RNN Based Auto-tuning of PID Control Gains in Hot Strip Looper Controller

Transcription

1 Memors of the Faculty of Engneerng, Okayama Unversty, Vol.40, pp.16-22, January, 2006 RNN Based Auto-tunng of PID Control Gans n Hot Strp Looper Controller Yoshhro Abe Dept. of Electrcal and Electronc Engneerng Okayama Unversty 3-1-1, Tsushma-Naka Okayama, Masam Konsh Dept. of Electrcal and Electronc Engneerng Dvson of Industral Innovaton Scence The Graduate School of Natural Scence and Technology Okayama Unversty 3-1-1, Tsushma-Naka Okayama, Jun Ima Dept. of Electrcal and Electronc Engneerng Dvson of Industral Innovaton Scence The Graduate School of Natural Scence and Technology Okayama Unversty 3-1-1, Tsushma-Naka Okayama, Tatsush Nsh Dept. of Electrcal and Electronc Engneerng Dvson of Industral Innovaton Scence The Graduate School of Natural Scence and Technology Okayama Unversty 3-1-1, Tsushma-Naka Okayama, (Receved December 14, 2005) In ths study, auto tunng of PID control gans n hot strp looper controller s made based on RNN model. Neuro emulator s employed to model the characterstcs of looper dynamcs. Combnng neuro emulator and RNN model, auto tunng system of PID control gans s constructed. As the nputs to RNN, plural evaluaton functons whch reflect ndvdual preference of human experts. Further, Self learnng mechamsm s embeded to RNN model whch enables adaptaton to the change n rollng chracterstcs. Through numercal experments, the effect of the proposed method s ascertaned. 1 Introducton In these years, plant control systems are beng hghly automated. But both the performances of plant and controller change wth the passage of tme, so t s necessary to tune them. Ths s why human experts ntervene and tune the control system to mprove the total plant performances. In ths paper, t s targeted especally to tune looper control gans n the threadng of hot strp mlls. In hot strp mlls, the looper heght control system s set up between stands. By use of them, the rollng operaton are kept to be stable. It s usual that the looper control system s governed by PID con- y-abe@cntr.elec.okayama-u.ac.p troller. Usually PID gans are determned by gan table accordng to a varety of strp thckness and hardness, but the optmal gan also vares wth tme due to the deteroraton of plant characterstcs. Now, t s often the cases that the skllful operators tune these gans. It s necessary to tune the PID gans approprately n order to stablze the looper s behavor. In the past, the looper control n the steady state has been wdely bult up as the control technque [1]. Contrary to ths, n threadng condton of the looper s behavor tends to be unstable, and then human expert never fal to ntervene and stablze the looper s behavorbytunng PID gans and nterposng manpulated varable of looper heght control system. In general, t s dffcult Ths work s subected to copyrght. All rghts are reserved by ths author/authors. 16

