Chapter 1. Introduction

Transcription

1 Chaptr 1 ntroucton 1.1 Avanc Mchatroncs Systs Montorng an Control of Dstrbut Paratrs Systs Most ngnr systs ar copos of chancal-lctrcallctronc-thral subsysts an hav fwr snsors (unr-snsng) than stats n for ontorng an control an, orovr, hav fwr actuators than grs of fro (unr-actuat). So of ths systs can b ol n a frst approaton as lup paratrs systs but, n gnral, rqur or copl approachs for propr sgn an opraton. Th focus n ths Avanc Mchatroncs tt s on th coputr bas -ntgraton, -ontorng an -control of systs that can b scrb as strbut paratrs systs. Th llustratons for strbut paratrs systs wll b acoustc fls, thro-ynac fls, agntc fls, vbratons n flbl structurs, tc. Th followng topcs wll b prsnt: - ovrvw of avanc chatronc systs: sgnals vrsus powr transsson, local snsng an actuaton n contnuous systs, cntralz vrsus local control - olng an control ssus for systs: ffort-flow olng, olng an sulaton of strbut paratrs systs, opn an clos loop control - nurcal solutons for nvrs probls usng rgularzaton an sngular valu coposton thos - ynac calbraton of snsors ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t. 1

2 Avanc Mchatroncs - transnt rspons of unr-actuat an unr-sns systs - actv vbraton control n flbl structurs - acoustc fls ontorng an control - thro-ynac fls n thral procss control - agntc fls n agntc lvtaton. Fgur 1.1 shows th schatc agra of a strbut paratrs chatronc syst. n Fg. 1.1 syst varabls ar asur by transucrs, sgnal conton an convrt fro analog to gtal for an transtt to a coputr. Th coputr prfors ral t ontorng an control as wll as sgnal analyss an has two typs of outputs, on for actuator coans an th othr for syst ontorng splay. D/A D R V E R S A C T U A T O R S S Y S T E M T R A N S D U C E R S Sgnal Contonng CONTRO Coans A/D C O M P U T E R Powr Supply MONTORNG Oprator D S P A Y Fg. 1.1 Schatc agra of a strbut paratrs chatronc syst Th coans ar thr oprator coans, or coput coans that ar shown appl n a clos loop control confguraton. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

3 ntroucton 3 Coputr output for control, aftr convrson fro gtal to analog for, sns coans to actuators. Control coans ar sgnals snt to rvrs that oulat th powr fro an trnal powr supply for th actuators. An avanc chatroncs approach has to ta nto account that physcal systs ar nhrntly strbut paratrs systs an that only so of ths systs can b rprsnt by a lup paratrs ol. up paratrs chatronc systs wr alray nvstgat tnsvly n svral chatroncs boos [1-9. Fgur 1.1 rfrs to a strbut paratrs syst that can rprsnt by partal ffrntal quatons [5, 44, 110. Nurous strbut paratrs systs ar systs. Eapls analyz n ths tt ar: acoustc, thral, flu, agntc systs an flbl structurs. 1. Sgnals vrsus Powr Transsson. up Paratrs Molng of Mchatronc Systs ntgraton of systs s achv transsson of sgnals an powr btwn subsysts. Dstrbut paratrs systs olng rqur olng of propagaton lays, bounars ffcts, 3D ntractons tc, whch ar not prsnt n a lup paratrs ol or n ts bloc agra countrpart. up paratrs systs, scrb by nar T nvarant (T) Ornary Dffrntal Equatons (ODE), ar rvw n ths scton, n orr to ntfy spcfc ns for olng strbut paratrs systs. Bloc agras contan varabls assocat to th unrctonal lns btwn blocs. Ths varabls can b sn as sgnals contanng th nforaton transtt fro th output of on bloc to th nput of anothr bloc. n control ngnrng sgnal flow graphs ar sots us as an quvalnt altrnatv for to bloc agras. What s portant n councaton systs s only th nforaton contan n th sgnals, not th powr transtt by th carrr of ths nforaton. n ths cas, blocs rprsnt transforatons appl to th ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

