Function approximation and digital linearization in sensor systems

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1 Fucto appromato ad dgtal learzato sesor sstems Já Šturcel, Mroslav Kameský Abstract The am of ths artcle s to epose the developed method for soluto of epermetall obtaed sesor characterstc appromato mcrocomputer. Ths process should fulfll demads gve b clams o data processg, for eample learzato of sesor characterstc. Errors the sstem were aalzed error of method, roudg error, fluece of ADC ad the approprate learzato method choosg techque was desged. The comparso of effcec of methods was eecuted several eamples. Ke words: smart sesor sstems, sesor sgal processg, sesor characterstc appromato, sesor characterstc learzato Itroducto Wth hgher qualt of each techcal elemet the cotrol sstem structure hgher demads are put o process value measuremets. Toda measuremet devces should be able to work wth correspodet metrologcal ad fuctoal qualtes. Ths s acheved b usg mcrocomputer the measuremet chael structure, fg.. measured sesor crcuts for value sesor autocalbrato compesato ADC MMP dsturbace sesor etc. Fg. Itellget measure elemet trasducer Smart sesor sstem (SSS s autoom dgtal sstem characterzed b prmar data processg (PSI, dagostc ad auto-calbrato fuctos ad commucato cotrol. The advatages of SSS are especall hgher qualt (better evaluato of orgal sgals obtaed from sesg elemet, better fuctoal qualtes, hgher safet, less demads o commucato sstem ad the more effectve performace of dgtal (local or total cotrol cetre. Because data from tellget measuremet elemet are prmar processed, the gve actual, accurate ad relable (clea formato about the state of cotrolled process (process value ad ths accelerates realzato of cotrol algorthm ad reacto to phscal values. Besdes acqurg of output sgal from sesor elemet ad AD coverso, there are mplemeted these fuctos of PSI to SSS partcular: learzato of sesor characterstcs, fltrato of comg sgal, reducto of measured values, dsturbace correcto, damc error correcto, drect measuremet, other computatos etc.. I these cases there are mplemeted mathematcal fuctos the mcrocomputer. Therefore accurate ad demads meetg realzato of these fuctos s mportat. Cosderg lmtatos of dgtal processg SSS, e.g. small memor, small computg capact, t s evtable to use appromatos ma cases. Appromato s especall ecessar, f the real mathematcal descrpto of realzed fucto s ukow. Usg appromato for dgtal learzato of sesor characterstcs wll bee dscussed. R R PC. Learzato of sesor characterstcs Natural feature of ma sesors s ther olear characterstc. I SSS dgtal learzato s fudametall used, but SSS producers usuall reduce the oleart aalog part of measuremet chael too. More mportat s to esure repetto of the aalog part. I the fg. there s outled process of learzato usg the verse sesor characterstc. mcrocomp. SSS =f( =k AMC =f( = /k Fg. Learzato of sesor characterstc Sesor characterstc s = f( the mcrocomputer t s embedded verse characterstc = f( so that t makes the whole characterstc to be lear =k. For the learzato mcrocomputer the appromato errors b dfferet appromato methods error of method ad also roudg error ad fluece of AD coverter should be aalzed.. Methods for mathematcal fucto appromato the mcrocomputer memor Cosder umber of values (argumets < < <...< ad fuctoal values = F(, = F(, = F(,..., = F(, where F s ukow real fucto. The task s appromato of the fucto F. Several methods ca be used for ths purpose. The method selecto depeds o our kowledge about the fucto besdes the fucto values gve argumets. Two ap- AT&P oural PLUS 6 3

