The proportional component generates an output signal that is proportional to the current error value (the difference between the desired setpoint and the measured process variable). Mathematically, this is represented as $P_{out} = K_p \times error$. The gain constant, $K_p$, determines the strength of the proportional response. A higher $K_p$ will cause the system to react more aggressively to errors, potentially leading to faster response times but also increasing the risk of overshoot and instability.

The integral component considers the accumulation of past errors. It integrates the error over time, meaning that even small errors, if persistent, will eventually lead to a significant corrective action. This is crucial for eliminating steady-state errors, where the system might settle slightly off the target due to constant disturbances or system biases. The integral term is calculated as $I_{out} = K_i \times \int error \, dt$. The integral gain, $K_i$, controls how quickly the integral term grows. While effective at removing steady-state error, an aggressive integral term can lead to 'integral windup' and cause the system to overshoot the setpoint.

The derivative component anticipates future errors by considering the rate at which the error is changing. If the error is decreasing rapidly, the derivative term will provide a counteracting force to prevent overshoot. Conversely, if the error is increasing rapidly, it will provide a stronger corrective action. This term is calculated as $D_{out} = K_d \times \frac{d error}{dt}$. The derivative gain, $K_d$, determines the influence of the rate of change. While the D term can significantly improve system stability and reduce overshoot, it is sensitive to noise in the sensor readings, which can lead to erratic control actions if not properly filtered.