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Adaptive control of sheet caliper on paper machines Li, Jing 1994

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Adaptive Control of Sheet Caliper on Paper Machines by Jing Li B. Eng. Tsinghua University, Beijing, China, 1989  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA December, 1994 © Jing Li, 1994  In  presenting this  degree  at the  thesis  in  University of  partial  fulfilment  of  of this thesis for  department  or  by  his  or  scholarly purposes may be her  representatives.  permission.  The University of British Columbia Vancouver, Canada  for  an advanced  Library shall make  it  agree that permission for extensive  It  publication of this thesis for financial gain shall not  DE-6 (2/88)  requirements  British Columbia, I agree that the  freely available for reference and study. I further copying  the  is  granted  by the  understood  that  head of copying  my or  be allowed without my written  Abstract  Caliper control is carried out in the calendering process in order to obtain even thickness of the sheet both in machine-direction and cross-direction. To control caliper, the local temperature of the calender rolls is manipulated. The relationship between temperature and caliper is nonlinear as heating and cooling must be represented by different linear processes. Both models feature timevarying parameters. The correlations among actuators across the roll are described in the interaction matrix of the C D model developed from mill data analysis. Generalized Predictive Control (GPC) is applied under system constraints in machine-direction (MD) and cross-machine direction (CD) control.  This control algorithm can handle the time-  varying parameters, couplings, and variable delays that are present in the caliper control system. The constrained optimal control signals of G P C are obtained by using the Lagrangian multiplier method. The on-line estimator is based upon Recursive Least Squares with Exponential Resetting and Forgetting factors. Simulations show satisfactory performance of constrained G P C in both M D and C D caliper control. The outputs track set-point changes closely and still have small variations even with parameter changes. Load disturbances added in the C D control system are rejected very well.  Table of Contents  Abstract  ii  List of Tables  v  List of Figures  vi  Acknowledgments  1  3  1  Introduction 1.1  2  viii  Background  1  1.1.1  Calendering  1  1.1.2  Caliper Measurement  3  1.1.3  Caliper Control  . . . 3  1.2  Motivation  4  1.3  Contribution  5  1.4  Thesis Outline  5  Caliper Control in the Calendering Process  6  2.1  Interaction in the Calendering Operation  6  2.2  The Calendering Equation  7  2.3  Caliper Variations and Actuators  9  2.4  Machine-direction and Cross-direction Data Separation  11  2.5  Machine-direction and Cross-direction Caliper Control  13  15  Caliper Control Strategy 3.1  Introduction to Generalized Predictive Control  15  3.2  Single-input Single-output Generalized Predictive Control  15  3.2.1  C A R I M A Plant Model and Output Prediction*.  15  3.2.2  The Predictive Control Law  16  3.2.3  The Control Horizon .  . . iii  18  3.3  4  5  6  Multivariable Generalized Predictive Control  20  3.3.1  C A R I M A Plant Model and Output Prediction . . .  20  3.3.2  The Multivariable Predictive Control Law  21  3.4  Generalized Predictive Control With System Constraints  22  3.5  Multivariable System With Variable Delays  25  Adaptive Machine-Direction Caliper Control  26  4.1  Machine-Direction Control Variables  26  4.2  Single-Input Single-Output Model  27  4.3  Parameter Identification  4.4  Constrained Generalized Predictive Controller  31  4.5  Simulation Results  32  4.6  Conclusions  44  . .•  29  45  Adaptive Cross-Direction Caliper Control 5.1  Cross-Direction Caliper Control Mechanism  45  5.2  Cross-Direction Model  46  5.2.1  Mill Data Analysis .  46  5.2.2  Multi-Input Multi-Output Model  52  5.3  Parameter Identification  57  5.4  Constrained Generalized Predictive Controller  5.5  Simulation Results  60  5.6  Conclusions  68  . . 58  69  Conclusions  72  References  iv  List of Tables  4.1 Time-varying parameters of the process  29  5.2 Information on the bump tests  47  v  List of Figures  1.1 The papermaking process. A. headbox; B. wire; C. pickup felt; D. top couch roll; E. wet press; F. pressure roll; G. drier; H. creeping doctor; I. cleaning doctor; J. calender; K. bottom felt; L. reel  1  1.2 Supercalender roll heated by Calcoil for CD caliper control  2  1.3 Accuray caliper sensor  3  1.4 Measurex caliper sensor  4  2.5 Smoothness/bulk relationship  6  2.6 Gloss/bulk relationship  6  2.7 Principle of high-frequency induction heating  10  2.8 The scanning process  11  "  2.9 MD and CD control of caliper on a supercalender  14  3.10 Set-point, control, and outputs in GPC.  17  4.11 Block diagram of adaptive MD caliper control  . .  27  4.12 Adaptive MD caliper control in the heating process  36  4.13 Estimation result for the adaptive MD caliper control in the heating process  37  4.14 Adaptive MD caliper control in the cooling process (operating around 50% of heating power).  38  4.15 Estimation result for the cooling process operating around 50% of heating power  39  4.16 Adaptive MD caliper control in the cooling process (operating around 80% of heating power)  40  4.17 Estimation result for the cooling process operating around 80% of heating power vi  41  4.18 Adaptive MD caliper control in heating and cooling process.  42  4.19 Estimation result for the heating and cooling processes  43  5.20 Cross-machine correlations among actuators and outputs  45  5.21 Block diagram of adaptive CD caliper control  45  5.22 Actuations during the mill trials  46  5.23 Raw data from mill trials  47  5.24 Spatial response to a bump-up test  49  5.25 Time response to a bump-up test  49  5.26 Spatial response to a bump-down test  51  5.27 Time response to a bump-down test  51  5.28 Extracted raw data of a bump-up test (heating process)  55  5.29 Simulated step response of the heating process  . 55  5.30 Extracted raw data of a bump-down test (cooling process)  56  5.31 Simulated step response of the cooling process  56  5.32 Adaptive CD caliper control in the heating process with A = 0.005  63  5.33 Adaptive CD caliper control in the heating process with A retuned  64  5.34 Adaptive CD caliper control in the cooling process  65  5.35 Adaptive CD caliper control in the cooling process without constraints 5.36 Adaptive CD caliper control in heating and cooling process.  vii  . 66 67  Acknowledgments  I wish to thank my supervisor Dr. Guy A . Dumont for his guidance. I also would like to thank the co-reader of my thesis, Dr. Michael Davies, for his advice. I want to thank all my friends at the Pulp & Paper Centre of the University of British Columbia. Special thanks go to Dr. Ye Fu who has given me a lot of help and valuable suggestions. I also would like to express my appreciation for the financial support provided by Dr. Dumont. I would also like to thank my husband, Lijun Wang, who has been very supportive in my pursuit of the Master's degree, and my parents who have taken excellent care of my loveliest 4-month-old, Michelle, to whom this thesis is dedicated.  viii  Chapter 1: Introduction  Chapter 1 Introduction  1.1 Background  1.1.1 Calendering Caliper is the quality used to describe the thickness of the paper sheet as it is manufactured. Caliper control is carried out during the calendering process which is the part of papermaking process that applies intense pressure on the paper through the nips of calender rolls. Figure 1.1 gives a simplified illustration of where the calendering process stands in the process of papermaking.  Figure 1.1: The papermaking process. A. headbox; B. wire; C. pickup felt; D. top couch roll; E. wet press; F. pressure roll; G. drier; H. creeping doctor; I. cleaning doctor; J. calender; K. bottom felt; L. reel [18].  The goal of calendering [24] is to meet the demands for paper with increasingly better and uniform qualities. Various types of calenders have been designed into paper machines to a) correct irregularities in sheet formation, b) improve the surface flatness or smoothness of the sheet and c) compact and densify the paper fibres. Mechanically there is very little difference in the various types of calenders as they normally stand in a vertical position, and have a complement of rolls, bearings, pressure systems, and electric controls. The types of calenders include breaker calenders, smoothing calenders, machine calenders, and supercalenders.  Chapter I: Introduction  Breaker calenders have a complement of three or four chilled-iron rolls and are located between the wet end and the first drier. This type of calenders is to smooth out the lumps in paperboard caused either by the lack of refining of the pulps or by the low-grade ingredients used in some fillers. They are also used to control sheet caliper. Smoothing calenders have been found to be very helpful in controlling wire marks in the sheet and in smoothing the base stock prior to coating. They have two or three rolls of the chilled-iron type. Most commonly placed between the last drier and the winder are the machine calenders and the supercalenders. The major difference between supercalenders [21] and all other types of calenders is that instead of having a full complement of chilled-iron rolls, supercalenders have fibre-filled rolls substituted in alternate positions. While machine calenders can have up to 10 rolls, supercalenders can have from 5 to 16 rolls. There are also soft nip calenders [12] that can be heated to a much higher temperature than ordinary supercalenders [30] [22].  Intot Air  Figure 1.2: Supercalender roll heated by Calcoil for CD caliper control [11].  The caliper of the sheet is changed by undergoing the pressure between nips. This pressure may be manipulated directly by the linear load applied on the calender stacks. Heating or cooling the rolls will increase or decrease their diameter, thus will also change the pressure between nips. 2  Chapter 1: Introduction  Calender rolls can be heated by different systems such as heated steam or oil, radiant hearing or induction heating. CD caliper control in this thesis aims at applying adaptive control to an induction heating system called Calcoil [29]. Figure 1.2 illustrates a Calcoil system on a supercalender. 1.1.2 Caliper Measurement  Caliper measurement is carried out by caliper sensors designed using the principle that the magnetic reluctance between a ferrite disk and a magnetic transducer is proportional to the square of distance between the two. Figure 1.3 and Figure 1.4 show two sensors of the same principle but with different design.  transoucer. floating d e t e c t o r head  reference plate  Figure 1.3: Accuray caliper sensor.  . In the Measurex gauge, both ferrite disk and transducer are in contact with the sheet while in the Accuray sensor, the transducer is mounted on a detector headfloatingon an air cushion.  1.1.3 Caliper Control  Three forms of disturbance are assumed to contribute to variations in paper sheet properties: machine direction (MD), cross direction (CD), and random residual noise. Sheet caliper is influenced by many factors in the calendering process or even before it, and shows each type of variation as variations in the caliper profiles.  