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problem8_sol

# 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 x + 0 . 1 0 0 0 . 1 u y = Cx = 1 0 0 1 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 = e 1 e 2 , so T = 1 2 1 - 1 1 1 and the modal form is z = ˜ Az + ˜ Bu, y = ˜ Cz, where ˜ A = TAT - 1 = - 0 . 5 0 0 - 0 . 1 , ˜ B = TB = 0 . 05 - 0 . 05 0 . 05 0 . 05 , ˜ C = CT - 1 = 1 1 - 1 1 . b) y ( t ) = ˜ Cz ( t ) = ˜ Ce ˜ At z (0) = ˜ Ce ˜ At Tx 0 = 1 2 1 1 - 1 1 e - 0 . 5 t 0 0 e - 0 . 1 t 1 - 1 1 1 x 0 , 1 x 0 , 2 = 1 2 1 1 - 1 1 e - 0 . 5 t - e - 0 . 5 t e - 0 . 1 t e - 0 . 1 t x 0 , 1 x 0 , 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 x 0 , 1 x 0 , 2 = y 1 ( t ) = 1 2 e - 0 . 5 t ( x 0 , 1 - x 0 , 2 ) + 1 2 e - 0 . 1 t ( x 0 , 1 + x 0 , 2 ) y 2 ( t ) = 1 2 e - 0 . 5 t ( x 0 , 2 - x 0 , 1 ) + 1 2 e - 0 . 1 t ( x 0 , 1 + x 0 , 2 ) c) Since all the eigenvalues are in the open left half plane, the system is (internally) exponentially stable, and since we have a minimal realization ( ( A, B ) completely controllable and ( A, C ) completely observable; clear by

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

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