Open Access Research of Strip Flatness Control Based on Legendre Polynomial Decomposition

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1 Sed Orders for Reprits to The Ope Automatio ad Cotrol Systems Joural, 215, 7, Ope Access Research of Strip Flatess Cotrol Based o Legedre Polyomial Decompositio Ya Liu, Yupeg Wag ad Jiagyu Li * School of Automatio Electrical Egieerig, Uiversity of Sciece ad Techology Beijig, Beijig, 183, Chia Abstract: A traditioal least squares has egatively a clear physical sigificace i the process of strip flatess patter recogitio ad it is too complex i calculatio to be suitable for idustrial cotroller to realize. I this paper we deduced the basic model of strip flatess patter recogitio based o Legedre polyomial. Uder this mode, there are three outstadig advatages: The itegral value alog the width directio of strip shape is, which satisfies the coditio of equilibrium i the residual stress. Each compoet of polyomial correspods to steer roller, bedig roller ad step coolig cotrol respectively, which makes the physical meaig clear. Operatioal calculatio is suitable for idustrial cotroller. The simulatio shows that calculatio time-cosumig dropped from average.12 s to average.2 s. Also practice i the real world had proved that this method is effective ad applicable. Keywords: Legedre polyomial, Least squares, Flatess recogitio, Flatess cotrol. 1. INTRODUCTION Flatess quality is a importat idex of plate ad strip products. Uder the coditio that the thickess cotrol techology is almost mature, flatess cotrol has already become the key part of process o high quality plate ad strip. Fie flatess ca ot oly improve coatig thickess ad uiformity, but also esure the plate ad strip deep drawig performace ad the follow-up processes. Furthermore, it ca reduce the probability of roller chage ad belt broke, ad icrease ru speed ad productio efficiecy. However there are may factors which ca ifluece the flatess, so the strip flatess cotrol is the core ad hard techology i the wide strip mill. The basic priciple of flatess cotrol is to esure the similarity betwee the sectio shape of etrace ad delivery [1]. The basic cotrol methods are steer roller, bedig roller ad step coolig cotrol [2]. Steer is used to adjust first order of strip sectio shape, ad bedig is used to adjust the secod order of strip sectio shape. The rest of high orders are performed by step coolig cotrol [3]. I these compoets, first order ad secod order which are weighted heavily are the foudatio of the strip shape quality. Though the weight of high orders is smaller, it is importat for high precisio cotrol o strip shape. Therefore, we eed to perform patter recogitio based o the features of strip shape cotrol devices to decompose each order precisely, so that we ca get high quality strip shape. Curretly, the widely used method of flatess patter recogitio ad flatess sigal processig is polyomial regressio aalysis based o least squares [4]. This method *Address correspodece to this author at the 3, Xueyua Road, Beijig, Chia. Postcard: 183; Tel: ; jiagyuosu@126.com uses the simplest least-squares method, which regresses the deviatio detected by flatess gauge ito the form of a polyomial. The algorithm is simpler, yet there are a lot of defects. The mai disadvatage is that the algorithm is complex i calculatio, ad the physical meaig is ot clear. Besides, the polyomial model caot be coverted to the regulatig variables of the flatess cotrol device, which makes it difficult to be used for cotrol directly. I additio, i order to ehace the accuracy, the orders of polyomial have to be icreased, which will produce more polyomials, further weakeig the physical meaig. The extra parameters go agaist the aalysis ad cotrol [5, 6]. Recet years, people have put forward to use Legedre polyomial as the basic model of patter recogitio to fit several commo strip shapes i the idustry. The orthogoal Legedre polyomial model overcomes the defects of complex computig ad uclear physical meaig [7]. I this paper, we maily discuss the followig problem: 1) Deduce the basic model of flatess patter recogitio based o Legedre polyomial. 2) Aalysis of shape cotrol algorithm by meas of Legedre polyomial. 3) Discuss the compariso results betwee least squares ad modified least squares method based o Legedre polyomial. 2. DERIVATION OF LEGENDRE POLYNOMIALS Legedre equatio as well as its solutio fuctio is amed by the Frech mathematicia Adrie-Marie Legedre. Legedre equatio is a kid of ordiary differetial equatio which physics ad other techical fields ofte ecouter [8, 9]. Whe tryig to solve three-dimesioal Laplace equatios i spherical coordiates or related partial differe / Betham Ope

