PI control ytem deign for Electromagnetic Molding Machine baed on Linear Programing Takayuki Ihizaki, Kenji Kahima, Jun-ichi Imura*, Atuhi Katoh and Hirohi Morita** Abtract In thi paper, we deign a PI controller for an electromagnet placed inide a molding machine. The eddy current inide the electromagnet are a patially-ditributed phenomenon that i difficult to capture by uing finite-dimenional ytem. Firt, we how that the theoretical model, expreed by partial differential equation, cannot perfectly capture the propertie of the actual machine. Second, we deign the PI control ytem via a loop-haping by directly uing the data of the frequency repone obtained from the experiment of the machine. The deign problem i reduced to a linear programing problem. I. INTRODUCTION In molding fabrication, machine which generate mold clamping force are called molding machine. In the pat, molding machine driven by oil preure were dominant. Recently, electric molding machine are widely ued to improve controllability and cycling time of job. Furthermore, electromagnetic molding machine, driven by electromagnet, are currently propoed [1]. In the machine, a uction force generated by the electromagnet i directly tranmitted to a mold in contrat to general electric molding machine, in which a thrut force generated by a rotary motor i tranmitted through amplification. A a conequence, the electromagnetic molding machine are capable of more precie molding fabrication. However, it i difficult to realize deired force repone uing heuritic tuning of controller. The reaon i that eddy current are patially ditributed in an iron core of the electromagnet. Thi mean that the ytem hould be treated a a ditributed parameter ytem [5], [1], [11], [13]. Therefore, the ytem i difficult to model in finite dimenion while keeping uitable propertie for control ytem deign. In thi paper, we firt overview a model expreed by partial differential equation (PDE) to analyze the property of thi ytem. However, from the analyi it i made clear that the theoretical model cannot perfectly capture the propertie of the actual machine due to a certain element of a time delay included in the machine. Therefore, we deign a PI control ytem via loop-haping by directly uing the data of the frequency repone obtained from the experiment of the machine. The deign problem can be reduced to a linear programing problem becaue the tructure of the controller i retricted to PI controller (or PI controller and filter) for indutrial application. *Graduate School of Information Science and Engineering, Tokyo Intitute of Technology; 2-12-1, Meguro ward, Tokyo {ihizaki,kahima,imura}@cyb.mei.titech.ac.jp **Sumitomo Heavy Indutrie, Ltd; 19, Natuhima, Yokouka city, Kanagawa {At Katoh,hrh morita}@hi.co.jp Thi paper i organized a follow. In ection 2, we explain the feature of an electromagnetic molding machine and the difficulty of the control ytem deign uing heuritic tuning. Furthermore, we introduce an expreion of the ytem a PDE and analytically characterize the tructure of the ytem in the frequency domain. Then, thi model i compared to the experimental data. In ection 3, we deign a PI control ytem via loop-haping by directly uing the experimental data of the machine. In addition, we how the time repone of the machine when uing the controller deigned optimally and dicu a trade-off relation between the performance and robutne of the controlled ytem. Finally, we examine the improvement of the performance of the controlled ytem when adding a firt order filter to PI controller. II. ELECTROMAGNETIC MOLDING MACHINE A. Summary of the molding machine Fig. 1 how a prototypical ytem of molding machine. Thi ytem conit of electromagnet, generating molding force, a center rod, tranmitting force generated at the electromagnet, a clamping mechanim, performing the molding and a frame, holding the whole ytem. The mold clamping mechanim conit of the mold and a load cell, which meaure molding force. Fig. 2 how the electromagnet placed inide the ytem. The electromagnet conit of an electromagnet core, a uction plate and a coil. They generate magnetic flux, hown in Fig. 2, by the current flowing in the coil. Sucking force are generated by the magnetic flux at a gap between the electromagnet core and the uction plate. Then, they are tranmitted to the mold located inide the mold clamping mechanim, which i connected to the uction plate throughout the center rod. B. Tuning of PI control ytem To control the electromagnetic molding machine, the repone of the molding force need to ettle in a hort time without overhoot. Fig. 3 how a block diagram of the PI control ytem. In thi figure, the electromagnetic molding machine, hown in Fig. 1, i repreented by the electro magnet block. The filter will be added to PI controller in order to examine the improvement of the performance (See ection III-C for detail). Fig. 4 how experimental reult of the tep repone of the molding force 1 when we vary a proportional gain of the control ytem. Generally, the convergent rate to a target value i larger a the proportional 1 In thi paper, the ampling time of the experiment were et at 1.5 khz.
