In this application we will see how the Bode plot can be used to design a PI controller using the Ziegler Nichols tuning rules for a first order process.

Consider a first order process with the following parameters: Gain (K)= 2, First order time constant (tau) = 3 sec and delay (t) = 2 sec

The transfer function is K e^-ts / (1 + s x tau). Accordingly, we have  a = 2, b = 0, g = 1, h = 3 and t = 2

To get the tuning parameters, we find the gain margin and the correspoding critical frequency (w_c) from the Bode Plot as shown below:

The ultimate gain K_u = 1 / 0.66 = 1.52
Ultimate period P_u = 2 x PI / w_c = 2 x PI / 0.95 = 6.61 s

The Z-N tuning is therefore, K = K_u / 2.2 = 0.7 and I = P_u / 1.2 = 5.5s