Based on a model of the CD basis weight profile, a system non-square interaction matrix with high dimensionality was analyzed by experimental studies and numerical computation

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(1)Journal of Korea TAPPI Vol. 51. No. 2, 2019, 76-87p ISSN (Print): 0253-3200 Printed in Korea. http://dx.doi.org/10.7584/JKTAPPI.2019.04.51.2.76. Multivariable Dimension-reduction and Synergic Control Strategy for Cross-directional Basis Weight of Papermaking Process Wen-juan Shan†and Wei Tang1 Received March 12, 2019; Received in revised form April 9, 2019; Accepted April 17, 2019. ABSTRACT In view of the characteristics of strong multivariable coupling in a cross-directional (CD) basis weight control system, a multivariable block-decoupling control strategy and a real-time decomposition algorithm are proposed in this paper. Based on a model of the CD basis weight profile, a system non-square interaction matrix with high dimensionality was analyzed by experimental studies and numerical computation. Along the diagonal of the interaction matrix, the matrix block method was adopted to reduce the system dimensions and convert it into a square system. The multivariable control system with high dimensionality is divided into several subsystems. According to the structure of each subsystem, a new constraint on the control variables was introduced to satisfy the triangular decomposition of the subsystem. Based on the multi-actuator parallel control strategy, the correlation between the subsystems was eliminated, and a high-dimensional multivariable decoupling control problem was transformed into a group single loop control general problem. The relative gain matrix of the new system was calculated to verify the decoupling effect in theory, and the real-time performance of the technique was demonstrated by results from an implementation on an actual project of a paper mill.. Keywords: Paper basis weight, cross-directional control, multivariable system, block matrix, dimension-reduction. 1. Introduction. challenge. The quality of the manufactured paper is evaluated in two dimensions: one is the average. Papermaking is a complex and sophisticated. sheet property profile along the sheet moving di-. manufacturing process. Keeping sheet quality uni-. rection while it is being manufactured, which is. form across the entire basis weight range is a great. referred to as machine directional (MD) control,. • Department of Light Industry Science and Engineering, Shaanxi University of Science & Technology, Xi’an, 710021, China 1 Department of Electrical and Information Engineering, Shaanxi University of Science & Technology, Xi’an, 710021, China † Corresponding Author: E-mail: [email protected]. 76. 펄프·종이기술 51(2) 2019.

(2) Wen-juan Shan·Wei Tang. and the other is the sheet profile across the width. widely used.8,9) At present, much research is fo-. of the paper machine, which is referred to as. cused on the structure and regulation principle of. cross-directional (CD) control.1) For any paper ma-. the hydraulic headbox with dilution actuators.10). chine, the sheet properties must be continuously. The adjustment of the CD profile by dilution wa-. monitored and controlled to guarantee that the. ter is superior to the traditional slice adjustment,. product quality specifications are satisfied along. and the pursuit of better paper quality has placed. 2,3). both the MD and CD.. It is generally recognized. new demands on CD control systems. In a CD pro-. that the CD control problem in web forming pro-. file control system, the number of sheet profile. cesses is much more challenging than the MD. measuring points may range from 200 to 2,000,. control problem. This is due to several difficulties. and the number of actuators could be up to 300.11). including the high dimensionality of the cross-. A large number of profile points and actuator zones. directional system, the high cross-directional spatial. will inevitably show coupling problems between. coupling among actuators, the uncertainty in the. adjacent dilution valves. When a CD actuator is. model, and the limited control freedom of the ac-. adjusted, spatial coupling typically exists between. tuators.4,5) The CD basis weight profile is a very. actuator zones and physical constraints are im-. important quality measurement of paper in the. posed to protect mechanical devices. In addition,. papermaking process. The objective of the CD con-. because of the spread of the flow, each actuator. trol system is aimed at reducing the CD variation. zone affects several adjacent zones on both sides.12). of the paper web as it is being