2 proaches are dstgushed. I ths paper Newto terpolato polomals (NIP are used wth respected odes.. Respectg of odes I ths case appromato cosstetl respects the gve values,,,..., argumets,,,...,, amed also odes or poles (fg.3. Ths s actuall terpolato, whch s appromato betwee odes. If the appromato fucto Fa s used outsde of the terval,, t s called etrapolato. Iterpolato (etrapolato s used, f errors values (acqured through epermet - measuremet could be eglected. For eample values are obtaed b meas of statstcs methods (mea value, whch s possble eas ad reproducble measuremets such as pressure or posto measuremets. Net eample s partcularl accurate measuremet. I ths case t s also recommeded to repeat the measuremet more tmes. Error s lmted wth dstace of odes or formato about the smoothess (shape of the fucto. B sesor characterstc appromato for ever terval, (or for ever two or three tervals depedg o polomal degree other appromato fucto (polomal s usuall emploed. The whole appromato fucto s composed from these partal fuctos. etrap. terpolato =F( Fg.3 Appromato wth respected odes F ukow fucto, Fa appromato fucto =Fa( etrapolato. Urespected odes Ths s the appromato, where fucto values obtaed b measuremet are accurate or the are ot respected because of other reasos ad appromato fucto could go ol ear to them as depcted the fg.4. The reaso of accurac ca be a ver dffcult epermet or too low umber of measuremets. B ths appromato method ol oe appromato fucto ca be emploed for the whole scale or for two or three tervals. There are several codtos used to decde about the best promt of two fuctos. The best kow codto s the least squares estmato. =F( Fg.4 Appromato wth ot respected odes F ukow fucto, Fa appromato fucto =Fa( 3. Newto terpolato polomal Iterpolato polomals are based o Lagrage polomals. If t s supposed that dfferece betwee poles s costat, t meas the step h = ( s costat, the otato of the polomal ca be adusted to the form of Newto terpolato polomal (NIP N ( defed as N ( = ( ( (...! h! h ( (...(! h where k s dfferece of degree k the ode : k k k k k = = ( (3 = Dsadvatage of polomal terpolato s parastc oscllato tedec whe usg hgher umber of odes (hgher degree of polomal. Ths s the reaso wh polomals of hgh degree are ot used sesor techque. For appromato of verse sesor characterstc t s sutable ol NIP of degree, or, fg.5. Appromato fucto arses as a combato of requred l polomals of degree m for odes ( N ( = Fa ( Fa = m, ;, =,,..., l If m>, umber of polomal l s l = (5 m h h Fg.5 Appromatos wth NIP m orgal NIP NIP NIP 3. NIP of degree Newto terpolato polomal of degree (NIP, fg.5 meas ol drectl measured values poles ( N ( F( Fa = ( (4 =, (6 Ths method s also called table method, where ever remembered value Fa ( = F( = s NIP of degree. Ever value wll be used for appromato o oe terval therefore small adaptato must be made the equato (4 (orgall appromato fucto Fa ( s vald for (, : ( N ( = Fa ( Fa =, ;, =,,..., l Fa ( = N ( = Fa ( ; = for = l, for (7 AT&P oural PLUS 6 4

3 Number of polomals s l=. 3. NIP of degree Newto terpolato polomal of degree s lear equato gve b two pots Fa ( = N ( = F, ( = F( ( ( F( ( It s terpolato wth lear fuctos (fg.5 Fa ( N ( = Fa ( =,,..., l ;, =, For ths case umber of polomals s l=. 3.3 NIP of degree Newto terpolato polomal of degree s square fucto solved from three pots (fg.5 ad t holds: ( ( ( ( Fa ( = N, = F F ( ( ( ( ( F( = ( ( = F F ( ( ( Fal appromato fucto s composed from l of these polomals, whle l=/: ( N ( = Fa ( Fa, =,,..., l ;, = 4. Errors of appromato mcrocomputer (8 (9 ( The bass for appromato process s requred accurac of appromato mcrocomputer. Accurate class of dgtal processg Ac C results from teded accurate class of whole dgtal measurg elemet (DME Ac DME ad of aalog measurg elemet (AME Ac AME, because for block scheme the fg.6 t ca be wrtte (equato for seral elemet placemet DME AME C Ac = Ac Ac ( DME Ac DME AME C * Ac AME Ac C Fg.6 Block scheme of dgtal measurg elemet whle accurate class of appromato s Ac C C = [%] (3 * FS where C s mamal appromato error ad * FS s full scale of mcrocomputer output value * (because b learzato the mcrocomputer mamal oleart error of AME s reduced, ths error must ot be volved Ac AME. Mamal absolute error of dgtal processg C wll be the put requremet. O the other had mal appromato error APP should be take to accout, whch s gve b mamal devato betwee appromato ad appromated fucto. For Newto terpolato polomals appromato error could be solved from the geeral relato for error of polomal terpolato (polomal of degree - ths equato s derved from statemet of Rolle from mathematcal aalss - [3] E M (4 m ( ( (... m ( m (! where M m could be determed as a mamal value of appromato fucto dervato F m ( ξ for the whole scale ξ (,. I the sesor techque verse sesor characterstc s usuall ukow therefore the aaltcal formulato of dervato F m of degree m s ukow too. The value M m could be estmated b meas of dfferece m F( [3]. Mamal value E correspods to the mamal error of mathematcal appromato APP : ( E( ( Fa( F( Ema = ma (5 APR = ma (6 ad requremet s: E ma APR (7 Accordg to ths the mamal appromato error APP could be estmated ad epresso betwee ths error ad the step h could be foud. If the accurac demads are hgh, t s better to use smulato. Net error whch should be take to accout s roudg error R. Ifluece of ths error o the overall error C has to be vestgated. As a lmtato for overall terpolato or etrapolato error b appromato of fucto F wth Lagrage polomal t could be declared value [3] R H E (8 = R The codto should be fulflled R C (9 If the mamal roudg error s dfferet from zero, for appromato error ca be wrtte ma ( E Ema APR = C H R = ( It meas that because of roudg error appromato must be desged wth smaller error so, that after cosderg both of these errors the appromato meets the requremet gve b the value C. It ca be show that our cases (NIP of degree, ad s H= ad appromato s proected wth error APR = C R ( The last error, whch s cosdered, s error caused b AD coverso AD. I the mcrocomputer there s realsed olear fucto. Ths fucto causes chage of mamal error of AD coverso (quatzato error from A to AD a for the relatve error δ' AD δ AD ( AT&P oural PLUS 6 5

