q We shall construct a D ÎRH¥ such that D(jw0 )= (1/s1) v1u1* and ||D||¥ £1. Indeed, for such D(s),
q det(I-M (jw0 )D(jw0 ))=det(I- U(jw0 )S(jw0 )V(jw0 ) (1/s1) v1u1* )
q
=1- u1* U(jw0 )S(jw0 )V(jw0 )v1 (1/s1) =0
q and
thus the closed-loop system is either not well-posed (if w0 = ¥) or unstable (if w0 Î R+). There are two different cases:
q (1)
w0 = 0 or ¥ : Then U and V are real matrices.
Choose
q D= (1/s1) v1u1* Î Rqxp
q (2)
0<w0 < ¥ : write u1 and v1in the following form
q
q
q where
u1i , v1j Î R
are chosen so that qi, fj Î [-p,0).