Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Modeling paper machine cross direction slice lip responses close to sheet edges Yang, David Wei 2005

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2005-0345.pdf [ 5.34MB ]
Metadata
JSON: 831-1.0092014.json
JSON-LD: 831-1.0092014-ld.json
RDF/XML (Pretty): 831-1.0092014-rdf.xml
RDF/JSON: 831-1.0092014-rdf.json
Turtle: 831-1.0092014-turtle.txt
N-Triples: 831-1.0092014-rdf-ntriples.txt
Original Record: 831-1.0092014-source.json
Full Text
831-1.0092014-fulltext.txt
Citation
831-1.0092014.ris

Full Text

MODELING PAPER MACHINE CROSS DIRECTION SLICE LIP RESPONSES CLOSE TO SHEET EDGES By DAVID WEI YANG B.Eng., National University of Singapore, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES ELECTRICAL AND COMPUTER ENGINEERING THE UNIVERSITY OF BRITISH COLUMBIA April 2005 © David Wei Yang, 2005 Abstract The design of paper machine cross directional (CD) control systems is normally based on actuator response models obtained in the centre rather than the machine edges. However process characteristics near the sheet edges are different from those in the centre of the sheet. Even with good CD control, the edge profile often shows the greatest variations from the target level. The application of control algorithms more suited to the centre of the machine reduces controller effectiveness at sheet edges. Control close to the sheet edge is normally carried out using a variety of assumptions concerning the expected response, however, these assumptions are often made for mathematical convenience rather than on the basis of physical principles. This work involves the comparison of edge response of the cross directional slice lip actuators with the response at the centre of the paper machine. The performance of a typical industrial CD actuator centre response model is tested and found to be inadequate when modeling the edge response. A new physical model, using surface wave theory of the slurry is investigated, tested and later modified in an attempt to better model the edge response. The new edge model is found to generate an improved edge response prediction, based on the centre response, but in general will be computationally expensive. ii Several variants of the proposed model are put to the test and one particular model with a fixed parameter is proposed, based on physical principles to replace the centre model when modeling the edge response of heavy grade papers. The performance of this new model is evaluated and found to be more effective at modeling edge response than the conventional model. In addition, this edge model can be applied in conjunction with typical industrial control software without difficulty. Superposition and linearity in actuator response are successfully evaluated and found to hold for the test responses. iii Table of Contents Abstract ii List of Tables vi List of Figures vii List of Abbreviations ix Acknowledgements x Chapter 1 Introduction 1 1.1 Introduction to the Paper Machine 1 1.2 Cross-Directional (CD) Control Systems on a Paper Machine 5 1.2.1 Basis Weight Control 6 1.2.2 Slice Lip Actuators 7 1.2.3 Approaches to CD Control 7 1.3 Problems near the Sheet Edges 9 1.4 Objectives and Contributions of the Work 12 1.5 Thesis Overview 13 Chapter 2 Cross Directional Slice Lip Response Models 15 2.1 Introduction 15 2.2 Existing Response Models used at the Edge 15 2.2.1 Cosine-exponential Model (CEM) 15 2.2.2 Other CD Response Models 18 2.3 Dispersive Wave Model (DWM) 19 2.4 Superposition and Linearity 24 Chapter 3 Data Collection and Model Identification 26 iv 3.1 Description of Mill and CD Control System 26 3.2 Bump Test Patterns 28 3.3 Modeling using Industrial Control Software 29 3.4 Reflection Wave Model (RWM) 33 3.5 Simplified Reflection Wave Model 34 3.6 Modeling Algorithm 36 Chapter 4 Parameter Estimation and Results Analysis 39 4.1 Comparison of Centre and Edge Responses using Cosine-exponential Model 39 4.2 Edge Response Modeling using Cosine-exponential Model 45 4.3 Edge Response Modeling using Reflection Wave Model 50 4.4 Edge Response Modeling using Simplified Reflection Wave Model 54 4.5 Performance Analysis 56 4.6 Superposition and Linearity Analysis 59 Chapter 5 Conclusions and Future Work 65 5.1 Conclusions 65 5.2 Future Work 67 Bibliography 70 Appendix Modeling Stages and Parameters 76 List o f Tables 4.1 Root Mean Square Error (RMSE) between actual responses and corresponding CRM using cosine-exponential modeling 44 4.2 Root Mean Square Error (RMSE) between actual edge responses and corresponding ERM using cosine exponential modeling 47 4.3 Comparison of RMSE from modeling using CRM and ERM for on-sheet edge response (actuator 3, 34) 47 4.4 Comparison of RMSE from modeling using CRM and ERM for off-sheet edge response (actuator 2, 35) 47 4.5 Comparison of model parameters from CRM and ERM using cosine exponential modeling for grade 1001b paper 48 4.6 Comparison of model parameters from centre model and edge models using cosine exponential modeling for grade 751b paper 48 4.7 RWM parameters from modeling on-sheet and off-sheet edge response for grade 1001b paper 52 4.8 RWM parameters from modeling on-sheet and off-sheet edge response for grade 751b paper 53 vi List of Figures 1.1 A simplified illustration of a modern paper machine 4 1.2 Diagram to illustrate typical spatially localized response from CD actuators 10 1.3 Diagram to illustrate actuator picketing of CD actuators at sheet edges 11 2.1 Illustration of cosine-exponential bimodal responses (Diagram used from [13] with permission from Honeywell) 16 2.2 Surface wave of slurry on wire 21 2.3 The basic concepts of dispersive wave theory for a = 1 in (2.6) 24 3.1 Water jet trims off paper strip at sheet edge in the sheet forming section 27 3.2 Mill's CD controller setup 28 3.3 Showing different bump patterns designed for different response analysis 30 3.4 Interlaced identification algorithm (Diagram used from [13] with permission from Honeywell) 31 3.5 Two typical scan profiles in a bump test at different scan time 32 3.6 Different locations of reflected wave at different time instance 38 4.1 Centre and edge response modeling for 1001b grade paper 41 4.2 Centre and edge response modeling for 751b grade paper 42 4.3 Centre and edge response modeling for 521b grade paper 43 4.4 Modeling the off-sheet edge responses (bump to actuator 2) using ERM 49 4.5 Edge response modeling of 1001b grade paper using the RWM 52 4.6 Edge response modeling of 751b grade paper using the RWM 53 4.7 Edge response modeling of 1001b grade paper using modified RWM with fixed shift, B 55 vii 4.8 Edge response modeling of 751b grade paper using modified RWM with fixed shift, B 56 4.9 Illustration of superposition in actuator response for 521b grade paper 61 4.10 Illustration of superposition in actuator response for 301b grade paper 62 4.11 Illustration of actuator response linearity for 1001b grade paper 63 4.12 Illustration of actuator response linearity for 751b grade paper 64 Chart 4.1 RMS error comparisons for 1001b grade paper 57 Chart 4.2 RMS error comparisons for 751b grade paper 58 List of Abbreviations 1. CEM Cosine-exponential model (Section 2.2.1) 2. DWM Dispersive wave model (Section 2.3) 3. CRM Centre response model - response model derived by modeling centre response profile using CEM 4. ERM Edge response model - response model derived by modeling edge response profile using CEM 5. RWM Reflection wave model - proposed new response model by combining dispersive wave theory and CEM for modeling edge responses (Section 3.4) 6. Simplified RWM RWM with one or two fixed parameters (Section 3.5) 7. CD Cross direction in paper machine 8. MD Machine direction in paper machine 9. MPC Model predictive control 10. SISO Single-input single-output 11. DCT Discrete Chebyshev Transform 12. RMSE Root mean square error ix Acknowledgments I would like to take this opportunity to thank the people who had played an important role in this work. First, I would like to thank my supervisor, Prof. Michael Davies from the University of British Columbia (UBC) for his continuous advice and guidance throughout the course of my graduate studies. Prof. Michael Davies admitted me into UBC graduate school and provided me with continuous support in various areas possible. I would also like to thank Prof. Guy Dumont of UBC and Dr. Greg Stewart from Honeywell Process Solutions. I have greatly benefited from the meetings and valuable discussions with them. I would like to thank some of the fellow students at the Pulp and Paper Centre, Stevo Mijanovic, Setareh Aslani, Benjamin Kan, Stephan Bibian, Tatjana Zikov and Quek Foo Lee for their friendship and technical discussions which inspired me in many ways. The assistances provided by Amor Lahouaoula of Honeywell Process Solutions during the mill visits are also greatly appreciated. I would like to thank all the friends that I have made during my stay in Vancouver especially Kelvin Tsang, Eunice Yung, Jack Chan, Jennifer Hu, Aileen Hua, Chen Yu Wang and Steven Ma who made my student life an enjoyable experience. Special thanks go to my girlfriend, Jessie Liu, for her love, patience and support in my research as well as my life. x Lastly, I would like to pay special tribute to my parents, Rong Huan Yang and Ju Fen Wei for their constant support, love and encouragement throughout my life. I would have never reached graduate school if not for them and I am truly indebted to them for all the things they have done for me. xi Chapter 1 Introduction 1.1 Introduction to the Paper Machine The word paper comes from Papyrus, a material used by the ancient Egyptians in an early attempt to make a writing sheet. It has been the most commonly used writing and drawing medium throughout the history of mankind. Early paper making was carried out in AD 105 by the Chinese court official, Ts'ai Lun using recyclable materials such as tree bark, bamboo and rags. Bamboo sticks were used by Chinese paper makers as the alternative to paper before Ts'ai Lun. The technique of papermaking was kept as a secret for five centuries before it reached Korea and Japan around 610AD. Later, the knowledge of papermaking also spread to Central Asia, Tibet and India [43]. After the Arabs acquired the technique of paper making in AD 750, paper mills were set up in Baghdad, Damascus and Cairo. The Arabs used mainly rags as the raw material for making paper due to the lack of fresh fibers. Their processing equipment was faulty and poorly designed. The secrets of paper making finally reached Europe in the 13th century through the export of paper from the Arab world. From then onwards, Italian papermakers improved on the paper making techniques by focusing on the preparation 1 process rather on the selection of raw material. It is the Germans who then set up paper mills to produce paper on a larger scale [43]. This mill-based papermaking technique spread throughout Europe in the 15th and 16th centuries and largely increased production and output paper quality. Technical progress continued on from the 17th to 19th century and it was in 1854 that the first chemical pulp was introduced. The greatest leap in paper manufacturing was in the 19th and 20 th century, when all work processes were fully mechanized. Rag substitutes, groundwood pulp and chemical pulp, were obtainable on an industrial scale. With further improvement in machinery and the introduction of automation, more paper and a greater variety of paper grades are produced better and faster [43]. The first paper mill in North America was built by William Rittenhouse in Pennsylvania in 1690. Later on, the first paper mill in Canada is constructed in the year 1803 at St. Andrews, Quebec [44]. James Crooks started Ontario's first paper mill in 1827 near Dundas and later followed with another in the Don Valley, Toronto. In 1840, development of the mechanical process to create pulp began and the first chemical process to create pulp was established in 1850. In 1894, the first pulp and paper mill in British Columbia was established at Alberni on Vancouver Island. By the year 1901 there were 53 pulp and paper mills in Canada employing about 6,200 people. Most of which were small mills making a wide variety of products such as: writing paper, newsprint, books, wrapping papers and building papers. Canada became the world's largest exporter 2 of paper in 1918. Today Canada accounts for 34% of the world's pulp and paper production, with exports to over 100 countries. Today, a sheet of paper is made primarily of pulp which consists of fibers extracted from wood structures. The two main pulping methods used to separate the fibers from the lignin are mechanical and chemical pulping. The fibres are delivered to the paper machine as low consistency water-born slurry. A simplified version of a typical fourdrinier type paper machine is shown in Figure 1.1. In the paper industry, the direction of sheet travel is known as machine direction (MD) and the direction across the machine perpendicular to the machine direction is referred to as cross direction (CD). Paper making starts with dissolving the pulp in a large storage tank known as the machine chest. Slurry is then diluted with a mixture of water and pulp and passes through several stages of screening before entering the headbox at a consistency of around 99.5% water and 0.5% fibers [9, 10, 11]. The headbox is a high pressure container that delivers the pulp mixture onto a wire mesh conveyor at a very high speed. At the same time, the headbox provides a uniform flow of pulp onto the wire by reducing fluctuations and irregularities across the machine. As the forming fibers passes over various dewatering elements, a significant amount of water is removed by gravity and suction through the wire resulting in the formation of a paper sheet. The second stage of dewatering occurs in the press section where the sheet is 3 first heated by steam boxes and then pressed to rid off excess water resulting an approximately 40% fiber concentration. Figure 1.1: A simplified illustration of a modern paper machine [9] The moisture content of the sheet reaches around 4% in the dryer section through rolling steam cans that use heat to evaporate the trapped moisture. In the final section of the paper machine, the thickness of the sheet and surface qualities are controlled by large Calendar rollers. The paper sheet is finally wound up onto a reel. Paper properties including basis weight, moisture and caliper are measured by moving sensors at different locations along the machine. The sensors travel across the paper sheet, which is moving at a high speed in the direction perpendicular to sensor action. Other sensors are used to 4 estimate sheet moisture and thickness. As a result, the measured sheet profile contains both MD and CD variations. A common approach is to consider the MD and CD control problems separately. 