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UNKNOWN_PARAMETER_VALUE(4) - CHAPTER 6 3—6-1 The...

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Unformatted text preview: CHAPTER 6 3—6-1. . The open-loop transfer function for the system is (5(3) ”(3.) : Kai-H) Wefirst locatetreopm—looppolesandzeromtmmluplane. Aroot locos exists on the negative real axis between -1 and -m. Since the open-loop transfer flmction involves two poles and one zero. there is a possibility that a. circular root loci exists. , The equatim for the root-locus brandxes can be obtained frcni the angle condition: K +13 ‘ (s -.= t/w‘aé +2) :1 whidioanbereurittenas ' (5-H - 2A == i/J‘a'ka-H) By substituting s =o-+ ju. we obtain (tr-rgw/ - 2 {w- I“? = :t /!0’(2k.+/) 2:; Vii—H) —25u'.’———=i_/80 (2w) an" (e, ) E”-‘4 = I "—33—— + mam) Taking the tangents of both sides of this last equation. - a..[t...-’(W)_ x’g—fl— =1“... [ red—— 1- 157: mm] which can be simlified to u __£,V_ ._....- H-l # 0” __ 0' to ._ W, ____a_;__, ._._U " ' -/+——-——-' 2+1 0. /_+-,£.-xo Hence NI from which we obtain -79.... This last equation is equivalent to 40:0 W ' _(H/)a +5“, 2.: / 'Ihesetmequationa are meequatims-forthe root loci for the-systen- The .- _ first equatim. w = 0, is the equation'for the real axis. unreal axis from-- . s=-ltos=-oooorrespmdetoaroot-locusforK;O. (flaminingparb ofthereal axis oorrespmdstoarootlocusforx<0.)- 'Inthepresentaysm, K is positive. The second equation is an equation of a circle with theoenter at cr= -1, y= 0 and the radius equal to 1. The root-locus diagram is shown below. 13—6—2. The open-loop transfer fixation isi INS-94‘ - . (ital/(é) =(Tfljg' 'Ihissyatemis sinilartotheoneinProblenB-S-l. mimimolmm polesandonezero., 'I‘heroot-loousplotinvolveeaoirmlarrootlocus._ A root-locus plot of the system is sham below. w, The opén-IOOp We: Winn K Qty/m}: :(HIXfi-r- {4: +5) hasthapolesatsaO,s=-1,s=-2;tjlandnozeros. Masmtoteahwa 319133145’and1135' finasynpbobesmtmthemgatiwmlajdsat 0', 8-1.25. MbnnchesoftlnmtlocicrossfinimghmnmdsatI-a :11. Mangleofdeparturafmtlncouplmcpoleintlnupparhausplm is +162°. amfiABprogmtOplottMrootlociandaayupbotu isgivanbnlav,to- gather with the resulting root-locus plot. ._ 96 "H? Hoot-locus plot “H" num-[O O O OH: den-[1 5 9 5. 0|: numa-[O 0 O 01]; done --[‘I 5 9.375 7.8125 2.44141; r - rlocustnumfion): Iot‘rr'J’ Id Current plot held ploflr,'o rlocuslnum.denal 2 -3 3]: axlslv}: axisi'sqtlare' I; —81_ 3-6—4. 1 mm program to plot; the root loci and asympbohes for the follow- Ing system: ' - _ . K 3(5-1- 6.5“)(5‘4- 9.55110) 669/76!) =- 13 given 132101: and the resultmg root—locus plot is shown below. Notethattheeqmtimfortheaaynphoteaia - __._..