IV Design of Discrete Time Control System by Conventional Methods

Transcription

1 IV Dig of Dicrt im Cotrol Sytm by Covtioal Mthod opic to b covrd. Itroductio. Mappig bt th pla ad pla 3. Stability aalyi 4. rait ad tady tat rpo 5. Dig bad o root locu mthod 6. Dig bad o frqucy rpo mthod 7. Aalytical dig mthod IV.. Itroductio Not: hr diffrt dig mthod ar itroducd: ) root locu bad mthod, ) frqucy rpo mthod i pla, 3) aalytical mthod. May covtioal cotiuou-tim ytm dig mthod ca b applid to dicrt-tim ytm.

2 IV.. Mappig bt th pla ad pla h complx variabl ad ar rlatd by th quatio j j j jk Not: frquci diffr i itgral multipl of th amplig frqucy, ar mappd ito th am locatio i th pla. j axi i th pla corrpod to. h itrior of th uit circl corrpod to th lft half of th pla. h xtrior of th uit corrpod to th right half of th pla. Not: ) th agl of vari from to a vari from to. a poit mov from j to j ( ) vari from to hi i th primary trip. ) Each othr trip ith rag of ill trac th pla i o circl. 3) Mappig bt pla ad pla i ot uiqu

3 Mappig bt th primary trip ad th uit circl a) cotat attuatio li i cotat map i to th circl of radiu 3

4 Sttig tim rlat to th rgio o th lft of, hich corrpod to th itrior of th circl ith radiu b) Cotat frqucy loci: Cotat frqucy locu, i th pla map ito th radial li of cotat agl i th pla. c) Cotat damp ratio li map ito a piral i th pla. 4

5 5 d j j, hr d d d j j hu d ad d hu th magitud of dcra ad th agl of icra liarly a d icra, th locu i th pla bcom a logarithmic piral. h rgio boudd by cotat frqucy li ad cotat attuatio li ad th mappig. Cotat li

6 Diagram of th orthogoality of th cotat ad mappig to th pla. Exampl 4..: pcify th rgio i th pla that corrpod to a dirabl rgio i pla. 6

7 IV.3. Stability aalyi Clod loop pul trafr fuctio ytm: C( ) G 4. R GH h tability of 4. may b dtrmid from th locatio of th clod loop pol i th pla, or th root of th charactritic quatio. P GH 0 4. Not: ) For th tabl ytm, all clod loop pol mut li i th uit circl i th pla. ) If a impl pol li at, th th ytm bcom critically tabl. Alo if a igl pair of cojugat complx pol li o th uit circl i th pla, th ytm i critically tabl. Ay multipl clod loop pol o th uit circl mak th ytm utabl 3) Clod loop ro do ot affct th abolut tability ad thrfor may b locatd ayhr i th pla. Mthod for ttig abolut tability: ) Schur-coh tability tt ) Jury tability tt 3) t bad o th biliar traformatio coupld ith th Routh tability critrio. 4) Liapuov tability aalyi Not: Both Schur-coh tability tt ad Jury tability tt may b applid to polyomial quatio ith ral or complx cofficit. Wh th cofficit ar ral, Jury tt ar much implr tha chur-coh tt. h Jury Stability t: P a0 a a a hr a 0 0 h tabl i giv a follo: 7

8 a a k 0 k b k,,,,, a0 ak b b k 0 k c k,,,,, b0 bk p p k 0,, 3 k qk p0 pk Stability critrio by th Jury t: A ytm ith th charactritic quatio P 0 rritt a P a0 a a a hr a 0 0 i tabl if th folloig coditio ar all atifid: ) a a0 ) P 0 3) P 0 for v 0 for odd 4) b b0 c c 0 q q 0 Exampl 4.: dtrmi th tability of folloig Ch quatio P

9 9 Stability aalyi by u of th biliar traformatio ad routh tability critrio Biliar traformatio dfid by : biliar traformatio map th iid of th uit circl ito th lft half of th pla. aum j if j j j j Not: oc traform 0 P ito 0 Q, it i poibl to apply th Routh tability critrio i th am mar. Exampl 4.3: dtrmi th tability of folloig Ch quatio uig biliar traformatio P

10 IV.4. rait ad tady tat rpo rait rpo pcificatio: ) Dlay tim t d th dlay tim i th tim rquird for th rpo to rach half th fial valu th vry firt tim. ) Ri tim t r h ri tim i th tim rquird for th rpo to ri from 0% to 90%, or 5% to 95%, or 0% to 00% of it fial valu. 3) Pak tim t p, th pak tim i th tim rquird for th rpo to rach th firt pak of th ovrhoot. 4) Maximum ovrhoot M p h maximum ovrhoot i th maximum pak valu of th rpo curv maurd from uity. Ct p C maximum prct ovrhoot = 00% C 5) Sttlig tim t th ttlig tim i th tim rquird for th rpo curv to rach ad tay ithi a rag about th fial valu of a i pcifid a a abolut prctag of th fial valu, uually %. Not: h trait rpo of a dicrt ytm to th Krockr dlta iput, tp iput, ramp iput, ad o o, ca b obtaid aily by u of MALAB. 0

