How does a PID controller combine proportional, integral, and derivative control?

Proportional Control (P):

The proportional control component is directly proportional to the error between the setpoint and the process variable. It produces an output that is proportional to the present error. The proportional action helps to provide stability and reduce steady-state error but does not consider past or future errors.

Integral Control (I):

The integral control component considers the accumulated error over time. It integrates the error and produces a control output that is proportional to the integral of the error. The integral action helps to eliminate steady-state error and corrects for any accumulated error, even if the proportional action alone cannot correct it. It has the ability to continuously adjust the control output to drive the error towards zero.

Derivative Control (D):

The derivative control component considers the rate of change of the error. It produces a control output that is proportional to the derivative of the error. The derivative action predicts the future trend of the error based on its current rate of change. It helps to provide damping, improve system response, and reduce overshoot and oscillations by anticipating the error's behavior.

Combining the three components, the PID controller output is calculated as follows:

Output = Kp * (P + Ki * I + Kd * D)

where:

Kp is the proportional gain that scales the proportional action.

Ki is the integral gain that scales the integral action.

Kd is the derivative gain that scales the derivative action.

P, I, and D are the respective proportional, integral, and derivative contributions calculated based on the error.

The gains (Kp, Ki, Kd) are tuning parameters that need to be adjusted to optimize the controller's performance for a given system.

By combining the three control actions, the PID controller can effectively respond to different aspects of the system's behavior. The proportional control provides an immediate response, the integral control eliminates steady-state errors, and the derivative control helps anticipate and counteract overshoot or oscillatory behavior. The combination of these actions allows the PID controller to achieve stable and accurate control in a wide range of systems.