Delay Analysis of Sampled-Data Systems in Hard RTOS

Transcription

1 Worl aemy of iene, Engineering an enology 4 8 Delay nalyi of ample-data ytem in Har O.. za,. lam, an.. Hain btrat In ti paper, we ave preente te effet of varying time-elay on performane an tability in te ingle-annel mltirate ample-ata ytem in ar real-time -Lin environment. e ampling ta reqire repone time tat migt eee te apaity of -Lin. o a traigt implementation wit -Lin i not feaible, beae of te lateny of te ytem an ene, ampling perio ol be le to anle ti ta. e bet ampling rate i oen for te ample-ata ytem, wi i te lowet rate meet all performane reqirement. -Lin i onitent wit it peifiation an te reoltion of te real-time i oniere. eon to aieve an effiient relt. e tet relt of or laboratory eperiment ow tat te mlti-rate ontrol teniqe in ar real-time operating ytem O an improve te tability problem ae by te ranom ae elay an aynronization. Keywor lti-rate, PID, -Lin, ample-ata, ervo. I. INODUION HE inreaing e of igital ompter i one of te mot important evelopment in ig-en embee ytem in reent year [,, 3, 4, 5, 6, 8, 9]. In te real-time ervo ompter ontrol loop, te ontrolle variable i ample /D at a itable ontant bae ampling-rate millieon in or ae an PID ontrol algoritm e ti information to ompte a new ontrol ation, wi i reontrte D/ at a fater rate integer mltiple of bae ampling rate an applie to te ontino ervo plant. ompleity arie wen te ranom elay are inerte into te ontrol loop. One ae elay i inerte between te ampler /D an te ontroller an te oter one i inerte between te ontroller an te ol D/ ing real-time -Lin trea program. ytem performane an be averely affete by te preene of ranom ae elay in te loe loop pat. In fat, mall or large ae elay an teir aoiate pae lag provie nwante oillation in te loe-loop repone, wi may even lea to intability. e reaon of elay an be e to te enor or atator arateriti, inffiient proeing pee, or ommniation lag et. e varying time elay in ompting an tranmitting te ontrol.. za i wit te Dept. of ompter iene an Engineering, Univerity, oaali, Daa-, anglae pone Et. 44; fa ; a.aza@bra.a.b.. lam, i wit te Dept. of E, Univerity, oaali, Daa-, anglae 5334@tent.b.a.b... Hain i wit te Dept. of E, Univerity, oaali, Daa-, anglae mi_bra@yaoo.om. otpt an it negative effet on real-time ontrol ytem are laifie into te elay an lo problem. e nonzero varying time elay orter tan te ampling interval yiel te elay problem an te lo of ontrol otpt to te atator or wen te varying time elay i eqal or longer tan te ampling interval. We implemente a PID ontroller algoritm in te - Lin environment for mlti-rate ampling ytem an t ynteize wit nown ae elay tat are tability an minimize ytem performane egraation. Or eperiment ow tat te effet of elay, even of onierable lengt in relation to te ample interval, an be effetively ontere ner real-time implementation. e bet ampling rate eletion for ample-ata ytem omprie many fator. e bai motivation i to minimize te ot fntion. Fat ampling rate in omparion to low ampling rate, improve omman ignal traing effetivene, an ereae enitivity to ranom plant itrbane, plant parameter variation, an mearement noie. II. ULI-E PLED-D D YE IN DIEE-IE DOIN Nyqit reontrtion teorem tate tat if te ignal i of limite banwit ten te original ignal an be reontrte eatly from te ample ignal if te ampling rate i iger tan twie te banwit of te ignal. t it i arer to analyze mlti-rate ample-ata ytem were ifferent ampler an ol wor at ifferent ampling rate. e ytem i perioi provie te ampling rate are relate by rational nmber an it an be tranforme to fit te tanar ingle rate ing lifting teniqe [7]. e ingle annel ample-ata feeba ytem i oniere in ti eperimental tet. e ontino-time plant G i eribe a follow [7] t t ww t y t y t D w t yw z t z t Dz t t For te ample-ata ontroller problem, te eqivalent irete-time repreentation of te plant ynami an average mearement are a follow 67

2 ŷ û w, te eqivalent irete-time ot fntion for te irete-time ontroller, K wit ampling time i given by, E Lt J, were, te ontant offet repreent te lower bon on te ample-ata performane. Now, we introe ranom ae elay in te plant inpt a well a in te mearement. e relting ample-ata plant i repreente by, y w wit te eqivalent irete-time ot fntion given by, E Lt J. For mall ae elay,,, w w,, an. Large ae elay, were plant an be moele by aing a elay blo oniting of elay, ea of eon, ten I I,, w w,, an. Now, we onier te problem of obtaining a tritly proper irete-time ynami ontroller for te ample-ata ytem wit ranom ae elay. e optimal irete-time LG ontroller tritly proper wit ampling time i given by [8, 9], ŷ D û ŷ were,, an. e poitive emi-efinite matrie an are te oltion to te following DE ] [ n V ] [ Were, te nonnegative efinite non-omogeneo term n V i te mearement noie ovariane []. e optimal ot of te above ample-ata ontroller i given by, ˆ opt tr J were Ŝ i te oltion to te following eqation, Worl aemy of iene, Engineering an enology

