Latn Amercan Apple Research 4:4-48(0) NEURAL COMPENSAION AND MODELLING OF A HO SRIP ROLLING MILL USING RADIAL BASIS FUNCION F.G.ROSSOMANDO, J. DENI F an A.VIGLIOCCO Insttuto e Automátca (INAU), Faculta e Ingenería. Unversa Naconal e San Juan (UNSJ). Av. San Martín 09 Oeste J5400ARL. San Juan Argentna, frosoma@naut.uns.eu.ar Laboratóro e Controle e Instrumentação, Unversae Feeral e Esprto Santo (UFES) C. Av. Fernano Ferrar s/n CEP 9060-970 Vtóra-ES-Brasl,.ent@ele.ufes.br Gerenca e Lamnacón, ERNIUM SIDERAR- San Ncolas-Pca e Bs. As.- Argentna, avglocco@ternum.com.ar Abstract In ths paper a Neural Compensaton Strategy for a hot rollng mll process s propose. he target of ths work s to bult a RBF-NN compensaton approxmaton for the classcal force fee forwar an spee controller. A strategy base on neural networks s propose here, because they are capable of moellng many nonlnear systems an ther neural control va RBF-NN approxmaton. Smulatons emonstrate that the propose soluton eals wth sturbances an moelng errors n a better way than classc solutons o. he analyss of the RBF-NN approxmaton error on the control errors s nclue, an control system performance s verfe through smulatons. Keywors hot strp mll, thckness evaton, neural compensaton, raal bass functons. I. INRODUCION he effcency of a hot strp mll can be ncrease f the amount of reecte materal s reuce. A strp s consere as reecte materal f t oes not meet the requrements set by the customer an, therefore, has to be sol as lower qualty prouct or has to be re-melte. hs last opton mples a tremenous amount of extra materal hanlng an hgh energy costs One way to mprove the effcency of the hot rollng mll s through a better fnshng mll control. In hot steel strp rollng processes, force fee forwar (FFF) control s necessary to reuce the effects of rap strp thckness varatons ue to temperature varatons an other factors such as support an bearngs eformaton as well as long term or slow varatons of thckness n the sx- stan fnshng mll (Huang et al. 004;Pttner an Marwan 006). he am of thckness control s to regulate the ext thckness from the roll gap an the spee output of the strp. Moern an avance control methos can be employe to ecrease the thckness stanar evaton an ecrease the thckness error by keepng the output spee of the strp constant. In ths approach, a complete moel for the system s evelope, whch s entcal to the process use n the facltes of ERNIUM SIDERAR-Argentna HSM, wth real ata an parameters prove by ths company. hs real-system moel s use as to bass from whch to bul a smulator for valatng new control strateges. 4 Some of the controllers esgne so far are base only on lnear moels of the HSM. For nstance, the controllers esgne n Rgler et al. (996) an Peersen et al. (995) compensate the roll gap ynamcs, whereas n Galvez et al. (003) the neural network base prectve control allows for overcomng the exstng tme elays n system ynamcs. In Mahfouf et al. (005) a neural compensaton for roll gap an roll spee s propose usng MLP-NN, but the results shown are base only on off-lne austment of the NN. he work Kug et al. (00) presents a nonlnear thckness controller that s acheve usng a ynamc feeback lnearzaton technque, an mplemente on HSM wth a hyraulc austment system. Hearns et al. (004) emphasze the mportance of the nteracton between loop length an ext thckness, an tre a performance trae off between mass flow control an gauge control by changng the output weghtngs, but the weghts can be change to ensure stable mll operaton at the expense of ext gauge control. Alvarez et al. (00) escrbe the esgn an mplementaton of a supervsory system for real-tme compensaton of uneven thckness on both ses of a rolle strp, snce t s base on vsual nspecton. An n Bouazza an Abbass (007) an off-lne moel-base controller an a process smulator are escrbe. In Rossomano an Dent (006) a lnear parameterzaton of two stans of a HSM s presente along wth the esgn of a thckness an looper arm controller base on ts entre moel. One avantage of ther controller s that ts parameters are rectly relate to the mll stan parameters. However, f the parameters are not correctly entfe or f the change wth tme because of, e.g. loa varaton, the performance of the controller can be severely affecte. In Lee et al. (007) the propose controller s mplemente on the conventonal control system, mprovng the performance of the system wthout any atonal cost. he mprovement was emonstrate by usng the real ata whch are collecte usng the propose control scheme n HSM, but t only austs the roll gap to reuce thckness varaton n the HSM. In Dng et al. (008), an aaptve threang technque was use to prect thckness an materal harness errors so as to mofy the setup of the remanng stans wthout the metal strp rollng n.