Fig. 3. PI control ytem High Force command Fig. 1. Prototype ytem Low Proportional Gain Fig. 4. Step repone of electromagnet ytem Fig. 2. Electromagnet ytem gain of PI control ytem i enlarged and when the gain reache ome threhold, overhoot of the repone will arie. However, in the cae of the electromagnetic molding machine, we can ee from Fig. 4 that the convergent rate doe not increae even if the gain i enlarged. From thee experiment, we can ee that the behavior of thi machine contradict the intuition of general ytem. Therefore, it can be anticipated that the performance of the machine will be hard to improve with the traditional tuning technique of controller [2]. To deign the control ytem ytematically, we firt derive the mathematical expreion of the ytem. C. Derivation of the expreion a partial differential equation The phyical ytem i uppoed to be axiymmetric a hown in Fig. 5. Therefore, we aim to derive relation of phyical quantity for a radial direction from baic law of phyic [3]. To thi end, we conider a micro region D along a path of the eddy current located at the radiu r. Here, Γ denote a radiu of the iron core. Uing magnetomotive force E M (t, r), magnetic reitance R M (r) and circumferential length l (r) = 2πr, the magnetic flux φ (t, r) generated in D i given by φ (t, r) = E M (t, r) R M (r) (1) and R M (r) = σ ( L l (r), σ := + d ), (2) µ µ µ where L, µ and µ denote the length of the flux path, the magnetic permeability in a vacuum and the relative magnetic permeability, repectively. Next, uing induced electromotive force E E (t, r) and electrical reitance R E (r), the eddy current i e (t, r) that arie in D i given by i e (t, r) = E E (t, r) R E (r). (3) Here, the electrical reitance i proportional to the circumferential length and i inverely proportional to the width of the path and therefore i given by R E (r) = ρ l (r) dr, (4) where ρ i a contant determined by the material. From the definition, the denity of the magnetic flux B (t, r) ha a relation of φ (t, r) B (t, r) = l (r) dr. (5) From Faraday law of electromagnetic induction, the induced electromotive force E E (t, r) i given by E E (t, r) = r l (ζ) B (t, ζ) dζ, (6) t i.e., the ummation of the time derivative of the magnetic flux. The magnetomotive force E M (t, r) in D i given a the ummation of the eddy current i e (t, r) that arie outide
D and the current in the coil i c (t). Therefore, it i expreed a Γ E E (t, ζ) E M (t, r) = dζ + N c i c (t), (7) ρl (ζ) r where N c denote the number of coil turn. Hence, organizing thee relation for B (t, r) yield B (t, r)= 1 σρ Γ r 1 ζ ζ ζ B (t, ζ) dζdζ + N c t σ i c (t). (8) Finally, the uction force F generated in the iron core i given by F (t) = 1 2µ Γ l (r) B (t, r) 2 dr. (9) Here, we define, repectively, the tate variable, the input and the output by χ (t, r) := B (t, r), υ (t) := i c (t) and Υ (t) := F (t) and then linearize (9) around an operating point ( υ, χ, Υ ). Note that the teady-tate value χ i contant and independent of the patial variable r. Then, we define the error from the operating point by (υ, χ, Υ) and replace the patial variable r/γ by r. Furthermore, by differentiation of (8) with repect to r and coordinate tranformation a r = ξ for convenience, we obtain ˆχ (t, ξ) t = 2α ξ ( ξ ˆχ (t, ξ) ξ ˆχ (t, ξ) = β υ (t) at ξ = 1 α ˆχ (t, ξ) ξ = at ξ = ξ Υ (t) = 1 ˆχ (t, ξ) dξ, ) at ξ (, 1) (1) where ˆχ (t, ξ) := χ ( t, ξ ), and α := 2σρ/Γ 2 and β := 2πρNc 2 υ/ (σµ ) R are poitive contant. Therefore, the dynamic of the phyical ytem are eentially identical to that of heat diffuion ytem with a diffuivity which i proportional to the patial variable [1], [11], [12]. The input-output tranfer function of thi ytem [6], [7], [8], [13] i given by ) where G () := β α J k (z) = m= are Beel function [9]. ( J 1 2 2α J 2α ( 2 2α ), (11) ( 1) m ( z ) 2m+k (12) m! (m + k)! 