REPETITIVE CONTROL FOR LINEAR TIME VARYING SYSTEMS. Zongxuan Sun. Research and Development Center General Motors Corporation Warren, MI 48090

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1 roceeigs of IMECE SME Ieraioal Mechaical Egieerig roceeigs Cogress of IMECE & Exosiio SME Ieraioal Mechaical New Orleas Egieerig Loisiaa Cogress November & Exosiio 7- November 7 New Orleas Loisiaa IMECE-3344 IMECE-3344 REETITIVE CONTROL OR LINER TIME VRYIN SYSTEMS Zogxa S Research a Develome Ceer eeral Moors Cororaio Warre MI 489 Ts-Chi Tsao Dearme of Mechaical a erosace Egieerig Uiversi of Califoria Los geles Los geles C 995 STRCT Reeiive corol ha asmoicall racs or rejecs erioic sigals has bee wiel se i ma alicaios. or liear ime ivaria ssem his roblem has bee horoghl sie a solve. This aer reses he aalsis a shesis of reeiive corol algorihms o rac or rejec erioic sigals for liear ime varig ssems. oh coios a iscree ime omai resls will be resee. ime varig ieral moel is embee i he feebac loo o esre asmoic erformace. I is show ha asmoic erformace ca be achieve wih a fiie imesioal coroller i he coios ime omai while i is ossible i he iscree ime omai. Simlaio resls emosrae he effeciveess of he roose algorihms.. INTRODUCTION Reeiive corol ha asmoicall racs or rejecs erioic sigals has bee wiel se i ma alicaios. To ame a few No-Circlar Trig rocess NCT for iso a camshaf machiig comer har is rive rac followig oical rig seel casig ec.. Earl wor o reeiive corol was iiiae b Ioe e. al. 98. Hara e. al. 988 resee he sabili aalsis of he ifiie imesioal reeiive corol ssem. Tomiza e. al. 989 resee he aalsis a shesis of he iscree ime reeiive corollers. Zero hase Error Tracig algorihm Tomiza 987 was se i he reeiive corol shesis. Tsao a Tomiza 994 resee a robs reeiive corol algorihm b sig filer which will r off he leaig scheme a high freqecies where moele amics rese. This is he raeoff bewee racig erformace a ssem robsess. To achieve asmoic racig erformace for corol las wih ow arameers Tsao a Tomiza 987 resee a aaive feeforwar zero hase error racig algorihm. S a Tsao resee he aaive reeiive corol algorihm a is alicaio o a Elecrohralic caor. Omaa e. al. 985 resee he reeiive corol for liear erioic ssem. Haso a Tsao 996 aresse he iscree ime reeiive corol for LTI ssem samle a a erioic rae. Less coservaive crieria have bee achieve comarig wih Omaa s resls. Tsaalis a Ioao 993 iscsse he ieral moe ricile base racig corol esig for liear ime varig ssems. Isire b he reeiive corol srcre S a Tsao 999 resee he oliear ieral moel ricile corol a reicive ieral moel corol for liear ssems wih oliear isrbace amics eseciall chaoic isrbaces i he iscree a coios ime omai resecivel. rher more S a Tsao resee he oliear ieral moel ricile corol for oliear ssems wih oliear isrbace amics. Corigh b SME

