Robust Mould Level Control

Transcription

1 5 American Control Conference June 8-1, 5. Portland, OR, USA ThA9.4 Robut Mould Level Control J. Schuurman, A. Kamperman, B. Middel, P.F.A van den Boch Abtract In the firt year of production ince, the cater of the Direct Sheet Plant in IJmuiden experienced untable ocillation in the control of the mould level. Thi paper report on the reearch that wa conducted to find the caue of and olution for thi problem. For that purpoe, a 1:1 cale water model wa ued. From thi cale model, it appeared that the ocillation were due to urface wave induced by topper movement. To model the effect of the urface wave, a imple tandard imulation mould level model wa extended by a heuritic wave model. Uing thi model, a robut table mould level controller wa deigned. Thi new controller ha proven to be ucceful ince it implementation in. I. INTRODUCTION He Direct Sheet Plant (DSP) of Coru, located in TIJmuiden, produce teel trip coil directly from liquid teel. Figure 1 chematically how the main procee in the DSP. In the DSP cater liquid teel i cat continuouly into a trand which i tranported via the tunnel furnace to the reduction mill. mould roll ladle topper tundih EMBr furnace mill coiler Figure 1 Schematical layout of the Coru Direct Sheet Plant. The cating proce play an important role in achieving good quality teel and a high production level. When Manucript received September 15, 4. J. Schuurman i with Coru Reearch, Development & Technology, PO Box 1., IJmuiden, Netherland (phone: ; fax: ; jan.chuurman@ corugroup.com). A. Kamperman i with Coru Strip Product, Direct Sheet Plant, PO Box 1., IJmuiden, Netherland ( arnoud.kamperman@corugroup.com). B. Middel wa with Coru Reearch, Development & Technology, and i now with Coru recruitment. P.F.A. van den Boch wa MSc tudent and i now with Océ Reearch N.V., P.O. Box 11, 59 MA, Venlo, the Netherland cating, liquid teel i poured into the tundih from the ladle; the teel flow through an opening in the bottom of the tundih and a Submerged Entry Nozzle into the mould. Above thi opening a topper i poitioned. The topper height (i.e. ditance between topper and opening) determine the liquid metal flow. The teel velocitie in the mould are reduced by the Electro-Magnetic Brake (EMBr). From the bottom of the mould the trand (with a olidified hell and liquid core) i pulled down by the roll. The mould level i meaured by a radiometric enor and fed back to the mould level controller which adjut the topper height in order to maintain the mould level at target level. Good teel quality and reliable cater operation require the mould urface to be flat and kept within a pecific window, and fluctuation to be reduced a much a poible. Fluctuation can reult in exceive rim formation and infiltration of lump of molten cating powder in the liquid teel that might be entrapped in the olidifying hell. Thi can reult in urface defect in the final trip product and/or irregular heat tranfer from the trand to the mould; the latter can caue crack in the hell that lead to a breakout. In cae of a breakout, liquid teel flow through the crack. After a breakout, cating i topped and the plant mut be repaired which can take 8-1 hour. To avoid thee problem the DSP require that mould level fluctuation are to remain maller than 3 mm, a much a poible. The mould level variation can be reduced by the EMBR and the mould level controller. After production tarted in, the cater uffered from inadequate mould level control, in particular during tartup of cating. Figure how an example of a tartup. The mould i filled up uing a PI controller, which i detabiliing and caue large mould level ocillation at a frequency of 3 rad/. At t = 17, control witche to PID with a Notch Filter (the detail of which will be decribed in ection III), but the ocillation remain. Thee ocillation, and ocillation at frequencie of 7 and 9 rad/, were occurring almot every time when tarting up at a cating width of 135 mm or above /5/$5. 5 AACC 4