2 D. Comparion with the experimental reult on the actual machine In thi ection, in order to verify the validity of the model expreed by PDE, we compare the frequency repone of the model (11) with that of the experimental reult of the machine [4]. In addition, we compare it with a firt order ytem for the examination. Fig. 6 how the Bode diagram of the experimental reult, the theoretical model and the firt order ytem. From thi φ (t, r) r i e (t) Γ dr D Fig. 5. N c Model of electromagnet i c (t) figure, we can ee that the lope of the gain characteritic of the theoretical model and the experimental reult i 1 [db/dec] at high-frequency. Generally, the lope of the gain characteritic of ytem with relative degree k i 2k [db/dec] at high-frequency. From thi fact, we can anticipate that the property of the actual machine i hard to even approximately expre in term of a general lumped parameter ytem coniting of rational function. On the other hand, we can alo ee from Fig. 6 that the phae characteritic of thee are different at high-frequency. Thi difference of the phae might be caued from certain element of the time delay included in the machine. From thi feature, we can anticipate that if we ue the theoretical model for the control ytem deign, the effect of thi time delay negatively affect the behavior of the controlled ytem. While, we can expect that the delay can be appropriately modeled a a time-delay ytem. However, it i quite difficult to deign the actual controller for uch model, compoed of PDE and, in addition, time-delay [8]. From thee examination, we can ee that the theoretical model expreed by PDE cannot perfectly capture the property of the actual machine. Therefore, deign method in which the data of the actual machine i directly ued are uitable for the control ytem deign of thi ytem. III. PI CONTROL SYSTEM DESIGN A. PI control ytem deign via Linear Programing In thi ection, we explain a deign technique for the PI control ytem, hown in Fig. 3, via a Linear Programing [14]. We define the PI controller by K () := k P + k I (13) and et aide the filter in Fig. 3. Furthermore, we denote the machine data of frequency repone by g n C and ω n R +, i.e., g n = G M (jω n ) for the input-output tranfer function of the machine G M (), which i not explicitly expreed in thi paper. Here, from the experiment of the machine, the data g n, ω n are obtained a Table I, in which ω 1,, ω 2 equentially denote the value of ω n from the mall one and alo g 1,, g 2 denote the value of g n
Experimental reult Theoretical model Firt order ytem Fig. 6. 1 [db/dec] Bode diagram of the machine according to ω n. In the ret of thi paper, we deign the control ytem baed on frequency haping of the open-loop tranfer function. Now, let u conider the following problem: Problem: Suppoe that data g n C and ω n R + for n = 1, 2,, N, (a i, b i, c i ) R 3 for i = 1, 2,, I 1 and (a I, b I ) R 2 are given. Furthermore uppoe that the et N i {n = 1,, N} for i = 1, 2,, I are given. Find k P and k I for which L I (jω ni ) := g ni K (jω ni ) maximize κ uch that a I Re [L I (jω ni )] + b I Im [L I (jω ni )] < κ, (14) ubject to all L i (jω ni ):=g ni K (jω ni ) for i = 1, 2,, I 1 atify for n i N i. a i Re [L i (jω ni )] + b i Im [L i (jω ni )] < c i (15) Thi frequency haping problem can be reduced to a linear programing problem, which can be olved efficiently by numerical computation, becaue the tructure of the controller i retricted to PI controller. Therefore, we can hape the frequency property of the open-loop tranfer function by uing the machine data obtained from the experiment. The phyical interpretation of each et and the parameter in thi problem and relation to the actual deign problem are explained throughout dicuion in the next ection. B. Loop haping by uing the frequency repone data In thi ection, we deign the PI control ytem and alo how the time repone of the deigned ytem. Here, we maximize κ with everal value of γ under the contraint that k P and k I atify N 1 = {1,, 12}, (a 1, b 1, c 1 ) = ( 1, 1, γ) N 2 = {1,, 2}, (a 2, b 2, c 2 ) = (, 1, κ). In thi optimization problem, we maximize κ under the contraint of the half plane for the loop tranfer function L i (jω ni ) at the frequency point ω ni, n i N i. The retriction boundarie are traight line, which are defined by the parameter (a i, b i, c i ) for i = 1, 2. To put it concretely, the firt retriction guarantee ome tability margin by giving ome γ and the econd retriction maximize the gain in the low- and middle-frequency range by maximizing κ. A a reult, we obtained k P =9.9, k I =8.43 