made.. The final result is that when a dilution valve is. The CD profile variations are generally found to. moved, it affects the CD profile value of the adja-. occur at a much slower pace than the MD varia-. cent measurement points. This leads to the control. tions. The CD profile variations can be traced to. system becoming a high-dimensional, multivari-. flow patterns inside the headbox. The modern hy-. able and strongly coupled system that is difficult to. draulic headbox is equipped with dilution valves for. control.. basis weight profile control. In the hydraulic head-. To address these issues, a multivariable dimension-. box with dilution actuators, the full-width CD ba-. reduction and synergic control strategy was ad-. sis weight is adjusted by injecting dilution water. opted for cross-directional basis weight of paper-. 6). upstream of the homogenizer. The task of the di-. making process. Based on a model of the CD basis. lution valves is to mix the dilution water as evenly. weight profile, a novel matrix block method was. as possible into the stock that flows from the inlet. developed to reduce the system dimensions. A. header, which creates a stock flow whose consis-. non-square system was converted to a low dimen-. 7). tency differs from that of the inlet header flow.. sional square system, and a multivariable system. The dilution valves are connected to electric actu-. with high dimension was divided into several sub-. ators that are controlled by the dilution profile. systems. According to the structure of subsystem,. control system. The control system is used to reg-. multiprocessor decomposition design technique was. ulate the local stock consistency to effect weight. employed to address the actuator coupling prob-. profile changes. This approach replaces the tradi-. lem. Then the one-to-one correspondence be-. tional approach that uses slice screws to control. tween the measurement points and the actuators. weight profiles. The hydraulic headboxes with di-. was solved by a new constraint on control vari-. lution actuators introduced by Metso, Voith and. ables. Based on the production requirements and. other papermaking equipment suppliers have been. synergic control strategy, a multiprocessor system J. of Korea TAPPI Vol.51 No.2 Mar.-Apr. 2019. 77.

(3) Multivariable Dimension-reduction and Synergic Control Strategy for Cross-directional Basis Weight of Papermaking Process. was designed and applied in a paper mill.. mill equipment. For CD control system, the CD basis weight profile are adjusted by dilution valves, which are installed on the hydraulic headbox. The. 2. Materials and Methods. operation of the hydraulic headbox is shown in Fig. 2. The dilution water regulating device is. 2.1 Materials. shown in Fig. 3, which includes a dilution water. The whole experiment process includes on-line. cone pipe and multiple dilution water valves with. measurement and engineering operations. The ex-. automatic control units. One interface of the dilu-. perimental process of data monitoring and calcu-. tion water valve is connected with the dilution wa-. lation is carried out in an on-line detection and. ter cone pipe, and the other interface is provided. control laboratory of paper quality. It equipped. with a water conveying hose that is connected to. with intelligent scanner, infrared quantitative. the pulping branch pipe. The amount of white wa-. sensor, microwave moisture sensor, PC worksta-. ter injected into the pulp is controlled by adjusting. tion and Siemens S7-300 PLC (Programmable. the opening of the dilution valve, thus realizing the. Logic Controller) control cabinet. The experimental. local fine-tuning of the cross-directional basis. of CD basis weight online detection and space-time. weight of the paper. For dilution valves, the typical. alignment algorithmic can be completed. The de-. width of each valve is approximately between. tection experimental platform is shown in Fig. 1.. 30 mm to 60 mm.11) When a dilution valve is ad-. Engineering operation is based on actual paper. justed, its impact on weight profile could spread to multiple adjacent zones. The high-precision valve positioner is shown in Fig. 4.. 2.2 Control method for the experimental device The process of CD control for a Fourdrinier machine equipped with a hydraulic headbox is shown in Fig. 5. The scanning frame as a basis weight on-line detecting device is installed between the. Fig. 1. On-line detection device for CD data.. Fig. 2. Hydraulic headbox with dilution water.. 78. 펄프·종이기술 51(2) 2019. calender and the winder of the paper machine, and. Fig. 3. Dilution water regulating device..