2 This aer reses he aalsis a shesis of reeiive corol algorihms o rac or rejec erioic sigals for liear ime varig ssems. oh coios a iscree ime omai resls will be resee. Necessar coiios o achieve asmoic erformace are firs erive base o he roose corol srcre. Isire b he iqe srcre of he ecessar coiios a se of sfficie coiios is he roose. The coroller coais a ime varig ieral moel i he feebac loo o esre asmoic erformace. Similar o he LTI reeiive corol esig i is show ha asmoic erformace ca be achieve wih a fiie imesioal coroller i he coios ime omai while i is ossible i he iscree ime omai. alical resls o he achievable ssem erformace wih fiie imesioal corollers i he coios ime omai are also resee. Simlaios have bee coce o emosrae he effeciveess of he roose algorihms. The res of his aer is orgaize as follows. Secio escribes he la a isrbace moels; Secio 3 reses he feebac corol esig; Secio 4 shows he simlaio resls a Secio 5 is he coclsio.. ROLEM DESCRITION efore roceeig o he roblem escriio we wol lie o iroce he followig efiiios Tsaalis a Ioao 993 o olomial iffereial oeraor DO a olomial iegral oeraor IO ha will be se o rerese he ssem moel a corollers. Defiiio : Le rerese he iffereial oeraor. The lef olomial iffereial oeraor DO is efie as: L a a a Similarl he righ DO is efie as: L a a a Defiiio : lef righ olomial iegral oeraor IO is efie as he oeraor ha mas he i o he zero sae resose of he iffereial eqaio where is he lef righ moic olomial iffereial oeraor DO. More secificall C Φ τ τ τ τ where Φ τ C are he sae rasiio marix he i a o marix resecivel corresoig o he observer coroller realizaio of he iffereial eqaio. Cosier he followig sigle i sigle o liear ime varig LTV la: where a are he i a o sigals resecivel. is he isrbace a i saisfies he followig liear ime ivaria LTI amic moel: Λ We sar wih he geeral form of isrbace amics as show above a laer we will al i o he erioic sigals. The followig assmios are mae o he la a he isrbace: ssmio : There exiss a iqe corol sigal sch ha. Le ssmio : The isrbace amic moel Λ oes have zeros i he righ half lae. ssmio 3: The isrbace is measrable b boe a smooh. To be secific aiserivaivesohe r h orer r are boe. The corol laws we are cosierig are o feebac. Deaile corol srcres will be resee i he followig secio. The corol objecive is o achieve lim for a iiial coiios of he la a isrbace. 3. OUTUT EEDCK CONTROL DESIN s show i igre he o feebac corol law is as follows: 3 N M The moivaio behi corol srcre 3 is ha we ee a self-exciaio mechaism i he feebac loo so ha i will rive he ssem o cacel o he ersise b boe isrbace oce he o becomes zero. I is well ow from he ieral moel ricile ha he isrbace moel ee o be icle i he feebac loo o esre asmoic erformace. To iser he ieral moel io he feebac srcre as show i igre we have he followig coiio: 4 s we ow if he la moel is liear ime ivaria LTI coiio 4 is boh ecessar a sfficie o achieve asmoic erformace rovie ha he close loo ssem is asmoicall sable. Obviosl his claim is o re amore Corigh b SME

3 3 Corigh b SME for he LTV ssems i reqires more coiios o achieve asmoic erformace. Or aroach o solve he above roblem is o aalze he close loo ssem firs fi o wha is he exra ecessar coiio eee o achieve asmoic erformace a he shesize he coroller base o solios of he ecessar coiios. 3. Necessar Coiios or smoic erformace I his secio we erive he ecessar coiios o achieve asmoic erformace for la wih corol laws 3 a 4. or oaio coveiece we will ro he oios a for he DO s a IO s se i he followig erivaio. Theorem : Cosier he liear ime varig la a he corol laws 3 a 4 i is ecessar o saisf he followig coiio o achieve asmoic erformace: 5 roof: s show i igre assme asmoic isrbace rejecio has bee achieve i.e. alog wih assmio we have: 6 So 7 Combiig 4 a 6 we have 8 Comarig 7 a 8 we cocle:. 3. Sfficie Coiios for smoic erformace Theorem shows ha he exra ecessar coiio reqire for he LTV ssem is coiio 5. closer loo a coiio 5 reveals a ieresig fac: his coiio is ohig b swaig bewee he oeraors. Obviosl his will alwas be re for liear ime ivaria oeraors b o for he geeral liear ime varig oeraors. Isire b he iqe srcre of coiio 5 we choose a 9 I is eas o verif he coiio 5 is saisfie. ow he corol law 3 becomes: M N The coiio 4 becomes: i.e. } { } { If we ca esig sch ha Λ X for some X wih boe coefficies he above eqaio becomes: X Λ Λ X ase o he isrbace moel he above coiio is obviosl re. Wih his we roose he followig heorem as sfficie coiios o achieve asmoic erformace. Theorem : Cosier he la a isrbace moels a ogeher wih he corol law he followig coiios are sfficie o achieve asmoic isrbace rejecio: Λ X N M X Λ is asmoicall sable. where M M a N N. roof: s show i igre 3 he o of he ssem is: rom we have So