4 4 Avanc Mchatroncs sgnal transtt, for apl lays, attnuaton or fltrng. On a councaton ln, sgnals can b transtt b rctonally. Bloc agra ols rprsnt only unrctonal transsson, fro th sgnat output of on bloc to th sgnat nput of anothr bloc an consquntly contan only a rct ol, fro caus to ffct. Th ol assocat wth a bloc corrspons only to th transfr fro th nput to th output. Ths ght b accptabl for sgnal transsson, but for powr transsson, whch s norally brctonal, ffort-flow ols ar rqur. nvrs ol, fro sr output to th rqur nput, s obtan by atr nvrson for squar T systs. nvrs ol for nonsquar T systs rqur psuo-nvrs. For non-lnar systs, no clos for soluton ght b avalabl for ol nvrson. n othr ngnrng systs, th powr transtt by th carrr bcos portant, an th quatons scrbng thr ynacs ar wrttn for varabls l forc an vlocty n chancal systs an voltag an currnt n lctrcal crcuts. Equatons usng ths varabls can also b us for bloc agra olng. Agan, whl powr oftn flows b-rctonally on a transsson ln, a bloc agra ol can rprsnt only a sngl rcton of th transsson. n fact, stat spac ols, transfr functon an bloc agra rprsntatons rqur th assgnnt of th rcton of th sgnal fro on coponnt of th ol to anothr. Eapl 1.1 Consr frst a spl chancal syst apl, shown n Fg. 1., copos of a ass, a sprng an a apr b an subjct to a forc nput F. Th vlocty v s assu th output. F v b Fg. 1. A ass-sprng-apr apl ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

5 ntroucton 5 Nwton s scon law gvs F (t) b (t) (t) Th abov ffrntal quaton can b wrttn usng v as varabl F v(t) b v(t) t 0 v(τ)τ for v( t ) (t) aplac transfor for zro ntal contons gvs 1 v(s) F(s) s b /s Du to th nput an output assgnnts, th sa syst s ol ffrntly whn th varabls F an v chang sgnaton. n ths cas, a spl nvrson of th transfr functon gvs th nvrs ol F(s) ( s b / s) v(s) n gnral, howvr, ol nvrson os not hav a clos for soluton, typcally for strbut paratrs systs. Ths rstrcts oularty an ntrchangablty to ouls wth ntcal nput an output assgnnt. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

6 6 Avanc Mchatroncs 1..1 Effort Flow Varabls an Two Port Mols Two port ols wr ntrouc for rprsntng coponnts of lctrc ntwors usng two trnals for ach port. Altrnatv nas for two port coponnts of a ntwor ar: four trnal ntwor or quarpol. Th two pol port ols hav assocat a currnt varabl an a voltag V varabl that prt th calculaton of th powr P V, transfrr through th port [8,9. Eapl 1. For an nuctanc-rsstanc -R crcut suppl by an al voltag E sourc (.. wth zro ntrnal panc), th crcut s shown n Fg Obtan th tr cuts agra. R E Fg. 1.3 A rsstanc-nuctanc R- crcut Rsstanc R an capactanc coponnts can b rprsnt as sparat lnts as a rsult of thr cuts (Fg. 1.4). R E Voltag nuctanc Rsstanc En Fg. 1.4 Thr cuts n R- crcut ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

7 ntroucton 7 Eapl 1.3 A rsstanc - nuctanc-capactanc (R C) srs crcut subjct to voltag V s shown n Fg Obtan Z(s) V(s) / (s) R E C Fg. 1.5 R--C crcut Soluton Th followng voltag rop quaton can b wrttn t V(t) R(t) (t) C (τ(τ) 0 aplac transfor of th abov quaton for zro ntal contons gvs V(s) (R s (C/s)) (s) Th panc Z(s) of th rsstanc nuctanc capactanc srs crcut s gvn by or Z(s) V(s) / (s) Z(s) R s C/s n th cas of sol boy chancs, fr boy agras rprsnt coponnts of a ult boy syst obtan by cuttng ach boy fro th syst an rprsntng bounary ffcts by local forc f an vlocty v whos prouct gvs th powr P f v. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