Latn Amercan Apple Research 4:4-48(0) In ths work, a new strategy that shows a goo performance uner nonlneartes an sturbances s presente, such as the col zones n the metal strp. he am of ths work s to bult a neural compensaton for thckness an spee varatons n the roll system. herefore, the ynamc controller s esgne on the bass of a nomnal moel an a RBF-NN compensaton controller capable of learnng the ynamc fference between the esre an the actual moels. Snce the RBF-NN compensates only for a esre moel error ynamcs, ts computatonal cost s sgnfcantly reuce as compare wth a whole NN ynamcs controller. An analyss s mae to stuy the effects of the RBF-NN approxmaton error on the control error when the entre control system, the aaptve ynamcs controller workng, s apple to a roll gap an spee control task. he paper s organze as follows: Secton II presents the overvew of the propose system, an shows the mathematcal representaton of the complete screw an spee system moel. he neural controller s scusse, n Secton III, as well as the corresponng error analyss (Secton III.A). Secton IV presents some smulaton results on the performance of the aaptve controller. Fnally, conclusons are gven n Secton V. II. PLAN MODELLING A. Process escrpton he rollng mll process conssts of pressng a strp metal between a set of rotatng rolls that reuces the thckness of the passng hot-metal strp. he mll housng, the back-up an work rolls an the roll gap austment system are the man machne sectons that partcpate n the rollng mll process; a fnshng mll secton s a set of rollng mll stans; Fgure shows a ynamc scheme of a smple rollng mll stan wth four rolls each one, two work rolls an two back up rolls, wth austment roll gap systems (screw). he rolls n contact wth the strp are the work rolls, they are the man responsble for the rollng mll physcal phenomena. Conserng a gven stan, the strp plate from the prevous stans (flat prouct or a col), s ntrouce n the gap between work rolls, whch s smaller than the thckness of the strp, ths gap s etermne by the poston of the screw. he work rolls rag an press the strp nto the gap to reuce ts thckness. hs strp has to leave an enter n the next stan untl the wshe thckness s reache an cole at the output of the last Stan. he fnshng mll process n ernum-serar conssts of 6 rollng mll stans (F5 to F0). he complete fnshng mll process s shown n Fg.. Fgure : Fnshng mll process Fgure : Mll stan ynamc moel In ths work, the mathematcal moel of the last Stan F0 of the rollng mll tran to evaluate the propose control technque s presente. he analytcal moelng use s ame at valatng through smulatons the obtane results. he results are compare wth the measure real values n the rollng mll stans controlle by the FFF technque (force fee forwar; Bryant, 973). o obtan an approxmate rollng mll stan moel, Alexaner (97) an Orowan (944) theores were use snce they are more relable to obtan the rollng mll force, an can be use n a we range of rollng mll contons. he mathematcal moel was ft an calbrate wth real values measure n the process, thus attanng a closer response to the real force value. hs moel s represente n a general form by Eq. (). S h t, h t, t, t, S,, t () an the rollng torque s expresse by: S h t, h t, t, t, S,, t, () where s the output strp tenson for Stan F0, s the nput strp tenson for Stan F0, h s the output strp thckness, h s the nput strp thckness, s the rolls gap for the Stan, s the frcton Coeffcent