1.2 Cross-Directional (CD) Control Systems on a Paper Machine Paper machine cross-directional CD control concerns the control of sheet property variations in a direction perpendicular to the sheet motion on the machine. Control of a measured profile is carried out by an array of actuators distributed across the machine (cross-direction). Among the important sheet properties controlled by CD control systems are: basis weight, sheet moisture content and caliper or sheet thickness [10, 12]. CD moisture actuators include steam boxes and rewet showers. Steam showers improve water removal by increasing the sheet temperature. The higher sheet temperature increases water fluidity and makes water removal in the press easier. The rewet showers remove over-dry streaks in the sheet by introducing more moisture. CD caliper control aims to smooth and even the sheet thickness in the dry end by changing the pressure exerted onto the paper sheet. CD caliper control uses induction heaters or cold/hot air showers. Eddy currents are induced in calendar rolls by high frequency current in coils and cause heat build up in the calendars. The high temperature causes the roll to increase in size and exert more pressure on the sheet. The opposite is achieved by reducing the temperature of the rolls. In this work, the focus is on basis weight control of the paper only. 5 1.2.1 Basis Weight Control Basis weight is a fundamental property of the paper sheet and is defined as the sheet weight per unit area [11]. It can be expressed in grams per square meter (g/m ) or pounds per ream (lbs/ream). One ream could be of 3000 or 3300 square feet. If considering 1 ream to be 3000 square feet, then 1 lb/ream is 1.6289 g/m2. Basis weight requirements range from the 35 g/m light weight newspaper grade sheet to the 300 g/m liner board grade. Basis weight includes the moisture content of the sheet; the corresponding property of the dry sheet is "Dry Weight" = basis weight - water weight. A primary goal in paper making is to maintain uniform dry weight and therefore a uniform distribution of fiber across the machine (CD profile) as the sheet is produced. Variability on the sheet can be due to disturbances which are introduced before the headbox or in the headbox itself that cause machine direction (MD) basis weight variations across the entire sheet. Fiber distribution across the sheet is controlled by CD basis weight actuators. Several approaches are used in order to control the fiber distribution across the sheet. The two primary methods are slice lip actuators, which adjust the stock flow from the headbox on to the fourdrinier by bending the slice lip, and dilution flow actuators, which adjust the stock dilution, and therefore the stock consistency across the slice lip. Only slice lip actuators are considered in this thesis, although it is likely that the approach will also be useful in considering edge effects on dilution flow headboxes. 6 1.2.2 Slice Lip Actuators A mixture of pulp and water is forced from the headbox out of a horizontal nozzle or slice which stretches across the machine. The slice has an adjustable top lip and a fixed bottom lip. The entire upper lip can be shaped by force actuators that are adjusted up and down according to sheet weight requirement to maintain an even basis weight profile across the sheet. The larger the slice lip opening, the more stock is pushed out of the slice at a particular location. Slice lip actuator responses are spatially localized, so that adjusting a single slice lip actuator results in changes across a range of neighboring measured locations. A typical response profile showing both the CD and MD components is illustrated in Figure 1.2. It follows that a good knowledge of the slice lip response is essential for a good CD basis weight control [1, 2, 3]. 1.2.3 Approaches to CD Control The three main approaches [47] to CD profile control in sheet processes are linear control, model predictive control and robust control. Linear profile control mainly includes linear quadratic optimal control and model inverse based control. In this approach, control actions are taken about a steady state target level and steady state models are adopted. It is often assumed that the process dynamics can be treated as a pure time delay. The linear quadratic optimal controller for two-dimensional models of the processes is developed in [50]. A draw-back to linear profile control is that many approaches do not take into account the problem of robustness, which can result to poor closed loop control performance in the presence of uncertainty. Linear control with 7 antiwindup compensation [47] can be used to deal with actuator constraints when there are large disturbances and sharp spatial variations across the sheet. Model predictive control (MPC) is a control method that can explicitly consider actuator constraints when calculating actuator movements. MPC applied to the paper machine [47] performs optimization of the performance objective functions taking into account the constraints on the actuator. To minimize the CD profile variance of sheet measurements, quadratic optimization problems can be solved but can be computationally expensive since these problems can be very large in a paper machine. Several approaches are available to reduce the computation complexity. One method is to express the control vector as a linear combination of low order basis functions and optimize the coefficients of the basis functions to shorten computation time [49]. Another approach deals with approximation of the MPC problem [47]. Actuator constraints are modified or even ignored to solve an unconstrained MPC problem. This speeds up MPC computations at the cost of closed-loop control performance. Models for sheet processes have a degree of uncertainty, and robust control attempts to address this mismatch. The large number of measurements and actuators make it desirable to decompose the large multivariable problem into a set of single variable problems by decomposing the measurement and control vectors using appropriate basis functions. In [5] is illustrated a method to design robust multivariable controllers based on only one single loop controller assuming circulant matrix models. The controller for 8 circulant symmetric and Toeplitz symmetric models can be decentralized. In [48], robust multivariable controllers are designed using the properties of unitary-invariant norms and the approach is applicable to circulant symmetric processes. The multivariable robust control problem can be reduced to a large number of single-input single-output (SISO) robust control problem. It is even possible to further reduce the problem to a single SISO robust control problem. 1.3. Problems near the Sheet Edges The design of paper machine cross machine control systems is normally based on actuator response models obtained in the centre rather than the machine edges [13]. However, it is well known that the process characteristics near the sheet edges are different from those in the centre of the sheet. Even with good CD control, the profile often shows the greatest variations from the target level near the edge of the sheet. This poor performance is due in part to the variability in the slurry caused by fluid dynamic edge effects such as the presence of deckle boards, coupled with uncertainty about the true edge responses. Deckle boards are commonly installed at the edges of the paper machine to contain fiber suspension on the fourdrinier. If the effects of deckle board on responses are not taken into account then the application of control algorithms more suited to the centre of the machine reduces controller effectiveness. Control close to the sheet edge is normally carried out using a variety of assumptions concerning the expected response, however, these assumptions are usually made for mathematical 9 convenience rather than on the basis of physical principles [11]. For example, it is often assumed that the C D processes are spatially invariant to facilitate the design of controllers. Spatial invariance indicates that process properties are unchanged with respect to shifts in spatial variable. -1 I 1 i 1 1 L I J -20 0 20 40 60 80 100 120 C D Posi t ion (a) C D component of a response profile due to a bump (centre bar) of a slice lip actuator 150 (b) M D and C D component of a response profile Figure 1.2: Diagram to illustrate typical spatially localized response from C D actuators 10 15 20 25 Actuator Position Figure 1.3: Diagram to illustrate actuator picketing of CD actuators at sheet edges The process characteristics are clearly different at the machine edges from the characteristics at the centre, thus the paper machine edges represent a disruption of the spatial invariance, but this fact is generally neglected. Current industrial practice uses methods derived from signal processing including zero padding, signal average padding and reflection padding to extend profile at the edges [45] and provide the control algorithms with a full set of data at the edges. These modifications can lead to instability of the control systems and therefore lead to unsatisfactory control at the edges [21]. An example is the well-known phenomenon of 'actuator picketing' at the edges, in which adjacent actuators tend to move in opposite direction [11,21]. Figure 1.3 shows an array of CD actuators in motion and illustrates the actuator picketing effect at sheet edges. The centre actuators exhibit a smooth contour while the edge actuators move in opposite direction from their immediate neighbors. This sign of instability at the edges must be avoided because of the interacting nature of neighboring actuators, tends to force these profile variations to propagate from the edge towards the centre of the sheet, affecting 11 overall profile and even destabilizing the whole control system. Therefore, the edge actuators of the CD control systems are often placed in open-loop and are then ineffective in maintaining sheet quality close to the edge and so in reducing the amount of trim required at the machine edges [5]. Some recent CD response modeling techniques [42, 46] adopt complex mathematical calculations or investigate the dynamics of CD actuators [25] and those will be discussed in Section 2.2.2 in detail. Since many current control schemes, for example MPC, explicitly include a process response model, they can easily be adapted to different process models, if for example a new model which is able to better represent the true CD responses at the edges, were to become available. 1.4 Objectives and Contributions of the Work The main focus of this work is to establish experimentally the nature of the slice lip edge response in comparison with the response at the centre of the machine. The second aim attempts to relate the edge response to a physical model that has been proposed on the basis of dispersive wave theory [9] and to compare it with a typical model used widely in the industrial practice. A third issue that will be discussed involves the assumptions of superposition in actuator response that are normally made when designing CD controllers. Similarly, the linearity of the response to a series of actuator steps of differing magnitude will be investigated. The paper describes mill trials in which data is gathered and used to model the edge actuator response. Subsequently, data are analyzed, results presented, and conclusions are drawn. Although it is likely that many of the results are more broadly useful, it is important to note that the results obtained in this work are based 12 on observation of a specific fourdrinier paper machine with slice lip actuators and water spray deckle boards. The main contributions of this work are: 1. The nature of edge response in comparison with the response at centre of the machine is successfully established and it is found that the edge response is significantly different from the centre response for some grades of paper. The edge effect is found to be more prominent in heavy grade papers. The response model derived from central responses is found to be acceptable in modeling edge response for light grade paper. 2. The new proposed physical model of edge actuator response, based on dispersive wave theory [3, 9], is validated. It is observed that the dispersive wave model is better than the centre response model at modeling the edge response. 3. Several variants of the response model based on dispersive wave theory [9] are introduced and tested. A modification of this approach is proposed, based on physical principles, to model the edge response of heavy grade papers. The modification allows the new edge model to be applied with minimum practical difficulty. 4. Superposition and linearity in the response of dry weight profile due to actuator movements are successfully evaluated and found to hold for the test responses. 