[freemint]

I assume you are talking about the pendulum. PID tuning is fundamentally restricted to holding set points and can't deal with non-linearites to reach those set points.

MPC if solved over a to small time horizon might fail to find a windup trajectory. But that isn't a restriction. However i would consider MPC overblown for this, as MPC is usually a series of optimal control problems.

By optimal control i assume you mean things approaches where you optimize a u() so it minimizes some integral. This approach does just that, only that u is not parametrized by polynomials/splines but by a neural network, taking state, time and other things into account.