3 Worl aemy of iene, Engineering an enology 4 8 ˆ [ ˆ ] i i epeially for te more retritive problem in wi te ontroller i tritly proper i.e.,., te D ample-ata eqivalent irete-time problem provie an optimal LG-type ample-ata reglator, wi epliitly aont for ranom ae elay.. PID ontroller Derive from te LG eglator for te n Orer ervo ytem in Freqeny Domain Proportional-Integral ontroller an tate etimation provie te element of a laial Proportional-Integral-Derivative PID ontroller for a eon-orer ervo ytem wit a ingle omman inpt, a ingle mearement, an a ingle ontrol []. t t n t t n t n t t t t, y t z t t t were, omman inpt vetor, y, repreent a eire vale of an otpt vetor y. It i ame tat itrbane are nmeare an tat te Kalman-y filter optimally etimate te tate pertrbation from te et point t t by integrating [], ˆ ˆ t t t ˆ ˆ t t n n n K z t ˆ t K were, K an K are optimal Kalman filter gain. e filter generate an etimate of te tate, ˆ a well, an it provie feeba information for PI ontrol law. Ignoring te initial onition of te integrator, te ontrol ignal i t F y t t I y t ˆ t t ˆ t ˆ t were, F, I are imply algebri fntion of te et-point enitivitie an te gain matri., te Laplae tranformation of te ontrol ignal i, F I y F I y y ˆ I I ˆ ˆ e tate etimator introe ynami between te mearement z an te etimate of te tate ˆ, bt te PID I arateriti ontaine in I i lear. i arateriti wol be retaine even if tere were no etimator i.e., if an were meare an fe ba iretly. III. EVO YE IN HD EL-IE eal ime ytem i an information ytem woe orretion oe not only epen on te logial otpt of te algoritm bt alo on te moment in time wen tee otpt orre., a real time ytem i not neearily fat, bt mt be arate in time. e eign of a real-time ytem goe trog everal pae Firt te ta to be performe an te temporal retrition tat mt be atifie are ientifie, eonly te oe i written an Finally te rntime of ea ta i meare an a elability tet i one to enre tat te ta will not mi it ealine wile te ytem i rnning. eal-time i ivie into two area ar an oft real-time. We will onentrate or iion to ar real-time ytem. ere i a wie range of real-time operating ytem available in te maret to pport any peifi real-time performane reqire by te ontrol appliation. e realtime performane i limite to te low-level interation wit te arware. la of ynronization an ranom elay appear in a natral way in Intrial itribte ompting an parallel proeing, were are ommniation i e wit a lot of enor an atator. In te ervo eperiment, ranom elay are generate by mean of real-time trea program intea of ing real intrial mlti-proeor ytem. e real-time ytem i implemente wit a ombination of Lin, -Lin, ata aqiition ar, ervo ytem, availability of ore oe an a tanar P.. Harware an oftware trtre e ervo eperiment ner -Lin environment inle a tanar Pentim laboratory P wit te feeba mole an a54 ata aqiition ar. We ave loae re at Lin 7. an te bai ytem ame p an e Vope for plotting te ytem repone. -Lin ome a a et of ifferent file from te ore verion..8. From te oftware point of view, a real-time trea program i evelope. It agt an interrpt from a lo, late te interrpt, rea te lo, waite te rigt amont of time for 69