1.5 Thesis Overview This thesis is organized as follows. A common industrial paper machine CD response model is illustrated in Chapter 2 which adopts a model that is widely used as the CD actuator process model [1, 13] in the industry. Several other CD response models are also briefly stated. A new response model based on surface wave theory of the slurry 13 proposed by Ghofraniha [9] is examined. An investigation of this model was carried out and the theory behind its construction was studied. Detailed analysis of this model however, is outside the scope of this thesis. An attempt is also made to validate the principle of superposition and linearity adopted in response modeling. In Chapter 3, a description of the mill CD control system organization and the paper machine used for data collection is presented. Bump test patterns, which are designed to isolate the problems at sheet edges and facilitate the investigation of superposition and linearity, are illustrated. The model based on surface wave theory [9] is modified to incorporate the typical CD response model [1, 13] derived using the centre actuator response. As a result, a new modified response model is proposed to model the edge responses. Several other variants of this new model are also suggested which aim to make an edge response model as simple as possible so that it can be applied in conjunction of typical industrial control software. It is important that the edge model is related in a simple manner to that used in the centre, so that a minimum number of new parameters need to be identified. Chapter 4 deals with the parameter estimation and results analysis of industry data. The models described in Chapter 3 are put to test using industrial data and their performance analyzed. Based on the results drawn, a new response model for the slice lip actuator at the edge is proposed. Finally, in Chapter 5, a summary of the thesis is provided. Conclusions and suggestions for future research are presented 14 Chapter 2 Cross Directional Slice Lip Response Models 2.1 Introduction The cross direction response model, combined with the CD alignment model and a dynamic model, predict the form of the basis weight profile resulting from a single movement of a slice lip actuator. Since CD control systems are spatially localized by nature, adjusting a single slice lip actuator results in changes across a range of neighboring units. This chapter considers some existing cross directional slice lip response models that are used widely in industry and research. Then a dispersive wave model, based on previous research, is derived and investigated. Finally, superposition and linearity issues in the process response are discussed. 2.2 Existing Response Models used at the Edge 2.2.1 Cosine-exponential Model (CEM) Many industrial paper machine CD control systems adopt a cosine-exponential model [13] as the CD actuator process model. One typical model is adopted in a Honeywell paper machine CD control system and is illustrated in the equation below. This industrial CD control system considered in this work is currently installed on many paper machines worldwide and will be referred to as the CEM model throughout the thesis. 15 g(x) =1/2 h [b(x - 8w) + b(x + 8w)] (2.1) b(x) = exp (-ax2 / w2) cos (nx I w) (2.2) In these two equations, x is the cross-directional coordinate with respect to the response centre, h is the height parameter; w is the width parameter; a is the attenuation parameter; and 5 is the divergence parameter. The height parameter, h, defines the magnitude of the response and width parameter, w, controls the separation of the main lobes. Attenuation, a, defines the degree to which negative lobes are present in the response. Divergence, 8, defines a bimodal response appearance with a trough in the middle which is typical for slice lip actuator response for heavy grades. For light grades of paper, however, divergence would be near zero. Influence of these parameters on the modeled response shape is illustrated in Figure 2.1, where responses from three different parameter sets are shown. Figure 2.1: Illustration of cosine-exponential bimodal responses (Diagram used from [13] with permission from Honeywell) diveraence 16 A typical response of a heavy grade paper, e.g., liner board, would have the bimodal response appearance with a trough in the middle due to its large divergence [13]. In general, it has larger width and has broader response. A light grade paper, such as telephone directory paper, has almost no divergence and therefore no trough in the middle. Its responses are slimmer and have smaller attenuation than heavy grade papers. Light weight papers tend to have narrower or more spatially localized responses. A telephone directory paper weighs about 35 grams per square meter whereas a book cover grade could weight 300 grams per square meter. Heavy grade paper has higher pulp concentration achieved with same headbox consistency and flow but lower machine speed. Therefore, a disturbance to an actuator would cause a wider effect in the CD response of a heavy grade paper. The response of light grade paper however is more centralized but with a high peak. It is a common practice to use the same CEM to describe both the CD response at the centre and at the edge of the machine. However, applying CEM to centre response and edge response separately is expected to yield very different model parameters. It is important to note that applying CEM to the centre response will yield the centre response model (CRM) and applying it to the edge response will yield the edge response model (ERM). Both CRM and ERM are of the same form but with very different parameters. Details will be explained in Chapter 4. 17 2.2.2 Other CD Response Models There is an abundance of research related to the cross directional control of basis weight profile. But specific results on paper machine CD response modeling are very limited and even fewer exist on response modeling at sheet edges. CD response modeling schemes are normally divided into different categories. One approach by Wellstead [46] adopts an empirical input and output approach and uses a matrix to represent the interaction in CD. A set of basis functions is used to represent this complex matrix as a combination of a small number of simple functions with known properties. Discrete Chebyshev Polynomials are used as the set of simple function and Discrete Chebyshev Transform (DCT) is performed to get their respective coefficients. The main advantage of this approach is that a small set of coefficient is able to represent a large set of process data points. However, the number of coefficients needed for accepted accuracy in process reconstruction is not fixed. The effects of insufficient coefficient include underestimating the height of principle lobe and inclusion of false side lobe in reconstruction. Some methods deal with the physical properties of the process. This includes the modeling of hydrodynamic behavior of the CD actuators [25] which leads to nonlinear partial differential equations and requires a good understanding of the process. Duncan used stiffness of slice lip to determine the shape of spatial response and Bernoulli differential equations to approximate lip deflection under a load. 18 A third method is proposed in [42] where Chen adopted two-dimensional (2D) response modeling for each CD actuator. A sequence of random probing is performed on an actuator and 2D sheet variations are measured. This approach treats the CD variations as a superposition of polynomials with MD dynamics and the resulting model includes both CD responses and MD responses. Although fairly accurate, it is computational expensive and offers tailor made response models. 2.3 Dispersive Wave Model (DWM) An attempt was made by Ghofraniha in [9] to predict the form of the response to be expected near the edge of the paper sheet using hydrodynamic principles. The model adopted in this paper is based on the physical model that was proposed by Ghofraniha. In [9], the disturbance generated by the deflection of the slice lip at the surface of the slurry is characterized as a surface wave. It is assumed that the change in the flow in machine direction (MD) are a minimal and the effect can be characterized as a two dimensional flow across the sheet. An illustration of this two dimensional wave is shown below in Figure 2.2. The amplitude of the wave is a function of CD position and time. The machine direction is actually the time axis and when projected out in three dimensions, and is perpendicular to CD position. Each slice of the wave at a particular time represents the shape of the slurry profile at a different MD location. Figure 2.2 shows the surface wave of the slurry on the wire at one MD position. The x-axis represents the CD positions and y-axis the slurry thickness. When a disturbance is generated by a slice lip, a 19 wave will be generated at the surface and move in both the positive and negative x-directions. The shape of this wave at the initial time resembles that of the lip. A description of DWM is considered as a surface disturbance of the following form: N = A 0 exp(-kd |x|) cos(kx ± wt) (2.3) which is a damped progressive wave of period 27i/w and amplitude Ao and traveling in the either the positive or the negative x-direction. There is also an energy dissipating term that is the damping factor kd. The variable x is the CD coordinate with respect to the response centre and t represents time in the MD direction. As mentioned earlier, this disturbance wave travels in both positive and negative x-directions. This implies that the wave is an even function of x. This progressive wave travels towards the edge with a spatial frequency (wave number) of k and time frequency of w. The main differences between Ghofraniha's CD response model and the CEM are the exponential term and the inclusion of bi-model response in the CEM. The DWM adopts |x| in the exponential term which results in a more pointed shape at the peak of the CD model rather than the smooth and curvy tip of the CEM resulting from the x 2 in its exponential term. Also, the bi-model response in CEM is more suited for modeling heavy grade papers. 20 1.5 S u r f a c e w a v e of s lurry c a u s e d by d i s t u r b a n c e E < CD =3 -1 .5 -40 0 C D P o s i t i o n 4 0 Figure 2.2: Surface wave of slurry on wire Edge effects by this wave on the Fourdrinier are the consequence of installation of deckle boards adjacent to the headbox and possibly incorrect installation of cheeking pieces inside the headbox. Checking piece is installed in the headbox and forms part of the side to seal the end of the slice opening. If the cheeking piece is misaligned, an edge wave could be started in the headbox. Deckle boards are used to contain fiber suspension at the edge of the wire so as to prevent the stock from flowing off from the sides. Conventional deckle boards can be attached to the headbox and sealed to the wire by a flexible rubber strip. There are also the water spray deckle boards and the curling of flexible wires installed at the edges to contain the stock. Al l three methods, when implemented, will result in edge wave reflections. Assuming a disturbance of an edge 21 slice lip actuator will generate a surface wave traveling towards the edge in the positive x-direction. At the edge, assuming a frictionless impact, the wave is reflected with the same amplitude and velocity in the opposite direction and a phase shift of 180 degrees [9]. The reflected wave will have the same form as the surface wave but will be traveling in the negative x-direction. N r = An exp(-kd x) cos(kx + wt) for x <0 (2.4) Assume superposition holds for the CD response profile. The combination of these two waves generates a standing wave pattern at the edges [9]. Nedge=N + N r = Ao exp(-kdxo) cos(kxo - wto) + Ao exp(-kdxi) cos(kxi + wti) (2.5) where ti = te + At e d g e and xi = x e + Ax ed g e- Ax e d g e is the distance from the generated disturbance to the edge while At e d ge is the traveling time of the incoming wave to the edge. In equation (2.5), parameters xo and t0 are the distance and time that the incoming wave meets the reflected wave from origin in positive x-direction. Parameters x e and te refer to the distance and time from the edge when the reflected wave meets the income wave. However, any surface variation of the response is assumed to be stopped at dry line and surface profile stay constant after dry line. 22 If the reflection at the edge is not frictionless, the reflected wave is assumed to be attenuated by a reflection coefficient a, 0 < a < 1, representing the fraction of energy loss at the impact. The surface profile will then be: Nedge = N + a N r = Ao exp(-kdxo) cos(kxo - wto) + aAo exp(-kdxi) cosfkxi + wti) (2.6) 0<a< 1 This concept is illustrated in Figure 2.3. The incoming disturbance wave is represented by the dashed curve. The reflected wave is represented by the dotted curve. The resulting edge response model is the superposition of these two curves and is drawn in solid. A common solution adopted widely in paper industry to reduce edge effects is to use an edge bleed. Edge bleed actuators are nozzles mounted at the side of the wire that inject a diluted stock mixture at high speed onto the slurry. The void areas on both side of the sheet caused by the edge waves are trimmed off by the trim squirts. A correct estimation of the location of the edge reflected wave can be made by adjusting the position of the trim squirts and reduce the amount of paper trimmed at the edge. 23 E < 3 CL Edge position C D Position Figure 2.3: The basic concepts of dispersive wave theory for a = 1 in (2.6) 2.4 Superposition and Linearity It is a common practice to assume superposition and linearity in actuator response when designing CD controllers. If bumps are made to two adjacent actuators respectively at different instances, two individual response profiles will be observed. Under the theory of superposition, the resulting summation of these two profiles should be the same as the profile generated by bumping the two adjacent actuators together with the same magnitude. This means that the summation response models from the two individually bumped adjacent actuators is the same as the response model from the two actuators 24 bumped simultaneously. Superposition is also a fundamental assumption in the dispersive wave analysis. In the case of linearity, the magnitude of the response profiles due to different bump magnitudes is considered. A linear relation implies that a bump of amplitude RA will generate a response whose amplitude is R times that produced by a bump of magnitude A. Both properties are essential and used widely in response modeling and CD controller design. However, it is not evident from physical principals that such assumptions will hold. The complexity of the hydrodynamics, and also the complex nature of the slice lip bending in response to the actuator changes give rise to concerns that neither superposition nor linearity need necessarily hold for the response. Those complexities led Ghofraniha to assume superposition and linearity in his model explicitly. Those assumptions are investigated experimentally in this work. Bump tests which allow comparison between the response of a single actuator and that resulting from simultaneous adjustment of two adjacent actuators will be designed. Similarly, the linearity of the response to a series of actuator steps of differing magnitude will be investigated. 