By Bruce A. Francis
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Extra info for A Course in H_infinity Control Theory
Let K=UV -1 be a right-coprime factorizafion. From (1) and defining D :=~IV-NU we have The two matrices on the left in (6) have inverses in RH~, the second by Lemma 1. Hence D-1 e RH~. Define Q :=-(XU-I'V)D-I, so that (6) becomes [_~ ~ ] IN U] =[10 -QDD] . (7) Pre-mulfiply (7) by [::] and use (1) to get Therefore (X -NQ )DJ " Substitute this into K= UV-I to get (2). e. Ge RH~. Then in (1) we may take N=~' =G X= r--1 y=0, in which case the formulas in the theorem become simply Ch. 4 39 X = - Q (I-GQ) -1 =-(I-QG)qQ.
X = A FX + Bv u =Fx +v y = CFX + D v . Evidently from these equations the transfer matrix from v to u is m ( s ) := [AF, B, F, I] (6a) and that from v to y is N (s) :=[AF, B, CF,D ] . e. G =NM -1 . Similarly, by choosing a real matrix H so that AH:=A +HC is stable and defining BH :=B +HD /¢/(s) := [AH, H, C, I] (6c) N(s) := [AH, BH, C, D ], (6d) we get G =1~1-1N. ) Thus we've obtained four matrices in RH~, satisfying (2). Formulas for the other four matrices to satisfy (3) are as follows: X (s) :=[AF, -H, CF, I] (7a) 24 Ch.
2 Stability This section provides a test for when a proper real-rational K stabilizes G. Introduce left- and right-coprime factorizations G = N M -1 -- ]IT'/-1]Q (la) K = UV-1 = ~ - 1 ~f. (lb) Theorem 1. The following are equivalent statements about K: (i) (ii) K stabilizes G, [0 I ] N V • RHo,,, 26 Ch. 4 -1 (iii) b[o i] e f' RH~ . The idea underlying the equivalence of (i) and (ii) is simply that the determinant of the matrix in (ii) is the least common denominator (in RH~) of all the transfer functions from w, v 1 , v2 to z, u, y; hence the determinant must be invertible for all these transfer functions to belong to RH~, and conversely.