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Unformatted text preview: MEEN 651 Control System Design Homework 3: Transfer Functions, State Space, Stability, and TimeDomain 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 statespace 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
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