8 8 Avanc Mchatroncs Eapl 1.4 Assu that a flbl horzontal ro s ln to an unfn arbtrary chancal systs by sphrcal jonts. Th ro, cut fro ths systs, gv th fr boy agra shown n Fg Obtan th ol. F 1 F v 1 v Fg. 1.6 Fr boy agra of a ro n Fg. 1.6 for ach cut th ntrnal forc F an th absolut vlocty v ar ntf. Assung th flbl ro rprsnt by a lup paratrs ol, shown n Fg. 1.7, th followng quatons can b wrttn F 1 (t) ( 1 (t) (t)) (v 1 (t) v (t)) b F (t) [( 1 (t) (t)) (v 1 (t) v (t)) b F 1 (t) Evn f thr s a sprng an a apr btwn th two forcs, th qualty F (t) F 1 (t) rflcts th fact that, n ths ol, th tvaryng forc chang F 1 (t) appl to n 1 appars transtt nstantanously at n, gvn that lup paratrs ols o not account for propagaton lay. For v (t) 1 1 v (t) (t) (t) ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

9 ntroucton 9 F 1 F b v 1 v 1 Fg. 1.7 Fr boy agra for th ro th aplac transfor gvs (s) v (s)/s 1 1 (s) v (t)/s Th two forc quatons gv v 1 (s) [1/( / s b) F (s) v (s) F 1 (s) F (s) Ths last two quatons gv th cut varabls at n 1, F 1 an v 1 functon only of cut varabls at n, F an v, an paratrs b an,.. npnnt of th ynacs of th systs to whch cut 1 an wr appl. up paratrs chancal systs can b scton by cuts nto subsysts ntrfac only by forc an vlocts fn wth rspct to th cuts. For a flbl torsonal shaft, wth cut paratrs torqu T an angular vlocty ω, th ol s structurally slar: ω 1 (s) [1/( / s b) T (s) ω (s) T 1 (s) T (s) ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

10 10 Avanc Mchatroncs A gnralzaton to a varty of ngnrng systs can bas on th two port coponnts that hav assocat a flow or through varabl f an an across or ffort varabl gvng th powr as th prouct (flow) (ffort) [8. Ths s th powr passng through th juncton of two coponnts assocat to a partcular port [9. n th cas of strbut paratrs systs, th ntractons ar too copl to b rucbl to quvalnt spl two-port ols. Th rcton of th powr flow n th juncton s brctonal as oppos to th bloc agra scrpton n whch sgnals hav unrctonal flow. Th sa scrpton, usng ffort-flow two pol ports, s sutabl for systs. Th thortcal bacgroun of ths scrpton can b foun n Haltonan ynacs for obtanng powr transfr quatons [8. Whl ffort-flow cuts prt to fn powr transfr btwn subsysts, Haltonan an agrang ynacs prt sultanous olng of systs, for apl of lctrochancal systs [ Nwton-Eulr an rchhoff Equatons for a M Elctro-Mchancal Syst Effort-flow rprsntaton of systs prts asy applcaton of Nwton-Eulr quatons of oton an rchhoff quatons for lctrc crcuts. Powr transfr consrvaton law at th convrsons of lctrcal an chancal nrgs prts to ntgrat th two ols n an jont lctro-chancal ol. Th splf agra of Prannt Magnt-Drct Currnt (PM-DC) otor s shown n Fg Th stator conssts of a par of agntc pols N-S. Th rotor conssts of cols of conuctng wrs connct through th sgnts of a collctor to a DC powr supply. Fgur 1.8 shows th cut fro a chancal loa (wth cut varabls torqu T an angular vlocty ω θ/) as wll as th cut fro a DC powr supply (wth cut varabls voltag u an currnt ). Th rotor s ol chancally as a rg boy wth a ont of nrta J an a vscous frcton coffcnt b accountng for th ar rag an vscous frcton n th lubrcat barngs. Th lctrc ol of th rotor s ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

