By Bruce A. Francis

ISBN-10: 0387170693

ISBN-13: 9780387170695

ISBN-10: 3540170693

ISBN-13: 9783540170693

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**Extra resources for A Course in H_infinity Control Theory**

**Example text**

The results of this chapter can be summarized as follows. The matrix G is assumed to be proper, with G 22 strictly proper. Also, G is assumed to be stabilizable. The formula (2) parametrizes all K's which stabilize G. In terms o f the parameter Q the transfer matrix from w to z equals T1-T2QT 3. Such a function of Q is called affine. In view of these results the standard problem can be solved as follows: First, find a Q in RH~, to minimize IIT1-T2QT311,o, i,e. solve the model-matching problem of Chapter 3.

46 Ch. 4 Notes 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 treatment is given in Nett (1985). As a general reference for the material of this chapter see Vidyasagar (1985a). The idea of doing coprime factorization over RHoo is due to Vidyasagar (1972), but the idea was first fully exploited by Desoer et al. (1980). The state-space formulas in Section 1 are from Nett et al. (1984).

It's desired to find a K which stabilizes G and makes the dc gain from w to z equal to zero (asymptotic rejection of steps). Give an example of a stabilizable G for which no such K exists. Find necessary and sufficient conditions (in terms of Ti, i =1-3) for such K to exist. The results of this chapter can be summarized as follows. The matrix G is assumed to be proper, with G 22 strictly proper. Also, G is assumed to be stabilizable. The formula (2) parametrizes all K's which stabilize G. In terms o f the parameter Q the transfer matrix from w to z equals T1-T2QT 3.

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