English: In a unity feedback configuration, an ideal PID compensator is used to enhance the step response of a system with the transfer function

${\displaystyle G_p=\frac{1}{s^2+1.33s+1}}$

The PID controller has the standard form K_p+K_i/s+K_ds. Therefore, the closed loop transfer function of the whole system becomes

${\displaystyle G=\frac{G_c{G}_p}{1+G_c{G}_p}=\frac{K_d{s}^2+K_{p}s+K_i}{s^3+(1.33+K_d)s^2+(1+K_p)s+K_i}}$

As the proportional, integral and derivative terms of the PID compensator are increased, the evolution of the step response, from the initial uncompensated response (corresponding to PID coefficents Kp=1, Ki=0, Kd=0) to the final desired response is depicted. The following effects of PID compensation can be readily observed:

• The proportional term increases the speed of the system. It also decreases the residual steady state error of the step response, but can not eliminate it completely.

• The integral term eliminates the residual steady state error of the step response, but adds undesired "oscillations" to the transient response (overshoot).

• The derivative term "damps out" the undesired oscillations in the transient response.