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By Bruce A. Francis

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Proper K the robust stability For a real-rational in RHoo. plants to consist of all strictly the same 1. 2a from v l' v 2 to u, y all belong to RHoo. We saw in Chapter 1 that robust stability is guaranteed by a small gain con- dition. Lemma nominal 1. 3 21 accomplished by G := [~ ~]. 2a); in Figure 1 K stabilizes G and makes the transfer matrix from w to z have Hoo-norm :S; 1. Notes and References The standard For treatments Pearson problem as posed in this chapter of the tracking (1984), and for robust is based on Doyle (1984).

On the left in (6) have inverses in RHoc,' 1. Hence D -lERHo". (6) the second by Lemma Define Q := -(XU-YV)D-l, so that (6) becomes [~~][~ Pre-multiply ~]~ [:'~D] (7) by [MNX Y] and use (1) to get u] [M Y] [MNV=NXOD' [I -QD] (7) Ch. 4 40 Therefore [u]V ~~ [(Y(X -MQ)D] -NQ)D . e. G ERHoo' Then in (1) we may take X=M=I, y=o, X=M=I Y=O, in which case the formulas in the theorem become simply K =-Q t 1 (1 -GQ = -(I -QG tQ 1 There is an interpretation . of Q in this case: -Q equals the transfer matrix from v 2 to u in Figure 1 (check this).

1. Theorem parametrized The of all (proper real-rational) = -MQ )(X G is (XQNt1(y t (2) -QM) (3) -NQ 1 Let's first prove equality (3). Let Q ERHoo' [oI K 's stabilizing by the formulas K = (Y Proof. QM)] [M Y -MQ] __ [ Equating -N M N X -NQ -- 1 the (1,2)-blocks on each side in (4) gives (X -QN)( Y -MQ ) = Or -QM )(X -NQ ) , which is equivalent to (3). 4 39 Next, we show that if K is given by (2), it stabilizes U := Y-MQ, V:= - ~ Define X --NQ .... U := Y -QM, G. 2). Also from (5) [~~r E RHoo So from Lemma 1 K stabilizes G.

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