Output Tracking Combing Output Redefinition and Non-causal Stable Inversion for Non-minimum Systems

Transcription

1 Op racking Combing Op Reefiniion an Non-caal Sable Inverion for Non-imm Syem Xeha Zhang Shan Li College of Conrol Science an Engineering Zheiang Univeriy Hangzho Abrac: A meho o achieve precie op racking for non-imm yem by combining op reefiniion wih non-caal able inverion i preene. Firly op reefiniion approach can make he yem reach Sable Iniial Sae afer a fixe ime perio. An opimal olion by a earch algorihm o imize he magnie of he reefine raecory eigne wih exponenial form i obaine. hen a precie op raecory racking i achieve via able inverion. In conra wih convenional able inverion he new meho can rece he pre-acaion ime. he imlaion of he propoe meho for a ip raecory racking of a one-link flexible maniplaor how ha he yem ae can be ranfe accraely o Sable Iniial Sae wih no vibraing op over he raniion proce. Key Wor: Op racking non-imm yem op reefiniion non-caal able inverion earch algorihm Sable Iniial Sae INRODUCION Op racking for non-imm yem raw many cholar aenion all he ime []. Chen an Devaia applie able inverion meho o hi problem fir [2][3]. Bayo olve he invere ynamical eqaion in freqency omain [4] however i wa ime cong. A able inverion approach wa p forwar in ime omain in [5]. All menione able inverion iaion reqire he informaion of he whole raecory which mean he calclaion i off-line. Zo came p wih a preview-bae able inverion o rack he op raecory making online calclaion poible [6]. Ye able inverion reqire pre- an po- acaion proce an he yem i limie o have no imaginary eigenvale [7]. A caal inverion via op reefiniion can be een in [8]. I can cancel he effec of he nable zero of he yem ealing eqally wih hyperbolic yem an non-hyperbolic yem an leaing o caal conrol. h he yem can ar fromanarbirary iniial ae. An approximae caal inverion conroller i eigne in [9]. he caal inverion conroller i a fee-forwar conroller combining wih a opimal feeback conroller o achieve precie racking. Compare wih he able inverion meho caal inverion oen nee pre-acaion proce. I i frher ie o achieve re-re op racking in []. o ge ri of he effec of nable zero he whole op raecory i reefine wih exponenial form. I i applicable in boh hyperbolic an non-hyperbolic yem wih no pre- or po-acaion proce being reqire. From [] we know he non-caaliy of able inverion lie a he Sable Iniial Sae. he pre-acaion proce can make he yem rn o he Sable Iniial Sae from ime negaive infiniy [2]. o have he ae move o an hi work i ppore by Naional Nare Science Fonaion of China ner Gran expece vale in a finie ime perio an opimal ae raniion meho i propoe [3]. I can be e o ge he Sable Iniial Sae [4]. In hi paper a new meho of op reefiniion ogeher wih non-caal able inverion o achieve precie op racking for non-imm yem i propoe. We can obain he eire Sable Iniial Sae afer a fixe ime perio conrol via op reefiniion. Uing a earch algorihm we ge he mo opimal olion of he reefine op in a family rack. hen a precie op racking i achieve from he Sable Iniial Sae hrogh non-caal able inverion. Coneqenly he magnie of he reefine raecory i imize. here are everal ifferen way o achieve he ieal Sable Iniial Sae for he inverion yem. In conra wih he available echniqe he new meho can move he ae o he eire vale preciely a a limie ime. I i well eigne wih exponenial form accompanying qie mall amplie a we expece. No ocillaion circmance can be fon here. o verify he effecivene of hi new approach imlaion rel are preene for a ip raecory racking of a one-link flexible maniplaor. 