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Adaptive paper coating weight control Chen, Yamei 1993

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Adaptive Paper Coating Weight Control by Yamei Chen M.A.Sc. University of British Columbia, 1993.  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBL^ October 1993 © Yamei Chen, 1993  In  presenting  degree  at  this  the  thesis  in  partial  fulfilment  of  University  of  British  Columbia,  I agree  freely available for copying  of  department publication  this or of  reference  thesis by  this  for  his thesis  and study. scholarly  or for  her  I further  purposes  (Signature)  Department of  e/^cfnC^-^  The University of British Vancouver, Canada  Date  DE-6 (2/88)  fvloi'.  iS )  1^  6^,^'^c..^^  Columbia  It  gain shall not  permission.  requirements that  agree  may  representatives.  financial  the  be  that  the  advanced  Library shall make  by  understood be  an  permission for  granted  is  for  allowed  the that  without  it  extensive  head  of  my  copying  or  my  written  Abstract  Paper coating is the process of applying coating colour or functional materials to the surface of paper. The goal of coating is to improve the surface quality of the paper. In this thesis, the bevelled-blade coating process is modelled based on the force equihbrium at the tip of the blade using fluid mechanic principles. The effects of the factors influencing coating weight are analyzed using simulation results. Mill trials have been carried out to investigate the dynamics of the coating process and the interaction of the cross machine profilers. Machine-direction (MD) and cross-direction (CD) variations were estimated using an algorithm previously developed at the University of British Columbia. Coating weight variations were studied and possible achievements were indicated. From the estimated cross-direction profiles, the amplitude and width of the profile response to the bump test were analyzed, and then used to build the interaction model of CD actuators. The goals of both the machine-direction and the cross-direction coating weight controls are to improve the uniformity of coating weight Due to the advantages of Generalized Predictive Control, both MD and CD control loops are designed using adaptive constrained GPC. The machine-direction coating weight process is modelled as a first-order system with a time-varying gain defined by the nonlinear relationship between coating weight and the control variable. The time-varying parameters are estimated by the recursive least-squares (RLS) method with exponential forgetting and resetting factors. The cross-direction coating process is also modelled as a first-order system with a nonlinear gain determined by the relationship between the local blade pressure and the coating weight Based on the interaction of CD actuators, the interaction matrix of the first-order model is defined as a band-diagonal matrix. Multi-input, single-output RLS estimators are developed to perform on-line parameter estimation. Constraints on the control signal are considered in response to the limits of the industrial settings. The optimal solutions to both constrained univariate and multivariable GPC are obtained using the Lagrangian multiplier method. Simulation studies are carried out to evaluate the performance of the adaptive constrained GPC. The results of the simulations show that the controllers can track the set-point changes closely, whUe rejecting the disturbance and handling the model plant mismatch weU.  Ikble of Contents  1  2  Abstract  ii  List of Tables  vi  List of Figures  vii  Acknowledgments  ix  Introduction  1  1.1 Background  1 1  1.1.2 Coating Weight Measurement  2  1.1 J Coating Weight Control  2  1.2 Motivation  3  1.3 Contribution  3  1.4 Thesis Outline  4  Coating Process Modelling  5  2.1 Blade Coaler Mechanism  5  2.2 Bevelled-blade Coater Modelling  8  23 3  1.1.1 Paper Coating  2.2.1 Mechanical Force  8  2.2.2 Dynamic Forces  9  2.2.3 Mechanism of Coating Weight Development  15  Simulation of the Coating Process  17  Coating Weight Variation and Mill IVial Data Analysis  25  3.1 Coating Weight Variation  25  3.1.1 Machine-Direction Variations  26  3.1.2 Cross-Direction Variations  26  3.1.3 Residual Variations  26 iii  3.2 Mill Trial Data Analysis  27  3.2.1 MD and CD Data Separation  4  27  3.2.1.1 Exponential Multi-Scan Trending  28  3.2.1.2 EIMC Algorithm  28  3.2.2 Mill Trial Data Analysis  30  3.3 Conclusion  36  Coating Weight Control Strategy: Constrained Generalized Predictive Control  38  4.1 Control Strategy Design for Coating Process  38  4.2 Introduction to Generalized Predictive Control  40  43  SISO GPC  41  4J.1 CARIMA Plant Model and Ouq)ut Prediction  41  4.3.2 The Predictive Control Law  43  4.3.3 The Control Horizon  44  4.4 Multivariable GPC  5  45  4.4.1 CARIMA Plant Model and Output Prediction  45  4.4.2 The Multivariable Predictive Control Law  46  4.5 Constraints  47  Machine-Direction Coating Weight Control  49  5.1 Machine-Direction Control Mechanism  49  5.2 MD Coating Process Model  50  53  52  Parameter Identification  5.4 Controller Design  53  5.5 Constraints for MD Control  53  5.5.1 Amplitude Constraints  54  5.5.2 Rate Constraints  54  5.53  55  Ou^ut Variation Constraints  5.5.4 Optimal Solution to Constrained SISO GPC iv  55  6  7  5.6 Simulations of Adaptive MD Coating Weight Control  56  5.7 Conclusions  63  Cross Machine Coating Weight Control  65  6.1 CD Control Mechanism  65  6.2 CD Coating Process Model  66  6.3 Parameter Identification for Multivariable System  69  6.4 CD Control Law Design  71  6.5 Constraints for CD Control  72  6.5.1 Actuator Adjustment Amplitude Constraints  72  6.5.2 Actuator Adjustment Rate Constraints  72  6.5.3 Adjacent Actuator Relative Position Limits  72  6.5.4 Bending Moment Limitation  73  6.5.5 Output Variation Constraints  74  6.5.6 Optimal Solution of Constrained Multivariable GPC  74  6.6 CD Coating Weight Control Simulations  75  6.7 Conclusions  82  Conclusions  84  A Quadratic Optimization by Lagrangian Multiplier Method  87  References  88  List of Tables  2.1 Simulation conditions for blade coating modelling  VI  17  List of Figures  2.1 Off-machine coater overview  5  2.2 Principle offloodednip blade coating  6  2.3 Bevelled blade and bent blade  7  2.4 Transfer of mechanical force to paper.  9  2.5 Coating flow at blade tip  10  2.6 Final coating layer development  16  2.7 Coating weight change with coater speed  19  2.8 Coating weight change with blade thickness  19  2.9 Coating weight change with blade tilt angle /?  21  2.10 Coating weight with blade wearing  22  2.11 Relationship between coating weight and blade pressure  23  3.1 Coating weight measurement by scanning sensors  27  3.2 Overlay of 53 scan raw data  31  3.3 Actuator bump test history  32  3.4 Local bump test  response  3.5 Cross machine-direction  response  33 34  3.6 Average CD profile power spectrum and accumulative contribution of variations  35  3.7 MD variations and its power spectrum  36  4.1 Control diagram of coating process  39  4.2 Set-point, control and outputs in GPC  40 vii  5.1 Mechanical blade loading  49  5.2 Diagram of adaptive MD coating weight control by blade pressure  50  5.3 Diagram of adaptive MD coating weight control by blade tilt angle  51  5.4 Machine-direction adaptive GPC with amplitude constraints  58  5.5 Machine-direction adaptive GPC with rate constraints  59  5.6 Machine-direction adaptive GPC with output constraints  60  5.7 Machine-direction adaptive GPC with model mismatch  61  6.1 Structure of cross-machine profiler.  65  6.2 Block diagram of CD adaptive coating weight control  66  6.3 Cross-machine correlations among actuators and measurements  70  6.4 CD coating weight control with amplitude constraints  78  6.5 CD coating weight control with actuator rate constraints  79  6.6 CD coating weight control with actuator adjacent position constraints  80  6.7 CD coating weight control with bending moment constraints  81  vm  Acknovt'ledgments  I would like to thank my supervisor Dr. Guy A. Dumont for his guidance. I wish to thank the co-reader of my thesis. Dr. Michael Davies, for his advice. I would also like to express my appreciation to Mr. Glen Woods, Mr. Rob Gilmaine and Mr. Stephen Chu for their effort and assistance in carrying out the industrial experiments. I am very appreciative of the advice of my colleagues in the Paper Machine Group at the Pulp and Paper Center. I would like to thank Dr. Guy Dumont and the University of British Columbia for providing financial support. Also thanks to my husband, Jiahua Zhou.  IX  Chapter J:  Introduction  Chapter 1 Introduction  1.1 Background 1.1.1 Paper Coating Coating is the process of applying colour or functional materials to the surface of a substrate. Coating technology has extensive applications in sheet manufacturing industries such as paper, steel and plastic industries. Paper coating and coating weight control are the main concerns of this thesis. The goal of coating has traditionally been to increase the quality of the paper. Depending upon the final goal, coating can be divided into two categories[l]: pigment and non-pigment coating. Pigment coating aims generally at improving the printing qualities of the paper. The appearance is improved by enhancing optical properties such as gloss, whiteness, and opacity. Most coatings contain three distinctive groups of materials[2] besides water: pigments, binders and additives. The most commonly used pigments are: kaolin clay, calcium carbonate, titanium dioxide, and alumina trihydrate. Binders are divided into two groups, natural and synthetic. Of the natural binders, starch is by far the largest group, with soy protein being a distinct second and casein a very distinct third. Of the synthetic binders, there are styrene-butadience latex, polyvinyl acetate latex and polyvinylacrylic latex. Coating additives are of a small but important portion of the coating materials, which perform specific fimctions, such as viscosity modification or lubrication, in the coating colour or in the finished coating layer. Printing properties are enhanced through the smaller and more even pore structure which results from the coating layer. Pigment coating can be carried out during the paper manufacturing or later off-machine. In both cases blade coaters are most popular. Non-pigment coating describes various converting processes such as extrusion coating, siliconization, laminating, and other processes aimed at giving fimctional properties, such as barriers for gases and liquids, non-adhesion, resistance towards grease and so on. These converting processes are carried out using roll coaters, air-knife coaters, or specialized equipments such as extruders and laminators.  Chapter I:  Introduction  1.1.2 Coating Weight Measurement Most measurements of coating weight are carried out by determining the absorption[3], reflection[4] orfluorescence[5]of atomic rays. Methods include: •  X-ray fluorescence: When an X-ray source is allowed to irradiate a coating, elements in the coating may emit X-ray themselves with a wavelength, frequency or energy characteristics of that element. If its proportion of the total coating weight is known, then the coating weight can be determined. This works quite well for titanium dioxide.  •  Differential Beta-ray absorption: This method is most used because of its short time constant which enables on-line profiling and rapid data collection. Beta-rays lose energy by producing ion pairs in an absorbing material and they are absorbed in the same way by cellulose, water and common coating materials. Thus they are a good means to measure mass, and are independent of the atomic number of the absorbing material. A differential combination of two beta-ray gauges is commonly employed in the industry to measure coating weight, one before the coater and one after. It is also common to employ two moisture gauges in the same positions, to correct for different moisture contents.  •  Differential ash: Coating elements or components have different absorption coefficients of gamma-ray. These may be determined experimentally for various weights of the pure native materials. These coefficients can be used to calibrate the composite coating absorption coefficient as the sum of the productions and their percentage in the coating. In most installations, atomic sensors are mounted on gauges which traverse the sheet. Because  the sheet moves quickly in the machine-direction while the sensors travel across the sheet more slowly, the sheet is actually sampled in a diagonal strip. Therefore the measurement is composed of both machine-direction and cross-direction information. Fast and precise coating weight measurement is a key factor for effective coating weight control. I.IJ Coating Weight Control Coating weight control aims at improving the uniformity of the coating layer. Coating weight uniformity can be improved indirectly by improving base sheet properties [6], and directly by  Chapter 1:  Introduction  controlling the coating station[7]. Coating weight control was first proposed by Mardon, et. al. in 1965. With the development of computer control algorithms, the control of coating weight grew rapidly in the 1980's[8][9][10][ll]. Coating weight control is usually carried out independendy in the machine-direction and in the cross-directioa While machine-direction control of coating weight is quite successful[12], crossdirection coating weight control is still far from maturity. But cross machine control techniques are growing rapidly with the increasing market demands. The use of induction heating on the calender stack before the coater has been claimed as a successful indirect cross machine coating weight control method[13]. Several installations of direct cross machine controls are reported in [14][15][16][14]. Technical trends for coating process control are explored in [17].  1.2 Motivation The coating process is influenced by so many factors[18] that a proper physical model must be derived from the complex industrial process for control purposes. In order to control the coating weight, the causes of the coating weight variations must be understood. The dynamics of the coating process and the couplings of the actuators also need to be comprehended. Due to the nonlinear nature of the relationships between coating weight and the control variables, adaptive control algorithms are desirable to track the system changes. The physical constraints on the system and the requirements of the product quality make it necessary to consider constraints in the control design. Thus a constrained adaptive control system is designed to achieve automatic coating weight control.  1.3 Contribution A beveUed-blade coating process is studied with focus on the interaction between the coating colour and the blade tip. A physical model is built based on the fluid mechanic principles. MUl trials were carried out in a paper mfll to investigate the dynamics of the coating process and the correlations between cross-directional profilers. Cross and machine-direction variations were  Chapter 1:  Introduction  analyzed and achievable performances were estimated. Trial reports were written and discussed with technical staff at the mUl. Cross and machine-direction coating weight controls are developed independently. For machinedirection coating weight control, an adaptive Generalized Predictive Controller with constraints is designed to achieve a fast and stable control of the nonlinear coating process. A multivariable Generalized Predictive Controller is developed for cross machine profiling. To make the system adaptive to the nonlinear nature of the process, a modified estimation method for application to the multivariable system is proposed using Recursive Least Squares. Constraints on actuating amplitude and rate, position difference between adjacent actuators and the bending moment of the blade are also taken into consideration.  