(4) Wen-juan Shan·Wei Tang. Fig. 4. Dilution profile actuator.. Fig. 5. Technological process for basis weight control in the paper machine. the basis weight sensors are installed in the frame. at one side of the sheet (for some time intervals at. probe box and scan paper sheet that moves. unpredictable time instants) for recalibrating itself.. through the opening of the frame. This system. From these sparse data, the whole profile can be. continuously obtains the instantaneous sample. reconstructed.. value of the basis weight along the machine direction and its CD profiles.. In a paper mill, the data of paper basis weight is obtained by QCS (Quality Control System) through. The dilution actuator setpoints are determined by. scanning sensors. The schematic diagram of the. the weight profile obtained from the scanning. on-line detection device for QCS data is shown in. weight sensor. The sheet typically moves at a ve-. Fig. 7. The data exchange between OPC (OLE for. locity on the order of 10 m/s, and the sensors typi-. Process Control) toolbox and configuration soft-. cally move approximately 10-1 m/s.13) As a result,. ware of WinCC (Windows Control Center) is carried. the actual measurement is taken from a zig-zag. out by MATLAB (Matrix Laboratory). Then the. path as shown in Fig. 6, where the angle ‘a’ may be. control algorithm is realized based on the data ob-. only approximately 1°.14) Moreover, the gauge stops. tained by MATLAB. The calculation results are fed. Fig. 6. Measurement path (result of sheet and traversing gauge movement). J. of Korea TAPPI Vol.51 No.2 Mar.-Apr. 2019. 79.

(5) Multivariable Dimension-reduction and Synergic Control Strategy for Cross-directional Basis Weight of Papermaking Process. mum basis weight were 142.1 g/m2 and 138.7 g/m2 in a measurement cycle. The average value of basis weight was 140.5 g/m2 and fluctuation deviation was 3.4 g/m2, the mean square deviation 2σ=1.721. It can be seen that the basis weight fluctuation range was significantly reduced, the product qual-. Fig. 7. Diagram of on-line detection device for QCS data.. ity was improved based on the control system. Based on implementation of the multivariable dimension-reduction and synergic control system. back to WinCC. S7-300 PLC is adopted for the. for cross-directional basis weight of the paper-. controller implementation.. making process, a decoupled CD system was realized in this project. In the decoupled CD system, each single loop controlled by 64 actuators has. 3. Results and Discussion. similar physical characteristics. It is first-order with a dead-time dynamic response of the process.. 3.1 Results. Each loop controller design is relatively simple.. For a project of 140 g/m2 of corrugated paper. Take a two-loop adjustment for example. For each. produced in a factory of Northeast China, the. single-loop design controller, the response curve. technological indexes for the CD control system as. under the action of the controller is shown in. follow:. Figs. 10 and 11.. The slice outlet width is 3,900 mm, the design. u1(t) and u2(t) represent the input signals of any. speed is 650 m/min, the number of dilution valves. two control valves, the corresponding control out-. is 64, and the installation distance of dilution valve. puts are y1(t) and y2(t). A good output of y1(t) can. is 60 mm.. be seen when the first step input of u1(t) applied to. The control algorithm was integrated into the. the system, and y 2(t ) is almost 0. The effect was. paper making process. As shown in Figs. 8 and 9,. found to be similar when the second input is used. the basis weight fluctuations after the decoupling. alone. The loops do not affect each other. It was. control were compared. The maximum and mini-. proven that the coupling relationship has been. Fig. 8. CD profile without using CD control system.. Fig. 9. CD profile of QCS under CD control system.. 80. 펄프·종이기술 51(2) 2019.

