hw2 - J.M. Gonalves c Winter 2004 CDS 213 - Robust Control...

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J.M. Gon¸ calves Winter 2004 CDS 213 - Robust Control Homework # 2 Date Given: February 5th, 2004 Date Due: February 19th, 2004 P1. Prove corollary 6.6 in DP p. 204. Also, prove that H is in the domain of the Riccati operator if and only if ( A, B ) is stabilizable and ( C, A ) has no unobservable modes on the imaginary axis. Also prove that Ker( X ) = 0 if and only if ( C, A ) has no stable unobservable modes (if you get stuck, and only after thinking about it for some reasonable time, you are allowed to look into theorem 12.4 in Zhou). P2. Do question 6.7, DP, p. 214. P3. Let A R m × m , B R n × n and C R m × n be given and consider the Sylvester Equation AX + XB = C for an unknown matrix X R m × n . Let M = ± B 0 C - A ² , N = ± B 0 0 - A ² . 1. Let the columns of ± U V ² C ( n + m ) × n be the eigenvectors of M associated with the eigenvalues of B and suppose U is nonsingular. Show that X = V U - 1 solves the Sylvester Equation. Every solution of the Sylvester Equation can be written
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This note was uploaded on 01/04/2012 for the course APH 183b taught by Professor List during the Fall '09 term at Caltech.

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hw2 - J.M. Gonalves c Winter 2004 CDS 213 - Robust Control...

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