LAG COMPENSATION:

Transfer function of Lag compensator:
s + 0.1
----------
10 s + 0.1
 

LEAD COMPENSATION:

Transfer function of compensator:
s + 4.41
--------
s + 18.4

LEAD - LAG COMPENSATION:

Lead Type Circuit: F_Lead(s) = K*N (s + b) / (s + bN)

Lag Type Circuit: F_Lag(s) =  (s + a) / (s + a/M)

Design Procedure for a Lead-Lag Compensator:

1. First we design the Lead part of the compensator then the Lag part of the compensator.
2. Plot open loop transfer function of uncompensated plant and decided how much gain and phase are required at the particular frequency of interest.
3. Add 6 deg to the extra phase lag to the value obtained from the open loop uncompensated system frequency response.
4. From Figure1 below decided  what value of N to select based on the total amount of phase required.
5. To obtain b so as to give the maximum phase at the desired frequency w_d use the formula

   w_d = b * √(N)

6. To calculate K use the formula K*√(N)*G(w_d*w_d)
7. To calculate the lag filter parameter a use a = 0.1*w_d
8. To calculate the lag filter parameter M use final value theorem and steady state error margin