Chapter I: Introduction  COMPUTER TOTAL SENSOR SUPPORT CALIBRATION STANDARDIZATION  -CORE WINOINOS - RUBBER BELLOWS  • FERRITE DISK - RUBBER BELLOWS UPPER AND LOWER HEAOSARF RETRACTABLE  Figure 1.4: Measurex caliper sensor. M D and C D control are carried out independently. The principle of M D caliper control is to change the overall stack load while C D control depends on distributing this load across the machine. M D caliper control is generally done by changing the linear load which has the biggest effect on thickness reduction; C D control is by adjusting the temperature profile of calender rolls. The roll temperature affects the nip pressure at any given point through the change in roll diameter. Successive developments of C D caliper control and its benefits are discussed in [7].  1.2 Motivation The nonlinear relationship between caliper and the control variables makes it necessary to represent the overall process by two models, one for heating and one for cooling. Due to many reasons such as limitations in the paper machine gauging system, lack of understanding between process and actuators, and that most current control strategies cannot handle the complexities of the process, caliper control has left a lot of opportunities for improvement. The changing process characteristics and the cross couplings of actuator responses make adaptive control a very appealing strategy. By proper tuning of the controller, it can also handle variable delays in the C D multi-input and multi-output model. It is also necessary to consider constraints in the control design since the actuators usually operate under limits of rate and magnitude.  Chapter 1: Introduction  Thus a constrained adaptive control system is designed to achieve automatic caliper control.  1.3 Contribution Industrial data from mill trials were studied so that the heating and cooling models could be identified for caliper C D control. Machine direction and cross-machine direction controls were developed and carried out independently.  A n adaptive Generalized Predictive Controller (GPC) with constraints is designed for  flattening M D variations. A multivariable G P C was developed to tackle coupling and variable dead time problems in C D control. On-line estimations by RLS with exponential forgetting and resetting factors are implemented for both M D and C D model identification.  1.4 Thesis Outline Chapter 2 discusses correlations among paper properties, and the calendering equation which describes major factors influencing bulk reduction in the calendering process. Caliper variations and actuators in M D and C D control are also discussed. Chapter 3 introduces the G P C strategy for both SISO and M I M O models.  Constraints are  arranged in the form to be used for obtaining optimal control signals by the Lagrangian multiplier method. Chapter 4 and 5 show applications of G P C in M D and C D caliper control respectively with simulation results. Chapter 6 concludes this research work and proposes future studies in this area.  5  Chapter 2:  Caliper Control in the Calendering Process  Chapter 2 Caliper Control in the Calendering Process  2.1 Interaction in the Calendering Operation  The thickness, roughness, gloss, porosity, and printing properties of paper are all changed in the calendering operation. However, because these properties are strongly correlated, it is hard to control any one of these variables independently. Bulk (thickness divided by basis weight) is the property that is used as independent variable in viewing the calendering operation, and the other properties are determined through their relationships with bulk. Figure 2.5 and Figure 2.6 show two of the surface property/bulk relationships for the top side of sheet [5]. E 3  6  c .c o> o  rx  5 4  3L  1  o.  1.0  1.2  1.4  1.6 1.8 2.0 Bulk (crrvg)  2.2  Figure 2.5: Smoothness/bulk relationship.  1.0  1.2  1.4  1.6 1.8 Bulk (cm/g)  2.0  Figure 2.6: Gloss/bulk relationship. 6  2.2  Chapter 2:  Caliper Control in the Calendering Process  Hence, uniform surface properties are possible given a level caliper profile under the precondition that there are no appreciable basis weight variations. It is worth mentioning that steam showers among several new calendering techniques have been proposed to alter the surface properties to bulk relationships, i.e. to decouple surface properties from bulk. Thus steam showers [14] as actuators for gloss, smoothness, or moisture control, can work with caliper control actuators on the calender for independent control of the properties to a limited extent.  2.2 The Calendering Equation When paper passes through the successive nips of a calender, it is difficult to predict how much thinner and smoother it will become, because this depends on a large number of process variables [28] and the properties of the paper. The nip load in each nip, the roll radii, the roll temperatures, the web speed, and the initial moisture content and bulk of the paper entering the stack all have an influence on the properties of the calendered paper [6]. How much each of these variables influences the paper also depends on the furnish composition and furnish properties. The calendering equation [24] accounts for the effects of all major calendering variables individually for each nip of a calender and facilitates the calculation of the reduction in bulk. Note that a change of thickness does not affect the basis weight. The calendering equation predicts the bulk reduction in each nip of a calender using readily determined empirical constants. It is of the form:  e = A + (j,Bi  (2.1)  where: e = the relative compression, defined as the change in thickness (caliper) divided by initial thickness H = the compression parameter defined later Bi = the initial bulk A = experimentally determined constant.  7  Chapter 2: Caliper Control in the Calendering Process  The relative compression may also be defined in terms of the initial and final bulks.  e=(B -B )/B i  f  (2.2)  i  Bi and B are the initial and final bulk. f  The compression parameter u is given by  u = ai + aiiiogL + aslogS + ajilogR + ag0 + a^M  (2.3)  where: ai = an initial constant, experimentally determined aL = a constant factor for the nip load L, kN/m as = a constant factor for the speed S, m/min aR = a constant factor for the roll radius R, m a# = a constant factor for the temperature 9, ° C aM = a constant factor for the moisture content M , % Once these constants have been determined for the paper being studied, they may be applied to each nip in succession by calculating a final bulk after each nip from the relative compression, then using it as the initial bulk for the next nip. The equation applies between the limit:  -A/fi < Bi <  (l-A)/2fi  (2.4)  Outside these limits:  B = Bi f  B  f  = (l-Ay/Au  for for  Bi < -A/u Bi>(l-A)/2/i  The constants must be determined experimentally. They are a function of the furnish properties. Among those machine variables, the linear load on the rolls has the greatest effect on bulk reduction. It not only affects the peak pressure which occurs on the nip centerline, but also the nip width, which affects the residence time. The peak pressure and the nip width can be calculated by:  (2.6)  8  Chapter 2:  Caliper Control in the Calendering Process  where •  b = the half-width of the nip, m;  •  F = the loading force per unit of width, N/m;  •  d = the diameter of each roll, contacting roll 1 and roll 2;  •  V = the Poisson ratio of each roll material;  •  E = the elastic modulus of each roll in Pa. The maximum pressure is given by: 2F  where F and b are defined above. The peak pressure is undoubtedly the predominant factor in calender finishing, although most of thetimethe linear load is reported, since other factors must be known before the pressure can be calculated. In machine-calendering and steel-steel nips, the effect of roll diameter alone on the amount of bulk reduction is only a slowly changing function.. Speed is also a variable that does not affect the bulk reduction too strongly. As speed goes up, the bulk goes up as well because of reduced residence time in the nip at higher speeds. 2.3 Caliper Variations and Actuators Caliper variations are present even before the sheet enters the calendering process. Variations in basis weight, moisture content, thickness, and temperature of the paper entering the calender can all contribute to caliper variations. The calender itself is another major source of disturbance. Calender rolls have a tendency to sag under their own weight, under the weight of the rolls above, and to deform when load is applied externally. CD variations in roll diameter can be caused by faulty grinding or by a poorly designed roll heating system. If a heated roll fails to maintain a constant roll temperature, this will lead to serious MD variations over time. The only actuators which are usually connected to the control computer are those used for CD profiling. MD caliper control is typically carried out by operators [5] [11]. On supercalenders, the 9  Chapter 2:  Caliper Control in the Calendering Process  crowning pressure applied on the journals of the variable-crown rolls is adjusted with either hydraulic or pneumatic pressure regulators. Variable-crown rolls are calender rolls with a larger diameter toward the middle of the roll than toward the edges to compensate for roll deflection. The most important recent advances in caliper control have been made in the development of new actuators to heat the calender rolls locally and to counteract C D variations. Traditionally, unconfined cold air jets were used to alter the calender roll diameter locally.  To extend the rather limited  operating range of these air systems, hot and cold air showers [20] were introduced using air nozzles. However, they did not always have enough power to provide satisfactory control of the profile. The new air actuator systems now use air issuing from confined jets [23] [13] at a constant flow rate. Air temperature can be heated as high as 300°C, and can be very close to the sheet for a longer time. These two factors contribute to the much greater efficiency of the new air actuators. The most radical departure from the traditional air jet actuator system has been the introduction of induction heaters called Calcoils [17]. Figure 2.7 illustrates the principle of A C induction heating. High frequency currents passing through an actuator coil create a varying magnetic field, which generates an alternating current in the roll. The dissipation of these currents heats the roll. C D control can also manipulate the curve shape of the C D zone-controlled variable-crown rolls on supercalenders, thus using V C rolls as actuators.  CALENDER ROLL  Figure 2.7: Principle of high-frequency induction heating [17].  Calcoils can heat a roll but not cool it. Cooling of the roll is accomplished by natural cooling, that is by heat transfer to the contacting paper and by convection to the air currents generated by the surface speed of the rolls. 10  Chapter 2:  Caliper Control in the Calendering Process  Figure 2.8: The scanning process.  2.4 Machine-direction and Cross—direction Data Separation Figure 2.8 shows how the scanning sensor system traverses the sheet in a zig-zag path. Note that the sheet travels at a much greater speed in the machine direction than C D velocity of the sensor. Data is collected by the scanner mounted on the frame to scan back and forth across the sheet. Such measurements, referred to as raw data taken at a typical sampling rate of 30 to 40 seconds, contain M D , C D and residual variations. M D variations are rapid variations and are assumed to have equal influence on all the C D positions. The C D profiles are considered nearly time-invariant. The following two methods are used for separating M D and C D information [15] [2].  (i) Exponential Multi-Scan Trending (EXPO) Exponential multi-scan trending is very simple and widely used in the paper industry. The filter is described by: y (t) n  = (1 - a)y (t n  - 1) + <xy (t) n  where •  y  n  is the estimated C D value at C D position n;  •  y  n  is the present measured deviation from the scan average at C D position n;  •  a is the weight on the measurement, a e [0, 1].  11  (2.8)  Chapter 2:  Caliper Control in the Calendering Process  The trended data are taken as the C D profile of the sheet, and the average of each scan is considered as the M D value.  (ii) E I M C Algorithm  The Estimation and Identification of Moisture Content (EIMC) algorithm was originally developed for moisture M D and C D profile estimations, but it can also be used for processing other scanned data such as caliper measurements. The idea [9] is to estimate M D and C D quantities alternately, using a Kalman filter for M D variation estimation and a Recursive Least Squares estimator with exponential forgetting and resetting factors (EFRA) for C D variation estimation.  The moisture variations can be modelled as:  Vk = P + (1 + Bp )u n  n  k  (2.9)  + v  k  where  •  j^k is the measured profile deviation from the reference level, at C D position n and time instant kT;  •  p is the profile deviation from the mean level in cross-direction at position n;  •  Vk models both sensor noise and neglected higher order terms, and is assumed to be a Gaussian  n  white noise with known variance R; •  Uk is the machine-direction disturbance at time kT,Ufc = u + M D , and £ is a zero mean stochastic process.  •  B is a constant and a function of the reference level.  12  u is the mean deviation in  Chapter . 2:  Caliper Control in the Calendering Process  The EIMC algorithm is expressed as:  +/?-*(^ -l) n  yvki  c" =  i  (1  2  k-i  y  + i y ) [i i]  (2.10)  i = Ff + i i : ( ^ - p ) n  f c +  fc  fc  where a, f3, 8, A are constants in the EFRA. Please see more details of this algorithm in [15]. The EFRA scheme will also be used later for process parameter identification in Chapter 4.  2.5 Machine-direction and Cross-direction Caliper Control  As discussed previously, paper properties are closely correlated with one another. A good caliper control scheme will serve to reduce variations of other properties of the end product, such as gloss, smoothness, etc. Quality control of caliper involves dealing with variations from machine-direction, cross-machine direction, and random causes. A caliper control algorithm will typically include the following steps: 1.  Separation of the machine-direction and cross-direction variations;  2.  Exponentialfilteringof the profiles;  3.  Mapping of the profile vector to the actuator location scale for CD control;  4.  Comparison of the profile with a target profile (usually the profile average value) to yield an error profile for CD control; comparison of the MD value (usually the scan average) with the set-point to yield an error signal for MD control;  5.  The error and the error profile are processed by MD and CD controllers separately.  13  Chapter 2:  Caliper Control in the Calendering Process  Figure 2.9 shows a proposed configuration of the numerous configurations of combined MD and CD control [4] on supercalender stacks. Configurations vary in control variables, type of rolls, type of actuators, and positions of control [16] on the calender stacks. Unwind  Filter  Figure 2.9: MD and CD control of caliper on a supercalender.  14  Chapter 3: Caliper Control Strategy  Chapter 3 Caliper Control Strategy  3.1 Introduction to Generalized Predictive Control Generalized Predictive Control or G P C [1] is a receding-horizon method which depends on predicting the plant output over several steps based on assumptions about future control actions, one of which is a "control horizon" beyond which all control increments become zero. G P C can be used either to control a simple plant with little prior knowledge or a more complex plant such as time-varying parameters, open-loop unstable, or coupled with variable dead-time. It is capable of stable control of processes which simultaneously have the above characteristics with proper tuning. Hence it is suited to high-performance applications such as caliper control in papermaking G P C includes an integrator as a natural consequence of its assumption about the basic noise model, which is shown later.  3.2 Single-input Single-output Generalized Predictive Control  3.2.1 CARJMA Plant Model and Output Prediction Consider a locally-linearized model [3]: A(q- )y(t) = B{q- )u(t-l) 1  + x(t)  1  (3.11)  where A and B are polynomials in the backward shift operator </ : -1  A(q~ ) = 1 + 1  0  l  + ... +  a q-  na  na  (3.12) B(q- )=b 1  + b q-  1  0  1  + ... +  b q-  nb  nb  x(t) is a disturbance term, which is given the form:  x(t) =  Ciq- )^)/* 1  (3.13)  where C(q~ ) = 1 + ciq~ + ... + Cn q~ , £(t) is an uncorrelated random sequence, A is the l  l  nc  C  differencing operator l - q . - 1  15  Chapter 3: Caliper Control Strategy  Thus Equation 3.11 becomes:  A{q- )y(t) = B{q- )u(t-l) l  + C{q- )m/^)  1  l  (3-14)  Equation 3.14 is the C A R I M A ( Controlled Auto-Regressive and Moving-Average) plant model. To derive a j-step ahead predictor of y(t+j), C(q~ ) is assumed to be 1 for simplicity, i.e. the 1  model is taken as:  (3.15)  A{q- )y(t) = B(q- )u(t-l)+at)/Mt) 1  1  Based on Equation 3.15, consider the identity:  l = E {q- )AA  .  + ->F (q- )  1  1  j  q  j  (3.16)  where Ej and Fj are polynomials defined given A ( r j ) and the prediction interval j . -1  Substituting for Ej(q~ )AA l  from Equation 3.15 gives:  y(t + j) = E BAu(t j  + j - l ) + F y(t) + E Z(t + j) j  j  (3.17)  As £ ? j ( g ) is of degree j — 1, the noise components are all in the future so that the predictor, - 1  given measured output data up to time t and any given u(t + i) for i > 1, is clearly:  y(t- j\t) = G Au(t + j - l ) + F y(t) r  where  j  j  (3.18)  = EjB.  Here a whole set of predictions is considered for which j runs from a given range in the future, defined by the minimum and maximum "prediction horizons".  3.2.2 The Predictive Control Law The objective of the predictive control law is to drive future plant outputs y(t+j) close to setpoints. G P C is capable of tracking both constant and varying future set-points. Figure 3.10 shows this receding-horizon approach for which at each sample-instant t, 16  Chapter 3:  Caliper Control Strategy  w set-point  t-2 t-1  t+1  t+N  t+NU  Time t Projected controls  u  Figure 3.10: Set-point, control, and outputs in GPC. (1) the prediction model of 3.18 is used to generate a set of predicted outputs with corresponding predicted system errors e(t + j) = w(t + j) — y(t + j\t); (2) further increments in control after some "control horizon" are assumed to be zero, to provide a suggested sequence of future controls u(t+j); (3) the first element u(t) of the sequence is asserted. The cost function of GPC is of the form: N  N,  3  J(N ,N ) 1  2  = E\  3  ly(t + J)-™(t  + J)} + 2  £  A(j)[Au(< + j - l ) ]  2  (3.19)  where: Ni is the minimum prediction horizon; N2 is the maximum prediction horizon, and A(j) is a control-weighting sequence. Denning f to be the "free response" prediction of y(t+j) assuming no control increments after t - 1, we can arrange the output predictions of 3.18 in the vector form: y = Gu + f  (3.20)  where the vectors are all N x 1; N is the "output horizon" and N = N : 2  17  Chapter 3: Caliper Control Strategy  y = [y(t+l),y(t + 2),...,y(t + N)]  T  (3.21)  it = [Au(t), Au(t + 1 ) , A u ( t + N- 1)]  T  f = [y(t + l\t), y(t + 2\t),.., y(t + N\t)f  ^iC? ) = (ffog + 3i q~ + • • • + gj+n-iq~*~ )', G is a lower triangular of dimension N x N: -1  -1  2  n  go  0  ... 0  31  3o  -  0  G=  (3.22)  L3^-l  9N-2  •••  3o.  The expectation of the cost function can be written as: Ji=^{J(l,JV)} (3.23)  = E^(y - w) (y — w) + Au w| T  T  with the reference sequence w = [w(t + 1), w(t + 2 ) , w ( t + N)] . T  Recall Equation 3.18 and minimize Ji assuming no constraints on future controls, we have the control increment vector: u = (G G + A7) G (w - f) T  1  (3.24)  T  Au(t) is thefirstelement of u, so the current control u(t) = Au(t) + u(t-l). 3.23 The Control Horizon  The control horizon is an important design parameter in GPC controllers. Instead of allowing future control actions to be "free", GPC assumes that after an interval NU < N , projected control 2  increments are zero, i.e.  s  Au(* + j - l ) = 0,  j > NU  (3.25)  The value NU is called the "control horizon"; If NU = 1, only Au(t) is considered, after which the controls are all taken to be equal to u(t). 18  Chapter 3:  Caliper Control Strategy  The use of NU < N significantly reduces the computational burden, for the vector u is then of dimension NU and the prediction equations reduce to: (3.26)  y = G\u + f  where: 3o  0  0  9l  30  0  3o  I-9N-1  9N-2  •••  isN x NU  (3.27)  9N-NU1  The corresponding control law is given by: « = (G^Gi + XI) Gl{w - f) l  (3.28)  and the matrix involved in the inversion is of the much reduced dimension NU x NU. So if NU = 1, this reduces to a scalar computation. In the case NU = 1, the actuations are generally smooth and sluggish. Larger values of NU, on the other hand, provide more active controls. Generally it is found that a value of NU of 1 is adequate for typical industrial processes. Several sets of rules for tuning the predictive controller are presented in [27]. For example, if the response to set-point changes is too slow, then the speed of the system can be increased by: 1. Decreasing A, if it is non-zero. 2.  Increasing NU, if NU is smaller than the order of A.  For more rules of thumb andfine-tuningrules in tuning the predictive controller in various types of closed-loop systems, please see [27]. 19  Chapter 3:  Caliper Control Strategy  3.3 Multivariable Generalized Predictive Control 3.3.1 CARTMA Plant Model and Output Prediction Multi-input and multi-output GPC [19] is able to handle processes that cannot be decoupled by linear state-variable feedback, and take into account constraints on the process inputs, outputs and their derivatives. If the process model is well identified and the control input is within the saturation domain, decoupling can be realized with precautions being taken in tuning the controller to accommodate influences of unstable zeros and the relativetimedelays of the different input-output channels in the MEMO system. The quadratic cost function is of the form: N-l  J=Y,  +J +  VMi* + J + !))*(»(' + J + 1) - VM(* + 3 +. !)) +Au(t + j) Q(j)Au(t  n  0 Q  .  + j)  t  where N is the output prediction horizon, Q(j) is the control weighting matrix, and yM(t+j+l) is the desired output or set-point. Consider the following multi-input, multi-output description of a linear process: A(q- )Ay(t)  = B(q- )Au(t-l)  1  (3.30)  1  where A{q- )=I- A q1  + ... +  l  r  1  A q-  n  n  (3.31) B(q- )=B q1  + ... +  1  1  B q-  n  n  •  y(t) and Au(t) are m x 1 output and incremental input vectors,  •  A,, B, are m x m constant matrices,  •  A = (1 - g " ) / . 1  To develop a predictor of the output, consider the Diophantine identity: / = EjAA  + q~ Fj  j = 1,JV  j  (3.32)  where Ej(q ) and Fj(q ) are polynomial matrices _1  _1  E (q- )=I 1  j  + Eiq-  1  + ... +  Ei_ q-i  +1  1  (3.33) Fjb- )-=*i 1  +  -+n<r  H  20  Chapter 3:  Caliper Control Strategy  These polynomial matrices are uniquely defined by A(q ), A, and the prediction horizon j. _1  Substituting 3.32 in the input-output relation 3.30, we obtain: y(t + j) = EjBAu(t + j - 1) + F (t)  (3.34)  jy  Defining G (q~ ) = E (q- )B(q- ), j  l  1  we have  1  j  y = Gf/ + f t  (3.35)  t  where U = [Au(<) . . . . Au(t + Ny =[y(t+l) y(t-r-2) T  ft=[fi(t) f (t)  T  G  (3.36)  T  N  - E <9"'') <  1  T  . . . f (t) '  T  2  fj(t) = (q&iq- )  T  . . . y(t + N) ]  T  t  T  T  1) ]  r  A  t  +  J~ ) + 1  j(l- )y(i)  F  1  This set of output predictions is given in a similar fashion as in the SISO case except that calculations here involve polynomial matrices. 