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Linear Algebra 3l

# Linear Algebra 3l - -1 1-1-2 1-2-2 6 Given ˙ x =" 3-2...

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ECE 531 LINEAR SYSTEM THEORY SPRING 2004 FINAL EXAMINATION May 10, 2004 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. Write down system equations for ˙ x = " 0 - 2 1 - 3 # x + " 3 2 # u y = h 1 - 1 i x after changing coordinates so that the “ A -matrix” is in Jordan form.

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2. For the system ˙ x = " - 2 1 1 - 2 # x + " 1 2 1 0 # u y = h 1 1 i x (a) is the system controllable? (b) Is it observable? (c) Is it minimal? (d) What is the transfer function?
3. Find a matrix P so that for the systems ˙ x 1 = " 2 3 1 1 # x 1 + " 1 1 # u y = h 2 4 i x 1 + h 3 i u and ˙ x 2 = " 10 - 3 23 - 7 # x 2 + " 3 7 # u y = h 2 0 i x 2 + h 3 i u we have x 2 = Px 1 .

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4. Find the controllable subspace of ˙ x = 1 3 - 1 2 - 1 4 2 3 0 x + 1 0 0 u y = " 2 1 0 1 2 3 # x
5. What is e At if A =

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Unformatted text preview: -1 1-1-2 1-2-2 6. Given ˙ x = " 3-2 10-6 # x y = h 1 1 i x ﬁnd (scalar) expressions for y ( t ) when (a) x (0) = " 1 2 # (b) x (0) = " 2 5 # (c) x (0) = " 3 7 # 7. Write down an controllable realization for G ( s ) = 1 ( s + 1)( s + 2) 3 s + 2 ( s + 3)( s + 4) s + 3 ( s + 1)( s + 2) 1 s + 2 8. Find K so that the eigenvalues of A-BK are λ =-2 ,-3 ,-4 if A = 1 1-4-5-6 B = 1...
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Linear Algebra 3l - -1 1-1-2 1-2-2 6 Given ˙ x =" 3-2...

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