(6) Wen-juan Shan·Wei Tang. Fig. 10. The step response curve after the adjustment. eliminated very well.. Fig. 11. The time response curve of the two input signals. The standard paper machine CD process model is given by. 3.2 Discussion. Y ( s ) = G ( s )U ( s )  G ( s ) = G0 g ( s ). In a CD profile control system, the number of sheet profile measuring points may range from 200. [1]. to 2,000, and the number of actuators could be up. where Y(s)∈Cm and U(s)∈Cn are the measurement. to 300. A large number of profile points and actu-. profile and the actuator set-point array, respec-. ator zones will inevitably show coupling problems. tively, and G0∈Rm×n is the spatial interaction cou-. between adjacent dilution valves. In addition, be-. pling matrix that describes the response of the. cause of the spread of the flow, each actuator zone. process. It also describes the mapping from the. affects several adjacent zones on both sides. The. actuators to measurements. m and n are the num-. final result is that when a dilution valve is moved,. bers of actuator arrays and controlled sheet prop-. it affects the CD profile value of the adjacent mea-. erties, respectively. G0 is a block-diagonal matrix. surement points. This leads to the control system. with its number of columns equal to the number of. becoming a high-dimensional, multivariable and. actuators and its number of rows equal to the. strongly coupled system. It is difficult to control.. number of profile points, where gij refers to the. To address actuator coupling issues, the mathe-. influence coefficient of the j-th actuator on the. matical model, coupling matrix and multiprocessor. i-th measurement point. For a typical paper ma-. decomposition design techniques should be dis-. chine, there could be as many as 600 individual. cussed.. actuators and 6,000 profile points if all CD actuator arrays and all controlled properties are included in. 3.2.1 Mathematical model of CD process. one control system.15) In other words, the dimen-. The mathematical model of a CD process consists. sions for the multiple-array process model G(s) in. of two parts: one part is the sparse interaction. Eq. 1 could be 6,000×600. g(s) is the first-order-. matrix across the entire width of the sheet, and. plus-time-delay dynamic response of the process.. the other part is the MD dynamics from the dilu-. The expressions of Y (s ), U (s ), G 0∈Rm×n, and are. tion water valve actuator to the measured profile.. g(s) as follows. J. of Korea TAPPI Vol.51 No.2 Mar.-Apr. 2019. 81.

(7) Multivariable Dimension-reduction and Synergic Control Strategy for Cross-directional Basis Weight of Papermaking Process.  g11 g12 g1n   y1 ( s )  u1 ( s )  g g   y ( s)  u ( s )  , Y( s ) =  2  , U( s ) =  2  , G 0 =  21 22              ym ( s )   un ( s )   g m1 g mn  1 g ( s ) = [2] e −τ s Ts + 1. tally, respectively. The attenuation parameter α changes the size of the negative lobes of the response. The divergence parameter β defines the presence of two maxima in the response and the distance between these two maxima.. 3.2.2 Coupling matrix analysis and identification The profile of the CD is similar to an inverted ‘S’ 16). shape,. In the experimental simulation, taking a dilution. The stationary CD profile shape of the. water hydraulic headbox with 64 dilution water. sheet is a two-dimensional wave, as shown in. valves as an example, the measurements and the. Fig. 12.. actuator form a large-scale interaction coupling. As the CD basis weight does not vary with time,. matrix with dimensions of 320×64. Based on such. then, g(x,t) can be simplified to g(x). For a contin-. an example, a large-scale interaction coupling. uous sheet-forming process, the relationship be-. matrix G0 can be described as Eq. 4. The parame-. tween the two-dimensional sheet variations g (x ). ters g1, …, g26 describe the spatial response of the. and CD control action to the j-th CD actuator can. process, where 26 is the half-width of the response. be formulated as Eq. 3.17). of one actuator and m represents the scanning space between adjacent actuators.. g ( x) =. r  e 2 . −α ( x + βξ ). ξ2. 2.  π ( x + βξ )  cos  +e ξ  .  π ( x − βξ )   cos   ξ   . −α ( x − βξ ). 2. ξ2. . [3]. It is shown that the spatial response of a single controller can be represented by the extension of a set of orthogonal functions, where the spatial coordinate χ is a scalar real number, γ is a gain parameter, and ξ is a width parameter. The parameters γ and ξ define the linear transformation of the response by stretching it vertically and horizon-.  g1 g  2  g3   g4  g5   g6 g  7  g  26 G0 =  0   0  0   0  0    0   0    0. g6 g5 g4 g3 g2 g1 g2 g 21 g 22 g 23 g 24 g 25 g 26 0 0 0. g11 g10 g9 g8 g7 g6 g11 g16 g17 g18 g19 g 20 g 21 g 25 g 26 0. 0. g 26 g 25 g 24 g 23 g 22 g 21 g 20 g1 g2 g3 g4 g5 g6 g10 g11 0. 0 0 0 0 0 g 26 g 25 g6 g5 g4 g3 g2 g1 g5 g6 0.                ∈ R m×n [4]              . The coupling matrix G0 can be calculated. G0 is a sparse band-diagonal matrix with a large number. Fig. 12. Schematic of a two-dimensional wave.. 82. 펄프·종이기술 51(2) 2019. of zero elements. As shown in Fig. 13, where the.

