## Info

Fig. 27.5. On/Off control with uj=0.0 and u2=0.2 GPM. Left: the measured variable responds fast to the set point change and presents large deviations. Right: the operating variable moves very frequently

### 27.2.2 PID Control

A PID controller produces an output signal that is a linear combination of (1) the error (Proportional action), (2) the integral of the error over time (Integral action), and (3) the time derivative of the error (Derivative action). This algorithm can be represented by the following equation:

Here, t represents the time, u(t) is the controller output, that is, the signal sent by the controller to the final control element, and e(t) is the controller input, corresponding to the error signal, that is, the set point minus the value of the measured variable. Figure 27.6 represents the terms involved in Eq. (27.2).

The contribution of each term (P, I, and D) to the total control action is shown in Fig. 27.7. Here we can observe that the contribution of the proportional action gets smaller when the error is reduced, therefore, pure proportional control cannot ensure that in the limit (steady state) the measured variable will reach the set point. On the other hand, integral action increases with time; therefore, a non-zero corrective action is applied even if the error disappears. When the system starts moving the derivative action is very high, helping the process to reach the set point sooner.