The Proportional-Integral-Derivative Response is a closed feedback loop that can be used to maintain a controlled amount of, in this case, liquid helium in a reservoir. This page is dedicated to providing an understanding of the mathematics behind the mechanism and hopefully help in the ability to change the PID gains in the Response VI controlling the level of LHe (for more information on the VI, please click this link to the LabVIEW Virtual Instrument Manual).
PID Theory
The PID control system is a common algorithm that begins with an error function with respect to time, e(t), where...
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The PID control utilizes three coefficients - the proportional (P), integral (I) and derivative (D) terms. Each term These coefficients can be any real number, but are usually positive. Each one is multiplied with the error function, and the products are then summed to create the new controller output, z.
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Since these three expressions are summed together, it stands to reason that if a coefficient is larger than the others, then that term would have more weight than the others. Likewise, if that coefficient is less than the others, it will matter less in the overall calculation of the controller output.
Elements of the PID Response
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A proportional feedback loop is also called a constant gain controller, and will be linearly proportional to the error. The P coefficient changes the output by taking the current error value alone into account. The magnitude of the P coefficient only changes how much the system reacts to the error - a too-large coefficient means it will overreact, whereas a too-small coefficient means it will under-react.
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Changing the PID coefficients to improve the feedback loop can be tricky, and mostly involves trial and error and seeing what works best with your given application. Before changing the values, however, it is important to store values that have been proven to work in case no better values can be found (this is why, when in the Response VI and Advanced Controls are enabled, a warning is presented to the user to inform them to only change the controls if they know what they are doing, as otherwise the default values should just stay as they are).
Resources
Here are a few resources that were used to help create this webpage! Please check if you have further questions.