11 ntroucton 11 gvn by th lup paratrs R an, th rotor wnng crcut fro th lctrcal cut (u, ) towars th chancal cut (T, ω). R u T θ Fg. 1.8 Th agra of a PM- DC otor Th convrson of th lctrcal nrgy fro th DC powr sourc nto th chancal nrgy suppl to th loa tas plac n th DC otor, n partcular n th lctroagntc fl of th ar gap btwn th stator an th rotor. Forcs appl on rotor cols ar gnrat as a rsult of th currnt flowng through th rotor wnng surroun by th agntc fl prouc by th PM of th stator. At th sa t, th so call bac lctrootv forc (bac..f.) ar nuc voltags n th ovng rotor wnng ovng n th agntc fl. Ths two ffcts n a PM-DC otor can b ol by sparatng th chancal subsyst an th lctrcal subsyst, ach bng ol by two port lnts, as shown n Fgs. 1.9 an 1.10, rspctvly. n th lft han s of Fg. 1.9, torqu coponnts ar rprsnt aroun a cross scton of th shaft. T r nots th torqu gnrat n th lctroagntc fl of th an actng on th rotor, whl U r rprsnts th bac lctroagntc forc (bac..f.) nuc by th agntc fl n th rotor wnng n oppost to th supply voltag u. Th torqu T, an angular vlocty ω ar th cut varabls towar th chancal loa, whl th voltag u an th currnt ar th cut varabls towar th DC powr supply. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

12 1 Avanc Mchatroncs Eapl 1.5 Obtan th ol quatons. Th fr boy agra an th two port crcut facltat th rvaton of th ol quatons. J ω/ T r - b ω - J ω/ T r ω T J b ω ω T Fg. 1.9 Fr boy agra for th chancal part of th DC otor R U r u Fg Two port (U r, ) an (u, ) crcut of th lctrcal part of th DC otor Two algbrac quatons rsult fro th lup-paratrs ol of th lctro-chanc convrson phnona T r U r ω whr [N/A s th torqu constant an [Vs/ra s th lctrcal constant. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

13 ntroucton 13 n cas of al convrson ffcncy, η 1, of th lctrcal powr U r nto chancal powr T r ω, whch gvs η (U r )/( T r ω) 1 T r ω U r Usng th abov two algbrac quatons, th followng rlatonshp s obtan ω ω such that, n approprat trc unts, n [N / A an n [Vs / ra, ar of qual valu Powr losss occur u to wnng rsstanc, agntc losss, frcton tc. n th cas of nglgbl losss, al powr convrson can b assu (η 1). For th chancal part, shown n th fr boy agra of Fg. 1.11, Nwton scon law gvs: ω J Tr - b ω T For th lctrcal part shown n Fg. 1.1, th voltag rop quaton gvs: u R U r Th last two ffrntal quatons an th two algbrac quatons rgarng th lctro-chanc convrson for a syst of four ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

14 14 Avanc Mchatroncs ffrntal-algbrac quatons contanng s varabls T, ω, T r, U r, an u. Ths syst of four ffrntal-algbrac quatons rprsnts th analytcal ol of th PM-DC otor. Th lnaton of ntrnal varabls T r an U r rsults n a ol ruc to two ffrntal quatons wth four varabls of th two cuts (T, ω) an (u, ): ω J b ω T u R ω Most DC otors hav nglgbl, such that th ol, for 0, s ruc to: ω J b ω T u R ω Ths quatons, obtan usng ffort-flow cuts, prt th trnaton of th lctrcal powr u an chancal powr T ω transfrr btwn ths subsysts agrang Equatons for a M Elctro-Mchancal Syst agrang quatons ar gvn by [11: qɺ r q r q r [U r for r 1,,..N ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

15 ntroucton 15 whr s ntc nrgy U s potntal nrgy q r s th gnralz coornat r s th gnralz forc corrsponng to th wor on by th gnralz coornat q r (or voltag n th cas of th lctrcal gnralz coornat) N s th total nubr of gnralz coornats n to copltly scrb n t th coponnts of th syst. For an lctrochancal syst wth on gnralz coornat for th chancal part an on gnralz coornat for th lctrcal part, agrang quatons for th chancal an lctrcal parts of th syst ar gvn by [9, 11: [U U F ɺ whr [U U V ɺ ar th lctrc an chancal ntc nrgs U U ar th lctrc an chancal potntal nrgs s th gnralz splacnt varabl (angular splacnt) ɺ v s th gnralz vlocty (angular vlocty) s th charg n capactv coponnts ɺ s th currnt F s th gnralz forc (sspatv an appl forcs or torqus) V s th voltag (sspatv voltag rop an appl voltag) Eapl 1.6 Obtan th ol for th DC otor usng agrang quatons. agrang quatons for a PM- DC otor ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

