Advanced Control Theory

Transcription

1 Advced Cotrol Theory Review of Cotrol Sytem

2 Outlie Simle feedbck cotrol Advced Cotrol Theory

3 Feedbck Sytem U V Y m Motly H - r : the referece iut - y m : the outut of the ytem - e : the cotrol error e = r y m - u : the cotrol commd Aume y m y e e Advced Cotrol Theory

4 O-Off Cotrol Simlet feedbck cotroller Simle d No rmeter to chooe Succeed i keeig the roce vrible cloe to the referece e.g. thermott. Tyiclly reult i ytem where the vrible ocillte d lot of witchig he u mx error u mi Advced Cotrol Theory

5 Advced Cotrol Theory Proortiol-Itegrl Cotrol Proortiol-Itegrl PI where i the itegrl time 0 T E U or dt t e t e t u i i t i i i T i

6 Advced Cotrol Theory PD or PID Cotrol Proortiol Derivtive PD Proortiol-Itegrl-Derivtive PID E U or dt t de t e t u d d E U or dt t de dt t e t e t u i d d t i 0

7 PID cotrol PID Term The I-term the itegrl of the error The P-term the roortiol of the error The D-term lier extroltio of the error Rereetig The Pt The Preet The Future Advced Cotrol Theory

8 Outlie Stblity Routh Hurwitz tbility tet. Advced Cotrol Theory

9 Stbility Cotrol ytem h my defiitio o tbility Stble i the ee of Lyuov, Aymtotic tbility. A tble ytem i dymic ytem with bouded reoe to bouded iut. - BIBO tbility i text Pole d tbility The ole of the cloed-loo trfer fuctio of give ytem re locted i the RHP the ytem become utble. Tet of tbility: Routh-Hurwitz Criteri, Root-Locu, Nyquit Stbility Criteri ll tet to check ole of the trfer fuctio i the RHP Advced Cotrol Theory

10 Advced Cotrol Theory Stbility Recll: trfer fuctio For imule iut r k k k k k k k k k k q j j j r r r q m m m m m m m m m b b b b z z z b b b b R Y r k k k t k k k t k q j t j t e t e e t y k j k j j i co

11 Stbility Pole d tbility Advced Cotrol Theory

12 Advced Cotrol Theory Stbility of Cotrol Sytem A cotrol ytem i tble if d oly if ll cloed loo ole lie i the left hlf le. Cloed loo G R H Y E b b b b H G G R Y m m m m

13 Routh-Hurwitz Stbility Criterio ecery d ufficiet criterio for the tbility of LTI cotrol ytem. Routh i 876, Hurwitz i 895 determie if there re y ole i the RHP. Chrcteritic equtio The deomitor of the cloed loo trfer fuctio 0 0 where the coefficiet re rel Check y root of Δ lie i RHP of the -le Advced Cotrol Theory

14 Advced Cotrol Theory Routh-Hurwitz Stbility Criterio Rewrite All coefficiet of Δ mut hve the me ig if ll the root re i the LHP All coefficiet of Δ mut be ozero t Stbility Criterio Note it i ecery coditio

15 Advced Cotrol Theory Routh-Hurwitz Stbility Criterio d Criterio t te: build rry formultio bed o orderig the coefficiet

16 Advced Cotrol Theory Routh-Hurwitz Stbility Criterio d te: form the rry h c c c b b b , b b b c b b

17 Routh-Hurwitz Stbility Criterio The umber of root with oitive rel rt i equl to the umber of ig chge i tht firt colum. <ecery & ufficiet cod.> t colum b c h 4 ditict ce of the t colum Advced Cotrol Theory

18 Advced Cotrol Theory Alictio to Cotrol Sytem Ex Y G G R Y

19 Outlie Root Locu Advced Cotrol Theory

20 Root Locu Stbility d triet reoe i cloely relted with the loctio of ole i -le How the ole of give ytem migrte bout the -le the rmeter re vried. Root locu method Ev i 948 Advced Cotrol Theory

21 Exmle DC motor with P cotrol J B T y = J B T θ T J B Oe Loo Trfer Fuctio: Cloed Loo Trfer Fuctio: J B J B Advced Cotrol Theory

