An Adaptive Control Algorithm for Multiple-Input Multiple-Output Systems Using Neural Networks

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1 An Adptve Control Algorthm for Multple-Input Multple-Output Systems Usng Neurl Networks JOSE NORIEGA Deprtmento de Investgón en Fís Unversdd de Sonor Rosles y Blvd. Lus Enns, Col. Centro, CP, Hllo, Son. MEXICO Abstrt: - Mny ndustrl proesses hve multple nputs nd outputs. Suh systems re known by the ronym MIMO. In order to properly ontrol suh proesses some lner tehnques lke multvrble ontrol, robust ontrol, et., hve been used []-[]. These tehnques requre known model of the ontrolled proess. However, mny ndustrl proesses re poorly known nd nonlner. Therefore, tehnque to model nd ontrol nonlner nd prtly known systems s needed. It s known tht rtfl neurl networks, prtulrly Mult-Lyer Pereptrons, prtly fulfll suh requrements. In ths pper the grdent desent optmzton rule s ombned wth trned neurl network model for the omputton of the ontrol vetor [5],[7]. The omputed ontrol vetor would drve the nonlner MIMO system outputs to the desred opertng pont. For ths purpose, qudrt ost funton s defned. The omputton of new ontrol vetor requres the prevous omputton of the grdent of the proess. Sne dret nformton of the grdent of the proess s not vlble, n urte ppromton of the systems grdent s omputed from the neurl network model. The generl performne of the ontroller s llustrted usng smultons. A smple mthemtl model of oupled tnks systems s used for ths purpose []. Key-Words: - Neurl networks, ontrol, stblty, modelng, optmzton. Introduton Conventonl proess ontrol requres the prevous deduton nd hrterzton of mthemtl lner model of the proess. The model must be urte nd relble n order to heve n eptble ontrol performne []-[]. To produe model, the ontrol engneer needs to norporte vlble dt. Ths my be dffult tsk sne n mny ses relble nformton of the proess my not be vlble or nomplete. In ddton, mny ndustrl proesses hve multple nputs nd outputs. Hene, the number of unknown prmeters nreses resultng n more omple modelng tsk. The estng ontrol methods ppled to MIMO systems lso need relble lner models of the proess. Some of these tehnques re LQR, H, frequeny domn tehnques (Nyqus, et. []. When n nlytl model of systems s not vlble, lterntve modelng tehnques my be ppled. Amongst these tehnques re rtfl neurl networks nd fuzzy log. Mny neurl network ontrol pprohes hve been tested [5]-[7],[9]. Shffmn [], llustrtes n dptve ontrol usng dret plnt ontrol neurl network usng generlzed predtve ontrol strtegy. It s demonstrted tht the proposed method drves the system urtely to the desred response. However, the trnng proess of the neurl network my be tme onsumng nd t s demonstrted only for sngle nput sngle output systems. Nrendr [7], ppled reurrent rtfl neurl networks to ontrol nonlner systems. Nrendr pproh to ontrol s shown to be effetve. The method hs the nonvenene of requrng long trnng tme for the reurrent neurl network ontroller. Nrendr lso llustrted tht the method ould be etended to MIMO nonlner systems. In ths work, mult-lyer pereptron neurl network hs been used for the modelng of MIMO proess. Ths s two lyer neurl network usng tnh() tvton funton. It s known tht ths type of neurl network re pble to ppromte the tme response of nonlner system to n ntpted ury [7],[]. The bkpropgton lgorthm hs been ppled for the trnng proess of the neurl network []. The grdent desent lgorthm s used to ompute the ontrol vetor [5]. In order to use the grdent desent rule, qudrt ost funton s defned frst. The ost funton onsst of the squre of the dfferene between the desred system output nd the neurl network output plus the squre of the dfferene of present nd pst ontrol vetors. It s ssumed tht the mnmzton of the ost funton on eh terton wll redue the ontrol error nd