2 level (mm) level (mm) time () 5 PI PID-NF time () Figure Mould level (upper graph) and topper poition (lower graph). Control witche from PI to PID-NF at t=17. Mould level fluctuation exceed mm. After a couple of breakout that were caued by thee ocillation, everal operational meaure were taken. Firt, it wa decided not to tart cating wider than 135 mm. Second, cating wa topped a oon a the ocillation appeared (to avoid a breakout). To olve the problem, reearch aimed at improving mould level control wa initiated. The problem of deigning a mould level controller i not new. In [1], [4] and [5] adaptive controller are preented that overcome the variation in the dynamic that are due to clogging and wear of the topper tip (or valve urface). Intead of adaptive control, fuzzy or H 8 control i applied in [], [3] and [6], for the ame reaon. In [7], the controller i extended by a feedback compenation for periodic diturbance. In all of the aforementioned publication, the dynamic of the mould level are decribed uing a imple ma balance, while in reality the dynamic are much more complicated due to urface wave. A will be hown in ection II of thi paper, the effect of urface wave could not be ignored in the deign of a controller for the DSP. Thi paper report on the reearch into the DSP mould level control problem. Section II report the effort to model the mould level dynamic. Section III decribe how a new controller wa deigned. The reult of thi controller are preented in ection IV. control loop i untable. To invetigate the caue of intability, a model of the cloed loop wa needed. The lumped mould level model that i uually preented in literature on mould level control (ee e.g. [1]-[7]), i baed on a imple ma balance (increae of ma per econd = net inflow of teel): 1 h = qin + d = P() qin + d (1) Mb with M the mould width (m) with value between m for the DSP, b =.7 m the mould thickne, the Laplace variable, h the mould level variation around teady tate (m), q in the inflow of teel into the mould (m 3 /), d the diturbance (m 3 /). The model decribed by (1) will be referred to a the Integrator model. The diturbance d in (1) can in principle be modeled a integrated white noie, although occaionally periodic diturbance may occur, a well a impule-like diturbance. The inflow q in i mainly determined by the topper height, which i controlled by a lave controller that adjut a hydraulic actuator. The relation between topper reference height and inflow can be modeled a: K qin = u = H() u () (.1 + 1) with K the topper gain mm /, u the topper reference height (m) and the Laplace variable. The topper gain value vary during cating. It value cannot be meaured directly, but it can be etimated indirectly from meaurement of the cater peed, mould level and topper poition. According to our etimation the value of K uually vary between m /. The mould level enor introduce dynamic and noie; the latter i due to the fact that the meaurement i baed on counting radiation particle. The enor dynamic can be modeled a τ e y = ( h+ n) = F( )( h+ n) (3) with y the meaured mould level (mm), n white noie (mm) with random ditribution and a variance of 1.5 mm, τ a time delay (). The time delay i believed to be introduced by the enor; it value i not certain, but i thought to be omewhere in the range Figure 3 how a block-diagram of the plant model, defined by (1), () and (3). d n II.MODELING OF THE MOULD LEVEL DYNAMICS The problem of mould level ocillation with increaing amplitude, a noted in the DSP, indicate that the feedback u H() topper actuator qin mould enor P () F () Figure 3 Block-diagram of mould level model. y 41

3 The control input u i adjuted by the mould level controller according to: u = C()( r y) with r the reference level (m) The controller C() ued in, conited of a PID controller according to.3.5 C ( ) = (4) Thi controller wa later extended by a notch filter N() placed in erie with the PID controller, according to N () = (5) The notch filter ha a notch at 7 rad/. The cloed loop model, decribed by equation (1)-(5) doe not explain the intabilitie a oberved in the DSP. Even if the gain K i increaed until intability occur (K > 3), the ocillation frequency i far from the oberved intability-frequencie (3, 7and9rad/). Hence, a more detailed model of the mould level dynamic wa needed. For that purpoe, ue wa made of a full cale water model of the cater. The open loop dynamic of the cale model were invetigated by varying the topper height inuoidally. In contrat to the DSP, the dynamic of the topper actuator and the enor, and the enor noie can be ignored. For the water cale model, the Integrator model become 9 y = u (6) Figure 4 how a Bode diagram of meaurement of the cale model and the Integrator model. phae (degree) gain (-) 4 meaured Integrator model frequency (rad/) Figure 4 Bode diagram of meaurement and Integrator model For frequencie below 1.5 rad/, the Integrator model i correct, but for higher frequencie the gain of the cale model how reonance peak. Thee peak appeared to be due to tanding wave in the fluid. In [8] the frequencie ω (rad/) of tanding wave are etimated at k kπ g ω k =, k = 1,,3,... W Thee frequencie agree well with the meaured reonance frequencie in the cale model. According to [9], the amplitude of the tanding wave, depend inuoidally on the poition of the enor, a hown in figure 5 for the cae of a tanding wave correponding to k=. level (not to cale) enor location ditance from middle of mould (m) Figure 5 Extreme urface line (olid and dahed) of tanding wave correponding to k=, and enor location. In the water model only the tanding wave correponding to k= and k=4 were oberved. To model thee tanding wave, the Integrator model wa extended by a tranfer function W(), according to h = Pw() qin + d 1 (7) Pw () = + W() Mb The tranfer function W() i defined by 4 AB k kωk W() = (8) k = + Bkωk+ ωk with π x Ak = Kk co M Bk = 1/( τω k k) x= ditance of enor from middle of mould (m) The parameter K k (-) and τ k () were fit to meaurement obtained from the water model; the value obtained were K = 6, K =, K = 15, τ = 8, τ = Figure 6 how how the model fit the meaurement. The agreement i remarkable, conidering that the extenion of W() wa not baed on phyic. 4

