By Prof. Bruce A. Francis (eds.)

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**Example text**

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). 1 is due to Youla et al. (1976) as modified by Desoer et al. (1980). Finally, see Minto (1985) for a comprehensive treatment of stability theory by state-space methods.

2, AFH 2 c H 2 . The converse is true too: if AFH2CH2, then F ~ H~. ) Example 4. This is the time-domain analog of the previous example. Recall that convolution in the time-domain corresponds to multiplication in the frequency-domain. Suppose F (s) is a matrix-valued function which is analytic in a vertical strip containing the imaginary axis and which belongs to L , . Taking the region of convergence to be this strip, let f (t) denote the inverse bilateral Laplace transform of F(s). Now define the convolution operator Ef from L 2 ( - ~ , ~ ) to L2(---~,'~) via Ch.

We shall require special coprime factorizations, as described in the next lemma. Lemma 1. For each proper real-rational matrix G there exist eight RH~-matrices satisfying the equations G =NM -1 =AT/-iN (2) Equations (2) and (3) together constitute a doubly-coprime factorization of G. It should be apparent that N and M are right-coprime and ~/and M are left-coprime; for example, (3) implies d-Pl N =I, proving right-coprimeness. It's useful to prove Lemma 1 constructively by deriving explicit formulas for the eight matrices.