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Unformatted text preview: MEEN 651 Control System Design Homework 3: Transfer Functions, State Space, Stability, and Time-Domain Response Solution Assigned: Tuesday, 16 Sept. 2008 Due: Tuesday, 23 Sept. 2007, 5:00 pm Problem 1) Given a linear time invariant (LTI) state space system described by: Cx y Bu Ax x = + = & with ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = γ β 1 A , ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = 1 β α B , and [ ] 1 = C , a. Derive the transfer function from u to y b. Determine the poles and zeros of the transfer function c. Determine the eigenvalues of the A matrix d. Determine values for α , β , and γ such that the system is stable e. Are there values for α , β , and γ such that there is a pole and zero with the same value? In this case, can you simplify the transfer function? Can you simplify the state-space system? What do you suppose this means about the dynamic system? Solution: a. The transfer function from D B A sI C s U s Y + − = − 1 ) ( ) ( ) ( [ ] ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − = − 1 1 1 ) ( 1 β α γ β s s s G ) )( ( ) ( γ β α − − + = s s s s G b. Zeros of the transfer function is α − = Z Poles of the transfer function are γ β , = P c. The eigenvalues of a lower/upper diagonal matrix are simply the elements on the diagonal. Hence the eigenvalues are...
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This note was uploaded on 09/20/2009 for the course MEEN 651 taught by Professor Staff during the Fall '08 term at Texas A&M.
- Fall '08