2 would therefore led to n stble ontrol of the MIMO system. For the mplementton of suh ontrol sheme t s neessry to rry out the omputton of the grdent of the system. Sne the system s ssumed to be poorly known nd nonlner, suh nformton s not redly vlble. Hene, n urte estmte of the grdent of the system s produed by omputng the grdent from the neurl network model. The results llustrted n ths work were generted usng smultons bsed on nonlner oupled tnks model []. The oupled tnks smulton suppled the requred dt used for on lne trnng of the neurl network model. The rest of the pper s orgnzed n fve setons. Seton two s foused on the desrpton of the neurl network model. Seton three desrbes the mthemtl development of the ontrol lgorthm from the proposed ost funton. Seton four llustrtes the smulton results for norml ontrol ondtons. Seton fve llustrtes the unstble operton due to poor seleton of the ontroller prmeters. Fnlly, seton s re the onlusons of ths work. Fgure, NARMAX frmework for predtve model Fgure, depts the orders of the system s n nd m. The model output s predton of the net output smple from the proess. The ury of the model depends on the trnng of the neurl network, proper hoe of n nd m nd proper seleton of trnng dt sets. Fgure llustrtes the nteronneton of the neurl network model nd the proess nputs nd outputs. Neurl Network Model Artfl neurl networks hve been suessfully ppled n the modelng of dynm systems [9]. To produe n urte predtve model, the dt suppled to the neurl network must be properly orgnzed. Ths mples the use of prevous knowledge of the proess. The NARMAX (Nonlner Auto Regressve Movng Averge wth exogenous npu frmework hs been ombned wth mult lyer pereptrons to produe nonlner dynml models [],[7]. The neurl network rhteture s embedded n the NARMAX frmework. Prt of the rhteture of the neurl network (number of neurons n the hdden lyer nd the number of nputs) s defned by the orders of the proess nd the number of nputs nd outputs. In ths work mult-lyer pereptron s trned usng the bkpropgton lgorthm. The trnng lgorthm requres pproprte etton to the nputs of the proess to produe menngful dt. As result of the trnng proess, the weghts of the neurl network would retn nformton of the response of the proess. The predtve neurl network model s llustrted n Fgure. Fgure, Predtve neurl network model nd proess The vrbles shown n Fgure re, the vetor of the proess nputs U ( u, u,..., u m ), the vetor of proess outputs Y ( y, y,..., y n ) nd the estmton of the proess outputs Yˆ ( yˆ, yˆ,..., yˆ n ). In generl the neurl network model s wrtten s ˆ o Y ( W (tnh( W P)) () Ths s two lyer neurl network where, P re the synpt weghts of the output lyer, the weghts of the nput lyer nd the nput vetor respetvely. o W, W y Control Algorthm The ontrol lgorthm s bsed on the dret pplton of the grdent desent rule to qudrt ost funton. A suessful mnmzton step of the qudrt ost funton would led to the

3 omputton of new ontrol vetor tht would redue the dfferene of the system output nd the desred system output. For ths purpose the qudrt ost funton s defned s. Where the error T T J [ Es QEs + E RE ] () E s s defned s. Es Yd ( Yˆ( () Ths s the dfferene between the desred proess output nd the tul neurl network output. The error E s. E U U ( t ) () Whh s the dfferene of the tul ontrol vetor nd the prevous ontrol vetor. It s neessry to ndte tht t s epeted tht the error E progressvely redues to ero durng the ontrol ton. Fnlly, Q nd R re postve semdefntve weghtng mtres. The grdent desent rule requres the omputton of the grdent of the ost funton. The omputton of the ontrol vetor s therefore epressed s. The result of Equton (7) s represented n mtr known s the Jobn. The Jobn of the neurl network model s. Yˆ( ( ( M m( ( ( ( M ( m n( L n( L ( O M ( n L m( (9) Fnlly, the ontrol vetor s omputed by ddng epressons from Equton (7) nto Equton (9) nd the resultng epresson s substtuted n Equton (5). As results, the ontrol vetor s omputed usng Equton (). ˆ( U ( U + η [ QEs + IRE ] () Here, the grdent of the ost funton s ombned wth Equton () nd the grdent desent rule s shown n Equton (5). Ths ontrol lgorthm s grphlly represented n Fgure n blok dgrm form. J U ( U η (5) Here, U ( u, u,..., u m ) s the ontrol vetor nd η, s the ontrol gn. The grdent of the ost funton s. J Yˆ( QEs + IRE () Where Y ˆ ( yˆ, ˆ,..., ˆ y yn ) s the neurl network vetor output. The grdent of the neurl Yˆ ( s epressed n the Equton (7). Y ( { W ˆ o seh ( W P( )}{( W P( )} (7) Here, the operton n between keys ndtes n element by element multplton for the two vetors. The nput vetor s. P ( y, y,..., y, u, u,..., u ) () ( n m Fgure, Blok dgrm of the ontroller Ths dgrm shows the ontrol loop nd the neurl network trnng loops. It s ssumed tht the trnng stge took ple prevously to the tvton of the ontrol stge. The weght mtres Q nd R re hosen to ompenste the mesured output vetor nd the mesured ontrol vetor. Suh ontrol lgorthm requres few omputtonl resoures for ts mplementton. It s lso esy to progrm n dgtl omputer. However, t s not possble to gurntee the stblty of the ontrolled proess usng ths lgorthm. Ths s due to the ft tht the grdent