25 Chapter 3 Data Collection and Model Identification 3.1 Description of Mill and CD Control System In order to model responses to actuator settings for a variety of positions across the sheet, tests were carried out on an operating paper machine in a Canadian mill. This paper machine produces a wide variety of paper with basis weight in the range of 37.8 to 151 lbs/3000 square feet (lbs/3 000FT2) and running at wire speeds of 601 to 2470 feet per minute (fpm). Sheet width at the reel is 163 inches. The paper machine uses motorized slice lip actuators to reduce the variations of the paper sheet cross-directional weight profile. The machine has 36 actuators distributed across the slice width of 3.44m. The total number of scanner measurement points, or data boxes, is 216. At the end of the Fourdriner table, just before press section, high pressure water jets are located at approximately 12 measurement points from either side of the machine edges and act as trimmers in the paper forming section. As a result, 19 cm wide sheet strips are trimmed off on both sides of the machine leaving only measurement points 13-205 as valid sheet profile measurements. Those measurement points will be referred to as 'on-sheet' measurements and their corresponding actuators, 3-34, are referred to as 'on-sheet' actuators. Similarly, actuators 1-2 and 35-36 which are placed in open loop during the tests are called 'off-sheet' actuators. An illustration is shown in Figure 3.1 below. 26 CD Direction Deckle board Trim jet Trimmed off section Scanning sensors Off-sheet measurements Paper sheet On-sheet measurements Slice lip actuators Trim jet Trimmed off section Off-sheet measurements MD Direction Figure 3.1: Water jet trims off paper strip at sheet edge in the sheet forming section The scanning sensors take 30 sec to traverse the paper sheet. Therefore, a set of new CD profile measurements is collected every 30 sec. The collected data is then updated onto the screen by the CD controller software after each scan and stored as Matlab files for reviews, calculations and diagnostics. The paper machine's slice lip actuators and the scanning sensors are all connected via industry network to an industrial CD control system. A simplified schematic of the paper machine CD control is shown in Figure 3.2. Based on the measured data, picked up by the scanning sensors and the desired setpoint 27 target value, the controller software generates a control signal and sends it to the slice lip actuator for appropriate actions. Slice lip actuator setpoints Target Industrial CD control system Paper machine Scanner data Figure 3.2: Mill's CD controller setup 3.2 Bump Test Patterns Bump tests were carried out on three grades of paper namely, 50 [lbs/ream], 75 [lbs/ream] and 100 [lbs/ream] grades, and for a variety of actuator locations. The CD controller is put to open loop during the test and no control action is applied. The magnitude of the actuator setpoint changes ("bumps") is in microns. The typical bump magnitude is around 200 to 400 microns which is high enough to observe an effect on profile response but low enough not to disrupt production standards and ensures overall paper quality. The bump test patterns are designed so as to facilitate the comparison of 28 edge actuator response and centre actuator response. At the same time, superposition and linearity in actuator responses can also be investigated. Four sample bump test patterns are illustrated in Figure3.3. Only the chosen actuators are bumped, the neighboring actuators are held fixed. Horizontal axis represents different actuator positions from 1 to 36. Based on the Figure3.3 (a) and (c), the two centre bumps are done to the same actuator but of different magnitude. This enables the investigation of linearity in actuator response. To investigate superposition, as illustrated in Figure3.3 (a), (b) and (c), two actuators next to each other are bumped individually at different times to obtain their individual responses and they are then bumped together for the combined response. As a result of trimming, only the responses from actuators, 3-34, can be completely measured as described in section 4.1. Note, however, that the response of off-sheet actuators will be seen extending into the on-sheet response area but those responses are incomplete. All bumps applied to edge actuators within actuator 3-34, e.g. actuator 3, are referred to as 'on-sheet' edge bumps and the resulting responses will be referred to as 'on-sheet' edge responses. Bumps applied to actuators 1-2 and 35-36 are 'off-sheet' bumps and their responses will be referred to as 'off-sheet' edge responses. 3.3 Modeling using Industrial Control Software The industrial control software used to control the mill's paper machine is a Honeywell paper machine CD control system that is currently installed on many paper machines worldwide. The overall process model consists of a Dynamic Model, CD Alignment 29 Model and a CD Response Shape Model. Only the CD response model is examined in this work. 35 2 1 17 18 (a) 1 1 35 2 I 1 18 (b) 34 1 3 1 17 (c) 1 1 34 (d) Figure 3.3: Showing different bump patterns designed for different response analysis 30 The software performs open loop identification of a CD process by applying an excitation signal to an input of the CD control system while the CD controller is in open loop. The identification algorithms [13] fit the model-based process response prediction against the data collected during a bump test. It uses every measurement point of the collected high-resolution data profile and computes the least-square estimates of the CD process parameters. The software [13] then determines model parameters that provide a minimum sum of the squared errors when fitting the model-based process response prediction against the data collected in a bump test. It achieves this by performing interlaced CD and time-response identification which the CD response is estimating by assuming the time-response is known and vice-versa. By repeating the process, accurate least-square estimates of the process CD and time-response parameters are obtained. The software provides a single CD process response profile to the bump estimated from the entire 2-D array of the collected data and offers a model-based prediction of the profile. Both are values essential in identification process and used to assist in the edge response modeling of this work. Bump Test Data Logged profiles Excitation: '': , • Bump profilo • Timing ". Estim ate C D response Estimate time response CD response the bum pi '• -'7- Vesfe;':*' Identify CD response rn odel: Response shape-M o d e / C D response; Time response in the bump test Identify time response model : .••/If'od.eV'Hi 'f&s'ppkser-Figure 3.4: Interlaced identification algorithm (Diagram used from [13] with permission from Honeywell) 31 As the raw response profile obtained after each sensor scan is very irregular and noisy, modeling with such data is difficult and could not guarantee correct results. Figure 3.5 shows two typical scan profiles in a bump test at different scan time. The raw scan data must be processed in order to arrive at a single CD response for modeling. The industrial software provides a single CD profile derived from the raw high resolution three dimensional bump responses using the interlaced identification as shown in Figure 3.4. The CD basis weight response was allowed to reach steady-state after each bump. The response was then processed over the total number of scans taken in the whole bump process in order to produce this single CD basis weight profile. This CD profile has 216 data bins in total but only bins 13-205 is valid due to trimming at the edges. Since the paper machine in which the bump tests were carried out was equipped with the Honeywell CD controller, it is convenient to use this single CD response profile for the response modeling sections in the Chapter 4. C D responses at different scan time 1 1 2 p i i i i i i i 107k 106\-20 40 60 80 100 120 140 160 180 200 C D position, [bins] Figure 3.5: Two typical scan profiles in a bump test at different scan time 32 3.4 Reflection Wave Model (RWM) Identification is a process of determining a model of a controlled system by processing its input and output data. The reflection wave model (RWM) is developed to analyze the edge responses for slice lip actuators. The RWM incorporates the concept of both the cosine-exponential model (CEM) in Section 2.2.1 and dispersive wave model (DWM) in Section 2.3. As described in Section 2.3, Ghofraniha carried out a fluid mechanical analysis of the stock flow and drainage on the fourdrinier, and concluded that a dispersive wave model best represented the predicted basis weight response. We have seen that according to his theory, the true edge response is characterized as a standing wave composed of a surface wave in the slurry traveling towards the edge of the machine and its reflected wave when that surface wave is reflected by the deckle board. By simplifying equation (2.6), the incoming wave and the reflected wave are separated by a spatial difference in x at the time of intersection. The reflected wave is simply a shifted and scaled version of the incoming wave. The RWM uses the CEM (2.2) derived from centre actuator response by industrial control software to describe the incoming surface wave. The new RWM, F(x), becomes: F(x) = f(x) + Af(x - B) 0<A< 1 (3.1) where f(x) is actually the centre response model (CRM) found from centre actuator responses, A is the reflected magnitude parameter and B is a horizontal shift parameter in 33 the CD direction. With reference to the dispersive wave theory, f(x) would correspond to the incoming wave generated by a bump of the actuator. The term Af(x - B) is the reflected version of this wave after it is reflected at the edge. Thus the edge model is a superposition of the incoming wave generated by a bump of an edge actuator together with its reflected wave which is scaled and shifted. The amplitude, A, of the reflected wave can be expected to vary depending on the energy loss due to edge boundary conditions, and the CD reflection shift is expected to depend upon the distance from the edge of fourdrinier. 3.5 Simplified Reflection Wave Model For practical use, it is desirable that an edge model be established as simply as possible, and that it be based on a modified version of the centre response. Examining the physical meaning behind the dispersive wave theory leads to another possible approach to simplifying the reflection wave model (RWM) further. A bump of the edge actuator is assumed to result in a cosine-exponential wave which travels towards the edge and is reflected at the deckle board to return and be combined with the original wave, so forming the resulting edge model. The distance between the bumped actuator and deckle board defines the parameter B, which accounts for the shift associated with the reflected wave in the RWM. The further the bumped actuator is from the deckle board, the longer it takes for the wave to travel until reflected. As a result, a larger shift parameter, B is associated with the longer distance between the bumped actuator and deckle boards. Since the position of deckle boards is always fixed, the value of B can be estimated based 34 on the physical location of the bumped actuator and its distance from the deckle board. Only the parameter A, the amplitude scaling, then needs to be found by non-linear curve fitting. This considerably simplifies the use of an edge model in practice. Based on the RWM (3.1), the spatial difference between the incoming wave and the reflected wave during the time of their impact is represented by B. Re-examine the dispersive wave theory in Section 2.3, surface variations at the edge of the paper start when an edge actuator is bumped and continue to the point of dry line where any surface variation of the response is assumed to be stopped. The positions of the reflected wave at the time of a bump and at the dry line are different. For example, at the initial time of a bump to an edge actuator, a surface wave in the slurry is generated and centres at position Pi ; with a reflected wave centres at position - P i . The spatial difference between the two waves is B. As the slurry travels down in the MD direction, the surface wave dissipates to the edge resulting to changes in the position of the reflected wave. At the position of dry line, surface variation is assumed to stop and the reflected wave is at a new position = (-Pi + wT) where w is the speed of wave dispersion or phase velocity and T is the time to dry line. The new spatial difference is then (B - wT). This concept is illustrated in Figure 3.6. Ghofraniha [3, 9] suggested a way to estimate w by a cross-correlation technique. The parameters A 0 , kd and k is calculated by comparing the shape of the slice lip at initial time of a bump with equation (2.3) when (t = 0). Then w is found by cross-correlate the shape of the output CD response with a normalized (2.3). 