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...
where the actual is the current level subtracted from the target level of a system
This error, which can be easily calculated in LabVIEW by simply using the value of the actual for that iteration, is what will variable that will help the PID controller determine what action, and how much of one, needs to be taken.
The PID control utilizes three coefficients - the proportional (P), integral (I) and derivative (D) terms. 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.
where z would be the valve position, and the expressions are (in order) the Proportional, Integral and Differential
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
PID Responses were created because simply using a proportional system system was insufficient in maintaining the target point over long periods of time. However, if an integral term was added to the proportional system, it maintained around the target for longer, and the same occurred when a derivative term was added. In order to gauge a better understanding of PID, we must understand these parts, so those systems will be explained below.
Proportional Response and the P Coefficient
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.
If the coefficient is large, the response will be much faster, however, it may increase too quickly and become unstable as the system keeps trying to over-correct itself. This instability is especially likely with higher-order error equations.
At the same time, if the coefficient is too low, the response time will be very slow and could undershoot - landing somewhere below the target output. Undershooting is more common when working with lower-order error equations.
In sum, the Proportional Response is helpful in increasing reaction time and eliminating a large error faster.
The I Coefficient
Since the Integral is of a higher order than the original function, it is more likely to become unstable with a high coefficient. A higher I term increases the amount of oscillating in the system and decreases damping.
Proportional-Integral (PI) Control
The main use of PI control is that it improves the target-tracking abilities of the Proportional feedback loop by reducing the likelihood that the system will settle on an incorrect target - in short, it provides more "oscillating" but also reduces stability
The D Coefficient
While the I coefficient increases the instability of a system by increasing "oscillating" and decreasing damping, the D coefficient makes a system more stable by decreasing the amount of oscillating and increasing the damping. This term increases the weight of the derivative term, which is of lower order than the original error function, therefore, it stands to reason that the amount of damping would increase as that is, in a way, lowering the order of the feedback function.
PID Gains and Coefficients
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.