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Finding Transfer Function from Step Response

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A bit of context:
I'm in a stundent project at my universtiy. We're desingning and building an Mechanical ventilator for Corona-Patients. I'm having some trouble with the control loop of the system.
The ouput of the control loop is controlling a proportional valve. The purpose of this is to reach the desired pressure (Setpoint) in less than 3 seconds. I'm using a PID control, yet it's not tuned correctly. That's why I decided to go to basics, and get the transfer function from a step response, to then simulate the different constants in MATLAB.
The following step response was achieved by opening the proportional valve fully, and waited till the Setpoint was achieved. There's a 1 bar relative pressure in the system. Any pressure above 25mbar flows out, through a PEEP valve. The proplem with the system is that we can't overshot, because we cannot get rid of this pressure as the patient is inhaling.
If someone could help me, calculate the transfer function of the following step response, I would be really gratefull.
Thanks in advance.

Accepted Answer

Ameer Hamza
Ameer Hamza on 23 Oct 2020
Edited: Arkadiy Turevskiy on 19 Jun 2024 at 20:17
You can do it using the system identification app: For this, you also need the toolbox:
Update by Kishen Mahadevan at MathWorks on 6/19/2024
Considering that the end goal here is to identify plant model (transfer function) for PID tuning, you can also directly use the PID Tuner app from Controls System Toolbox that lets you automatically tune the gains of PID controller for your plant model, including identifying new plant model from response data all within the same app. Identifying new plant model step requires System Identification Toolbox to obtain a plant from measured or simulated system response data.
Please refer to this page for more information on PID tuner - Tune PID controllers - MATLAB (
Please refer to this example to see how to Interactively Estimate Plant Parameters from Response Data and tune PID controller gains - Interactively Estimate Plant Parameters from Response Data - MATLAB & Simulink (
Ameer Hamza
Ameer Hamza on 23 Oct 2020
I am glad to be of help! You can find many examples in the documentation.
Max Mai Tobon
Max Mai Tobon on 25 Oct 2020
The transfer function I got was. H (s) = 847.4 / ( s + 757.2)
I'm now trying different constants for my PID control. Tomorrow I'll test the results.
Thanks again

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More Answers (1)

Ameer Hamza
Ameer Hamza on 23 Oct 2020
Edited: Ameer Hamza on 23 Oct 2020
Another approach can be something like this. Consider the transfer function of the system is
and the transfer function of input step function is
The transfer function of output will be
which in time-domain become
Here you have y(t). You can use fmincon() to estimate a and b.
Max Mai Tobon
Max Mai Tobon on 29 Oct 2020
Oh, that was the problem! I didn't know I had to download an extra Toolbox.
It's working now.
Thanks a lot for your help! Very appreciated.

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