2005_mid_sol

2005_mid_sol - MS&E201 Dynamic Systems Professor Edison Tse...

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page 1 of 2 Professor Edison Tse April 25, 2005 Midterm Solution Problem 1. (5 points each) 1) True 2) False 3) False 4) True Problem 2. (30 points) (a) [15 points] By substituting the explicit expressions for C(t) , G(t) and I(t) into Y(t) , we get the following equation: 1 ) 2 ( ) 1 ( ) 1 ( ) ( + - - - + - = t Y t Y t Y t Y αβ α Let ) ( ) ( 1 t Y t X = and ) 1 ( ) ( 2 - = t Y t X . The system can be transferred to: b t X A t X t X t X t X t X + - = + - - - + = = ) 1 ( 0 1 ) 1 ( ) 1 ( 0 1 ) 1 ( ) ( ) ( ) ( 2 1 2 1 β (b) [15 points] To find the eigenvalues of the above transformed system, we set 0 ) 1 ( ) det( 2 = + + - = - λ I A , and by solving this equation we find that the eigenvalues are complex when 0 4 ) 1 ( 2 2 < - + = , i.e., 2 ) 1 ( 4 + < . Now, since the eigenvalues are complex, for the system to be
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This note was uploaded on 06/16/2010 for the course MS&E 201 taught by Professor Edisontse during the Spring '08 term at Stanford.

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2005_mid_sol - MS&E201 Dynamic Systems Professor Edison Tse...

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