‘$__...__._ (-7 to. 299)” _ - r 5“ 1- /. I: -' +- a. #:375- ._s-"+ man/:75: : ”mm/iv 64$) ”4(3) :- 95 "H. Float-locus plot ""- num-IO 0 0 0'1]; den—[1 1.1 10:3 5 0]: name-IO O 0 O 1]: dana - [‘1 1.1 0.45375 0.0531574 0.0057191]: r - rlocusfnurnmen): plollr.'-'I hold Current Plot held platin'oi rlocuslnumafional v - [-5 5 -5 5]: axlsM: axisI'square'i: grid . titlef'Plot of Root Loci and Aawnptotes Problem 8-8-4?! MMMWMW [Prom-M4! _ 3-6-5. A mm_ program to plot the root loci and asymptote: for the system . 5' J: -76 {) (SH-25+ 29(F'L1'z5' 9) is shown below. The resulting root—isms plot. is also m below. The ' ' root loci cross the imaginary axis at a) = 1-, 1.87. This point is obtained by - solving the following equation: . [(jw)‘+ a,’w+z][0w)‘+ ij r9] 1- 1< = 09““- 7a." +/o+ k) +j(—sr-w’+/¢.w) =0 By equating the imaginary part equal to zero, we obtain a}: 1; 1.8708. By eqmtingtherealpartequaltozaro. wget'thegainvalue atthecroeaing point to be 9.25. ' ' _ 96 .*"' Roof-locus plot H". _. Hum-=10 O 0 O1}: den-l1-4 11 14 10!: name:- 10. 0 O _O 1]: dens - I1 4 B 4 I]: r - rlocuslnumden}; plottr.'-'l hold Conant plot held ploflr.'o'l ' rlocustnuma.donal v - [-6 4 -5 5]:sxistvl;axis{'squara'l; _ grid . titlel'Plot of Root Loci and Asymptotes {Problem B-B-sl'l Phlofflodlndlflm mm ‘---m -n-MV‘- --M-- --m-- . 'A---h‘ “0L4 -‘ N (a) ' .5 G MM' 0 1'.» lb .‘s i1!- - 4 -33.. 3-6-6. . (”104:z +(z-r-JK): +/é+/0K s‘+25 +/o /+ 6(I)//(3) : The omracteristic equation 0+ KJS‘+ (awn): +/a+’ xak=o has two roots at 8 $=_ #3; 1i {K HH’HP Il'i-‘K Ifuewritee=Xd=jL thatis {If-3" 5"; VK‘+/‘?-A'.+? /+K_’ — l-H: — x==- I #4 1.9: +7 /0 ( kw)" X Y (Ii-K) + (l-HC)‘ (/+K)' / ' This indicates that the root loci are on a' circle about the origin of radius 50'. 3-6-7. The opm—loop transfer: function . 54-04.) GM/ffiF-‘f “ . S‘(s-r3.6) . haatmzeroats=-O.2.andthedoub1epolesats=0andasinglepoleat . s = -3.6. The asymptotee have anglesof 1-90“. The asynptotes meet on the. real axis at 0"; u -1.7.- The breakaway or break—1n points are located-at to 0,3 = 41.43155. and a = 4.6685. A mm program plot is shown below. The resulting root-lows plot is 81'an on the nut page. num :- [0 O ‘l 0.2]: - den - [1 3.6 0_ 0]; rlocuslnumdan} ' v -= [-6 -2 -4 4|; axlsM: axist'squara') grid - titlot'Hoot-Locu's Plot {Problem 3-8-73'1' -5 4 .5 4 -a -2 -1 o 1 mm; _ 3-6-3. The open-loop transfer mm. 60-) ”(1-) '_-_-. __———K(’+ " 5') s’+s‘+-l ; has the poles at s = 0.2328 g: 1 0.7926 and s = 4.4656; -0.5. A mm program the to root-locus plot-.13 shown on nextpage. 