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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 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 , then the 

Proportional-Integral Response and the I Coefficient

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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

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