(8) Wen-juan Shan·Wei Tang. matrix G0. Along the diagonal, the coupling matrix can be divided into q×q block matrices. Thus, G0 can be partitioned as G2   G1 G0 =  0   0 . G1T G2 . 0 G1T G1 0. 0    0   G2 G1T  G1 G2  q×q. [5].  m   320  where q =  =  = 64 ([ ] is the  2 p + 1  2 × 2 + 1 rounding up function, for example, [h] represents. Fig. 13. The value of coupling matrix.. the smallest integer that is no less than h). G1 is an upper triangular (2p+1)×(2p+1) matrix, and G2 is a. non-zero data reflect the coupling coefficient of. symmetric (2p +1)×(2p +1) matrix. The structures. each CD position and the strip width reflects the. of G1 and G2 are given by Eq. 6.. strength of the coupling. 0 0 g p g1  0 0    G1 =  g p  ,   0 0   0 0 0  ( 2 p +1) ×( 2 p +1). 3.2.3 Dimension-reduction and decomposition For non-square high-dimensional coupling systems, it is necessary to adopt a suitable method to reduce the system dimension and convert it into a square system for decoupling. According to the.  g0   G2 =  g p   0 . strip coupling width of the above interaction matrix, G0 is standardized as shown in Fig. 14, where. p is the coupling width, which indicates the number of affected actuators on the adjacent side when. gp g0 gp. 0   gp    g 0  ( 2 p +1)×( 2 p+1). [6]. adjusting an actuator (p=2 in this paper), and 2p+1 Based on such a division, an m ×n -dimensional. is the strip width. For simplifying the analysis, a blocking-based. CD control system is divided into q groups, and. triangulation strategy is adopted for the coupling. each group contains 2p +1 subsystems, where the model of the subsystem can be expressed as Eq. 7. Y j (t ) = g (z −1 )[ G2 u j (t − τ) + G1u j −1 (t − τ) + G1T u j +1 (t − τ) ] 1. . 2. 3. + E (t ). [7]. For subsystem Yj(t), the symmetric parts of 2 and 3 in expression Eq. 7 correspond to the paper CD. Fig. 14. The standardized incidence matrix.. basis weight boundaries. The block matrix G1,G1T is J. of Korea TAPPI Vol.51 No.2 Mar.-Apr. 2019. 83.