4 M M coiio we ge M M XΛ M XΛ N M N N XΛ M XΛM N X Λ Sice Λ ogeher wih coiio we have lim. Remar: Coiio is he ime varig ieral moel while coiio secifies he sabilizig coroller. Remar: oher imora observaio is ha a secial case of he roose corol esig is Ieral Moel Corol IMC b leig N a M. Remar: Tsi a Holmberg 995 resee he ieral moel ricile a ieral moel corol ogeher IMCT for he iscree ime liear ime ivaria LTI ssem. The objec was o rovie a simlifie esig meho for isrbace rejecio er he ieral moel corol srcre. Two LTI olomial eqaios ee o be solve alhogh he are o ecessar coiios o achieve asmoic erformace. Tsaalis a Ioao 993 resee a corol esig for liear ime varig ssems o rac a class of measrable sigals. Two LTV olomial eqaios ee o be solve. Coiios a are similar o he LTV olomial eqaios erive b Tsaalis a Ioao 993. However he are esige o asmoicall rejec measrable isrbaces a as we have show he are he ol obvios solios of he ecessar coiio 5. So alhogh we call hem sfficie coiios he are acall close o ecessar coiios. Usig similar aroaches we ca exe heorem o he case of liear ime varig la wih liear ime varig isrbace amics. The he la a isrbace moels a become: 3 Λ 4 The corol law becomes: 5 N M Theorem 3: Cosier la a isrbace moels 3 a 4 ogeher wih he corol law 5 he followig coiios are sfficie o achieve asmoic isrbace rejecio: X Λ 6 X Λ M N 7 is asmoicall sable. where N N. roof: s show i igre 3 followig similar erivaios i heorem we have: M M coiio 6 we ge M M XΛ N N M XΛM N XΛ Sice Λ ogeher wih coiio 7 we have lim. Remar: The mai cosrai of coiio 6 is ha he orer of he isrbace moel Λ ca be higher ha he la orer i.e. he orer of. 3.3 Reeiive Corol Desig I his secio we will al heorem o boh coios a iscree ime omai reeiive corol esigs. The objecive of reeiive corol is o rac or rejec measrable erioic sigals where he erio is ow b he amlie a hase are ow. The Lalace omai rereseaio of he erioic isrbace moel is: st Λ s e 8 where T is he erio. Sice he above isrbace moel is ifiie imesioal here is o fiie imesioal coroller which ca solve coiio associae wih he isrbace moel 8. Defie Λ Λ ± jw a Λ ± jw L jw 9 ± 4 Corigh b SME

5 where π w T Sbsie 9 io we ge X Λ Theorem 4: Cosier he la moel isrbace moel 8 a he corol law if a saisf he followig coiios: X Λ X Λ M N is asmoicall sable. The close loo ssem will be asmoicall sable a he sea sae o is: a M X Λ c 3 where c T b as roof: T e jw >. a we ow he close loo ssem is asmoicall sable. Similarl o heorem we ca ge: M XΛ M N X Λ So M XΛM N X Λ Λ Λ c e where c Λ T > T jw c e e jw jw. Λ > c assmio a is erivaives o he r h orer are boe. The c O r. r Obviosl c is boe a hs well efie. > The we cocle M X Λ c which roves coiio 3. > X b Sice a are sricl sable boh M a Λ are boe. rom a we have: M X Λ c C c where C is a boe osiive cosa. > > Sice c O r r c is boe a lim c. > So we have lim C lim c > We he cocle as. Remar: Obviosl c is he coefficie of he orier series of he erioic isrbace. To miimize he righ ha sie of ieqali 3 oe ca alwas choose Λ sch ha c is miimize for he remaiig harmoics. Now le s al he above resls o he iscree ime omai reeiive corol esig. I his case we ca relace he iffereial oeraor wih he ime avace oeraor q. The erioic isrbace moel becomes: N Λ q q 4 where N is he erio. Obviosl we ca solve he roblem wih a fiie imesioal coroller sice he isrbace moel 4 is ol fiie imesioal. Theorem 5: Cosier he iscree form of la a he corol law ogeher wih he isrbace moel 4 he followig coiios are sfficie o achieve asmoic isrbace rejecio: q q q q X q Λ q 5 X q Λ q M q q q N q q 6 is asmoicall sable. roof: Similar o heorem. Omie. 5 Corigh b SME

6 4. SIMULTION RESULTS Cosier he followig coios ime sigle i sigle o liear ime varig ssem: 7 where a a > a b b. The measrable isrbace has he followig form: si ω α 8 where ω is ow a α are ow. ase o 8 we choose Λ as: 3 b Le 3 a a solve he above eqaio we ge: a a& a 33 a& a&& a a& 3 Choose a.5si a b he 3 3 a 33 become:.5cos.5si 34.5si 35 3 Λ jω jω ω Now we are rea o esig he feebac corollers escribe i o saisf he coiios a. The are esige i he followig wo ses: Se : Desig o saisf coiio. Choose X a ω lg hem io eqaio : 3 6 8si 4cos 8 si 4cos si cos si 6 4si cos 8 si 4cos si cos si si cos 4si cos si 8 si 4cos si cos si I is eas o verif ha 8 si 4cos si cos si > for all. 36 a b 9 Solve he above eqaio we ge: a a& 3 b a 3 b Obviosl is a sricl sable coroller. Se : Desig N M o sabilize he close loo ssem accorig o coiio. Sice X Λ is seco orer we choose followig form: as he N M 3 The he close loo olomial iffereial oeraor becomes: X Λ M N 3 So 36 is well efie. lso we ca verif ha si cos 4si cos si 3 > for all. 8 si 4cos si cos si So is a sricl sable coroller. Now we have obaie all he arameers for he liear ime varig corollers. The close loo ssem se for he simlaio is show i igre 3. Drig he simlaio we π choose a α for he isrbace moel 8 a 6 igre 4 shows he erioic isrbace. igre 5 shows he la o a corol sigal. s reice b heorem a 4 asmoic erformace has bee achieve. igre 6 a 7 show he ime varig corol arameers for N a M resecivel. 5. CONCLUSIONS This aer reses he aalsis a shesis of reeiive corol algorihms o asmoicall rac or rejec erioic sigals for liear ime varig ssems. oh coios a iscree ime omai resls have bee rovie. Necessar coiios o achieve asmoic erformace are firs erive base o he roose corol srcre. Sfficie coiios are he roose i he form of wo Diohaie eqaios. I 6 Corigh b SME

