A PID controller combines proportional, integral and derivative actions:

$$u(t)=K_Pe(t)+K_I\int_0^t e(\tau),d\tau+K_D\frac{de}{dt}.$$

Its ideal transfer function is

$$C(s)=K_P+\frac{K_I}{s}+K_Ds.$$

The three terms address different aspects of the response. Proportional action reacts to present error, integral action removes persistent error, and derivative action can improve damping by reacting to rapid change.

The gains interact because all three terms modify the same closed-loop characteristic equation. Tuning one gain while ignoring the others can therefore change rise time, overshoot, stability and actuator effort simultaneously.

Practical PID controllers usually include derivative filtering and protection against integral windup. The useful abstraction is not that PID is universally optimal, but that it provides three simple mechanisms for shaping tracking and transient behavior with little model complexity.