Chapter 22. PID Controllers
The Transfer Function of the PID Controller
As we saw in Chapter 4, the output uPID(t) of a PID controller in terms of its input e(t) is given by
Notice that uPID(t) is linear in e(t). (Both integration and differentiation are linear operations.) The linearity of the PID controller is one reason for its popularity.
We can take the Laplace transform of this expression term by term to obtain the dynamic response of a PID controller in the frequency domain. Using Table 20-1, we find without difficulty that
The expression within brackets is the transfer function of the controller in frequency space. It has a particularly simple form—in fact, just the factors s and 1/s are often used to signify the corresponding terms (see Figure 22-1).
The Canonical Form of the PID Controller
As we have just seen, the transfer function of a PID controller is
This is the form ...