By Bruce A. Francis

ISBN-10: 0387170693

ISBN-13: 9780387170695

ISBN-10: 3540170693

ISBN-13: 9783540170693

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**Example text**

The Toeplitz operator with symbol F, denoted OF, maps H 2 to H z and is defined as follows: for each g in It 2, ®Fg equals the orthogonal projection of Fg onto H2. Thus OF = FIzAF IH2. The relevant commutative diagram is Figure 2. As a concrete example consider the scalar-valued function F (s) = 1~(s-l) in RL=. For g in H2 we have Fg=gl +g2 gl~H~-, g2~H2 g 1(s) =g (1)/(s-I) g2(s) = [g (s)-g (1)]/(s-I). Thus OF maps g to g 2. Example 6. For F in L= the Hankel operator with symbol F, denoted FF, maps H2 to H~ and is defined as FF := HIAF IH2.

Exercise 1. In Figure 4 suppose G (s) = - . Consider a controller of the form s (s-l) K= -Q 1-GQ where Q is real-rational. Find necessary and sufficient conditions on Q in order that K 41 Ch. 4 stabilize G. 5 Closed-Loop Transfer Matrices Now we return to the standard set-up of Figure 1, Chapter 3. 1 gives every stabilizing K as a transformation of a free parameter Q in RH**. 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.

The following equations are not hard to derive: 11~11= sup {llqbxll : Ilxll< 1 } = sup {IIq~xll : Ilxtl= 1 }. Such a bounded linear function is called an operator. Example 1. The Fourier transform, denoted Y, is an operator from L2(-o%~ ) to L 2. 1 says that its norm equals 1. Example 2. 3 the direct sum L2 ( - ~ , ¢~) = L2 (--0%O] (~L2 [0,~). Each function f in L2(-oo, o~) has a unique decomposition f = f l + f 2 with f i e L2(-oo,O] and f2~ L2[O, oo): 48 Ch. 5 f l ( t ) = f ( t ) , f 2 ( t ) = 0 , t<0 f l ( t ) = 0 , f2(t)=f (t), t > 0 .

### A Course in H_infinity Control Theory by Bruce A. Francis

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