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

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.