## A Course in H∞ Control Theory by Prof. Bruce A. Francis (eds.) PDF

By Prof. Bruce A. Francis (eds.)

ISBN-10: 3540170693

ISBN-13: 9783540170693

ISBN-10: 3540472002

ISBN-13: 9783540472001

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**Additional resources for A Course in H∞ Control Theory**

**Sample text**

By controllability and observability Wc is surjective and Wo is injective. Hence tPotP¢ has rank n, so FF does too. [] We need another definition. Let (I) be an operator from X to X, a Hilbert space. A complex number X is an eigenvalue of q) if there is a nonzero x in X satisfying 56 Ch. 5 CI~x= ~ . Then x is an eigenvector corresponding to %. 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.

1 Hankel Operators Let X and Y be Hilbert spaces and let qb be a linear function from X to Y. The function ~ is bounded if there exists a real number a such that II@xII

Theorem 1. If F ~ RL,~, then F F has finite rank. Proof. e. F2~ RH~. Since FF=FF1, we might as well assume at the start that F is strictly proper and analytic in Re s <0. Introduce a minimal realization: F(s)=[a, B, C, 0]. The operator FF and its time-domain analog have equal ranks. As in Example 7 the latter operator equals WoWc. By controllability and observability Wc is surjective and Wo is injective. Hence tPotP¢ has rank n, so FF does too. [] We need another definition. Let (I) be an operator from X to X, a Hilbert space.

### A Course in H∞ Control Theory by Prof. Bruce A. Francis (eds.)

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