3.3.2 The Multivariable Predictive Control Law  The quadratic criterion can be rewritten as: J(t) = (yt - VMtfivt - VMt) + AujAAut  (3.37)  where diag(\,A)  A=  yMt = yu(t + i) ,-,yu(t T  +  N) '  T ]T  (3.38)  and 3.35 is substituted for y , Au are defined as U. t  t  If there is no constraint on Au , the solution minimizing J(t) can be explicitly found, using t  21  Chapter 3: Caliper Control Strategy  i  leading to: Au = (G C7 + A ) T  _ 1  t  xG (y r  M t  -f)  (3.40)  As with the SISO GPC, a "control horizon" NU can be introduced to reduce the size of G and used as a "tuning knob" for the controller.  3.4 Generalized Predictive Control With System Constraints GPC is particularly effective when constraints on inputs, outputs, and their derivatives exist. Changes in set-point or in the process dynamics lead to control inputs, which may be characterized by fast variations which often cannot be applied to the process due to the physical limitations of the actuators. The explicit form of control signal mentioned earlier can no longer be applied because the condition on the criterion derivative is generally no more satisfied. Constraints on u and y may be of the following type: max ^ ( 0  u  ^  u  min  u  (3.41) Vmax > V(t) > Vmin &y ax m  > Ay(t) >  -Ay max  where u , u i„, max  and Ay  max  m  y  and y i„  max  m  are respectively the vectors upper and lower bounds; Au  max  are bounds on their increments. Au is iteratively computed using standard quadratic t  programming algorithms with all input-output constraints rearranged in the form of : C Au uy  t  > ip  uy  (3.42) where C  = [Ci, —C\,Ci, —C2,C$, —Cz,C±, -C4]  T  uy  (3.43)  22  Chapter 3: Caliper Control Strategy  with  Ci = J,  (Nn x Nn)  [I , (Nn x Nn)  C = 2  (3.44)  C =G 3  C =G4  G  1  where N is the prediction horizon and the number of inputs and outputs in the system are assumed to be the same and equal to n. V>i = [ A u  ,...,  m a i  Au ] max  i/> (t-l) =  [(u i -u(t-l)),...,(u -u(t-l))]  ip3(t-l) =  [(u -u(t-l)),...,(u -u(t-l))]  2  m  n  min  max  max  (3.45)  Mt) = [(ymin~ h(t)),...,(y -f (t))] min  N  V^O) = [(Vmax ~ /l(<))> • • • , (Vmax ~ /iv(<))] Ve(<) = [(-Aymax - fi(t) + y(t)), (-Ay  - Af (t)),...,  max  xl> (t) = [(Ay 7  - h(t) + y(t)), (Ay  max  max  2  (-Ay  - A/ (<))>•.., (Ay  max  2  max  -  -  Af (t))] N  Af (t))] N  where the fi(t) have been previously given. For the calculations of the above results, please see [8]. The system can be either single-input single-output or multi-input multi-output to use the results presented in this section. At each sampling time, the control signal should minimize J(t) while satisfying the constraints. After the constraints have been arranged into the above form, the cost function of GPC should be rewritten in the quadratic form in order to use the Lagrangian multiplier method to get the optimal control signals [2]. The cost function of GPC can be rewritten as: J = u (G G T  T  + Al)u + 2(f-y t) Gu-r(f-yMt) (f-yMt) T  T  M  (3.46)  thus is in line with the standard form for quadratic programming: min | ^x Hx + c a; j T  T  23  (3.47)  Chapter 3: Caliper Control Strategy  subject to constraints Ax < b. The inequality constraints can be converted to equality constraints by adding a vector of slack variables Ax-6  + <5 = 0  (3.48)  2  Then we can define a Lagrangian function L(x,u) = \x Hx T  + c x + u(Ax-b T  + 6) 2  (3.49)  where ui. is weight vector. The stationary solution x* of Equation 3.49 satisfies: dL(x) |i=x* — 0 dx X dL(x) *.=0 du dL(x) |x=x* 0 06 ] x =  (3.50)  =  where 0 denotes a zero vector. To optimize the problem of constrained GPC as described previously using the above optimization technique, substitute: x=u H = G G + AI T  c = G (f T  (3.51)  y t) M  while substitutions for the inequality constraints are: A —Guy (3.52) b = -lp y U  There are software packages such as the Optimization Toolbox of Matlab [10] that provide solutions of x*. 24  Chapter 3:  Caliper Control Strategy  3.5 Multivariable System With Variable Delays The multivariable predictive control law can also be expressed as [26]: A«(r)  'VM(t+l)  Au(t + 1) R  VM(t +  •  2)  =S  \_Au(t + NU - 1) J  -V(g- )A«(«-l)-J,( - )j,(*) 1  9  1  (3.53)  lUM(t + N)]  where R and S are constant matrices and V ( q  _1  ) and L ( q  _1  ) are polynomial matrices of  appropriate dimensions. R is a symmetric matrix and can be guaranteed non-singular by suitable choices of N and N U . This multivariable G P C strategy does not require an exact knowledge of the plant delay, provided B( q ) is chosen long enough to accommodate the various plant delays in different channels. -1  ' The sufficient but not necessary condition for choosing N and N U is:  N-NU>d,  •max-  I  (3.54)  where dmax is the maximum forward shift in the plant model's interactor matrix. The multivariable G P C control law has also been demonstrated to be independent of input-output pairing if output and control horizons are selected to be the same for each channel. This is a very good feature for controlling industrial processes where measured data are often poorly mapped to actuators.  25  Chapter 4:  Adaptive Machine-Direction Caliper Control  Chapter 4 Adaptive Machine-Direction Caliper Control  4.1 Machine-Direction Control Variables As discussed in previous chapters, there are many kinds of actuators for caliper control and also many ways to place them in different configurations according to paper grades, type of rolls, and operating conditions. M D caliper control is typically carried out by overall stack loading, incoming weight, or number of nips engaged. The nip load can be adjusted by changing the crowning pressure of the variablecrown rolls on the supercalender with either hydraulic or pneumatic pressure regulators. The changes are usually made with reference to tables or graphs showing the appropriate crowning pressures. It is interesting to note that since the rolls are usually heated internally, roll temperature control is important to both M D and C D profile of caliper. But in current practice, it is preferable not to use roll temperature for caliper M D control since constantly higher roll temperatures are usually desired and the time response to temperature change is slow. Instead, the linear load is adjusted manually with reference to graphs or tables. However, dynamic roll temperature control is used for M D caliper control in this study. Some reasons for this are: (1) The experiments carried out to provide data from the mill trials were designed so that heaters across the roll were bumped by the same percentage at the same time. Thus the effect was the same as changing overall roll temperature by use of one big actuator resulting in changing the overall stack load. The fitted model thus carries information about the dynamic characteristics of the process related to temperature changes in a certain time-frame of calendering; hence a worthwhile candidate for adaptive M D variation control study. (2) Roll temperature control, if used in conjunction with linear load control, may be considered as fine-tuning in reaction to M D variations during steady state conditions. This study of the role of temperature can assess the value of an adaptive roll temperature control algorithm for M D caliper control;  26  Chapter 4:  Adaptive Machine-Direction Caliper Control  (3) It is appropriate to choose temperature change as the control variable in this study since other M D control means such as number of nips engaged, linear load, etc. can be related to temperature changes. For example, a higher roll temperature requires fewer number of nips for the same linear load, or a decrease in roll temperature results in the similar way as relieving in the linear load; thus the G P C application in this study may be a reference for applications where other control variables are used, once the dynamic model relating these control variables with caliper is established.  The following sections focus on MD caliper control with discussions on simulation results.  4.2 Single-Input Single-Output Model  MD Estimator  \  Set-point*9~ )  MD Controller  u  PROCESS  Filter  Figure 4.11: Block diagram of adaptive MD caliper control A Canadian mill provided a M D caliper control model through their mill trials during which the actuators along one roll were bumped up or down simultaneously. The effect was the same as loading or relieving the overall stack. Thus the data were used to derive a M D caliper control model with roll temperature being the control variable. The trials show that the system is nonlinear if we consider one process involving both heating and cooling. These two processes—heating and cooling show different time constants according to the direction of set-point change, which implies two models depending on the directions of change in heating levels of the actuators.  27  Chapter 4: Adaptive Machine-Direction Caliper Control  (i) Heating Process Model For heating, the dynamics are nearly an integration process, and can be approximated as:  {  0.5fco, 0 < t < tl,tl = 125mms, ko tl<t< t2, t2 = 250mins, (4.55)  1.5fc t2 < t < t3, t3 = SOOmins, k = -0.00107/xm/% 0  0  where the process gain ki is a negative value which indicates that if the heating level goes up, the caliper will go down. This model which features time-varying ki was obtained from a typical set of mill trial data by the industry; however, it might not be representative for general calendering processes. It can be written in discrete-time form with zero order hold input and sampling interval T =30 s  seconds as:  y(t)=y(t-l)  + bu(t-l)+t(t)/A  (4-56)  where •  y is the M D caliper;  •  u is the system control variable representing heating power applied on the roll;  •  £ is a white noise sequence. b is time-varying. During the simulation period, it changes three times.  (ii) Cooling Process Model For cooling, the dynamics are approximately a first order process:  y[ j s  u(s)  ( T = 10mins,kp =—0.2fim/%, 0 < t < tl = 200mins, TS + 1' | _ 5 T  m z n S 5  k = -0.05fim/%, p  tl<t<t2  = 500mm*,  ( 4  '  5 ? )  which is also discretized in the simulation:  y(t) = ay(t - 1) + bu(t - 1) + £(r)/A 28  (4.58)  Chapter 4: Adaptive Machine-Direction Caliper Control  where both a and b are time-varying, u and y and f are defined as same as in the cooling process. Table 4.1 summarizes the values of time-varying parameters in both heating and cooling processes, a is the pole of the discrete-time model, a = e~ l ; T is the sampling time of one scan; r is the T,  T  s  time constant of the system. Table 4.1 Time-varying parameters of the process. Heating Process  Cooling Process  Scan Number  b  Scan Number  a  b  0 — 249  - 0.0160  0 — 400  0.9512  - 0.0098  250 — 499  - 0.0321  401 —1000  0.9018  -0.0047  5 0 0 — 1000  - 0.0481  Therefore the process is modelled as a linear system with time-varying time constant and gain. The system gain is negative, because heating up reduces the caliper while cooling down makes it larger.  