for the Stan, S s the Yel stress for the Stan an s the emperature of the strp. B. Screw system moelng he mplemente mathematcal moel consers last stans process, In ths case, the thckness austment by RBF-NN control technque s mae, usng the mathematcal moel expresse by the next equatons. he output thckness of each stan s a functon of the gap rolls an the stan stretchng. h t t E S (3) where h s the output strp thckness of the stan, s the Rolls gap of the stan, s s the Rollng mll force of the stan an E(.) s the Elastcty functon of the stan. here s a slght fference between the roll gap (wthout mll force) an the thckness of the rolle prouct, cause by the elastcty effect of the mll structure. he elastcty functon can be expresse by: 4
F.G.ROSSOMANDO, J. DENI F, A.VIGLIOCCO S S S E C C C. (4) 3 he elastcty coeffcents of Eq. (4) have a etermne value epenng on the locaton of the stan mll. he able I shows the elastcty coeffcents for each stan. he motor prouces the followng torque, whch s proportonal to the motor current: ms t kmsims t, (5) where k ms : s the motor torque constant, I ms s the c motor current an ms s the electrc motor torque. he electrc counter force s proportonal to the motor velocty ms t ems t kes (6) where θ ms s the angular poston of the screw motor, e ms s the electromagnetc counter-force an k es s the electrcal constant of the motor. he electrc equaton of DC motor s expresse by: Ims t Ls RI s ms t ems t Vs t. (7) he mechancal equatons of the DC motor, accorng to Newton s n Law, are ms t ms t ms t Jms JB B... B... K t t (8) B ms t K t t P B ms where J ms s the mechancal nerta of the motor (constant), J B s the nerta of the shaft an B mr s the vscous frcton of the motor. he necessary torque to move the screw ( p ) s gven by the followng formula: f S me me sgn H P F t me sgn H F t able : Elastcty Coeffcents Stan C [mm/ton ] C [mm/ton] C 3 [mm] F5-5.07e-007 0.003. 0.0757 F6-6.3338e-007 0.0034 0.70 F7 -.0553e-006 0.0056 0.0859 F8 -.67e-006 0.0050 0.4 F9 -.84e-006 0.0044 0.56 F0 -.537e-006 0.0053 0.984 Fgure 3: Mll force acton on the screw (9) where F(t) s the resultant of the forces apple n the recton of the screw axs, expresse n tons; me s the 43 able : Moel parameters Symbol Parameter Unt a n r Gear rato 56.3. p p Screw ptch 3.897 mm/ra. J B Inerta of the screw referre to the motor shaft 60 kgm J ms Inerta of the rotor referre to 6 kgm the motor shaft L s Electrcal nuctance of the 0.000 Hy motor R s Electrcal Resstance of the 0.0937 ohms. DC Motor K es Electrc-mechancal constant 4.3 of the motor M Weght of the bearngs an 0 ton + 5 ton of the backup rolls M Weght of the bearngs an ton. + 7ton. of the work rolls K el Elastcty nex of the stan 436 ton/mm. F0 (lnear approx. of E) J mr Inerta of the rotor referre to 33 kgm the motor shaft (Roll system) J gb Inerta of the gear box referre to the motor shaft 3093 kgm (Roll system) J r Inerta of the roll referre to 74.4 kgm the motor shaft L r Electrcal nuctance of the 8e-5 Hy motor (Roll system) R r Electrcal Resstance of the 0.00547 ohms. DC Motor (Roll system) K er Electrc-mechancal constant 3 of the motor (Roll system) n gb Gear rato of roll system Stan F5 5,6 Stan F6 3,7 Stan F7,90 Stan F8,00 Stan F9,00 Stan F0,00 Mean ameter of the splacement screw, gven n meters (m); α s the angle of the screw s helx; μ s the frcton coeffcent between the splacement screw an the nut an H s the lea of the screw s helx. Functon f s (θ ) states whether the force F(t) s requre to open or to close the roll gap (see Fg. ). he relaton between torque an mll force s shown n Fgure 3. he translaton movement of the system that nvolves the vertcal movement of the backup rolls s rule by the followng equaton. t t F t M B ca E h (0) where M s the Mass of the backup rolls, work rolls an bearngs (M an M n able I), B ca s the vscous frcton coeffcent between the Stans an bearngs of the backup rolls; E - (.) s the elastcty nverse functon of the stan. Γ s can be erve from Eq. (3) an replacng n Eq. (0) an the resultant of the forces apple n the recton of the screw axs s: t t F t M B ca. () S he relaton between the angular splacement of the shaft an the lnear vertcal splacement of the backup rolls s gven by the followng expresson:
Latn Amercan Apple Research 4:4-48(0) pp t t () nr where p p s the ptch of the screw, n r s the ransmsson rato of the gear box. he system responsble n charge of postonng the rolls n the esre gap s known as AGC (Automatc Gage Control) (Dark et al., 989; an Gunzburg, 989). he AGC control loop s close by the feeback system of the gap, an the controller s a PI type (proportonal ntegral), whose parameters are auste to obtan a fast uner ampe response of the gap (γ). (Gunzburg, 989) shows the control system of the gap (AGC). he gap controller s a PI controller (AGC control) an can be efne by the followng equaton: t Vs t satkis.... 0 ms u (3) KPs. ms t u t where θ u (t) s the nput poston of the screw system, sat s the saturaton functon, V s (t) s the output voltage, K Is s the ntegral gan an K Ps s the proportonal gan. he total process, n whch the propose control shoul be apple, s consttute by the AGC system an by the loa moel of the rollng process. he loa moel epens on the output thckness an at the same tme the AGC system s affecte by the rollng loa (Γ s ). he parameters to bul the moel were prove by ERNIUM SIDERAR an are shown n able I C. Man rve control system In orer to evelop the roll spee response moel, t s necessary to ve the mll nto several mechancal components couple by a flexble shaft lnk, an then evelop the moton equatons for the nvual components usng Newton s law. In ths paper, a moel of three masses (the motor nerta, the reucton gear an the roll mll) structure was ntally aopte for the stan mll, whose smplfe mechancal confguraton s represente as: mr t mr t mr t Jmr B mr...... Kmr mr t gb t Jgb gb t Bgb gb t Kmr mr t gb t... ngb ngb (4) K gb... gb t r t ngb K gb J r r t B r r t gb t r t M t ngb ngb ngb where J mr, J gb, J r are the mechancal nerta of the motor, gear box, an the flexble shaft, respectvely; B mr, B gb, B r are the vscous frcton of the motor, gear box an the flexble shaft, respectvely an K mr, K gb : are the elastcty of the flexble shaft lnk of the motor an the gear box, respectvely. he DC electrc motor ynamcs s the same as that represente by Eqs. (5),(6) an (7). Equatons parameters are k mr, k er, L r an R r. he motor spee controller s a PI type (proportonal ntegral Spee control) an s etermne by a smlar Eq.