2 Yoshhro ABE et al. MEM.FAC.ENG.OKA.UNI. Vol.40 to automatze human s acton because of flexblty and dversty of hs human operatons not only n hot strp rollng but also n other obects. In ths paper, we try to automatze tunng of PID gans by human. That s, the method of usng recurrent neural networks (RNN) [2][3][4] s proposed. Because human can memorze the past tunng results whether old or new, human s tunng acton depends on past tunng experences. RNN s a smple mechansm generatng ts outputs dependng on the past data. Frstly, neuro emulaton s studed to emulate a dynamc behavor of hot strp rollng. The obtaned neuro emulator s combned wth PID controller to smulate the Looper heght control dynamcs human tuned PID gans of looper control system. Then, methods to update the RNN model by usng learnng algorthm wll be proposed. 2 Looper Heght Control n Hot Strp Rollng The looper angle s determned by the dfference of strp veloctes between adacent rollng stands. In the followng, the mathematcal model of looper control n hot strp rollng, strp thckness control and looper heght control are descrbed. 2.1 Mathematcal Model for Hot Rollng The fundamental equatons for hot rollng process are descrbed as follows. P = σ qr 0(H h )Q (1) F = 1 r s 2 arctan H h π h h 8 R 0 ln h R 0 (2) r à r h π Q = H h 2 arctan H h h s s! R 0 ln h φ + 1 R 0 ln H π h h 2 h h 4 (3) ³ R 0 = R 1+ cp (c = )(4) H h h = s + P (5) s M h φ = tan F (6) f = 1 2 φ2 R 0 ³ 2R 0 1 h (7) b = h R 0 (1 + f ) 1 (8) where, P s rollng force, σ s flow stress, H s entry strp thckness, h s ext strp thckness, R s roll radus, s s roll gap, f s forward slp rato, b s backward slp rato, M s mll modulus, and subscrpt denotes th stand. Usng f and b led by solvng above equatons, ext strp speed v f and entry strp speed v b can be obtaned as follows v f = V R (1 + f ) (9) v b = V R (1 + b ) (10) where V R s the rollng velocty of th mll. 2.2 Mathematcal Model for Looper System As shown n Fg. 1, the dstance between mlls s L = L 1 + L 2,ext strp speed of th mll s v f,and entry strp speed of ( +1) th mll s v b+1. Assumng L 1 = L 2 and θ = θ 0, the strp loop angle θ can be obtaned as follows sec θ = 1+ 1 L Z t 0 (v f v b+1 )dt (11) where the ntegral s started when the strp gets threads ( +1) th mll. The strp loop fgure can be approxmated as sosceles trangle. Thus, and looper angle α s gven by sn α = 1 ³ L tan θ + δ (12) l 2 Fg. 1: Looper and nterstands schematcs Varables v f,v b, θ 0, θ, α, δ,l,l,l 1,L 2 are explamed n Fg. 1 17

3 January 2006 RNN Based Auto-tunng of PID Control Gans n Hot Strp Looper Controller 2.3 Mass Flow Model for Rollng velocty Mass flow rule s used n order to determne set pont of rollng velocty. Mass flow rule can be descrbed as h o 1V R1 (1 + f 1 )=h o 2V R2 (1 + f 2 ) (13) where h o s the desred value of strp thckness at th stand. 2.4 Strp Thckness Control Model Gauge meter equaton s wrtten by 3.2 Inter Stands Looper Dynamcs The problem treated n ths paper conssts of two stands and one looper faclty between them. The looper angle s controlled by changng rollng speed of the upstream stand. In the followng, constructon of the problem for hot strp looper heght control system wll be stated. Fg. 2 shows two stands hot strp mll system wth controllers. The block dagram showng dynamc characterstcs of nter stands looper are gven n the lower part of Fg. 3. h = s + P (14) M where h, s, P and M are ext strp thckness, roll gap, rollng force and mll modulus. The control scheme of Gauge Meter AGC(Automatc Gauge Control) s gven by h = h h o (15) s = k h (16) where k s AGC gan. AGC s started to operate soon after the head end of strp arrves at the next stand. Fg. 2: Hot Strp Mll 3 Problem descrpton of Looper Control System 3.1 PID Control of Looper Angle PID control s descrbed as ³ m(t) = K p e(t)+ 1 Z T de(t) e(t)dt + T d (17) dt where m(t) s manpulated varable and n ths case, e(t) s error and means looper devaton, Kp s proportonal gan, T s ntegral tme, and Td s dervatve tme of PID control. In smulator, PID control s used wth velocty form algorthm and lagged dervatve. u n = u n 1 + K p h(e n e n 1 )+ T s e n + μ n T (18) η n = μ n 1 + μ n (19) μ n = (γβ 1)μ n 1 + β(e n e n 1 ) (20) β = T d T s + γt d (21) where T s s samplng tme nstance, e n s error at samplng nstance nt s, e n 1 s error at samplng nstance (n 1)T s.and 1 γ s dervatve gan. Fg. 3: Block Dagram of Hot Strp Mll wth Controllers In ths paper, based on the model n Fg. 3, numercal smulatons are carred out to generate looper dynamcs for the constructon of gan tunng system. 4 Auto Tunng of PID Gans by RNN The Constructon of Gan Tunng System s shown n Fg. 4. the upper part of Fg. 4 shows real plant nvolvng hot strp looper and PID controllers. The lower part of Fg. 4 shows the proposed gan tunng mecha- 18