16 16 Avanc Mchatroncs For th DC otor shown n Fg. 1.8, agrang quatons ar [U U F θɺ θ θ [U U V ɺ or, tang nto account that an θɺ ω ɺ [U U F ω θ θ [U U V whr (ω) J ω / U 0 F(ω, ) b ω T () / U 0 V(, ω) u R ω ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

17 ntroucton 17 Partal rvatvs ar Jω ω 0 θ [U U 0 θ 0 [U U 0 such that, for, agrang quatons rsult as follows (J ω) b ω T ( ) u - R Ths ar th sa as th quatons rv for th sa DC otor usng Effort-Flow rprsntaton of systs an Nwton-Eulr quatons of oton an rchhoff quatons for lctrc crcuts. Dsspatv coponnts ar th sspatv voltag rop R an th sspatv gnralz forc, n ths cas th sspatv racton torqu b ω. ω ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

18 18 Avanc Mchatroncs n, agrangan ynacs approach os not rqur ffort-flow cuts nthr for th chancal subsyst nor for th lctrcal subsyst, an no ntrnal varabls wr fn for such cass. For th ntrfac btwn lctrcal an chancal subsysts, appl torqus T (trnal loa torqu) an (otor torqu) an appl voltags u (trnal voltag) an ω (nuc voltag) ha to b howvr ntf an ths rqurs n fact th fnton of th ffort-flow cut at ths ntrfac. Eapl 1.7 Fgur 1.11 shows a plungr solno consstng of a solno of nuctanc (), pnnt of th splacnt of th plungr fro th non-nrgz poston 0. Th oton of th plungr along s u to th plungr nuc forc, caus by th solno currnt. Th currnt flows n th lctrc crcut R-() subjct to th appl trnal voltag u(t). On th chancal s, th plungr of ass M conssts of a flbl ro wth stffnss coffcnt support by a lubrcat barng wth vscous frcton coffcnt b. Obtan th ol usng agrang quatons. M b () R u Fg Th agra of a plungr solno ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

19 ntroucton 19 agrang quatons n ths cas ar ɺ [U U F whr an such that ɺ v [U U ɺ v ɺ [U U F V an [U U (ω) M v / U / F b v () / U 0 V u(t) R V ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

20 0 Avanc Mchatroncs Partal rvatvs ar v M v () [U U () 0 [U U 0 such that agrang quatons for th chancal gnralz coornat an for th lctrcal gnralz coornat rsult as follows M v () b v () u(t) R Th tr ( / ) () / corrspons to th poston pnnt forc appl by th solno on th plungr, whl th tr / () corrspons to th poston pnnt voltag rop on th solno nuctanc. n rfrnc [11 can b foun othr apls of agrangan ynacs for an lctrochancal syst n whch thr s a poston ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

21 ntroucton 1 pnnt capactanc an for an angular poston pnnt utual nuctanc. n ths scton, th sa quatons of oton of an lctro-chancal syst wr obtan usng two approachs ffort-flow cuts wth Nwton-rchhoff ynacs an agrangan ynacs. Th lttr approach s partcularly ntrstng u to th ln to Haltonan ynacs an yapunov stablty analyss for systs [9. Eapl 1.8 Fgur 1.1 shows an lctrochancal syst copos of a sprng, wth sprng coffcnt, an a col of raus ρ, wth ont of nrta J an wth N turns n whch flows a currnt / [11. Th angular poston of th col wth rgar to th horzontal plan s θ an vars fro 0 to 180. Th col s subjct to a agntc fl prouc by a solno wth n turns n whch flows a currnt q/. Th angular splacnt of th col s u to th nuc torqu rsultng th solno currnt an col currnt. Rsstancs of th col an of th solno ar R an r, rspctvly. Th col s subjct to a voltag U(t) whl th solno s subjct voltag u(t). Slf-nuctancs of th col an l of th solno ar constant,.. npnnt of th angular poston θ of th col. Th utual nuctanc M(θ), btwn th statc solno an th rotatng col, s pnnt of th angular poston θ of th col M (θ) nn π ρ n N sn θ whr nn s a charactrstc constant of th col. Th col s support by a lubrcat barng wth vscous frcton coffcnt B. Obtan th ol usng agrang quatons. agrang quatons n ths cas ar [U U F θ ɺ ϑ θ ɺ [U U V ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