(3) (wth K Ir an K Pr, the ntegral an proportonal gans, respectvely); but the nput spee s ω u (t). D. hckness Austment Dynamc he whole system ynamc of the stan s represente n the state space as shown n Eq. (5) an eq. (6) kes BB KB Ims ms t ms t t Jeq Jeq J eq ms t Bca n S t r 0 K B t fs ms t t ms t M ps M M ms t t t Rs kes sat Ims ms t KIss t KPs ms t u t t Ls Ls L s I ms t ms t u t s t mr t ker Bmr Kmr Imr mr t mr t gb t mr mr mr mr t J J J mr t gb t gb mr gb gb gb t n K B K mr t gb t gb t gb t r t Jgb Jgb J gb r t r t gb t I mr t K gb B n r gb gb t r t r t t0 r t Jr Jr Jr r t Rr k I er mr sat mr t KIrr t KPr mr t u t Lr Lr L r mr t u t An the output vector s efne by: ps h ES to n (6) v r r where o r (t) an o s (t) are auxlary varables to calculate the ntegral acton, J eq s the total nerta of the system (J ms +J B ), v s the spee output of strp, r work roll raus an 0 s the elay tme (one step tme)- he torque of rollng τ(.) an the loa force Γ(.) are elaye one samplng tme ( 0 ) to avo algebrac loops. Expressng Eq. (5) an Eq. (6) n compact form. t t, t y hx t x f x u where: x I u an (5) (7) ms ms ms mr mr gb gb r r Imr u y h u v. A basc problem wth the thckness control s the ffculty to measure the thckness when the plate s beng rolle. Instea, the thckness s estmate an the uncertanty ntrouce here s reflecte n the stablty analyss. Some conseratons on the moel are: ) he frcton coeffcent n the moel, the yel stress tenson an the stresses forwar an backwar from the strp are consere constants to ths analyss. 44
F.G.ROSSOMANDO, J. DENI F, A.VIGLIOCCO ) he sturbances of the process cross through the mathematcal moel affect the output value of the plant. ) he man conseraton s that the control s apple when a strp exsts nto the stan. hs means that the prevous moment to the entrance of the strp n the last stan s not consere. v) he sample tme for purpose of control s o =0.04 sec. to esgn the elay tme Assumpton: he rollng loa s relate to the strp's temperature, nput an output thckness, the frcton coeffcent, the yel stress an stresses forwar an backwar from the strp. Any varaton n sa varables s consere a sturbance. hese sturbances are boune an affect the process outputs. he whole process consere n ths work s shown on Fgure 4. III. PROCESS CONROL A. Error ynamcs approxmaton A smple control strategy, whch s smlar to the wellknown moel reference aaptve control (MRAC) scheme, s propose n Fg. 4. hs scheme s aopte n ths stuy because t has the avantage of generatng the esre nput sgnal wthout requrng that the network be trane ntally offlne. Moreover, the outputs of the neuro controller u N whch has fault tolerant ablty through on-lne learnng only epen on the past an present value nputs, outputs an esre nput reference of the system. he selecton of the moel shoul not greatly excee the ynamcs of the system uner control whle the sturbances are consere constant, otherwse large control efforts mght be requre. Assumng all the states of the system are accessble, the state error e s efne as e=x-x. he error ynamcs for the overall system s, e x x fx, u fx, u (8) ey yy hxhx Eq. (8) can be expane n the esre reference usng aylor seres. fx, u e xxoxx... x xx uu fx, u u u Ou u u x x uu hx ey xxoxx... x xx (9) where O. represents hgher orer terms. Substtutng xx fxu u xx by B, fxu, x by A, uu, uu Fgure 4: Rollng mll control process scheme hx x x x by C, an neglectng all the hgher orer terms. he Eq. (9) can by expresse by: e AeBuu (0) ey Ce Now applyng the KYP (Kalman-Yacubovch-Popov) Lemma (Vyasagar, 993) n Eq (0). Lemma : Let Z(s) = C(sI A) - B be a x transfer matrx such that Z (s)+ Z (-s) has normal rank, where A s Hurwtz, (A,B) s stablzable, an (C,A) s observable. hen Z(s) s Strctly Postve Real (SPR) f an only f there exst symmetrc postve efnte matrces P an Q such that: PA A P Q () PB