hw9s - MEEN 651 Control System Design Homework 9...

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MEEN 651 Control System Design Homework 9: Controllability/Observability Solutions Assigned: Tuesday, 28 Oct. 2008 Due: Tuesday, 4 Nov. 2008, 5:00 pm Problem 1) Problem 7.5
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Problem 2) Problem 7.6
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Problem 3) Problem 7.14 (by hand, do not use Matlab)
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Problem 4) Problem 7.26
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Problem 5) We use the notation A > 0 to denote a positive definite matrix (i.e. symmetric matrix with all positive eigenvalues), and A 0 to denote a positive semi-definite matrix (i.e. symmetric with all eigenvalues greater than or equal to zero). Let A and B be two n x n real matrices which are positive definite. Show that A B if and only if B -1 A -1 . Solution) 0 A and 0 B assume 1 1 RB A exists 0 Ax x T and 0 Bx x T Let’s assume B A 0 B A 0 ) ( x B A x T 0 ) ( 1 x B A I A x T (1) 0 ) ( 1 x I AB x T (2) 0 1 Bx xAA Ax x T (3) Adding (2) and (3), 0 1 1 + Bx x Bx AB x Bx AA x Ax x T T T T 0 ) ( 1 1 + Bx A B A x Bx x Ax x T T T (4) (First two terms Bx x Ax x T T are always true)
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hw9s - MEEN 651 Control System Design Homework 9...

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