22 Advced Cotrol Theory Exmle Cloed loo ole re Cloed loo ole Nturl frequecy & Dmig rtio J B J, B J R Y 4, J J B B

23 Exmle For give lt, ie. J,B i fixed, how will ffect the loctio of ole? Im Re, B B J 4J B 4J 0 two rel root B 4J 0 double root B 4J 0 comlex cojugte root cf J, B J Advced Cotrol Theory

24 Root Locu Pth of ole trced out i -le ytem rmeter vrie from 0 to T G G Chrcteritic equtio i G 0 Cloed loo ole tifie G G G k i - le Advced Cotrol Theory

25 Root Locu Mgitude coditio Agle coditio G G k k 0,,, - Im G, 3, 5, Re G Advced Cotrol Theory

26 Additio of Oe-loo Pole d Zero Addig ole to the LHP uhe rt of the locu ito the RHP. Addig zero to the LHP ull rt of the locu ito the LHP. Advced Cotrol Theory

27 Root locu of multivrible Root locu for vrible, Chrcteritic eq.: + + = 0 R Y = 0, =vrible = cot, = vrible + = = 0 Advced Cotrol Theory

28 Outlie The cocet of the frequecy reoe Bode digrm Advced Cotrol Theory 8

29 Cocet of Frequecy Reoe Frequecy reoe: the tedy tte reoe of the ytem to iuoidl iut I frequecy reoe method, we vry the frequecy of the iut igl over certi rge d tudy reultig reoe Bic ricile For lier ytem, ie wve iut ie wve outut with me frequecy i tedy tte Note: mgitude d he my be differet Advced Cotrol Theory 9

30 Advced Cotrol Theory Cocet of Frequecy Reoe Proof G i t X t x j b j j j X X G Y j t j t j t j j e be e t y 30

31 Cocet of Frequecy Reoe G Y / X x t X i t Iut Amlitude Frequecy y t Y i t Outut Phe Amlitude Shift ubtitute = jω X Xe t X j Xe X j Xe j t j t Xe t co t j i t X co t j i t Y j G j X j Y G j X Y X G j t Im[ G Re[ G j] j] G j Advced Cotrol Theory 3

32 Cocet of Frequecy Reoe For iuoidl iut Mgitude: mlitude rtio of the outut iuoid to the iut iuoid G j Y j X j Phe: he hift of the outut iuoid with reect to the iut iuoid G j Y j X j A oitive he gle of G jω i clled he led, egtive i he lg Advced Cotrol Theory 3

33 Uefule of Frequecy Reoe The trfer fuctio c be determied exerimetlly from iut d outut igl without detiled modelig Ex: Mgitude db HllSeor Reoe Fit ex fitted Phe degree ex fitted Frequecy Hz Advced Cotrol Theory 33

34 Bode Digrm Bode digrm: grh of the trfer fuctio of lier, time-ivrit ytem veru frequecy, lotted with log-frequecy xi, to how the ytem' frequecy reoe. Coit of lot: Mgitude of G jω geerlly X: logcle Y: db cle Mgitude db HllSeor Reoe Fit ex fitted Phe of G jω geerlly X: logcle Y: lier cle Phe degree Frequecy Hz ex fitted Advced Cotrol Theory 34

35 Proertie of the Bode Digrm Bode digrm Geerlly, mgitude i lotted i decibel G j 0log0 G j db Why log cle i mgitude? Ue of log i mgitude covert multilictio ito um ytem reoe: G G 0log G 0 0log0 log G 0 0 Wht bout Phe? Nturlly tifie G G G G By kowig the bode digrm of ech block, we c build more comlex ytem eily by dditio Advced Cotrol Theory 35

36 Outlie Polr lot Nyquit lot Nyquit tbility criterio Advced Cotrol Theory

37 PolrNyquit Plot Polr lot: The locu of the mgitude of G jω v. the he of G jω o olr le ω goe from 0 to Good: it deict the frequecy reoe chrcteritic of ytem over etire frequecy rge i igle lot Bd: the lot doe ot clerly idicte the cotributio of ech idividul fctor of the oe-loo trfer fuctio Advced Cotrol Theory