(9) Multivariable Dimension-reduction and Synergic Control Strategy for Cross-directional Basis Weight of Papermaking Process. located at the boundary of the interaction matrix, and G2 is located on the main diagonal of the interaction matrix. When the effect controlled by G2 is non-zero and the other parts of the control effect are zero, G0 can be converted to a main diago-.  Y1 ( s )   G11 G12  Y ( s )  G G22  2  =  21       Y ( s ) G  64   n1 … . … G1n   u1 ( s )    u ( s)  1 -τ s  e × 2 ×   Ts + 1     … Gnn  64×64 u64 ( s )  [10]. nal matrix. Then, the system can be simplified as a main diagonal system, and the coupling can be re-. For the new low-dimensional square system, the. duced. Based on the input variable adjustment, the. interaction matrix with a number of dimensions. decomposition algorithm is adopted to eliminate. equal to the number of actuators and a one-to-. the control effect of sub-blocks 2 and 3 in this pa-. one correspondence between the CD measuring. per.. point and the actuator is solved. The new interac∼ tion matrix G 0 can be given as Eq. 11.. At one moment, each actuator rotates in a sequence of Eq. 8, [u1 , 0,, 0], [0, u2 , 0,, 0],  , [0,, 0, ui , 0,, 0], , [0,, 0, u2 p+1 ] [8] Cyclically input the control variables in chronological order. Actuators belonging to different groups can act simultaneously. During the period from t to t +2p +1, efficient control inputs [u 1,u 2,…,uq] (q =64) (such as Eq. 9).  1 0.7 0.3 0.7 1 0.7   0.3 0.7 1  G0 =  0.3 0.7  0.3    . 0.3 0.7 1 0.7 0.3. .     0.3  0.7 0.3 [11]  1 0.7 0.3  0.7 1 0.7  0.3 0.7 1  64×64. ∼ Moreover, the new interaction matrix G 0 is a. Toeplitz symmetric matrix. G 0 Toeplitz f [1, 0.7, 0.3, 0 0] = = ( f ), where G 0 Toeplitz = = ( f ) f [1, 0.7, 0.3, 0 0].. are applied to the 320×64 CD system. [u1 , u2 ,, u64 ] 1 5  0 =    0 . . 1 5. 0 0.  0 0 0 0  1 1 0 0  5 5   1 1 0 0  5 5  320×64. In the case of ignoring the edge effect, the entire paper web and the mechanical structure are sym-. [9]. metrical.18,19) Based on the control input, the new ∼ diagonal interaction matrix G 0 represents the new ∼ structure of the system. G 0 is given by Fig. 15. ∼ Where the details of G2 in the matrix of G 0 can be described as Eq. 12.. Based on these control inputs, the original high-dimensional non-square CD control system (320×64) can be converted to a new low-dimensional CD system (64×64), which can be described as Eq. 10.. ~. Fig. 15. The matrix of G 0.. 84. 펄프·종이기술 51(2) 2019.

(10) Wen-juan Shan·Wei Tang. 0   1 0.7 0.3 0 0.7 1 0.7 0.3 0    G2 =  0.3 0.7 1 0.7 0.3    0 0.3 0.7 1 0.7  0 0.3 0.7 1   0. This result indicates that the original large CD system was converted to a new CD square system [12]. based on the steps of system dimension reduction and block alignment calculation. For the new system with 64 dimensions, there were 64 actuators mapping 64 regions, which means that the cou-. Next, the coupling characteristics of the new. pling between each region was eliminated.. system were studied by the relative gain, and for the new CD system, the relative gain matrix of the system was calculated. The interception part of the. 4. Conclusions. calculation results was shown in Fig. 16. According to Fig. 16, the main diagonal element. This paper presents a novel control strategy that. of the relative gain matrix is 1 (marked with a red. efficiently addresses the issue of the strong cou-. line), the other elements are approximated to 0. pling in the large number of cross-direction actu-. after taking the truncation error, and the matrix. ators with which sheet-forming processes are. 20,21). can be approximated as a 64×64 unit matrix,. equipped. The method employs a new block decou-. as shown in Eq. 13.. pling algorithm for the MIMO (multi-input. 1 0 Λ= 0  0. 0 1 0 0. . 0 0    1 64×64. multi-output) process, the matrix block method and model reduction were applied in the interaction [13]. coupling matrix of dimensions of 320×64, and the original 320×64 non-square CD control system can be converted to a new low-dimensional CD. where Λ is the relative gain matrix. According to. system with dimensions of only 64×64. The algo-. the relative gain characteristics, the relative gain. rithm decouples a high-dimension multivariable. matrix of the decoupled system must be a unit. system into several low-dimension subsystems. In. matrix.22,23) This conclusion also applies to multi-. this algorithm, a new constraint on the control in-. variable systems.. put was constructed to satisfy the triangular de-. Fig. 16. The part of the relative gain array. J. of Korea TAPPI Vol.51 No.2 Mar.-Apr. 2019. 85.

