problem8_sol - EE221A Problem Set 8 Solutions - Fall 2011...

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EE221A Problem Set 8 Solutions - Fall 2011 Problem 1. BIBO Stability. a) First write this LTI system in state space form, ˙ x = Ax + Bu = " - ( β + f C ) V C β V C β V H - ( β + f H ) V H # x + " f C V C 0 0 f H V H # u, = ± - 0 . 3 0 . 2 0 . 2 - 0 . 3 ² x + ± 0 . 1 0 0 0 . 1 ² u y = Cx = ± 1 0 0 1 ² x where x := ( T C ,T H ) T ,u := ( T C i ,T H i ) T . This has two distinct eigenvalues (so we know it can be diagonalized) λ 1 = - 0 . 5 with eigenvector e 1 = (1 , - 1) and λ 2 = - 0 . 1 with eigenvector e 2 = (1 , 1) . Let T - 1 = ³ e 1 e 2 ´ , so T = 1 2 ± 1 - 1 1 1 ² and the modal form is z = ˜ Az + ˜ Bu, y = ˜ Cz, where ˜ A = TAT - 1 = ± - 0 . 5 0 0 - 0 . 1 ² , ˜ B = TB = ± 0 . 05 - 0 . 05 0 . 05 0 . 05 ² , ˜ C = CT - 1 = ± 1 1 - 1 1 ² . b) y ( t ) = ˜ Cz ( t ) = ˜ Ce ˜ At z (0) = ˜ Ce ˜ At Tx 0 = 1 2 ± 1 1 - 1 1 ²± e - 0 . 5 t 0 0 e - 0 . 1 t ²± 1 - 1 1 1 ²± x 0 , 1 x 0 , 2 ² = 1 2 ± 1 1 - 1 1 ²± e - 0 . 5 t - e - 0 . 5 t e - 0 . 1 t e - 0 . 1 t ²± x 0 , 1 x 0 , 2 ² = 1 2 ± e - 0 . 5 t + e - 0 . 1 t - e - 0 . 5 t + e - 0 . 1 t - e - 0 . 5 t + e - 0 . 1 t e - 0 . 5 t + e - 0 . 1 t ²± x 0 , 1 x 0 , 2 ² = y 1 ( t ) = 1 2 e - 0 . 5 t ( x 0 , 1 - x 0 , 2
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problem8_sol - EE221A Problem Set 8 Solutions - Fall 2011...

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