Chapter 7 Control Systems Design by the Root Locus Method

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1 haptr 7 ntrl Sytm Dgn by th t Lu Mthd 7. Intrdutn! Prfrman Spfatn: h rqurmnt mpd n th ntrl ytm ar plld ut a prfrman pfatn, whh gnrally rlat t auray, rlatv tablty, and pd f rpn.! Sytm mpnatn: Sttng th gan th frt tp n adjutng th ytm fr atfatry prfrman. In many pratal a, hwvr, adjutmnt f th gan aln may nt prvd uffnt altrnatn f th ytm bhavr t mt th gvn pfatn. A rdgn r addtn f a utabl dv alld mpnatn. h dv alld a mpnatr.! Sr mpnatn and fdbak (r paralll) mpnatn # Sr mpnatn: Fgur 7- (a). h mpnatr G () plad n r wth th plant. $ Smplr than fdbak mpnatn. $ Frquntly rqur addtnal amplfr t nra th gan and/r t prvd latn. # Fdbak mpnatn: Fgur 7- (b). It fd bak th gnal() frm m lmnt() and pla a mpnatr n th rultng nnr fdbak path.! mpnatr # Lad mpnatr: If a nudal nput appld t th nput f a ntwrk and th tady-tat utput ha a pha lad wth th am frquny, thn th ntwrk alld a lad ntwrk. h mpnatr havng a haratrt f a lad ntwrk a lad mpnatr. # Lag mpnatr: h mpnatr wth a pha lag haratrt. # Lag-lad mpnatr: Pha lag ur n th lw-frquny rgn and pha lad n th hgh-frquny rgn.! Dgn prdur: # St up a mathmatal mdl f th ntrl ytm # Adjut th paramtr f a mpnatr by tral and rrr ung mputr mulatn, n rdr t mt th dgn pfatn. # ntrut a prttyp and tt th pn-lp ytm. If ablut tablty f th ld-lp aurd, th dgnr l th lp and th prfrman f th ld-lp ytm. If th prfrman nt atfd, g t th frt tp, agan. 7-

2 7. Prlmnary Dgn ndratn! W aum that th plant gvn and unaltrabl. h dgn prblm bm th f mprvng ytm prfrman by nrtn f a mpnatr. mpnatn f a ntrl ytm rdud t th dgn f a fltr wh haratrt tnd t mpnat fr th undrabl and unaltrabl haratrt f th plant.! t-lu apprah t ntrl ytm dgn: In th dgn, th rt l f th ytm ar rhapd thrugh th u f a mpnatr that a par f dmnant ld-lp pl an b plad at th drd latn. (Oftn, th dampng rat and undampd natural frquny f a par f dmnant ld-lp pl ar pfd.)! fft f th addtn f pl: h addtn f a pl t th pn-lp tranfr funtn ha th fft f pullng th rt lu t th rght, tndng t lwr th ytm rlatv tablty and t lw dwn th ttlng f th rpn (Fgur 7- ).! fft f th addtn f zr: h addtn f a zr t th pn-lp tranfr funtn ha th fft f pullng th rt lu t th lft, tndng t mak th ytm mr tabl and t pd up th ttlng f th rpn (Fgur 7-). 5.8 ltrn ntrllr (rvw)! Opratnal amplfr: Fr th rut n Fgur 5-4, w hav ( ) ( ) 0 Whr th nput and may b d r a gnal and th dffrntal gan r vltag gan. h magntud pf apprxmatly 0 5 ~0 6 fr d gnal and a gnal wth frquny l than apprxmatly 0Hz. (h dffrntal gan dra wth th gnal frquny and bm abut unty fr frqun f MHz ~50MHz.) In th dal OP amp, th nput mpdan nfnty (n urrnt nput) and th utput mpdan zr.! Invrtng amplfr: Fr Fgur 5-44 lt u btan th utput vltag

3 7- Sn nly a nglgbl urrnt flw nt th amplfr, Sn ( ) 0 0 and ff, mut b almt zr. Hn w hav r If, thn th p-amp rut at a a gn nvrtr.! Nn-nvrtng amplfr: Fr Fgur 5-45(b), whh quvalnt t Fgur 5-45(a), w hav fr >>. hu Sn and hav th am gn, th p amp rut nn-nvrtng. xampl 5-) Fr Fgur 5-46 lt u dfn Sn th urrnt flwng nt th amplfr nglgbl, w hav Sn 0, w hav akng th Lapla tranfrm, ( ) dt d ( ) dt d dt d () () () ()

4 h p-amp rut hwn n Fgur 5-46 a frt-rdr lag rut. S abl 5-.! Impdan apprah fr btanng tranfr funtn: Fr th rut hwn n Fgur 5-47, w hav () () () () xampl 5-4) h mplx mpdan () and () fr th rut n Fgur 5-46 ar Hn, h tranfr untn! Lad r lag ntwrk ung pratnal amplfr: Fr th rut n Fgur 5-48, w hav whr () () dv dt () I() I () V() () () I() () () () () α 4 7-4

