By Bruce A. Francis

ISBN-10: 0387170693

ISBN-13: 9780387170695

ISBN-10: 3540170693

ISBN-13: 9783540170693

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Similarly, a left-coprime factorization has the form G=~I-I~[ where/q and/~ are left-coprime. Of course implicit in these definitions is the requirement that M and ~ / b e square and nonsingular. 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.

The eigenvalues of F ) F F are real and nonnegative and the largest of them equals llFFFrll. This theorem together with (4) says that IIFFII equals the square root of the largest eigenvalue of F~F F. So we could compute IIFFIIif we could compute the eigenvalues of F~FF. How to do this latter computation is the last topic of this section. We continue with the notation introduced in Example 7 and the proof of Theorem 1. The self-adjoint operators ~c~P~* and ~oWo map C n to itself, Thus they have matrix representations with respect to the standard basis on C n.

Hence (2) ~=XM~ + YN~. But from Figure 2 M~= [:] [zt , Ng= y-v2 Substitution into (2) gives " Ch. 1. 2. With internal signals 28 Ch. 4 Hence the three transfer matrices from w, v 1, v2 to ~ belong to RH~. A similar argument works for the remaining three transfer matrices to rl. [] Proof of Theorem I. We shaU prove the equivalence of (i) and (ii). e. its inverse exists as a rational matrix. We have [0 1]N V = ,0 i] ioOl 0 I (3) G21 G22 Now Io:] is nonsingular because both M and V are. Also, since G22 is strictly proper, we have that 0 I G21 G22 1 is nonsingular when evaluated at s =~,: its determinant equals 1 at s =~.

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A Course in H_infinity Control Theory by Bruce A. Francis

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