2 BASIC SYSEM FRAMEWORK For a linear non-imm SISO yem x ( = Ax( + B( y ( = Cx ( ( n n Where x R A R B R C R y i a calar. he relaive egree of he yem i r in he rivial cae i i ame ha r < n ricly [4]. ( r r ( r y = CA x+ CA B (2 From eqaion (2 we have ( r ( r r = ( CA B ( y CA x (3 Sbiing eqaion (3 ino eqaion ( coneqenly ( r x = Ax + By ( ( /6/$3. c 26 IEEE 65

2 ( ( Where ( r ( r A = A B CA B CA B = B( CA B h we have he eqivalen yem ( r x = Ax + By ( y = Cx (5 here exi a marix making he coorinae ranformaion ξ = ξ x x = (6 Sbiing eqaion (6 ino (3 ( r ( r r = ( CA B ( y CA x ( r ( r r ξ = ( CA B ( y CA ( r ( r ξ = ( CA B ( y Qξ Q (7 ( r = Cy + Cξ + C y ξ Where ( r ( Cy = ( CA B ( r Cξ = CA B Qξ ( r C = ( CA B Q. Defining he exernal ae vecor ( r ξ = y y y (8 he original yem can be wrien in he new coorinae a ξ ξ = A + By (9 Frher we ge he inernal ae eqaion We name a inernal ae vecor = A+ By ( ( n r ( n r ( n r ( n r ( R A R B R ( r n y r R. An ( can be frher ecompoe. For marix A here exi inverible marix V we make he linear ranformaion V = ( hen we ge J A JBy = + (2 J A i a real Joran form of A marix J A J B can be ivie ino block a A S - B JA = V AV = A JB = V B = B where A S i a iagonal marix wih iagonal elemen ha are he eigenvale of AS wih negaive real par an A i a iagonal marix wih iagonal elemen ha are he eigenvale of A wih poiive real par. A a rel he inernal able an nable ae ifferenial eqaion can be wrien a ( ( ( r = A + By ( (3 ( ( ( r = A + By ( (4 Definiion : Invere yem Eqaion ( ogeher wih eqaion (7 compoe he invere yem ( = A( + By ( (5 ( = Cy y ( + Cξξ ( + C ( In which he reference raecory become he new inp an he original inp ( will become he new op. Definiion 2: he reference op raecory he reference op raecory y ( i efine on ime inerval [ f ]. Meanwhile he reference op raecory i reqire o be fficienly mooh which mean ( i y ( L L i = r. 3 NON-CAUSAL SABLE INVERSION 3. Sable Inverion Proce o olve he invere yem (5 we le y = y ξ = ξ ( r an ξ = [ y... y ]. We have he inernal ae ifferenial eqaion of he yem a eqaion (3 an (4 how he following bone ynamic can be calclae. A ( τ ( r ( ( - ( ( ( A τ r ( = e B y τ τ (6 = e B y τ τ (7 I can be eaily inegrae of he able par in forwar ime an he nable par in backwar ime. hen we ge V = (8 Becae ξ i alreay known from eqaion (7 we have (9 Remark he non-caal able inverion can be regare a he only precie echniqe for op racking of non-imm yem. Non-caaliy mean he yem oen ar from he acal iniial ae b a eire iniial vale exiing in he nable manifol. We call he ieal iniial ae of he yem Sable Iniial Sae []. 3.2 Sable Iniial Sae he eire Sable Iniial Sae can be obaine a - r ( x ( = ( ( = V ( (2 Wih eqaion (7 we ge he eire iniial vale of nable ae ( ( ( A τ r ( = e B y τ τ ( h Chinee Conrol an Deciion Conference (CCDC