1113 mm is at s -_ loci. in am below. The resulting 96 «In. Hoot-locus plot “we! mun-[001 den-[1 1 0 rlocuslnumfien} v - [-3 3 -3 3)::xistvlmxlsl'squme'} grid tillel'floot-Locus Plot 0.5]: 1]: (Problem 8-8-8?! 3-6—9. The opal-loop transfer fuming qfiyfly) g _K_(§"'_’L 5(3‘ M: 441/) : thapolasata-O.s=-2-;t"jJ?andtha_zaroata--9. 112W +90°andnaettlnrealaxisat 03:2.5. .maoonplexbnndns -1maginaryaxis ats=ij4.45.j Mmleofdeparturefrmfln. ample: pole inthe upperhalt a plane is -16.5'. - ' g E Thedaminant closed-loop polasm‘vingthadamphg ratio is 0.5 can'be locatedasflninterqecbimofthemoblociandlmesfrmfinodgihhafing' anglesiw‘. ‘I'hedeeireddaminantcloeed—looppolesaretumdtobaat' . Sz-loSiJ2.J‘fi 'I'rnthird p016 1i ates—1. Thegatnmue corresponding'totheaedcminanb closed-loop poles is K I: 1. AMATIAB program to 19191:.“ m. 'helaur. ‘fnnmultingroot-locusplobisalmmthenextpa. “.3 96 "‘" Root-locus plot "'" num - [O O 1 9]: den - [1 4 1‘!_ 0]: flocuslnumden) hold ' Current plot held a: - l0.-3J: v - 10.5.1961: Enemy}: . v”; [-15 5 -10 10]: axialvl: ulat'squm'l a . ‘ tlllePfioot—Locus Plot of Gull-Ha] :- Kla+91flsif2+4¢+11rl Rod-lam Plat daisy-Ki) u NIW!) 3-6-10. AmMprogzmtoobtainarmt-lomsplotofthegim-m is slumbelaw. 'I'heresulting root-locus plot is shown mttgnmctpage. % C"... Hoot-loo". pbt {III-I num - [O O O 2 2]; don-[1 7 .10 0 0]: numa - l0 .0 0 'Il: dam :- [0.5 8 6 4]: r - rlocustnumfien]: '01-": I, Id held Current I plottt.‘ 'og‘o rlocuslnumafienal v”: [-10 10 -10 10]:axia{v);axla('square'l: _ title! Plot of Root Loci and Asympmtea (Problem 3-8-10)? -37- mummwm mad-1g ml amt-lemmamtmonginmbeohuimdbyenmmantonm‘ immunnprogrmintothecmputer. mmtingmot-lmplotmrtha don The V - grid title % I...‘ Bootm nut-II... hum-lo o o 2 21: I [I 7 10 O 013 uslnum.danl [-3 3 -3 3]; axislvl; axlsi'aquara'): (float-Locus Plot. near the Origin [Prom-m 3-8-10!" ' Mmdxforstabilitycanbeddtemihadbymofmmstabnity criterion. Since the closed-loop transfer functicn is ((4') __ 21: (3+ /) . R6) _' s“'+ 7s" +4453?- + 2K:,+ 2!: the maracteristic equation for the system is‘ 3‘+7:’+/a:‘+2mr + 21:" =0 Tha Routh array of coatflcients because as {allow-a: 5" ‘ / m -- 2K. 3’ 7 2k ' " to: stability: we require 70>2K #z—¢k>o . K>a Thus. thgramcotxforntahflity il 'I/a.,¢> K 2- 0' 3.5.11. The duncterictic cquntion for tho syctu is fl+$r+ts+Kao Ifltic'aetequalto 2. thanttndnracteristic equaticnhocm 3"!“ 9'5‘4'35' +2:=0 -flacloaed-looppolecmlocaudufouaus _89.. s = 43.2887 See the following mm program for finding the closed-loop poles. p a [1 4 8 213 route!” -1 .8557 + 1.8869! 