2 24 The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 Liu et al. tial equatios), the problems will be dow to solve Legedre equatio. Legedre equatio is as follows: 1! x 2 )y! 2xy + +1)y = 1) Which is 1! x 2 ) d 2 px) dx 2! 2x dpx) + +1) px) = 2) dx Amog them, is ay real umber. Set solutio as the form of series, we get: cc!1)a x c!2 + c1+ c)a 1 x c!1 +{ k + c + 2)k + c +1)a k+2![k + c)k + c +1)! +1)]a k }x k+c = Obviously: cc!1)a = cc +1)a 1 = k + c)k + c +1)a k+2![k + c)k + c +1)! +1)]a k = 5) y = px) = x c a + a 1 x + a 2 x 2 +!+ a k x k +!) = a k x k+c 3) Differetial termwise, we ca obtai: dy = k + c) ak x dx k = k+ c 1 d 2 y dx = k + c)k + c!1)a 2 k xk+c!2 Brig the two differetial results ito the Legedre equatio: 1! x 2 ) k + c)k + c!1)a k x k+c!2!2xk + c)a k x k+c!1 + +1) a k x k+c = = k + c)k + c!1)a k x k+c!2!x 2 k + c)k + c!1)a k x k+c!2!2x k + c)a k x k+c!1 + +1) a k x k+c = k + c)k + c!1)a k x k+c!2! k + c)k + c +1)a k x k+c + +1) a k x k+c The we ca come to the coclusio: k + c)k + c!1)a k x k+c!2![k + c)k + c +1)! +1)]a k x k+c = Separate the items before ad the items after k=1 i 4), after fiishig reductio we get the followig formula:! 4) a k+2 = The we ca get recursive formula: k! )k + +1) a k + 2)k +1) k 6) Therefore, we ca represet Legedre polyomial solutios with the lowest terms. yx) = a y + a 1 y 1 a y = a [1! +1) x 2 + 2!!1) + 2) a 1 y 1 = a 1 [x! x 3 3!!1)! 3) + 2) + 4) + x 5!] 5!! 2) +1) + 3)!] 4! Obviously, the solutio is composed of odd ad eve umber terms. a k = After fiishig recursive formula, we get: k + 2)k +1) k! )k + +1) a k+2 7) Through equatio 7), whe K = - 2 as the first item, brig it ito the recursive formula!1) a!2 =! 22!1) a The, we set the highest item 1!3!5!2 1) a =! a!2 = = 2)! =, 1, 2 ) 2 2!) We ca get the followig expressio:!1)!22!1) 2)! 2!) =!1) 2! 2)! 2 2!1)!! 2)! a!4 =!1) 2 2! 4)! 2 2!! 2)!! 4)!! a!2s =!1) s 2! 2s)! 2 s!! s)!! 2s)!

3 Research of Strip Flatess Cotrol The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 25 s =,1,2![ 2 ] [ 2 ] is the greatest iteger less tha 2. Whe is a eve umber, we ca calculate that: 2 2! 2s)! a y =!1) s 2 s!! s)!! 2s)! s!2s s= Whe is a odd umber, we ca calculate that:!1 2 2! 2s)! a 1 y 1 =!1) s 2 s!! s)!! 2s)! s!2s s= After combiig the two formulas, it ca be expressed as: [ 2 ] 2! 2s)! y = px) = 2 s!! s)!! 2s)! x!2s 8) s= The, accordig to the geeral term formula, we ca solve the Legedre polyomial. 3. THE MODIFIED ALGORITHM METHOD There are six kids of commo egieerig strip shapes. They are left waves, right waves, middle waves, bilateral waves, quartered waves ad edge waves. Whe establishig the mathematical model, we eed to satisfy the coditio of equilibrium i the residual stress. The so-called equilibrium i the residual stress is that the itegral value alog the width directio of strip shape is after rollig. The formula is as follows. 1 δ xdx ) = 9) 1 The Legedre polyomial will coverge to ifiite series whe the itegral is from -1 to 1. I order to make some certai items of Legedre polyomial become the basic model of flatess patter recogitio, we choose the first, secod ad fourth as the basic model, they are: Left waves: y 1 = p 1 x) = x Right waves: y 2 =! p 1 x) =!x Middle waves: y 3 = p 2 x) = 3x2!1 2 Edge waves: y 6 =! p 4 x) =! 35x4! 3x The stress distributio curve of strip flatess is show below i Fig. 1). Both X-axis ad Y-axis uits are ouitized. X-axis expresses legth from ceter to edge)/ total legth) Y-axis expresses ceter thickess - edge thickess)/ ceter thickess) The liear combiatio of flatess basic model is:! x) = a 1 p 1 x) + a 2 p 2 x) + a 4 p 4 x) We detect a set of discrete data y i,! i ) through measurig roller, i = 1,2,!,. is the umber of poits we measure. y i is the abscissa, ad. σ i is model deviatio value i y i Based o the least squares method, we ca get the followig formula: mi [ p )! i ] 2 1) Covert ito the multivariate fuctio: ,2,4 2 j j i δ 11) i i= 1 j= 1 φ a, a, a ) = [ a p x ) ] Differetiate the above formula: 1,2,4! = 2 [ a!a j p j ) i ]p k ) = k=1, 2, 4) 12) k 1,2,4 j= 1 j=1 Set c jk =! p j ) p k ), b k = p k )! i k=1,2,4) Brig the above formula ito formula 12): cjk. aj = b k=1, 2, 4) 13) k Writte i matrix form: C A = B 14)! Ad, A = a 1 a 2 a 4!, B = b 1 b 2 b 4!, C = c 11 c 12 c 14 c 21 c 22 c 24 c 41 c 42 c 44 Calculate the coefficiets of matrix C ad matrix B. Bilateral waves: y 4 =! p 2 x) =! 3x2!1 2 Quartered waves: y 5 = p 4 x) = 35x4! 3x