22 Avanc Mchatroncs Solno Col J θ u(t) U(t) B Fg. 1.1 Th agra of rotatng sprng col an solno syst whr qɺ q [U q U v θ ω q such that ω θ [U θ U F ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

23 ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t. ntroucton 3 V U [U v U [U q q an ω J (ω) θ U ω -B F sn(θ) Ν n ρ π l M(θ ) l n N or snθ α l U 0 V U(t) R v u(t) r whr th constant α s α nn π ρ n N

24 ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t. Avanc Mchatroncs 4 Partal rvatvs ar cosθ α snθ α θ θ 0 0 q θ U [U θ snθ α snθ α l 0 0 q 0 U [U ω ω ɺ cosθ ω α snθ α ɺ ɺ cosθ ω α snθ α l ɺ ɺ

25 ntroucton 5 such that agrang quatons for th chancal gnralz coornat θ an for th lctrcal gnralz coornats an q rsult as follows or J ɺ ω α cosθ θ B ω ɺ α ɺ snθ α ω cosθ U(t) R ɺ α ɺ snθ α ω cosθ u(t) r J ɺ θ B θɺ θ α cosθ ɺ R U(t) α ɺ snθ α ω cosθ ɺ r u(t) -α ɺ snθ -α ω cosθ Ths thr nonlnar ffrntal quatons wth varabls θ(t), (t) an (t) rprsnt th ol of th syst fro Fg. 1.1, gvn th nputs U(t) an u(t),.. th rct probl. n practcal applcatons, on of th nputs, U(t) or u(t), can b hl constant. For thr or vanshng, th frst quaton gvs th qulbru poston θ 0. n th frst quaton, for th rotatonal chancal subsyst, th tr T α cos θ rprsnts th torqu prouc by th agntc fls ntracton of th solno wth th col, whch s zro whn th col an th solno ar prpncular,.. whn θ 90, or whn th two agntc fls ar paralll. As a rsult, th angl θ shoul b lt to th oan 90 ε < θ < 90 ε whr ε can b obtan fro th conton that au assbl currnts a an a prouc a nu rqur torqu T n to b abl to rotat th J-B- chancal syst,.. T n α a a cos θ ε ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

26 6 Avanc Mchatroncs n scon quaton, for th ovng col, th trs αɺ snθ αω cosθ rprsnt th nuc voltags n th col u to th t varyng currnt an u to th col angular vlocty. Slarly, th trs αɺ snθ α ωcosθ rprsnt nuc voltags n th solno u to th t varyng currnt an u to th col angular vlocty. Eapl 1.9 Fgur 1.13 shows a capactanc wth a ovng top lctro of ass an wth a gap X, whr X s th gap. Th qulbru poston of th top lctro s 0, whn no voltag s appl to th capactanc an th sprng s strtch by g / to countrbalanc top lctro wght g. Th botto lctro s sttng on a f lctrc nsulator. Th top lctro can ov vrtcally wth th splacnt, as a rsult of th t varyng voltag appl to th lctros fro a voltag sourc wth U(t) connct through wrs wth rsstanc R an nuctanc [11. Th top lctro s connct to th ovng botto n of a sprng wth sprng coffcnt. Th sprng has th top n connct to a f nsulator. Assu that th structural apng coffcnt s b. Th capacty of th t varyng gap capactanc s gvn by A C() X whr s th lctrc constant an A s th cross-sctonal ara of th capactanc. Obtan th ol usng agrang quatons. agrang quatons n ths cas ar [U U F ɺ ɺ [U U V ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

27 ntroucton 7 whr v f fra nsulator R U(t) X C nsulator f fra Fg Dagra of a syst consstng of a capactanc an a sprng such that [U U F v ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

28 ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t. Avanc Mchatroncs 8 V U [U an v (v) () U v -b F () A X - () U U () (X - )/( A c) Partal rvatvs ar 0 ) v ( A A ) (X [ U [U v ) v ( v v 0