11 Stady-Stat Error Aalyi: rcall h loop trafr fuctio i ritt i gral form: k ( i ) i Gc ( ) G( ) Q N ( p ) M k N h trm i th domiator rprt a pol of multiplicity N at th origi. h umbr of itgratio i oft idicatd by lablig a ytm ith a typ umbr that i imply i qual to N. Not: a th typ umbr icra, th tady tat rror rduc, hovr, tability problm aggravat. Cocpt of tatic rror cotat ca b xtdd to th dicrt-tim cotrol ytm. h dicrt ytm loop trafr fuctio i ritt i gral form: k C( ) R B k ( i ) i N Q A N ( p ) M k k B hr cotai ithr a pol or a ro at =. h th ytm ill b claifid a A typ N ytm. R + E R() - E * G P () C() C() b t H() Coidr abov ytm: t r t bt from fial valu thorm, hav lim k lim E 4.3 k

12 for abov ytm cofiguratio : dfi th hav: ad G GH G p Z ad C R G Z G GH p H E R B R GH E E R GH ubtitut ito 4.3, hav lim 4.4 k lim E lim k GH Ca : Uit tp iput: r ( t) u( t) R( ) R E lim lim lim GH GH W dfi th tatic poitio rror cotat thu K p limgh K p 4.5 Not: h tady tat rror i rpo to a uit tp iput bcom ro if rquir that GH ha at lat o pol at =. Ca : ramp tp iput: r ( t) tu( t) R( ) K p, hich

13 lim E lim GH GH lim GH W dfi th tatic vlocity rror cotat K thu v lim 4.6 K v Not: th tady tat rror i rpo to a ram tp iput bcom ro if rquir that GH ha at lat doubl pol at =. K v, hich Ca 3: Uit acclratio iput: r ( t) t u( t) R( ) 3 lim E lim lim 3 GH GH GH W dfi th tatic acclratio rror cotat K lim thu a a 4.7 K Not: h tady tat rror i rpo to a uit acclratio iput bcom ro if hich rquir that GH ha at lat thr pol at =. Summary: K, abl of tady tat rror yp umbr Stp iput Ramp iput Acclratio iput 0 A ifiity ifiity K p 3 or mor A K V ifiity A K a a 3

14 Static rror cotat for typical clod-loop cofiguratio of dicrt tim cotrol ytm 4

15 Rpo to diturbac: N R + - G D G C (a) N + - G C G D aum th rfrc iput i ro, or R 0 C G N G G E, hich ca b obtaid from th quivalt modl i b. R C C D G G G D (b) G E lim lim N GG D Not: o rduc th diturbac impact, ca icra th gai G N G D 5

16 IV.5. Dig bad o root locu mthod Not: h root locu mthod dvlopd for cotiuou-tim ytm ca b xtdd to dicrttim ytm ithout modificatio, xcpt that th tability boudary i chagd from th j axi i th pla to th uit circl i th pla. Agl ad magitud coditio charactritic quatio may hav ithr of folloig form: G H 0 ad 0 0 F Agl coditio: 80k Magitud coditio: F GH hich ca b ritt a : F k 0,, Gral procdur for cotructig root loci. ) Obtai th charactritic quatio F th rarrag th quatio: K 0 m p p p 0 ) Fid th tartig poit ad trmiatig poit of th root loci. A K icra from 0 to ifiit, a root locu tart from a op loop pol ad trmiat at a fiit op loop ro. Locat th pol pi ad ro i o th -pla ith lctd ymbol. By covrio, u x to dot pol ad o to dot ro. Rmark: Loci bgi at th pol ad d at th ro. h umbr of parat loci i qual to th umbr of pol. Root loci mut b ymmtrical ith rpct to th horiotal ral axi bcau th complx root mut appar a pair of complx cojugat root. 3) Dtrmi th root loci o th ral axi. Root loci o th ral axi ar dtrmid by op-loop pol ad ro lyig o it. h root locu o th ral axi alay li i a ctio of th ral axi to th lft of a odd umbr of pol ad ro. 4) Dtrmi th aymptot of th root loci. 6

