ELEC 372 LECTURE NOTES, WEEK 6 Dr. Amir G. Aghdam Concordia University
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1 ELEC 37 LECTURE NOTES, WEE 6 Dr mir G ghdm Cocordi Uiverity Prt of thee ote re dpted from the mteril i the followig referece: Moder Cotrol Sytem by Richrd C Dorf d Robert H Bihop, Pretice Hll Feedbck Cotrol of Dymic Sytem by Gee F Frkli, J Dvid Powell d bb EmmiNeii, Pretice Hll utomtic Cotrol Sytem by Frid Golrghi d Bejmi C uo, Joh Wiley & So, Ic, Stedytte error i feedbck ytem Coider the followig uity egtive feedbck cotrol ytem d ume tht the overll ytem i tble (thi i the mi umptio) E ( R ( G ( Y ( I thi block digrm: We hve: R( Y ( E( : Ope loop trfer fuctio : Referece iput : Output igl : Error igl Y ( R( E( R( Y ( R( ume tht ll of the pole of vlue theorem, we will hve: R( re locted i the LHP Uig the fil Lecture Note Prepred by mir G ghdm
2 e R( lim e( t) lim E( lim t It i deired to fid the tedytte error due to differet referece iput: Step iput: e r ( t) u( t) R( lim lim p, p : lim p i clled the poitio cott or poitio error cott or ttic poitio error cott or teperror cott For zero tedytte error due to tep referece iput, we mut hve which implie tht G ( mut hve t let oe itegrtor p Note tht the tbility of the cloed loop ytem eure tht the fil vlue theorem c be ued i thi ce cloed loop ytem with N itegrtor i the ope loop trfer fuctio, ie, p(, where p ( d q ( re polyomil i N q( d p ( ) d q ( ) i clled type N ytem (fctor out from the polyomil i the umertor d deomitor, if ecery, d implify them oe term i the omitor oly) For zero tedytte error due to tep referece iput we eed t let type ytem Rmp iput: r ( t) tu( t) R( e lim lim lim [ ] v, v : lim v i clled the velocity cott or velocity error cott or ttic velocity error cott or rmperror cott Lecture Note Prepred by mir G ghdm
3 3 For zero tedytte error due to rmp referece iput, we mut hve which require G ( to hve t let two itegrtor v For zero tedytte error due to rmp referece iput we eed t let type ytem 3 Prbolic iput: r ( t) t u( t) R( 3 e 3 lim lim lim [ ], : lim i clled the ccelertio cott or ccelertio error cott or ttic ccelertio error cott or prbolicerror cott For zero tedytte error due to prbolic referece iput, we mut hve which require G ( to hve t let three itegrtor For zero tedytte error due to prbolic referece iput we eed t let type 3 ytem The followig tble ummrize the reult for the tedytte error: Iput / / / 3 / i Type /( p) / v / 3 i i /t i : lim t i I other word, to hve zero tedytte error for tble cloed loop ytem with referece iput of the form give i the bove tble, the ope loop trfer Lecture Note Prepred by mir G ghdm
4 4 fuctio G ( mut hve t let my itegrtor the umber of iput pole i the origi Exmple 6: Coider the followig cloed loop cotrol ytem: R ( ( Y ( 4 Deig ( uch tht: i) There i o tedy tte error for the tep iput ii) The percetge overhoot for the tep iput i P O 5% Solutio: Uig ( d chooig the prmeter uch tht the cloed loop ytem i tble, coditio (i) will be tified With thi cotroller, we will hve: Y ( R( 4 For tbility, we mut hve poitive vlue for For coditio (ii), o the other hd, we mut hve ζ 7 Thi implie tht: ζω 4, ζ 7 ω 857 ω 863 (, < 863 I geerl, oe c ue the iterl model priciple for the tedy tte lyi of uity feedbck cotrol ytem follow Iterl model priciple: Coider the followig uity feedbck cotrol ytem: E ( R ( G ( Y ( Lecture Note Prepred by mir G ghdm
