# PID controllers for beginners

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 KP, KI e KD 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 KP, KI e KD parameters by acting on special knobs of the regulator calibrated at the factory.

The phases of optimal regulation are:

• set KP=0, KI=0 e KD=0 and the adjustment ring is closed;
• the value of the KP parameter is gradually increased, after excluding the derivative and integral actions, until the system reaches the limit of stability;
• the value of KP=KPmax is measured, for which the response of the system to the unit step is an oscillation of constant amplitude;
• the pulsation values are measured ωc and the Tc period of the persistent oscillation;
• adjust the other knobs so that the parameters KP, KI and KD take the values shown in the following table. 