3 o are he racking accracy x( m eqal o he eire ae x ( for all ( hen ( = ( m be flfille. I can be prove ha he iniial coniion ( = ( in able inverion cae i only nee o be coniere. he proof i preene a follow. From (5 A( A( τ ( r = + A A Aτ = + ( ( e e B y ( τ τ e ( e ( e B y ( τ τ (22 Sppoing he inernal nable ae reach i eire iniial vale ( ( ( + A τ r ( = e B y τ τ (23 Sbiing eqaion (23 ino (22 we ge + A A A( τ ( = e ( e e B y ( τ τ Aτ ( r + e By ( τ τ A Aτ = e ( e By ( τ τ + A ( τ ( r = e By ( τ τ = ( A o able inner ae A ( τ ( ( - (24 = e B y τ τ (25 Accoring o efiniion 3 we know y = ( r y = for [ f ] h ( =. hen A A( τ ( r = + A ( τ = e By ( τ τ A ( τ = e B ( - y τ τ ( ( e e B y ( τ τ reling (26 = ( Remark 2 For any ime > once he inernal nable iniial ae vale i preciely achieve he Sable Iniial Sae can be reache. Coneqenly he precie op racking can be realize. he op racking or epen on he or beween he acal ae vale an Sable Iniial Sae a. 4 MEHODS O OBAIN SABLE INIIAL SAE 4. Op reefiniion meho he main iea of op reefiniion meho i bae on reefining a new raecory wih fncion o ha a bone conino inp ignal can be obaine while canceling he effec of nable zero a he ame ime. A a rel he iniial ae of he yem can be convere o Sable Iniial Sae. We re-efine he op fncion over [ ] i y h h m 2 y = = = ( r ξ = [ y... y ] = c e m c e m (27 (28 Sbiing y ξ ino he yem. Sppoe he yem r = 2 n = 4. he eqaion (3 (4 can be rewrien a = a + b y ( a < (29 = a + b y ( a > (3 where h i he nmber of he exponenial fncion erm eere by he rericion coniion e in eire ae an op raecory. Conan m i choen in avance wih a earch algorihm o fin he mo opimiic vale. A la c i only neee o be coniere. Firly we inroce he earch algorihm o fin he opimal conan m. Regaring m max an m a he variable accoring o eqaion (3 all he conan m ( =... r are variable relaive o m max an m an m m. max ( m mmax ( (... m = mmax + = r ( r (3 he opimm principle i o have he malle racking or 2 = ( i y y y (32 An we e he reference op o ge he Sable Iniial Sae y = [ i ] hen i y = ( y 2 (33 hen y i a co fncion merely relae o variable m max an m. he plane m max an m i mehe a fir. hen a each noe he vale of i calclae. Finally m max an m which give he imal vale for y are elece a he be pair. hen he correponing m i obaine. Seconly he proce o olve he invere yem employing he new reefine raecory i lie a follow. he olion of eqaion (29 i compoe of he general olion ( of he homogeneo ifferenial eqaion g = a an a pariclar olion ( p of (29. hen he general olion i = ( + ( where g p a ( g = e ( p = e = m a A la we have a = e + = m a e m Similarly o olve eqaion (3 = ( + ( m g p (34 ( h Chinee Conrol an Deciion Conference (CCDC 67

4 becae a > = a ( g = e ( p = e m a = m (36 o make re i able coniion m be aifie hen p m e = m a = ( = (37 he new raecory ha o aify he rericion coniion originae from he yem ae an op raecory.ha mean y y y hol aify he iniial an final coniion a ime i. o be menione pariclarly he bon coniion a ime m be he Sable Iniial Sae meaning ( = ( ( = ( (38 he nmber of bon coniion i h = 2n+ 2. o implify he marix eqaion a ZC = B c (39 where Z i a conan marix C i he vecor o be olve B c i he bon vale. Once we ge C from (27 we can ge y y frher from (35 an (37 we have hen x. A la he op of he invere yem ( r ( r r = ( CA B ( y CA x ( r = C y + C ξ + C (4 y ξ 4.2 he pre-acaion meho If he pre-acaion proce work from ime negaive infiniy he Sable Iniial ae can be obaine preciely. Apparenly hi i no pracical in acal iaion [2]. he inner nable ae or δ ( converge o zero a ime ecreae o negaive infiniy lim A ( e δ ( = (4 In pracical iaion he yem ar from a finie momen p p [ ] reling he ae or δ A ( ( p = e δ ( p (42 Remark 3 Pre-acaion meho i an approximae meho o reach Sable Iniial Sae ring a finie ime inerval. 