1.4 Thesis Outline Chapter 2 outlines the coating process model. Chapter 3 discusses the nature of coating weight variations and analyzes the variations using the industrial data. Control strategies for machine and cross-direction coating weight control are selected and Generalized Predictive Control algorithms are introduced in Chapter 4. Adaptive control methods and simulation results for machine-direction and cross-direction controls are given in Chapters 5 and 6respectively.Chapter 7 concludes this research work and proposes future studies in this area.  Chapter 2:  Coating Process Modelling  Chapter 2 Coating Process Modelling  2.1 Blade Coater Mechanism There are two types of blade coaters: off-machine and on-machine. The only diflference is that the on-machine coater is part of a paper machine while the off-machine coater is a separate station. Scanners with weight and moisture measurement  Unwindei Winder  Coating station No. 1 Coating station No.2  Figure 2.1: Off-machine coater overview. A typical off-machine coater is illustrated in Figure 2.1. It has two coating stations to permit coating of both sides, three scanner gauges, two drying sections, a winder and an unwinder. The base sheet is imwoxmd and measured before the first coating station. After being coated on the first side, the sheet passes through a drying section and then is measured again before passing through the second coating station. The final sheet, with coating applied to both sides, is dried and measured before reaching the winder. The woimd sheet is ready for shipping or may possibly be processed by a supercalender for improved gloss and other surface properties. When the differential beta-ray absorption method is employed for coating weight measurement, both a beta-ray sensor for basis weight measurement and an infrared sensor for moisture measurement are usually present. The dry coating weight is determined horn the difference between basis weight and moisture measurements between the scanners before and after coating.  Chapter 2:  Coating Process Modelling  After each scan of the sensor across the sheet, measurements are compared with the taiget value. Control is carried out based on the dry coating weight deviation from the target value. Corrections are calculated by the computer and adjustments are made as necessary.  PAPER  Figure 2.2: Principle offloodednip blade coating.  A close view of afloodednip inverted-blade coating station is shown in Figure 2.2. When paper passes through the coating station, the coating colour is applied to the paper surface by an applicator roll. Extra colour is then metered off by a blade pressed against the paper and backing roU. Coating weight uniformity is achieved by manipulating the blade load or the blade angle. Today's blade coaters arc becoming more and more sophisticated, and there are numerous types of construction. They differ in the mode of application of the coating material and in the levelling and coating weight adjustment mechanism. The applicators can be divided, according to the pressure prevailing between the paper and the colour, into the following groups: roll applicators, fountain applicators, jet applicators, and contact applicators. Depending upon the time that the paper takes to travel from colour applicator to the metering blade, the coaters are classified as short dwell and long dwell coaters.  Chapter 2:  Backing Roll  Backing Roll  Bevelled Blade  Bent Blade  Coating Process Modelling  Figure 2J3: Bevelled blade and bent blade.  The blade coating process can also be characterized by the type of blade, i.e. beveUed-blade coaters and bent-blade coaters as shown in Figure 2.3. In bevelled-blade coating, the forces acting on the paper are smaU and the coating weight is dependent on the surface roughness volume of the base sheet. The surface roughness volume is defined as the void volume of the paper surface in relation to a reference plane pressed against the surface[19]. The coating colour is deposited in the cavities of the paper, giving a smooth surface. An increased blade pressure diminishes the void volume of the paper surface and therefore reduces the coating weight Longer dwell time before the metering allows more colour to fill the pores of the paper surface and the coated surface is more smooth. For a very short dwell time, the surface changes from, being smooth to being covered by a contoured coating layer of constant thickness. In bent-blade coating, the amount of coating depends on an equilibrium between the high dynamic forces from the colour and a high blade pressure. Under low blade pressure, an increasing blade pressure will decrease the coating weight Under an increasing blade pressure, a maximum coating weight can be reached, after which the coating weight decreases again. Although the beveUed-blade coating is described as stiff-blade, straight-blade, or trailing blade, the metering blade in its mounting is actually significantly deformable. In bent-blade coating, which is also called low-angle-blade, or flexible blade coating, the blade is more flexed and has a smaU contact angle with the paper surface. As Eklund[20] and others had observed, there is a continuous progression from stiff- toflexible-bladebehaviour in coating operatioa  Chapter 2:  Coating Process Modelling  Since beveUed-blade mode is most used in high speed low-weight coating, the physical model of beveUed-blade coating process is studied as followed. The model can be extended to flexible-blade coating with consideration given to the deflection of the blade.  2.2 Bevelled-blade Coater Modelling The coating weight, or rather the thickness of the colour layer passing through the gap between the blade tip and the base paper, is determined by the state of equilibrium of forces acting on the blade and paper. In the early 1970's, Tiirai[21] and Hayward[22] proposed that the balance between the blade load and the hydrodynamic force of the colour fluid determines the coating weight Kahila[23] and Eklund[20] claimed that other flow dynamic forces are also of importance. An elastohydrodynamic interaction among the blade, liquid and the loading is suggested as a key factor of coating weight development by Pranckh and Scriven[24] in 1990. Using a different approach. Savage gave a mathematical model for coating process in 1982[2S]. The coating process modelling in this thesis is aimed at obtaining a first principle model for coating weight control. The model is approached by studying the force equilibrium on the blade tip. A detailed analysis of the forces is given as foUows. 2.2.1 Mechanical Force The mechanical pressure is applied on the blade to load it so that its edge presses against the backing roU in a desired way. The blade pressure is transferred to the paper as illustrated in Figure 2.4. The relationship between F^ and F is[23]: F^=\l where xi and X2 are blade dimensions, W is deflection of the blade tip.  +  2Va;2/  2x2  F + —^  (2.1)  Chapter 2:  Coating Process Modelling  Figure 2.4: Transfer of mechanical force to paper. D is stiffness index of the blade, expressed by Z> = ^^fi^-v)^' E is Young's modulus of elasticity, d is blade thickness, and V is a material-dependent factor, the value is 0.3 for steel. The force FQ acting perpendicular to the base can be obtained from FQ = F^cosf  (2.2)  where 7 is the blade angle. 2.2.2 Dynamic Forces There are two distinguishable stages in the blade coating process. First, the colour is applied to the paper surface, and then excess colour is metered off by a blade. The colour flow near the blade tip is demonstrated in Figure 2.5. The interactions between the colour and the blade are associated with two regions: In region II, the colour flow strikes the blade. A major proportion of this flow changes direction and flows down along the blade. In region I, the narrow blade tip gives access to no more than a very thin layer of colour. This yields the ultimate coating layer on the paper surface.  Chapter 2:  A Ri  Coating Process Modelling  u  1 .--•Jt.  mU  Figure 2.5: Coatingflowat blade dp. Because of interactions between the blade and the colour flow, dynamic forces acting on the blade arise. They can be distinguished as impulse forces and hydrodynamic forces. Impulse Forces: When the excess coating colour strikes the blade and changes direction, the change of momentum induces impulse-type forces. The forces can be described as one towards the blade, Fi, and the other towards the paper surface, Ri. If the amount of colour which passes through the blade tip is neglected and the cross-section of the flow is assumed to be constant before and alter the direction change, the average velocity of the flow is constant based on the consistency of the mass flow rate. If ftictional losses are also disregarded, momentum conservations in both vertical and horizontal directions can be obtained: iiisin7 = mU — (—mi7cos7) (2.3)  Fi cos 'f — Ri = mil sin 7 — 0 where •  m is mass flow doctored by the blade,  •  f/ is the colour velocity, i.e. the speed of paper,  •  7 = a -)- /3, and  •  a is the blade bevel angle, /? is the blade tilt angle, 7 is the blade holding angle.  10  Chapter 2:  Coating Process Modelling  From Equation 2.3 we have Fi = mU(l + cos 7)/sin 7 (2.4) Ri = rhU(2 cos^ 7 4- cos7 — l ) / sin7 iZ,- is the force that the colour flow applies on the paper and backing roll in region I as indicated in Figure 2.5. It is not of concern here because it can hardly change the coating layer buUt up. Fi, however, is important because it is the force that the colour applies to the blade. Thus the impulse force Fi acting on the blade is proportional to the production of mass flow rate and velocity of the colour. This force is also influenced by the blade angle. When the massflowand paper speed are kept constant, the impulse force will not change if the blade angle is unchanged. Hydrodynamic Force: The effect offiictionalflowmay become significant when shear stresses significantly affect the liquid equilibrium. This situation arises when liquid enters into the wedge shaped space (see Figure 2.5), where relative velocity differences between the wall and liquid layer are large. In this case, the viscosity of the liquid becomes an influencing factor. When the Reynolds number of the flow is much smaller than one, the colour flow can be approximated as a lubrication flow. The hydrodynamic force generated by thisflowcan be obtained by the lubrication approximation to the Navier-Stokes equations. As illustrated in Figure 2.5, hydrodynamic forces may develop in both areas I and II. The angle between the blade and paper offers a condition for the development of such a viscosity dependent hydrodynamic force. However, the direction of the flow must be towards the narrow edge of the wedge. Therefore this pressure can only develop in a close proximity to the blade wedge where the coating colour no longer flows downwards along the blade. In order to apply lubrication theory in region II, the Reynolds number must be much less than 1. At low coater speeds the Reynolds number is very small. Lubrication theory is then applicable and the hydrodynamic force is significant enough to be considered. But at high coater speed the Reynolds number is greater than 1, lubrication theory cannot be applied and the hydrodynamic force is not a major force in this area. Because most modem coaters run at high speed, the hydrodynamic forces generated in area II are usually negligible. 11  Chapter 2:  Coating Process Modelling  Another area where a hydrodynamic force may develop is the narrow gap beneath the blade tip (see Figure 2.5). In most blade coating applications, the blade tip is not parallel to the paper surface and consequently the hydrodynamic force arising beneath the blade tip prevails. Assuming the coating colour is a Newtonian flow, Navier-Stokes' equation can be used is:  where •  /7 is the pressure distribution xmder the blade tip.  •  X and Y directions are indicated in Figure 2.5. Z is a direction perpendicular to both X and Y directions and is towards the paper in Figure 2.5. Ux, Uy, Uz 3X0, the flow speeds in X, Y and Z directions respectively,  •  /o is colour density,  •  n IS colour viscosity, and  •  9x1 9y, 9z is the components of the gravitational acceleration. Because /? is small, the flow under the blade can be considered as a two dimensional laminar  flow between parallel straight walls. The flow is two-dimensional because of symmetry along the blade width. To maintain such a flow, the velocity along the Y and Z directions must be zero, i.e. Uy = Uz = 0. Also all partial derivatives withrespectto z must vanish. Theflowcontinuity equation  f. + ^ + « i OX  oy  oz  becomes dUx  12  =0  (2.6)  Chapter 2:  Coaling Process Modelling  Since the coater operates at constant speed, —^ = 0. Neglecting the gravitational forces, the Navier-Stokes equations reduce to:  „=*+/£'• dx  dy"^  0=-^  (2.8) dy  The first line of equation 2.8 can then be rewritten as $dx = / .dy^  (2«  Integrating equation 2.9 with respect to y yields  ''• = GI)T+''^+'^ where ci,  (^-'W  C2 are constants of integration.  Applying the following boundary conditions: when y = 0, Ux = U, and (2.11) when y = h, Ux = 0, to equation 2.10 gives C2 = U,  (2.12)  where h is the distance between the bevelled edge of the blade and the sheet at the position x on the X axis. Then Ux = -7—TT flax z  7 n  Hdx\2  2V 13  y+ u (7 ^'^\ h  Chapter 2:  Coating Process Modelling  Over a unit width of the blade, the flow rate can be expressed as: h  Q= I U^dy 0  (2.14) 0  Hdx\  12)  2  Because the flow rate is constant at any cross section, equation (2.14) can be rewritten as dp dx  Q- Uhl2 /1 = 6Ufi -h^/12fi U^  2Q/U\  h^ )  (2.15)  The coating thickness, h, at position x can be calculated by (2.16)  h = hi - "nC^i ~ ^o) Define variables xi and n as: x\ = x/B  (2.17) n = hi/fiQ  where B is the projection of the blade thickness to the paper surface, and ho is the height of the blade tip outlet. Substitute equation 2.17 into equation 2.16, to obtain: h — ho(n — nxi + xj)  (2.18)  Replacing h in equation 2.15 gives dp dxi  2Q  6fj.UB  hi  (n — nxi + Xi)  Uho(n — nxi + xi)  (2.19)  and integrating this equation we have:  /IQ  (n — nxi + xi){n — 1) I 14  Q Uho{n —nxi + xi) + c  (2.20)  Chapter 2:  Coaling Process Modelling  For boundary conditions p{l) = 0 and p(l) = 0, we obtain:  ''-  hi  ( l - n ) ( l + n)  ^^-^'^  Thus K.i) = 6/.^^ (n-lWl-xQ ^0 (« — nari + xi) (n + 1)  ^2.22)  The total force applied on per unit length of the blade is: 1  Fh = B fpdxi  (2.23)  Therefore F.=  '"""' l „ ( n ) - ^ ( " - ' ) n+ 1 ftj(n - 1)'  (2.24)  or in another form 2(n - 1)  •  ''  tan2/3 In (n) - ^ n + 1^^  (2.25)  Similar results can be obtained in area II (Figure 2.5) if the lubrication theory is applicable. 