Slovak University of Technology in Bratislava Institute of Information Engineering, Automation, and Mathematics PROCEEDINGS

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1 lovk Unverst of Technolog n Brtslv Insttute of Inmton Engneerng, Automton, nd themtcs PROCEEDING of the 8 th Interntonl Conference on Process Control Hotel Ttrs, Ttrnská Lomnc, lovk, June 4 7, IBN Edtors:. Fkr nd. Kvsnc Doležel, P., Tufer, I., reš, J.: Pecewse-Lner Neurl odels Process Control, Edtors: Fkr,., Kvsnc,., In Proceedngs of the 8th Interntonl Conference on Process Control, Ttrnská Lomnc, lovk, 96 3,. Full pper onlne:

2 8th Interntonl Conference on Process Control June 4 7,, Ttrnská Lomnc, lovk Po-We-8, 43.pdf Pecewse-Lner Neurl odels Process Control P. Doležel* I. Tufer** J. reš*** * Unverst of Prdubce, Fcult of Electrcl Engneerng nd Inmtcs, Deprtment of Process Control, Nám. Čs. legí 565, 53 Prdubce (Tel: ; e-ml:petr.dolezel@upce.cz.) ** Unverst of Prdubce, Fcult of Electrcl Engneerng nd Inmtcs, Deprtment of Process Control, Nám. Čs. legí 565, 53 Prdubce (e-ml:vn.tufer@upce.cz.) *** Insttute of Chemcl Technolog, Deprtment of Computng nd Control Engneerng, Techncká 5, 66 8 Prgue 6 (e-ml: jn.mres@vscht.cz) Abstrct: There s ntroduced n lgorthm whch provdes pecewse-lner model of nonlner plnt usng rtfcl neurl networks, n ths pper. Tht pecewse-lner model s precse nd ech lner submodel s vld n some neghbourhood of ctul plnt stte. Ths model cn be used plnt control desgn. There s presented n exmple t the end of ths pper, where defned nonlner plnt s controlled v Pole Assgnment technque usng pecewse-lner neurl model nd control response s compred to dt obtned b common PID controller.. INTRODUCTION Artfcl Neurl Network (ANN) s populr methodolog nowds wth lots of prctcl nd ndustrl pplctons. As ntroducton t s necessr to menton pplctons s mthemtcl modellng of boprocesses n ontgue et l. (994), Texer et l. (5), predcton models nd control of bolers, furnces nd turbnes n Lchot et l. () or ndustrl ANN control of clcntons processes nd ron ore processes n Dwrpud, et l. (7). Theree, the m of the contrbuton s to expln how to use ANN wth pecewse-lner ctvton functons n hdden ler n process control. To be more specfc, there s descrbed technque of controlled plnt lnerzton usng ANN nonlner model. Obtned lnerzed model s n shpe of lner dfference equton.. ANN FOR APPROXIATION Accordng to Kolmogorov's uperposton Theorem, n rel contnuous multdmensonl functon cn be evluted b sum of rel contnuous one-dmensonl functons, see Hecht- Nelsen (987). If the theorem s ppled to ANN, t cn be sd tht n rel contnuous multdmensonl functon cn be pproxmted b certn three-lered ANN wth rbtrr precson. Topolog of tht ANN s depctured n Fg.. Input ler brngs externl nputs x, x,, x P nto ANN. Hdden ler contns neurons, whch process sums of weghted nputs usng contnuous, bounded nd monotonc ctvton functon. Output ler contns one neuron, whch processes sum of weghted outputs from hdden neurons. Its ctvton functon hs to be contnuous nd monotonc. o ANN n Fg. tkes P nputs, those nputs re processed b neurons n hdden ler nd then b one output neuron. Dtflow between nput nd hdden neuron j s gned b weght w j,. Dtflow between hdden neuron k nd output neuron s gned b weght w,k. Output of the network cn be expressed b followng equtons. P j = w j, x + w () j x x x 3 x P ( j ) j w, w,p Input Ler Fg.. w w w Hdden Ler w, w, Three-lered ANN w, = ϕ () w Output Ler 96