5 5 ume tht the cloed loop ytem i tble ume lo tht R ( i rtiol fuctio of d h o pole i the LHP The the tedytte error e i zero if d oly if the pole of G ( iclude ll pole of R ( For the effect of diturbce i the tedy tte, oe c ue the followig exteio of the iterl model priciple Coider the followig uity feedbck cotrol ytem: R ( ( G ( Y ( D ( ume tht the cloed loop ytem i tble ume lo tht G ( h o zero i the RHP or o the j ω xi, d tht D ( i rtiol fuctio of with o pole i the LHP The, the effect of the diturbce d (t) o the output y (t) will go to zero t, if d oly if the pole of ( iclude ll pole of D ( Note tht i the ce of referece iput, we wt the output to pproch the iput t o tht the tedytte error goe to zero but i the ce of diturbce iput, we wt the output to go to zero i the output i elimited by time) Exmple 6: Coider the followig cotrol ytem: D ( t (o tht the effect of diturbce R ( ( Y ( Deig cotroller ( uch tht: i) The tedytte error for the tep referece iput i zero Lecture Note Prepred by mir G ghdm
6 6 ii) The effect of the tep diturbce iput o the output goe to zero t iii) The percetge overhoot for tep referece iput i % Solutio: Coditio (i) d (ii) c be tified by uig ( with poitive vlue for (ote tht oly for poitive vlue of the cloed loop ytem will be tble) For thi cotroller, we will hve: Y ( R( ζω ω ζ ω 78 ( I order to tke the hitory of the error ito ccout (ot jut the tedy tte error), oe c ue oe of the followig performce idice: Itegrl of the qure of the error :ISE T e ( t) dt Itegrl of the bolute mgitude of the error :IE Itegrl of time multiplied by the bolute error : ITE Itegrl of time multiplied by the qured error : ITSE T e( t) dt T T t e( t) dt te ( t) dt T i ufficietly lrge fiite umber d i uully coveiet to chooe it the ettlig time t ISE d IE re ofte ued i the optiml cotrol of prcticl ytem The tbility of lier feedbck ytem The RouthHurwitz (RH) tbility criterio: The RH criterio i ued to fid the umber of root of give polyomil i the RHP (icludig the j ω xi Give polyomil q(, we form the Routh tble follow: Lecture Note Prepred by mir G ghdm
7 b c : b b 3 : 3 3 b b : Number of ig chge i the firt colum of the Routh tble i equl to the umber of the RHP root of q ( Exmple 63: Coider the followig polyomil: 4 3 q ( Fid the umber of root of the equtio q ( i the RHP Solutio: The Routh tble for thi polyomil i follow: There re two chge of ig i the firt colum d o two of the root re locted i the RHP The root of thi polyomil (obtied by uig MTLB) re i fct locted t 878 ± j 46 d 878 ± j 8579 Lecture Note Prepred by mir G ghdm
8 8 Note tht the RH method oly give the umber of RHP root of polyomil ot the exct loctio of the root To check the tbility of ytem, it i ufficiet to kow if there re y root i the RHP but for reltive tbility, it i importt to kow how fr the root of tble ytem re from the imgiry xi The RH method doe ot give uch iformtio Exmple 64: Coider the followig ecod order equtio: Uder wht coditio re ll of the root of the bove equtio i the LHP? Solutio: The Routh tble for thi equtio i follow: For ll of the root to be i the LHP,,, d mut hve the me ig Thi reult c be ued for ll ecod order equtio Exmple 65: Coider the followig third order equtio: 3 Uder wht coditio re ll of the root of the bove equtio i the LHP? Solutio: The Routh tble for thi equtio i follow: 3 Lecture Note Prepred by mir G ghdm
