sol4.09 - ECE521 Linear Systems Fall 2009 Homework 4: Due...

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Unformatted text preview: ECE521 Linear Systems Fall 2009 Homework 4: Due Never These are some practice problems that may help your preparation for the upcoming midterm exam. Problem 1 : Consider the continuous system: ˙ x ( t ) = parenleftbigg 1 0 2 3 parenrightbigg x ( t ) + parenleftbigg 1 2 parenrightbigg u ( t ) y ( t ) = [ c 1 c 2 ] x ( t ) Under what conditions on [ c 1 c 2 ] is the system observable? The observability matrix is parenleftbigg c 1 c 2 c 1 + 2 c 2 3 c 2 parenrightbigg We need c 2 negationslash = 0 and c 1 c 2 negationslash = c 1 + 2 c 2 3 c 3 which means that c 1 negationslash = c 2 . Problem 2 : Consider the continuous system: ˙ x ( t ) = 0 1 − 1 − 2 0 − 1 0 x ( t ) + 1 1 u ( t ) y ( t ) = [1 1 − 1] x ( t ) • Find bases for the reachable but unobservable subspaces for the system. The reachibility and observability matrices are Q R ( A, B ) = 0 0 0 1 0 0 1 0 0 , Q O ( A, B ) = 1 1 − 1 − 1 1 − 1 − 1 − 1 1 It is seen that the reachable subspace is spanned by the vector [0 1 1] T the nullspace of Q O ( A, B ) is is spanned by the same vector. Thus [0 1 1] T is the bases of reachable but unobservable subspace.subspace....
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sol4.09 - ECE521 Linear Systems Fall 2009 Homework 4: Due...

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