P B C () In orer to evelop an algorthm to aust u so that the system becomes asymptotcally stable, n spte of parameter varatons an moelng errors. Now conserng uff u u ff (3) where u ff s the correcton sgnal (roll gap an spee) from the prevous Stan an u s the set pont of the Stan (constant). Conserng the correcton sgnals an makng, uun u ff (4) hus, the total control sgnal to the stan mll s the sum of the FFF austment, spee correcton an the RBF- NN controller sgnals, from Eq. (4) an usng Eq. (0), the error ynamcs can be approxmate as: e Ae BuN uff Bu (5) B. RBF-NN Parameterzaton Settng the RBF-NN s nputs as ζ where y, y, y, un, u, u N N can be approxmate by a RBF-NN through on-lne learnng, m un w exp c n (6) w ξ, c, n where w (m x ) an ξ (m x ) are optmal parameter vectors of weghts w an raal bass functons ξ, respectvely; c an η are optmal parameter vectors of centers c an whs η, respectvely; an ε n s the approxmaton error. 45
Latn Amercan Apple Research 4:4-48(0) However, the optmal parameter vectors are unknown, so t s necessary to estmate the values. Defne an estmatve functon. m uˆ ˆ ˆ ˆ N w exp c n (7) wˆ ξˆ, cˆ, ˆ n where ŵ an ˆξ are estmate parameter vectors of w an ξ, respectvely; an ĉ an ˆ are estmate parameter vectors of c an η, respectvely. Wth the RBF-NN compensaton, usng Eq. (7) an replacng u by ˆ N u N, the control nput vector u from (4) s, m uwˆ exp ˆ ˆ c nuff (8) wˆ ξˆ nu ff Defnng w w wˆ an ξ ξ ξˆ, an replacng Eq. (3) an Eq. (8) n Eq. (5) the error ynamcs may be wrtten as, ˆ ˆ ˆ e Ae B w ξ w ξ wˆ ξ w ξ n Bu ff (9) ˆ B w ξ wˆ ξ represents the learnng error ψ l where an conserng ˆ ˆ ff w ξ u an w ξ nto n. herefore, a RBF-NN compensator s use for on-lne learnng of. he Eq. (9) can be expresse by u ff ˆ e Ae B w ξ wˆ ξ n (30) Usng an approxmaton for the functon ξξ, c, ξ ˆ, c ˆ, ˆ. Ïn orer to eal wth ξ, the aylor s expanson of ξ s taken about c c ˆ an ˆ. ˆ ξ, c, ξ, cˆ, ˆ Ξ c Φηδ, c, (3) where δ enotes the hgh-orer arguments n a aylor s seres expanson, an Ξ an Φ are ervatves of ξ, c, wth respect to an η at c ˆ,ˆ. hey are expresse as : ξ, c, Ξ c c cˆ η ηˆ (3) ξ, c, Φ η c cˆ η ηˆ Equaton (3) can be expresse as ξξ c Φη δ, c, (33) From (33) the hgh-orer term δ s boune by δ, c, ξξ c Φη ξ Ξ c Φη (34) c 3 η where κ, κ, an κ 3 are some boune constants ue to the fact that RBF an ts ervatve are always boune by constants (the proof s omtte here to save space). Substtute (33) nto (30), t can obtan that ˆ e Ae B ˆ w ξ w Ξ c Φηδ n (35) ˆ e Ae Bw ξ Bwˆ Ξ cbwˆ Φη ε where the uncertan ε Bwˆ δ n, s reasonable to assume ε s boune by a constant ε H, an sup ε (36) H t t C. Stablty Analyss an neural parameters austment o erve the stable tunng law, the followng Lyapunov functon s chosen, V epe trw w c cη η (37) where P s an x symmetrc postve efnte matrx, an Θ, Λ are m x m non-negatve efnte matrces. he ervatve of the Lyapunov functon s gven by, V tr epe epe w w c c η η (38) From eq. () b, Q A P P A snce A s Hurwtz, the Lyapunov functon can always be foun an has a unque soluton an replacng Eq. (30) nto Eq. (38), ˆ V eqe ξ wb Pe c ΞwB ˆ Pe ηφwb ˆ Pe... ε BPetr w w c c η η (39) Substtutng Eq. () n Eq. (39) ˆ V eqe ξ we ˆ ˆ y c Ξwey ηφwe y... ε BPetr w w c c η η (40) Notng n eq. (39), tr w w w w (4) ˆ ξ we y ξ w ey Eq. (40) becomes, V eqe ε BPec wey c... η Φwe η w e tr w w y y (4) (43) where w s the -th column of matrx w an e y s the - th row of vector e y, If c, η an w are selecte as, w ˆ ξe y,... (44) c ˆ wey (45) η ˆ Φwey (46) hen Eq. (43) becomes, V eqe ε BPe (47) V can be emonstrate negatve accorng to Eq.(47), V mn Q e e P B H (48) Let B H, t can be shown rectly that V s negatve H when e P a H Q (49) mn 46
