• Note that we can write
• A+BR-1D*C= A+B(g2I-D*D)-1D*C= A+B(I-DD1)-1DC1
• with D=D*/g, D1 =D/g, and C1
=C/g. Then ||D||<1 and
• ||C1(sI-A)-1B+D1||¥ = g--1 ||G||¥ <1.
• Hence by small gain
theorem, A+B(I-DD1)-1DC1 is stable for all D with ||D||<1. Thus A+BR-1D*C is stable for all g- such that ||G||¥ <g-.
• (iii) Þ (iv) follows from
the fact that the ARE
•X(A+BR-1D*C)+(A+BR-1D*C)*X+XBR-1B*X+C*(I+DR-1D*)C=0
• can be regarded as a
Lyapunov equation with
• A1 := A+BR-1D*C,
Q := XBR-1B*X+C*(I+DR-1D*)C
• Hence X ³ 0 since A1 is stable and Q³0.
• (v) Þ (i): Assume D = 0 for simplicity. Then
there is an X³0
• XA+A*X+XBB*X/g2+CC=0
• and A+BB*X/g2 has no jw-axis eigenvalue.