38 Advced Cotrol Theory Geerl Polr Plot l = 0 Tye 0 Plot trt o the oitive rel xi with tget erediculr to the rel xi. Termil oit ω= i t the origi Curve i tget to oe of the xi which oe deed o reltive degree, odd give jω-xi m m m m b j j j b j b T j T j j T j T j j G l

39 Geerl She of Polr Plot l = Tye ω = 0 Plot trt t ifiity with gle -90, rllel to Im ω = Plot ed t the origi tget to oe of the xi l = Tye ω = 0 Plot trt t ifiity with gle -80, rllel to -Re ω = Plot ed t the origi tget to oe of the xi Every free itegrtor dd 90 o of he d rotte low frequecy ortio of Nyquit lot Advced Cotrol Theory

40 Advced Cotrol Theory Geerl Polr Plot Arrivl gle to origi: determied by -m m m m m b j j j b j b T j T j j T j T j j G l

41 Polr Plot of Stdrd Trfer Fuctio Advced Cotrol Theory

42 Polr Plot of Stdrd Trfer Fuctio Advced Cotrol Theory

43 Motivtio Nyquit tbility criterio: grhicl techique for determiig the tbility of ytem. bed o Cuchy theorem o fuctio of comlex vrible oly eed the olr lot of the oe loo ytem # of ech tye of right-hlf-le igulritie mut be kow. c be lied to ytem defied by o-rtiol fuctio, uch ytem with dely. retricted to lier, time-ivrit ytem. Advced Cotrol Theory

44 Comlex Mig i the -le For comlex fuctio F, y oit i the -le c be rereeted i the F le log i t ole of F Ex: j Im F j Im Re Re - le F - le Advced Cotrol Theory

45 Comlex Mig i the -le A cotour drw i the comlex le, ecomig but ot ig through y umber of zero d ole of fuctio, c be med to other le F l e by the fuctio F. F Coforml mig gle reerved Advced Cotrol Theory

46 Comlex Mig i the -le Comlex rtiol fuctio F Coforml mig gle reerved The re ecloed by cotour i the re to the right the cotour i trvered i the clockwie directio Advced Cotrol Theory

47 Cuchy Theorem If cotour i the -le ecircle Z zero d P ole of F d doe ot through y ole or zero of F d the trverl i i the clockwie directio log the cotour, the correodig cotour i the F-le ecircle the origi of t he F-le N=Z-P time i the CW F 0.5 N Z P 0 Advced Cotrol Theory

48 Cuchy Theorem Ex. F N Z P 3 Advced Cotrol Theory

49 Cuchy Theorem F N Z P 0 Advced Cotrol Theory

50 Cocet of Nyquit Stbility Criterio Ue Cuchy Theorem to determie tbility Drw Nyquit cotour tht ecircle the etire RHP Cotour goe log j xi, the circle bck with ifiite-rdiu hlfcircle If y ole or zero re i the RHP, they how u ecirclemet of F t origi ume o j xi ole for ow Advced Cotrol Theory

51 Cocet of Nyquit Stbility Criterio Let F=+GH For the ytem to be tble, ll zero or root of F mut lie i LHP. The umber of utble zero of F i thu Z N P Whe the ytem i oe-loo tblep=0, the Z=N Advced Cotrol Theory

52 Nyquit Stbility Criterio R + - E G Y H Exmiig tbility of F=+GH i the me exmiig the CW ecirclemet of -+j0 by GH cotour Advced Cotrol Theory

53 Nyquit Stbility Criterio R + - E G Y H Aume -m 0. If GH h k ole i the RHP d the, goe from - to +, GjHj mut ecircle -+j0 k time i the CW directio for tbility Advced Cotrol Theory

54 Ue of Nyquit Stbility Criterio I ummry Z N P Z : # of zero of +GH i RHP i.e Cloed-loo ole P : # of ole of GH i RHP Oe-loo ole N : # of CW ecirclemet of the -+j0 by GH For tble ytem Z=0, Oe-loo tble lt: P=0 N=0 A feedbck ytem i tble if d oly if the cotour i GH le doe ot ecircle the -,0 oit. Oe-loo utble lt: P 0 N=-P A feedbck ytem i tble if d oly if, for the cotour i GH le, the umber of couterclockwie of the -,0 oit i equl to the umber P of ole of with oitive rel rt. Advced Cotrol Theory