F.G.ROSSOMANDO, J. DENI F, A.VIGLIOCCO Now conserng w w w, c c c η η η an w 0, c 0, η 0 the tunng rules are, w ˆ ξe y,... (50) c ˆ wey (5) η ˆ Φwey (5) he negatveness of the Lyapunov functon s guarantee, resultng n the overall system to be stable. IV. SIMULAION RESULS In orer to make the control technque evaluaton collecte real ata of the rollng mll of the steel col were use, wth N o 98 6 lamnate n the ERNIUM SIDERAR plant n Argentna. he obectve of the smulaton shown n ths secton s to utlze the propose control strategy to reuce varatons of the parameters an moelng errors as well as ecrease any possble sturbance. I n ths smulaton the number of RBF neurons s equal to fve (m=5). he sturbance sgnals are shown n fgure 7 an fgure 8 shows the smulate output strp thckness wth RBF-NN correcton an the strp thckness wth conventonal control (FFF). nput thckness [mm] 4.8 4.7 4.6 4.5 0 50 00 50 00 50 tme (sec.) 950 Output Spee (m/s.) Fgure 7: Output spee of the strp for the RBF-NN compensaton Output hckness (mm.) 7 6.5 6 5.5 5 4.5 4 0 50 00 50 00 50 me (sec.) 3.87 3.86 3.85 3.84 3.83 3.8 3.8 3.8 3.79 3.78 Output Spee Espesor RBF-NN FFF Desre RBF-NN FFF Desre 3.77 0 50 00 50 00 50 me (sec.) emperature [ o C] 940 930 90 90 0 50 00 50 00 50 tme (sec.) Fgure 8: Output thcknesses for the RBF-NN compensaton an FFF. 50 00 Loa Force real estme Fgure 5: Input Dsturbances u NN (mm.) 0. 0. 0-0. Gap Compensaton -0. 0 50 00 50 00 50 me (sec.) Spee Compensaton Loa Force (tons) 50 00 050 000 950 50 00 50 00 50 tme (sec.) u NN (m/s) 0 - - 0 50 00 50 00 50 me (sec.) Fgure 6: Gap an spee compensaton from RBF-NN Fgure 9: Estmaton of the loa force from the rollng moel an the measure loa force. he stues on control smulaton nvolve parameter varatons, moelng error an sturbance reecton cases (temperature an nput thckness varatons), where the output thckness an output spee are controlle usng the RBF-NN roll gap an spee compensaton as the manpulate varable. he RBF-NN control varaton error s lower than 60 m an, n the case of conventonal control (FFF), t reaches up to 00 m. 47
Latn Amercan Apple Research 4:4-48(0) Fgure 8 shows the output spee strp for stan F0, beng the output expecte reference spee equal to 5.45 m/s. he same results for fferent ata collecte n the rollng mll process were obtane an the thckness error n the case of the RBF-NN control s always smaller than the error of the control FFF, whch s the actual control technque use n the plant. Fgure 9 shows the loa varaton of the actual rollng loa measure n the process, compare to the estmate loa by the control moel, n whch the estmate loa by the moel has a smlar behavor to the actual process. he error between them coul be cause by an error n the measurements, or because some of the nvolve varables were estmate. V. CONCLUSIONS he HSM moel presente n ths artcle s calbrate wth real values measure n the rollng mll process an constructe wth parameters prove by the frm ERNIUM SIDERAR. In ths way, a ynamc moel for the last rollng mll stan s obtane wth a behavor whch s also smlar to the real process wth non-lneartes that was consere. Lyapunov's stablty theory s use to erve a stable tunng rule to upate all parameters n the RBF-NN, thus ensurng