4 gain (-) M w =1.5m M w =1.m TV n SV d G () = WRV h n WRV h d with = (1 ), = (1 ), = (1 ) T L L S L R CF L L = P HCF v 1 v V () n Vd () d n phae (degree) frequency (rad/) frequency (rad/) Figure 6 Comparion of meaurement from the cale model and Integrator model (1) extended by (6). The cloed-loop model defined by (1)-(8) i untable if the topper gain K exceed a value around or larger. Depending on the mould width, the level can how undamped ocillation correponding to the frequencyω and ω 4. A thee frequencie were alo oberved in the DSP, the tanding wave eemed to explain the intabilitie, except for the ocillation at ω 1. Thee ocillation indicate that in the DSP the k=1 wave doe occur, even though it doe not occur in the cale model. The fact that thee ocillation ometime do occur, and ometime not, indicate that the parameter K and K k are not certain. Uing the model preented in thi ection, the mould level controller wa redeigned. III. CONTROLLER DESIGN A argued in ection II, the mould level dynamic can be decribed by (1) - (6), but with parameter that vary with time and that are not certain. The H loop haping deign technique wa applied ince uncertaintie can be taken into account in a ytematic way, a will be hown next. Figure 7 how the nominal plant model, modeled by equation (1), extended by weighting filter V n (), V d () and W h (). The H controller wa obtained by minimiing γ in the inequality G () γ (9) with G() defined by u H() topper actuator qin mould enor P () F () controller C () W () h Figure 7 Block-diagram of the model ued in the deign of the H controller. In fact, G() decribe the tranfer function from v 1 and v to z 1 and z, i.e.: z v 1 1 G () z = v The following implication wa ued in chooing the weighting filter (in particular the filter W h ()): G () γ Gij γ i, j= 1, (1) The weighting filter gain of Vn () wa choen almot 1 ince the noie level i contant, except for high frequencie (above 1 rad/) where the gain ha been increaed to avoid numerical computation problem. The weighting filter Vd () wa choen a a near integrator ince diturbance d can be conidered a integrated white noie (ee ection I). An exact integrator could not be ued a the ytem i then not tabiliable; to avoid thi, the pole of Vd () i not choen at zero, but lightly negative. The tranfer function Wh () i choen uch that robut tability i guaranteed if γ < 1 i achieved. To explain how thi tranfer function ha been choen, the mall gain theorem i tated here firt. Theorem 1 (Small Gain Theorem) Conider the interconnection hown in Figure 8. Aume that both the nominal ytem B() and the perturbation () are table, then the cloed loop i table for all perturbation () if and only if B < 1 y z 1 z 43

