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:

$m(t)=K_{p}e(t)+K_{D}\frac{de(t)}{dt}+K_{I}\int_{0}^{t}e(t)dt$

$M(s)=K_{P}E(s)+K_{D}sE(s)+K_{I}\frac{1}{s}E(s)$
$G(s)=\frac{M(s)}{E(s)}=K_{P}\left ( 1+\frac{K_{D}}{K_{P}}s+\frac{K_{I}}{K_{P}}\frac{1}{s}\right)$

High and low frequency gain limitation

As seen, the ideal PID controller is an improper system, because $\left|G(j\omega ) \right|\to \infty$ 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:

$G(s)=\frac{R_{6}}{R_{5}}+\frac{R_{Cl}}{R_{l}}\frac{1}{1+sR_{Cl}C_{l}}+\frac{sR_{D}C_{D}}{1+sR_{CD}C_{D}}$

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.

Output signal and transfer function

High and low frequency gain limitation

PID Controllers Design

The Ziegler-Nichols method

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