4.3 Parameter Identification In view of the nonlinear model in terms of both time constant and gain, on-line estimation of process parameters should have the ability to be self-adjusting in order to track the time-varying parameters. It is known that the basic recursive least squares algorithm (RLS) has optimal properties when the parameters are time-invariant, and the process noise is white; however, it is not sufficient for a model with time-varying parameters since the algorithm gain converges to zero before it can track the changed parameters of the process. There are several ways of giving tracking abilities to R L S , e.g. constant forgetting algorithm, constant trace adjustment, covariarice resetting, covariance modification. A l l of them have some ability for estimating time-varying parameters, but each has obvious hmitations. Finally, exponential forgetting and resetting algorithm (EFRA) [25] is chosen for the adaptive caliper control. Its main features are: •  exponential forgetting and resetting; 29  Chapter 4: Adaptive Machine-Direction Caliper Control  •  an upper bound for P, i.e. a non-zero lower bound for P";  •  an upper bound for P", i.e. a non-zero lower bound for P.  1  1  This algorithm contains four constants that can be adjusted according to the behaviour of the estimation. By making particular choices for the constants in the algorithm, it reduces to many of the algorithms mentioned above. The algorithm is expressed as the following:  a -a 6 k  p, _ 1 R  ~ ~ 6k  k-i<t>k _  aP  l  +  IT^WT*  , _ <*Pk-i<t>k<t>ZPk-i  (4-59) B  I  _  g  p  2  where •  §k = ^a, SJ is a vector of parameters to be estimated,  •  0fc = [ y{k ~ 1)  •  P is the covariance matrix,  •  a adjusts the gain of the least squares algorithm; typically a e [0.1, 0.5];  •  (5 is a small constant which is directly related to the minimum eigenvalue of P; typically  —  —  1)] is vector of measured input and output, T  a  P G [0, 0.01];  •  A is the usual exponential forgetting factor; typically A € [0.9, 0.99];  •  6 is a small constant that is inversely related to the maximum eigenvalue of P; typically 6 e [0, 0.01].  EFRA, like all identification algorithms, depends largely on whether the excitation contains rich input signals for a fast tracking of process parameters. In the simulations, a = 0.5, (3 = 0.005, 6 = 0.005, A = 0.95. 30  Chapter 4: Adaptive Machine-Direction Caliper Control  4.4 Constrained Generalized Predictive Controller A cost function for the MD caliper control is considered to be: N  N  3  J(t) = E \ ^  3  + £  [y(t + j) - w(t + j)}  2  A[Ati(* + j - l ) ]  . 1 2  J  i=M  (4.60)  As discussed in Chapter 3, J(t) can also be expressed as: Jj = (Gu + / - wf(Gu + f-w)  + \u u T  (4.61)  which can directly give a control signal if there are no constraints on the input and output of the system, and when arranged in the form of = u(G G + \l)u + 2(f - wfGii + (/T  w ) (f-w)' T  (4.62)  which is ready to be used for the Lagrangian multiplier method, a control signal under constraints can be obtained through numerical optimization.. As this thesis studies a practical industrial process, system constraints are unavoidable. In caliper control, the amplitude and rate of change of actuators must be constrained.  Amplitude Constraints Since heaters are the actuators for MD control, there is a certain maximum to their heating power and a limit of temperature to which they can heat up the roll or sheet. Active cooling as provided by cold showers is not available with heaters, rather natural cooling takes place after the heating level is reduced. So the farthest they can go in the cooling direction is being turned off. Hence max  (4.63)  u  is applied in the simulation where u = [u(t) u(t + 1) ... u(t + NU-  l)f  (4.64)  in which NU is the control horizon. If NU > 1, the amplitude constraints apply not only on the current control signal, but also on future controls. 31  Chapter 4:  Adaptive Machine-Direction Caliper Control  The upper bound of heating is chosen to be 75% of its power, and the lower bound is 0%. This is assumed to be the normal operating range of the actuators during all time.  Rate Constraints Rate constraints on actuators limit the change of the control signal at every scan. When significant changes occur in the process, for example, a grade change, or a parameter change, it is necessary to ensure a smooth action by the actuators in order to avoid a break. Even during steady state operation, actuators can be protected if frequent drastic actions are avoided by using constrained signals. Hence  -Au  m a i  < u < Au max  (4.65)  is also applied in the simulation where  u = [Au(t) Au(t+1) ... Au(t + NU-l)f  (4.66)  The bound is assumed to be 4% of heating level during the entire simulation. The rate constraint is also imposed on future control increments if N U > 1. These two kinds of constraints are put together to form the system constraint. Following methods discussed in Chapter 3 and by using the Optimization Toolbox of Matlab, an optimal control signal is produced at each sampling time.  4.5 Simulation Results The adaptive M D caliper control is carried out in three sections of simulations — the heating process, the cooling process and simulations involving both heating and cooling. Whether the process is a heating or cooling process depends on the relative levels of set-point. If the set-point is at a lower level than the previous set-point of caliper, then the closed-loop system will be in a heating process in order to track the new set-point because the overall effect of the closed-loop control is to bring the heating power to a higher level. The control signals will fluctuate around the elevated level. If the new set-point is, on the other hand, at a higher level than the previous 32  Chapter 4: Adaptive Machine-Direction Caliper Control  one, then the overall control signals willfluctuatearound a lower heating level, which indicates the process will be a cooling process. The on-line estimator keeps abreast of set-point changes. It first compares the current set-point with the previous one, i.e.: f <  0  heating  >  0  Cooling  SPcurrent ~ SPpreviousl  -=  (4.67)  0 unchanged  then it either turns to estimate the parameters in the correspondingly changed process model or keeps estimating parameters for the same model in the case of unchanged set-point. In industrial practices, product variances are evaluated by the 2a value, a is the standard deviation of the output compared with the reference level. In the following simulations, 2a is also used to evaluate the variances in the system output. The minimum variance of the output depends on the variance of the noise in the system. A very close value to that of the noise means a very small output variance as a result of good closed-loop control.  Heating Process The simulation time is 600 scans which is long enough to encompass the time-varying characteristics of the model given previously. Figure 4.12 shows the output performance, control signals and parameter estimation. Setpoint is 78 microns before scan 200, and is 77 microns between scan 200 and 400. It then changes further to 76 microns after scan 400. The output shows a very good performance in dealing with the gain changes at scan 250 and 500. Variations due to the parameter change are kept to a minimum. The set-point changes are followed closely, though slower at the start-up. The response is smoother than the output controlled by unconstrained control signals. The 2a value for the steady-state output is 0.21 which is very small compared with 0.20 of the noise. The comparison between constrained and unconstrained control signals shows that, the unconstrained control signal exceeds the amplitude constraint of 75% at the beginning. The result of this on the output performance is a slower response. When the constrained and unconstrained signals 33  Chapter 4: Adaptive Machine-Direction Caliper Control  differ at other scans, it is because of the rate constraint. The result of the rate constraint on actuators is bigger variations in the output. The estimation tracks parameter changes very closely, but does not converge very fast to the true values especially at the beginning. There are many factors that can influence the estimation. The richness of input signals is always the most important. So it is necessary in real industrial processes to have open-loop parameter estimation under PRBS signals for several scans at the start-up before doing the closed-loop estimation. Please refer to Table 4.1 for the true values of the parameters. The process noise also adds fluctuations and offsets in the estimation, particularly when the variance is not small compared with parameter values, which is the case in this simulation. The choices of the constants a, f3, 6, A and the initial values of the covariance matrix or the parameters can all influence the behaviour of the estimator. Thanks to the robust features of GPC, the output still pursues the set-point very closely.  Cooling Process In Figure 4.14, the set-point is 77 microns in the first 200 scans, and increases to 78 microns until scan 500, then it changes to 79 microns. The output shows a much smoother transition at the first two set-point changes with the constraints on control. However, at scan 400, the system time constant and gain changed, the output deviates from the set-point after the process changed parameters. This is primarily due to the constraints on actuators, which can be seen from the control signals. The control signal, unlike in the heating process, is constrained both in rate and amplitude. Even before the set-point change at scan 500, the heating level has already reached the lower bound to adjust to the parameter change, but unsuccessfully. Further set-point changes which require more cooling operation are beyond the system capacity. Looking at the unconstrained control, it indicates negative heating for the process from scan 400, i.e. active cooling such as cold showers, cold air jets are required. Figure 4.16 shows that if the actuator has been working around 80% of its heating power instead of 50%, the output, showing a relatively large variation after the change, at least managed to get back  34  Chapter 4:  Adaptive Machine-Direction Caliper Control  to the set-point without having deviated much after parameter changes at scan 400. Its performance also improves a great deal after scan 500 though much of the controls are bounded. Figure 4.14 indicates that it is necessary for real systems to have a combination of actuators to accommodate all kinds of paper grade changes and parameter changes. In Figure 4.14 and Figure 4.16, the estimator gives the same parameter estimations which show the tracking ability of the estimator more clearly than in the heating process.  Heating and Cooling Process In Figure 4.18, the simulation time is 400 scans. Starting with the first 100 scans with the cooling process, heating and cooling switches every 100 scans. The set-point changes by 1 um at scan 1 and at scan 100. The set-point changes by 2 urns at scan 200 and at scan 300. The output performance is very satisfactory in tracking model changes, with variations kept to the same level as in either heating or cooling. The unconstrained control signal exceeds amplitude constraint at two points — scan 200 and scan 300, in cooling and heating respectively. The control signal is only rate constrained at other times. If the control horizon is reduced to one step, the control will be more sluggish but is generally acceptable for typical plant models including the ones in the simulation. Estimation of the parameter a shows that the true value a = 1 is not estimated in the heating process. As in the two previous processes, it is seen from both parameters a and b estimations that constraints on actuators have less impact on the estimation result from the unconstrained case than they do to the output.  