First nd t si a s 1 1 0s 2 1 1 s2 1 si

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Unformatted text preview: e−2t2 � �� e−t1 −t1 − e−2t1 e 0 e−2t1 � e−t1 −t2 0 = −t1 −t2 − e−t1 +2t2 + e−t1 +2t2 − e−2t1 −2t2 ] e−2t1 −2t2 [e � � e−(t1 +t2 ) 0 = e−(t1 +t2 ) − e−2(t1 +t2 ) e−2(t1 +t2 ) = �(t1 + t2 ) � Problem 3 1. (a) The characteristic equation for the matrix A is: det(�I − A) = 0 �� �2 + 4� + 4 = 0 So the system has repeated eigenvalues at � = −2. Since these eigenvalues lie in the left-half plane, the system is stable. (b) � s + 1 −1 sI − A = 1 s+3 ��1 � 1 1 2+ 1 s+3 � (sI − A)−1 = = (s+2) −1 s+2 (s + 2)2 −1 s + 1 (s+2)2 � � (1 + t)e−2t te−2t � �(t) = −2t −te (1 − t)e−2t � 3 1 (s+2)2 1 −1 + s+2 (s+2)2 ⎤ (c) (1 + t)e−2t te−2t � (t) = �(t)� (0) = x x −te−2t (1 − t)e−2t � �� � � −2t � 1 e = −e−2t −1 2. First find �(t): � sI − A = s + 1 −1 0s + 2 � � � �1 1 s+2 1 � (sI − A) = = s+1 0 s+1 0 (s + 1)(s + 2) � −t −t � e e − e...
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This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

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