5 z1 () B () z Figure 8 Interconnection of a table tranfer function B() and an uncertain perturbation block () Figure 9 how the plant model of the DSP, extended by a perturbation block () that repreent the uncertainty. u H() topper actuator z1 () z enor P () F () nominal mould model controller C () Figure 9 DSP model extended by perturbation block () (additive uncertainty model). The mould dynamic i thu modeled by the o-called additive uncertainty model P () = P () + () 1 with () the perturbation block and P () = the Mb nominal mould model with M =.15 m, b =.9 m. According to the Small Gain Theorem, the condition for robut tability i that the tranfer function from z to z 1 in Figure 9 i table and that it product with () ha infinity norm maller than 1. The tranfer function B () from z to z1 i given by: H() C() F() B () = 1 L ( ) Hence, for robut tability, the product of thi tranfer function with the perturbation block () mut have an infinity norm maller then one: HCF 1 L 1 If the H deign ha been completed, inequality (1) i atified, i.e.: CF Wh Vd γ 1 L Hence, by chooing () H () P () P () H () Wh > = (11) Vn() Vn() robut tability i guaranteed if γ <1 Condition (11) i equivalent to y P( jω) P ( jω) H( jω) Wh( jω) > = Perr( jω) ω > V ( jω) n (1) with ω the radial frequency (rad/). The weighting filter W h ha been choen uch that condition (1) wa atified for different mould dynamic model of P() including the wave dynamic, i.e. defined by equation (7), but for width M between 1 and 1.5 m. Figure 1 how graph of the gain of W h and gain of Perr ( jω ). A can be een, the weighting filter i choen omewhat conervative at higher frequencie to be on the afe ide. At low frequencie the gain of the weighting filter eem to be inufficient, but that appeared not be the cae in imulation. gain frequency (rad/) Figure 1 Graph of the gain of the weighting filter W h (olid line) and the gain of Perr ( jω) a given in (1) for variou mould width (dahed line). To achieve γ <1, the gain of the weighting filter V d wa varied. The reulting H controller (of 1th order) wa implemented a a erie connection of 5 tandard econd order tranfer function in the GE control oftware of the DSP. IV. RESULTS The H controller wa firt teted in December and ha been in ue ever ince. Figure 11 how the mould level and topper poition during a tartup of cating in 3. The mould i filled firt, during which a PI controller i ued. A oon a control witche to the H controller, the controller top reacting to the ocillation, and they dampen. The H controller ha been teted for all mould-width and appeared to function atifactorily. Since it implementation, untable 44

6 ocillation have not been oberved anymore. Becaue of it reliability, the limitation on mould-width ha been abolihed, production ha increaed ignificantly and the number of breakout per month ha reduced. level (mm) [5] D. Lee, J.S. Lee, T. Kang, Adaptive fuzzy control of the molten teel level in a trip cating proce, Control Eng. Practice, Vol.4, No. 11, p , 1996 [6] D. Lee, Yeongub, S. Lee, High performance hybrid mold level controller for thin lab cater, Control eng. Practice, Vol. 1, No. 3, p , 4. [7] T.J. Manayathara, Rejection of unknown periodic load diturbance in continuou teel cating proce uing learning repetitive control approach, IEEE Tranaction on control Sytem Technology, Vol. 4, No. 3, 1996 [8] H. Lamb, Hydrodynamic, 6th ed. Cambridge: Cambridge Univ. Pre, 193. [9] P.F.A. van den Boch, Mould Level Control at the DSP Cater, MSc thei, Eindhoven Univerity of Technology, level (mm) PI H time () Figure 11 Mould level (upper graph) and topper poition (lower graph). Control witche to H at t=17. V.FUTURE RESEARCH The dynamic of the mould level, in repone to topper movement are not yet fully undertood. The phyical cale model ha helped to clarify the fact that tanding wave can occur, but it i not yet clear what caue them, and how they are influenced by cater peed and the EMBr. Further reearch i required to increae our undertanding of the mould level dynamic. With more knowledge of the ytem, the controller can be tuned more tightly, leading to improved control. ACKNOWLEDGMENT The author would like to thank Dr. S. Weiland and the reviewer for their contribution. REFERENCES [1] T. Heketh, D.J. Clement, R. William, Adaptive Mould Level Control for continuou lab cating, Automatica, vol. 9, no. 4, p , [] R. Keyer, Improved mould-level control in a continuou teel cating line, Control Eng. Practice, Vol. 5, No., p31-37, [3] T. Kurokawa, T. Kondo, T. Mita, K. Liu, M. Sapei, Development of mold level control in continuou cating by H control theory, nd IEEE conf. on control Application, p , Sep [4] M.A. Barron, A. Aguilar, J. Gonzalez, E. Mendelez, Model-baed control of mold level in a continuou teel cater under model uncertaintie, Control engineering Practice, Vol. 6, p ,