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To 22 on G 22 are dropped. The relevant block diagram is Figure 1. Bring in a doubly-coprime G = NM-1 factorization of G , = M-1jy (1) ('Ii. 1. 1. -, Diagram for controller parametrization eh. 4 38 and coprime factorizations (not necessarily doubly-coprime) of K , uv-1= V-I U . 1; the proof is omitted. Lemma 1. The following are equivalent statements r about K : K stabilizes G , (i) (il) [~~ E RH", , lr _U]-1 [-N M (iii) E RH 00' The main result of this chapter is the following. 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).

Insert and introduce the factoriza- two new signals ~ to get Figure 2. Lemma 1. The nine transfer belong of (i) and (ii) is simply matrices to RHoo iff the six transfer in Figure matrices 1 from w, in Figure 2 from v l' V 2 to z, w, v V 2 l' u, y to ~, 'Yf belong to RHoo' Proof. (If) This direction follows immediately from the equations which in turn follow from Figure 2. matrices X and Y such that YN = I. Hence ~= XM E + YN ~. (2) 28 Ch. = Substitution [ y -v z 2 ] . into (2) gives Hence the three transfer A similar argument matrices from w , v l' V 2 to f.

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)).

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