35 For the paper machine on which the bump tests were carried out, the total number of slice lip actuators is 36 and total number of scanner points across the sheet (bin) is 216. Therefore the number of bins per actuator is 6 assuming an ideal scanner-actuator alignment at the initial time of a bump. For an 'off-sheet' edge bump at actuator number 2 which corresponds to bins 7-12, the bump profile peaks at bin 9.5. As the paper is trimmed at bin 13, values from bin 7-12 are invalid and therefore only a portion of the bump effect can be picked up by sensor. At the time of the bump, the reflected wave peaks at bin -9.5. So the spatial difference is 19 bins in this case. Similarly, for an 'on-sheet' edge bump at actuator number 3 which corresponds to bins 13 - 18, the bump profile peaks at bin 15.5. And since the paper is trimmed at bin 13, the whole effect of the bump could be observed in the data. At the time of the bump, the reflected wave peaks at bin -15.5. So the spatial difference is 31 bins. The difference between spatial difference of 'on-sheet' and 'off-sheet' edge responses could be found as 31-19 = 12 bins. Given the values of w, the speed of wave dispersion, and T, time to dry line, the value of parameter B in RWM (3.1) can be estimated. 3.6 Modeling Algorithm Matlab is the tool used in this work and the Optimization Toolbox is used extensively for edge response modeling. A non-linear least square curve fitting function in the Optimization Toolbox is used to identify the edge response parameters from the test data. That is, given an input data, 'xdata', and the observed output 'ydata', the algorithm 36 determines coefficients Q that best-fit an equation F(Q, xdata) to 'ydata'. Both 'ydata' and Q are 1-dimensional matrices and 'xdata' is 2-dimensional. Given a predefined form of equation F(Q, xdata), a set of parameters, Q, for the function could be found using the non-linear least square curve fit. To avoid underdetermined systems, it is required that the number of equations, i.e., the number of input and output data sets collected, be at least as great as the number of unknown parameters, Q, in F(Q, xdata). In other words, length(ydata) > length(Q). Real variables are assumed therefore the user-defined function must only return real values. The algorithm adopted is a subspace trust region method and is based on the interior-reflective Newton method described in [17]. The algorithm aims to achieve min il y^ata — F(Q, xdata) \\. Q In this work, the single CD response data collected from the Honeywell control software from the bump tests are taken as observed output, 'ydata' in the non-linear least square curve fitting function described above. The total scanner bins (1 - 216) are taken as the input, 'xdata' and the various response models discussed in the previous sections are used as F(Q, xdata) with a set of model parameters, Q, to be determined. After applying the least square algorithm, those model parameters which best fit the equation F(Q, xdata) or response model to the collected response data are determined. 37 Edge position / j \ i Reflected wave / ] Surface wave = \ B Bumped actuator 3 *"' \ P l \ If P1 1/ -60 0 60 CD Position, [bins] (a) Estimated surface wave and reflected wave due to a bump to actuator 3 at t CD Position, [bins] (b) Estimated surface wave and reflected wave due to a bump to actuator 3 at t = T line Figure 3.6: Different locations of reflected wave at different time instance Chapter 4 Parameter Estimation and Results Analysis 4.1 Comparison of Centre and Edge Responses using Cosine-exponential Model Analysis was first carried out to establish the relationship between the edge response and that at the centre of the machine. The parametric model tested is the cosine-exponential model (CEM) used frequently in modern paper machine controllers, and described in Section 2.2.1. Bump tests identical to Figure 3.3(b) and (d) were carried out on the paper machine, which is equipped with 36 CD slice lip actuators, and the corresponding CD responses of 216 data bins were collected using Honeywell's industrial CD control software. The basis weight response used for modeling in this thesis is the single CD response provided by the software as described in Section 3.3 and because the paper is trimmed at the edges, only bins 13-205 have valid values. Since the edge actuators and the centre actuators are bumped simultaneously in most of the bump tests, it is not appropriate to use the complete CD response as a whole to produce accurate individual response models. The generated model for the centre 39 response would be biased due to the effects from the edge response. In order to perform accurate modeling of the centre response and edge response, data from the single CD response test is sectioned to include only the bump response corresponding to an individual actuator of interest. For centre response modeling, the CD response from a bump to a centre actuator is sectioned and modeled using the CEM to arrive at the centre response model (CRM). Figure 4.1 illustrates CD response modeling of a bumped test response on the grade 1001b sheet. In Figure 4.1(a), the irregular data line shows the sectioned response profile after a bump was performed on the centre actuator. The solid line represents the best-fitting CEM using the non-linear curve fitting technique in Matlab. This solid line forms the CRM for this particular bump test. Figure 4.1(b) illustrates the response profile of a bump to the edge actuator which is 'off-sheet', and the corresponding model drawn using the parameters taken from the CRM in Figure 4.1(a). Similarly, Figure 4.2 and Figure 4.3 illustrate the corresponding modeling results for grade 751b and 521b sheets. It is observed that the centre responses for all three grade papers are modeled very well by the CRM. However, the models drawn using the parameters taken from the CRM did not model the edge response well. Al l the CRM fail to reflect the true behavior of the actual edge responses especially towards the edge of the sheet and the peak of the bumps. This failure however, has been found to lessen as the weight of the paper grade decreases judging from the modeling results of other bump test responses. Edge effect in 521b light grade paper is observed to be negligible. 40 CD response profile (centre) for grade100#/3000FT2 from bumping actuator 18 - • - Centre response Predicted model (centre) 56 66 76 86 96 106 116 CD Position, [bins] 126 136 146 156 (a) CD response profile (offsheet edge) for grade100#/3000FT2 from bumping actuator 2 -0.2 - • - Edge response Predicted model (centre) -V \ - —V -i i ,061 i i i i i 1 i 1 1 12 17 22 27 32 37 42 47 52 57 CD Position, [bins] (b) gure 4.1: Centre and edge response modeling for 1001b grade paper CD response profile (centre) for grade75#/3000FT2 from bumping actuator 18 CD Position, [bins] (a) CD response profile (offsheet edge) for grade75#/3000FT2 from bumping actuator 2 -0.4 - * - Edge response Predicted model (centre) - \ \ \ - \ i i . 0 61 1 i 1 i i 1 i 1 1 12 17 22 27 32 37 42 47 52 57 CD Position, [bins] (b) Figure 4.2: Centre and edge response modeling for 751b grade paper CD response profile (centre) for grade52#/3000FT2 from bumping actuator 18 2 1.5 1 0) I 0.5 E < 0 I -0.5 - • - Centre response Predicted model (centre) i i 55 65 75 85 95 105 115 125 135 145 155 CD Position, [bins] (a) CD response profile (offsheet edge) for grade52#/3000FT2 from bumping actuator 35 - • - Edge response — Predicted model (centre) 81 1 i 1 1 ! L J ! L _ 155 165 175 185 195 205 CD Position, [bins] (b) Figure 4.3: Centre and edge response modeling for 521b grade paper Table 4.1: Root Mean Square Error (RMSE) between actual responses and corresponding CRM using cosine-exponential modeling RMSE from Modeling of Bump Position Centre response (actuator 18) On-sheet edge response (actuator 3, 34) Off-sheet edge response (actuator 2, 35) Test Bump 1 on 1001b paper 0.13405 0.1537 0.1778 Test Bump 2 on 1001b paper 0.11246 0.113 0.2978 Test Bump 3 on 1001b paper 0.12217 0.1342 0.1542 Test Bump 4 on 751b paper 0.08593 0.0916 0.1607 Test Bump 5 on 751b paper 0.08636 0.139 0.1449 Test Bump 6 on 521b paper 0.04341 0.0683 0.0947 A table of the Root Mean Square Error (RMSE) between the actual response of various bump tests and their corresponding models is presented in Table 4.1. The error between the actual response and the corresponding model is calculated and squared before the mean value is computed and rooted to arrive at the RMSE. The first column represents the RMSE between the centre responses from bumping actuator 18 and their corresponding CRMs after curve fitting. The second column shows the RMSE between on-sheet edge responses from bumping edge actuator 3 or 34 and the corresponding models. Lastly, the third column shows the RMSE between off-sheet edge responses from bumping edge actuator 2 or 35 and the corresponding models. From the results, it is observed that fitting the CRM against the edge response generates greater RMSE and produce different RMSE for on-sheet and off-sheet edge responses. The RMSE generated between on-sheet edge responses and CRM are relatively similar to the centre response with values in between 0.001 to 0.05. However, RMSE between the off-sheet edge response and CRM is between 0.03 and 0.18. This increase of RMSE from on-sheet to off-sheet edge response modeling is because off-sheet bumps are closer to the edge than 44 on-sheet bumps and therefore, more effect from the reflected wave is included in the final response. Based on the analysis, which was consistent over several tests, it is clear that the CRM is effective at modeling the centre response, but that the edge response behaves differently. The CRM is not adequate for predicting behavior near the edge. It is also evident that edge effects seem to be more prominent in heavy grade papers such as 1001b grade as compared to the lighter grades paper. 4.2 Edge Response Modeling using Cosine-exponential Model In order to better understand the true behavior of slice lip actuator responses at the edges and to investigate the effectiveness of the cosine-exponential model (CEM) at modeling those edge responses, modeling was applied to the edge responses directly using the CEM (2.1 & 2.2) described in Section 2.2.1. For edge response modeling, the response of a bumped edge actuator is sectioned from the complete single CD profile response and modeled using the CEM to arrive at the edge response model (ERM). The nonlinear curve fitting function was again used as the modeling algorithm as in section 4.1, however in this case, all the four parameters in CEM are derived based on the edge response instead of the centre response. Only bump test results from grade 1001b paper and the lighter 751b grade papers were used for modeling since edge responses from 52 lb grade paper exhibited very little edge effect. Some graphical results in Figure 4.4(a) and (b) show 2 example of edge response modeling for grade 1001b paper and the lighter 751b. Both graphs show the off-sheet edge response of bumping actuator 2. As shown in the figure, the ERM is able to model the edge responses very well and to keep the RMSE 45 low compared when the centre response model (CRM) is applied to the edge responses in section 4.1. The ERMs are observed to be very different from the CRMs with different model parameter values. The detailed results of the RMSE between the edge responses and the ERMs are presented in Table 4.2. A significant reduction of RMSE between the off-sheet ERMs and the process responses can be observed in Table 4.2 as compared with Table 4.1. The comparison of RMSE between modeling edge response using CRM and ERM for both on-sheet (actuator 3 or 34) and off-sheet edge actuator (actuator 2 or 35) are illustrated in Table 4.3 and Table 4.4. The biggest improvement can be observed for modeling off-sheet edge response where reductions of error are at least 44%. This is to be expected since a set of new parameters for the edge model are generated. In fact, additional degrees of freedom are introduced into the optimization Although both CRM and ERM shares the same form as the CEM, different parameter values are obtained by curve fitting the respective centre and edge response. An example of these different values compared to the parameters from the CRM is presented in Table 4.5 and Table 4.6. There seems to be little connection among the sets of parameters of CRM, parameters of off-sheet ERM and on-sheet ERM. Unfortunately, due to this lack of consistency in the new parameters, little conclusion can be drawn about the true nature of the edge responses and no physical explanation could be found. 