PID controller ($u(t)=k_{p}e(t)+k_i{\int }_0^{t}e(\tau )d\tau +k_d\dot{e}(t)$)

An ideal PID controller combines proportional control, integral action, and a term proportional to the derivative of the tracking error. With $k_d=0$, it gives proportional-integral control. The integral stores past error, while the derivative responds to its current rate of change. Practical implementations must account for measurement noise, which an ideal differentiator can amplify. ^[ref].