qProof: Assume g = 1. System is stable iff
det(I-M D) has no zero in the closed right-half plane for all D ÎRH¥ and ||D||¥ £1.
• (Ü ) det(I-M D) ¹0 for all D ÎRH¥ and ||D||¥ £1 since
•|l(I-M D) | ³ 1-max| l(M D) | ³ 1- ||M||¥ >0
• (Þ ) Suppose ||M||¥ ³1. There exists a D ÎRH¥ with ||D||¥ £1 such that det(I-M (s)D(s)) has a zero on the imaginary axis, so the
system is unstable. Suppose w Î R+È {¥} is such that s1(M(jw0 )) ³ 1. Let M(jw0 )= U(jw0 )S(jw0 )V*(jw0 ) be a singular value decomposition with
• U(jw0 )=[u1,u2,…,up], V(jw0 )=[v1,v2,…,vp],
•S(jw0 )=diag[s1, s2,…]