A PID controller consists of the proportional, integral, and derivative blocks connected in parallel. It brings together the characteristics of its building blocks and can therefore be used in all applications.

## Output signal and transfer function

The output signal m(t) and the transfer function G(s) of the regulator, placed e(0)=0, are respectively equal to:

## High and low frequency gain limitation

As seen, the ideal PID controller is an improper system, because when the pulsation ω tends to infinity and when ω=0. To limit the gain, in high and low frequency, the real PID regulator represented in the figure is used, whose transfer function is:

## PID Controllers Design

The designer must calculate the value of the K_{P}, K_{I} e K_{D} coefficients so that they meet the specifications of the frequency response (phase and gain margin, bandwidth, etc.) and those of the time response (full error, settling time, delay time, etc.).

## The Ziegler-Nichols method

Among the many methods available, this method is quite widespread in the industrial field. It consists in obtaining the optimal values of the K_{P}, K_{I} e K_{D} parameters by acting on special knobs of the regulator calibrated at the factory.

The phases of optimal regulation are:

- set K
_{P}=0, K_{I}=0 e K_{D}=0 and the adjustment ring is closed; - the value of the K
_{P}parameter is gradually increased, after excluding the derivative and integral actions, until the system reaches the limit of stability; - the value of K
_{P}=K_{Pmax}is measured, for which the response of the system to the unit step is an oscillation of constant amplitude; - the pulsation values are measured ω
_{c}and the T_{c}period of the persistent oscillation; - adjust the other knobs so that the parameters K
_{P}, K_{I}and K_{D}take the values shown in the following table.