyshev) and s > α (Legendre)—together with a precise account of analytic inputs: exponential decay of the truncation error down to an algebraic floor O(N^{−(3/2−α)}) imposed by the projection of the singular Caputo images. Third, a five-criterion comparative study is conducted within a nonlinear, constrained NMPC framework applied to ship dynamic positioning, encompassing control accuracy (ISE, IAE, ITAE, ITSE), KKT system conditioning, per-step computational cost, spectral convergence rate, and robustness to fractional-order uncertainty. Across four ship dynamic-positioning scenarios and a fractional Van der Pol benchmark (25 Monte Carlo realisations each, identical settings except the polynomial basis), the two bases do not perform equivalently: the shifted-Legendre basis reduces the integral tracking error (ISE) by 11–45% (all p<10⁻⁶, large effect sizes), lowers the per-step CPU time by 6–22% and the number of interior-point iterations by 8–24%, and improves robustness to fractional-order mismatch. Its operational matrix is additionally better conditioned than the Chebyshev one ( κ ratio ≈ 0.7 across N ). The spectral condition number of the assembled Karush–Kuhn–Tucker (KKT) system, by contrast, does not favour either basis, indicating that the Legendre advantage originates in approximation quality rather than KKT conditioning. The shifted-Legendre basis is therefore recommended for real-time onboard fractional-order NMPC implementation.