4 desent rule my get stuk n lol mnm [9]. In ddton, the grdent desent rule s n slow onvergng lgorthm. Therefore, the omputton of the ontrol vetor my not be pproprte for proesses wth rpd response. Smultons The performne of the ontrol lgorthm desrbed heren s llustrted by smultons, usng nonlner oupled tnks model. Two tests re presented. Frst, the response of the ontrol lgorthm s smulted for prevous defned vlues of the desred response vetor. Seond, the ontrol lgorthm s tested for dsturbnes. Also desrpton of the mthemtl model of the nonlner oupled tnks s presented. y y ρ p + g ( k + k + + u ) u u p u (). Nonlner oupled tnks model The nonlner oupled tnks proess onsst of pr of tnks hvng onstnt trnsversl re nd oupled through tube wth onstnt flow resstne. In ths se, the frst tnk s heted by n eletr heter. For our purpose the lqud ontned n the tnks s wter, therefore densty ρ s nd the spef het p s. The nputs to the proess re: temperture of the nput flow T e ( ndted by u, nput flow rte F e ( ndted by u nd nput het Q e ( denoted by u. The proess outputs re: the level of wter n tnk L ( ndted by y nd wter temperture n tnk T s ( ndted by y (. It s ssumed tht both tnks re well solted to redue het leks. It s lso ssumed tht the temperture of the nput flow s not ontrollble nd t s used s dsturbne nput durng smultons. Therefore, ths proess onsst of two nputs nd two outputs. The oupled tnks proess s llustrted n Fgure. Here the prmeters nd re the orrespondng res of trnsversl seton of tnks nd respetvely, k nd k re the nverse of the flow resstnes between the two tnks n t the flow output n tnk respetvely nd g s the elerton due to grvty. The proess outputs re y nd y whh represent the level n tnk nd wter temperture n tnk respetvely. For our smulton purpose the prmeters re set to 5,, k y k. Therefore, the prmeters of the set of dfferentl equtons re.9 g ( k + k.9 ).9 () To smulte the response of the proess onventonl numer ntegrton lgorthm s used. Sne the ontrol lgorthm s dsrete, the ntegrton lgorthm s rrnged to ompute the sttes on eh desred smple tme. The results of the ntegrton proess re fed to the neurl network nd to the ontroller s llustrted n Fgures nd. Fgure, Coupled tnks proess The nlytl model of the oupled tnks proess s desrbed by set of four dfferentl equtons for the sttes nd two epressons for the outputs of the proess []. These equtons re llustrted below.. Control of the proess The ontrol lgorthm n Equton () requres few djustments n order to be ppled to the oupled tnks. Frst, t s neessry to defne mtres Q nd R. In ths se both mtres re of sze nd the elements re