3 8th Interntonl Conference on Process Control June 4 7,, Ttrnská Lomnc, lovk Po-We-8, 43.pdf = w, + w (3) ( ) = ϕ (4) In equtons bove, φ (.) mens ctvton functons of hdden neurons nd φ (.) mens output neuron ctvton functon. As t s mentoned bove, there re some condtons pplcble ctvton functons. To stsf those condtons, there s used mostl hperbolc tngent ctvton functon (eq. 5) neurons n hdden ler nd dentcl ctvton functon (eq. 6) output neuron. ( j ) = tnh (5) j = (6) entoned theorem does not defne how to set number of hdden neurons or how to tune weghts. However, there hve been publshed mn ppers whch re focused especll on grdent trnng methods (Bck-Propgton Grdent Descend Alg.) or derved methods (Levenberg-rqurdt Alg.) see Hkn (994). 3. YTE IDENTIFICATION BY ANN stem dentfcton mens especll procedure whch leds to dnmc model of the sstem. ANN hs trdtonll enjoed consderble ttenton n sstem dentfcton becuse of ts outstndng pproxmton qultes. There re severl ws to use ANN sstem dentfcton. One of them ssumes tht the sstem to be dentfed (wth nput u nd output ) s determned b the followng nonlner dscrete-tme dfference equton. ( k) = ψ [ ( k ), ( k n), k ), k m)], m n In equton bove, ψ(.) s nonlner functon, k s dscrete tme nd n s dfference equton order. The m of the dentfcton s to desgn ANN whch pproxmtes nonlner functon ψ(.). Then, neurl model cn be expressed b (eq. 8). ( k) = ψˆ [ ( k ), ( k n), k ), k m)], m n (7) (8) In (eq. 8), ψˆ represents well trned ANN nd s ts output. Forml scheme of neurl model s shown n Fg.. It s obvous tht ANN n Fg. hs to be trned to provde s close to s possble. Exstence of such neurl network s gurnteed b Kolmogorov's uperposton Theorem nd whole process of neurl model desgn s descrbed n detl n Hkn (994) or Tufer et l. (8). 4. PIECEWIE-LINEAR ODEL As mentoned n secton, there s recommended to use hperbolc tngent ctvton functon neurons n hdden ler nd dentcl ctvton functon output neuron n ANN used n neurl model. However, f lner sturted ctvton functon (eq. 9) s used nsted, ANN fetures st smlr becuse of resemblng courses of both ctvton functons (see Fg. 3). j > j = j j < j The output of lner sturted ctvton functon s ether constnt or equl to nput so neurl model whch uses ANN wth lner sturted ctvton functons n hdden neurons cts s pecewse-lner model. One lner submodel turns to nother when n hdden neuron becomes sturted or becomes not sturted. Let us presume n exstence of some dnmc neurl model whch uses ANN wth lner sturted ctvton functons n hdden neurons nd dentc ctvton functon n output neuron see Fg. 4. Let us lso presume m = n = mkng process eser. ANN output cn be computed usng eqs. (), (), (3), (4). However, nother w ANN output computng s useful. Let us defne sturton vector z of elements. Ths vector ndctes sturton sttes of hdden neurons see (eq. ). z = > < Then, ANN output cn be expressed b (eq. ). (9) () z - Hperbolc Tngent Lner turted F. z -.5 k) z - z - (k) Fg.. Neurl model Fg. 3. Actvton functons comprson 97

4 8th Interntonl Conference on Process Control June 4 7,, Ttrnská Lomnc, lovk Po-We-8, 43.pdf z - (k-) w, w w, ( k) = ( k ) ( k ) + + b u~ ( k ) + b u~ ( k ) + c + + ( b + b ) u (3) Equton (3) becomes constnt term free, f (eq. 4) wll be stsfed. k) z - z - z - Fg. 4. w w (k-) k-) w,4 Input Ler Hdden Ler w, Output k-) Ler w w, Pecewse-lner neurl model (k) c u = (4) b + b It s obvous tht mentoned procedure cn be extended nto n order of dfference equton. Whole lgorthm of pecewse-lner neurl model usge n process control s summrzed n followng terms.. Crete neurl model of controlled plnt n m of Fg. 4.. et k =. ( k) = ( k ) ( k ) + () + b k ) + b k ) + c where:, = w,, = w,, 3 w, b b c = =, 4 w, = w + w, z + ( ( z ) w, w ) Thus, dfference equton () defnes ANN output nd t s lner n some neghbourhood of ctul stte (n tht neghbourhood, where sturton vector z sts constnt). Dfference equton () cn be clerl extended nto n order. In other words, f t s desgned neurl model of n nonlner sstem n m of Fg. 4, then t s smple to determne prmeters of lner dfference equton whch pproxmtes sstem behvour n some neghbourhood of ctul stte. Ths dfference equton cn be used then to the ctul control cton settng due to n of clsscl or modern control technques. If chosen control technque requres model n m of dfference equton wth no constnt term (c = ), (eq. ) cn be trnsmed n followng w. Let us defne u ~ ( k ) = u ( k ) u () where u s constnt. Then, (eq. ) turns nto 3. esure plnt output (k). 4. Determne prmeters, b nd c of dfference equton (). 5. Trnsm (eq. ) nto (eq. 3). 6. Determne u ~ ( k ) ccordng to some chosen control technque usng lner plnt model n m (eq. 3). 7. Trnsm u ~ ( k ) nto k) usng (eq. ) nd perm control cton. 8. k = k +, go to EXAPLE Demonstrtve nonlner controlled sstem s defned b dfference equton (5). ( k) =.5 ( k ).8 ( k ) +. k ) +.5 k ) +. ( k ) k ) [ ( k ) ] (5) There re defned the boundres of nput k) to ntervl <;3>. ttc chrcterstc of the sstem s fgured below (Fg. 5) Fg u ttc chrcterstc of the sstem 98