4.3 he opimal ae o ae raniion (OS he opimal ae o ae raniion meho can rani an original ae o he eire one in a finie ime. B he op amplie can be opimal. In hi way he Sable Iniial Sae can be achieve from he acal aring ae. A a bone ime inerval we can fin he inp o an yem ae x o o ranfer he yem ae from one ae x( a ime o anoher ae x( 2 a ime 2 ( 2 aring he conrol inp energy i he malle. he proce i name a he opimal ae o ae raniion. he inp eign formla i preene a ( ( ( = R Be G [ x e x ] Τ A ( 2 A( 2 ( 2 2 (43 R i an arbirary poiive efinie ymmeric real marix. A an B are given yem parameer. G ( 2 i conrollable an inverible an name a grammian marix. 2 Τ A ( 2 τ Τ A ( 2 τ G( 2 e BR B e = τ (44 We e OS meho o achieve precie ae raniion from ( i o (. 5 SIMULAION RESULS A linearize one-link flexible maniplaor moel i e for or imlaion [5].he en-effecor anglar poiion i aken a he op for he yem. he eaile yem parameer are hown a follow. x ( = Ax( + B( y ( = Cx ( where A = B = he eire op raecory i efine a y C = [.5 ] D =. [ 5] 5 (.4( 5 e (53] e (32] (2 26] = 5 (.4(2 where i = = 5 f = 26. he relaive egree r = 2 he raniion marix.5.5 = he yem can be ranforme o have he able an nable ifferenial eqaion a (29 (3. hen we can ge he conan a a =.4665 b =.248 a =.546 b = Op reefiniion meho o ge he Sable Iniial Sae he imlaion rel of he one-orer earch algorihm i hown a Fig.. Wih ifferen vale of m max an m Fig. how he or beween he reefine op an he reference op. hen we can ge he imm or ner conan vale m max = m =. he op i reefine on ime region [5] Afer he reefiniion proce he yem achieve he Sable Iniial h Chinee Conrol an Deciion Conference (CCDC

5 Sae X = [ ] i. Wih he Sable Iniial Sae he able inverion can preciely rack he eire op ring ime [526]. he imlaion rel can be hown a Fig. 2 Fig. 3. be een a Fig 4. Boh he wo echniqe work ring ime inerval [5]. I i clear ha he reefiniion meho ha a beer racking effec. Fig. 5 how he racking effec when pre-acaion ar from ime = 5 a mch beer racking effec which frher confirm he pre-acaion proce ime hol be a long a poible. 5 v. m max m.2 he reefine op he preacaion op he eire op m m max -.5 op (eg m - v. m max m m max Fig.. Error beween reefine he op an he reference op ver mmax an m ime ( Fig. 4. he op conra beween op reefiniion an pre-acaion meho a whole ime inerval.2 acal op eire op.2 acal op eire op.8.8 Op raecory (m Inp (N.m ime ( Fig. 2. Op racking combine wih op reefiniion an able inverion Fig. 5. Pre-acaion racking effec wih ime inerval Comparion wih he opimal ae raniion echniqe he opimal ae o ae raniion work ring ime perio [5] an a comparion beween he correponing op an he reefine op can be een a Fig 6. While he whole op racking effec conra i hown in Fig ime ( Fig. 3. he inp ignal of op reefiniion an able inverion proce 5.2 Comparion wih he pre-acaion echniqe We have verifie ha he racking accracy i merely relae o he or beween he acal iniial ae an he Sable Iniial Sae. he imlaion rel emonrae a big racking or when he pre-acaion proce ime i only 5. heoreically he pre-acaion ime hol be infinie. A racking rel conra beween he pre-acaion meho an op reefiniion meho can op (eg he reefine op he OS op ime ( Fig. 6. he op conra beween op reefiniion an OS proce a ime inerval [5] 26 28h Chinee Conrol an Deciion Conference (CCDC 69