4 26 The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 Liu et al. a) Left waves b) Right waves c) Middle waves d) Bilateral waves e) Quartered waves Fig. 1). The stress distributio curve of strip flatess. f) Edge waves C =! p 1 ) p 1 )! p 1 ) p 2 )! p 1 ) p 4 )! p 2 ) p 1 )! p 2 ) p 2 )! p 2 ) p 4 )! p 4 ) p 1 )! p 4 ) p 2 )! p 4 ) p 4 ) B = p 1 )! i p 2 )! i p 4 )! i Usig Matlab, We ca calculate the results directly by left except.

5 Research of Strip Flatess Cotrol The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 27 Table 1. The first group of test data uit: mm) Actual shape LS fittig curve Fig. 2). Compariso of the actual shape curve ad the least squares fittig curve Table 1). A = C \ B 15) We ca get dispesable mold, quadratic form, biquadratic form so as to achieve the purpose of cotrollig strip shape. 4. SIMULATION I this sectio, we discuss the compariso results betwee least squares ad modified least squares method based o Legedre polyomial. The modified LS ca reflect the physical meaig of the cotrol variables better ad it has faster computig speed i theory. So we compare the two algorithms through some data obtaied from idustrial productio. We maily complete the followig tasks: 1) Test accuracy of the modified method. 2) Compare operatio speed of these two kids of algorithms. The simulatio platform we used is Itel Core-2 due 2.2GHz CPU ad Matlab Two groups of data are obtaied from the real world. The first group of data are show i the Table 1, a total of 33 group detectio uits. Compariso of the actual shape ad the least squares fittig curve are show i Fig. 2). Compariso of the actual shape ad the modified least squares fittig curve are show i Fig. 3). Compariso of the least squares fittig curve ad the modified least squares fittig curve are show i Fig. 4).

6 28 The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 Liu et al. The secod group of data are show i the followig Table 2, a total of 33 group detectio uits. Compariso of the actual shape ad the least squares fittig curve are show i Fig. 5). Compariso of the actual shape ad the modified least squares fittig curve are show i Fig. 6). Compariso of the least squares fittig curve ad the modified least squares fittig curve are show i Fig. 7). From the results of these two groups data, we ca come to a coclusio that the modified least squares curve fits more closely to the actual shape. Accuracy is similar to the least squares.

7 Research of Strip Flatess Cotrol The Ope Automatio ad Cotrol Systems Joural, 215, Volume Actual shape Modified LS fittig curve Fig. 3). Compariso of the actual shape ad the modified least squares fittig curve Table 1) LS fittig curve Modified LS fittig curve Fig. 4). Compariso of the least squares fittig curve ad the modified least squares fittig curve Table 1). Table 2. The secod set of test data uit: mm)

8 21 The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 Liu et al. Table 2. cotd Actual shape LS fittig curve Fig. 5). Compariso of the actual shape ad the least squares fittig curve Table 2). - Actual shape Modified LS fittig curve Fig. 6). Compariso of the actual shape ad the modified least squares fittig curve Table 2). Table 3. The performace compariso of two kids of algorithm Ru times) Mea Square Error Group LS Group average Group Modified LS Group average.3.137

Research of Strip Flatess Cotrol The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 211 - LS fittig curve Modified LS fittig curve - -1 -.5.5 1 Fig. 7).