35  Chapter 4:  Adaptive Machine-Direction Caliper Control  Set-point and Output 79.5 79 — set-point — without constraints  78.5  78 hi 5  77.5  o 77h 76.5 76 75.5  100  200  300 Time (Scan)  400  500  600  500  600  Control Signal 80  unconstrained constrained  35  100  200  300 Time (Scan)  400  Figure 4.12: Adaptive MD caliper control in the heating process. 36  Chapter 4:  Adaptive Machine-Direction Caliper Control  Estimation of b  -0.01  -0.02  -0.03  -0.04 h  -0.05 h  -0.06  -0.07 h  -0.08  300 Time (Scan)  400  500  600  Figure 4.13: Estimation result for the adaptive MD caliper control in the heating process.  37  Chapter 4: Adaptive Machine-Direction Caliper Control  Set-point and Output 80 r 79.5 h 79 h  - set-point - without constraints  i  1  78.5 h  76.5 h 76  u  100  200  300 Time (Scan)  400  500  600  Control Signal  I a I  •frr  unconstrained constrained I \  i  IF TM l , !1 1  -20 h  ill i ' 1  -40  1  r  I  I ' > l II II  1  I  -60 r 100  200  300 Time (Scan)  400  500  600  Figure 4.14: Adaptive MD caliper control in the cooling process (operating around 50% of heating power).  38  Chapter 4:  Adaptive Machine-Direction Caliper Control  Estimation of a  300 Time (Scan)  400  500  600  400  500  600  Estimation of b  -0.05  100  200  300 Time (Scan)  Figure 4.15: Estimation result for the cooling process operating around 50% of heating power.  39  Chapter 4: Adaptive Machine-Direction Caliper Control  Set-point and Output 80 79.5 • set-point - without constraints 78.5 o  78  (0 o  77.5  76.5 76  100  200  300 Time (Scan)  400  500  600  Control Signal I  •  I  I  80  -  60  „  40 -  I  — without constraints -  ID  I  (0 x  j  20  Oh  1 , ) .  -20  -40  i  i  100  200  i  300 Time (Scan)  i  400  t  1 500  600  Figure 4.16: Adaptive MD caliper control in the cooling process (operating around 80% of heating power).  40  Chapter 4: Adaptive Machine-Direction Caliper Control  Estimation of a On  1  1  1  1  1  -0.1  -  -0.2  -  -0.3  -  -0.4 -0.5  -  -0.6 -0.7  -  0  100  200  300 Time (Scan)  400  500  600  Estimation of b On  ,  -0.04-  J'  -0.045 -  |  -0.05' 0  \  ,  r  j  •—— 100 1  1  200  1  300 Time (Scan)  1  400  1  500  1  600  Figure 4.17: Estimation result for the cooling process operating around 80% of heating power.  41  Chapter 4:  Adaptive Machine-Direction Caliper Control  Set-point and Output 801  751  0  1  1  50  1  1  100  r—  1  1  150  i  200 Time (Scan)  i  250  i  300  i  350  I  400  Control Signal  Time (Scan)  Figure 4.18: Adaptive MD caliper control in heating and cooling process.  42  Chapter 4:  Adaptive Machine-Direction Caliper Control  Estimation of a 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 • - - Without Constraints  -0.8 -0.9 -1  100  150  200  250  300  350  400  250  300  350  400  Estimation of b  -0.01  -0.02  -0.03  -0.04 h  -0.05 h  -0.06  -0.07  Figure 4.19:  150 200 Time (Scan)  Estimation result for the heating and cooling processes.  43  Chapter 4:  Adaptive Machine-Direction Caliper Control  4.6 Conclusions The machine-direction caliper control is modelled in two parts — heating and cooling, each is a linear model with time-varying parameters. The heating process is an integration process which is open-loop unstable, and the cooling process is a first-order process with a very large and timevarying time constant. The control variable is the heating power of the actuators. The parameters are estimated by the recursive least squares with exponential forgetting and resetting factor. As industrial processes have physical constraints on actuators, the controller is designed as a constrained generalized predictive controller. The constraints are on amplitude and rate of change. Simulations are of three different processes: heating, cooling, and heating with cooling. The controllers were tuned with N\ — 1, N  2  = 3,NU = N = 3, A = 0.001. Except for the case shown 2  in Figure 4.14, overall the output shows prompt tracking of parameter and set-point changes with variations kept to a minimum. Figure 4.14 is a reminder that sometimes an unsatisfactory product may be produced due to the inadequacy of the physical system or due to a particular operating range.  44  Chapter 5: Adaptive Cross-Direction Caliper Control  Chapter 5 Adaptive Cross-Direction Caliper Control  5.1 Cross-Direction Caliper Control Mechanism As discussed in Chapter 2, C D control is carried out by manipulating the roll temperature across the width of the paper sheet. The Calcoil is the actuator used in the mill trials discussed in this thesis. These actuators are heaters mounted along one roll, each 75-100 mm wide. They vary sheet or roll heating and thereby localized stack load without changing the overall stack load. 1  u  2  Q \  \  \  3  \  v  \ \  \  / /  V \  / \ /  \  V \ V  V A  /  6 1  / / / /  \  / /  /  / \ \  \ \ \ \ \  \' /  /  I  /  '\  v  \  6  \ \  /  \  \  \  \  Q o  / /  \  V  /  \  / \ /  /  /  /  A / \ / \ / \ / \ / \  '  /  \  /  /  /  \  \ \ \  6 O 6 6 O 2  3  4  5  6  o 6 n-1  Figure 5.20: Cross-machine correlations among actuators and outputs  CD Estimator  Target profile  +  CD Controller  n  \1  / /  \  \  n-1  Q \  /  . \  \  V  /  \  \/ «  /\ / * / \ / \ / \ / \ / v  5  Q O Q  9 / / / /  4  u  PROCESS  Figure 5.21: Block diagram of adaptive CD caliper control. 45  Filter  n  Chapter 5:  Adaptive Cross-Direction Caliper Control  Figure 5.20 is an illustration of couplings in C D caliper control. If one actuator moves to correct irregularity in the measurement of its corresponding zone, it will not only affect this one zone, but also the neighboring zones. Viewing the couplings from the measurement side, one measurement at a certain C D position will change even if its corresponding actuator does not move, but because its neighboring actuators have control signals. Thus it is important to take into consideration the couplings in C D modeling and control. With good estimations of the parameters in the C D model, G P C should allow decoupling in the closedloop system. Figure 5.21 is an illustration of the C D control loop.  5.2 Cross-Direction Model  5.2.1 M i l l Data Analysis  Scan Number  Figure 5.22: Actuations during the mill trials. 46  Chapter 5: Adaptive Cross-Direction Caliper Control  Scan Number  Figure 5.23: Raw data from mill trials. Table 5.2 Information on the bump tests.  s  On the CalCoil  c Zone 11  Zone 20  Zone 31  10  | 25%  T 25%  36  t 25%  | 25%  T 25% 1 25%  a n  f: bump up ; J.: bump down.  Data was obtained from the same mill which provided the MD caliper control model. Trials were carried out with individual actuators bumped up or down. Information on the bump tests provided by the mill are summarized in Table 5.2. The actions taken on the actuators are shown on Figure 5.22. The small bumps beyond zone 40 were actions 47  Chapter 5: Adaptive Cross-Direction Caliper Control  on another kind of actuator for other experimental purposes. Those data are not of concern in this study. Figure 5.23 shows the data of caliper in response to the bump tests. From scan 10 to scan 35, the Calcoil at zone 11 was bumped down by 25% from 31% to 6% of heating power. Figure 5.23 shows caliper at zone 11 and neighboring zones increased in response to this bump test. During the same period of time, actuators at zone 20 and zone 31 were bumped up. Increased heating decreased caliper around these zones. From scan 36 on, heating level of Calcoils at the above zones was switched back by 25% from 6% to 31%. After studying the raw data to pick out the most useful information, two bump tests are chosen for determining a CD model. The basic principle in choosing the data that contains more usable information about the process is to plot the output increments of zones neighboring the bumped zones on both sides from scan 7 to scan 35 or from scan 35 to scan 59, and compare the plots in order to see which ones show more predominant responses to the impulse control actions for either heating or cooling. The two data sets chosen are: the bump-up test at zone 31 at scan 10, and the bump-down test at zone 11 also at scan 10. These two processes both start from a steady-state.  (i) Heating Process Model Data was extracted from scan 9 to scan 35 in time and from zone 27 to zone 35 in space, to study the couplings among zones in the heating process. Figure 5.24 shows the data in spacial correlations. It is the response in the cross machine direction. The response drops most at zone 31 where the actuator was bumped up, and less on its two neighbors — zone 30 and zone 32, but remains almost unchanged beyond this neighborhood. This means the bump-up on one actuator of the Calcoil primarily influences its adjacent neighbors on each side. And the influence is symmetric according to the centre zone. The amplitude of this influence can be estimated to be one half of the amplitude of the centre zone.  48  Chapter 5: Adaptive Cross-Direction Caliper Control  30  31 Zone Number  34  32  35  Figure 5.24: Spatial response to a bump-up test. 80  r  78  Zone 33  Zone 31 68 h  66  L  10  15  20 Scan Number  25  30  Figure 5.25: Time response to a bump-up test.  49  35  Chapter 5:  Adaptive Cross-Direction Caliper Control  Figure5.25 shows the data in time response. Seen again are many of the characteristics shown in Figure 5.24, but also with time delay information. The time responses of zone 30 and 32 are very similar to each other and are estimated to have two more scans in time delay than zone 31 which is the zone central to the control action. The C D model is therefore expected to have variable delays on different input-output channels, but with the features of G P C discussed in Chapter 3, this should be of no problem in yielding good outputs as long as the controller is properly tuned.  (ii) Cooling Process Model Data was extracted from scan 9 to scan 35 in time and from zone 27 to 35 in space for studying the cross-machine direction relations between actuators in the cooling process. Analysis follows the same way as for the heating process. Figure 5.26 shows the spatial response in the cooling process. Caliper increases most at zone 11 where the actuator was bumped down.. Two more zones at each side of the centre zone also produce much increased caliper than rest of the zones, which means the input influences at least five zones with symmetric characteristics to the centre zone. The gains of the neighboring zones can be estimated by the basic least squares method. Figure 5.27 further verifies what is observed in Figure 5.26, except that the responses of zone 9 and zone 13 are not quite symmetric. This can be contributed by the noise in the system during the trial; in the following proposed model it is assumed that the second most adjacent actuators to the bumped-down zone have symmetric characteristics. Figure 5.27 also shows the different time delays at different zones. Zone 10 and 12 have one more scan in delay than zone 11, and zone 9 or zone 13 have another one more scan in delay than zone 10 or 12.  50  Chapter 5:  10  Adaptive Cross-Direction Caliper Control  14  13  11 12 Zone Number  15  Figure 5.26: Spatial response to a bump-down test. 80  1  i  i  i  79  -  78 77 Zone 11 Zone 10  In  Zone12  «j o  -\, /  /<.  \  s  \  s  - - J—  t-'-J^^Z y^^^^  74  "  ' /  ^ /  /  *  —I  73 Zone9 Zone13  72 71 i  70  10  15  i  .  