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hw3_sol

# hw3_sol - Question 1 We are given the system x(t = Ax(t...

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Question 1 We are given the system ˙ x ( t ) = Ax ( t ) + Bu ( t ) , (1) y ( t ) = Cx ( t ) + Du ( t ) . (2) To relate y (0) to the state x (0), substitute t = 0 into (2) (note that u (0) and all derivatives of u ( t ) evaluate at t = 0 are zero). This yields y (0) = Cx (0). To relate the d k dt k y (0), note that differentiating (2) yields d k dt k y ( t ) = C d k dt k x ( t ) + D d k dt k u ( t ) , d k dt k y (0) = C d k dt k x (0) . (3) From (1) we have that d k dt k x (0) = d k - 1 dt k - 1 d dt x (0) , = A d k - 1 dt k - 1 x (0) , = A 2 d k - 2 dt k - 2 x (0) , . . . = A k x (0) . (4) Combining (3) and (5), we have that d k dt k y ( t ) = CA k x (0) . Vectorizing, we have y (0) ˙ y (0) ¨ y (0) . . . y ( n - 1) (0) = Cx (0) CAx (0) CA 2 x (0) . . . CA n - 1 x (0) = C CA CA 2 . . . CA n - 1 x (0) . Question 2 By substitution we have ¯ O n = ¯ C ¯ C ¯ A ¯ C ¯ A 2 . . . ¯ C ¯ A n - 1 , = CT - 1 CAT - 1 CA 2 T - 1 .

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hw3_sol - Question 1 We are given the system x(t = Ax(t...

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