Lecture 7 - SISO Loop Analysis

Transcription

1 Lctr 7 - IO Loop Anal IO ngl Inpt ngl Otpt Anal: tablt rformanc Robtn EE39m - prng 5 Gornvk ontrol Engnrng 7-

2 ODE tablt Lapnov mathmatcal tablt thor - nonlnar tm tablt fnton frt rct mtho xponntal convrgnc con mtho: Lapnov fncton gnralzaton of nrg paton Lapnov xponnt omnant xponnt of th convrgnc for a nonlnar tm for a lnar tm fn b th pol x x & x δ f x, t ε x A -ax -A -ax t EE39m - prng 5 Gornvk ontrol Engnrng 7-

3 x& Ax x B D tablt: pol H H haractrtc val tranfr fncton pol l.h.p. for contno tm nt crcl for ampl tm I/O mol v. ntrnal namc I A B D Imag Imag z Ral H N g n K g D p pn g Ral EE39m - prng 5 Gornvk ontrol Engnrng 7-3

4 EE39m - prng 5 Gornvk ontrol Engnrng 7-4 tablt: clo loop h tranfr fncton pol ar th zro of Watch for pol-zro cancllaton! ol fn th clo-loop namc nclng tablt Algbrac problm, ar than tat pac m : ID controllr k k k D D I τ : lant - ; [ ]

5 powr amp control voltag EE39m - prng 5 Gornvk rvomotor Exampl Motor controllr nor poton tpont h control goal to track th poton tpont U ID control τ D. k k I kd τ - D mol Jx && bx& ci LI& RI LF x ranfr fncton G F I J J.c, F I.c ontrol Engnrng 7-5. c G

6 rvomotor Exampl ID controllr: k I D D ; k ; k.; τ. tablt [ ] - Imag p 5 5 >> fback,*; >> >> >> [z,p,k] zpkata; >> plotp EE39m - prng 5 Gornvk Ral p ontrol Engnrng 7-6

7 tablt For lnar tm pol crb tablt almot, xcpt th crtcal tablt For nonlnar tm lnarz aron th qlbrm mght hav to look at th tablt thor - Lapnov Orbtal tablt: trajctor convrg to th r th tat o not - th tmng off paccraft FM, 3-D trajctor wthot arcraft arrval tm EE39m - prng 5 Gornvk ontrol Engnrng 7-7

8 rformanc N to crb an analz prformanc o that w can gn tm an tn controllr What th prformanc nx? hr ar all man conflctng rqrmnt Engnr look for a raonabl tra-off k k, k D k D Optmzr, rformanc I τ D lant mol k ki m EE39m - prng 5 Gornvk ontrol Engnrng 7-8

9 rformanc: Exampl lctng optmal b n th Watt govrnor - HW Agnmnt b Optmzr rformanc lant mol, gvn b ampng b rformanc nx: trannt ca rat v b m EE39m - prng 5 Gornvk ontrol Engnrng DAMING b

10 rformanc - pol ta tat rror: t tranfr fncton at. tp/pl rpon convrgnc, omnant pol a mn { R p } n j j at A omnant xponnt aton! Fat rpon pol far to th lft ma la to pakng fat rpon low rpon EE39m - prng 5 Gornvk ontrol Engnrng 7-

11 rformanc - tp rpon tp rpon hap charactrzaton: ovrhoot ttlng rror nrhoot ttlng tm r tm ta tat rror EE39m - prng 5 Gornvk ontrol Engnrng 7-

12 EE39m - prng 5 Gornvk ontrol Engnrng 7- rformanc - qaratc nx Qaratc prformanc rpon, n frqnc oman { π π π t t t J t E ~ ~ For t a zro man ranom proc wth pctral powr Q For Q, th jt arcval thorm π Q t t t E J t [ ]

13 rvomotor xampl tp rpon LOED LOO EOIN E REONE.8.6 >> fback*,; >> tp >> Qaratc nx J t t t t >> fback*,; >> t.; >> tp,:t:.; >> J m-.^*t J.8 EE39m - prng 5 Gornvk ontrol Engnrng 7-3

14 ntvt trbanc rfrnc lant ontrollr otpt - rror trbanc Fforwar lant F rfrnc otpt - rror L Fback ntvt << for L >> goo for an frqnc for L << nvr ntabl can b ba for L - rngng, ntablt EE39m - prng 5 Gornvk [ ] F FF L Fforwar ntvt ontrol Engnrng 7-4

15 EE39m - prng 5 Gornvk ontrol Engnrng 7-5 ranfr fncton n control loop v n v n v n ontrollr lant - v trbanc fforwar rfrnc otpt control rror n no

16 EE39m - prng 5 Gornvk ontrol Engnrng 7-6 ranfr fncton n control loop ntvt omplmntar ntvt No ntvt Loa ntvt [ ] [ ] [ ] [ ] v n v n v n v n

17 EE39m - prng 5 Gornvk ontrol Engnrng 7-7 ntvt rqrmnt Dtrbanc rjcton an rfrnc trackng << for th trbanc << for th npt no v Lmt control ffort << conflct wth trbanc rjcton whr < No rjcton << for th no n, conflct wth trbanc rjcton v n v n v n

18 rvomotor xampl - ntvt Otpt trbanc ENIIVIY NOIE ENIIVIY 3 Otpt no 5 Magnt B - - Magnt B Frqnc ra/c Frqnc ra/c tpont trackng OMLEMENARY ENIIVIY - LOAD ENIIVIY Fforwar -3 Magnt B - - Magnt B EE39m - prng 5 Gornvk Frqnc ra/c -7 3 ontrol Engnrng 7-8 Frqnc ra/c

19 Robtn A controllr work for a mol. Wll t work for a ral tm? an chck that controllr work for a rang of ffrnt mol an hop that th ral tm covr b th rang Uncrtant t lant Fback controllr t EE39m - prng 5 Gornvk ontrol Engnrng 7-9

20 Robtn Atv ncrtant Mltplcatv ncrtant t t t t onton of robt tablt < EE39m - prng 5 Gornvk onton of robt tablt < mall Gan horm: loop gan < tablt ontrol Engnrng 7-

21 Nqt tablt crtron - γ t G t Homotop roof G tabl, hnc th loop tabl for γ. Graall ncra γ to. h ntablt cannot occr nl γgw for om γ. G 8 < a ffcnt conton btlt: r.h.p. pol an zro Formlaton an ral proof ng th agrmnt prncpl, ncrclmnt of - tabl ntabl tabl a γ EE39m - prng 5 Gornvk ompar agant mall Gan horm: ontrol Engnrng 7-

22 Gan an pha margn - L Im L Loop gan L [ L ] gan margn - 8 ϕ m /g m R L Nqt plot for L at hgh frqnc L << gc pha margn EE39m - prng 5 Gornvk ontrol Engnrng 7-

23 Gan an pha margn Bo plot gan croovr pha croovr /g m Im L gan margn - 8 ϕ m R L gc gc pha margn EE39m - prng 5 Gornvk 8 ontrol Engnrng 7-3

24 rvomotor xampl Imagnar Ax Gan croovr at 7 ra/ pha margn Nqt Dagram Ral Ax gan margn Magnt B ha g Bo Dagram gan margn 9B pha margn 57 g ha croovr at 399 ra/ -5 3 Frqnc ra/c EE39m - prng 5 Gornvk ontrol Engnrng 7-4