the local stablty of the overall system. he results show the strp thckness varaton obtane wth the two control technques. he RBF-NN scheme of the controlle process evaluate by computatonal smulaton shows a smaller thckness varaton than the real system force fee forwar FFF that was taken for comparson presents greater thckness sperson than the RBF-NN compensaton. Smulaton stues base on HSM moel emonstrate that the propose scheme can tolerate parameters varatons, an ths scheme has the avantage that t can be use n parallel, for system-performance verfcaton uner real operaton contons, allowng for a future probable complementng of the system's force fee forwar (FFF) n HSM stans. REFERENCES Alexaner, J.M., On the heory of Rollng, Proc. R. Soc. Lonon, A. 36, 535-563 (97). Alvarez, J.C., A.B. Díez, D. Alvarez, J.A. González an F. Obeso, hck unevenness compensaton n a hot rollng mll havng automatc gage control, IEEE transactons on nustry applcatons, 38, 559-564 (00). Bouazza, S.-E. an H.A. Abbass, Moel Base Control System for Hot Steel Strp Rollng Mll Stans, Asan Journal of Informaton echnology, 6, 46-53 (007). Bryant, G.F., Automaton of anem Mlls, he Iron an Steel Insttute Carlton House errace, SWY5BD, Lonon (973). Dark, O., H. Mabush, I. Degawa an H. Nakamura, Progress of AGC utlzaton technques at plate 48 mll of ota works, Nppon Steel echncal Report, 4, 38-47 (989). Dng, J., X.L. Hu, J.M. Jao, G.F. She an X.H. Lu, Applcaton of Aaptve hreang echnque to Hot Strp Mll, Int. Journal of Iron an Steel Research, 5, 9-3 (008). Gálvez, J.M., L.E. Záratell an H. Helman, A moelbase prectve control scheme for steal rollng mlls usng neural networks, J. Braz. Soc. Mech. Sc. & Eng., 5, ISSN 678-5878 (003) Gnzburg, V.B., Steel Rollng echnology: heory an Practce, Marcel Dekker. New York an Basel (989). Hearns, G., P. Reeve,.S. Blkhu an P. Smth, Multvarable gauge an mass flow control for hot strp mlls, IFAC Automaton n mnng, mneral an metal proc. (MMM'04), France, 3-34 (004). Huang, M., X. Fang, J. Wang an S. Gu, Roll eccentrcty compensaton control for strp rollng mlls base on wavelet packet e-nosng theory, Intellgent Control an Automaton, 4, 3565-3569 (004). Kug, A., K. Schlacher an R. Novak, Nonlnear Control n Rollng Mlls: A New Perspectve, IEEE transactons on nustry applcatons, 37, 394-40 (00). Lee, Y.K., Y.J. Jang an S.W. Km, Aaptve Feeforwar Automatc Gauge Control n Hot Strp Fnshng Mll, ISIJ Int., 47, 444-45 (007). Mahfouf, M., Y.Y. Yang, M.A. Gama an D.A. Lnkens, Roll Spee an Roll Gap Control wth Neural Networks compensaton, ISIJ Internatonal, 45, 84-850 (005). Orowan, E.. he Calculaton of Roll Pressure n Hot an Col Rollng, Proc. Inst. of Mechancal Engneers, 50, 40-67 (944). Peersen, L.M., Moelng an entfcaton of a hot rollng mll, Amercan Control Conference, Seatle, Washngton, 3674-3678 (995). Pttner, J. an A.S. Marwan, Control of a contnuous tanem col metal rollng process, Control Engneerng Practce, 6, 379-390 (007). Rgler, G.W., H.R. Aberl, W.A. Stauer, K. Astlener, an K.H. Wenberg, Improve Rollng Mll automaton by means of avance control technques an Dynamc Smulaton, IEEE transactons on nustry applcatons, 3, 599-607 (996). Rossomano, F.G. an J. Dent Flho, Moellng an control of a hot rollng mll, Latn Amercan Apple Research, 36, 99-04 (006). Vyasagar, M., Nonlnear Systems Analyss, Prentce- Hall, N.J. (993). Receve: December 6, 009 Accepte: September 7, 00 Recommene by Subect Etor: José Guvant