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lecture_31 - 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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H ( s ) u ( t ) = A s i n ( w t + q ) y ( t ) = A | H ( j w ) | s i n ( w t + q + Ð H ( j w ) ) s s L i n e a r S y s t e m | H ( s ) | j M I | H ( j M ) | i s a " s l i c e " o f t h e | H ( s ) | s u r f a c e a l o n g t h e i m a g i n a r y a x i s o f t h e s - p l a n e . Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 31 1 Reading: Nise: 10.1 Class Handout: Frequency Response and Bode Plots 1 Sinusoidal Frequency Response (continued) In Lecture 30 we saw that the steady-state response of a linear system with transfer function H ( s ) to a sinusoidal input u ( t ) = A sin( ωt + θ ) is y ss ( t ) = A H ( ) sin ( ωt + θ + H ( )) | | where H ( ) = { H ( ) } + 2 { H ( ) } | | 2 H ( ) = arctan � { H ( ) } � { H ( ) } We note that H ( ) = H ( s ) s = , | that is H ( ) is H ( s ) evaluated along the imaginary axis of the s -plane. 1 copyright c D.Rowell 2008 31–1
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1.1 The Frequency Response of Systems with Zeros If a system has a transfer function N ( s ) H ( s ) = D ( s ) the frequency response function is N ( ) H ( ) = . D ( ) For complex a and b , a/b = a / b and ( a/b ) = b , so that | | | | | | a N ( ) 2 { N ( ) } + 2 { N ( ) } (1) | H ( ) | = | | D ( ) | | = 2 { D ( ) } + 2 { D ( ) } H ( ) = N ( ) D ( ) = arctan � { N ( ) } arctan � { D ( ) } (2) � { N ( ) } � { D ( ) } Example 1 Find and plot the frequency response of s + 5 H ( s ) = s + 10 The frequency response function is + 5 H ( ) = + 10 and N ( ) ω 2 + 25 | H ( ) | = | | D ( ) | | = ω 2 + 100 H ( ) = N ( ) D ( ) = arctan ω arctan ω 5 10 The following MATLAB commands were used to plot the frequency response: w=0:.2:100; sys=zpk(-5,-10,1) y=freqresp(sys,w); plot(w,squeeze(abs(y))) plot(w,squeeze(angle(y))) which produced the following plots: 31–2
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0 2 0 4 0 6 0 8 0 1 0 0 0 2 0 4 0 6 0 8 0 1 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 2 0 1 8 1 6 1 4 1 2 1 0 8 6 4 2 0 F r e q u e n c y R e s p o n s e P h a s e ( d e g ) F r e q u e n c y R e s p o n s e M a g n i t u d e f r e q u e n c y ( r a d / s ) f r e q u e n c y ( r a d / s ) We note
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