P, I, D controllers and role of PI, PD

P, I and D controllers

• Proportional (P): $u(t) = K_p e(t)$, $G_c(s) = K_p$. Output is proportional to the present error. Raising $K_p$ speeds up the response and reduces (but does not remove) steady-state error; too much gain gives overshoot and instability.
• Integral (I): $u(t) = K_i \int e\,dt$, $G_c(s) = K_i/s$. Output depends on the accumulated past error. It adds a pole at the origin and removes steady-state offset, but adds 90° of phase lag and slows the response.
• Derivative (D): $u(t) = K_d\, de/dt$, $G_c(s) = K_d s$. Output depends on the rate of change of error (future trend). It adds damping, but is never used alone because it gives no output for a constant error and amplifies noise.

Role of the PI controller

$G_c(s) = K_p + \dfrac{K_i}{s}$, a pole at the origin and a zero at $-K_i/K_p$.

Role of the PD controller

$G_c(s) = K_p + K_d s$, a zero at $-K_p/K_d$. For $G = \dfrac{\omega_n^2}{s(s+2\zeta\omega_n)}$ with PD, the characteristic equation becomes $s^2 + (2\zeta\omega_n + K_d\omega_n^2)s + K_p\omega_n^2 = 0$, so the effective damping ratio increases.

So PI is chosen to improve steady-state performance and PD to improve transient performance. A PID controller combines both.