P
Proportional
Produces an output proportional to the current error. A larger gain reacts faster but can overshoot and oscillate around the setpoint. Alone, it never fully eliminates steady-state error.
Kp · e(t)
Understanding the PID Controller
A PID controller is the most widely used feedback control mechanism in industry. It continuously calculates an error value — the difference between a desired setpoint and a measured process variable — and applies a correction based on three terms.
The control law
u(t) = Kp·e(t) + Ki∫e(t)dt + Kd·de(t)/dt
where e(t) = setpoint − process variable
I
Integral
Accumulates past error over time, eliminating residual steady-state error. Too much integral action makes the system sluggish and prone to overshoot and windup.
Ki ∫ e(t) dt
D
Derivative
Predicts future error based on its rate of change, adding damping and reducing overshoot. Sensitive to noise, so it is often filtered or used sparingly.
Kd · de/dt
How they work together
The three terms combine to balance responsiveness, accuracy, and stability. Proportional gives speed, integral removes offset, and derivative adds damping. Tuning means finding the right trade-off for your system.
Where it's used
PID controllers govern cruise control, drone stabilization, 3D printer hotends, HVAC systems, chemical reactors, and motor speed control — anywhere a system must hold a target value.
A quick tuning intuition
- Increase Kp until the system responds promptly, but before it oscillates.
- Increase Ki just enough to erase steady-state offset.
- Increase Kd to tame overshoot — watch out for noise amplification.
Ready to see it in action?
Live PID Simulator
Adjust the gains and setpoint. The simulation restarts from t = 0 whenever you change a value.
Setpoint
Process variable
Controller output
t = 0.00 s
Drag sliders to retune live