4 Yoshhro ABE et al. MEM.FAC.ENG.OKA.UNI. Vol.40 The methods of trang connectng weghts n neuro emulator, λ (2) and λ(1) and descbed n the followng equatons. λ (1) = λ (1) + λ (1) (22) λ (2) = λ (2) + λ(2) (23) Fg. 4: Constructon of Gan Tunng System nsms. As shown here, neuro emulator s employed to generate emulated looper angle dynamcs. The results of emulaton are estmated n estmator whose outputs are used for the nputs to RNN model. RNN generates gans for PID controller. The parameters n neuro emulator are updated such that whose output becomes smlar to the looper angle of hot rollng mlls. 4.1 Neuro Emulator model The structure of neuro emulator model s shown n Fg. 5. = η E 1 1 λ (2) λ (2) (24) λ (1) E 1 = η 1 λ (1) (25) E 1 λ (2) =(ˆα[N] α[n])y (2) [N] (26) E 1 λ (1) =(ˆα[N] α[n])λ (2) [N](1 y (2) [N])y (2) [N]y (1) [N] (27) E 1 = 1 2 (ˆα[n] α[n])2 (28) when, E 1 s dfference between α and ˆα, η 1 s gan of trang, y (1) s the output of th neuron n the nput layer, y (2) s the hdden of th neuron n the output layer. Thus, the looper dynamcs can be represented by neuro emulator. As show n Fg. 6, looper dynamcs can be replaced by neuro emulator. In the smulaton for gantunng, neuro emulator s connected to looper controller as show n Fg. 7. Fg. 6: Replacement of looper dynamcs by Neuro Emulator Fg. 5: Structure of Neuro Emulaton model The nputs to a neuro emulator are tme seres data of looper nputs, devaton of rollng speed, and past Looper angles. These data are stored n data base of looper dynamc characterstcs and used for dentfcaton and trang of connectng weghts n neuro emulator. Fg. 7: Connectng method of neuro emulator wth controller 19

5 January 2006 RNN Based Auto-tunng of PID Control Gans n Hot Strp Looper Controller 4.2 RNN Model ThestructureofRNNmodelsshownnFg.8. The network has three layers. The frst layer s the nput layer, whch consst of two parts. The one s the plan layer, whch represents nput evaluaton value I (;1 5) respectvely. The thrd layer s the output layer, whch s descrved as follows. s k = 3X =0 w (2) x(2) ; k =1 3 (35) x (2) 0 = 1 (36) g k = s k ; k =1 3 (37) where w (2) s the weght for the connecton between th neuron n the hdden layer and k th neuron n the output layer. Control perfermances are evaluated quantatvely by usng evaluaton equatons (38) to (42). Fg. 8: Structure of RNN Model The other s the context layer at number of rollng of N 0,whchsexpressedby c k = N 0 X d N 0 m g k [m] ;k =1 3 (29) g 1 : K p, g 2 : T, g 3 : T d (30) where d s a postve constant and less than 1, whch means forgettng effect and N 0 s the rollng number ndcatng Nth 0 precedng nstant. As shown n Fg. 8, the context layer s the layer that the past output data of RNN s stored. Then the output of nput layer s ( x (1) I f 1 5 = (31) c 5 f 6 8 The second layer s the hdden layer, whch can be expressed as s = x (1) 6X =1 w (1) x(1) ; =1 6 (32) 0 = 1 (33) x (2) 1 = 1+e s ; =1 6 (34) where w (1) s the weght for the connecton between th neuron n the nput layer and th neuron n the hdden layer. I 1 = I 2 = I 3 = Z T1 α 1 dt (38) T 0 αo Z T α 1 dt, (39) T 1 αo Z 1 T t α 1 dt (40) T T 1 T 1 αo I 4 = K α over α under α o K = 10 (41) t I 5 = t : settlng tme (42) T T 1 Evaluaton value I s(;1 5) n eqs.(38) to (42) are shown n Fg. 9. Fg. 9: Evaluaton value In the learnng process of RNN model, weght prameters are optmsed such that each evaluaton value approaches to ts expected value. In the learnng, Sum of the dfferences of evaluaton values Is from ther expected values I o s Method of PID Gans tunng by RNN The structure of RNN model s updated refferng dfferences of evaluaton values wth ther amed values. 20