29 ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t. ntroucton 9 ) v ( 0 A X A X / [ U [U v v ɺ ɺ agrang quatons for th chancal gnralz coornat θ an for th lctrcal gnralz coornats an q rsult as follows v b A v ɺ R U(t) A X ɺ Th followng two scon orr nonlnar ffrntal quatons wth unnowns (t) an (t) an q(t) rprsnt th ol of th syst fro Fg. 1.13, gvn th t varyng nput voltag U(t). F() b ɺ ɺ ɺ U(t) C() R ɺ ɺ ɺ whr C() s th t varyng gap pnant capactanc wth

30 30 Avanc Mchatroncs A C() X an F() A s th charg pnant forc appl by th ovng lctro to th botto n of th sprng. Th two nonlnar ffrntal quatons wth varabls an prt to ol th ffct of t varyng trnal voltag U(t) on th splacnt (t) of th ovng top lctro,.. a rct probl. 1.3 ocal Snsng an Actuaton n Spatally Contnuous Systs Spatally contnuous systs, can b ol usng thr ffort-flow cuts or agrangan ynacs. Ths ols ar n for th sgn of systs or for thr ral-t ontorng an control. Contnuous systs can b ol wth lup paratrs ols or wth strbut paratrs ols, pnng on th accptabl lvl of accuracy an olng ffcults. n both cass, th nubr of nputs can b lowr than th nubr of grs of fro, rsultng n unr-actuaton or lowr nubr of outputs than stats, rsultng n unr-snsng. Th ssu of local snsng an actuaton has to b nvstgat n both cass. Control of ths systs can b thr opn loop or clos loop. Unr-actuaton an unr-snsng hav consquncs on th prforanc of both typs of systs, but s a partcularly ffcult probl to solv for strbut paratrs ols [18. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

31 ntroucton up Paratrs Mols wth Unr-Actuaton an Unr-Snsng up paratrs ols for lnar cas can b wrttn n th for of lnar ornary ffrntal quatons (ODE): whr X(t) / A(t) X(t) B(t) u(t) G(t) w(t) y(t) C(t) X(t) D(t) u(t) X(t) n-vctor of stats wth gvn ntal contons (0) u(t) -vctor of nputs w(t) -vctor of sturbancs y(t) p-vctor of outputs A(t), B(t), G(t), C(t), D(t) t varyng atrcs. up ols for nonlnar cas can also b wrttn n th for of lnar ornary ffrntal quatons (ODE): X(t) / F(X(t), u(t), w(t)) y(t) H(X(t), u(t)) whr F an H ar nonlnar functons. Th nubr of stats, n, s fnt an, consquntly, lup paratrs ols whch ar a splf rprsntaton of contnuous systs. Crtanly, spatal rsoluton s n th forr cas lt. Unr-actuaton rsults fro fwr nputs than th nubr of grs of fro N,.. < N, an unr-snsng fro fwr outputs p than th nubr of stats,.. p < n. A contnuous syst woul hav nfnt valus for n an N, consquntly, fnt nubr of actuators an snsors wll always rsult n ths cas n unr-actuaton an unr-snsng. Gvn th coplts of strbut paratrs ols, unr-actuaton an unr-snsng ssus ar asr to b analyz usng n a frst approaton lup lnar ols rprsnt by ornary ffrntal quatons (ODE) wth t nvarant (T) paratrs. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

32 3 Avanc Mchatroncs 1.3. Dstrbut Paratrs Mols wth Unr-Actuaton an Unr-Snsng Dstrbut paratrs ols can b ta a larg varty of athatcal fors. A gnrc for s: δx(, y, z, t) / δt F(X(, y, z, t), X(, y, z, t), X(, y, z, t),..., w(t)) subjct to bounary contons an ntal contons Whl, output quaton s G(X( b, y b, z b, t),u(t)) 0 (X(, y, z, 0),u(0)) 0 H(y(, y, z, t), X(, y, z, t), u(t)) whr s th partal ffrntaton oprator, wth rgar to, y, z, varabls an th functon G an th subscrpt b rfr to bounary contons, whl th functon fns ntal contons. t can b obsrv that control varabls u(t) appar n ths cas only n th bounary contons, a typcal cas n practc whr th contnuous syst s actuat only fro spcfc syst bounars. Slarly, th outputs y ar typcally asur n so spcfc ponts, y, z. Ths ltatons rgarng local actuaton an snsng pos spcfc challngs to th sgn an prforanc of controllrs an for th ntgraton of spatally contnuous systs. 1.4 Cntralz vrsus ocal Control ocal snsng an actuaton of systs wth larg or nfnt nubr of stats s ln also to th ssu of cntralz vrsus local control. A fnt nubr of actuators can b controll thr at th actuator locaton or usng a cntralz control for all actuators. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

