By Prof. Bruce A. Francis (eds.)
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Extra resources for A Course in H∞ Control Theory
Note that D22=0 because G2z is strictly proper. It can be proved that stabilizability of G (an assumption from Section 3) implies that (A,B2) is stabilizable and (C2,A) is detectable. Next, find a doubly-coprime factorization of G22 as developed in Section 1. For this choose F and H so that AF :=A +B2F, A H :=A +HC2 are stable. Then the formulas are as follows: M2(s) = [AF, Bz, F, I] Ch. 4 43 N2(s)=[AF, B 2, C 2, 0] itS/2(s) = [AH, H, C2, I] N2(s)=[AH, B2, C2, 0] X2(S)=[AF,-H, C2,I] Y2(s) = [AF,-H, F, 0] X2(s) =[All, -B 2, F, I1 Y2(s)=[A H, -H, F, 0].
From (1) and defining D :=~IV-NU we have The two matrices on the left in (6) have inverses in RH~, the second by Lemma 1. Hence D-1 e RH~. Define Q :=-(XU-I'V)D-I, so that (6) becomes [_~ ~ ] IN U] =[10 -QDD] . (7) Pre-mulfiply (7) by [::] and use (1) to get Therefore (X -NQ )DJ " Substitute this into K= UV-I to get (2). e. Ge RH~. Then in (1) we may take N=~' =G X= r--1 y=0, in which case the formulas in the theorem become simply Ch. 4 39 X = - Q (I-GQ) -1 =-(I-QG)qQ. There is an interpretation of Q in this case: -Q equals the transfer matrix from v2 to u in Figure 1 (check this).
From (1) we have I'o ;] [io-,O]--, so that X-NQJ= I . - (4) Equating the (1,2)-blocks on each side in (4) gives ( 2 - Q ~ X r - M Q ) = ( f ' - Q ~ t X X - N Q ~, which is equivalent to (3). Next, we show that if K is given by (2), it stabilizes G. 2). Also from (5) So from Lemma 1 K stabilizes G. 38 Ch. 4 Finally, suppose K stabilizes G. ~. Let K=UV -1 be a right-coprime factorizafion. From (1) and defining D :=~IV-NU we have The two matrices on the left in (6) have inverses in RH~, the second by Lemma 1.