9 9 For ll of the root to be i the LHP, we mut hve: >, > > > Specil ce: I two pecil ce, the Routh tble termite premturely Specil ce : Firt elemet of row i zero, but the etire row i ot zero I thi ce, we replce the zero i the firt colum with ε > d cotiue the tble to the ed The, we let ε d cout the chge of ig i the firt colum Exmple 66: Fid the umber of RHP root of the followig equtio Fid the umber of root of the equtio q ( i the RHP Solutio: The Routh tble for thi equtio i follow:, ε 3 ε 3 ε 3 For ε the firt colum of the Routh tble will be follow: 3 Lecture Note Prepred by mir G ghdm
10 Sice there re two ig chge i the firt colum for ε, the equtio h two root i the RHP The root of thi equtio (obtied by uig MTLB) re i fct locted t 957 ± j 9 d 457 ± j 98 Specil ce lwy reult i t let oe root i the RHP other pproch for pecil ce : Root of d q ( L q ( L do ot chge hlf ple I other word, q ( d q ( hve the me umber of RHP root To verify thi clim, defie: We will hve: q ( x) q ( / x q ( x) L x ( x L x x x x x x x ) Sice the rel prt of d of q ( d q ( re the me x / hve the me ig, the umber of RHP root Specil ce : The etire row i zero I thi ce, we idetify uxiliry equtio from the coefficiet of the row right bove the zero row d proceed by differetitig thi equtio Exmple 67: Coider the followig equtio: Ue the RH method to fid the umber of RHP root of thi equtio Solutio: The Routh tble for thi equtio i follow: Lecture Note Prepred by mir G ghdm
11 ux eq' : d( 8 d ( 4 4 Note tht to proceed with the tble, we replce the zero row with the coefficiet d( of 8 Sice there i o chge of ig i the firt colum, there re o d root i the RHP Root of the uxiliry equtio ( re lwy the root of the origil polyomil For itce, i Exmple 67, ± j re the root of the uxiliry equtio d lo the root of the origil equtio Therefore, lthough thi equtio doe ot hve y RHP root, it doe hve root o the imgiry xi Root of ( re lwy ymmetricl with repect to the origi For exmple they c hve oe of the followig form: Im{} ple Im{} ple Im{} ple j ω σ j ω σ j ω jω Re{} σ σ Re{} σ j ω σ j ω Re{} For the ce whe the root of the uxiliry equtio re locted t ± jω (o repeted root, d ll other root re locted i the LHP, the output of the correpodig ytem due to zero iput (d ozero iitil coditio will hve udmped iuoidl ocilltio with the frequecy ω, t icree Lecture Note Prepred by mir G ghdm
12 Sice the root of polyomil c be eily obtied by uig computer, the RouthHurwitz criterio i more ueful i fidig the depedecy of the RHP pole to the prmeter of the equtio Exmple 68: Coider the followig cloedloop ytem: r (t) G C ( y (t) ( )( ) Deig the cotroller G C ( uch tht: i) the cloedloop ytem i tble; ii) the tedytte error due to tep iput i zero Solutio: I order to meet the give deig pecifictio, we mut hve tble cloedloop ytem with t let oe itegrtor i the forwrdpth trfer fuctio So, we will chooe G C ( d will try to et the vlue of uch tht the root of the chrcteritic equtio re ll i the LHP The chrcteritic equtio of the cloedloop ytem uig the bove cotroller i: The Routh tble will be follow: 3 3 So, we hd ee before, the coditio for the tbility of the cloedloop ytem i: >, > < < Lecture Note Prepred by mir G ghdm
13 3 For, we will hve the ecod pecil ce d the correpodig uxiliry equtio will be Therefore the cloedloop ytem will hve two root t ± j d the other root (which c be obtied by dividig the origil equtio by the uxiliry equtio) will be t Thi implie tht for, the ytem will ocillte with the frequecy ω rd/ec Lecture Note Prepred by mir G ghdm
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