qSuppose the controller
implemented in the system (or plant G22) is actually
• K=kKopt ,
• with a nominal value k=1. Then the closed-loop
system A-matrix becomes
•
•
•
• The characteristic
polynomial has the form
•
• With a1= a+b-4+2(k-1)ab, a0=1+(1-k)ab
• - necessary for
stability: a0>0 and a1>0.
• -a>>1 and b>>1 and k¹1 Þ a0
» (1-k)ab, and a1 » 2(k-1)ab
• -a>>1 and b>>1 (q and s), the system is unstable for arbitrarily small perturbations in k in either direction.
Thus, by choice of q
and s the gain margins may be made
arbitrarily small.