By Bruce A. Francis
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Extra info for A Course in H_infinity Control Theory
In general an operator may not have any eigenvalues! For the remainder of this section let F~ RL~. The self-adjoint operator I'~FF maps Hz to itself and its rank is finite by Theorem 1, This property guarantees that it does in fact have eigenvalues. We state without proof the following fact, Theorem 2. 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.
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. 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.
It's easy to prove that its norm equals 1. In the same way we have L2 = H~- @H2. The orthogonal projection from L 2 onto H~- will be denoted 1~1 and from L 2 onto H 2 by I-I2 . Example 3. Let F ~ L~ and define the function AF from L 2 to L 2 via AFg :=Fg. Thus the action of AF is multiplication by F. Obviously Ap is linear. 2 says that IIAF[[= [IFII~, so A F is bounded. This operator is called a Laurent operator and F is called its symbol; so A/: is the Laurent operator with symbol F. A related operator is AF I H2, the restriction of AF to H 2, which maps tt 2 to L 2.