4.8557 - 1.8669! 4.1.2887 % Gill. Root-locus plot '0‘!" num - [0 0 0 1]: den :- [1 4 8 0|; rlocustnumfien} axist'square'l odd - _ tltlet'Root-Lacus Plot of GM - Krista-“'2 4434-8]?! mmmuom wit-(mean 3-6-12. The open-loop transfer Emotion for the system is . . _'_ __J£-:z“0_____ . 6(r)fl(~") -—- (33+ 2“. 2x311. 2.; +9) A possible MMIAB program to plot a root-lows diagram is shown below. ‘Ihe resulting root-locus plot is also shown below. nun - [0 0 0 l I]: don- [l 4 .ll 14 10]; K1 - 0:0.1:2; K2 - 2:002:25; K3 - 25:05:10; K4 - 10:1:50; KS - 50:5:800; K - [K1 K2 K3 K4 K5]; "Wilma-1K); mm . V- [-8 2 -5 Sufism. Moot-locus Plot of 0(3) - K(s-I-1)I[(s"2+28+2)(s"2+2s+5)]') mamas) H ”I W) s -------'4-- ‘ 1,, III-II“..- -------L‘-- ' --------“- III-Inna 4 4 4' 3-6-3. the span—loop transmi- function is 91m by . Kari-0. (It?) . (s) 1% =.— —-———_—__.__._...___ Q ) .r * +3.:fo/s’4- 7. uzrfl The equation for the asmtotee may be obtained as " KI . _ 53+ (3:3-{49/+ Alli?) s‘+ - . - k . (S + 33¢” +0-(léZ)-' . 3 . .K, (s+/.3§ 9.03 K s’+ F. to (355+ nit/5‘8"? 2-3325" 6a (’7 ”4(5) = ll gee.wemtertIefo11w1ngmntoreanddenomimmintheWm.m nun=[0 o 0.1 40.6667] den=[1 3.3401 7.0325 0-. o] Forflnasynptobas: _, much) 0 .0 -1] dean-at; 4.0063' 5.3515- 2.3325] AmnABpx-ogramtoplotthe root loci and asymptotes is give-11mm. .1112 resulting root—locus plot is sham on the'next page. _ . % II... Root-Ix“ plot I...‘ nurn - [0 O 0 1 0.6657]: den - [1 3.3401 7.0325 0 0]: name - [0 0 0 1]: dena - [1 4.0085 5. 3515 2. 3826]: K1 - 0: 1: 50: K2 In 50: 5. 200: K - [K1 K2]: 1' - rlocusinmmdenJQ: a :- rlocusmummdenmlo: ' . plottt.‘ 0') v - [-6 2 -4 4!: existvl: axlfl'equare'l hold Current Plot held plot“:- grid fltlat'floot-Locua Plot (Problem 3-5-13” xlebell'fieel Axts') - ylabelt'lmag Axis'] -92.. K‘éfi“ +02. w‘)t+w‘(l+2-fi‘)‘ ' -l = [0*(a~-+/)— 42"] +w"(/+¢a~+ $44M) 3 [9(Hl) 440‘] +612" [Munro-rm] +w" = K2 I mamutantgainlociforxsl:2.5:10 aMZOonflnaplanaaxnahom '-93- 3-5-15. mm(s+1)1nflnfaadtornrdtramfarfimb1mandflnm -Is+1)1nflnfeedhadctnnafermtimmlead1m. mm duncberiatic equation is __ A15“) / k _ H- éfiw’s)" '/ + Sufi-25+!) 5+! 5(5‘1-251-0’) =0 '11:: open-100p poles ofG(a)I-I(s) is at a = O and s = —1 1:15. The following mmProgramproducestheroot-locusplotshomthnextpage. % I'll. mm p‘ot I'll-I num- [O 0 0 1]: den - [1 2 6 0]: flocuslnumdanl Wan'un : Divide by zero 5"; [- -3 _-4 4];axia(vl:uls('squaro'l tiflel'floot—Locus Plot [Problem 3-8-15?! 