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Linear Algebra 1

Linear Algebra 1 - is P = C z C-1 x(where the C matrices...

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ECE 531 LINEAR SYSTEM THEORY SPRING 2001 FINAL EXAMINATION May 9, 2001 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. What is the transfer function for ˙ x ( t ) = " - 3 1 - 5 2 # x ( t ) + " 1 1 # u ( t ) y ( t ) = " 2 1 0 2 # x ( t )

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2. Write down closed form expressions for e At and A k given A = " - 2 - 1 2 - 5 #
3. If A = " λ 1 0 λ # what is e At ?

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4. Find a realization for the input-output system for which Y ( s ) = 2 s + 2 1 s + 3 s + 6 s + 3 0 U ( s )
5. Given ˙ x ( t ) = " 2 - 1 0 1 # x ( t ) + " 1 1 # u ( t ) y ( t ) = h 1 0 i x ( t ) (a) find another state space system with the same transfer function but which is not controllable; (b) find another state space system with the same transfer function but which is not observable.

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6. Let ˙ x = Ax + bu y = cx and ˙ z = Fz + gu y = hz be two single-input, single-output systems, both minimal, for the same transfer func- tion. Show that the coordinate transformation P taking the x system to the z system ( z = Px ) is

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Unformatted text preview: ) is P = C z C-1 x (where the C matrices are the controllability matrices for the respective systems). 7. Given ˙ x ( t ) = -2 1-3 2-2 2-1 x ( t ) + 1 1 u ( t ) y ( t ) = h 1 1 1 i x ( t ) for which of the following initial states is it possible to ﬁnd u ( t ), 0 ≤ t ≤ 4, so that x (4) = 0? (a) x (0) = [1 2 1] T (b) x (0) = [1 1 0] T (c) x (0) = [1 1 1] T (d) x (0) = [1 3 2] T 8. For ˙ x ( t ) = -1-2-3 1 1 x ( t ) + 1 u ( t ) y ( t ) = h 1 5 6 i x ( t ) (a) ﬁnd K (in terms of p 1 , p 2 and p 3 ) so that the poles of the v to y system achieved by u = v-Kx are at-p 1 ,-p 2 and-p 3 ; (b) ﬁnd K so that the transfer function of the v to y system achieved by u = v-Kx is G ( s ) = 1 s + 1...
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