2.2.3 Mechanism of Coating Weight Development During the coating process, bofli mechanical and dynamic forces act on the blade together. The blade can be considered as a suspended beam kept in balance by the forces applied on it For coating weight development, the paper surface roughness is a key element, because the coating colour is deposited in the cavities of the paper surface. Since paper is compressible, its physical properties are subject to change as loading and pressure are altered. Therefore the amount of coating colour that has access to the space between the base paper and the blade is determined not only by the surface roughness volume of the base sheet, but also by the load exerted by the blade. Furthermore, the paper is in contact with water from the coating colour for a certain time before it meets the blade. The interaction between moisture and base paper takes place at this time. The paper fibers are swollen by the water and the paper compressibility is no longer that of dry paper. In bevelled blade coating, an increased blade pressure diminishes the surface volume and the coating 15  Chapter 2:  Coating Process Modelling  weight The coating colour is deposited in the cavities of the paper giving a smooth surface, and the longer the dwell time before the blade, the smoother the surface. For a very short dwell time, the surface changes from being smooth to being covered by a contoured coating layer of constant thickness. The mechanism can be summarized as: an excess amount of coating colour is applied to the paper by the applicator. The blade meters off most of the excess colour and the residue on the paper surface forms the final coating layer. When the forces acting on the blade tip are balanced in the Y direction, the hydrodynamic force is determined by i^/,cos/3 = ( i ^ ^ - F , ) c o s 7  (2.26)  Because forces in liquid are transferred omnidirectionally, the hydrodynamic force that the colour flow applies to the blade is also applied to the paper and backing roll. In most cases, this force is so great that the rubber covered backing roll deflects and the paper is compressed. The higher the force, the more the paper is compressed. As a consequence, the paper surface volume wiU be less and the coating weight will be lighter. Low loading of the blade results in low hydrodynamic force and expansion of paper surface volume. This may cause poor coating weight profile.  Figure 2.6: Final coating layer developmenL 16  Chapter 2:  Coating Process Modelling  Table 2.1: Simulation conditions for blade coating modelling. Coater Velocity U Blade Bevel Angle a  30°  Colour Viscosity fj, Colour Mass Flow rti  1500 mPa«s 4 kgAns  Solids Content S Colour Density p  65%  Blade Thickness T  3° 635 fim  Blade Nip Pressure Fz  2.5 kN/m  Blade Tilt Angle /?  12m/s  1200 kg/m^  It is interesting to point out that the final wet coating layer is not exactly the height of the blade tip outlet. For example, if a linear speed distribution under the blade tip is assumed, the average speed V imder the blade is V = lu  (2.27)  Since the flow rate passing the blade and deposited on the sheet is the same, we have V-ho = U-h 1 h = -ho  (2.28)  The final coating layer is thus only half of the height of the blade tip outlet. This situation is illustrated in Figure 2.6.  23 Simulation of the Coating Process Coating weight development depends on the mechanical force applied to the blade, and on the impulse and hydrodynamic forces generated by the coating colour flow. It is also influenced by many factors such as blade pressure, blade bevel angle and blade tilt angle, colour viscosity, coater speed and colour mass flow towards the blade. Simulations will be carried out so that only one factor changes at a time, and the coating weight change due to this factor will be clearly displayed. Simulation conditions are given in Table 2.1. Under these conditions, the Reynolds number in area II, as illustrated in Figure 2.5, is greater than 1. In this case, the hydrodynamic force generated in area 11 is negligible. Therefore the forces acting on the blade are: the mechanical force holding the blade against paper, the impulse force generated by the colour flow, and the hydrodynamic force generated beneath the blade tip. 17  Chapter 2:  Coating Process Modelling  When the above forces balance each other, the wet coating thickness at the outlet of the blade tip, ho, can always be calculated under one changing element. In order to calculate ho, the NewtonRaphson method[26] is employed to solve the following nonlinear equation: f{ho) = Fh cos /3-iF,-  Fi) cos 7 = 0  (2.29)  The Newton-Raphson algorithm solves the nonlinear/('jc^=0 equation in five steps: 1. Use incremental search to find a interval [xi, X2], which satisfies f(xi) • f(x2) < 0. 2. Use linear interpolation to obtain a better estimate: _ 3:1/(0:2) -  X2fixi)  X3 = — 7 7 — X  77—X—  f{X2) -  (2-30)  fixi)  3. Calculate /(xa) and its derivative  _ f(x3 +  h)-f(x3)  (2.31)  h If function f(x) is differentiable, and /t is a very small incremental in x. 4. Find point X4 where tangent line passing through [2:3, 7(2:3)] intersects the X-axis, i.e.  /(.,) = ^i(M  (2.32)  X4 — X3  So 3:4 = 2:3 -  (2.33)  f'{X3) 5. Replace X3 by X4 and repeat steps 3 to 5 until X4 — X3 X4  where e is the required accuracy.  18  <£  (2.34)  Chcpter 2:  Coaling Process Modelling  Once ho is available, the coating weight can be foimd. As indicated in Equation 2.28, if the force distribution under the blade tip is linear, the final wet coating layer on the paper surface will have half the height of ho. Therefore the coating weight can be expressed as: Coating Weight = -hopS  (2.35)  Under the conditions in Table 2.1, the impulse force can be calculated by Equation 2.4 as 162 NIm. Calculated from the force balance Equation 2.26, the hydrodynamic force is 1.96 kNIm. Thus the coating weight generated in this situation can be obtained from Equation 2.25 as 18.9 glm^. It is evident that the hydrodynamic force is much greater than the impulse force and is the dominant force to balance with the blade load during the coating process. Effect of colour viscosity: In most circumstances, the coater speed and colour viscosity are constant for the same grade of paper. A change is needed only if a different type of coating is applied. An increase of the colour viscosity implies that more colour will be applied to the paper by the applicator. Thus the mass flow is increased and the impulse force is increased. When the blade load is constant, the impulse force will push the blade up and the coating weight will increase with increasing colour viscosity. 24  1  j  j  1  24  j  22  22  i: r  —\.—j—j.—\.^^^^^^..—  20  j.  ]  j.  1-.^^.  \.  1  \^..\.  ni  — \ — 1 — \ ^ ^ — \ —  £16  i:  \  •§ 12  o O  10  s  ".  6  8  10  12  14  §3>0  15  Coater Speed (m/s)  300  400  500  600  Blade Thickness (p.m)  Figure 2.7: Coating weight change with coater speed.  Figure 2J: Coating weight change with blade thickness.  19  700  80  Chapter 2:  Coating Process Modelling  Effect of Coater Speed: High coater velocity yields higher coating weight as indicated in Figure 2.7. As the speed of the coater increases, if other conditions shown in Table 2.1 arc constant, the impulse force and the hydrodynamic force generated by the colour flow arc increased. Under a constant mechanical load, the coating colour under the blade tip tends to lift the blade due to the increased forces that it generates. As a consequence, the coating weight is increased. Effect of Blade thickness: The blade geometry is a very important factor in determining the final coating weight. Figure 2.8 shows the influence of blade thickness on coating weight It may be seen that thicker blades yield heavier coating weight In fact the blade thickness has a direct influence on the surface area upon which the blade presses against the paper. Under the same blade pressure, which is denoted by force per unit length of the blade, the force per unit area that the blade applies to the paper is different with blades of different thicknesses. For thicker blades, the force is distributed over a larger area and the paper is less compressed than for a thin blade. Consequently, the coating weight given by the thick blade is greater than that given by the thin blade. Effect of Blade Angle: The blade tilt angle /? has a significant influence on the hydrodynamic force. Figurc 2.9 shows coating weight versus /? for different blade bevel angles a. It is interesting to see that the coating weight first increases with 13, and after reaching a maximum then declines. Simulations show that maximum coating weight happens at different values of j5, depending on the running conditions. The curves in Figure 2.9 show that the blade with greater bevel angles give flatter response than those with smaller bevel angles. In practice, the blade usually works at a small tilt angle, about one to three degrees. In this range, the coating weight increases sharply with fi. Thus coating weight control can be achieved by changing the blade angle. In Figure 2.9, /? changes from 1° to 20°. However, a high /3 value is rarelyreportedeven in trial reports. The shapes of the curves in Figure 2.9 imply a change in the sign of the system incremental 20  Chapter 2:  Coating Process Modelling  Figure 2.9: Coating weight change with blade tilt angle (3.  gain when the tilt angle /3 increases. This is not desirable for control system design using a linear system model. However, the system can be operated in an approximately linear range of fi value, for example near 2° or 6°.  It is very important to understanding the role that the blade angle plays in the coating process, because blade wear is a significant problem in the coating process. When wear occurs, assuming the blade is held at the same angle 7, the equivalent blade tilt angle /? increases while the equivalent honed blade angle a decreases. Figure 2.10 shows the influence of wear for the blades with different initial honed angles OQ. For aU new blades with different initial bevel angles OQ. the initial blade tilted angle ^ is assumed to be 0.5°. It can be seen that initially the coating weight increases, but with more wear, the coating weight tends to decrease. Starting with different initial /? values will result in different curves and trends. 21  Chapter 2:  Coating Process Modelling  35  30  25  •220  I  15  10 5-  2  3  4  5 P  6  7  8  9  _i  I  10  Figure 2.10: Coating weight with blade wearing. Effect of Blade Pressure: The high pressures and high viscosity of the colour flow at temperature 60° C, lead to rapid blade wear. Unfortunately, this blade wear is always uneven. Several effects may cause this uneven wean uneven profile of the base sheet, nonuniformity of the coating colour, uneven loading of the cross-direction actuators, or possibly just the nonunifomiity of the material from which the blade is made. A steel blade is nonnally used for 8 hours before it is badly worn, and a ceramic blade for 32 hours. The effects of blade wearing make coating weight control more complicated and make automatic cross machine control necessary. Figure 2.11 demonstrates how mechanical force influences coating weight As expected, coating weight declines exponentially as the blade pressure increases. This matches the mill observations. It can also be explained in terms of a typical deflection of a steel beam in response to the force applied on it 22  Chapter 2:  4  6 8 10 Blade Nip Pressure (KN/m)  12  Coating Process Modelling  14  Figure 2.11: Relationship between coating weight and blade pressure. In practice, when a bevelled blade is used, the excess load applied on the blade is usually between 20 to 40 PU ( pound per inch of length), which is equivalent to 3.5 to 7.0 kNIm. Qianges in these forces significandy change the coating weight and blade pressure can therefore be used as a control mechanism. Figures 2.9, 2.10 and 2.11 also show that the blades with different bevel angles yield different coating weights. Under the same blade pressure, a blade with greater bevel angle gives lower coating weight than that with smaller bevel angle. Assuming the same blade thickness, the length that the blade projects on the paper surface B (see Hgure 2.5) is: B = T cos /?/ sin a  (2.36)  As a is increased, B is reduced and less coating colour flows into the wedge of the blade tip. The same amount of hydrodynamic force can be developed in a narrower gap and thus thefinalcoating weight is decreased. 23  Chapter 2:  Coating Process Modelling  Some important conclusions for bevelled blade coating can be drawn from the above simulations: Coating weight increases with increasing coater speed and blade thickness, A\^th increasing blade tilt angle, coating weight first increases then decreases. Higher blade load yields less coating weight  24  Chapter 3:  Coating Weight Variation and Mill Trial Data Analysis  Chapter 3 Coating Weight Variation and Mill IVial Data Analysis  3.1 Coating Weight Variation Variations in coating weight arise fixjm either the base sheet or the coating process. Base sheet variation directly affects coating weight in the blade coating process. Since the wet coating layer thickness is determined by the gap between the blade tip and the paper surface, basis weight variation of the base sheet is mapped to coating weight variations. Thus the base paper roughness plays a very important role in determining the coating weight The moisture content is also an important factor because it influences the compressibility of base sheet. In principle, base sheet variations can be compensated for in the coating process, but this is not always effective in practice. Therefore the uniformity of the coating weight relies heavily on good base paper properties. As the coating process model derived in the previous chapter indicates, coating weight varies with: Blade pressure, Coater speed or paper speed. Coating colour viscosity and density. Speed at which the applicator applies the colour to the paper, and Blade geometry, which includes the blade angle and thickness. Any changes in the above variables may cause significant variations in the coating weight The solids content of the coating colour may not influence the wet coating layer thickness significantly but will influence the final dry coating weight In the paper industry, the coating weight variations are usually described by their 2-a values, where a is the standard derivation of the measurement of interest Defining x as the average of the measurements, the range x±2a contains 96% of the data in the case of a Gaussian distribution, and is therefore taken as a measure of the range within which almost all measurements can be expected to lie. 25  Chapter 3: Coating Weight Variation and Mill Trial Data Analysis  Coating weight variations affect the print quality and gloss of the final paper, and in extreme cases, variations could cause breaks and other runnability problems during printing. The variations are classified as: machine-direction (MD), cross-direction (CD), and random or residual (R) variations.  3.1.1 Machine-Direction Variations The machine-direction variations arc those variations in the direction that the web travels. The wavelength of the variation could vary finom several millimeters to several hundred kilometers. Variations of all wavelength happen at the same time. In the coating process, machine-direction variations may be created by the changes of the machine speed, applicator roll speed, colour viscosity, colour solid content, global blade load and blade angle, or base sheet properties.  3.1.2 Cross-Direction Variations The cross-direction variations are defined as those across the sheet. Cross-direction variations may be caused by the wearing of the blade tip, the uneven local blade loading, and the poor profile of the base paper. CD variations are usually much slower than MD variations. On-machine traversing systems do not give a true CD profile because the signal also contains the MD and residual variation. However, with most of the MD and the residual variations filtered out, these signals can be used for CD process control.  3 . U Residual Variations Residual variation remains after CD and MD have been subtracted from total variation. The cause of the random variation may be poor properties of the base paper, machine vibration, and other unidentified disturbances. It usually contains high frequency variations. 