46 Table 4.2: Root Mean Square Error (RMSE) between actual edge responses and corresponding ERM using cosine exponential modeling RMSE from Modeling of Bump Position On sheet edge response (actuator 3, 34) Off sheet edge response (actuator 2, 35) Test Bump 1 on 1001b paper 0.12927 0.098 Test Bump 2 on 1001b paper 0.07776 0.09723 Test Bump 3 on 1001b paper 0.07582 0.08062 Test Bump 4 on 751b paper 0.07963 0.05569 Test Bump 5 on 751b paper 0.08283 0.0743 Table 4.3: Comparison of RMSE from modeling using CRM and ERM for on-sheet edge response (actuator 3, 34) RMSE from Modeling of Using CRM Using ERM % Reduction Bump Position of error Test Bump 1 on 1001b paper 0.1537 0.12927 16.9% Test Bump 2 on 1001b paper 0.113 0.07776 31.2% Test Bump 3 on 1001b paper 0.1342 0.07582 43.5% Test Bump 4 on 751b paper 0.0916 0.07963 13.1% Test Bump 5 on 751b paper 0.139 0.08283 40.4% Table 4.4: Comparison of RMSE from modeling using CRM and ERM for off-sheet edge response (actuator 2, 35) RMSE from Modeling of Bump Position Using CRM Using ERM % Reduction of error Test Bump 1 on 1001b paper 0.1778 0.098 44.8% Test Bump 2 on 1001b paper 0.2978 0.09723 67.3% Test Bump 3 on 1001b paper 0.1542 0.08062 47.7% Test Bump 4 on 751b paper 0.1607 0.05569 65.6% Test Bump 5 on 751b paper 0.1449 0.0743 48.7% 47 Table 4.5: Comparison of model parameters from CRM and ERM using cosine exponential modeling for grade 1001b paper Bump Test 1 2 Model Centre On-sheet Off-sheet Centre On-sheet Off-sheet (Bumped actuator positions) response (actuator 18) edge response (actuator 34) edge response (actuator 2) response (actuator 18) edge response (actuator 34) edge response (actuator 2) Bump 200 300 200 200 200 300 Magnitude (Micron) Height, h 2.16 2.75 1.40 1.48 1.44 1.63 Attenuation, a 1.65 1.69 0.99 1.65 1.55 1.11 Width, w 22.84 24.61 19.51 25.09 25.11 19.46 Divergence, 5 0.37 0.35 0.41 0.35 0.38 0.38 Table 4.6: Comparison of model parameters from CRM and ERM using cosine exponential modeling for grade 751b paper Bump Test 1 2 Model (Bumped actuator positions) Centre response (actuator 18) On-sheet edge response (actuator 34) Off-sheet edge response (actuator 2) Centre response (actuator 18) On-sheet edge response (actuator 34) Off-sheet edge response (actuator 2) Bump Magnitude (Micron) 200 300 200 200 200 300 Height, h 1.89 3.19 1.43 2.23 1.83 2.41 Attenuation, a 1.66 1.43 1.02 1.79 1.51 1.13 Width, w 21.24 19.22 15.14 18.88 18.05 17.46 Divergence, 8 0.31 0.36 0.46 0.36 0.32 0.38 48 49 In examining the results, it is observed that, for the same grade of paper using the same response model and modeling technique, different slice lip actuator response models were developed at the edge of the paper and the centre of the sheet. There is little connection between the models. There is always a small hump close to the peak of the response and generally this has not been modeled well. This confirms the finding in Section 4.1 that the actuator response behaves differently at the edge from the centre of the sheet. Although the ERM model seems to work well at the sheet edges, each model is tailored to its particular edge response with four free parameters thus providing a well-fitted model in every case. This is not only time consuming but also offers no physical insight into to the behavior of the actuator at the edges. 4.3 Edge Response Modeling using Reflection Wave Model Following Ghofraniha's [9] conclusion that a dispersive wave model best represented the predicted basis weight response, the RWM, as illustrated in Section 3.4, was applied to analyze the CD responses from both on-sheet and off-sheet edge bumps for grade 1001b and 751b papers. Responses from grade 521b paper are not modeled as they exhibit little edge effect. This model adopts the CRM as the incoming wave generated by a bump of an edge actuator, and its scaled and shifted reflection as the reflected wave term. The model, F(x), used is: F(x) = fix) + Af(x - B) 0 < A < 1 (3.1) 50 where f(x) is the CRM generated from modeling the centre bump responses in Section 4.1, A is the reflected magnitude parameter and B is a horizontal shift parameter in the CD direction. The amplitude, A, of the reflected wave is an indicator of energy loss due to edge boundary conditions, and the CD reflection shift is expected to depend upon the distance of a bumped actuator from the deckle boards at the edge of fourdrinier. The edge responses of interest are sectioned from the single CD bump responses and non-linear curve fitting is applied to arrive at the desired models. Figure 4.5 illustrates the RWM applied to model the response of a bump to off-sheet edge actuator 2 for 1001b grade paper. The irregular line represents the edge response data and the dotted line is the CRM obtained from modeling the centre response in Section 4.1. The dashed curve represents the shifted and scaled version of the CRM as the reflected edge wave and the sum of the dashed curve and dotted line is the final RWM, which is shown as a solid black curve. Similarly in Figure 4.6, the RWM is used to model the response of the same bump position for 751b grade paper. Modeling analysis was carried out on all response data, and the results show consistently that the RWM is effective at modeling the actuator edge response. The RWM is able to follows the edge response closely. Although the RWM is effective at modeling the edge response, it is computationally expensive since it has to estimate a set of three new parameters for each model response. Since the models are tailor-made to each edge response, it is to be expected that the additional parameters will result in a 51 good match. Moreover, the resulting parameters show little consistency between the 100 and 75 grade papers, at least for the given limited sets of bump test data, as shown in Table 4.7 and Table 4.8. Table 4.7: RWM parameters from modeling on-sheet and off-sheet edge response for grade 1001b paper Bump Test 2 Model On-sheet edge Off-sheet edge On-sheet edge Off-sheet edge response response response response (actuator 3) (actuator 2) (actuator 3) (actuator 2) Bump 300 200 200 300 Magnitude (micron) A 0.4147 1 0 1 B -16.1277 -15.636 -14.4186 -17.71 CD response profile (offsheet edge) for grade100#/3000FT2 from bumping actuator 2 CD Position, [bins] Figure 4.5: Edge response modeling of 1001b grade paper using the RWM 52 Table 4.8: RWM parameters from modeling on-sheet and off-sheet edge response for grade 751b paper Bump Test 2 Model On-sheet edge Off-sheet edge On-sheet edge Off-sheet edge response (actuator 3) response (actuator 2) response (actuator 3) response (actuator 2) Bump 300 200 200 300 Magnitude (micron) A 0.028 0.88724 0.4763 0.52146 B -11.6 -14.20 -9.945 -13.83 CD response profile (offsheet edge) for grade75#/3000FT2 from bumping actuator 2 o) 0.2 -0.2 -0.4 -0.6 | l 5 h Edge after trirr \ —•- Edge response data — Centre response model Reflected Wave — Reflection wave model \ ; \ \ \ \ m * \ A. * \ * \ * : \ \ : \ . \ * '•• \ » *i M V \ 1 I \ - \ ••• \ \ \ \ 1 \; \ * V ft :* \Y\ ; t lr\\ ; » \\ * \ \ . , . . \ \ * \ %s / \ \ \ ~~- -j \ A ; \ i i i ; l I I i I i I I i I i I 6 11 16 21 26 31 36 41 46 51 56 CD Position, [bins] Figure 4.6: Edge response modeling of 751b grade paper using the RWM 53 4.4 Edge Response Modeling using Simplified Reflection Wave Model As described in Section 3.5, an attempt to simplify the edge response modeling using RWM requires an examination of the physical meaning behind the dispersive wave theory. Modification is done to the RWM which requires one or more parameters (A and B) in the RWM to be fixed. Three modified RWM are investigated and tested. The first modified RWM deals with shift parameter B and the distance between the bumped actuator and deckle boards. With a fixed position deckle board, the value of B is estimated based on the physical location of the bumped actuator. The distance between the incoming wave and the reflected wave or shift parameter B is estimated from knowledge of the position of the deckle board with respect to the bumped actuator. As a result, only parameter A, the amplitude scaling, need to be found using non-linear curve fitting. With this technique, physical insight can be applied to make the modeling easier. This model is tested on grade 1001b and 751b papers and some results are shown in Figure 4.7 and 4.8. Judging from the values of B in section 4.3, shift parameter B, for 'off-sheet' edge response, can be fixed to be -14 bins. Consequently, shifts B for 'on-sheet' edge response is fixed at -26 bins since the difference between shifts is 12 bins. Same edge responses used for modeling in Section 4.3 are used and applied with the modified RWM with fixed B. Based on the modeling results, it is observed that the modeling of the edge responses is less satisfactory compared to when none of the parameters was fixed. This is 54 anticipated since the modified RWM has one less degree of freedom. However, there is still a significant improvement over the use of the centre response model. CD response profile (offsheet edge) for grade100#/3000FT2 from bumping actuator 2 CD Position, [bins] Figure 4.7: Edge response modeling of 1001b grade paper using modified RWM with fixed shift, B Other tested models involve fixing the amplitude A as A = 1 to simulate an energy lossless impact of an incoming wave and allowing shift parameter B to be estimated. Lastly, a model with fixed parameter A and B is also tested. These two models serve as extreme scenarios to better gage the range of estimation on parameter B. 55 C D response profile (offsheet edge) for grade75#/3000FT2 from bumping actuator 2 1 1 1—1_—i ! 1 1 1, i i i = n C D Posit ion, [bins] Figure 4.8: Edge response modeling of 751b grade paper using modified RWM with fixed shift, B 4.5 Performance Analysis A measuring criterion is necessary in order to analyze the accuracy of fit obtained with the various edge models and parameter choices. As a result, the Root Mean Square Error (RMSE) between the edge responses and the tested models in the previous sections were calculated to determine the goodness of fit. The error between the actual edge response and the corresponding model is calculated and squared before the mean value is computed and rooted to arrive at the RMSE. The results are consolidated and presented in Charts 1 and 2 below. Chart 1 is the performance of edge response modeling on grade 1001b paper whereas Chart 2 shows results for grade751b paper. The vertical columns of the table below represent different models used, and the horizontal row shows the 56 different bump tests. The first column on the left represents the use of CRM derived based on modeling centre response using CEM and the second column shows the ERM derived when modeling edge responses with CEM directly. The third column is the RWM and the fourth column is the modified RWM with shift parameter fixed. Two other variants of the RWM considered and tested are illustrated as well. They are fixing scaled parameter A and estimating shift parameter B, and fixing A and B together. They are shown in column five and six respectively. Grade 1001b paper RMSE compare (off-sheet edge response modeling) Models Chart 4.1: RMS error comparisons for 1001b grade paper 57 Grade 751b paper RMSE compare (off-sheet edge response modeling) • Bump test 1 • Bump test 2 rfl l l -u - Center response model (CRM) Edge response model (ERM) Reflection wave model (RWM) RWM(fixedA.getB) RWM(fixedB.getA) RWM(fixedA&B) D Bump test 1 0.1607 0.055 0 093 0.095 0.093 0.095 • Bump test 2 0.1449 0.074 0.091 0.113 0.091 0.136 Chart 4.2: RMS error comparisons for 751b grade paper It can be observed that very high RMSE are generated when the response model CRM derived from the centre response is used to model the edge response. The best fit is the ERM based on the limited sets of data, but this performance is to be anticipated. Since each model is tailor-made to its particular edge response with four free parameters, a well fitted model is to be expected in every case. The RWM provides a good edge model when tested, as it is indicated by the second lowest RMSE. The performance of the RWM and its modified version with fixed parameter B are relatively on par. Although the resulting fit is less good when shift parameter B is determined based on the location of the deckle board prior to modeling, there remains a significant improvement over the use of the CRM. In conclusion, although the modified RWM with fixed B did not produce 58 the best results compared to other cases, its simplicity and ease of derivation based on physical meaning make it a good edge response model for slice lip actuators in practice. 4.6 Superposition and Linearity Analysis The bump test sessions were also used to investigate the extent to which superposition and linearity in actuator response are borne out in practice. In order to establish superposition, bump tests have deliberately been arranged to allow comparison between the response of a single actuator and that resulting from simultaneous adjustment of two adjacent actuators. Figure 4.9 below illustrates one such example for the 521b paper. Actuator 10 and 11 are bumped individually at different instance with same magnitude. The two individual response models are in dashed lines and their resulting summation model is the solid line. Both actuators are then bumped together with same magnitude. The response model from the simultaneous bump of the adjacent actuators is shown as a line with data points. The RMSEs between the individual bump responses and their models are 0.0719 and 0.0828. The RMSE between the response from simultaneously bumped actuator 10 and 11 and the resulting response model is 0.0757. In comparison, the RMSE between the response from simultaneously bumped actuator 10 and 11 and the summation model, which is derived by adding response models for actuator 10 and 11, is 0.08722. It was found that superposition holds for actuator responses in this example. Another example of superposition is illustrated in Figure 4.10. In this case, the resulting summation of individual response models, from actuator 42 and 43, does not 59 match closely to the response model from simultaneous bump of the same adjacent actuators. The RMSEs between the individual responses and resulting models are 0.218 and 0.189. The RMSE between the response from simultaneously bumped actuator 42 and 43 and the resulting response model is 0.192. In comparison, the RMSE between the response from simultaneously bumped actuator 42 and 43 and the summation model, which is derived by adding response models for actuator 42 and 43, is 0.2197. Superposition is found to hold in actuator response based on the test results Linearity was also tested for centre machine responses and some typical results are shown below. Bump tests were conducted at the same actuator location but with different magnitudes at different instances. The example shown in Figure 4.11 is for 1001b Grade paper. The dotted line represents the centre response from a bump of magnitude 200 microns. The solid line represents the response from a bump at the same location but with magnitude 300 microns. The dashed line is the scaled version of the 200 micron bump response model with 3/2. RMSE between the magnitude 200 micron bump response and its model is 0.17 and RMSE between the magnitude 300 micron bump response and its model is 0.227. In comparison, the RMSE between 300 micron bump response and the scaled 200 micron bump response model with 3/2 is 0.226. Another test is carried out in edge response in Figure 4.12. RMSE between the magnitude 200 micron bump response and its model is 0.037 and RMSE between the magnitude 300 micron bump response and its model is 0.0566. The RMSE between the scaled model and magnitude 300 micron 60 bump response is 0.0728. It is concluded that linearity holds for actuator response in the collected test results. C D response profile (superposition) for grade52#/3000FT2 Response from bumping actuator 11 - • - Response from bumping actuator 10 Response from bumping actuator 10&11 Superposition response 10 20 30 40 50 60 70 80 90 100 110 CD Position, [bins] (a) Respective response profiles C D response profile (superposition) for grade52#/3000FT2 1.5 E < -0.5 I 1 1 1 - • - Response model(actuator10&11) - - - Response model(actuatorlO) Response model(actuator11) — Superposition response model M 3 t a i a i S ' $ i .