17 h loci procd to th ro at ifiity alog aymptot ctrd at A ad ith agl A. h umbr of ro M i l tha th umbr of pol by N=-M. th N ctio of loci mut d at ro at ifiity. h ctio of loci procd to th ro at ifiity alog aymptot a K approach ifiity. h liar aymptot ar ctrd at a poit o th ral axi giv by pol of ro of ( pi ) P P i i A ( ) ( ) M M h agl of th aymptot ith rpct to th ral axi i k 0 A 80, k=0,, (-M-), M M ( ) 5) Dtrmi th brakaay poit o th ral axi (if ay). I gral, th tagt to th loci 0 at th brakaay poit ar qually pacd ovr 360. W may valuat dk 0 d 6) Dtrmi th agl of dpartur of th locu from a pol ad th agl of arrival of th locu at a ro, uig th pha agl critrio. h agl of locu dpartur from a pol i th diffrc bt th t agl du to all othr pol ad ro ad th critrio of 80 0 (k ), ad imilarly for th locu agl of arrival at a ro. i 7) Fid th poit hr th root loci cro th uit circl 8) Ay poit o th root loci i a poibl clod-loop pol. 7

18 G ith ro of H F ad th domiator of Cacllatio of th pol of if G H G ad umrator of H hav commo factor th th corrpodig op loop pol ad ro ill cacl ach othr, rducig th dgr of th charactritic quatio. Exampl 4.4 H i th fdback loop : H c ad G b a c d G c a a ( ) H Pol c b c d d b ha b caclld. Exampl 4.5 H i th fd forard loop : H c ad G b a c d G c a a ( ) H Pol c b c d d b ha b caclld. Exampl 4.6 Sytm ha loop gai K KG 0 G, dra th root locu. Not: amplig im ill affct th K valu. Somtim, it ill affct th pol ad ro locatio a ll. Effct of amplig Priod o trait rpo charactritic Not: Icra th amplig priod ill mak th ytm l tabl ad vtually ill mak it utabl. 8

19 9 h damp ratio of th clod loop pol ca b aalytically dtrmid from th locatio of th clod loop pol i th pla. d j j, hr d d d j j hu d ad d

20 Exampl 4.7 (B-4-8) Coidr th digital cotrol ytm ho i figur, plot th root loci. Dtrmi th critical valu of gai K for tability. h amplig priod i 0. c hat valu of gai K ill yild a dampig ratio of th clod loop pol qual to 0.5? ith gai K t to yild 0. 5, dtrmi th dampd atural frqucy d ad th umbr of ampl pr cycl of dampd iuoidal ocillatio. R + - K C 0

21 IV.6. Dig bad o frqucy rpo mthod G j Not: o obtai th frqucy rpo of d to oly ubtitut for. h j fuctio G i commoly calld th iuoidal pul trafr fuctio. It i priodic, ith th priod qual to. Exampl 4.8 (B-4-8) Coidr ytm dfid by th tady tat output. G ith iput Ai k. Obtai G j j co j i co j i i ta G j co i G j co hu th tady tat output: x k A co i i k ta i co Biliar traformatio ad th pla Not: traform map th primary ad complmtary trip of th lft half of th pla ito th uit circl i th pla, covtioal frqucy rpo mthod, hich dal ith th tir lft half pla, do ot apply to th pla. traformatio ill olv th problm:

22 , hr i th amplig priod. th ivr traform: Not: th primary trip of th lft half of th pla i firt mappd ito th iid of th uit circl i th pla ad th mappd ito th tir lft half of th pla. vari from 0 j alog j axi i th pla vari from, alog th uit circl vari from 0 alog th imagiary axi i th pla. pla primary trip: pla frqucy 0 v Although pla rmbl th pla gomtrically, th frqucy axi i th pla i ditortd. h fictio frqucy v ad th actual frqucy ar rlatd a follo: j jv j ta v ta j j

23 Folloig figur ho th rlatiohip bt v ad G b b b m m 0 m, hr m a a tak th traformatio th, m m 0 G Nyquit tability ad bod diagram ca b applid. Not: high frqucy aymptot of th logarithmic magitud for G j ad G jv may b diffrt. Advatag of bod diagram approach: ) lo frqucy aymptot of th magitud curv i idicativ of th tatic K p rror cotat Kv or Ka ) Spcificatio of th trait rpo ca b tralatd ito tho of th frqucy rpo i trm of pha margi, gai margi, badidth, ad o forth. 3) Dig i implr mar. 3

