Limiting transfer function of PID to upper and lower bounds

Question

I have a question about Limiting the output of a transfer function in my system model.
(Im fairly new to control theory and control systems)
I have a model where a PID controller is directly controlling the acceleration of a quadcopter. The quadcopter is "trying" to fly at a set hight.
Currently, I have the following:

My question is: how can I limit the acceleration (the PID output) in the system?
I've tried the solution from question 133541. But that didn't work out for me. Mathematica tells me the transfer function of the PID controller cant be converted into a nonlinearStateSpaceModel? I don't really know why or how as I'm new to the field.

Thank you in advance!

Suba Thomas · Accepted Answer

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]]

Cesareo · Answer

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.

Limiting transfer function of PID to upper and lower bounds

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