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Unformatted text preview: EE 141 Fall 2007 Homework 5 1. Calculate the characteristic polynomial of the matrix A = 1 1 a b a b For what values of a is the matrix A stable? (10 points) Solution:  I A  =  1  1 a  b a  b = 2 (  b ) 2 a (  b ) a ( (  b ) a ) = 2 (  b ) 2 2 a (  b ) + a 2 = ( (  b ) a ) 2 = f ( ) The real part of the eigenvalues of A has to be negative to have A stable. We could compute the roots of f ( ), but it is much faster to use RouthHurwitz (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: (  b ) a = 2 b a = 0 b > Unstable regardless the value of a b < 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 = X n =0 t n n sinh( t ) = e t e t 2 =...
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This note was uploaded on 03/18/2008 for the course EE 141 taught by Professor Balakrishnan during the Fall '07 term at UCLA.
 Fall '07
 Balakrishnan

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