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e483h4_07

# e483h4_07 - Simon Fraser University School of Engineering...

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Simon Fraser University School of Engineering Science ENSC 483 - Modern Control Systems Spring 2007–Assignment 4 1. Find the modal matrix and the Jordan Canonical Form of the following matrices: A = - 1 0 - 1 1 1 3 0 0 1 0 0 0 0 0 2 1 2 - 1 - 1 - 6 0 - 2 0 - 1 2 1 3 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 - 1 - 1 0 1 2 4 1 B = - 1 - 2 3 0 - 2 3 0 2 3 0 0 2 - 4 0 1 - 2 1 - 1 - 4 - 1 0 0 - 1 0 0 0 0 0 0 0 0 5 - 7 - 1 2 - 4 1 - 2 - 7 - 1 0 - 1 1 0 - 1 0 0 0 1 0 0 3 - 4 0 1 - 3 1 - 1 - 4 - 1 0 1 - 1 0 1 - 1 - 1 - 1 - 1 0 0 - 1 1 0 0 0 0 - 1 1 0 0 0 0 0 0 0 0 0 - 1 0 0 4 - 5 0 2 - 3 1 - 2 - 5 - 2 Hint: A has all of its eigenvalues located at λ = 1, whereas eigenvalues of B are all at λ = - 1. 2. Let z be an arbitrary n dimensional vector such that P [ z Az · · · A n - 1 z ] is nonsingular. Use the Cayley-Hamilton Theorem to show that the solution of the linear equation problem P x = A n z is x = [ - a 0 - a 1 · · · - a n - 1 ] T where a 0 , a 1 , · · · , a n - 1 are the coefficients of the

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e483h4_07 - Simon Fraser University School of Engineering...

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