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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online