4 Worl aemy of iene, Engineering an enology 4 8 te enoer lat to ettle an rea te poition. Wit te poition nown, it alle te PID rotine to etermine te otpt voltage to rive te motor. e lo interrpt wa arbitrarily et to me. mong te two important real-time ie, one i te ability to lat te enoer in te interrpt ervie rotine an te oter one i te ability to ompte te repone witin a reaonable perio of time millieon o tat no interrpt i mie. eal-time ata aqiition framewor of mlti-rate ample-ata ytem i preente in -Lin environment. e -Lin failitie for ta anling are bai. ere i rt_ta_init, wi reate an tart a ta. e ta ize an priority an be peifie. Lin itelf i rn a a real-time ta wit te lowet priority. e ta i et p to rn at perioi interval by rt_ta_mae_perioi. e rt_ta_wait faility blo te alling ta. e ta are rn ing a imple preemptive eler. e primary mean of ommniation between te realtime ta an te Lin proee i te FIFO. e rtf_reate faility reate a FIFO of a eire ize. Data i enqee onto te FIFO by rtf_pt, retrning an error if te FIFO i fll. imilarly, rtf_get eqee ata from te FIFO, retrning an error if te FIFO i empty. e mot obvio e for ti FIFO eme i ata treaming. In a ata aqiition appliation, for eample, a real-time ta ol be et p ing rt_ta_init an rt_ta_mae_perioi to aqire ample from an I/O boar at fie interval. i ta wol en it ata to a Lin proe ing rtf_pt. e Lin proe wol be in a loop, reaing ata from te FIFO an perap writing te ata to i, ening it over a networ, or iplaying it in an X-winow. e FIFO wol erve a a bffer, o te Lin proe ol operate witot real-time ontraint. imple real-time ernel wit a priority eling eme wa implemente for te ata aqiition ar wit ifferent ampling an ol rate. a are aigne to ea mole ring te initialization tep. e flow of ata to an from te ata aqiition ar i implemente wit real-time ta trog real-time FIFO. preemptive eme wa implemente ing interrpt-bae teniqe to anle tree real-time ta. e inter-ta ommniation among te tree real-time ta i one wit are memory. e -ernel reeive a fie et of ta at te time of initialization an ea of te ta a a priority level aigne in te preemptive eme. e real-time ta ommniate wit non real-time Lin an te ata aqiition ar an te - FIFO avoi meage loe. e non real-time GUI program ing GK GIP oolit in te Lin environment provie te ig level interfae between te er an te eperiment. e et of ta e.g., ata logging, iplay an GUI et. aigne to it i non time-ritial. In te ata aqiition eme, te real-time ta elivering to Lin a low rate of relt an final ata trog are memory an irlar bffer., Lin i ligtly loae aving PU reore for non time-ritial ta iplay, ata logging et... eal-ime PID ontroller ynronization between ontroller an plant an lo of information e to te ranom elay are a potential ae of ytem intability. One poible oltion to ti problem involve ing mlti-rate teniqe an alo -Lin operating ytem i neee to perform ontrol ta an time management of ti in of ytem. In ti regar, a mltirate ample-ata ontroller an be implemente wit ifferent freqenie in te ampler an ol. In fat, a low freqeny i applie to te ampler /D to provie te ontroller wit te neeary information to tae it eiion an employ a fat freqeny to te ol D/ to apply ontrol ation., in te real-time ta, te priority of te fat freqeny to te toa mt be iger tan te low freqeny to te ampler. In eleting tee freqenie - Lin reoltion. eon mt be oniere o tat te fat freqeny to be ig enog to aieve reqire ontrol peifiation an low freqeny to be low enog to avoi te lo of information e to te ranom elay. lti-rate ontroller generate iger freqeny irete ontrol ignal in it otpt an te lower freqeny feeba ignal in it inpt. In PID ontroller [9], te tranfer fntion of a ifferentiator i, bt ti i neirable beae it magnifie any noie wi may be introe by mearement for eample, qantization effet in te /D onverter. oneqently, a ifferentiator i approimate by, wi i a ifferentiator an a low-pa filter. i approae te tre ifferentiator a i mae mall.. e atration fntion enre tat te tate pae eqation are nonlinear. We ave to eal wit two firt-orer tranfer fntion, wi will give two tate. e Proportional part of te ontroller i e K e For te Integrator Part i at e i i n te ifferentiator part e e, were e an e e final tate-pae ytem omprie two tate eqation t at e i i i an e i t an an otpt eqation at at e i e e final atration blo get ae jt to enre tat even wen te otpt of te ifferentiator i large, an may be ae to an alreay atrate integration term, te ontrol otpt remain bone an a a range itable for te D/ p 7

