MT 604 Soln 2008 - Name Problem#1 2 3 4 ECE 604 LINEAR...

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Name: ________________________________ ECE 604 LINEAR SYSTEMS Midterm Exam Thursday, October 30, 2008 7:00 P.M. - 9:00 P.M. The exam is open book and open notes. You must show your work to receive partial credit. The problem values are: Problem #1 20 Points Problem #2 20 Points Problem #3 15 Points Problem #4 15 Points Problem #5 30 Points Total 100 Points Consistent with the ECE Honor Code, you are asked to read the following voluntary statement carefully and sign it before beginning your work in each exam booklet: I have not given or received unauthorized aid on this exam. ___________________________________ ___________ Signature Problem #1: _______ 2: _______ 3: _______ 4: _______ 5: _______ Total: _______
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Name: ________________________________ Problem 1: (20 points) Consider the linear system: ! x t ( ) = Ax t ( ) ; x 0 ( ) = x 0 (1) where A = 0 0 1 0 ! 1 0 0 1 0 " # $ $ $ % & a) (10 points) What are the eigenvalues of A ? b) Compute the transition matrix for system (1). a) Note that a simple transformation (switching the 2 nd and 3 rd states) results in an upper triangular matrix: P = 1 0 0 0 0 1 0 1 0 ! " # # # $ % & & & A ( PAP ) 1 = 1 0 0 0 0 1 0 1 0 ! " # # # $ % & & & 0 0 1 0 ) 1 0 0 1 0 ! " # # # $ % & & & 1 0 0 0 0 1 0 1 0 ! " # # # $ % & & & = 1 0 0 0 0 1 0 1 0 ! " # # # $ % & & & 0 1 0 0 0 ) 1 0 0 1 ! " # # # $ % & & & = 0 1 0 0 0 1 0 0 ) 1 ! " # # # $ % & & & Thus, the eigenvalues are ! = 0, 0, " 1 { } . b) The transformation used in part a could also be used here, but the computation of e A t using the Laplace transform method is straightforward: e A t = L ! 1 s I ! A ( ) ! 1 { }
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Name: ________________________________ = L ! 1 s 0 ! 1 0 s + 1 0 0 ! 1 s " # $ $ $ % & ! 1 ( ) * * + * * , - * * . * * = L ! 1 1 s 2 s + 1 ( ) s s + 1 ( ) 1 s + 1 0 s 2 0 0 s s s + 1 ( ) " # $ $ $ $ % & ( ) * * + * * , - * * .
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