Linear Algebra 2

Linear Algebra 2 - G s = 1 s 1 s 2 s 3 s 1 s 2 1 s 2 3 s 2...

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ECE 531 LINEAR SYSTEM THEORY SPRING 2002 FINAL EXAMINATION May 15, 2002 Name: This is a closed book, closed notes exam. Please show all work on the attached pages (use the backs of the pages if needed) and indicate your ﬁnal answer clearly. Each problem is weighted equally. 1. Consider ˙ x = Ax + Bu with A 4 × 4 and B 4 × m . If A = 2 1 0 0 0 2 0 0 0 0 2 1 0 0 0 2 what is the minimum value of m for which it is possible to choose B to make the system controllable? Give an example of such a B for that value of m .

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2. If ˙ x = " 2 3 1 1 # x + " 1 1 # u y = h 2 4 i x + 3 u and z = " 1 2 2 5 # x write down the equivalent state space system with state z .
3. Find the unobservable subspace of ˙ x = 1 2 2 3 - 1 3 - 1 4 0 x + 1 2 2 1 3 0 u y = h 1 0 0 i x

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4. Write down an observable realization for

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Unformatted text preview: G ( s ) = 1 ( s + 1)( s + 2) s + 3 ( s + 1)( s + 2) 1 s + 2 3 s + 2 ( s + 3)( s + 4) 5. What is the transfer function for ˙ x = "-2 1 1-2 # x + " 1 2 1 0 # u y = h 3 1 i x 6. Given ˙ x = " 3-2 1 # x y = h 1 1 i x ﬁnd (scalar) expressions for y ( t ) when (a) x (0) = " 1 1 # (b) x (0) = " 2 1 # (c) x (0) = " 3 2 # 7. What is e At if A = 2 1 0 0 0 2 0 0 0 0 3 1 0 0 0 3 8. Find K so that the eigenvalues of A-BK are λ =-1 ,-2 ,-3 if A = 1 1-1-2-3 B = 1...
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Linear Algebra 2 - G s = 1 s 1 s 2 s 3 s 1 s 2 1 s 2 3 s 2...

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