1 3 and κ=1.95 when γ =.1 and k P = 19.8, k I = 1.69 1 4 and κ = 3.9 when γ =.2. Fig. 7 and Fig. 8 how Nyquit plot of L i (jω ni ), which are enlarged in Fig. 9 and Fig. 1 around the origin, and that of the theoretical model (11) when uing the controller obtained here. In thee figure, the olid line and the broken line denote the Nyquit plot of L i (jω ni ) and that of the theoretical model, i.e., G (jω)k (jω), repectively. In addition : and denote the range ω n1 for all n 1 N 1 and ω n2 for all n 2 N 2, repectively. From thee figure, we can ee that the gain are maximized in the low- and middle-frequency range while all L i (jω ni ) atify the retriction and alo that the obtained κ become maller when we give larger γ. Furthermore, the phae of L i (jω ni ) delay more than that of the theoretical model G (jω)k (jω). Therefore, we can anticipate that if we ue the theoretical model for the control ytem deign, we cannot guarantee the tability margin of the controlled ytem appropriately. Furthermore, we can obtain a trade-off relation between κ and γ, i.e., guarantee of the gain in the low- and middlefrequency range and the robutne of the cloed ytem, a hown in Fig. 11. That i to ay, we cannot improve the convergence rate without loing ome of the tability margin. Conequently, we can theoretically how the performance limitation of the PI controller for thi ytem, which i hard to determine by heuritic tuning of the controller. In addition, Fig. 12 how the experimental reult of the time repone for the controller obtained from olving the following problem: N 1 = {1,, 16}, (a 1, b 1, c 1 ) = ( 4.6, 1, γ) N 2 = {1,, 2}, (a 2, b 2, c 2 ) = (, 1, κ). A a reult, we obtained k P = 3.28, k I = 33.5 and κ =.6 when γ =.12 and k P = 6.56, k I = 66.2 and κ =.12 when γ =.24. We can ee from thi figure that both experimental reult converge to the target value and alo that the convergence rate improve a the value of γ i larger. C. Examination for the improvement of the performance In ection III-B, we howed the performance limitation when uing the PI controller only. Beide, in thi ection we examine the improvement of the performance when adding a firt order filter to PI controller.
TABLE I DATA SET OF FREQUENCY RESPONSE OF THE MACHINE ω n [rad/ec].6283 1.257 1.885 3.142 5.27 g n.136 +.1j.1291 +.144j.1272 +.182j.1226 +.247j.1156 +.317j 6.283 12.57 18.85 31.42 5.27.1111 +.386j.97 +.452j.77 +.473j.64 +.445j.46 +.48j 62.83 125.7 188.5 314.2 52.7.4 +.385j.244 +.317j.17 +.278j.99 +.244j.15 +.26j 628.3 1257 3142 6283 1257 -.15 +.188j -.76 +.19j -.79 +.j -.18 -.5j.26 -.3j (a 1, b 1, c 1 ) (a 1, b 1, c 1 ) Imaginary Axi ω n2 1 ω n1 (a 2, b 2, c 2) Imaginary Axi 1 ω n1 GK (Theoretical model) GK ω n2 (a 2, b 2, c 2) Real Axi Real Axi Fig. 7. Nyquit plot (γ =.1) Fig. 8. Nyquit plot (γ =.2) In the ret of thi ection, we olve the ame frequency haping problem a thoe in ection III-A for the controller of K () := k P + k I + k F T + 1, (16) in which the filter i added to (13) and T denote the time contant of the filter. Here, when T = 3 and γ =.2, we maximize κ under the contraint that k P, k I and k F atify N 1 = {1,, 12}, (a 1, b 1, c 1 ) = ( 1, 1,.2) N 2 = {1,, 2}, (a 2, b 2, c 2 ) = (, 1, κ). Fig. 9. Enlargement of Fig. 7 Fig. 1. Enlargement of Fig. 8 A a reult, we obtained k P =19.3, k I = 3.32 1 7, k F = 9.98 1 8 and κ = 6.31. Fig. 13, in which the denotation are ame a thoe in Fig. 7 or Fig. 8, how Nyquit plot of L i (jω ni ) and the theoretical model G (jω)k (jω) when uing the controller obtained here. From thi figure, we can ee that the gain in the low- and middle-frequency range i bigger than that in Fig. 8 while keeping the ame tability margin, i.e., γ =.2. That i to ay, the performance of the controlled ytem improve by increaing the order of the controller. Moreover, the Nyquit plot of the theoretical model i ditant from that of L i (jω ni ). Furthermore, Fig. 14 how the obtained value of κ when varying the time contant T. Thi figure indicate that the performance will improve ignificantly by chooing the time contant between 2 and 3 (max κ 9). However, ince the obtained gain are too large to implement, we need to practically deign controller taking thi point into account.
