ps4 - EE 482 Reading assignment: Chapter 5. Problem 16: (20...

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EE 482 PROBLEM SET 4 DUE: 7 March 2008 Reading assignment: Chapter 5. Problem 16: (20 points) If E ( s )= b m s m + b m - 1 s m - 1 + ··· + b 0 s n + a n - 1 s n - 1 + + a o . has n distinct poles ( p i ± = p j for all i ± = j ), then E ( s ) can be expressed using the partial fraction expansion E ( s c 1 1 s - p 1 + + c n 1 s - p n where the constants c i are given by c i =( s - p i ) E ( s ) | s = p i . Alternatively, the constants c i can be evaluated using the relationship c i = N ( s ) D ± ( s ) ± ± ± ± s = p i , where N ( s ) is the numerator polynomial N ( s b m s m + b m - 1 s m - 1 + + b 0 and D ± ( s ) is the derivative of the the denominator polynomial D ( s s n + a n - 1 s n - 1 + + a 0 with respect to s . To understand how this second approach for calculating c i works, consider the third-order transfer function E ( s N ( s ) ( s - p 1 )( s - p 2 )( s - p 3 ) = c 1 1 s - p 1 + c 2 1 s - p 2 + c 3 1 s - p 3 . Show that N ( s ) D ± ( s ) ± ± ± ± s = p i = N ( s ) ( s - p j )( s - p k ) ± ± ± ± s = p i where i ± = j , i ± = k and j ± = k . Use the last result to derive c i = N ( s ) D ± ( s ) ± ± ± ± s = p i .
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Problem 17: (30 points) 1. (10 points) Consider the phase-lead compensator with transfer function representation
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ps4 - EE 482 Reading assignment: Chapter 5. Problem 16: (20...

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