(11) Multivariable Dimension-reduction and Synergic Control Strategy for Cross-directional Basis Weight of Papermaking Process. composition of the subsystem, and a parallel control structure was designed to implement the algo-. of large scale sheet and film processes, Journal of Process Control 2(11):149-177 (2011).. rithm. Based on the multi-actuator parallel control. 6. Kang, G. B., Chen, K. F., Liu, J. A., and Li, J.,. strategy, a decoupled multivariable system was. Numerical simulation of dilution water applied. achieved. The mathematical values for the real CD. to headbox with perforated rolls, Journal of. actuators were computed and implemented in the. South China University of Technology (Natural. plant. The solution from the decoupled system was. science edition) 38(9):57-62 (2010).. then projected back to the original space for im-. 7. Fan, H. M., Liu, J. A., and Zhang, C., The. plementation in a real paper mill. Simulation ex-. Influence of velocity and inject angle of dilu-. amples were included to illustrate the method, and. tion water in headbox on the effect of dilution. it was shown that the coupling effect can be sig-. water addition, China Pulp & Paper 32(5):26-. nificantly eliminated and the system real-time. 30 (2013).. performance can be improved.. 8. Gheorghe, C., Lahouaoula, A, and Chu, D. L., Multivariable CD control with adaptive alignment for a high-production linerboard ma-. Acknowledgement. chine, Journal of Science & Technology for Forest Products and Processes 2(5):32-40. This work was partially supported by Shaanxi. (2012).. Science & Technology Co-ordination & Innovation. 9. Hu, W. J., Tang, W., Liu, W. B., and Wang, M.. Project (2016KTCQ01-35). We sincerely thank for. X., Design and simulation of fuzzy controller. the funding of the project.. for CD basis weight, Control and Instruments in Chemical Industry 12(44):1441-1453 (2011). 10. Ohenoja, M. and Leiviskä, K., Multiple prop-. Literature Cited. erty cross direction control of paper machine, Modeling, Identification and Control 32(3):. 1. Tang, W., Wang, M. X., and Li, M. H., The. 103-112 (2011).. advanced control strategies and decoupling al-. 11. Morales, R. M. and Heath, W. P., The robust-. gorithms of headbox, Transactions of China. ness and design of constrained cross-direc-. Pulp and Paper 21(1):108-114 (2006).. tional control via integral quadratic con-. 2. Cegrell, T. and Hedqvist, T., Successful adaptive control of paper machines, Automatica 2(11):53-59 (1975).. straints, IEEE Transactions on Control Systems Technology 19(6):1421-1432 (2011). 12. Chen, S. C., Naimimohasses, R., and Zehnp-. 3. Dumont, G. A., Application of advanced con-. fund, A., Improving reel-building with multi-. trol methods in the pulp paper industry - A. variable CD control, Paper Conference and. survey, Automatica 3(22):143-153 (1986).. Trade Show 2012, Atlanta, pp. 984-1002. 4. Backström, J. and Backer, P. A., Benefit analysis of model predictive machine direc-. 13. Amma, M. E., and Dumont, G. A., Identifica-. tional control of paper machine, Pulp and Pa-. tion of paper machines cross-directional mod-. per Canada 3(21):141-149 (2008).. els in closed-loop, Proceedings of Interna-. 5. VanAntwerp, J. G., Featherstone, A. P. and Braatz, R. D., Robust cross-directional control. 86. (2012).. 펄프·종이기술 51(2) 2019. tional Conference on Modeling, Identification and Control, Cairo, USA, pp. 3-9 (2013)..

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