20 Scan Number  25  30  Figure 5.27: Timeresponseto a bump-down test.  51  35  Chapter 5:  Adaptive Cross-Direction Caliper Control  The above analysis shows that apart from the differences between the heating and cooling processes that are present in the MD  model, there are also differing correlations among zones in  the cross-machine direction. The main differences in this aspect are: •  the number of outputs that can be affected by interactions from one input: 3 in the heating process, 5 in the cooling process;  •  how much is each output influenced by the input: for heating, the gain of the response on most neighboring zones is 50% of that on the centre zone; for cooling, it is 63.3% on the nearest zones, and 31.65% on the second nearest zones;  • time delay of each influenced output: the neighboring two zones have 2 more scans in delay than the centre zone in the heating process; in the cooling process, the nearest zones have 1 more delay than the centre zones, and the second nearest zones have 2 more scans of delay than the centre zone.  5.2.2  Multi-Input Multi-Output Model  Summarizing the characteristics displayed by the raw data, a CD model is proposed as the following:  Ay(t) = Bu(t - 1) + e(t)/A  (5.68)  where •  B is called the interaction matrix which contains the information about couplings;  •  A and B are n x n polynomial matrices, where n is the number of actuators in the system;  •  y and u are n x 1 vectors representing all the outputs and inputs;  •  e is a white noise sequence;  •  A = 1-q-. 1  The values of elements in A and B are based on the MD model given in Chapter 4 and the above data analysis. 52  Chapter 5: Adaptive Cross-Direction Caliper Control  (i) CD heating model  1 - q-  0  0  1 - q-  1  b ~  b ~  b  2  iq  0  b ~  2  b ~  :  0  iq  0  :  1  0  1 1  sfe(<) v»(<) o  1  (5.69)  •«l(t-l)"  0  u (t - 1)  e (i) 2  2  b  2  iq  2 / 1 (<)'  1.  1-q-  0  0  2  iq  1  1  0 • b  0  :  o  M~ 0  6 -  6  2  ag  /A  •  u (t- 1)  e»(*)  0  0  n  i  2  +  •  1  where the zero input element and a column of "l"s in B are used to prevent a badly scaled or singular matrix; bi = b/2. (ii) CD cooling model  •1 — aq  0  1  0  1 - aq'  0 •  0 b q--l  • b  x  b b q-  2  2  -l  hq'  ;  0  . 0  hq-  hq~  0  2  1  '•.  0  0  1 — aq'-1  0  b q~  2  ;  0  hq'  o  r  0  2  2/2(0  *  1  "«l(*-l)"  • (ty  u ( « - 1)  e (t) 2  2  b  2  :  hq-  +  ;  2  hq-  1  b  1.  1  b q~  2  2  hq'  1  53  (5.70)  ei  « „ ( * - 1)  e»(«)  0  0  /A  Chapter 5:  Adaptive Cross-Direction Caliper Control  where •  a and b are time-varying as in the M D model.  •  bi = b/1.58;  •  b  2  = bi/2; or b  2  = b/3.16.  Figure 5.29 and Figure 5.31 are the simulated step responses in heating and cooling respectively. Comparison between these two figures and Figure5.28 and Figure 5.30 shows that the overall model is very close to reality.  54  Chapter 5:Adaptive Cross-Direction Caliper Control  Chapter 5: Adaptive Cross-Direction Caliper Control  **1  Figure 5.30: Extracted raw data of a bump-down test (cooling process).  i ] 3  35  7  Figure 5.31: Simulated step response of the cooling process.  56  Chapter 5: Adaptive Cross-Direction Caliper Control  5.3 Parameter Identification The number of actuators on a calender roll in the cross direction was 80 for the system in the mill trials discussed previously. If there were no knowledge about the system correlations, multivariable parameter identification would mean require 80 x 80 estimations of the elements in aninteraction matrix. Obviously this is difficult to achieve on-line within reasonable time. Fortunately there is usually some knowledge about the system that can help reduce the number of parameters to be estimated. The parameter identification method used in this study is based upon the following known features of the interaction matrix in the C D model:  •  B is a band matrix; only the non-zero elements need to be estimated;  •  the band is symmetric according to the main diagonal of the matrix;  •  in the heating process, bi = b/2; in the cooling process, bi = b/1.58 and b = b/3.16; such 2  relationships can be assumed to be time invariant. •  If there are not serious edge effects, bi and b in the first or last row need not be identified 2  separately.  These features are nearly time-invariant compared with the time-varying parameters of the process in machine-direction, and so can be updated by off-line estimation. Depending on how many of these features are utilized, the identification can be relatively simpler. If only the first feature is to be used to simplify identification, then the algorithm which is RLS with exponential resetting and forgetting factor can be employed in the following way [2]:  ek = Vkn  * P  _ !  p  _n  <j>lh-\ <*Pk-l<f>k  *-  1+  .  T T ^ p W r *  aP -i<f>k<f>ZPk-i , k  where  57  (5.71) o  r  ,-2 p  Chapter 5: Adaptive Cross-Direction Caliper Control  •  6=  hi b\\ 612  ^  0=  a  n  6(„_i)  and fa = [—y\(k — 1) u\(k — 1) u (k — 1)] , for the first row; 2  *. "j X"  < N n  6„„  m  and ^ = [-y (k - 1) w - i ( ^ - 1) u„(k - 1)] , for the last row. n  n  The independency of the input signals is a necessary condition for this method, and it is very important to have enough information from the input signals. If all of the known features are utilized, it would mean that only the values of a and b are to be estimated. Note that in M D control, the M D estimator takes the average of each scan as the M D value and gives estimations of a and b. In this way, the parameters in the C D model may adopt the estimations out of the on-line M D estimator, while the coupling relations are estimated by off-line batch data. This method is acceptable as long as the coupling relations among actuators change only very slowly with time.  5.4 Constrained Generalized Predictive Controller The cost function for the above multivariable system is: JMGPC  = [GU + f - «;]  [GU + /-«;]  +  U AU T  (5.72)  where w is the target profile. For N U = 1:  'f/= A (tf  Au (t)  ..  T  Ul  2  w = [w(t + 1 ) w(t + 2 ) T  f = [f (t + 1 ) f (t + 2 ) T  Au {t) ] T  n  • • w(t + N ) ]  T  T  2  • f(t + N ) ] T  T  T  (5.73)  2  fA A=  A and  w ( t + j ) = [wi(t+j) w ( t + j ) 2  . . . w (t+j)] n  f(t+j) = [fl(t+j) f (t+j) ... fa(t+j)] 2  j = l, ... 58  N  2  (5.74)  Chapter 5: Adaptive Cross-Direction Caliper Control  n is taken as 20 in the simulations instead of 80 as in the real process for sake of reducing simulation time. The choice of N  2  is different in the heating and cooling simulations because of the variable delays  in the system. Recall the sufficient condition for choosing N and N U in [26] to ensure closed-loop 2  stability, N needs to be equal or larger than N U + maximum delay - 1. The maximum delay term in 2  both models is 3 and N U is 1, which indicates N should be at least 3 in both processes. Moreover, 2  for the heating process which is open-loop unstable, simulations show that it is necessary to set N  2  = 4 in order to give a smoother control. So in the simulations, N = 3 in the cooling process, and 2  N  2  = 4 in the heating process. The choice of A will be different for the two processes, as will be discussed with simulation  results. As in the M D control, C D caliper control also faces constraints on actuators in amplitude and rate of change.  Amplitude Constraints The control signals for all actuators at all times are bounded as:  Umin  < u(t) <  U  m  a  (5.75)  x  where u(t) = [ ( t ) u (t) . . . u ( t ) ] Ul  2  T  n  (5.76)  and each control signal in u is bounded in the range of [5%, 75%] in terms of the heating level of the actuators.  Rate Constraints Rate constraints are also imposed on all actuators at alltimesby: -Au  m  a  x  < U <Au max  where each control increment in U is bounded between - 4 % and +4%. 59  (5.77)  Chapter 5: Adaptive Cross-Direction Caliper Control  As in M D control, theses two kinds of constraints are imposed on all actuators at any time in the system, and an optimal control signal on the basis of G P C is produced by using the Optimization Toolbox in Matlab [10] at every sampling time.  5.5 Simulation Results Simulations of C D caliper control performance are also carried out in three parts — heating process, cooling process and process of both heating and cooling. The estimator decides which model parameters to estimate on the same principle as in M D control, that is, to compare current set-point with the previous one:  {  <  0  heating  >  0  cooling  =  0 unchanged  (5.78)  Heating Process The simulation time is 600 scans including two points at which parameters change. Set-point is 78 microns from scan 4 to scan 349, and is 73 microns from scan 350 to 600. From scan 80 to 90, a load disturbance of 0.3 microns is present from zone 10 to 12. b changes at scan 250 and 500. Please Refer to Table 4.1 for details of process parameters. Figure 5.32 shows the system output in three dimensions. Note in this simulation, A in the G P C controller is not re-tuned during the process and is constant at A = 0.005. Before the parameter change at scan 250, the output profile is very smooth and handles the load disturbance successfully. But after the first time of parameter change, the system output becomes somewhat oscillatory, even more so after the second time of parameter change. The constrained controller is tuned with A = 0.005 during the whole simulation. Although the control signals are oscillating after the parameter changes, they are bounded in rate of change and amplitude. The oscillations are a direct result of the interactions in the system. If we only take the caliper variations in time to view at any one of the zones from the 3-D response, we can see that the output  60  Chapter 5: Adaptive Cross-Direction Caliper Control  at any one zone is oscillating with time, too. This is in contrast to what we see in the MD control where there are no concerns of interactions with the tuning parameter of the controller, A, is equal to 0.001. With the interactions in CD control, A = 0.001 — 0.005 is not efficient in handling the variations introduced by time-varying parameter changes. So it becomes a problem of eliminating the oscillations intimeby proper tuning of the GPC controller. One good feature of GPC is that it is relatively easy to tune according to output performances. In the above simulation, oscillations almost double in amplitude each time after the parameters change; it turns out that by doubling the value of A at each point of parameter change, the output variance is well kept to a much lower level without the oscillations. Figure 5.33 shows the improved output and the constrained control signals. The average la value for the improved output after thefirsttimeof parameter change is down from 0.20 to 0.06, and down from 0.35 to 0.075 after the second time of parameter change. The comparison of these two simulation results indicates that with on-line parameter identification in order to detect parameter changes in the system, it is not only possible but sometimes also necessary to tune the adaptive controller in order to get outputs with less variations. Control signals respond to the load disturbance in the same direction, i.e. when the sheet is thicker in some area, the heaters will increase heating power to suppress it. Because of the interactions in the system, actuators adjacent to the thicker area have to move with those actuators in the area with load disturbance. Figure 5.33 or Figure 5.32 shows that the system successfully keeps the effect of the load disturbance in a limited area.  Cooling Process Simulationtimeis 500 scans during which the set-point changes at scan 250 from 73 microns to 78 microns, and the parameters a and b change at scan 400. A load disturbance is present from zone 10 to 12 and from scan 80 to 90, which is the same disturbance as in the heating process. Figure 5.34 shows the system output and control signals under rate and amplitude constraints. 