55 Ue of Nyquit Stbility Criterio Z N Z : # of zero of +GH i RHP i.e Cloed-loo ole P : # of ole of GH i RHP Oe-loo ole N : # of CW ecirclemet of the -+j0 by GH P Followig cerio oible No ecirclemet of - Sytem i tble if there re o ole of GH i RHP Otherwie utble CCW ecirclemet of - Sytem i tble if # of CCW ecirclemet = # ole of GH i RHP Otherwie utble CW ecirclemet of - Utble ytem Advced Cotrol Theory

56 d order ytem Ex: G H 00 0 R + - E G H Y Nyquit lot # ecirclemet: N = 0 # of Pole i RHP: P = 0 Z = N + P = 0 tble Advced Cotrol Theory

57 Pole/Zero o the j -Axi For ce with j -xi ole? Nyquit th mut ot through ole or zero of GH ue emicircle with the ifiiteiml rdiu j e... e G e H e e j j j e e... e j j j j Advced Cotrol Theory

58 Outlie Reltive tbility Cocet of gi & he mrgi Advced Cotrol Theory

59 Cocet of Reltive Stbility I deigig cotrol ytem, the ytem eed to be tble. Prcticlly, it i ecery tht the ytem hve dequte reltive tbility due to ucertity Reltive tbility: R How cloe the ytem i to itbility? + - E G H Y How to coider reltive tbility?. Gi & Phe Mrgi. Seitivity TF: S Advced Cotrol Theory

60 Cocet of GM d PM Cocet of Gi d he mrgi the criticl ce of itbility G j H j 0 G j H j G j H j R + - E G H Y G j H j 80 Gi mrgigm: How much gi c icree before itbility whe he lg i fixed t G j H j 80 Phe mrgipm: How much he lg c be dded before itbility whe gi i fixed t G j H j Advced Cotrol Theory

61 Gi Mrgi i Bode Digrm Gi mrgi Mgitude of the recirocl of the oe-loo trfer fuctio GH evluted t the frequecy of he gle of -80 deg. he croover frequecy 50 Bode Digrm Gm = 9.93 db t 4.0 rd/, Pm = 04 deg t rd/ Mgitude db 0-50 Gi Mrgi GM G j H j Phe deg Frequecy rd/ ω π : Phe croover frequecy Advced Cotrol Theory

62 Gi Mrgi Exmle Ex. G 0, H R + - E G H Y GM=8.7dB rd/ Bode Digrm Gm = 8.7 db t.69 rd/, Pm = 55.9 deg t rd/ 50 3 Ste Reoe Mgitude db Phe deg Amlitude G + G 4G + 4G 8:574G + 8:574G 0G + 0G Frequecy rd/ Time ecod Advced Cotrol Theory

63 Gi Mrgi Exmle Ex. G, H 5 R + - E G Y H Advced Cotrol Theory

64 Phe Mrgi i Bode Digrm Pi mrgi Defied t 80 o lu the he gle of the oe-loo trfer fuctio GH evluted t the frequecy of uity gi gi croover frequecy 50 Bode Digrm Gm = 9.93 db t 4.0 rd/, Pm = 04 deg t rd/ Mgitude db 0-50 Phe deg Phe Mrgi ω c : Gi croover frequecy Frequecy rd/ PM 80 G jc H jc Advced Cotrol Theory

65 Gi Mrgi i Nyquit Plot I Nyquit lot, reltive tbility c be determied by the ditce from -+j0 cotour Advced Cotrol Theory

66 Advced Cotrol Theory Gi d Phe Mrgi i Nyquit Plot Gi chge i Nyquit Plot T j T j j T j T j j G b

67 Coditiolly Stble Sytem Sytem with multile gi or he croover frequecie coditiolly tble ref. Root locu The he mrgi i meured t the highet gi croover frequecy Gi mrgi i meured t lowet he croover frequecy Advced Cotrol Theory

