Control Systems. Lecture 8 Root Locus. Root Locus. Plant. Controller. Sensor

Transcription

1 Cotol Syt ctu 8 Root ocu Clacal Cotol Pof. Eugo Schut hgh Uvty Root ocu Cotoll Plat R E C U Y - H C D So Y C C R C H Wtg th loo ga a w a ttd tackg th clod-loo ol a ga va Clacal Cotol Pof. Eugo Schut hgh Uvty

2 Chaacttc Equato: Root ocu 0 Th oot o of th chaacttc quato a th clod-loo ol of th fdback yt!!! Th clod-loo ol a a fucto of th ga Wtg th loo ga a b a b a b a b a Th clod loo ol a gv dtctly by th oluto of: b 0, 0, a b 0, a Clacal Cotol Pof. Eugo Schut hgh Uvty 3 Root ocu { } oot{ d u } R o wh va fo 0 to otv Root ocu o fo 0 to - gatv Root ocu > 0: $! #!" 80 Magtud codto Pha codto < 0: $! #!" 0 Magtud codto Pha codto Clacal Cotol Pof. Eugo Schut hgh Uvty 4

3 Root ocu by Chaacttc Equato Soluto Exal: R E U Y 0 - Clacal Cotol Pof. Eugo Schut hgh Uvty Y R 0 Clod-loo ol: 0 0 0, ± 5.5 ± ± > < 0 5 Root ocu by Chaacttc Equato Soluto W d a ytatc aoach to lot th clod-loo ol a fucto of th ga ROOT OCUS Clacal Cotol Pof. Eugo Schut hgh Uvty 6 3

4 4 Clacal Cotol Pof. Eugo Schut hgh Uvty Pha ad Magtud of a Taf Fucto 0 0 a a a b b b b Th facto, - j ad - k a colx ub: k j k k j j k j,, 7 Clacal Cotol Pof. Eugo Schut hgh Uvty Pha ad Magtud of a Taf Fucto Now t ay to gv th ha ad agtud of th taf fucto: [ ], 8

5 Root ocu by Pha Codto Exal: R E U Y o 3 blog to th locu? Clacal Cotol Pof. Eugo Schut hgh Uvty 9 Root ocu by Pha Codto [ ] o 3 blog to th locu! Not: Chck cod locu_hacodto. Clacal Cotol Pof. Eugo Schut hgh Uvty 0 5

6 Root ocu by Pha Codto o 3 W d a ytatc aoach to lot th clod-loo ol a fucto of th ga ROOT OCUS Clacal Cotol Pof. Eugo Schut hgh Uvty Bac Pot: Root ocu { } oot{ d u } R o wh va fo 0 to otv Root ocu o fo 0 to - gatv Root ocu 0 a b 0 Nub of bach ub of o-loo ol R bg at o-loo ol 0 a 0 R d at o-loo o o aytot # b 0 0 "! > 0 R ytcal about R-ax Clacal Cotol Pof. Eugo Schut hgh Uvty 6

7 Root ocu Rul : Th bach of th locu tat at th ol of ad of th bach d o th o of. : od of th doato of : od of th uato of Rul : Th locu o th al ax to th lft of ad odd ub of ol ad o. I oth wod, a tval o th al ax blog to th oot locu f th total ub of ol ad o to th ght odd. Th ul co fo th ha codto!!! Clacal Cotol Pof. Eugo Schut hgh Uvty 3 Root ocu Rul 3: A, of th clod-loo ol aoach th o-loo o, ad - of th aoach - aytot wth agl ad ctd at π l l, l 0,,, b a ol o α, l 0,,, Clacal Cotol Pof. Eugo Schut hgh Uvty 4 7

8 Root ocu Rul 4: Th locu co th jω ax loo tablty wh th Routh cto how a tato fo oot th lft half-la to oot th ght-half la. Exal: , ± j5 Clacal Cotol Pof. Eugo Schut hgh Uvty 5 Exal: Root ocu Clacal Cotol Pof. Eugo Schut hgh Uvty 6 8