#/ \ \ I nfe*^ Mr* 1 1 1 1 i 1 1 10 20 40 50 60 70 C D Position, [bins] 100 (b) Respective response models Figure 4.9: Illustration of superposition in actuator response for 521b grade paper 61 CD response profile (superposition) for grade30#/3000FT2 220 240 250 260 CD Position, [bins] 270 280 (a) Respective response profiles CD response model(superposition) for grade30#/3000FT2 — Response model(actuator 43) — Response model(actuator 42) Superposition response model Response model(actuator42&43) 230 240 250 260 C D Position, [bins] 270 280 (b) Respective response models Figure 4.10: Illustration of superposition in actuator response for 301b grade Linearity test: CD response profile (actuator 18) for grade100#/3000FT2 T ! r 60 70 80 90 100 110 120 130 140 150 CD Position, [bins] (b) Respective response models Figure 4.11: Illustration of actuator response linearity for 1001b grade paper Linearity test: C D response profile (actuator 2) for grade75#/3000FT2 Response C from mag200 bump Response D from mag300 bump Response C x 3/2 20 30 C D Position, [bins] (a) Respective response profiles Linearity test: C D response model (actuator 2) for grade75#/3000FT2 1 i Response model C (mag200 bump) model C x 3/2 — Response model D (mag300 bump) V \ \ \ \ \ \ : « Edge after trinr \ / * \ / f \ y y i i j i 10 20 30 C D Position, [bins] 40 50 (b) Respective response models gure 4.12: Illustration of actuator response linearity for 751b grade paper Chapter 5 Conclusions and Future W o r k 5.1 Conclusions This work has focused on the effectiveness of a single basis weight response model, derived using responses from centre of the paper machine, at modeling the edge responses. For cases in which the centre response model is inadequate, a procedure is proposed to replace this model with a new edge model, based on surface wave theory. The performance of this new model is tested and it is found to be more effective at modeling edge response than the conventional model. A typical industrial paper machine CD control system is discussed in Chapter 2 which adopts a cosine-exponential model (CEM) form for the response. This model is widely used as the CD actuator response model, although it is known to have limitations when modeling edge responses. An investigation of the new response model based on surface wave theory of the slurry was carried out and the theory behind its construction was studied. For this model, the disturbance generated by the deflection of the slice lip at the surface of the slurry is characterized as a surface wave. A disturbance of the edge actuators would generate a surface wave which travels towards the edge of the sheet and 65 is reflected by the deckle boards. The edge response could be modeled as the superposition of those two waves. In Chapter 3, a description of the mill CD control system arrangement for the paper machine used for data collection is given. It is found that a common practice in the mill is to place the edge actuators in open loop due to the poor CD control performance at the sheet edges. This poor edge control contributes to a significant amount of paper being trimmed off at the edges. A better edge response model would help to improve that situation. Bump test patterns are carefully designed to isolate the problems at sheet edges and facilitate the comparison of edge actuator response and centre actuator response. At the same time, superposition and linearity in actuator responses were investigated. The dispersive wave model proposed by Ghofraniha [9] as described in Chapter 2 is modified to incorporate the CEM derived using the centre actuator response as the incoming surface wave. With the superposition of its reflected wave, the reflection wave model (RWM) is generated. It is desirable that an edge model be established as simply as possible, so that it can be applied in conjunction of typical industrial control software without difficulty. Examining the physical meaning behind the dispersive wave theory leads to another possible approach to simplifying the RWM. Since the value of shift parameter B in the RWM is related to the physical location of the bumped actuator, it can be estimated based on the distance between the actuator and deckle board. Only the parameter A, the amplitude scaling, then is found by non-linear curve fitting. The centre response model (CRM) required in the dispersive wave model can be easily found using a 66 commercial CD control system by Honeywell. As a result, simple modification is made to incorporate the new edge response model. Various response models described in Chapter 3 are tested in Chapter 4. Several conclusions can be drawn from the results, which are applicable to basis weight responses to slice lip actuators. The paper edge response is significantly different from the centre response, and use of the CRM for the entire sheet is likely to compromise controller performance. The edge effect is more prominent in heavy grade papers. The RWM can be used to generate an improved edge response prediction, based on the centre response, but in general will be computational expensive. A RWM with fixed shift, although not as accurate, is proposed as a useful compromise approach, based on physical principles, to improve representation of edge behavior. The results show that edge response model (ERM) with estimation of four parameters gives the most accurate fit and smallest error, as anticipated. However, for the modified RWM with fixed B, the errors in edge response modeling are still a significant improvement over use of the CRM. Therefore it is recommended that such a model replace the CRM when controlling the edge response of heavy grade papers. Both superposition and linearity were found to hold, to reasonable accuracy, for the bump test results. 5.2 Future Work Some possible research directions from the work presented in this thesis could be in the following areas. 67 The most obvious extension of this work would be to assess the extent to which the better edge model can improve control performance. The modified dispersive wave model with fixed shift has been proposed as better in modeling edge response for heavy grade papers. It is a more accurate identification of CD processes near sheet edges and therefore when applied with CD control algorithm should result in a better CD control. A new CD controller which uses separate models for centre response and edge response could be designed. Application of the new CD model to this CD controller and its application in CD slice lip actuator control would be a desirable extension to this thesis. One of the limitations of the dispersive wave model is the exclusion of the drag/rush phenomenon on the wire. There is typically a speed difference of 4-7 fpm between the slurry and the wire with rush describing the case in which the slurry is faster than the wire, and drag describing the if the situation in which the wire is faster than the slurry. This phenomenon is important to good formation of the sheet. In such case it would be necessary to model the drainage on the wire and the new model would become a 3 dimensional wave with a varying drainage parameter. Investigation on the effect of rush/drag on the drainage characteristics of the wire is a good extension to the proposed response model. There is also a need to consider paper shrinkage and alignment models of the paper machine in order to better model the edge response. Uniform shrinkage of paper in the paper making process would result in misalignment of CD actuator and corresponding 68 scan data. Also, the importance of identifying nonlinear shrinkage paper increases with the introduction of CD actuators with very narrow spacing, such as are ones in headbox dilution control systems, where spacing can be only 35 mm. As the number of actuators increases, even a little increase of the shrinkage at the edges relative to the centre of the sheet can result in the response centres of the edge actuators being shifted more than an actuator zone. An element worth consider in the model is the variation of parameters in the surface wave equation due to changes in depth of the slurry. A variable depth results in a variable wave number or spatial frequency in the equation (2.3). Since spatial frequency is a variable, the dispersive wave equation can not hold. 69 Bibliography 1. D. Gorinevsky, "Performance Analysis of Cross-Direction Process Control Using Multivariable and Spectral Models", IEEE Transaction on Control Systems Technology. Vol. 8, No. 4, July 2000. 2. D. Gorinevsky and M . Heaven, "Automated Identification of Actuator Mapping in Cross-Directional Control of Paper Machine", Proc. of the American Control Conference, Albuquerque, New Mexico, June 1997. 3. J. Ghofraniha, M . S. Davies and G.A. Dumont, "CD Response Modeling for Control of a Paper Machine", Proc. of the 4th IEEE Conference on Control Applications, Albany, New York. ppl07-112, September 1995. 4. S. Mijanovic, G.E. Stewart, G.A. Dumont, and M.S. Davies, " H o , Robustification of a Paper Machine Cross-Directional Control System", Proc. of American Control Conference, Arlington, Virginia, pp2203-2209, June 2001. 5. G.E. Stewart, "Two Dimensional Loop Shaping Controller Design for Paper Machine Cross-Directional Processes", PhD Thesis, University of British Columbia, BC, Canada, 2000. 6. J. Ghofraniha, M.S. Davies and G.A. Dumont, "Modeling of Sheet Formation Including Edge Effects on Paper Machines", IEEE International Conference on Systems, Man and Cybernetics, Vancouver, pp3363-3367, October 1995. 7. J. Ghofraniha, M.S. Davies and G.A. Dumont, "Cross Direction Response Identification and Control of Paper Machine Using Continuous Wavelet Transform", IEEE/ACC97(ID 1-97101E), ppl483-1488, June 1997. 70 8. J. Fan, "Model Predictive Control for Multiple Cross-Directional Processes: Analysis, Tuning, and Implementation", PhD Thesis, University of British Columbia, BC, Canada, 2003. 9. J. Ghofraniha, "Cross Directional Response Modeling, Identification and Control of Dry Weight Profile on Paper Machines", PhD Thesis, University of British Columbia, BC, Canada, 1997. 10. G. Gavelin, "Paper Machine Design and Operation - Descriptions and Explanations", Angus Wilde Publications, Vancouver and Bellingham, 1998. U . S . Mijanovic, "Paper Machine Cross-Directional Control Near Spatial Domain Boundaries", PhD Thesis, University of British Columbia, BC, Canada, 2004. 12. G.A. Smook, "Handbook for Pulp & Paper Technologists", Angus Wilde Publications, Vancouver and Bellingham, 1994. 13. Intellimap 3 User Manual, Honeywell, 2004. 14. S. C. Chen, "Cross Machine Profile Control for Heavy Weight Paper", EUCEPA, Stockholm, pp 77-94, 1990. 15. H. Karlsson and L. Haglund, "Optimal Cross-Direction Basis Weight and Moisture Profile Control on Paper Machines", The Third International Pulp & Paper Process Control Symposium, Vancouver, BC, Canada, 1983. 16. S. Mijanovic, G.E. Stewart, G.A. Dumont, and M.S. Davies, "A controller perturbation technique for transferring closed-loop stability between systems", Automatica, 39(10): 1783{1791, October 2003. 17. The Mathworks, Optimization Toolbox, 2001. 71 18. D.M. Gorinevsky and C. Gheorghe, "Identification tool for cross-directional processes", IEEE Trans, on Control Systems Technology, 11(5):629-640, September 2003. 19. P. Wellstead and W.P. Health, "Two Dimensional Control Systems: Application to CD and MD Control Problem", Control Systems Conference, Whistler, BC, Canada, 1992. 20. W. Westermeyer, "Modeling Flow in the Headbox Slice and on the Fourdrinier Wire with Regard to Cross Direction Basis Weight Profile of the Paper Web", Das Papier 4i(77j,pp591-601, 1987. 21. S. Mijanovic, G.E. Stewart, G.A. Dumont, and M.S. Davies, "Stability-preserving modification of paper machine cross-directional control near spatial domain boundaries", Proc. of IEEE Conference on Decision and Control, pp 4113-4119, Las Vegas, Nevada, USA, December 2002. 22. S. Duncan, W. Heath and P. Wellstead, "Relating Different Approaches to Cross-directional Control", XIVIMEKO World Congress, Tampere, Finland, June 1997. 23. P. Wellstead and M . Waller, "Modeling Paper Machine Cross Direction Profiles", XIV IMEKO World Congress, Tampere, Finland, June 1997. 24. X. Hong, C.J. Harris, M . Brown and S. Chen, "Sparse Model identification using orthogonal forward regression with basis pursuit and D-optimality", IEE Proc Control Theory Appl. 151(4), pp90-98, 2004. 72 25. S. Duncan, J. Allwood, W. Heath and K. Corscadden, "Dynamic modelling of cross-directional actuators: implications for control", IEEE Trans, on Control Systems Technology, 8, pp667-675, 2000. 