PID control operation with block diagram

A PID controller computes its control output from the error e(t) = r(t) - y(t), the difference between the set-point and the measured output, as the sum of a proportional, an integral and a derivative term.

u(t) = Kp·e(t) + Ki·∫ e(t) dt + Kd·de(t)/dt

Block diagram

          +--> [ Kp * e ] --------+
          |                       |
 r -->(+)-+--> [ Ki * integral ]->(+)-u-> [Plant] --+--> y
       ^ -|                       |  ^                |
       |  +--> [ Kd * d/dt ] -----+  d (disturbance)  |
       |                                              |
       +--------------[ Sensor ]<---------------------+

Operation of each term

• Proportional (P): output is proportional to the present error. Larger Kp gives a faster response but more overshoot, and a P-only controller leaves a steady-state offset.
• Integral (I): sums the past error, so it keeps increasing the output until the error is zero. It removes steady-state error but can add overshoot and slow oscillation (and integral wind-up).
• Derivative (D): reacts to the rate of change of error, predicting future error. It adds damping, reduces overshoot and improves stability, but amplifies sensor noise.

PID tuning

Tuning means choosing Kp, Ki, Kd so that the response is fast, with small overshoot and zero steady-state error.

1. Manual method: set Ki = Kd = 0; raise Kp until the output oscillates a little, then reduce it. Add Kd to damp overshoot. Add Ki slowly to remove the offset.
2. Ziegler-Nichols: raise Kp until sustained oscillation (ultimate gain Ku, period Tu), then use Kp = 0.6 Ku, Ti = Tu/2, Td = Tu/8.
3. Software tools / auto-tuning using a plant model or step-response test.

Check the result with rise time, overshoot, settling time and steady-state error.