26  Clujpter 3:  Coating Weight Variation and Mill Trial Data Analysis  Machine direction (MD) • •• • *  f-  • / /  •  •  ' ' Measurement patii  kT  •  (k-1)T  Figure 3.1: Coating weight measurement by scanning sensors.  3.2 Mill lY-ial Data Analysis Several mill trials have been carried out on an off-machine coater in a paper mill. The purpose of the trials was to investigate the dynamics of the coating process, and the response of the crossdirection profiling actuators.  3.2.1 MD and CD Data Separation The coater scanning sensors collect scanning data. The sensors arc mounted and guided to scan slowly back and forth across the sheet, taking measurements at certain intervals of time. Due to the fast movement of the sheet in machine-direction, the measurement points forms a zig-zag pattern on the sheet, as shown in Figure 3.1. Such measurements, referred to as raw data, contain MD, CD and Residual information. MD variations are rapid variations and are assumed to have equal influence on aU the CD positions. The CD profiles are considered nearly time-invariant. There are several methods available [27][28][29] to process scanned data and two of them are discussed in the following sections. 27  Chapter 3:  Coating Weight Variation and Mill Trial Data Analysis  3^.1.1 Exponential Multi-Scan Trending Exponential multi-scan trending is widely used in the paper industry.  In its basic form,  exponential profile trending, is very simple and can be described by a single mathematical parameter — the trend factor. The average of each scan is considered as the MD value. The trended data are taken as the CD profile of the sheet. The filter is described by: j/„(0 = (1 - a)j/„(< - 1) + aj/„(0  (3.1)  where 2/„ is the estimated CD value at CD position n, yn is the present measurement deviated from the scan average at CD position n, a is the weight on the measurement, a £ [0,1]. The advantage of this method is that it is straightforward and is easy to implement The only tuning factor is a. a close to 1 means fast dynamics is adopted to the CD profile while a close to 0 means CD profile displays only slow variations, a values between 0.2 and 0.5 are mostiy used in paper making and coating. 3.2.1.2 EIMC Algorithm The Estimation and Identification of Moisture Content (EIMQ algorithm [29] estimates the CD and MD profiles fiom the zig-zag pattern provided by the scanning sensor. Though it was originally developed for moisture, it has since been extended for application to other scanned data, such as basis weight and coating weight The moisture model for this algorithm is y^ = p"(l + Bp^)uk + vu  (3.2)  where t/jf is the measured profile from the reference level, at the cross position n {I < n < N), and at time instant kT\ N is the total number of positions at which the measurements are taken; T is the sampling interval between the measurements are taken, and ^ is an integer. 28  Chapter 3: Coating Weight Variation and MM Trial Data Analysis  p" is the profile deviation fmva the reference level in cross-direction at position n. By N  definition X) P" = 0n=l  Uk is the machine-direction disturbance at time kT, and £ is a constant and a function of the reference level. Vk models both sensor noise and the neglected higher order terms, and is assumed to be a Gaussian white noise process with known variance. The machine-direction disturbances are described by  (3.3)  w* = « + 6  where u is mean value of MD disturbance, and ^ is  (3.4)  6+1 = " 6 + Wk  a is a known constant and w is a zero-mean Gaussian white noise process with known variance.  Combining equations 3.2, 3.3 and 3.4 into a state-space form, we have Xk+\ = Axk + Wk  (3.5) y? = p " -H C^xk -H Vk  where '1 0"  u Xk  , A=  =  .6.  '0 " (3.6)  , Wk =  0 a  .'^k.  For cross-direction profile estimation, a least-squares parameter identifier with resetting and forgetting factor is utilized. The machine-direction disturbance is estimated by a Kalman filter. 29  Chapter 3:  Coating Weight Variation and Mill Trial Data Analysis  Details of the algorithm may be found in [29].  n" = vT-i/A -1r'?:Z'L + {nfv;:_. + ^ - ^(n-i)^ -n  'n  l+M  ^-1  C " = ^1 + 5"^") [11]  (3.7)  F = A- KkC" Xk+i = Fxk + Kkivt -  f)  3.2^ Mill IVial Data Analysis Several bump tests have been performed on an off-machine coater. During the course of the bump tests, the CD profilers were adjusted so as to further press the blade towards the paper. Coating weight decreases were expected at and near the bumped actuator zone. The goal of the trials was to determine the coating process dynamics and to investigate couplings between the CD actuators. The scan data collected from the coater were divided into MD and CD profiles using EIMC algorithm. One thing to note is that, in practice, an actuator zone usually contains more than one measurement point. These measurement points are usually called data boxes. On the particular coater tested, one actuator zone contains three data boxes. Finely scaled data boxes give more information than the mapped actuator zones, and so the trial data were recorded and analyzed in the data boxes. In some cases there can be difficulty mapping actuator zones to data boxes since the acmator is not necessarily aligned centrally with a single data box. One typical case is studied below. 30  Chapter 3:  50  100  Coating Weight Variation and Mill Trial Data Analysis  150  200  Data Box Number  Figure 3J: Overlay of S3 scan raw data. Figure 3.2 shows the overlay of the raw coating weight data of S3 scans. Deviations from the taiget value arc shown. Values close to zero therefore indicate good coating quality. The average 2-0 value for all measurements is 0.53. Note that because one actuator zone contains approximately three data boxes, the offset around data box #222 results from a bump test we carried out on the actuator #75. The raw data were decoupled into cross-direction and machine-direction variations, using the EIMC algorithm. The estimated CD and MD profiles approximate the true profiles but they contain errors, because the MD and CD separation algorithms need time to estimate the true profiles. For instance, if a is chosen as 0.2 for exponential trending method, it will take 10 scans toregister90% of a step-wise profile change. Thus it is a significant response time from the control point of view. But a low a value is still used because lai^e control actions are not desirable in most paper-making and coating processes. 31  Chapter 3:  Coating Weight Variation and Mill Trial Data Analysis  The time constant of the EIMC algorithm is determined by its tuning parameters. The desired time constant can be obtained by the right combination of the parameters.  Data Box Number  Scan Number  Figure 33: Actuator bump test history.  The three dimensional plot of CD profile time trajectory and its contour lines are shown in Figure 3.3. It demonstrates the drop of coating weight and recovery of coating weight during the bump test between scan 43 to scan 47.  32  Chcpter 3:  44  Coating Weight Variation and Mill Trial Data Analysis  46 Scan Number  48  Figure 3.4: Local bump test response.  Figure 3.4 traces the history of the responses of the data boxes corresponding to the adjusted actuator. The actuator was moved towards the paper at scan 43 and moved back to its original position at scan 47. It can be seen that data boxes #222 and #223 are in the center of the response since they have the most significant coating weight drops. It is not hard to imagine that, if the time constant of the estimator is zero, the real coating weight would follow the actuator adjustment almost immediately. Indeed, in mill trials, visible strips of the coating weight changes were observed immediately after the actuators had been adjusted. It takes a very short time for the sheet to travel fixim the coating station to the scanning sensor since the coater speed is about 12m/s. Therefore the process dynamics are much shorter than a scanning period which is usually 25 to 30 seconds. 33  Chapter 3:  215  Coating Weight Variation and Mill Trial Data Analysis  220 Data Box Number  230  Figure 3.5: Cross machine-direction response.  Figure 3.5 shows the response in the cross-machine-direction. The response has the greatest amplitude in the center and less on both sides. The width of the response is about the space that 8 data boxes cover. If one actuator zone consists of three data boxes, the response covers three neighboring actuator zones. This indicates that the movement of one actuator influence only its nearest neighbors on each side. The amplitude of the responses of the neighboring zones is smaller than that of the center one. Negative response at both sides of the bell shaped response to basis weight actuators of the type [30] seen on paper machines are not seen on coaters. Since the coating colour is more viscous and has higher solid content than paper machine fiber and water suspensions, no wave action can be expected. The upper part of Figure 3.6 is the Fast Fourier Transform of the average CD profiles. The horizontal axis is the frequency of the variation in terms of the inverse of the number of data boxes. 34  Chester 3:  2.2.5  T  •(0  §  Q  T  y  j___4  2  CocUing Weight Variation and Mill Trial Data Analysis  [ Wavelength of 16 Dat« Boxes  0.1 0.2 0.3 0.4 1/{Wavelength in Number of Data Boxes}  c o  0.5  3 X3  sioo c  1 '  o O 80 v> en  1 Wav^ength of 16 Da4 Boxes  B 60  -;1 •  S 40 20  ^3 E  0.  1  1 _>—^"^  •T^*-—I  0) Q.  i  ^  '  ^^ / / ^  1 1 r \  3  >^  —  r' —  _  1  1 1  1 1 1  1 1 1  \  1  .  0.1 0.2 0.3 0.4 1/{Wavelength in Number of Data Boxes}  0.5  O  Figure 3.6: Average CD profile power spectrum and accumulative contribution of variations. The higher the FFT power spectrum density, the more variance at this frequency. The highest peak indicates that CD variations are concentrated at a wavelength of 33 data boxes. The lower part of Figure 3.6 demonstrates a cumulative percentage contribution of variations with increasing frequency. As found from Figure 3.4, the response of the bump test covers 8 data boxes. Therefore one actuator can successfully adjust a coating weight variation with a wavelength of 16 data boxes. Faster variations will need finer actuator spacing. Therefore, in this particular case, variations with wavelength of 16 data boxes or longer are controllable. As indicated in the lower part of Figure 3.6, 32% of peak to peak variation can be reduced if perfect control is achieved. Machine-direction variations for 52 scans are shown in the upper part of Figure 3.7. The plot shows that the amplitude of machine-direction variations is not as laige as those of CD variations. This implies relatively good MD control and MD product consistency. The lower section of this 35  Chapter 3:  Coating Weight Variation and Mill Trial Data Analysis  I 0.5 2 0 -t  7  ri  10  7  1—r—7—t—r-7  20  -\  30 Scan Number  r-  —rr  40  50  ^0.15  10  20 30 40 Time in Number of Scans  50  60  Figure 3.7: MD variations and its power spectrum, figure shows the FFT of MD variations. It is seen that there are several peaks below 25, so that most variations have wavelength shorter than 25 scans.  3.3 Conclusion Variations in coating weight are generated both by the factors which determine the coating weight and the disturbances in the coating process. The variations are classified as MD, CD and residual variations. Raw data collected from the scanning sensors contain not only CD but also MD and residual variations. In order to control the MD and CD coating weight separately, the MD and CD information should be decoupled. Two data separation methods, which are exponential multi-scan trending and EIMC, are introduced in this chapter. Mill trial data are analyzed to investigate the dynamics of the coating process and the correlations of CD actuators. It should be mentioned that mill trial analysis is made based on a limited number of trials and a limited time frame. Although the test results are qualitative due to the mechanical structure 36  Chapter 3:  Coating Weight Varicaion and Mill Trial Data Analysis  of the actuator screw, but all bump tests seem to have reasonable responses and are quite consistent. Under the assumption that coating weight control will be implemented in a similar way to these trials, some typical characteristics of the process can be deduced from these trials and observations: •  The process has a fast dynamics and its time constant is very short comparing to the scan period. The response of one actuator adjustment covers about 8 data boxes, which implies that the nearest neighbors on each side are influenced. The amplitude of the response is dependent on the amplitude of the adjustment as well as the initial position of the actuator screw.  •  W^th current profiling actuator arrangement, CD variations with period of 8 data boxes or longer are controllable. The peak to peak variations can be reduced by 30% to 65%. Machine-direction variations are controllable by manipulating the global blade pressure and blade angle.  37  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  Chapter 4 Coating Weight Control Strategy: Constrained Generalized Predictive Control  4.1 Control Strategy Design for Coating Process A common approach to coating weight control is to divide the control into two loops: machinedirection control and cross-direction control. Machine-direction control considers control of the coated paper quality over time, while the cross-direction control is aimed at the control of the CD profile uniformity. The machine-direction control is usually carried out by globally manipulating the blade pressure or the blade angle across the entire sheet. CD coating weight control isrealizedby changing local blade load through actuators which are mounted on the blade beam across the sheet. As observed in the mill trials[10][31], the coating process dynamics arc fast. Thus the system dynamics of coating stations and sensors are very fast compared to the scan period. Because the coater runs at high speed and the delay is short enough to be ignored. In order to control the coating process in both machine-direction and cross-direction, the MD and CD data must be decoupled from the raw measurement data. As discussed in Chapter 3, most data separation methods have significant time constants amounting to several scan periods. Considering the coating station, sensors and the data separator all together as a plant, it was found that the system dynamics of this plant can be described as afirst-ordersystem without delay. The control system can be designed as two adaptive control loops: machine-direction and crossdirection loops as illustrated in Rgure 4.1. The adaptive controls are employed to cope with the time-varying parameters of the coating processes. The control of these loops wiU be carried out independently. Both MD and CD control loops are simulated by a first-order plant without delay, with a parameter identifier, the controller itself, and constraints of the practical plant included. The control and estimation take place either once a scan or at a desired faster rate, depending upon the requirement accuracy of the control. The study on the data separator is outside the scope of this thesis. For details, refer to [29] and [27]. 38  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  XyTl  JVlly \^uiiduauii.  ML/ (^Oxiuuiici L  MD Estimator  1'  0SV  MD and CD Separator  Plant  r  MD CD  . CD Estimator  1  rT) Pnr>»''-'n»»-  Figure 4.1: Control diagram of coating process. Adaptive Generalized Predictive Control (GPC) is chosen for both MD and CD control: singleinput single-output (SISO) GPC is used for MD coating weight control loop, and Multivariable GPC is employed for CD coating weight control. Adaptive control is proposed because paper coating is a nonlinear and time-varying process as discussed in Chapter 3. GPC is employed because: •  It can be made relatively insensitive to prior knowledge of the system delay. This is an advantage for the coating process. Although the delay of the process is not significant, it is unknown and may still have influence on system identification and controller design.  •  It can handle non-zero mean noise.  •  It can control non-minimum phase systems, quite likely to occur in practice.  •  It has an inherent integrator so that it handles both set-point changes and load disturbances very well.  •  Long range prediction helps it to smooth the control action and enables the smooth start-up and grade change of paper or coating colour. This helps avoid breaks and increase machine uptime. 39  Chapter 4: Coating Weight Control Strategy: Constrained Generalized Predictive Control  Adaptation is used to track the time-varying parameters.  4.2 Introduction to Generalized Predictive Control  ur set-point Predicted output  Time t Projected controls  Figure 4.2: Set-point, control and outputs in GPC. The Generalized Predictive Control (GPQ algorithm of Qarke et al. [32] is a long-range, multi-step predictive control algorithm. Its predictive control law allows a smooth approach from the current output y(t) to the set-point Ur, as indicated in Figure 4.2. GPC has five important features: •  The use of a Controlled Auto-Regressive Integrated Moving Average (CARIMA) model for the plant introduces an integrator in the controller, and provides offset-free control for load disturbances and set-point changes.  •  The long-range prediction over a finite horizon greater than the dead-time of the plant and at least equal to the model order is used. It provides increasedrobustnessin presence of unloiown or time-varying dead-time, as well as non-minimum phase behaviour.  •  As will be discussed later, the recursive solution of the Diophantine equation (Equation 4.3) makes this scheme simple to implement and effective. 40  Chapter 4:  •  Coating Weight Control Strategy: Constrained Generalized Predictive Control  The consideration of weighting of control increments in the cost function ensures the offset-free rejection of non-stationary disturbances as a result of the inherent integral action.  •  The choice of a control horizon, after which projected control increments are taken to be zero, cuts the computation time significantly. With these features, GPC can be used to control not only a simple plant with little prior  knowledge, but also complex plants with non-minimum phase response, unstable open loop response, or time-varying dead-time. GPC also copes well with unmodelled dynamics and overparameterized models. In practical applications, it is necessary to consider constraints on GPC. The physical setup of the equipment and working conditions usually constrain the controller away from extreme tuning.  4.3 SISO GPC  4.3.1 CARIMA Plant Model and Output Prediction The Controlled Auto-Regressive Integrated Moving Average (CARIMA) process model is defined as: A{q-^)y{t) = B{q-^)u{t - 1) + C{q-^)mi^  (4-1)  where f (0 is an unconelated random noise sequence with zero mean. A, B and C are polynomials in the backward shift operator q~^ A{q-^) = 1 + aiq-^ + • • • + a„.<?-"B{q-^) =ho + hq-^ + ••• + br^.q-^"  (4.2)  C{q-^) = Co + cxq-^ + ••• + CnA'""' Since industrial processes are subject to disturbances from different sources acting at different times, the C polynomial is expected to vary with time. However, algorithms developed for the purpose of identifying C are subject to conveigence problems or computationally burdensome. Therefore the identification of the noise model is difficult. An alternative approach is to use a design polynomial T to represent prior knowledge of the process noise[33][34]. To improve the estimation oi A and B 41  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  polynomials and to improve the noise rejection of the controller, 1/T is usually selected as a low pass filter so that the model is intended to match the process in the low frequency. Particularly, if T=C, the Recursive Least Squares estimator using the filtered data will give unbiased A and B estimations. To derive aj-step ahead predictor of y{t-\- j) based on equation 4.1, consider the Diophantine equation: T = Ej{q-')AA  + q-^Fj{q-')  (4.3)  where Ej and Fj are polynomials uniquely defined given A and the prediction interval j . Ej is of order j-1 and Fj is of order «„. Using Equation 4.1 and 4.3, we can find the optimal predictor, given the output measurements up to time t and u(t+j) for j>l, is Ty(t + j) = GjAu{t + j - l ) + Fjy{t)  (4.4)  y(t + j) = GjAuf(t  (4.5)  or  where Gj = EjB. Let f{t-\-j)  + j - l ) + Fjyf{t)  u^, and y^ are the quantities filtered by  1/T[q~^).  be the component of y(t + j) composed of signals known at time t. In order to  determine f{t + j) and express y{t+j)  in terms of unfiltered present and future control actions,  consider the identity: Gj = GjT + q-^Gj  (4.6)  where the polynomials Gj and Gj are of degree j-1 and Taax(nf, — I, tie — 1), respectively. The coefficients of G are those of G where the initial identity T = I. Substituting equation 4.6 into 4.5 gives: y(t + j) = GjAu{t + j-l)  + f(t + j)  (4.7)  where f(t + j) = Fjyf{t) + GjAuf{t 42  - 1)  (4.8)  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  Thus the predicted output can be spitted into two parts: •  GjAu(t + j — 1), the prediction of the plant output which depends on the future control signals yet to be determined; and  •  f(t+j), the predictive output depends on the past known controls and measured outputs, assuming future control signals unchanged, i.e. Au{t -f ji — 1) = 0, j = 1, 2 • • •. f(t+j) corresponds to the free-response of the process.  4.3.2 The Predictive Control Law Suppose a future set-point sequence Ur{t + j) is available. A smooth approach to the future set-point is chosen as w(t) = y{t) (4.9) w(t + j) = aw(t -I- i - 1) -I- (1 - a)urit + j)J  = 1, 2 • • •  where a w 1 is for a slow transition from the current measured variable to the real set-point Ur- The objective of the predictive control law is to drive the future plant outputs y(t+j) close to the set-point Ur in a smooth approach as shown in Figure 4.2. The cost function to minimize for SISO GPC is:  E [2/(* + i) - ^(i + J)? + E A(i)[A^(i + i - 1)]' \  (4.10)  where Ni is the minimum costing horizon, A'2 is the maximum costing horizon, X(j) is a control weight sequence. Ni is usually set to 1 or to the plant dead-time k if it is known. N2 is typically set to the rise time of the plant. Thus it is called the output horizon. X(j) is usually set to a constant A for simplicity. With Equations 4.7and 4.9, the cost function in Equation 4.10 can be rewritten as: JGPC  = [ G U -I- f - w]^[Gii -H f - w] -I- Au^u 43  (4.11)  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  Minimizing this cost function gives the control signal: ii = [ G ' ^ G + AI] ~ ^ G T ( W -  (4.12)  f)  where w = [w{i + Ni) i(;(t + Ni + 1) • • • w{t + N2)]^ f = [/(t + Ni) / ( t + N i 4-1) • • • /(f+N2)]^ u = [Aw(^) Au{t + 1) • • • Au{t + N2and G is a matrix of dimension (A'2 — A^i + 1) x ^^2 : '9N,-i  •••  go 0  0  •••  (4.13)  l)f 0T  G =  (4.14)  where gi is the value of the step response of the CARIMA model at the ith sampUng interval. 4.3.3 The Control Horizon The real power of GPC approach lies on the assumptions made about the future control actions. Instead of allowing them to be free, the control increments after an interval NU < N2 are assumed to be zero, i.e. Au(t + j - l ) = 0,j>  NU.  (4.15)  The value NU is called control horizon. A large value of NU implies that the control action is potentially more active while a small value implies that the control is more conservative. The introduction of the control horizon leads to a significant reduction of the computation. As ii is of dimension NU, the prediction equation is reduced to y = Giu-|-f where  u = [Au(t)  Au(t+1)  y = [Ay(t-tNi)  (4.16)  ••• Au(t +  Ayit + N^ + I)  NU-l)f  ••• Ay{t + NU)f (4.17)  w = [w{i + A^i) f = [ / ( i + iV:)  «;(^ + A^i + 1) /(f + iVi + l) 44  ...  w{t + NU) ] ^  f{t + NU)Y  Chapter 4:  and  Gi  Coating Weight Control Strategy: Constrained Generalized Predictive Control  go  0  0  51  50  0 is N X NU.  =  (4.18)  50  .9Ni-l  9N2-2  •••  gN2-NUi  The corresponding control law is then given by u = [GIG-L + AI] ~^GJ'(W - f)  (4.19)  If NU=l, which is the case for most simple plant, the calculation of control signals reduces to a scale computation.  4.4 Multivanable GPC 4.4.1 CARIMA Plant Model and Output Prediction A CARIMA model for a n-input m-output system can be described by[35] [36]:  A{q-')y{t) = B{q-')u(t  - 1) + Ce(t)/A  (4.20)  where A and C are diagonal polynomial matrices of dimension n x n, fi is a polynomial matrix of dimension n \ m u is the input vector of dimension n x I y is the output vector of dimension n x I e is a noise vector of dimension n x I This multivariable CARIMA model has all the properties the SISO CARIMA model has and it can represent most multivariable processes. Without destroying the generality of Equation 4.20, C is assumed to be 1 for the simplicity of development. Thus we have:  Ay{t) = Bu(t - 1) + e(t)/A 45  (4.21)  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  Solving the Diophantine equation / = EjAA + q-'^Fj  (4.22)  where / is an identity matrix of dimension n x n, £• is a (j"l)th order polynomial whose coefficients are n x n matrices, F is a Tiath order polynomial whose coefficients are n x n matrices, we have the predicted output y(^ + J) = GjAu{t + i - 1) + Fjy{t) = GjAu(t  + i - 1) + GjAu{t + j-l)  = GjAu{t + j-l)  + Fjy{t)  (4.23)  + f (t + j)  where Gj = EjB, and •  f is the predicted output depending on the known controls and is also called free-response;  •  GjAu{t + j — 1) is the prediction of the output depending on the control signals yet to be determined.  4.4.2 The Multivariable Predictive Control Law The cost function to be minimized by MGPC is JMGPC =  J2 fy(^+•?') - ^(*+i)]^[y(^+j) - w(<+j)] (4.24)  + J2 ^"(* + i - l)^AAu(i -H J - 1) Assuming control horizon is NU, the above cost function can be rewritten as min JMGPC  =  GU-l-f-w  GU-f-f-w  -j-U^AU  (4.25)  The predicted outputs are Y = G U -I- f 46  (4.26)  Chapter 4: Coating Weight Control Strategy: Constrained Generalized Predictive Control  and the control signals are -1.  Au(i) = [/„ 0 • • •] [G^G + A] ' G ( W - f)  (4.27)  where w is the set-point vector, A is the diagonal control weighting matrix as •Ai  (4.28)  A= Am, and U = [Au(f + A^i - 1) Y = [y{t + Ni)  f=[/l(f+l)  Au(/ + A^i)  •••  /\n{t + NU - l)f  y(i + iVi + l) ••• y{t + NU)f Wn{t+l)  ••• Wi{t + NU)  ••• fn{t+l)---  Mt + NU)---  Wn(t + NU)y  (4.29)  fn{t + NU)Y  A = matrix of NU block diagonal A matrix If NU is set to the same value for all channels, the G matrix is of dimension n(A^2 — N\ + l) x (m-NU):  G=  SN,-1  • ••  go  SN,  •  gl  LgNa-l  0 •••  0  go 0 •••  0  0  (4.30)  •••  gN,-NU  where gj is a submatrix of dimension n x m. The elements of gj are the step responses of ith sampling interval of each individual input-output relation. 4.5 Constraints Physical constraints are always present on a real process. Constrained control problems of SISO systems are studied in references [37] and [38]. Constrained multivariable systems are studied in [39] and [40]. The most common constraints on an industrial process are constraints on amplitude and/or the change rate of the manipulated variables. For a SISO system, they could be (4.31)  47  Chapter 4:  Coating Weight Control Strategy: Constrained Generalized Predictive Control  where u = [u{t) u{t+\)---  u{t +  NU-l)Y  u = [Au{t) Au{t + 1) • • • Au{t + NU-  1)Y  (4.32)  Umin , Umax, ^Umiji, and Aumin are of dimension NU X 1 The constraints on the output should be mapped to the constraints on the input Aw, i.e. ymin S 2/ S ymax  (4.33)  Successful control of the output depends heavily upon the exactness of the process model. A poor process model will not produce desired constraints on the output. Constraints on the multivariable systems are handled in the same fashion as for SISO systems. The only difference for a multivariable system is that each polynomial element in a SISO system becomes a matrix and the scalars must be replaced by vectors. The solutions to the constrained GPC can be obtained through various optimization methods. Detailed solutions of constrained SISO and Multivariable GPC will be discussed in Chapter 5 and 6 respectively.  48  Chapter 5:  Machine-Direction Coating Weight Control  Chapter 5 Machine-Direction Coating Weight Control  5.1 Machine-Direction Control Mechanism  Figure 5.1: Mechanical blade loading.  The goal of machine-direction coating weight control is to improve the uniformity of the coating layer in response to machine-direction variations. The MD control on the blade coater is usually carried out by manipulating the global blade load and blade tilt angle. Figure 5.1 demonstrates a typical mechanical blade loading system. At the set-up position, the blade is at a position just in contact with the backing roll. The blade pressure is zero at this position. The angle formed by the tangent line of the backing roll and the blade is called set-up angle. When the blade clamp moves towards the left, the blade is physically bent around a pivot near the support beam. The greater the bend in the blade, the higher the resulting blade loading. A change of the blade load may alter the blade angle. Consequently, another change of the blade load should be made to compensate for the change of the blade angle. 49  Chapter 5:  Machine-Direction Coating Weight Control  The blade tilt angle can be adjusted in the same way as the blade load. The blade can be turned around to achieve different blade tilt angle. Automatic blade tilt angle control can be accomplished by encoders, stepping motors, etc.  In some coater installations, both blade pressure and blade angle controls are available. In this case, the blade pressure is used as a fine adjustment of coating weight while the blade angle is used as a coarse tuning. For instance, as a bevel blade wears, the blade pressure may have to be increased beyond the workable range. However, if the blade angle changes by only a small value, the blade can again work well under a reasonable pressure.  5.2 MD Coating Process Model  Un(t-1)  MD Constraints  MD Controller ,I  MD Estimator  y'(t-i  y'(t-1l ,>  1  y(t)  TS+1  u(t-1) u(t-1)  Figure 5.2: Diagram of adaptive MD coating weight control by blade pressure.  50  Chapter 5:  Machine-Direction Coating Weight Control  Un(t-1) MD Constraints  MD Controller a  MD Estimator  y'(t-i  V'(t-1^ i' / ~ " - \  >•  u(t-1)  1 TS+1  y(t)  u(t-1) Figure 5.3: Diagram of adaptive MD coating weight control by blade tilt angle. As discussed in Chapter 4, the system to be controlled in machine-direction can be considered as a coating process and a data estimator. The coating process is described as a static gain process. The MD and CD data separator can be described as a first-order system corresponding to the time constant of the data estimations. Therefore, as illustrated in Figure 5.2, the plant is a first-order system with a nonlinearly changing static gain. If the blade pressure is the control variable, the static gain of the system is determined by the nonlinear relationship between the blade pressure and coating weight (Figure 5.2) . If the blade angle is used to control the coating weight, the process gain is determined by the relationship between the blade tilt angle and the coating weight (Figure 5.3). In Figures 5.2 and 5.3, r represents the time constant of the first-order system. The system can be modelled as: (1 -f- aq-^)Ay{t)  = bAu{t - 1) + ^(t)  (5.1)  where u is system control variable which represents either the blade pressure or the blade tilt angle, y is the MD coating weight, and ^ is a white noise sequence.  