Limiting transfer function of PID to upper and lower bounds

The core problem is that NonlinearStateSpaceModel does not support descriptor systems.

When there are pure derivative terms, the state-space representation will need a singular descriptor matrix. This is supported by StateSpaceModel but not by NonlinearStateSpaceModel.

Through@{StateSpaceModel, NonlinearStateSpaceModel}@TransferFunctionModel[s, s]

A workaround is to do away with the derivative term or use a filter on the derivative terms. (These are done in applications where there is large sensor noise or abrupt reference changes.)

SystemsModelSeriesConnect[TransferFunctionModel[kp + ki/s + kd s/(s + α), s], 
 NonlinearStateSpaceModel[{{}, Clip[u]}, {}, u]]

Due to the double pole representing the copter, a simple PD will suffice. Follows a layman procedure to find an acceptable solution. The procedure consists in searching through a minimization procedure, the nearest response regarding a reference response to a step. Here the reference response is given by

stepref = InverseLaplaceTransform[(a^2 + b^2)/((s + a)^2 + b^2)/s, s, t]

the actual step response is obtained as follows:

PID = kp + ki/s + s kd;
COPTER = 1/s^2;
model = COPTER PID/(1 + COPTER PID)
stepresponse = InverseLaplaceTransform[model/s, s, t];

then follows the minimization procedure

parms = {a -> 2, b -> 2};
tmax = 4;
n = 20;
stepref0 = stepref /. parms;
tab = Sum[Abs[stepresponse - stepref0], {t, 0, tmax, tmax/n}];
sol = NMinimize[{tab, kp > 0, ki > 0, kd > 0}, {kp, ki, kd}]
stepresponse0 = stepresponse /. sol[[2]]

Follows a plot showing in blue the reference response and in red the response found.