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lec08_print - MAE143b Linear Control - Theory and...

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MAE143b Linear Control - Theory and Applications Lecture 8, Thursday Aug. 27, 2:00-4:50pm Prof. R.A. de Callafon callafon@ucsd.edu CONTENTS OF THIS LECTURE Basic Properties and role of Feedback ( chap. 4 ) Control design via sensitivity function – P control design for 1st order system – P control design for 2nd order system – PD control for 2nd order system Tracking : motivation for integral control – P control design for 2nd order system – PD control for 1st and 2nd order system PID control – PID control for 2nd order system – properties & proper implementation MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 8, Page 1 Feedback systems - general equations G C y u r 1 r 2 negative feedback + + + Recap - important equations: y = Gu and u = r 1 + C ( r 2 y ). Transfer function from r 1 ,r 2 to y : y = Gu = Gr 1 + GC ( r 2 y ) (1 + GC ) y = Gr 1 + GCr 2 y = G 1+ GC r 1 + GC 1+ GC r 2 and u = 1 GC r 1 + C GC r 2 Simple rule: the TF of a single-loop negative feedback system is given by forward TF divided by 1 plus the loop TF MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 8, Page 2
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Feedback systems - comparing open- & closed-loop For comparison, consider the following open-loop and closed- loop (negative feedback) systems: C G H w + + u r C G H w + + u + r open-loop control closed-loop control yy e For the open-loop system we have: y ( s )= G ( s ) C ( s ) r ( s )+ H ( s ) w ( s ) For the closed-loop system we have y ( s G ( s ) C ( s ) 1+ G ( s ) C ( s ) r ( s H ( s ) G ( s ) C ( s ) w ( s ) e ( s 1 G ( s ) C ( s ) r ( s ) H ( s ) G ( s ) C ( s ) w ( s ) MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 8, Page 3 Role of feedback - disturbance rejection For open-loop system: y ( s H ( s ) w ( s ). For closed-loop system: y ( s H ( s ) G ( s ) C ( s ) w ( s ) Evaluating Bode response for disturbance rejection , we see: | H ( ) | | G ( ) C ( ) | < | H ( ) | , provided 1 | G ( ) C ( ) | < 1 The sensitivity function plays important role: S ( s ):= 1 G ( s ) C ( s ) , | S ( ) | = 1 | G ( ) C ( ) | Assuming the closed-loop system is stable - for disturbance re- jection at a particular frequency ω we need | S ( ) | < 1 If indeed | S ( ) | < 1, closed-loop control does a better job at disturbance rejection then open-loop control ! MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 8, Page 4
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Role of feedback -t rack ing With tracking error e ( s )= y ( s ) r ( s ) for closed-loop system: e ( s 1 1+ G ( s ) C ( s ) r ( s ) H ( s ) G ( s ) C ( s ) w ( s ) Evaluating Bode response for tracking error rejection , we see: | e ( ) | | r ( ) | < 1 , provided 1 | G ( ) C ( ) | < 1 Again the sensitivity function plays important role: S ( s ):= 1 G ( s ) C ( s ) , | S ( ) | = 1 | G ( ) C ( ) | Assuming the closed-loop system is stable - for perfect tracking at a particular frequency ω we need | S ( ) | =0 < 1 If indeed | S ( ) | , closed-loop control does a perfect job at tracking ! MAE143b, UCSD, Summer II 2009, R.A. de Callafon – Lecture 8, Page 5 Sensitivity function - general shape Sensitivity function | S ( ) | = 1 | 1+ G ( ) C ( ) | important to study disturbance rejection! 10 -2 10 -1 10 0 10 1 ω b General shape of amplitude of | S ( ) | : Small at low frequencies (disturbance rejection + tracking) This occurs when | G ( ) C ( ) |→∞ as ω 0. This means, either sys- tem/plant G or controller C should have a ‘large gain’ at low frequencies.
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This note was uploaded on 05/27/2010 for the course MAE 143B taught by Professor Paoc.chau during the Spring '06 term at UCSD.

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lec08_print - MAE143b Linear Control - Theory and...

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