33 ntroucton 33 ocal controllrs us collocat actuators an snsors, hav th avantag of asr sgn an tunng an tn to prouc prctabl local syst bhavor, but ar not optal for th syst as a whol. Morovr, ynac couplngs n th syst can rsult n nffcnt or unstabl syst bhavor. Cntralz control can b sgn optally, but suffrs fro unavoabl splfcatons of th syst ol on whch thy ar bas an rqurs oftn a prohbtvly larg nubr of sgnal transssons [19. Ths ssus ar crtcal for contnuous systs strbut ovr a larg ara or for foratons. Probls 1. Consr th syst shown n Fg. 1. but wth a vscous frcton btwn th ass M an th groun, wth vscous frcton coffcnt B. Obtan v(s) gvn F(s).. For th syst shown n Fg. 1.5, obtan th four cuts rprsntaton. 3. For th fr boy agra shown n Fg. 1.7, consr that th ass of th ro s not nglgbl an that s concntrat qually at th two ns of th agra as M 1 an M. Obtan th quatons for v 1 an F 1 functon of v an F. 4. For th DC otor shown n Fg.1.8, assu that th shaft s flbl, such that n th fr boy agra fro Fg. 1.9 a torsonal sprng coffcnt s n srs wth th ont of nrta J. a. Obtan th ol wth two ffrntal quatons for th cut varabls (T, ω) an (u, ) b. Vrfy that th sa ol s obtan usng agrang quatons. 5. Assu that th plungr solno fro Fg has th plungr of ass M connct by a sprng, wth sprng coffcnt, to a ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.

34 34 Avanc Mchatroncs rght han s rg wall. Obtan th agrang quatons of oton. 6. For lctrochancal syst shown n Fg. 1.1, th utual nuctanc btwn th statc solno an th rotatng col s M (θ) nn π ρ n N sn θ. Th col, of ont of nrta J, actuats a flbl shaft support at on n by a lubrcat barng wth vscous frcton coffcnt B. Th shaft, wth torsonal stffnss coffcnt, has a loa wth a ont of nrta J, an has tslf a nglgbl ont of nrta, rlatv to th two n onts of nrta. Obtan agrang quatons for ths syst. 7. Consr th syst shown n Fg. 1.13, whch conssts of a capactanc wth a ovng top lctro of ass an wth a gap X -, whr X s th gap for th qulbru poston 0, whn no voltag s appl to th capactanc, an th sprng s strtch by g / to countrbalanc top lctro wght g. Th botto lctro s sttng on a f nsulator. Th top lctro s ovng vrtcally wth th splacnt, as a rsult of th t varyng voltag appl to th lctros fro a voltag sourc wth U(t) connct through wrs wth rsstanc R an nuctanc. Th top lctro s connct to th ovng botto n of a sprng wth sprng coffcnt an n paralll wth a apr wth apng coffcnt b. Th sprng has th top n connct to a f nsulator. Th capacty of th t varyng gap capactanc s C() c A / (X - ), whr c s a constant pnnt of th nsulator btwn th lctros. Obtan th ol usng agrang quatons. 8. For a ult-dof lnar lup paratrs chancal syst, th syst s consr unr-actuat f: a. thr ar fwr actuators than th nubr of stats b. thr ar as any actuators as th nubr of stats c. thr ar as any actuators as th nubr of grs of fro. ADVANCED MECHATRONCS - Montorng an Control of Spatally Dstrbut Systs Worl Scntfc Publshng Co. Pt. t.