4.3147 + 2.17541 --o.a147 - 2.1754: 4.3700 I'l‘hufi the closed-loop pole- at. located at n - 4.3147 3 1 2.1754, a . -o.37os 3-6-16. M the 5m slum in HE 6—6553}: hmmprogramtoplotéroob-lm diagramforfllesystanalmminfigurefi 6—6501) is slum in mm Program (a). The resulting root-1m plot is shown in Figure (I) (see next page). ' . ' ' . -95.. % MA'I‘LAB Program (a): _ man! «[0 1 -1]; dull =[1 6 8]; K1 - 0:001:50; K2=50:0.5:1000; K-[Kl- K2]; mmmmlx) grid unmet—Low Plot ofG(s) - K(s-l)l(s"2+63+l)‘) xlabchRealeif') 'illf'l Am') . For the gag: sham in 019E 6-6590]: 'A'MMIABpmgramwproduceamot-locusplobofthesyatemalminrigum 6-6500) is given in mm Program (1:) on the neat. page, _ ' mama-[0 —l ' l];' m2 - [l 6 8]; K! - 011.0150; K2 - 50:0.5:1000; _ K - [Kl K2]; Mnmnldenflc) xlnbeK’leAfis‘) mm? %mm Prom (b): V'I—S 8.-8 ”’13:“va I. fi'id Moot-Lows Plot of 6(a) 5- K(l-s)_l(s"2+6¢+c)') The resulting root-locus plot. is shown Notethattheeqmtions forunerootloci forbotheystemearethe-same._ Myaregivenby a2[(o-—/)‘+ w‘—/5'] = 0 mi: equatidn is equivalmt to I _ area W (m-I)‘+w“.=/§. 'Ihei'iret. equation (a): O) is the equation for the real-aids. 'me second equation" is the equation for the circle with center at. (1.0) and the radiue equaltolfi. _ -1heequatim£orthehreakawayorbreak—inpointaieobtainedfm dK/dszo. Forbethsyatem.fl1eaolutimsfordfi/ds=0m 3:16.973 , .5-42-3’73 For system (a): K a ‘0.254 :02 8 = “2-873 . 'Ihisnenhsthatthere arenobteak-myorbreakhin points Erma). The rootlociadetmlymtherealaxie. ('Ihe‘rootiocieadatbebnemaa-Z, andsslandbetweensa-‘iandea-oo.) " For System (1:): _ - . K = 15.746 for s a: 4.873 K '= 0.254 _ for a = 4.873 -97- Helios.s=—2.87Bands=4.873areactualbreakawuyand respectively. The root loci imrolves the circul locus where the center'of thecircie is at (1 OJandtheradiusequalto J15. The loci also existentherealaxis,froms=-2tos =-4an‘dfroms=1toa=oe. break-1n mints ; root 3-46-17. 2K .45- . f“) " m e The characteristic equation for the closed-loop system is 2K .-_. 1+ ——-—- e.- ” = 0 AW 5‘ ‘l' / The angle conditim is { ,WKH, -g/m-+/ =1/80'(2£ H), '81” _/_" “as! **J”-gea;wefmw =-4£d radians = - 229.23) degrees The angle condition hecanes .—227 2w -- Zs+p.P/- =:t /Jo'(zk+/) Fork: O,theroot-1ocuswplotcan.beobtainedaesMnbe1ov. The mitotic condition states that Since . . ' 16""! =' le‘wl'lc'w‘“! = e __ The magnitude condition beau-as as [mas-9H [—- 2K 8 "W” "merootlocusorosmtlnjvaxisatw-OJQZT. Bysubstitutmga'lo. - a) = O. 3927 into this last, equation, we obtain the critical gain Kc as 15011”; l/WZJ’e3727)-+/l= 21c.= e" |/ +33%”! szkc ' A or. I_801ving£orkc.vegat ' I Kc=19.64 guinea gain for stability is 19.64. m the nobility range to: - gain K is ' 19.64>K>0 ...
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