Zeigler-Nichols Rule explained with example

When to use the Zeigler-Nichols

Tuning rules work well when the controller required is analog, system is linear, monotonic and a response that is dominated by a single-pole exponential “lag”.

Actual plants are unlikely to have a perfect first-order lag characteristic. Higher-order poles will introduce an extra phase shift, phase shift matters a lot to loop stability. You can’t depend upon the single “lag” pole to match both the amplitude roll-off and phase shift accurately.

So the ZN model presumes an additional fictional phase adjustment that does not distort the assumed magnitude roll-off. At the stability margin there is a 180 degree phase shift around the feedback loop which is the Nyquist’s stability criteria. A first order lag can contribute no more than 90 degrees of that phase shift.

To summarize, then, the Ziegler-Nichols rule assumes that the system has a transfer function of the following form:

= (Ke^-sT)/ (a+s)

The model matches the system response at frequencies 0 and at the stability limit.

How do you measure response parameters?

Given the magnitude and phase open-loop response curves of the plant, you can fit the assumed model in the following manner.

1. The ratio of output level to input level at low frequencies determines the gain parameter K of the model.
2. Observe the frequency F0 at which the phase passes through -pi radians (-180 degrees). The inverse of this frequency is the period of the oscillation, T0.
3. Observe the plant gain Kc that occurs at the critical oscillation frequency Fu. The inverse of this is the gain marginK₀.

Example:

The given transfer function is:

G_p = 1/s(s+11.6+5.8i)(s+11.6-5.8i)

Zeigler-Nichols Compensation

MATLAB CODE

num=1;

den=conv([1 0],conv([1 11.6+5.8i],[1 11.6-5.8i]));

figure

rlocus(num,den) %frequency response of the system

T0=(2*pi)/13;  % Measuring the period T0 at the stability limit

Kp0=3900;      % Measuring the gain margin Kp0 for loop stability

Ti=0.5*T0;     % Calculating the tuning parameters

Td=0.125*T0;  

kp=0.6*Kp0;

ki=kp/Ti;

kd=kp*Td;

numgc=kp*[Td*Ti Ti 1]; %Numerator of PID transfer function 

dengc=[Ti 0];          %denominator of PID transfer function

gc=tf(numgc,dengc);    % PID controller’s transfer function

gp=tf(num,den);

gf=gc*gp;

t=gf/(1+gf);           %Closed loop transfer  function

figure

step(t)

 Root Locus