  We develop a one-Newton-step-per-horizon, online, lag-$L$, model predictive
control (MPC) algorithm for solving discrete-time, equality-constrained,
nonlinear dynamic programs. Based on recent sensitivity analysis results for
the target problems class, we prove that the approach exhibits a behavior that
we call superconvergence; that is, the tracking error with respect to the full
horizon solution is not only stable for successive horizon shifts, but also
decreases with increasing shift order to a minimum value that decays
exponentially in the length of the receding horizon. The key analytical step is
the decomposition of the one-step error recursion of our algorithm into
algorithmic error and perturbation error. We show that the perturbation error
decays exponentially with the lag between two consecutive receding horizons,
while~the algorithmic error, determined by Newton's method, achieves quadratic
convergence instead. Overall this approach induces our local exponential
convergence result in terms of the receding horizon length for suitable values
of $L$. Numerical experiments validate our theoretical findings.