68 Prcticlly ued Mrgi The gi d he mrgi of ytem re idictio of how cloe the ytem i to itbility Phe Mrgi: he leg of 30 or or 45 PM Gi Mrgi: gi ucertity of 6 db GM Advced Cotrol Theory

69 Mximum of eitivity Seitivity TF: S G H R + - E G Y H Mximum of eitivity TF: M or S Advced Cotrol Theory

70 Outlie Led-Lg cometor deig i frequecy domi Advced Cotrol Theory

71 Frequecy Domi Deig Frequecy reoe roch c imoe the triet reoe erformce idirectly Bode digrm: ueful for the frequecy domi ecifictio Nyquit lot: ueful for ytem reltive tbility lyi. Advced Cotrol Theory

72 Led Cometor Arorite imrovemet i triet reoe d mll chge i tedy-tte ccurcy G c T T c c 0 T T Polr lot C = jt Gc j c jt α determie the mximum he gle φ m which occur t ω m Zero:- T Pole: - T i m Advced Cotrol Theory

73 Led Cometor Bode digrm C =, α=0. G c j m jt jt T G c j 0. jt j0. T Led cometor: High- filter Gol of the led cometor: to rovide ufficiet he-led gle to offet the exceive he lg i the ucometed ytem Advced Cotrol Theory

74 Led Cometor The mximum he led gle φ m occur t ω m To icree he mrgi, mgitude of the cometed ytem to cro 0 db t thi frequecy ω m but eed to cre for mgitude ditortio dded by led cometor 50 Bode Digrm Gm = 9.93 db t 4.0 rd/, Pm = 04 deg t rd/ Mgitude db 0-50 Phe deg Phe Mrgi PM 80 G jc H jc ω c : Gi croover frequecy Frequecy rd/ Advced Cotrol Theory

75 Led Cometor The mgitude of the led cotroller t: m T G c j jt jt j j m T Thi i the mout tht the led cometor will hift the mgitude lot. To hve the gi croover oit 0 db t the right oit, we hve to mke ure tht the geometric frequecy me fll t the oit where the oe-loo ucometed ytem i 0 log 0 α below 0 db. If i borbed ito the cotrol gi, the the reviou mximum mgitude would be Advced Cotrol Theory

76 Led Cometor Deig Ste 0: Aume followig led cometor Oe-loo Plt: G Cometor: G c T T c c T T Loo Gi of Cometed Sytem: T T GC G G G T T with G G Advced Cotrol Theory

77 Led Cometor Deig Ste Ste : Determie to tify ttic error cott or v Ste : Uig thi, drw Bode digrm of G, d evlute the he mrgi Ste 3: Determie the ecery he gle eeded to meet deig ec. Advced Cotrol Theory

78 Led Cometor Deig Ste Ste 4: Determie by uig φ m =required he led gle + 5~deg Fid the frequecy where: i Select thi frequecy the ew gi croover frequecy, where the mximum he hift occur Ste 5: Determie the led cometor m c T m G 0log0 - T - T Advced Cotrol Theory

79 Led Cometor Deig Ste Ste 6: Solve for c Ste 7: Check the gi mrgi. If it i ot tifctory, oe my hve to iterte. Advced Cotrol Theory

80 Exmle Nyquit Advced Cotrol Theory

81 Lg Cometor A recible imrovemet i tedy-tte ccurcy t the exee of icreig the triet-reoe time G c T T c c T T Polr lot C = Zero:- T Pole: - T G c j c jt jt Advced Cotrol Theory

82 Lg Cometor Bode digrm C =, β=0 G c j 0 jt j0t G c j c jt jt Led cometor: Low- filter Gol of the led cometor: to rovide tteutio i high frequecy rge to give ytem ufficiet he mrgi Advced Cotrol Theory

83 Lg Cometor Deig Ste Ste 0: Aume the followig lg cometor Cometor: G c T T c c T T Loo Gi of Cometed Sytem: G c T T G G G T T with G G Ste : Determie to tify ttic error cott. Advced Cotrol Theory