9 Root ocu Dg dag vald by th Root ocu: Hgh latv dg: Fo - 3 w hav clod loo tablty du to aytot Nou ha o: Thy attact clod loo ol to th RHP Not: Chck cod ootlocu. Clacal Cotol Pof. Eugo Schut hgh Uvty 7 Root ocu Vt foula: Wh th latv dg -, th u of th clod loo ol cotat a clod loo ol b a b a b a b a Clacal Cotol Pof. Eugo Schut hgh Uvty 8 9

10 0 Clacal Cotol Pof. Eugo Schut hgh Uvty Pha ad Magtud of a Taf Fucto Exal: o Clacal Cotol Pof. Eugo Schut hgh Uvty Root ocu- Magtud ad Pha Codto { } { } u d oot o R wh va fo 0 to otv Root ocu o fo 0 to - gatv Root ocu > 0: < 0: [ ]

11 Clacal Cotol Pof. Eugo Schut hgh Uvty Root ocu Slctg fo dd clod loo ol o Root ocu: If o blog to th oot locu, t ut atf th chaacttc quato fo o valu of Th w ca obta a o o o Clacal Cotol Pof. Eugo Schut hgh Uvty Root ocu Exal: o o o o ytf,oly[- -5] o-34 [,POES]locfdy,o Ug MATAB:

12 Exal: o 7 5 o 4.06 Root ocu 5 o 3 4 o 7 5 Wh w u th abolut valu foula w a aug that th ot blog to th Root ocu! Clacal Cotol Pof. Eugo Schut hgh Uvty 3 Exal: Root ocu - Coato 5 Ca w lac th clod loo ol at o -75 oly vayg? NO. W d a COMPENSATOR. 5 D 0 5 Clacal Cotol Pof. Eugo Schut hgh Uvty Th o attact th locu!!! 4

13 Root ocu Pha lad coato Pu dvatv cotol ot oally actcal bcau of th alfcato of th o du to th dfftato ad ut b aoxatd: D, > D, > 5 Pha lad COMPENSATOR Wh w tudy fqucy o w wll udtad why w call Pha ad to th coato. How do w choo ad to lac th clod loo ol at o -75? Clacal Cotol Pof. Eugo Schut hgh Uvty Exal: Root ocu Pha lad coato D, > 5 Pha lad COMPENSATOR.04 t u choo ? Clacal Cotol Pof. Eugo Schut hgh Uvty 6 3

14 Exal: Root ocu Pha lad coato D 0 5 Pha lad COMPENSATOR o Clacal Cotol Pof. Eugo Schut hgh Uvty 7 Root ocu Pha lad coato D, > Pha lad COMPENSATOR Slctg ad a tal a o ocdu. I gal: Th o lacd th ghbohood of th clodloo atual fqucy, a dtd by -t o ttlg t qut. Th ol lacd at a dtac 5 to 0 t th valu of th o locato. Th ol fat ough to avod odfyg th doat ol bhavo. Th xact oto of th ol a coo btw: No uo w wat a all valu fo Coato ffctv w wat lag valu fo Clacal Cotol Pof. Eugo Schut hgh Uvty 8 4

15 Exal: Root ocu Pha lag coato D l l D l What ca w do to ca? Suo w wat D, < 0 5 W choo: Pha lag COMPENSATOR Clacal Cotol Pof. Eugo Schut hgh Uvty 9 Exal: Root ocu Pha lag coato D o Clacal Cotol Pof. Eugo Schut hgh Uvty 30 5

16 Root ocu Pha lag coato D, < Pha lag COMPENSATOR Slctg ad a tal a o ocdu. I gal: Th ato o/ol cho bad o th o cotat cfcato. W ck ad all to avod affctg th doat dyac of th yt to avod odfyg th at of th locu tg th doat dyac Slow tat du to th all alot caclld by a all. Th ato o/ol caot b vy bg. Th xact oto of ad a coo btw: Stady tat o w wat a lag valu fo / Th tat o w wat th ol lacd fa fo th og Clacal Cotol Pof. Eugo Schut hgh Uvty 3 Root ocu - Coato Pha lad coato: Pha lag coato: D, < D, > W wll why w call ha lad ad ha lag to th coato wh w tudy fqucy o Clacal Cotol Pof. Eugo Schut hgh Uvty 3 6