51  Chapter 5:  Machine-Direction Coating Weight Control  The system gain is defined by 6/(1 + a). As discussed in Chapter 4, the time constant of the plant is mainly defined by the time constant of the raw data separator. The model is a linear approximation of the process around operation point.  5.3 Parameter Identification Because of the nonlinear relationships between the coating weight and the blade pressure or the blade angle, and of blade wear, the system modelled in Equation 5.1 has a time-varying incremental gain. To avoid a complex nonlinear control, an adaptive control is employed to control this nonlinear process as a linear system with a time-varying gain. Thus an on-line parameter estimation method, such as recursive least squares (RLS) algorithm, is needed in the control system design to estimate the incremental gain. The basic recursive least squares algorithm[41] is known to have optimal properties when the parameters are time invariant. But the algorithm does not work well to track time-varying parameters since the basic algorithm gain eventually converges to zero. RLS with exponential resetting and forgetting factors is an improved algorithm[42], because it tracks time-varying parameters and has •  exponential resetting and forgetting factors,  •  an upper bound for the estimator gain matrix P, and  •  an upper bound for the inverse of the gain matrix P~^. The exponential forgetting and resetting algorithm (EFRA) may be expressed as: efc = Vk - 4>kh-\ aPk-\4>k ^k = Ok-l + , . ,Tr^ T^k l + cgPk-.cfk^"  where Sk =  ah  is a vector of estimated parameters,  <j)k = [—Ar/(A; — 1) Au(A; — 1)] is a vector of measured input and output data, P is the so-called covariance matrix, 52  (5.2)  Chapter 5:  Machine-Direction Coating Weight Control  •  A is the exponential forgetting factor, and A G [0.9, 0.99],  •  a adjusts the gain of RLS algorithm, typically a G [0.1, 0.5],  •  /3 is a small constant directly related to the minimum eigenvalue of P, typically (5 G [0.1, 0.01], and  •  (5 is a small constant that is inversely related to the minimum eigenvalue of P, typically 6 G [0.1, 0.01]. This improved RLS algorithm ensures the stability of the estimator by frequently resetting the  covariance matrix to a multiple of the identity matrix. The exponential forgetting factor also helps the estimator to converge rapidly when the parameters are time-varying. So this algorithm incorporates both exponential data forgetting, and exponential resetting towards an unprejudiced treatment of new data.  5.4 Controller Design The cost function of the MD control loop can be written as: JGPC  = [Gil + f - w]^[Gu + f - w] + Aii'^u  (5.3)  which gives the control signal a = [G^G + A7]~^G(w - f)  (5.4)  where G,w, f, ii are defined in Chapter 4. Usually only the present control signal u(t) is of concern. For a simple first-order plant without delay, the control horizon is usually defined as 1, i.e. NU=\. But a higher control horizon is also applicable when active control action is desired. When NU=l, the calculation of the control law is reduced to a scalar computation.  5.5 Constraints for MD Control Machine-direction control is carried out by manipulating the blade pressure F or the blade tilt angle /3. In either case, there are several mechanical constraints on the adjustments. These constraints will be described in the following sections. 53  Chapter 5:  Machine-Direction Coating Weight Control  5.5.1 Amplitude Constraints Every blade has a load limitation. The global blade pressure is adjusted by bending the blade, and the position of the blade clamps determine the blade load. Physical limits are usually set on the blade position in order to ensure a stable operation. The tilt angle is also usually limited to a certain range. The constraints are usually set up for the blade angle 7. For example, if the blade angle can only change from 20° to 60°, given the blade bevel angle as 45°, then the tilt angle is limited to 0° to 15°. A negative tilt angle means the blade works on its heel. This condition is not desirable because it yields poor coating surface. The amplitude constraints can be described by: ^min S '^ S ^max  (5.5)  where u = [u{t)  u{t+l)  •••  u{t + NU-1)  f  (5.6)  ^min, and Umax are the lower and upper limits of the control signals respectively. They have the same dimension as u, i.e. NU x 1. 5.5.2 Rate Constraints The blade position and the blade angle adjustment are time consuming, depending on the mechanical driving system. There are other cases of rate limits that are not related to the speed limit of the driving system, for example, when a grade change is carried out on line, a smooth transition is desired because a sudden change may break the sheet. The rate constraint can be interpreted as a constraint on the input signal rate of change: -AUmax  < U < AUmax  (5.7)  where u = [Au{t)  Au{t+1)  •••  Au{t + NU-l)Y'  AvLmax is the upper limit of the change rate and is of dimension NU x 1. 54  (5.8)  Chapter 5:  Machine-Direction Coating Weight Control  5.5.3 Output Variation Constraints The constraints on the output variations can be mapped to the constraints on the input signals. Because y = Giii + f  (5.9)  where y=[y(< + l)  y{t + 2)  •••  yit + NU)f  (5.10)  The constraints on the predicted output J max  (5.11) may be rewritten as ymm - f < G l U < ymaa; - f  (5.12)  where y^m and ymax are the lower and upper bounds of the output variations. Because the estimated outputs are dependent on the model and the parameter estimation, the successful control of the output depends greatly on the exactness of the process modelling and parameter estimation. 5.5.4 Optimal Solution to Constrained SISO GPC If none of the above mentioned constraints is violated, the control signals derived from the GPC algorithm are the optimal solution. When one or more above constraints is violated, these solutions are no longer feasible. The optimal solutions then have to be found under constraints. The cost function of GPC could be rewritten as min JGPC = u ^ ( G ^ G + A/)u + 2(f - w)^Gii + (f - w)^(f - w)  (5.13)  Which is a quadratic object function with respect to ii. The control signals can be obtained by solving the quadratic optimization problem with constraints. The Lagrangian multiplier method[43] provides a means to solve the quadratic problem: mm < -x^Hx  + (Fx > subject to Ax < b,  X  55  (5.14)  Chapter 5:  Machine-Direction Coating Weight Control  where the Hessian matrix H and vector c are the set of coefficients of the quadratic objective function. The matrix A and vector b are the coefficients of the linear constraints. The vector x is a set of independent variables. Comparing the GPC cost function in Equation 5.13 with Equation 5.14, we have H = G ^ G + XI (5.15) c = G^(f - w) The GPC cost function is to be minimized subject to •  A,  -Ai  ' ^max - A2u{t  - 1)  - U m m + A2u{t  - 1)  INUXNU  (5.16)  VL < -INUXNU  Gi  Ymin ~ *  -Gi  Jmax ~r  where  A, =  1  0  0  1  1  0  1  1  1  1  1  1  A2 = [l  1  (5.17)  •••  •••  J  NUxNU  (5-18)  IYNUXI  and INU-XNU is an identity matrix of dimension NU x NU. A solution can be obtained through the Optimization Toolbox in Matlab. A detailed solution to a quadratic optimization problem using the Lagrangian Multiplier method is provided in Appendix A.  5.6 Simulations of Adaptive MD Coating Weight Control Machine-direction coating is simulated by the process described in Figures 5.2 and 5.3. The process is simulated by: (1 + aq-^)y(t)  = (1 + a)f{u{t - 1)) + d{t) + (l + 56  cq-^)v{t)//\  (5.19)  Chapter 5:  Machine-Direction Coating Weight Control  where •  a = —e~'^'l'^, T, is the MD sampUng rate in terms of one scan, r is the time constant of the process. Based on the discussion in Chapter 3, the time constant is defined as three times of the sampUng interval in the simulation. Therefore the pole of the open-loop plant is 0.7.  •  d{t) represents the unmeasurable load disturbance of the process. The load disturbances of Ig/m?' are added between scan 140 and 170, and between scan 240 and 270.  •  As c is defined as -0.5, the system is actually under the disturbance of coloured noise. f{u{t — 1)) describes a nonlinear relationship between coating weight and the control variable  u. The control variable could be either the blade pressure or the blade tilt angle. Simulation results using blade pressure control are demonstrated in this chapter. Adaptive coating weight control by the blade tilt angle can be carried out in the similar way. In the simulation, the nonlinear relationship between coating weight and the blade pressure is determined by the simulation conditions given in Table 2.1. The control variable is defined as the blade load which corresponds to the value of blade pressure in kN/m, i.e. when the blade load is 5, the blade pressure is 5 kN/m. As adaptive control is applied, parameter estimation is carried out on-line by RLS with a forgetting factor A=0.99, and resetting factors a = 0.3, (5 = 0.01, and 6 = 0.01. The evaluation of the coating process control is performed with an adaptive SISO GPC whose tuning parameters are set to NU = l,Ni chosen as T = 1 -  = 1, N2 = 3, and the control weighting A=0.01. The design polynomial T is 0.8q-^.  Because the coating process is always bounded in practice, all simulations demonstrated in this chapter are under the amplitude constraints. The amplitude constraints are 10 and 0.6 for the upper and lower amplitude bounds respectively. Figure 5.4 shows the adaptive GPC coating weight control in machine-direction under amplitude constraints. Figure 5.4.a shows the changes of the coating weight set-point and a very good control tracking of the output. The controller rejects the load disturbances very effectively. The controller 57  Chapter 5:  Machine-Direction Coating Weight Control  (a) Set-point Change and Output  50  250  300  350  250  300  350  1  1  150 200 Time (scan)  100  (b) Control signal 15 •D (C O -J  constrained  10  unconstrained  CD  •D  CO  50  100  150  200  Time (scan) (c) parameter estimation 0  fl  ,  1  '  1  I  1-0.5  I -,7  1  f'^"  ^  f,-A  I  '  —  J  -  Estimated a  (0 Q.  — -1.5.  Estimated b 1  1  50  100  1  1  150 200 Time (scan)  1  1  250  300  Figure 5.4: Machine-direction adaptive GPC with amplitude constraints. 58  350  Chapter 5:  Machine-Direction Coating Weight Control  (a) Set-point Change and Output  50  100  150 200 Time (scan)  250  300  350  300  350  300  350  (b) Control signal constrained unconstrained  150 200 Time (scan)  (c) parameter estimation  -1.5i  Estimated b I 50 100 -J  L  150 200 Time (scan)  250  Figure 5.5: Machine-direction adaptive GPC with rate constraints. 59  Chapter 5:  Machine-Direction Coating Weight Control  (a) Set-point Change and Output  150 200 Time (scan)  350  (b) Control signal constrained unconstrained  150 200 Time (scan)  350  (c) parameter estimation T  CD  1  1  1  1  1  1  1  > - ir  —*y  /  1-0.5  J'  CD  +-* 0)  /  i -1  ^z  Estimated a  CO  Q.  Estimated b  -1.5  1  1  50  100  1  150 200 Time (scan)  1  1  250  300  Figure 5.6: Machine-direction adaptive GPC with output constraints. 60  350  Chapter 5:  Machine-Direction Coating Weight Control  (a) Set-point Change and Output  50  100  150 200 Time (scan)  250  300  350  250  300  350  1  I  (b) Control signal  100  150 200 Time (scan)  (c) parameter estimation 1  1  1  1  )  CD ' - - -^.VN . .  "(0-0.5 >  1  '" -  1  »  £5  CO  0--1.5  —  ^Vv—-s^  /y  -^—-^-x  ^  ,  -  A ^  Estimated a  1  /  J  —  -  Estimated b 1  1  50  100  1  1  150 200 Time (scan)  1  1  250  300  Figure 5.7: Machine-direction adaptive GPC with model mismatch. 61  350  Chapter 5:  Machine-Direction Coating Weight Control  rejects the second load disturbance between scan 240 and 270 with greater effort because the system gain is greater when coating weight is lighter. In Figure 5.4.b, the control solution given by unconstrained GPC is indicated by the dashed line. The solid line represents the optimal solution of constrained GPC, which is used as control signal to the process in the simulation. It can be seen that the control signals are bounded in amplitude in this case. When the grade changes, the active control action is constrained. The estimated pole is biased from its true value. The estimated open-loop model is unstable at this moment. However, the closed-loop is stable. After enough excitation is obtained from the input signal, the parameter estimation converges again. Estimated parameters are shown in Figure 5.4.c. The estimated pole a is biased due to the disturbance of the coloured noise. The changing static gain of the process is reflected in the estimated parameter b when the grade changes. Since the static gain of the system is determined by the nonlinear relationship between the coating weight and the control variable, the gain of the system changes quickly under active control actions. Therefore it is hard to achieve fast parameter convergence using the RLS estimation method. This increases the bias of the estimation. However, with the data filtered by the low pass filter l/T, the estimated parameters are close to their true values. Nevertheless, with the biased parameters, the system performance is stiU good with the help of the features of GPC. Controller rate constraints are added to the simulation demonstrated in Figure 5.5. The rate constraints can be written as  - 0 . 8 < Au{t) < 0.8  (5.20)  The control signal in Figure 5.5.b indicates that the control is less oscillatory than that in Figure 5.4. A smoother transition during the grade changes is evident, which is desirable in industrial processes. The process output also shows improved tracking with less overshoot. In fact, smooth output can be realized by GPC with a greater A^2- The prediction horizon A^2 is set to small value in order to demonstrate the effects of the constraints of the control signals. 62  Chapter 5:  Machine-Direction Coating Weight Control  Figure 5.6 shows a simulation case where the control signal is constrained in amplitude and rate, and the process output is also constrained as well, i.e. 0.6 < u{t) < 10 - 0.8 < Au(f) < 0.8  (5.21)  5 < y(i + l) < 21 Comparing Figure 5.6 with Figure 5.5, the maximum output is 22.6 g/m?' in Figure 5.5 and 22.1 g/m? in Figure 5.6. The success of the control depends greatly on the exactness of the modelling and parameter estimation. Usually when the control is active, the output offsets violate the output constraints. However the estimated parameters are usually biased from the true values because of the nonlinearity of the system gain. Therefore the output constraints do not always work well. Figure 5.7 demonstrates the system dynamics and control activity in presence of a model mismatch. The plant has a fractional delay of O.ST^. The model used in the estimation and the control is still a first-order system without delay. The amplitude and rate constraints on the control signals are the same as those indicated in Figure 5.5. It seems that the output follows the set-point changes well. However, compared with Figure 5.5, the estimated parameters are more biased from the true values and the output contains more variation when the set-point changes.  