lec08_print

# lec08_print - MAE143b Linear Control - Theory and...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 17

lec08_print - MAE143b Linear Control - Theory and...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online