## A cource in H Control Theory by Bruce A. Francis

By Bruce A. Francis

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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)). problem CHAPTER 4 STABILITY THEORY In this chapter model-matching matrix it is shown how the standard problem. The procedure problem can be reduced to the is to parametrize, via a parameter Q in RHoo, all K 's which stabilize G .

The matrix G is Also, G is assumed to be stabiIG. In terms of the Q the transfer matrix from w to z equals T1-T2QT3• Such a func- '8 which stabilize tion of Q is called affine. e. solve the model-matching Q into (2). 3. Then obtain a controller K by substituting and References The material of this chapter is based on Doyle (1984). Earlier relevant references are Chang and Pearson (1978) and Pernebo (1981); a more general treat- ment is given in Nett (1985). As a general (1985a). reference for the material The idea of doing coprime factorization of this chapter see Vidyasagar over RHov is due to Vidyasagar (1972), but the idea was first fully exploited by Desoer et at.

Lemma 3. 1. Proof of Theorem 2. 1, that the six transfer from w, v from v v l' l' 2 V 2 to ~, 'f/ belong to RHoo' to ~, 'f/ do. 2 and write the corresponding equations: (5) V'f/=[O I]N(. By left-coprimeness (6) there exist matrices Rand T in RHoo such that (7) Po,t-multiply (7) by [~]w to get (8) Ch. 4 36 Now subtract (8) from (5), rearrange, and define (9) (10) to get (11) Also, rearrange (6) and define (12) to get + . (13) The block diagram corresponding to (11) and (13) is Figure 1. By Lemma 3 and Vr/ = l' v 2 ~1 v2 K stabilizes G 22 we know that the transfer matrices in Figure 1 the fact that from v [0 I]N to ~1' r] v 2 belong to RHoo.