4 Errors would be equal f a lear fucto s mplemeted. Process of error chage s show the fg.7 (as are labeled accordg to the fg.6. Somewhere sde of measuremet scale error mght be reduced but mamal error s alwas bgger due to the oleart. It ca be estmated b meas of frst dervato of appromated fucto. * Depedece of appromato error from put value s depcted the fg.9. Ths could be draw f the appromated fucto (orgal s kow. I the mcrocomputer the value each pot s rouded, frst AD coverter ad the output (DAC or dspla. The dfferece betwee mathematcal appromato ad fucto mplemeted mcrocomputer could be see the fg.. The error fucto forms evelope of the mathematcal appromato error fucto, fg.. ' AD N ' AD AD AD Fg.7 Ifluece of olear fucto o the error of AD coverso Resumg all metoed errors t was foud out that the sum of all errors fluecg appromato the mcrocomputer must be smaller tha demaded error APR R ' AD C (3 5. Realzato Fucto appromato was mplemeted to a mcrocomputer. Fg.8 shows NIP of degree emploed for sesor characterstc learzato. The pots ths graph are the odes ad ever le betwee two odes s oe polomal. The scale of put value dcates usage of -bt ADC. Fg. Comparso of mathematcal ad mplemeted appromato N Fg.8 Iverse characterstcs appromato usg NIP of degree Fg. Error of appromato the mcrocomputer detals Cocluso Fg.9 Error of appromato wth NIP of degree AT&P oural PLUS 6 Appromato wth Newto terpolato polomal has bee theoretcall descrbed. For appromato wth polomal the mcrocomputer error aalss has bee made. Three maor error sources has bee cosdered ad cluded the fal equato (3 for appromato error. Uderstadg of error sources s the basc part for appromato desg whch s the ma part of dgtal sesor 6

5 characterstc learzato. The eamples of errors from the appromato desg have bee demostrated. [8] Sobotka, Z.: Questos ad aswers mcroprocessors ad mcrocomputers, Applcatos. Alfa, Bratslava, Czechoslovaka, 988 Ackowledgemet Ths work was supported b proect VEGA /53/3. Refereces [] Šturcel, J.: Dagostcs ad autocalbrato smart sesor sstems. 3rd Iteratoal Smposum Mechatrocs, Kočovce, Slovaka, [] Kameský, M.: Sgal processg smart sesor sstems Graduato theses. FEI STU, Bratslava, Slovaka, [3] Hrubý, F., Kaňovský, J., Krečíček, J., Starý, J.: Iterfacg mcrocomputer wth evromet. ČSVT, Praha, Czechoslovaka, 984 [4] Rečaová, Z., Reča, B., Oleček, V.: Numercal methods. Alfa, Bratslava, Czechoslovaka, 983 [5] Mklíček, J.: Numercal methods. VUT, Bro, Czechoslovaka, 98 [6] Křížek, M.: Numercal methods. VŠSE, Plzeň, Czechoslovaka, 983 [7] Kaukč, M.: Numercal aalss I, The basc problems ad methods. MC Eerg, Žla, Slovaka, 998. Assoc. prof. Já Šturcel, PhD. Slovak Uverst of Techolog Bratslava Facult of Electrcal Egeerg ad Iformato Techolog Isttute of Cotrol ad Idustral Iformatcs Ilkovčova Bratslava Tel.: 4//69678 E-mal: a.sturcel@kar.elf.stuba.sk Ig. Mroslav Kameský Slovak Uverst of Techolog Bratslava Facult of Electrcal Egeerg ad Iformato Techolog Departmet of Measuremet Ilkovčova Bratslava Tel.: 4//69393 E-mal: mroslav.kamesk@stuba.sk AT&P oural PLUS 6 7

Fourth Order Four-Stage Diagonally Implicit Runge-Kutta Method for Linear Ordinary Differential Equations ABSTRACT INTRODUCTION

Fourth Order Four-Stage Diagonally Implicit Runge-Kutta Method for Linear Ordinary Differential Equations ABSTRACT INTRODUCTION Malasa Joural of Mathematcal Sceces (): 95-05 (00) Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method for Lear Ordar Dfferetal Equatos Nur Izzat Che Jawas, Fudzah Ismal, Mohamed Sulema, 3 Azm Jaafar