5 8th Interntonl Conference on Process Control June 4 7,, Ttrnská Lomnc, lovk Po-We-8, 43.pdf.5 w.5 u w u.5.5 w, u, w, u, k Fg. 6. Control response wth PID controller Frstl, sstem s controlled wth PID controller tuned b trl nd error more sophstcted tunng methods fl to brng better permnces becuse of sgnfcnt nonlnert of the plnt. Control response (Fg. 6) shows serous lck of qult. For lower vlues of controlled vrble (k), control permnce osclltes uncceptbl, whle hgher vlues of (k), control permnce s too dmped. Then, pecewse-lner neurl model s used control. Neurl model s desgned ccordng to nmton descrbed n secton 4. Detled descrpton of the process s not referred here, becuse t s stndrd well-known procedure. Certn control technque, whch cn use sstem model n m of (eq. 3), hs to be determned. In ths demonstrton, Pole Assgnment control technque (PA) of Algebrc Control Theor s used. In smple words, ths control technque determnes controller prmeters so tht whole closed control loop behves s some defned stndrd. In one ts verson, PA uses control loop shown n Fg. 7. Controlled sstem should be descrbed b polnomls A(z - ), B(z - ), where polnoml prmeters re equl to dfference equton prmeters used lner model of the controlled sstem. Both feedwrd nd feedbck prt of controller re defned b polnomls P(z - ), Q(z - ), R(z - ), whch cn be determned b solvng of severl dophntne equtons. tndrd control loop behvour hs to be chosen. Whole procedure of PA s descrbed n detl n book edted b K. J. Hunt (993). tndrd ths demonstrton s defned s dscrete frst order sstem wth unt gn nd denomntor ( -.665z - ). Control permnce s shown n Fg. 8. Compred to Fg. 6, there comes cler mprovement. w (k) Fg. 7. R(z - ) P(z - ) k) YTÉ Q(z - ) P(z - ) Pole Assgnment Control Technque (k) k Fg. 8. Control Response wth PA Controller nd Pecewse-Lner Neurl odel 6. CONCLUION The pper s focused on usge of neurl network wth lner sturted ctvton functons n process control. Neurl model wth such neurl network wthn s sutble controller desgn usng n of huge set of clsscl or modern control technques. As exmple, there s presented control of nonlner dscrete plnt usng Pole Assgnment technque. Comprson to control permnce provded b PID controller proves gret mprovement. ACKNOWLEDGENT The work hs been supported b the funds No nd No of nstr of Educton of the Czech Republc, No. EB 83 of nstr of Educton, cence, Reserch nd port of the lovk Republc nd of nstr of Educton of the Czech Republc nd No. GFEI6/. Ths support s ver grtefull cknowledged. REFERENCE Dwrpud,., Gupt, P. K. nd Ro,.. (7). Predcton of ron ore pellet strength usng rtfcl neurl network model, IIJ Interntonl, Vol. 47, No. pp. 67-7, IN Hkn,. (994). Neurl Networks: A Comprehensve Foundton. Prentce Hll. New Jerse. IBN Hecht-Nelsen, R. (987). Kolmogorovʼs mppng neurl network exstence theorem. In: Proc 987 IEEE Interntonl Conference on Neurl Networks. Vol. 3, pp. -3. IEEE Press. Hunt, K. J., Ed. (993). Polnoml methods n optml control nd flterng. Peter Peregrnus Ltd. tevenge. IBN Lchot, J. nd Grbovsk,. (). Applcton of rtfcl neurl network to boler nd turbne control, Rnek Energ, Vol. 6, No. IN ontgue, G. nd orrs, J. (994). Neurl network contrbutons n botechnolog, Trends n botechnolog, Vol., No 8. pp. 3-34, IN

6 8th Interntonl Conference on Process Control June 4 7,, Ttrnská Lomnc, lovk Po-We-8, 43.pdf Tufer, I., Drábek, O., edl, P. (8). Umělé neuronové sítě zákld teore plkce (), CHEgzín, vol. XVII, ssue, pp IN Texer, A., Alves, C. nd Alves, P.. (5). Hbrd metbolc flux nlss/rtfcl neurl network modellng of boprocesses, In: Proceedngs of the 5th Interntonl Conference on Hbrd Intellgent stems, IEEE Computer ocet, Los Almtos. IBN