5 Q R () of the output of the oupled tnks proess. In ths eperment the mtres n Equton () tke the sme vlue s n the eperment desrbed n seton.. The dsturbne s n brupt hnge from T e ( 5 to T e (.75. Fgure llustrtes the smulted response of the ontrolled proess Entrds The elements n the two mtres were hosen s result of sequene of smulton tests. The objetve of these tests ws to produed n eptble response of the ontrolled proess. A ontrol gn of η.5 ws seleted. Lke n the se of the weghtng mtres, the vlue of the ontrol gn ws seleted for the purpose of llustrtng n eptble ontrol performne, not n optml ontrol performne. The desred outputs of the proess were defned s T s.5 (ndted by y ) nd F s.5 (ndted by y ) for the temperture of wter n tnk nd wter level n tnk respetvely. Fgure 5 llustrtes the response of the ontrolled proess u u u Entrds Respuest del Proeso y y Fgure 5, Response of the ontrolled proess It n be seen tht the response of the proess rehes the desred outputs wth smll overshoot. The nputs u nd u re the proess nputs nd the nput u s n unontrollble nput used for the smulton of dsturbnes. In Fgure 5 the ontrolled proess s unffeted by perturbtons nd therefore, the nput u remns onstnt.. Response to nput dsturbnes Some smulted dsturbnes were ppled to the ontrolled proess to study ts performne. The smulted dsturbnes onssted of the ntroduton of n brupt hnge of the temperture of the nput u of the proess. It ws ssumed tht the hnge n the temperture would produe noteble hnge u u Respuest del Proeso 7 5 y Fgure, Response to dsturbne It n be seen tht the dsturbne produes noteble hnge n the outputs of the proess. However, the ontrol lgorthm produes pproprte ontrol vetors to ompenste the effet of the dsturbne. It s evdent tht the effet of the dsturbne s lrger t the output y. 5 Stblty The stblty of the ontrol lgorthm depends on the pproprte seleton of the elements of Q nd R nd the hosen ontrol gn η. In order to llustrte the stblty of the proess two smulted ses were studed. Frst, the effet of the prmeters of Q nd R s depted on the frst smulted results. In ths se both mtres were hosen to drve the proess to n unstble operton. Seond, the ontrol gn η ws djusted to lrge vlue suh tht the response of the proess rehed n unstble regme. 5. Unstble response due to Q nd R The mtres Q nd R weght the effet of the nputs nd outputs of the proess n the ost funton. Adjustng these mtres llows to heve n eptble response of the ontrolled proess. Nevertheless, settng those mtres to norret vlues my use n unstble response of the proess. To llustrte ths, both mtres hve been djusted to drve the proess to n unstble ondton. The resultng djusted mtres re u y

6 Q R () In ths se the ontrol gn s kept onstnt t η.5. The smulton results re llustrted n Fgure 7. u Entrds Respuest del Proeso u u u y y Entrds u Fgure, Unstble response due to poor η u Respuest del Proeso y y Fgure 7, Unstble response due to poor Q nd R It n be seen tht the response of the ontrolled proess s unstble. In order to hose pproprte vlues for Q nd R the ontrol engneer requres to rry out prevous sequene of eperments. 5. Unstble response due to η A seleton of npproprte ontrol gn my led to n unstble response or to n slow ontrolled response. A sequene of eperments wll be requred n order to selet ontrol gn to produe desred ontrolled response. In ths se, the purpose of the smulted eperments s to demonstrte the effet of lrge ontrol gn over the ontrolled proess stblty. For ths purpose the weghtng mtres n Equton () re kept unhnged. The ontrol gn s set to η. to drve the proess to n unstble response. The result of the smulton s shown n Fgure. Agn, the response of the ontrolled proess s oslltory but ths tme t s only due to poor hoe of the ontrol gn. Conluson The ontrol lgorthm desrbed n the prevous setons hve been tested under smulted ondtons. It hs been demonstrted tht the lgorthms perform stsftorly under smulted opertng ondtons. The ontrol lgorthm s esy to mplement n omputer softwre nd requres few omputtonl resoures. Further trls would be requred to vldte the lgorthm under epermentl ondtons. On the other hnd, the stblty of the ontrol lgorthm ws smply demonstrted usng smultons. It s however neessry to develop n stblty nlyss for ths lgorthm. The seond method of Lypunov my be used for ths purpose. It s neessry to remember tht t s not possble to gurntee the stblty of the ontrol lgorthm. The grdent desent rule my get stuk n lol mnm. As result the requred ontrol vetor my not be omputed urtely. Fnlly, the present ontrol lgorthm wll be ombned wth fult deteton nd dgnoss lgorthm to produe fult tolernt ontrol. 7 Aknowledgments The uthor wshes to epress hs grttude to CONACYT for the support tht mde ths work possble, through grnt I-A. Also, support from Deprtmento de Investgón en Fís de l Unversdd de Sonor s grtefully knowledged.

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