7 is show ha asmoic racig ca be achieve wih a fiie imesioal coroller i he coios ime omai while i is ossible i he iscree ime omai. REERENCES Daviso E. J. 976 "The Robs Corol of a Servomechaism roblem for Time-Ivaria Mlivariable ssems" IEEE rasacios o omaic Corol Vol. No racis.. a Woham W. M. 976 "The Ieral Moel ricile of Corol Theor" omaica Vol. No.5-E Haso R. D. a Tsao T.-C. 996 Discree-Time Reeiive Corol of LTI Ssems Samle erioic Rae roceeigs of IC 3 h Trieial Worl Cogress Sa racisco C.3-8. Hara S. Yamamoo Y. Omaa T. a Naao M. 988 Reeiive Corol Ssems: New Te Servo Ssem for erioic Exogeos Sigals IEEE Trasacios of omaic Corol Vol. C-3 No Ioe T. Naao M. a Iwai S. 98 High ccrac Corol of a roo Schroro Mage ower Sl roceeigs of he 8 h Worl Cogress of IC.6-. Mahawa. a Lo Z. Reeiive Corol of Tracig Ssems wih Time-Varig erioic Refereces Ieraioal Joral of Corol Vol. 73 No..-. Omaa T. Hara T. a Nao M. 985 Reeiive Corol for Liear erioic Ssems Elecrical Egieerig i Jaa Vol S Z. a Tsao T.-C. 999 Rejecio of Disrbace wih Noliear Damics roceeigs of he merica Corol Coferece Sa Diego C S Z. Tracig Corol a Disrbace Rejecio wih licaios o No-Circlar Trig for Camshaf Machiig h.d. Disseraio Uiversi of Illiois a Urbaa-Chamaig. S Z. a Tsao T.-C. Corol of Liear Ssems wih Noliear Disrbace Damics roceeigs of he merica Corol Coferece rligo V S Z. a Tsao T.-C. Disrbace Rejecio for Noliear Ssems roceeigs of he merica Corol Coferece chorage lasa Tomiza M. 987 "Zero hase Error Tracig lgorihm for Digial Corol" SME rasacios o Joral of Damic Ssems Measreme a Corol March Vol Tomiza M. Tsao T.-C. a Chew K.-K. 989 "alsis a Shesis of Discree-Time Reeiive Corollers" SME rasacios o Joral of Damic Ssems Measreme a Corol Se. Vol Tsaalis K. a Ioao. 993 Liear Time-Varig Ssems reice Hall. Tsao T-C. a Tomizaa M. 987 aive Zero hase Error Tracig lgorihm for Digial Corol SME Joral of amic ssem measreme a corol Vol Tsi Y. Z. a Holmberg U. 995 Robs Sochasic Corol Usig Ieral Moel ricile a Ieral Moel Corol Ieraioal Joral of Corol Vol. 6 No Woham W. M. 976 "Towars a bsrac Ieral Moel ricile" IEEE rasacios o omaic Corol November Vol. SMC-6 No Corigh b SME

8 ref - - Ieral Moel igre. loc Diagram for he O eebac Corol - igre. loc Diagram for smoic Disrbace Rejecio ref - - Ieral Moel igre 3. Reeiive Corol loc Diagram 8 Corigh b SME

9 erioic Disrbace Time Seco igre 4. The erioic isrbace 6 la O a Corol Sigal Time Seco igre 5. la o a corol sigal 9 Corigh b SME

10 Time Varig Corol arameers: a Time Seco igre 6. Time varig corol arameers: a Time Varig Corol arameers: a Time Seco igre 7. Time varig corol arameers: a 3 Corigh b SME