9 Research of Strip Flatess Cotrol The Ope Automatio ad Cotrol Systems Joural, 215, Volume LS fittig curve Modified LS fittig curve Fig. 7). Compariso of the least squares fittig curve ad the modified least squares fittig curve Table 2). Fig. 8). Flatess detectio ad cotrol results. The performace compariso of two kids of algorithms is show i Table 3. Through the above simulatio results, we ca see, the modified LS improves a lot i speed ad avoids complex calculatio effectively. Although the mea square error compared with the ordiary method is a little poor, it has little impact i egieerig applicatio. The research results have bee successfully applied i a 185mm productio lie i a alumium foil factory, achievig fie cotrol effect. Its flatess detectio ad cotrol results are show i Fig. 8). Fig. 8) is a real cotrol iterface. The cotiuous curve expressed settig flatess, which is decided by supplied strip. Each bar is value of flatess detectig roller. If every bar s summit is coicidece o curve, it meas etrace sectio shape is the same with delivery sectio shape. Fie flatess will be gotte. CONCLUSIONS 1) The modified least squares method based o Legedre polyomial satisfies the coditio of equilibrium i the residual stress, so it is more suitable tha LS as the basic model of flatess patter recogitio. 2) The modified least squares method based o Legedre polyomial has obvious improvemet i speed, ad avoids complex calculatio effectively, which is more suitable i idustrial field. Although Mea square error compared with the ordiary method is a little poor, the step coolig method is geerally used to elimiate the high order error, so it doest affect the fial cotrol results. 3) The modified least squares method based o Legedre polyomial ca obtai three kids of cotrol quatity of strip shapes directly, ad the physical meaig is clear. This is the most promiet place compared to LS. LS ca oly stay o the theoretical level; however, modified LS ca be used i idustrial productio. CONFLICT OF INTEREST The authors declare that there is o coflict of iterests regardig the publicatio of this article. ACKNOWLEDGEMENTS The authors gratefully ackowledge the support Fudametal Research Fuds for the Cetral Uiversities No. FRF-SD-12-8B) ad Beijig Key Disciplie Developmet Program No. XK18537). REFERENCES [1] J. Hua, ad Z. Zhou, Polyomial regressio ad mathematical model of flatess defect for cold strip, Iro Steel, vol. 27, p. 27, March [2] X. Zhou, ad G. Wag, Orthogoal polyomial decompositio model for the flatess of cold-rolled strip, Iro Steel, vol. 32, p. 46, August 1997.

10 212 The Ope Automatio ad Cotrol Systems Joural, 215, Volume 7 Liu et al. [3] H. Zhou, Research o itelliget methods of flatess recogitio predictio ad cotrol simulatio, Yasha Uiversity, Yasha, Heibei, Chia 25. [4] X. Zhag, H. Liu, GA-BP model of flatess patter recogitio ad improved least-squares method, Iro Steel, vol. 38, pp. 29, October 23. [5] Y. Su, C. Tog, ad K. Peg, Cold rollig productio automatio techology, Beijig: Metallurgical Idustry Press, 28. [6] L. Fu, ad H. Wag, A comparative research of polyomial regressio modelig methods, Math. Stat. Maag., vol. 21, p. 48, Jauary 24. [7] H. Di, X. Zhag, X. Liu, G. D. Wag, Orthogoal polyomial decompositio ad mathematical model for measured sigals of this strip flatess i cold rollig, Iro Steel, vol. 3, p. 33, September, [8] H. Yu, ad S. Fag, Research o Legedre recursive formula, J. Sichua Uiv. Sci. Eg., vol. 21, p. 27, February 28. [9] F. Zhou, S. Li, W. Yu, L. Luo, A efficiet method of computatio of Legedre momets, Chi. J. Comput., vol. 23, o. 8, pp , August 2. Received: November 6, 214 Revised: December 5, 214 Accepted: December 13, 214 Liu et al.; Licesee Betham Ope. This is a ope access article licesed uder the terms of the Creative Commos Attributio No-Commercial Licese by-c/4./) which permits urestricted, o-commercial use, distributio ad reproductio i ay medium, provided the work is properly cited.

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

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