By Bruce A. Francis
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Extra resources for Course in H-infty Control Theory
But by (10) and (12) those from w to vl, v2 belong to RH=. Hence those from w to ~I, rl belong to RH=. Finally, we conclude from (9) that the transfer matrix from w to ~ belongs to RH~.  Exercise 2. Suppose G t t =G 12=G21=G 22. Prove that G is stabilizable. 4 Parametrization This section contains a parametrization of all K's which stabilize G 22. To simplify notation slightly, in this section the subscripts 22 on G22 are dropped. The relevant block diagram is Figure 1. Bring in a doubly-coprime factorization of G, G =NM -1 =,(,/-1~ [ _ ~ ~ 1 [NM Y3 = I , (1) and coprime factorizations (not necessarily doubly-coprime) of K, K=UV_ 1 = ~-1~/.
F2~ RH~. Since FF=FF1, we might as well assume at the start that F is strictly proper and analytic in Re s <0. Introduce a minimal realization: F(s)=[a, B, C, 0]. The operator FF and its time-domain analog have equal ranks. As in Example 7 the latter operator equals WoWc. By controllability and observability Wc is surjective and Wo is injective. Hence tPotP¢ has rank n, so FF does too.  We need another definition. Let (I) be an operator from X to X, a Hilbert space. A complex number X is an eigenvalue of q) if there is a nonzero x in X satisfying 56 Ch.
1 that K stabilizes G.  Exercise 1. Prove equivalence of (i) and (iii) in Theorem 1. Hereafter, G will be assumed to be stabilizable. Intuitively, this implies that G and G22 share the same unstable poles (counting multiplicities), so to stabilize G it is enough to stabilize G22. 1 the four transfer matrices from v i and v 2 to u and y belong to RH~, T h e o r e m 2. K stabilizes G i f f K stabilizes G22. The necessity part of the theorem follows from the definitions. 1. L e m m a 3. 2 from v 1, v2 to ~, rl belong to RH,~.
Course in H-infty Control Theory by Bruce A. Francis