61  Chapter 5:  Adaptive Cross-Direction Caliper Control  It is seen that two zones from each edge suffer from offsets after the parameters change at scan 400. This is due to the same reason as discussed in cooling process M D control that there is not enough cooling because of the constraints on actuators. This can be verified in Figure 5.35 where it shows output and control without any constraint. On account of the interactions and proper C D control, the system tries to solve the insufficient cooling problem within itself, and manages to push the variances to the ends. As discussed in M D control, there are two ways to avert the problem. One is to apply cold showers or cold air jets to the roll or sheet. If this method is to be used, only the edges where the variances are concentrated need to be treated . Another way is to allow room for cooling, i.e. to require the system to work at a much higher level of heating before the parameters change. This in fact suggests coordinated C D / M D control. Load disturbance is well rejected with its effects quite limited in time and space. The average 2<r value for the steady-state output is 0.06 which is fairly small compared with 0.02 of the noise.  Heating and Cooling Simulation time is 250 scans. From scan 4 to scan 149 the system is in the heating process, and as the set-point increases from 67 microns to 71 microns, the system works in the cooling process. Load disturbance is present during both the heating and the cooling processes — from zone 6 to 8 scan 50 to 60, and from zone 10 to 12 scan 190 to 200. The load disturbance is 0.9 um during the heating process in this simulation which is three times that in the previous processes. Figure 5.36 shows the output and constrained control signals. The output tracks set-point changes very quickly.  As a result the system deals with model  change successfully. Although during the heating process when the load disturbance is so large that the actuators must work at its upper hmit, the system still succeeds in confining the effects of the disturbance within a very small area. The variations are kept to a minimum level.  62  Chapter 5:  Adaptive Cross-Direction Caliper Control  Output  Scan Number  600  0  Actuator Zone  Control Signals  Scan Number  600  0  Actuator Zone  Figure 5.32: Adaptive CD caliper control in the heating process with A = 0.005.  63  Chapter 5:  Adaptive Cross-Direction Caliper Control  Output  Scan Number  600  0  Actuator Zone  Control Signals  Scan Number  Figure 5.33:  600  0  Actuator Zone  Adaptive CD caliper control in the heating process with A retuned.  64  Chapter 5:  Adaptive Cross-Direction Caliper Control  Output  Figure 5.34: Adaptive CD caliper control in the cooling process.  65  Chapter 5:  Adaptive Cross-Direction Caliper Control  Output  Control Signals  Actuator Zone  0  0  Scan Number  Figure 5.35: Adaptive CD caliper control in the cooling process without constraints.  66  Chapter 5:  Adaptive Cross-Direction Caliper Control  Output  Scan Number  250  0  Actuator Zone  Control Signals  Scan Number  250  0  Actuator Zone  Figure 5.36: Adaptive CD caliper control in heating and cooling process.  6 7  Chapter 5:  Adaptive Cross-Direction Caliper Control  5.6 Conclusions Cross-direction caliper control is carried out by actuators such as Calcoil along the calender roll. The CD control model is derived based on the MD model and mill trials data analysis. The interaction matrix of the heating model is necessary because one input affects three outputs with different delays in the heating process. The cooling model allows one input to affect five outputs with different delays. Depending on how much is known about the correlations among actuators and outputs, the multivariable CD. estimator can be developed in different ways. A relatively simple form with a complete n x n estimator or a very simple implementation as used in MD  control. The basic  identification algorithm is Recursive Least Squares with Exponential Forgetting and Resetting. The behaviour of the estimator can directly influence the performance of the system output when parameters are changing with time, so it is always important to have rich input signals for the estimator. Controllers are designed on the principles of multivariable GPC with constraints and variable delays. When the constrained control signals are not too far away from what the system requires in rate and amplitude, the output can track the set-point changes fast with very small steady-state variances. Variances resulting from parameter changes can be further reduced if the adaptive controller is re-tuned, i.e. resetting A when the on-line estimations tell a change in parameters. The system also shows a strong ability in rejecting load disturbances.  68  Chapter 6: Conclusions  Chapter 6 Conclusions Caliper control is of great importance in the paper machine calendering process, since it represents the last chance to correct any irregularities in the paper properties before it is reeled up as the end product. Good reel-building requires a uniform hardness and uniform surface and printing properties, and they can be realized in a large part by achieving a good caliper profile in the sheet. Traditionally the control algorithms are simple PIDs, which are not able to deal with the complex characteristics of the caliper control process efficiently since they include time-varying process parameters, non-linearity, variable delays and coupling. As developments are being made in producing more effective calender rolls and actuators, and some improved signal processing methods are also available, it is necessary to consider advanced control algorithms for the calender stacks in order to produce paper products of higher quality. This thesis is mainly concerned with application of the Adaptive Constrained Generalized Predictive Control algorithm to caliper control. Simulations based on models developed from industrial data were carried out, and the results show that this control method is effective in handling the complexities mentioned above in the process. Following is a summary of the ideas and results presented in the thesis: Sheet caliper can be influenced by many factors both during and before the calendering process. Factors such as calender stack load, number of calender nips, stack temperature, and incoming sheet characteristics which include thickness, moisture, temperature, viscoelastic properties, etc. all affect thefinalvalue of caliper. Any variations in those factors will introduce variations into sheet caliper. Variations are classified as machine-direction, cross-machine direction and residual variations. The goal of caliper control is to reduce MD and CD variations as much as possible, and it is carried out in MD and CD independently. Machine-direction Control 69  Chapter 6:  Conclusions  Data from mill trials reveal the nonlinear relationship between caliper and roll temperature as the control variable. Consequendy the dynamic MD model is described using two processes — heating and cooling, each with a linear model. The heating process isfirst-orderintegration withtime-varyinggain; the cooling process is firstorder withtime-varyingtimeconstant and gain. The parameters are estimated by recursive least squares method with exponential forgetting and resetting factors. Considering the limitations in industrial installations, constraints are imposed on control signals in amplitude and rate of change at alltime.Optimal solutions of the constrained control signals are given by Matlab Optimal Control software. Simulation results show that the output variance is kept very small despite changing parameters of the process. Output tracks set-point change very well, and the switch between heating and cooling models is smooth and fast. As a result of the constraints on actuators, for example, insufficient cooling of the roll, the output sometimes fails to achieve the set-point level. So it is then necessary to consider using a combination of actuators if the operating range of the heaters cannot accommodate the needs. Cross-direction Control Data from mill trials are studied and the correlations between neighboring actuators are identified. The CD model contains all control zones and their corresponding measurements. The heating or cooling dynamics of each corresponding input-output channel have the same characteristics of the MD model, while the interaction matrix is found to be a band diagonal matrix, different for heating and cooling. The interaction matrix also includes variable delays on different channels. So the prediction horizon and the control horizon of the GPC controller are chosen properly to ensure stability of the closed-loop system. The parameters in the CD model can be estimated on-line by a multi-input, single output algorithm, or just by the MD estimator if the CD correlations are considered to be time-invariant, 70  Chapter 6:  Conclusions  thus can be estimated off-line. •  All actuators are subject to constraints on amplitude and rate of change at all time.  •  Simulations show that CD variations are kept low; the output tracks grade changes closely; load disturbance is very well rejected with its effects confined to a very small area on the sheet.  •  Variations resulting from the changing parameters can be reduced by re-tuning of the GPC controller.  •  Constraints on actuators can result in more variations on the edges of the sheet after a considerable change in parameters. It is usually due to insufficient cooling as Calcoils can only heat but not cool the roll. Thus, it requires additional cooling to be applied only on the edges.  The flexible features of the GPC algorithm have shown their strengths in the caliper control simulations. There is good reason to believe that GPC, if applied in the real industrial process, will solve some of the tough problems that current controls do not. This should encourage more research in areas like: •  better process/actuator understanding;  •  better measurement and signal processing;  •  establishing dynamic models that relate caliper to control variables other than the roll temperature especially for MD caliper control;  •  coordinated CD/MD control of stacks;  •  coordinated control of a combination of actuators;  •  parameter estimators that can track parameter changes faster and give better estimates;  •  incorporation of automatic adjustment of A in the GPC controller.  71  References  [I]  R. R. Bitmead, M. Gevers, and V. Wertz. Adaptive Optimal Control, chapter 2. Prentice Hall, 1990.  [2]  Y. Chen. Adaptive paper coating weight control. 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A. Dumont, M. S. Davies, K. Natarajan, C. Lindeborg, F. Ordubadi, Y. Fu, K. Kristinsson, and I. Johnsson. An improved algorithm for estimating paper machine moisture profiles using scanned data. In 30th IEEE CDC, 1991.  [10]  A. Grace. Optimization Toolbox for use with MATLAB. The MATHWORKS Inc., 1990.  [II] E. M. Heaven. Caliper control challenges. Technical report, Measurex Devron, Inc., 500 Brooksbank Avenue, North Vancouver, B.C. Canada V7J 3S4, 1992.  [12]  H. Hess. Soft calenders — can they replace supercalenders and still meet technological requirements? In Finishing & Converting Conference, pages 79-90, 1991.  [13]  J. D. Higham. Therma-jet — a new high performance caliper actuator. Paper Age, January, 1985.  [14]  K. K. Hilden and D. Sawley. Calender steam showers — an effective new way of hot calendering. Tappi Journal, 70(7):87-91, 1987.  [15]  I. M . Johnsson. Estimation and identification of moisture content in paper. 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