6 Yoshhro ABE et al. MEM.FAC.ENG.OKA.UNI. Vol.40 Detals are descbed n the followng. w = w (1) = w (1) + w (1) (43) w (2) = w (2) + w(2) (44) w (2) = η 2 w (2) w (1) = η 2 w (1) (45) (46) 5X (I m Im) o I m x (2) (47) g k step4;back propagaton Eqs.(43) to (48) are used n back propagaton to update the RNN model. 5 Numercal Experments Accordng to teraton progress, PID gans smultaneously approach to certan covergence values. These convergent behavors are shown n Fg. 10. As shown here, PID gans K p,t and T d are tuned to ther optmal values durng teratons. = w (1) 3X ³ 5X (I m Im) o I m g k k=1 where, E 2 s defned by E 2 = 1 2 w (2) (1 x(2) )x (2) x (1) 5X (I m Im) o 2 (49) (48) η 2 s gan of trang. 4.3 Tranng of RNN Model The RNN model s traned by the back propagaton algorthm (BP). BP algorthm s descrbed as follows. step1;ntalzaton Fg. 10: Varaton of PID gans durng tunng Fg. 10 shows offlne gan tunng results. Vavatons of loooper angle durng tunng are shown n Fg. 11. The looper angles shown here are calculated by neuro emulator. As shown here, looper behavor becomes perferable durng teratons. The end condton of learnng s determned, and ntal value of weght parameters w (1) sandw(2) saredetermned by generatng random number between 0 and 1. step2;nput and preprocessng The evaluaton values, I [N 0 ](;1 5) at N 0 th rollng are nput to the plan layer and PID gans at prevous rollng number of rollng N 0 are nput to the context layer. Where nput to the context layer s 0 at N 0 =1. step3;forward propagaton By Eqs.(29)to(37), the output x (2) of hdden layer and the output g (3) k [N 0 + 1] of output layer are calculated. Fg. 11: Varaton of Looper Angle(ˆα) durng tunng At the same tme, estmaton values reach to ther obectve values as shown n Fg. 12. That s, by the effect of selt learnng of RNN model, optmzatons of combnng weghts of RNN model are consdered to be made. 21

7 January 2006 RNN Based Auto-tunng of PID Control Gans n Hot Strp Looper Controller In ths paper, auto gan tung method of PID controller for hot strp looper heght control s proposed. Neuro emulator s newly employed to emulate the dynamcs of looper heght dynamcs durng rollng. RNN wth self learng functon s used to generate PID gans. As for the nputs to RNN model, evaluaton functons of control performances are adopted whch enables reflecton of the human preferences. Throught numercal experments, the applcablty of the proposed method s confrmed. Combnaton of a neuro emulator and RNN s consder to be powerful tool of PID gan tunng for other control system. Bblography [1] S. H.Asada et al, Adaptve and Robust Control- Method wth Estmaton of Rollng Characterstcs forlooperanglecontrolathotstrpmll, ISIJ Internatonal, Vol.43, No.3(2003), pp Fg. 12: Varaton of Evaluaton value durng tunng Fg. 13 shows smulated looper dynamcs for each PID gans obtaned by RNN model comparng emulated result n Fg. 11. As shown here, wth excese of tunng, mprovement of control performance s attaned. Apperently, smulated results show the effectveness of the proposed gan tunng method by RNN model. [2] I.M.Mutaba et al,, Applcaton of Neural Network to Generaton of Robot Arm Traectory, AROB9th, pp ,2003 [3] Lakhm C.Jan et al, INDUSTRIAL APPLICA- TIONS of NEURAL NETWORKS, The CRC Press, 1999 [4] Chan H. Dagl, ARTIFICIAL NEURAL NET- WORKS for Intellgent Manufacturng, Chapman Hall, 1994 [5] S. Imao et al., Human Model for Gan Tunng of Looper Control n Hot Strp Rollng, tetsu-to- Hagane,vol.90,No. 11,pp , 2004 Fg. 13: Varaton of Looper Angle by tunng(α o = [30deg]) 6 Concluson 22