24 Rvi pha lad, pha lag, pha lag-lad: ) Pha lad i commoly ud for improvig tability margi. h pha lad compatio icra th ytm badidth. hu th ytm ha fatr pd to rpod. It may b ubjctd to high-frqucy oi du to it icrad high frqucy gai. ) Pha lag compatio rduc th ytm gai at highr frquci ithout rducig th ytm gai at lor frquci. h ytm badidth i rducd ad th ytm ha lor rpo. Stady tat accuracy ca b improvd. High frqucy oi ca b attuatd. 3) Somtim, pha lag compator i cacadd ith a pha lad compator. Lo frqucy gai ca b icrad, th badidth ca b maitaid. PID cotrollr a a xampl. (PD a lad ad PI a lag) Dig Procdur i th pla. ) Firt obtai G, th traform G to G through, ( hould b proprly cho) rul of thumb i to ampl at th frqucy 0 tim that of th badidth of th clod loop ytm) ) Subtitut jv ad plot th bod diagram for G jv 3) Rad th bod diagram tatic rror cotat, th pha margi, th gai margi. 4) Aum th lo frqucy gai of th cotrollr G D i uity, dtrmi th ytm gai by atifyig th rquirmt for a giv tatic rror cotat. Uig covtioal dig tchiqu to dig G D 5) raform th G D to G D through th biliar traformatio giv by 6) Raliatio th pul trafr fuctio G D by computatioal algorithm. 4

25 Exampl 4.8 (B-4-5) Uig th bod diagram approach i th pla, dig a digital cotrollr for th ytm ho i figur. h dig pcificatio ar that th pha margi b 50 dgr, th gai margi b at lat 0 db, ad th tatic vlocity rror cotat Kv b 0 c -. h amplig priod i aumd to b 0. c. aftr th cotrollr i digd, calculat th umbr of ampl pr cycl of dampd iuoidal ocillatio. R + - G D K 0.5 C() 5

26 IV.7. Aalytical dig mthod Dig of digital cotrollr for miimum ttlig tim ith ro tady tat rror Dfi th traform of th plat that i prcdd by th ro-ordr hold a G Z GP h op loop trafr fuctio bcom G D G (fd forard) Dfi th clod loop pul trafr fuctio C R G G G, or GD F 4.8 G D Sic it i rquird that th ytm xhibit a fiit ttlig tim ith ro tady tat rror, th ytm mut xhibit a fiit impul rpo. Hc, F mut b of th folloig form: F Not: a N 0 a a N, hr N, i th ordr of th ytm. F mut ot cotai ay trm ith poitiv por i. olv 4.8 ill hav F G D G F 4.9 Not: Both F ad G D mut b phyically raliabl ) h ordr of th umrator of G D mut b qual to or lor tha th ordr of th domiator. (Othri, th cotrollr rquir futur iput data to grat currt iput. ) L ) If th plat G p ivolv a traportatio lag, th th digd clod loop ytm mut ivolv at lat th am magitud of th traportatio lag. (Othri, th clo loop ytm ill rpod bfor th iput i giv) 3) If G i xpadd ito a ri i, th lot -por trm of th ri xpaio of F i mut b at lat a larg a that of G. 4) I additio to th phyical raliability coditio, atttio hould b paid to th tability. W mut avoid cacllig a utabl pol of th plat by a ro of th digital cotrollr. Similarly, th digital cotrollr pul trafr fuctio hould ot ivolv utabl pol to cacl plat ro that li outid th uit circl. 6

27 G Aum G ivolv a utabl pol G, hr a a G G G G D C D a F R G G G D GD, thu F mut hav a a ro. a a F a G G Not: ic i F a ro. Zro of of D 7 G D hould ot cacl utabl pol of G G li iid th uit circl may b caclld ith pol of G D G that li o or outid th uit circl mut b icludd i F Dig proc: R C R F, all utabl pol mut b icludd. Hovr, all ro a ro. E 4.0 for uit tp iput R for uit ramp iput R for uit acclratio iput: R 3 I gral, E P R, ubtitut ito 4.0 q R C R F P F q to ur that th ytm rach tady tat i a fiit umbr of amplig priod ad maitai ro tady tat rror, E mut b a polyomial i ith a fiit umbr of trm., cho th fuctio F to b th form q F N, hr N i a polyomial i th, E P N, hich i a polyomial i oc hav G G F, th ubtitut ito 4.9 to gt F F G F q N D ith a fiit umbr of trm. G D ith a fiit umbr of trm.

28 For a tabl plat G P, th coditio that th output ot xhibit itramplig rippl aftr th ttlig tim i rachd my b ritt a follo: C( t ) co ta t, for tp iput C( t ) co ta t, for ramp iput C( t ) co ta t, for acclratio iput Exampl 4.9 (B-4-8) Coidr th cotrol ytm ho i figur. Dig a digital cotrollr G D uch that th ytm output ill xhibit a dadbat rpo to a uit tp iput (that i, th ttlig tim ill b th miimum poibl ad th tady tat rror ill b ro; alo th ytm output ill ot xhibit itramplig rippl aftr th ttlig tim i rachd) th amplig priod i aumd to b c. R + - G D C() 8