By Prof. Bruce A. Francis (eds.)
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Additional info for A Course in H∞ Control Theory
The objective in this section is to find the transfer matrix from w to z in terms of Q. In the previous section we dropped the subscripts on G22; now we must restore them. Bring in a doubly-coprime factorization of G22: o2= =N2M~ t =f~If~2 -Y2 M2 Y2 -IV2 M2 N2 X2 =I. (1) Then the formula for K is K = (Y2-M2Q)(X2-N2Q) -1 (2a) (2b) Nowdefine T1 :=Gll +G12M2Y2G21 (3a) T2 := G 12M2 (3b) T 3 :=M2G21. (3c) Theorem 1. The matrices Ti (i =1-3) belong to RH~,. With K given by (2) the transfer matrix from w to z equals T1-T2QT3.
Then M and N are right-coprime iff there exist matrices U and V in RH~ such that Proof. (It') Define where a question mark denotes an irrelevant block. Then IX YI N =I, so M and N are right-coprime. (Only if) Define and bring in matrices G, H, and Fx as per Lemma 1. Since F is left-invertible in RH~ (by right-coprimeness), it follows that Ch. 4 31 is left-invertible in RH~ too. But then it must have the form with F~ 1 ~ RIt~. Defining K:=G[ 0 we get Thus the definition gives the desired result, that is inverfible in R t L .
Note that D22=0 because G2z is strictly proper. It can be proved that stabilizability of G (an assumption from Section 3) implies that (A,B2) is stabilizable and (C2,A) is detectable. Next, find a doubly-coprime factorization of G22 as developed in Section 1. For this choose F and H so that AF :=A +B2F, A H :=A +HC2 are stable. Then the formulas are as follows: M2(s) = [AF, Bz, F, I] Ch. 4 43 N2(s)=[AF, B 2, C 2, 0] itS/2(s) = [AH, H, C2, I] N2(s)=[AH, B2, C2, 0] X2(S)=[AF,-H, C2,I] Y2(s) = [AF,-H, F, 0] X2(s) =[All, -B 2, F, I1 Y2(s)=[A H, -H, F, 0].
A Course in H∞ Control Theory by Prof. Bruce A. Francis (eds.)