Exercise 3.9 Consider again the distillation process  with the decoupling controller but also include output gain uncertainty b i That is let the perturbed loop transfer function be Ls GKinv 
s  b 
b
G  

G


z L

where L has no dynamics for the distillation example The closed loop poles of the perturbed system are solutions to detI  Ls detI  ksL  or equivalently det
s k
I  L sk
 trLsk  detL 
For k
we see that instability occurs if and only if the trace andor the determinant of L are negative Since detL
for any gain error less than  instability can only occur if trL  Evaluate trL and show that with gain errors of equal magnitude the combination of errors which most easily yields instability is with b 
b


 Use this to show that the perturbed system is unstable if
r 


where  gg

 detG is the 
element of the RGA of G In our case   and we get instability for    Check this numerically