5.7 Conclusions The machine-direction coating weight control is modelled as a first-order system with a timevarying gain which is defined by the nonlinear relationship between coating weight and the control variable. The control variable can be either the blade pressure or the blade tilt angle. The timevarying parameters are estimated by the recursive least squares method with exponential forgetting and resetting factors. As most industrial processes have physical constraints, the adaptive GPC is designed with constraints on its control signals. If the constraints are not violated, the control signal calculated from the unconstrained GPC is applied to the system. When any constraint is violated, an optimal solution is obtained from the constrained GPC using the Lagrangian multiplier method given by Appendix A. 63  Chapter 5:  Machine-Direction Coating Weight Control  Simulations show the constrained adaptive GPC copes well with coating weight set-point changes. The controller also rejects load disturbances very effectively. With control rate constraints, the control signal is damped and the output offset is reduced. Constraints on the system output can be mapped to the control signals. Thus the output constraints are realized by limiting the control signals. The successful handling of the constraint on the output relies on the exactness of the process model and thus on the good behaviour of the parameter estimation.  64  Chapter 6:  Cross Machine Coating Weight Control  Chapter 6 Cross Machine Coating Weight Control The goal of cross-direction control is to maintain a uniform profile of coating weight. Successful CD coating weight control helps the quality of the coated paper and consequently improves the printing quality. A poor cross-direction control may cause uneven coating and lead to sheet breaks.  6.1 CD Control Mechanism  Backing Roll  Figure 6.1: Structure of cross-machine profiler. CD control is carried out by actuators across the sheet. They are typically spaced at intervals of 50mm to 100mm across the blade supporting beam. A simplified structure showing a CD coating weight profiler is demonstrated in Figure 6.1. Pressure on the blade is adjusted by varying the screw position. The screw position determines the air pressure in the profile tube, and consequently the local pressure on the blade. Although coating weight profilers may have different mechanical structures, in all cases the pressure is manipulated by the screw position. In order to control the CD profile automatically, the actuators must be automated. The actuators may be automated by: electronic motors, thermalrods, hydraulic motors, thermal-hydraulic systems, and hydraulic stepper drives. 65  Chapter 6:  Cross Machine Coating Weight Control  6.2 CD Coating Process Model  TS+1  ''''"^"^^ffc/^m; '°  CD Estimator  CD Constraints  CD Controller  Figure 6.2: Block diagram of CD adaptive coating weight control. Just as for machine-direction coating weight control, the cross-direction coating process can be described as a first-order system with a nonlinear gain function. However, as shown in Figure 6.2, the control system must be designed for a multivariable process. At each CD position, the static gain of the system is determined by the slope of the nonlinear relationship between the blade pressure and coating weight. Thus the coating process for cross-direction control can be modelled as: Ay{t) = Bu{t - 1) + e(<)/A  (6.1)  where A is a diagonal first-order polynomial matrix of dimension n x n:  \l-aiq-^ 0  0  •••  1 - 02?-^  0 :  A =  (6.2) :  0  •••  •••  0  0  l-a„g-l.  where n is the number of actuators a coater has in its cross-direction. 66  Chapter 6: Cross Machine Coating Weight Control  B is the interaction band matrix of actuators: r^n bi2 0 0 621  ^22 ^23 0  0  ^32 ^33  &34  (6.3)  B =  0  •••  K(n-l)  Knlnxn  e ( 0 = [ei(/)  62(0  •••  en{t)f  y{t) = [yi{t)  y2{t)  •••  yn{t)f  U{t) = [ui(t)  U2(t)  •••  Unit)f  (6.4)  e is a vector composed of n uncorrelated noise variables, u and y are the input and output vectors respectively. The output is the CD profile decoupled from the raw measurement data. The input is defined as the actuator screw position. The screw position of the actuator is directly related to the blade pressure and is the easiest to obtain. In practice, the number of measurement points is usually greater than the number of actuators. For control and display purposes, the measurement data are often mapped to the corresponding actuator zones. There are several mapping methods available. The simplest way is to take the average of the measurements in the zone. After mapping, we will have the same number of actuators as that of mapped measurement outputs. The interaction matrix shown in Equation 6.3 is band-diagonal because one actuator only influences one adjacent actuator on each side, as indicated in Chapter 3. If all actuators have the same mechanical structure and have the same dynamic response, then the interaction matrix can be written as: bo  bi  bi  bo 61 0  0  61 bo  0  0 0  61  B =  (6.5)  ^1  67  ^O^nxn  Chapter 6:  Cross Machine Coating Weight Control  where fei is a fraction offeo-For example, 61 = 0.36oBut the interaction matrix described by Equation 6.3 represents the interactions among the CD actuators more precisely. End effects at the edges of the sheet can be represented since, unlike other actuators, the actuators at both ends of the sheet receive influence from only one side. Another advantage to Equation 6.3 is related to the blade wear condition and its initial position. Uneven blade wear causes the difference in local actuator response. Unfortunately, the blade wear is always too serious to neglect. Different initial positions of the actuators imply that they give different initial local blade pressures. According to the nonlinear curves in Figure 2.11, the coating weight change rate is smaller under a high blade pressure than that under a low blade pressure so that at a high blade pressure the amplitude of actuator response will be less than that at a low blade pressure. The condition number of the interaction matrix is another issue to be considered in control system design. In GPC controller design, the interaction matrix has to be inverted in order to obtain the control signals. When the matrix is poorly scaled or singular, the inversion cannot be carried out. One way to prevent a badly scaled or singular interaction matrix is to add a zero input element and a column of 1 in the interaction matrix as following: -1 -aiq-'^  0  0  1 - 0 2 5- 1  0  1  yiii) y2{t)  1 0  0  0  1 - ttnq ^  Ij 0  bn  612  0  0  0  hi  &22  ^23  0  0  632  633  634  1  \Mi)'  Mt)'  U2{t)  62(0  + u„(t)  (6.6)  /A e„(<)  0 I 0 K(n-1) bnn 1 Then the interaction matrix is always of full rank and singularity should be no problem. 0  In summary, the CD coating process can be modelled as a first-order system without delay. Its interaction matrix is a band diagonal matrix. 68  Chapter 6: Cross Machine Coating Weight Control  6.3 Parameter Identification for Multivariable System Parameter identification for multivariable systems is much more complicated and time consuming than that for SISO systems. On-line parameter identification is very necessary but difficult for crossdirection coating weight control. Because a coater may have 80 actuators in cross-direction. Then the interaction matrix is 80 by 80 and it consists 6400 elements. It is impossible to estimate all these elements on line with an acceptable speed. However, for a band interaction matrix, only those nonzero elements need to be estimated and it can be done on line. To avoid getting into the complication of general multivariable system identification, a solution to multivariable system identification for this type of process by a modified recursive least squares method is proposed as follows. For each row of the matrix in Equation 6.1, we have (1 - a i 9 - i ) j / i ( i ) = hiui{t (1 - aiq~'^)yi{t) (1 - anq~^)yn{t)  -l)  + bi2U2{t - 1)  = fe(j_i),Uj_i(< - 1) -|- biiUi{t - 1) + fei(j+l)"*+l(* " 1) = \n-l)nUn-l{t  (6-7)  " 1) + KnUn{t " 1)  where 2 < i < n — \. Each equation above is a multi-input single-output system. Providing the input series are independent from each other, we can apply RLS identification method on each multi-input single-output system independently. For example, if RLS with exponential resetting and forgetting is used, the estimator for the system described by the first row is:  tik = t>k-l + , . , r „  —^k  l + 4>lPk-i^k^^  p _ l p A  0'Pk-l<l>k<PkPk-l l + ((>j^Pk-i<^k  r.j  (6.8) rp2  where Ok = [ai  in  hif  (6.9)  and h = [-yi{k - I)  u^{k-l) 69  U2{k-l)f  (6.10)  Chapter 6:  Cross Machine Coating Weight Control  For rows between 2 and n-1, the identifiers are similar to the first one. However the number of parameters to be estimated is different, i.e. Ok = [a-i b^i-i)i  hi  (6.11)  bi(i+l) ]  and ^fc = [ - y i ( ^ - l )  Uj-i(A;-l)  Ui{k-1)  Ui+i{k-l)]^  (6.12)  For the last row, the estimator is in the same fashion as the first estimator but ^  Ok = [«re  ^  ^  \n-l)n  Kn ]  rp  (6.13)  and <Pk = [-yn{k - l)  w„_i(A;-l)  Unik-1)]^  (6.14)  Because the independency of the input signals is a necessary condition for this method, it is very important to have enough information from the input signals.  Cross Machine Actuators  y  1  2  3  4  Coating Weiglit Measurements Figure 6.3: Cross-machine correlations among actuators and measurements. The above identification method can be interpreted geometrically as demonstrated in Figure 6.3. Any inner mapped measurement point receives the influences not only from its corresponding actuator but also from the two adjacent actuators. At each end of the sheet, the measurement point receives influence from only one side, besides the influence from its corresponding actuator. When 70  Chapter 6:  Cross Machine Coating Weight Control  the input signals are independent series, the interaction coefficients can be estimated recursively with reasonable speed. When the responses to all the actuators are considered identical, the interaction matrix is similar to that in Equation 6.5. In this case, only one response needs to be measured in order to identify the two unknown parameters. Alternatively some points are estimated and the average values from those estimations will be used as the system parameters.  6.4 CD Control Law Design As derived in Chapter 4, the cost function for the multivariable system described in Equation 6.1 is: min JMGPC  =  GU+f-w  G U + f - w + U-'AU  (6.15)  and the control signals are Au(<) = [7„0 •••][G^G + A]  -1,  'G(w-f)  (6.16)  Au„(f)]^  (6.17)  where Au(^) = [Aui(<)  Au2(0  •••  As NU=\ is usually enough for a practical plant, the controller for the CD coating weight control is based on NU=1. So U = [Aui{t)  Au2{t)  w=[wi{t+l)  •••  W2it + 1)  f=[/l(^+l)  f2{t+l)  Aun{t)f •••  •••  Wn(t+l)f  (6.18)  fn{t+l)f  and A = A where fAi A =  (6.19) An  71  Chapter 6: Cross Machine Coating Weight Control  6.5 Constraints for CD Control 6.5.1 Actuator Adjustment Amplitude Constraints Actuators have their physical limits. For example, the stroke of a coating weight profiler screw is normally a few millimeters. Control signals must thus have upper and lower limits as: "mm < u(<) < Umax  (6.20)  where U{t) = [ui{t)  U2{t) •••  Ur,{t)f  (6.21)  and VLmin and Umax are the lower and upper bounds of the screw positions respectively. They are of the same dimensions as u(0, i.e. « x 1. 6.5.2 Actuator Adjustment Rate Constraints The drive system for actuator adjustment always has a time constraint. If this time constraint is significant compared to the control period, rate constraints on the control action arise. Rate constraints may also be needed to avoid abrupt coating weight change in the process. As with to MD control, rate constraints on CD actuator adjustment can be expressed as: -^Mmax < Au(<) < Au^ax  (6.22)  where Au(^) = [Aui(^)  Au2(i)  •••  Au„(<)]^  (6.23)  and AUfftax is a vector consisting of limitations on actuator adjustment rate. It has the dimension of « x 1. 6.5.3 Adjacent Actuator Relative Position Limits The coating blade is usually very flexible and deflects when adjacent actuators apply different forces on it. To prevent the blade from being damaged, the position difference between adjacent actuators must be limited, i.e. -^Umax < Ui+l{t) - Ui{t) < 72  (6.24)  Chapter 6:  Cross Machine Coating Weight Control  where Su^ax is the maximum position difference between adjacent actuators. This constraint can also be expressed as (6.25)  max  where  ' 1  -1  0  ;  0  1  -1  j  0  (6.26)  p—  0  1  -1  0  1  nxn  6^max consists of the position constraints for adjacent actuators. 6.5.4 Bending Moment Limitation To prevent plastic deformation of the blade, the bending moment of the blade should be limited. The bending moment of the blade beam is proportional to the second moment of the blade setting. Therefore the bending limits on the blade can be written as: •Ubmax < Ui-i(t)  - 2ui{t)  + Ui+i{t)  < Ubma  (6.27)  where Uhmax is the allowable bending limit. With all the actuators taken into consideration, the constraints can be written in a matrix form -Ubmax  where  < AbU(t)  - 1 1 0  0  0  1 - 2 1 0 0 Ab =  0  (6.28)  < Ubn  0  1 - 2 1 0  1 - 2  1  1 0  (6.29)  -2  1  0 1 -1 The elements of Ubmax are the bending moment limits for the blade. 73  Chapter 6:  Cross Machine Coating Weight Control  6.5.5 Output Variation Constraints In the same fashion as MD control, the constraints on the output can be mapped to the control signals. Since the predicted system output is: Y = GU + f  (6.30)  When NU=l, Y = yW = bi(0  y2(t)  •••  ynit)f  (6.31)  Therefore the constraints on the outputs Ymin < Y < Ymax  (6.32)  Ymin - f < G U < y „ a x " f  (6.33)  can be interpreted to  where Ymin and ymax are the lower and upper limits of the coating weight outputs respectively. The controllability of the outputs obviously depends on the exactness of the process model and the parameter estimation. 6.5.6 Optimal Solution of Constrained Multivariable GPC In case no constraint is violated, the solution to the unconstrained GPC is both feasible and optimal. When one of the constraints is violated, this optimal solution is no longer feasible. The problem is then to minimize the cost function  min JMGPC  =  GU + f - w  GU+f-w  + U^AU  (6.34)  subject to constraints on U. The above cost function can be rewritten in a standard quadratic form as: lamJMGPC = U^(G^G + A / ) U + 2(f - w)^GU + (f - w)^(f - w) 74  (6.35)  Chapter 6:  Cross Machine Coating Weight Control  Since NU is considered to be 1 in the CD coating weight control system, we have U =  Au(t).  Therefore the above discussed constraints on Au(t) can be transferred to U. These constraints are summarized as: Umax - U(< - 1)  •'raXra ~J-ny.n  -Umax  + u(< - 1)  J-nycn ~InXn  A,  ^^max  - ApU(t  - 1)  U<  (6.36) - ^ U m a x + ApU(t  - 1)  Ab  Ubmax - AbU{t - 1)  -Ab  - U b m a x + A u ( « - 1)  G  Ymax ~ I  -G  ~ymin  "T I  The constrained quadratic problem can be solved by Lagrangian Multiplier method as provided in Appendix A. Solutions were obtained using the Optimization Toolbox in Matlab.  6.6 CD Coating Weight Control Simulations The cross-direction coating process is simulated as a first-order multivariable system with a nonlinear relationship between the input and the output variables. The final coating weight at each actuator zone is determined by the relationship between coating weight and the local blade pressure, and the interactions of the actuators. Since the local blade pressure is determined by the screw position of the actuator, the screw position is defined as the control variable corresponding to the blade pressure. The plant to be controlled can be described by: (1 -I- aq-^)y{t)  = {l + a)*B*  F{u(t - 1)) -|- e ( 0  (6.37)  where (6.38)  75  Chapter 6:  Cross Machine Coating Weight Control  y is the CD profile of coating weight, u contains the actuator screw positions, n is the number of actuators across the sheet, n is 20 in the simulation. e is a n X 1 vector containing white noise. a = -e~^»/^. Based on the discussions in Chapter 3, the time constant of a CD data estimator is usually several sampUng periods. Taking the integral of the coating process and the CD profile estimator as the system to be controlled, its time constant r is defined as three times of sampling period T, which is one scan. B is the interaction matrix of actuators. To avoid unbalanced edge effects, in the simulation B is defined as:  B =  1  0.6  0  0  •••  0.3  1  0.3  0  •••  0  0.3  1  0.3  0 •  0-  (6.39)  0.6  1 -J 20x20  i^ is the nonlinear function of the screw position at each actuator zone as indicated in Figure 6.2. The parameters of the model described in Equation 6.1 are estimated on-line using the method  discussed in the third section of this chapter.  