κ : Performance Fig. 11. γ : Robutne Trade-off relation between κ and γ Imaginary Axi ω n2 1 ω n1 (a 1, b 1, c 1) GK (a 2, b 2, c 2) Force [tf] Fig. 12. γ =.24 γ =.12 Force command Experimental reult Time [ec] Experimental reult of time repone κ : Performance Real Axi Fig. 13. Nyquit plot of the deigned ytem IV. CONCLUSION Spatially-ditributed phenomena are important to indutrial application, while the mathematical treatment of them are complex. In thi paper, we howed that the theoretical model expreed by PDE wa not necearily uitable for the control ytem deign becaue the model could not perfectly capture the property of the actual machine. Therefore, in thi paper we deigned a PI control ytem by directly uing the experimental reult via the frequency haping. The deign problem wa reduced to a linear programing problem. Conequently, we theoretically howed the performance limitation, which i hard to determine by heuritic tuning of the controller. REFERENCES [1] H. Morita, A. Kato, T. Yamamoto and T. Shibata: WO publication, WO29/28488 [2] Y. Nozaka: Indutrial Control Sytem and Control Apparatu, CORONA PUBLISHING CO., LTD (1995) [3] D. K. Cheng: Field and Wave Electromagnetic Second Edition, ADDISON-WESLEY PUBLISHING COMPANY (1992) [4] L. Ljung: SYSTEM IDENTIFICATION: Theory for the Uer, PREN- TICE HALL PTR (1987) [5] R. Curtain and H. Zwart: An Introduction to Infinite-Dimenional Control Theory, Springer (1995) [6] R. Curtain and K. Morri: Tranfer function of ditributed parameter ytem: A tutorial Automatica 45 111-1116 (29) [7] R. Padhi and S. F. Ali: An account of chronological development in control of ditributed parameter ytem Annual Review in Control 33, 59.68 (29) Fig. 14. T : Time contant Change of κ with T [8] N. Abe and A. Kojima: Control in Time-delay and Ditributed Parameter Sytem, CORONA PUBLISHING CO., LTD. (27) [9] N. W. McLachlan: Beel Function for Engineer Second Edition, OXFORD UNIVERSITY PRESS (1955) [1] J. Crank: The Mathematic of Diffuion, OXFORD UNIVERSITY PRESS (1973) [11] W. M. Deen: Analyi of Tranport Phenomena, OXFORD UNIVER- SITY PRESS (1998) [12] H. S. Carlaw and J. C. Jaeger: Conduction of Heat in Solid, OXFORD SCIENCE PUBLICATIONS (1986) [13] M. Krtic and A. Smyhlyaev: Boundary Control of PDE A Coure on Backtepping Deign, Society for Indutrial and Applied Mathematic Philadelphia (27) [14] M. Kunze, A. Karimi and R. Longchamp: Frequency domain controller deign by linear programming guaranteeing quadratic tability In Proceeding of the 47th IEEE Conference on Deciion and Control, page 345-35, Cancun, Mexico, December (28).