84 Lg Cometor Deig Ste Ste : Fid C for G Check PM d GM to ee if they meet ec If ot, fid the frequecy where G jw = -80 deg + required PM Required PM = ecified PM + 5~ deg. for fety mrgi Thi frequecy i the ew gi croover frequecy C Ste 3: Chooe the corer frequecy =/T of the zero We wt to chge the mgitude lot without chgig the he lot t the ew croover frequecy Therefore, chooe the zero t /T to be roud decde below the ew corer frequecy C Advced Cotrol Theory

85 Lg Cometor Deig Ste Ste 4: Determie d the ole loctio... We ow exmie th 0 db. G j c to fid out how much it i greter 0dB G j log c 0 0 Chooe d the the ole i t /T Ste 5: Form the lg cometor... The ctul cometor gi c Advced Cotrol Theory

86 Led-Lg Cometor Whether to ue the Led or Lg cotroller deed o the ture of your lt Led for imroved triet erformce Lg for imroved tedy-tte erformce whe you eed imroved triet erformce d tedy-tte trckig led-lg cometor G c c T T T T, Advced Cotrol Theory

87 Led Lg Cometor Phe led dd he t the ucometed gi croover frequecy thereby icreig the he mrgi Phe lg rovide tteutio llowig icree i gi t low frequecy to imrove tedy-tte erformce c =, γ = β = 0, T = 0T. c =, γ = β. Advced Cotrol Theory

88 Led Lg Cometor Qulittive ytem reoe y t y t y t yt y t y t y t yt ucometed led lg led-lg Advced Cotrol Theory

89 Outlie PID tuig rule Modifictio of PID cotrol Advced Cotrol Theory

90 PID Cotrol Mot of idutril cotroller re of the PID or it vriety How to tue the cotrol gi? few heuritic tuig rule Ziegler-Nichol tuig rule c be ued to tue cotroller without modelig Advced Cotrol Theory

91 Zeigler-Nichol Tuig Rule Zeigler Nichol tuig rule I the 940', Ziegler d Nichol devied two emiricl method ued for o-firt order lu ded time itutio ueful for the cotrol ytem where the dymic re ot reciely kow. roughly get 5% mximum overhoot i te reoe..4 Ste Reoe. Amlitude Time ec. Advced Cotrol Theory

92 Zeigler-Nichol Tuig Rule- t Method t te: Perform uit te reoe tet of lt oe-loo yt d te: Fid T, L from the reoe my ue roximtio model yt time dely Y e U T L 3 rd te: Ue tble to fid cotrol gi G c Ti T d Advced Cotrol Theory

93 Zeigler-Nichol Tuig Rule- d Method t te: Let T i =T d =0 d ue Prootiol cotrol. Icree from 0 to criticl vlue cr t the outut exhibit the utible ocilltio yt yt d te: Fid P cr from the reoe 3 rd te: Ue tble to fid cotrol gi G c Ti T d Advced Cotrol Theory

94 Zeigler-Nichol Tuig Rule If the lt doe t coform to method or method, my hve to ue other et of tuig rule. There re my tye of lt for which the Ziegler-Nichol rule re ot rorite The overhoot c be betwee 0% d 60% o te reoe. 5% w jut verge reoe The Ziegler-Nichol rule rovide trtig oit for PID gi electio. Uully oe h to do ome fie tuig to obti the deired erformce. Advced Cotrol Theory

95 Modifictio of PID: Filtered Derivtive If the et-oit, or referece igl to the PID cotroller chge with te fuctio, the error igl goig ito the cotroller will lo chge with te. The derivtive term ocited with thi te chge will be imule fuctio. Set-oit ick Thi imule c imrt dmgig hock lod to the lt. Ue Td T d ited of T d Advced Cotrol Theory

96 Modifictio of PID: PI-D Cotrol Aother wy to void thi d mooth out the derivtive ctio i to ut the Derivtive-term i the feedbck loo. PI-D cotrol Derivtive ct oly o the feedbck, ot the error. U R Td Ym Ti Ti Y m Advced Cotrol Theory

97 Modifictio of PID: I-PD Cotrol Aother wy to void thi i to ut the roortiol d derivtive-term i the feedbck loo. I-PD cotrol U R Td Ym Ti Ti Y m Advced Cotrol Theory