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e483h4s_07

# e483h4s_07 - 1 ENSC 483 HW#4 Solutions by M Saif Simon...

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ENSC 483 HW#4 Solutions by M. Saif 1 Simon Fraser University School of Engineering Science ENSC 483 - Modern Control Systems 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. Solution: Consider matrix A : λ 1 = λ 2 = ··· = λ 7 = 1 . ρ ( A - I ) = 3 = γ ( A - I ) = 4 dim N 1 = 4 v 1 v 4 v 6 v 7 ρ ( A - I ) 2 = 1 = γ ( B + I ) 2 = 6 dim N 2 = 6 v 2 v 5 ρ ( B - I ) 3 = 0 = γ ( B - I ) 3 = 7 dim N 3 = 7 v 3 Based on the above information the largest Jordan Block is of 3 × 3 size, and there are four JBs in all. So the size of the blocks are 3 × 3 , 2 × 2 , 1 × 1 , and 1 × 1 . Now to ﬁnd a chain of generalized eigenvectors of length three, pick v 3 such that ( A - I ) 3 v 3 = 0 , and ( A - I ) 2 v 3 6 = 0 . One such vector is v 3 = 0 1 0 0 0 0 0 ; then v 2 = ( A - I ) v 3 = 0 0 1 0 0 0 - 1 ; v 1 = ( A - I ) v 2 = - 1 0 1 - 1 0 0 0

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ENSC 483 HW#4 Solutions by M. Saif 2 Now pick a chain of generalized eigenvectors of length 2. So pick a v 5 ∈ N 2 ,v 5 / ∈ N 1 and independent of v 2 such that ( A - I ) 2 v 5 = 0 but ( A - I ) v 5 6 = 0 . v 5 = 0 0 0 1 0 0 0 ; then v 4 = ( A - I ) v 5 = 1 0 - 1 1 0 0 1 Now ﬁnd two more linearly independent (from one another as well as v 1 and v 4 eigenvectors in N 1 . These are v 7 = 0 0 - 1 - 2 1 0 0 ; and v 8 = 1 3 1 0 0 1 0 Now M = £ v 1 v 2 v 3 v 4 v 5 v 6 v 7 / , and ˆ A = M - 1 AM = 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
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e483h4s_07 - 1 ENSC 483 HW#4 Solutions by M Saif Simon...

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