Self Tuning Controller Design

The graphs showing the variation of Kp and $\tau _p$ are shown below:

Figure 9.9: Variation of $K_p$ with temperature

Figure 9.10: Variation of $\tau _p$ with temperature

1. PI Controller using Ziegler Nichols Tuning:
   

   L = 6
   R = (0.016$\times$T-0.114)/(66.90-0.415$\times$T) where T is the temperature
   Kc = 0.9(66.90-0.415T)/6(0.016T-0.114)
   = (60.21 - 0.3735T)/(0.096T - 0.684)
   $\tau _i$ = 3 $\times$ 6 = 18
   

2. PID Controller using Ziegler Nichols Tuning:
   

   L = 6
   R = (0.016 $\times$ T-0.114)/(66.90-0.415 $\times$ T) where T is the temperature
   K = 1.2(66.90-0.415T)/6(0.016T-0.114)
   = (80.28 - 0.498T)/(0.096T - 0.684)
   $\tau _i$ = 2 $\times$ 6 = 12
   $\tau _d$ = 0.5$\times$6 = 3
   

3. PI Controller using Direct Synthesis ($\tau_{cl}$ is taken as $\tau _p$/2):
   

   K = 2/(0.016$\times$T-0.114)
   $\tau _i$ = (66.90-0.415$\times$T) where T is the temperature