Assume piecewise linear measured disturbances for RungeKutta(4) in MPC/MHE ? #469

The simple evaloutput and updatestate! API is not able to handle anything else than ZOH inputs (since the inputs of the next time step cannot be fed to updatestate! right now). But the internal MPC and MHE predictions are able to. The RungeKutta(4) solver is the only case in which the measured disturbances are not discretized using first-order hold (FOH). It is well known that FOH is more adequate for sampled continuous signal in MPC, which is usually the case for measured disturbances. Right now, in the package:

• Continuous linear model are discretized using Tustin method for the measured disturbances (similar to FOH)
• TrapezoidalCollocation discretizes the measured disturbances with FOH
• OrthogonalCollocation discretizes the measured disturbances with FOH

but:

• MPC/MHE with SingleShooting or MultipleShooting and RungeKutta(4) solver discretizes the measured disturbances with ZOH.

In comparison to other MPC packages, 99.9% of times they just assume ZOH for all the inputs with classical explicit RK4. But it is still an inconsistency in MPC.jl in comparison to other discretization methods, and FOH can theoretically improve the rejection of measured disturbances for MPC. If I change this in the package it comes with two caveats:

1. It will still be ZOH discretization for the plant model simulation (see first sentence), so a small plant-model mismatch during simulations
2. It won't be applicable to UKF and EKF, because of the simple evaloutput and updatestate! API

About point 1: we can clearly see the effect of the plant-model mismatch around t=10 h in the plot here (the MHE+MPC internally use FOH but the plant model is simulated with ZOH for the measured disturbance).