# A cource in H Control Theory by Bruce A. Francis

By Bruce A. Francis

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E cighl-oop,ime, lhen the mal,ix [~] can he filled out to yield a 'qua,e matcix which is invertible in RHoo. Lemma 2. Let M and N be RHoo-matrices Then M and N are right-coprime with equal number iff there exist matrices that Proof. (If) Define where a question mark denotes an irrelevant block. Then of columns. U and V in RHoo such Ch. 4 32 so M and N are right-coprime. -coprimeness), is left-invertible with F 1-1 G Defining [F ~[{ ~ ] , we get Thus the definition gives the desired result, that M [N U as per Lemma 1.

4 Parametrization This section contains a parametrization simplify notation of all K's slightly, in this section the subscripts which stabilize G 22. To 22 on G 22 are dropped. The relevant block diagram is Figure 1. Bring in a doubly-coprime G = NM-1 factorization of G , = M-1jy (1) ('Ii. 1. 1. -, Diagram for controller parametrization eh. 4 38 and coprime factorizations (not necessarily doubly-coprime) of K , uv-1= V-I U . 1; the proof is omitted. Lemma 1. The following are equivalent statements r about K : K stabilizes G , (i) (il) [~~ E RH", , lr _U]-1 [-N M (iii) E RH 00' The main result of this chapter is the following.

1 says that its norm equals 1. Example 2. 3 the direct sum Each function f in Lz(-oo,oo) f lELz(-oo,O] and f zELz[O,oo): has a unique decomposition f = f 1 +- f z with Ch. 5 49 f l(t) = f (t), fl(t)=O, The function f nal projection f z(t) = 0, fz(t)=f(t), --+ f 1 t>O. from Lz(-oo,oo) of Lz(-oo,oo) t <0 to Lz(-oo,O] is an operator, onto L2(-oo,0]. the orthogo- It's easy to prove that its norm equals 1. In the same way we have The orthogonal projection from Lz onto Hi will be denoted ITI and from Lz onto Hz by ITz.