The forgetting and resetting factors are set to  the same values for all estimators. The forgetting factor is 0.97, and the resetting factors are a = 0.3, /3 = 0.01, and S = 0.01. Controller tuning parameters are A^i = 1, A^2 = 3, and NU = 1. The control weighting factor is set to the same value for every actuator, i.e. A=0.01. In the simulation, the nonlinear function F is determined by the relationship between the blade pressure and the coating weight under the simulation conditions given in Table 2.1. The control variable, which is the actuator screw position, is assumed to be proportional to the local blade pressure, i.e. a screw position of 5 refers to a local blade pressure of 5 kN/m. The coating process simulated in this chapter has 20 CD profilers. Each is assumed to have the same structure, physical constraints and dynamic responses. Each profiler has a stroke of 10. 76  Chapter 6:  Cross Machine Coating Weight Control  The control variables is limited to a range of screw position from 0.8 to 10. To help the estimated parameters to converge faster at the start-up, the parameter estimators work under open-loop with pseudo-random binary input signals till scan number 7. The closed-loop simulation starts at scan 11 and no more external excitations are added to the plant thereafter. All signals passed to the parameter estimators are filtered by low-pass filters. The transfer function of an individual filter can be written as: F(s) = —^—T ^^ 0.3s-M  (6.40)  We can see that the time constant of the coating process is ten times that of the filter. Therefore the filter can filter out the unwanted high frequency noise and still keep the dynamics of the plant. The controller is turned on at scan number 8. The target coating weight is 14 g/m?' from the beginning to the 40th scan. From scan 41 to 100, coating weight set-point changes to 20 g/m?. Between scan 60 and 80, a load disturbance of 1 g/m?, covering actuator zones 9, 10 and 11, is introduced to the process. With an adaptive multivariable GPC, control performances are evaluated under various constraints. Figure 6.4 shows that the system output converges rapidly to the target coating weight at the start-up. The control variables or rather the CD profiler screw positions are bounded at the start-up to avoid extreme control action beyond the physical limits of the actuators. When a grade change occurs at scan number 40, the output follows the coating weight change closely. With a small prediction horizon iV2 value, offsets can be seen when set-point change. The control is active to reinforce the convergence of the parameter estimations at the grade change. CD profiles after the grade change show that the controller rejects the load disturbance successfully. Apart from the peak and the valley at the beginning and the end of the load disturbance, the coating weight profile quickly converges to the set-point. Control signals respond to the load disturbance in the same direction, i.e. when the load disturbance arises the actuators in the area press more towards the sheet in order to reduce the coating weight. Because of the interactions of the actuators, not only the actuators in the corresponding area but also those in the neighboring zones have to work harder. That is why we see a wider range of actuator moves and the profile distortions. 77  Chapter 6:  Cross Machine Coating Weight Control  System Output ''~~~~~  fScan;  100  Control signal --'~~:^~  15 10 ^ o ^ e 100  Figure 6,4: CD coating weight control with amplitude 78  constraints.  Chapter 6:  Cross Machine Coating Weight Control  System Output ''~~-~.  100  Control Signal \/AA '  \ 1  \\  1 1 1  r  ~-  1 1 1 1  '^ (Scan^  100  Figure 6.5: CD coating weight control with actuator rate constraints. 79  -J I I I  Screw Position OO  N3  O)  CO  _^Coating Weight (g/m^)  0\  n o o O C  eg  2 cfo' B" «-* O o B  o 00  o c  o n o >a o  I p\ o^  o a  o  o s 3  I  I  Screw Position oo  Coating Weiglit (g/m^) ;=i 7^ z^ N> oo  I* n o  o  o  5'  o3" o o 3  O  cr  s Q.  3' OQ 3 o  I p\  O 3 on  I5  J7  f  Chapter 6:  Cross Machine Coating Weight Control  Besides the amplitude constraints on the control signal. Figure 6.5 shows a case in which change rate constraints are added to the control action. The rate constraints on the control signal are written as: - 1 < Aui{t) < 1, where l<i<  20.  (6.41)  In this case, the control signal cannot move at the optimal rate, therefore the transition is slower and smoother. Compared to Figure 6.4, the output converges to the set-point more slowly but the offset of the output at the grade change disappears. It also can be seen from the actuator screw positions that the blade retrieves a flat profile more slowly than the previous case. Figure 6.6 shows the system performance when the constraints on the position difference of the adjacent actuators are added to the previous two constraints. These constraints can be written as: - 3 < ui+i{t) - ui{t) < 3, where 1 < « < 20.  (6.42)  With the relative position constraints, the adjacent actuators are bound together. When the disturbance arises, the actuators cannot makes the optimal move because of the adjacent actuator position. Therefore the coating weight profile loses its smoothness due to the insufficient control effort. Figure 6.7 demonstrates a case when the constraints on the blade bending moment are further added to the previous constraints. The bending moment constraints are expressed as: - 3 < Ui_i{t) - 2ui{t) + ui+i(i) < 3, where l<i<  20.  (6.43)  These constraints should improve the blade profile and protect the blade from being over bent. It is obvious that during the course of the load disturbance, the blade profile is flatter while the coating weight profile displays more offset.  6.7 Conclusions Cross-direction coating weight control is carried out by the coating weight profilers across the sheet. The screw position of the profiler determines the local blade pressure and further the coating weight in the corresponding area. Thus the CD coating process is modelled as a first-order system with a nonlinear gain, where the gain is determined by the relationship between the actuator screw 82  Chapter 6: Cross Machine Coating Weight Control  position and the coating weight. Based on the interaction of CD actuators, the interaction matrix of the first-order model is defined as a diagonal band matrix. As the system can be considered as a time-varying linear system, the recursive least squares method is applicable in the parameter estimation. Improved multi-input single-output RLS estimators are developed to carry out the parameter estimation. The advantages of this method is that it avoids the complexity of multivariable identification and it also reduces a great deal of computation. Optimal solutions of constrained multivariable GPC can be obtained by Lagrangian Multiplier method as given in Appendix A. Simulations show that the adaptive controlled system follows the grade change closely and rejects the load disturbance well. With amplitude constraints on the control variable, the blade is protected from damage. The change rate constraints make the transitions of the output smoother and slower. Constraints on the position difference of adjacent actuators and bending moment constraints improve the blade profile, however, the smoothness of the coating weight profile is sacrificed in the same process. Constraints on the output can be mapped to the control signals and the success of the control depends greatly on the behaviour of the parameter estimators and the accuracy of the model.  83  Chapter 7: Conclusions  Chapter 7 Conclusions Weight control for the paper coating process is of great importance. Good control of coating weight can improve not only the runnability of both the coating and the printing processes, but also the quality of coated paper at reduced cost. The ideas and the results presented in the thesis are aimed at achieving an automatic control of paper coating weight. They are summarized as follows. In the bevelled-blade coating process, the thickness of the coating layer (and so coating weight), is influenced by the equilibrium of forces acting on the blade tip. The forces are the mechanical force supplied by the actuators and the hydrodynamic and dynamic forces generated by the coating colour flow. Thus coating weight is determined by coater speed, blade pressure, blade geometry, coating colour viscosity and density, and the speed at which colour is applied on the base paper. Simulations reveal the relationships between coating weight and the factors affecting coating weight development. Among the factors influencing coating weight, the blade pressure and the blade tilt angle are most likely to be the manipulating variables for coating weight control. Any variations of the factors influencing coating weight will introduce variations in the final coating weight.  The variations are classified as machine-direction, cross-direction and random  variations. The goal of coating weight control is to minimize the variations of coating weight as much as possible. In order to control coating weight independently in machine and cross-directions, measured scanning data must be decoupled. Among many separation methods, multi-scan trending and EIMC are used as effective means. Several mill trials were carried out to investigate the dynamics of the coating process and the correlations of responses to CD actuators. Observations and data analysis show that the coating process has fast dynamics and its time constant is very short compared to the scan period. Bump tests on the CD profilers also show that a CD coating weight actuator usually affects its adjacent actuator zones. Significant reduction of variations can be achieved with the automation of CD actuators and good CD control. MD and CD coating weight controls are developed independently. Due to the flexibility of Generalized Predictive Control, both MD and CD control loops are designed using constrained GPC. 84  Chapter 7: Conclusions  The machine-direction coating weight process is modelled as a first-order system with a timevarying gain which is defined by the nonlinear relationship between coating weight and the control variable. The control variable can be either the blade pressure or the blade tilt angle. The timevarying parameters are estimated by the recursive least-squares method with exponential forgetting and resetting factors. To deal with industrial processes subject to complex disturbances, a design polynomial Tis introduced to improve the behaviour of the parameter estimator and the noise rejection of the controller. Considering the constraints of the industrial installations, amplitude and change rate constraints on the control signals are handled by the software. Constraints on the output can be also realized by mapping the constraints to the control variable, although good results can be achieved only if the process model and estimated parameters are precise. Optimal solutions are obtained for the constrained GPC using the Lagrangian multiplier method. Simulations show that the MD controller tracks grade changes and rejects the load disturbances well. The cross-direction coating process is modelled as a first-order system with a nonlinear gain, where the gain is determined by the relationship between the local blade pressure and the coating weight. Based on the interaction of CD actuators, the interaction matrix of the first-order model is defined as a band-diagonal matrix. Multi-input, single-output RLS estimators are developed to carry out the parameter estimation. The advantages of this method is that it avoids the complexity of multivariable identification and also reduces a great deal of computation. The optimal solutions to constrained multivariable GPC can be obtained by the Lagrangian multiplier method. Simulations show that the adaptive controlled system follows the grade change closely and rejects the load disturbance well. With amplitude constraints of the control variable, the blade is protected from permanent damage. The change rate constraints make the transients of the output changes smoother and slower. Constraints on the position difference of adjacent actuators and the bending moment improve the blade profile. However, the coating weight profile smoothness deteriorates. Constraints on the output can be mapped to the control signals and the success of the control depends greatly on the behaviour of the parameter estimators and the accuracy of the model. As either the blade pressure or the blade tilt angle is considered as the control variable for MD coating weight control in this thesis, more variables can be involved such as coater speed, and colour 85  Chapter 7:  Conclusions  viscosity. Then MD control system will be a complex and multivariable system. However, it will better control the process by involving all elements which may generate coating weight variations. The studies of the bevelled-blade coating process in this thesis can be extended to other types of coating processes in the future. The core of the work would be modelling the coating process, such as bent-blade coating. Once the process model is obtained, the adaptive control algorithm developed in this research could be applied to most linear or nonlinear processes without much difficulty. Improvement can also be carried out on the parameter estimators. Usually the good behaviour of an adaptive control is limited to slow varying systems because the recursive least-squares method does not work well with fast varying systems. In the coating process, fast disturbances are always present and push the output away from the set-point. The static gain of the system linear model can change very fast, and to track this change closely, the parameter estimator must be improved.  86  Appendix A  Quadratic Optimization by Lagrangian Multiplier Method  A quadratic problem with inequality constraints can be expressed as: min < -x^Hx  + (Fx > subject io Ax <h  (A.1)  where the Hessian matrix H and the vector c are the set of coefficients of the quadratic objective function. The matrix A and vector b are the coefficients of the linear constraints. A is of dimension m X n and b is of dimension w x 1. The vector x is a set of independent variables and is of dimension « x 1. The inequality constraints can be converted to a group of equality constraints by adding a vector of slack variables Ax-b  + 6^ = 0  (A.2)  where '5' = [<5f SI  ...  Slf  (A3)  Then we can define a Lagrangian function L(x,uj) = -X'^HX  + CFX + U(AX  -b + S^)  (A.4)  where the weight factor w is defined as: u; = [wi UJ2 •••  ujm]  (A.5)  The stationary solution x* of Equation A.4 satisfies dL{x) dx ' ^=^' ^  0  I .=.. = 0  (A.6)  where 0 denotes a zero matrix of dimension w x 1. The minimum of the quadratic function can be obtained by substituting these solutions into the index. 87  References  [I]  D. E. Eklund, "Review of surface application," in Fundamentals of Papermaking, vol. 2, (Cambridge, UK), pp. 833-870, Trans, of the 9th Fundamental Research Symposium, 1989.  [2]  L. G. Andersson, The Coating Process, pp. 9-19. TAPPI Press, 1993.  [3]  J. Gloeckner and L. 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