lecture_19

# lecture_19 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 19 1 Reading: Nise: Chapter 4. 1 System Poles and Zeros Consider a system with transfer function N ( s ) H ( s ) = . D ( s ) If we factor the numerator and denominator polynomials and write H ( s ) = K ( s z 1 )( s z 2 ) . . . ( s z m ) ( s p 1 )( s p 2 ) . . . ( s p n ) where p 1 , p 2 , . . . , p n are the roots of the characteristic polynomial D ( s ), and are known as the system poles , z 1 , z 2 , . . . , x m are the roots of the numerator polynomial N ( s ), and are known as the system zeros . Note that because the coeﬃcients of N ( s ) and D ( s ) are real (they come from the modeling parameters), the system poles and zeros must be either (a) purely real , or (b) appear as complex conjugates and in general we write p i , or z i = σ i + i . Example 1 Find the poles and zeros of the system 5 s 2 + 10 s G ( s ) = s 3 + 5 s 2 + 11 s + 5 5 s ( s + 2) = ( s + 3)( s 2 + 2 s + 5) 5 s ( s + 2) = ( s + 3)( s + (1 + j 2))( s + (1 j 2)) 1 copyright c D.Rowell 2008 19–1
s - p l a n e Â { s } Á { s } j w s o o x x x - 3 - 2 - 1 - j 2 j 2 s = s + j w so that we have (a) a pair of real zeros at s = 0 , 2 and (b) three poles at s = 3 , 1 + j 2, and s = 1 j 2. The system poles and zeros completely characterize the transfer function (and there- fore the system itself) except for an overall gain constant K : m ( s z i ) G ( s ) = K i =1 n i =1 ( s p i ) 1.1 The Pole-Zero Plot The values of a system’s poles and zeros are often shown graphically on the complex s -plane in a pole-zero plot. For example, the poles and zeros of the previous example are drawn: where the pole positions are denoted by an

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