5 Worl aemy of iene, Engineering an enology 4 8 onverter. In te ompter ontrolle ervo ytem, te motor aft poition, meare wit a enor, i te ontrolle variable. e above onventional PID ontroller algoritm a been eigne to rea ertain loe-loop peifiation. In orer to obtain a goo ontino beavior, a ampling perio of me i oniere to be witin -Lin reoltion apaity. e PID ontrol, from or point of view, i a fntion tat tae a nmber of parameter -- te poition of te ervomotor -- an retrn te vale of te ontrol ignal tat a to be applie to te ervomotor -- te voltage tat mt be fe to te ervomotor. e teory bein te eign of PID algoritm, wi by te way i etenive, ame tat te omptational time i negligible, tat i, te time from reaing te poition of te ervomotor ntil te time wen we at i very mall. Uner normal irmtane ytem allow for a mall elay time. noter arateriti of ti type of ontrol i tat it mt be eete perioially, in oter wor, te PID algoritm mt be eete reglarly. If te time between two onetive all to te PID fntion i too large, ten te ervomotor may rea an ntable onition. Fig. epone to te tep inpt wit very large ae elay IV. EUL FO HE LOOY E Initially, we aven t introe any elay in te ytem wit ingle-rate N= ontroller. e tep repone of te ervo ytem reflet te perfet tability wit reaonable pea overoot an ettling time. en, we introe mall ae elay 5me into te ytem ing -Lin real-time trea program an oberve te repone an fon no ignifiant effet on tability an te tep repone alo remain nange., te mall ae elay wit in te bonary oen t ae any lo of information in te ontrol loop, wi i ignifiantly reponible for te intability. Wen, we introe large ae elay 5me into te ontrol loop, it inflene te tability in ytem repone by introing more oillation to ettle own into teay tate Fig.. e pea overoot inreae to.9 a well. If te pper limit of te elay i more tan te oen ampling perio, ome of te ample/ol will be lot ring te proe. We ave oniere te pper limit of large ae elay 5me in or ar real-time eperiment. two to tree ample/ol may be lot in ea time. Fig. epone to te tep inpt wit large ae elay Fig. 3 epone to te tep inpt for mlti-rate ontroller wit large ae elay e ranomne of very large ae elay ae te irreglarity in it beavior by loing ome ample/ol, wi yiel potential intability a own in Fig.. e tep repone Fig. of te ingle-rate ontroller wit very large ae elay me in te loop provie very ig pea overoot, more oillation to ettle own an t treaten tability. De to te retrition of performane peifiation, we imply an t inreae te ampling perio to avoi te lo of information. Paper [] propoe tat wit iger priority of ol D/ ta, te ae elay an be ree till te pper limit of ae elay i le tan te eire ampling perio. oneqently, te priority of ampler ta migt be ree an te ampling perio of te ampler ol be inreae to overome loing ample an o te oberve elay i ontant wi an be ompenate., te freqeny of te ol D/ i iger tan te ampler freqeny an o a mlti-rate ontroller i to be employe. epone ing mltirate ontroller i own in Fig. 3, were te pper limit of large ae elay i oniere a 5me. e tep repone for te mlti-rate ontroller ner large ae elay wit no lo of information an aynronization i oberve. It i lear tat te intability inflene Pea overoot an oillation iappeare ompletely ner ar real-time - Lin operating ytem. 7

6 Worl aemy of iene, Engineering an enology 4 8 V. ONLUION For real-time implementation of mlti-rate ampling ytem, te poition ontrol of ervo ytem wit real-time PID ontrol algoritm ing -Lin ar real-time trea program i oniere. o meet all te reqire performane for te ample-ata ytem, te bet ampling rate i oen. e ingle annel mlti-rate ontroller i ompoe by a low ampler operating at a low freqeny an a fat ol operating at a ig freqeny. Ea of te low ample i onverte into a grop of N fat ample Hol an t only one of ea grop of N ample an be e a information feeba to te ontroller ine enor operate at low freqeny. lti-rate ontrol teniqe in -Lin an olve te tability problem ae by te ranom ae elay an aynronization. e relt ing te mlti-rate ontroller i almot imilar to te fat ingle-rate one. EFEENE [] V. aanova & J. alt, ltirate ontrol for an I environment. Part I te ranom ae elay, Proeeing of te n IF worop on Linear ime Delay ytem,, nona Italy. [] J. alt & P. lberto, ltirate ontroller eign by rate eompoition, Proeeing of te 39t onferene on Deiion an ontrol,, yney, tralia. [3].. lan, eal-ime ytem an oftware, Wiley,. [4]. Wittenmar,. atian & J. Nilon, nalyi of ime Delay in ynrono an ynrono ontrol Loop, Proeeing of te 37t onferene on Deiion an ontrol, ampa, Floria, U, 998, pp [5] K. J. trom &. Wittenmar, ompter-ontrolle ytem, ir eition, Prentie Hall, 997. [6]. K. a, G. Feng &. Heet, ltirate aaptive optimal ontrol wit appliation to motor, ompter Elet. Engng, Vol. 3, No., 997, pp [7].. za &. Heet, H -Optimal ontrol Of lti-rate ampleata ytem, Proeeing of merian ontrol onferene, norage, laa, U,, pp [8] D.. erntein, L. D. Davi &. W. Greeley 986, e Optimal Projetion Eqation for Fie-Orer ample-data Dynami ompenation wit omptational Delay, IEEE ranation on tomati ontrol, Vol. -3, 986, pp [9].. za,. Heet &. Eaton, eal-time Implementation ltirate ampling ytem in -Lin Environment, Proeeing of e fort International onferene on ontrol an tomation, ontreal, anaa, 3, pp [] F.. obert, 994, Optimal ontrol an Etimation, Dover Pbliation, In. New Yor,