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20075ee141_1_hw5_solutions

20075ee141_1_hw5_solutions - EE 141 Fall 2007 Homework 5 1...

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EE 141 Fall 2007 Homework 5 1. Calculate the characteristic polynomial of the matrix A = 0 0 1 0 0 0 0 1 a 0 b 0 0 a 0 b For what values of a is the matrix A stable? (10 points) Solution: | λI - A | = λ 0 - 1 0 0 λ 0 - 1 - a 0 λ - b 0 0 - a 0 λ - b = λ 2 ( λ - b ) 2 - λa ( λ - b ) - a ( λ ( λ - b ) - a ) = λ 2 ( λ - b ) 2 - 2 ( λ - b ) + a 2 = ( λ ( λ - b ) - a ) 2 = f ( λ ) The real part of the eigenvalues of A has to be negative to have A stable. We could compute the roots of f ( λ ), but it is much faster to use Routh-Hurwitz (yes, here is useful as well), and see that we just have a seconde order equation. To have roots with negative real part we must have all coefficients greater than 0: λ ( λ - b ) - a = λ 2 - - a = 0 b > 0 Unstable regardless the value of a b < 0 a < 0 to have a stable system 2. Let A be a matrix such that A 2 = Identity. Show that e At = I cosh t + A sinh t regardless of dimension. (20 points) Solution: e t = n =0 t n n sinh( t ) = e t - e - t 2 = n =0 t 2 n +1 (2 n + 1) cosh( t ) = e t + e - t 2 = n =0 t 2 n (2 n ) e At = I + At + t 2 2 A 2 + t 3 3 A 3 + . . . = I + At + t 2 2 I + t 3 3 A + . . .
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