26. S. Duncan, and K. Corscadden, "Minimizing the range of cross-directional variations in basis weight on a paper machine", Proc. of the IEEE International Conference on Control Applications, ppl49-154, Sept 1996. 27. S. Duncan, and G. Bryant, "The design of dynamics for cross-directional controllers in papermaking", Proc. of the IEEE International Conference on Control, Mar 1994. 28. A. Kjaer, P. Wellstead, and W. Heath, "On-line sensing of paper machine wet-end properties: dry-line detector", IEEE Trans, on Control Systems Technology, 5(6), Nov. 1997. 29. A. Wills and W. Heath, "Analysis of steady state performance for cross-directional control", IEEProc. on Control Theory Appl, 149, pp433-440, 2002. 30. W. Heath, "Orthogonal functions for cross-directional control of web forming processes," Automatica, 32, ppl83-198, 1996. 31. J. Akkermans, A. Wills and W. Heath, "Robust cross-directional control of paper making machines with saturating actuators", Control Systems 2004, Quebec, June 2004. 32. D. Gorinevsky and M . Heaven, "Performance-Optimized Applied Identification of Separable Distributed-Parameter Processes", IEEE Trans. On Automatic Control, 46(10), Oct 2001. 73 33. S. Chen and R. Subbarayan, "Identifying temporal and spatial responses of cross machine actuators for sheet forming processes", IEE Proc. Control Theory Appl, 149(5), Sept 2002. 34. A. Rigopoulos and Y. Arkun, "Reduced Order Cross-Directional Controller Design for Sheet Forming Processes", IEEE Trans. Control Systems Technology, 11(5), Sept. 2003. 35. R. Vyse, J. King, M . Heaven and S. Pantaleo, "Consistency Profiling - A New Technique for CD Basis Weight Control", Pulp & Paper Canada, 97:9, pp 62-66, 1996. 36. S. Duncan, W. Heath, A. Halouskova, and M. Karny, "Application of basis functions to the cross-directional control of web processes", UKACC International Conference on Control, 427(2) , Sept. 1996. 37. J. Kniivila and S. Nuyan, "Focusing on the edges of profile control", Control Systems 2004, Quebec, June 2004. 38. S. Aslani, M.S. Davies, G.A. Dumont and G.E. Stewart, "Detecting Aliasing between Cross and Machine Direction Variations by Variable Sampling Rate", Control Systems 2004, Quebec, June 2004. 39. B. Kan, G.A. Dumont and J. Ball, "Real Time Paper Machine Data Wavelet Analysis", Control Systems 2004, Quebec, June 2004. 40. A.R. Taylor and S. Duncan, "Multivariable Control Loop Performance Assessment for Cross-Directional Control", Control Systems 2004, Quebec, June 2004. 74 41. D. Yang, M.S. Davies, G.A. Dumont and G.E. Stewart, "Modeling Paper Machine Cross Direction Slice Lip Responses Close To Sheet Edges", Control Systems 2004, Quebec, June 2004. 42. S. Chen and R. Subbarayan, "Cross-machine direction (CD) response modeling with two-dimensional sheet variation measurements", Proc. of American Control Conference, June 1999. 43. http://www.paperonline.org/history 44. http://www.stamps.org/care 45. G. Strang and T. Nguyen, "Wavelets and Filter Banks", Wellesley Cambridge Press, 1996 46. P. Wellstead and M . Waller, "Modeling Paper Machine Cross Direction Profiles", Proc of International CD Symposium 1997, pp. 168-73, June 1997, 47. A. Featherstone, J. VanAntewerp and R. Braatz, "Identification and Control of Sheet and Film Processes", Springer-Verlag London Limited, 2000 48. R. Braatz and J. VanAntwerp, "Robust Cross-directional Control of Large Scale Paper Machines", Proc of IEEE International Conf. on Control Applications, pp. 155-160, 1996 49. A. Rigopoulos, Y. Arkun, and F. Kayihan, "Full CD Profile Control of Sheet Forming Processes using Adaptive PCA and Reduced Order MPC Design", Proc of ADCHEM97, pp. 396-401, 1997 50. W. Health and P. Wellstead, "Self-tuning Prediction and Control for Two-dimensional Processes", Int J. of Control, 62:65-107, 239-269, 1995 75 Appendix Modeling Stages and Parameters A.l Stage One Initial edge response modeling was done using cosine-exponential model: g(x) =1/2 h [ b(x - 8w) + b(x + 5w)] b(x) = exp (-ax2 / w2) cos (7ix / w) h - height parameter w - width parameter a - attenuation 8 - divergence, bi-model response A.l. l . For Grade 100#/3000FT2 Bump Test 2448 Model (actuator no.) Imap (18) Centre (18) On sheet (34) Off sheet (2) Bump Magnitude (microns) 200 200 300 200 Height, h 1.9366 2.166 2.7541 1.0166 Attenuation, a 1.2013 1.0293 1.5813 0.0046 Width, w 22.8092 21.8499 24.3721 30.2334 Divergence, 8 0.3995 0.39788 0.3613 2.35 RMS error 0.1703 0.1584 0.1299 (0.1543) 0.18437 (0.34) Variance 0.02377 0.022 0.01714 (0.02409) 0.02379 (0.03235) Bump Test 1407 Model Imap Centre On sheet Off sheet (actuator (18) (18) (34) (2) no.) Bump 200 200 200 300 Magnitude (shifted by 0.2) (microns) Height, h 1.76 1.4701 1.2633 1.5836 Attenuation, a 1.0508 2.5194 0.2333 0.7257 Width, w 18.53 27.0012 30.1683 23.1173 Divergence, 8 0.38979 0.3232 0.3999 0.38209 RMS error 0.15864 0.11706 0.12018 0.13014 (0.1964) (0.3656) Variance 0.01699 0.01358 0.0142 0.01541 (0.013) (0.09074) Bump Test 1447 Model (actuator no.) Imap (18) Centre (18) Off sheet (35) Bump Magnitude (microns) 200 200 200 Height, h 1.4232 1.4184 1.3129 Attenuation, a 0.768 0.76726 0.0198 Width, w 27.24 27.3028 39.35 Divergence, 8 0.37639 0.37578 0.425 RMS error 0.16635 0.13039 0.11995 (0.22552) Variance 0.0209 0.017 0.013788 (0.0211) 77 Bump Test 1659 Model (actuator no.) Imap 01) Centre (11) Bump Magnitude (microns) 300 300 Height, h 3.23 2.8216 Attenuation, a 1.4988 1.5021 Width, w 22.8092 23.9008 Divergence, 8 0.3995 0.3665 RMS error 0.2 0.1765 A.1.2. For Grade 75#/3000FT2 Bump Test 2429 Model (actuator no.) Imap (18) Centre (18) On sheet (34) Off sheet (2) Bump Magnitude (microns) 200 200 300 200 Height, h 2.312 1.7316 3.1955 0.726 Attenuation, a 1.1914 1.0728 1.3929 0.9503 Width, w 18.6058 21.3869 19.1684 24.8023 Divergence, 8 0.3895 0.32329 0.3705 0.0366 RMS error 0.13736 0.11416 0.08 (0.0933) 0.12389 (0.2186) Variance 0.0131 0.01047 0.00649 (0.00855) 0.00938 (0.02644) 78 Bump Test 1302 Model (actuator no.) Imap (18) Centre (18) On sheet (34) Off sheet (2) Bump Magnitude (microns) 200 200 200 300 Height, h 2.1096 2.177 1.8457 1.122 Attenuation, a 1.0703 0.93721 3.8615 0.72667 Width, w 17.9466 18.0299 21.1219 25.8317 Divergence, 8 0.39987 0.39644 0.2619 0.04 R M S error 0.1745 0.1712 0.14152 (0.24251) 0.22288 (0.3297) Variance 0.01369 0.01575 0.01413 (0.0197) 0.0173 (0.0215) A.1.3. For Grade 52#/3000FT2 Bump Test 1837 Model Imap Centre On sheet Off sheet (actuator (18) (18) (3) (35) no.) Bump 200 200 200 200 Magnitude (microns) Height, h 1.8384 1.6850 1.418 1.6545 Attenuation, a 1.2024 1.1852 1.0074 0.96717 Width, w 16.9942 18.45 18.0066 13.2462 Divergence, 8 0.14707 0.098 0 0 R M S error 0.121 0.11 0.1788 0.07873 (0.1982) (0.1356) Variance 0.01149 0.0091 0.0143 0.006 (0.0101) (0.0174) 79 A.2 Stage Two A third DC term is added into the model to take out the DC offset. And the resulting model becomes: g(x) =1/2 h [ b(x - Sw) + b(x + 8w)] + C b(x) = exp (-ax2 / w2) cos (ux / w) h - height parameter w - width parameter a - attenuation 8 - divergence, bi-model response C - a constant number A.2.1. For Grade 100#/3000FT2 Bump Test 2448 Model Imap Centre On sheet Off sheet (actuator no.) (18) (18) (34) (2) Bump 200 200 300 200 Magnitude (microns) Height, h 1.9366 2.1695 2.7515 1.4037 Attenuation, a 1.2013 1.6501 1.6933 0.99428 Width, w 22.8092 22.8405 24.6134 19.5135 Divergence, 8 0.3995 0.37299 0.3567 0.41157 DC shift -0.136 -0.022 -0.255 RMS error 0.1703 0.13405 0.12927 0.098 Edge RMS error (0.1543) withdc (0.34) with dc (use Imap para) (0.1537) no dc (0.1778) no dc Variance 0.0237 0.0181 0.017 0.00983 (0.02409) (0.0323) Bump Test 1659 Model (actuator no.) Imap (11) Centre (11) Bump Magnitude(microns) 300 300 Height, h 3.23 2.8216 Attenuation, a 1.4988 1.5021 Width, w 22.8092 23.9008 Divergence, 8 0.3995 0.3665 RMS error 0.2 0.1765 80 Bump Test 1407 Model Imap Centre On sheet Off sheet (actuator no.) (18) (18) (34) (2) Bump Magnitude 200 200 200 300 (microns) Height, h 1.76 1.4836 1.4499 1.6346 Attenuation, a 1.0508 1.6543 1.5589 1.1175 Width, w 18.53 25.099 25.1128 19.4646 Divergence, 8 0.38979 0.35346 0.38309 0.38209 DC shift 0.066728 -0.1737 -0.1276 RMS error 0.15864 0.11246 0.07776 0.09723 Edge RMS error (0.1964) with dc (0.3656) with dc (use Imap para) (0.113) no dc (0.2978) no dc Variance 0.01699 0.01277 0.00616 0.00967 (0.013) (0.0907) Bump Test 1447 Model Imap Centre Off sheet (actuator no.) (18) (18) (35) Bump Magnitude 200 200 200 (microns) Height, h 1.4232 1.512 1.0912 Attenuation, a 0.768 1.818 3.116 Width, w 27.24 27.0155 26.9796 Divergence, 8 0.37639 0.35023 0.35155 DC shift -0.13053 -0.22345 RMS error 0.16635 0.12217 0.08062 Edge RMS error (0.2255) with dc (use Imap para) (0.1542) no dc Variance 0.0209 0.015 0.0066 (0.02427) 81 A.2.2. For Grade 75#/3000FT2 Bump Test 2429 Model Imap Centre On sheet Off sheet (actuator no.) (18) (18) (34) (2) Bump 200 200 300 200 Magnitude (microns) Height, h 2.312 1.8996 3.1966 1.4392 Attenuation, a 1.1914 1.6639 1.4361 1.025 Width, w 18.6058 21.2455 19.2262 15.1458 Divergence, 8 0.3895 0.31466 0.36877 0.46 DC shift -0.1141 -0.0119 -0.1438 RMS error 0.13736 0.08593 0.07963 0.05569 Edge RMS error (0.0933) with dc (0.2186) with dc (use Imap para) (0.0916) no dc (0.1607) no dc Variance 0.01311 0.00745 0.00646 0.00317 (0.00855) (0.0264) Bump Test 1302 Model Imap Centre On sheet Off sheet (actuator no.) (18) (18) (34) (2) Bump 200 200 200 300 Magnitude (microns) Height, h 2.1096 2.2364 1.8306 2.4117 Attenuation, a 1.0703 1.7938 1.5146 1.1323 Width, w 17.9466 18.88 18.058 17.4632 Divergence, 8 0.39987 0.3663 0.32599 0.3838 DC shift -0.2057 0.16866 -0.2926 RMS error 0.1745 0.08636 0.082839 0.0743 Edge RMS error (0.2425) with dc (0.3297) with dc (use Imap para) (0.139) no dc (0.1449) no dc Variance 0.01369 0.0075 0.0069 0.00564 (0.0197) (0.0215) 82 A.3 Stage Three Then, Reflection wave model was applied. For every edge bump made, a standing wave is generated. This wave travels towards the edge and gets reflected by the edge. The model used is: F(x) = f(x) + A * f(x - B) where f(x) is the centre response model A * f(x - B) is the reflected wave generated by the bump A is the scaling factor and B is the shift and F(x) is the edge response model The resulting model was improved as compared to stage one & two. However, there is the issue of small DC component in the response data which resulted in dc error in the final model fit. A third DC term is added into the model to take out the DC offset. And the resulting model becomes: F(x) = f(x) + A * f(x - B) + C, where C is a constant number A.3.1 For grade 100# /3000FT2 Bump Test 24 48 1407 1447 Model On sheet Off sheet On sheet Off sheet Off sheet (actuator no.) (34) (2) (34) (2) (35) Bump 300 200 200 300 200 Magnitude (microns) A 0.4147 1 0 1 0.80244 B -16.1277 -15.636 -14.4186 -17.71 -21.5693 C 0.022 -0.19039 -0.16 -0.093 -0.1222 RMS error 0.12694 0.1079 0.11297 0.14874 0.1271 Variance 0.0164 0.0119 0.013 0.02263 0.01647 83 A.3.2 For grade 75# /3000FT2 Bump Test 2429 1302 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 0.028 0.88724 0.4763 0.52146 B -11.6 -14.20 -9.945 -13.83 C 0 -0.077 0.2205 -0.2378 RMS error 0.093 0.09388 0.1049 0.0917 Variance 0.0083 0.009 0.01122 0.0086 A.4 Stage Four The modeling is then repeated first with fixed A = 1, amplitude to see effect it has on the shifts, B. A.4.1 For grade 100#/3000FT2 Bump Test 2448 1407 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 1 1 1 1 B -17.47 -15.636 -21.95 -17.71 C 0.084 -0.19039 -0.08 -0.093 RMS error 0.1663 0.1079 0.111 0.1487 Variance 0.02819 0.0119 0.0126 0.02263 A.4.2 For grade 75#/3000FT2 Bump Test 2429 1302 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 1 1 1 1 B -15.16 -14.54 -11.325 -18.96 C 0.1392 -0.07 0.257 -0.25 RMS error 0.2226 0.09516 0.1365 0.1129 Variance 0.0505 0.0092 0.019 0.013 A.5 Stage Five Based on wave reflection theory, determine the values of shifts, for on-sheet and off-sheet edge responses. F(x) = f(x) + A * f(x - B) + C • Total number of actuators = 36, total number of bins = 216, number of bins/actuator = 6. • For off-sheet edge (actuator no 2 = bins 7-12), a bump peaks at bin 7 and trimmed at bin 13. At the time of a bump, reflected wave centres at -7 bin. So B = -14 bins For on-sheet edge (actuator no 3 = bins 13 - 18), a bump peaks at bin 13 and trimmed at bin 13. At the time of a bump, reflected wave centres at -13 bin. So B = -26 bins • Therefore, the difference between shifts B for on-sheet and off-sheet edge responses = 26 - 14 = 12 bins. Also, Shift B for on-sheet edge response > shift B for off-sheet response. Try fixing shift, B, for off-edge response to be -14 bins. So shifts B for on-sheet edge is fixed at -26 bins. 85 A.5.1 For grade 100#/3000FT2 Bump Test 2448 1407 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 0 0.99 0.49 1 B -26 -14 -26 -14 C -0.012 -0.188 -0.1319 -0.0789 RMS error 0.1537 0.1149 0.0975 0.208 Variance 0.024 0.01352 0.00968 0.0442 A.5.2 For grade 75#/3000FT2 Bump Test 2429 1302 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 0 0.8808 0 0.52481 B -26 -14 -26 -14 C 0 -0.0764 0.198 -0.2385 RMS error 0.0933 0.09395 0.139 0.09179 Variance 0.00855 0.009 0.0197 0.0086 A.6 Stage Six The magnitude, A, is fixed at 1 and shift, B, is fixed at 14 (on-sheet edge) or 16 (off-sheet edge). A.6.1 For grade 100#/3000FT2 Bump Test 2448 1407 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 1 1 1 1 B -26 -14 -26 -14 C 0.06645 -0.187 -0.1 -0.0789 RMS error 0.2354 0.1149 0.1143 0.208 Variance 0.05646 0.01352 0.0133 0.0442 A.6.2 For grade 75#/3000FT2 Bump Test 2429 1302 Model On sheet Off sheet On sheet Off sheet (actuator no.) (34) (2) (34) (2) Bump 300 200 200 300 Magnitude (microns) A 1 1 1 1 B -26 -14 -26 -14 C 0.0518 -0.0667 0.2163 -0.1863 RMS error 0.15116 0.09558 0.1818 0.1368 Variance 0.02328 0.0093 0.0337 0.0191 87 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0092014/manifest

Comment

Related Items