6 A he imlaion rel emonrae ring he ame ime inerval o obain he Sable Iniial Sae he amplie of reefine op i alway maller han he one from OS. Frhermore here i no ocillae phenomenon fon in he reefine raecory meho. op (eg he reefine op he OS op he eire op ime ( Fig 7. he op conra beween op reefiniion an OS meho a whole ime inerval We can raw he conclion from he imlaion rel ha ring he ame ime perio he op reefiniion meho ha he mo eire op amplie a we hope. Afer he reefiniion proce he Sable Iniial Sae can be preciely obaine reling an accrae op racking a he following ime. 6 CONCLUSIONS A new meho o achieve precie op racking for non-imm yem by combining op reefiniion wih non-caal able inverion i preene. he main iea amon o reefine he op raecory o Sable Iniial Sae. hen able inverion meho can realize accrae op racking. he op reefiniion proce can rani he yem o he eire ae vale from he acal ae a a limie ime perio. Beie a earch algorihm o fin he mo opimal rack wih he lea amplie i inroce here. Compare wih he available echniqe or approach can rece he pre-acaion ime wih qie imal op amplie. he imlaion rel ppor or heory here. Jornal of Dynamic Syem Mearemen an Conrol Vol [6] Zo. Q. an S. Devaia Preview-Bae Sable-Inverion for Op racking of Linear Syem Jornal of Dynamic Syem Mearemen an Conrol Vol.2 No [7] S. Devaia Approximae Sable Inverion for Nonlinear Syem wih Nonhyperbolic Inernal Dynamic IEEE ranacion On Aomaic Conrol Vol.44 No [8] M. Benoman an G. Le Vey Sable Inverion of SISO Nonimm Phae Linear Syem hrogh Op Planning: An Experimenal Applicaion o he One-Link Flexible Maniplaor IEEE ranacion On Conrol Syem echnology Vol. No [9] X. Wang D. Chen Op racking Conrol of a One-Link Flexible Maniplaor via Caal Inverion IEEE ranacion On Conrol Syem echnologyvol.4 No [] M. Vakil R. Foohi an P.N. Nikifork Caal en-effecor inverion of a flexible link maniplaor Mecharonic Vol [] Y. Zhang S. Li Pre-acaion an opimal ae raniion bae precie racking for maximm phae yem Aian Jornal of Conrol DOI:.2/ac.265. [2] Q. Zo Opimal preview-bae able-inverion for op racking of noimm-phae linear yem Aomaica Vol [3] F. Lewi V. Syrmo Opimal Conrol John Wiley & Son Inc. hir Avene New York 995. [4] S. Devaia Nonlinear imm-ime conrol wih pre- an po-acaion Aomaica Vol.47 No [5] A. De Lca L. Lanari an G. Ulivi Nonlinear reglaion of en-effecor moion for a flexible robo arm New ren in Syem heory Genova Ialy Jl REFERENCES [] A. Iiori an C. Byrne Op Reglaion of Nonlinear Syem IEEE ranacion on Aomaic Conrol Vol.35 No [2] D. Chen Op racking conrol of nonlinear nonimm phae yem Proceeing of he 33 r Conference on Deciion an Conrol [3] S. Devaia D. Chen an B. Paen Nonlinear Inverion-Bae Op racking IEEE ranacion On Aomaic Conrol Vol.4 No [4] E. Bayo A Finie-Elemen Approach o Conrol he En-Poin Moion of a Single-Link Flexible Robo Jornal of Roboic Syem Vol. 4 No [5] D. Kwon an W. Book A ime-domain Invere Dynamic racking Conrol of a Single-Link Flexible Maniplaor h Chinee Conrol an Deciion Conference (CCDC