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4 25 (6d) we get G =M-1 N. ) (This can be derived as above by starting with G (s )T Thus we've obtained four matrices in RHoo satisfying (2). Formulas for the other four matrices to satisfy (3) are as follows: x (s):= Y (s ) [AF' -H, := [AF' -H, CF, I] (7a) F , 0] (7b) X(s):= [AH' -BH, Y(s):= [AH' -H,F,O]. The explanation F, I] (7c) (7d) of where these latter four formulas come from is deferred to Sec- tion 4. Exercise 1. Verify that the matrices in (6) and (7) satisfy (3). Example 1. As an illustration of the use of these formulas, consider the scalar-valued example G(s)= s-1 s(s-2) A minimal realization is G (s) = A ~ [~ ~], [A, B, C, D] C = [-1 1], H ~ D [~] = 0.

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). example see Vidyasagar stabilization (1985b) and Wang and see, for example, Kimura (1984). Lemma 1 is due to Doyle and Stein (1981) and Chen and Desoer (1982). There are several other examples of the standard weighted sensitivity problem problem, for example, the (Zames (1981)) and the mixed sensitivity (Verma and Jonckheere (1984), Kwakernaak (1985)).

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. Finally, suppose [( stabilizes Let K in RHoo' G. vVe must show K satisfies (2) for some Q l = UV- be a right-coprime factorization. ] [M u] [I ~¥U J'V] = -N M The two matrices N V 0 D . 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.

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