5 7. Lad mpnatn! Lad mpnatr: Fgur 7-4 hw an ltrn rut ung pratnal amplfr. h tranfr funtn fr th rut btand a fllw: whr () () α # Lad ntwrk: > ; Fgur 7-5(a) # Lag ntwrk: < ; Fgur 7-5(b) 4! Lad mpnatn thnqu bad n th rt-lu apprah: ndr a dgn prblm n whh th rgnal ytm thr untabl fr all valu f gan r tabl but undrabl trannt-rpn haratrt. In uh a a, th rhapng f th rt lu nary n th brad nghbrhd f th jω ax and th rgn n rdr that th dmnant ld-lp pl b at drd latn n th mplx plan. h prblm may b lvd by nrtng an apprprat lad mpnatr n aad wth th fd-frward tranfr funtn.! h prdur fr dgnng a lad mpnatr fr th ytm hwn n Fgur 7-6 by th rt-lu mthd may b tatd a fllw:. Frm th prfrman pfatn, dtrmn th drd latn fr th dmnant ld-lp pl.. By drawng th rt-lu plt, artan whthr r nt th gan adjutmnt aln an yld th drd ld-lp pl. If nt, alulat th angl dfny φ. h angl mut b ntrbutd by th lad mpnatr f th nw rt lu t pa thrugh th drd latn fr th dmnant ldlp pl.. Aum th lad mpnatr G () t b G () α ( 0 < α < ) Whr α and ar dtrmnd frm th angl dfny. dtrmnd frm 7-5

6 th rqurmnt f th pn-lp gan. 4. If tat rrr ntant ar nt atfd, dtrmn th latn f th pl and zr f th lad mpnatr that th lad mpnatr wll ntrbut th nary angl φ. If n thr rqurmnt ar mpd n th ytm, try t mak th valu f α a larg a pbl. A larg valu f α gnrally rult n a largr valu f v, whh drabl. 5. Dtrmn th pn-lp gan f th mpnatd ytm frm th magntud ndtn. xampl 7-) h rt-lu plt fr th ytm n Fgur 7-7(a) hwn n Fgur 7-7(b). h ld-lp tranfr funtn bm () 4 () 4 h ld-lp pl ar latd at (Fgur 7-7 (b)) ± j hu ξ 0.5 ω n h dampng rat dtrmn th angular latn f th pl, whl th dtan f th pl frm th rgn dtrmnd by th undampd natural frquny (Fgur 7-8). It drd t mdfy th ld-lp pl that r ξ 0.5 ω n 4 ± j If th rgnal ytm ha th pn-lp tranfr funtn G(), thn th mpnatd ytm wll hav th pn-lp tranfr funtn. G () G() G() Nt that thr ar many pbl valu fr and α that wll yld th nary angl ntrbutn at th drd ld-lp pl. In th prnt ytm, th angl f G() at th drd ld-lp pl (S Fgur 7-0) 7-6

7 4 j ( ) 0 hu, f w nd t fr th rt lu t g thrugh th drd ld-lp pl, th lad mpnatr mut ntrbut φ0 at th pnt. Frt, draw a hrzntal ln pang thrugh pnt P, th drd latn fr n f th dmnant ld-lp pl. h hwn a ln PA n Fgur 7-9. Draw al a ln nntng pnt P and th rgn. Bt th angl btwn th ln PA and PO. Draw tw ln P and PD that mak angl ±φ/ wth th btr PB. h ntrtn f P and PD wth th ngatv ral ax gv th nary latn fr th pl and zr f th lad ntwrk. By fllwng th dgn prdur, w dtrmn th zr and pl f th lad mpnatr, a hwn n Fgur 7-0, t b 90 0 r r at -.9, Pl at α 0.57 hu th pn-lp tranfr funtn f th mpnatd ytm bm G () G() ( ) ( )( 5.4) whr 4. h rt-lu plt fr th mpnatd ytm hwn n Fgur 7-0. h gan valuatd frm th magntud ndtn a (.9) ( )( 5.4) 8.7 j It fllw that G () G() 8.7(.9) ( )( 5.4) h lad mpnatr ha th tranfr funtn G () (.9) If th ltrn rut ung pratnal amplfr a hwn n Fgur 7-4 ud a 7-7

8 th lad mpnatr, thn th paramtr valu f th lad mpnatr ar dtrmnd frm () () 0.85 In Fgur 7- w hav arbtrarly hn 0µF and 0Ω.! mparn f tp rpn f th mpnatd and unmpnatd ytm: S Fgur

ELEC 372 LECTURE NOTES, WEEK 11 Dr. Amir G. Aghdam Concordia University

ELEC 372 LECTURE NOTES, WEEK 11 Dr. Amir G. Aghdam Concordia University ELEC 37 LECTURE NOTES, WEE Dr Amir Aghdam Cncrdia Univrity Part f th nt ar adaptd frm th matrial in th fllwing rfrnc: Mdrn Cntrl Sytm by Richard C Drf and Rbrt H Bihp, Prntic Hall Fdback Cntrl f Dynamic