When can bopt(P)
be small

Recall
(*b*opt(*P*)) 2 £*|N**0*(*s*)*N**z*(*s*)|2 +*|M**0*(*s*)*N**p*(*s*)|2 " Re(s)>0

Let *s=re*jq and note that *N**z*(*z**i*)=0 and *N**p*(*p**j*)=0. Then the bound can be small if

q|*N**z*(*s*)| and |*N**p*(*s*)| are both small for some
*s*. That is, |*N**z*(*s*)|»0 (i.e., *s* is close to a right-half plane zero of
*P*) and |*N**p*(*s*)|»0 (i.e., *s* is close to a right-half
plane pole of *P*).

This is possible if *P(s)* has a right-half plane zero near a right-half
plane pole. (See Example 16.1.)

q*|N**z*(*s*)| and *|M**0*(*s*)| are both small for some s. That is, *|N**z*(*s*)|»0 (i.e., *s* is close to a right-half plane zero
of *P*) and *|M**0*(*s*)|»0 (i.e., |*P*(*j**w*)| is large around w*=|s|=r* ).

This is possible if |*P*(*j**w*)| is large around w*=r* where *r* is the modulus of a right-half plane zero of *P *(See Example16.2)