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I have a basic question because I'm trying to understand right now a concept that I thought it was obvious. Looking at this video he is going to feedback the variable state x with the input of the system, which is a force f.

Now, if I'm correct it is only possibile to feedback variables which share the same units, so I expect to drive a meter through an input variable which is a meter and the difference will be then feed into the PID. Is the example in the video just to show up how to use simulink? Or I m wrong?

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On the video you've linked the reference(x desire) is distance and the feedback variable is also distance, what is different is the input to the system which is force. The magic happens inside the PID where the fine tuning of the gains, which are commonly regarded as abstract values, makes the system behave the way you want.

So you can only compare variables of the same units to input the error to the PID, the PID will take care of calculating the right value no matter what units the input to the system is in.

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Keep in mind that the PID gains themselves can have units too. So if you have a position error, multiplied by a gain of the appropriate units, you can get a force.

$$ \begin{align} e &= x - x^d \\ u_f &= -k_p e - k_d \dot{e} \end{align} $$ If $x$ and $e$ have units of [m] and $u_f$ needs to have units of [N], then we get $k_p$ with units of [N/m] and $k_d$ with units of [N/(m/s)]

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