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Control of color in dyed paper Bond, Tracy 1988

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C O N T R O L O F C O L O R IN D Y E D P A P E R Tracy Bond B.Sc. McGill University, 1970 B.A.Sc. The University of British Columbia, 1979  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF A P P L I E D  SCIENCE  in T H E F A C U L T Y O F G R A D U A T E STUDIES DEPARTMENT OF ELECTRICAL  ENGINEERING  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH COLUMBIA  April 1988  © Tracy Bond, 1988  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Electrical Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  Date:  Abstract  T h i s paper will examine and evaluate via computer simulations different methods, both adaptive and non-adaptive, for the feedback control of the color of dyed paper.  The  objectives are to maintain the paper color at a desired setpoint despite disturbances such as addition of recycled dyed paper (i.e.  broke), and to perform color changes  as smoothly as possible. T h e dynamics of a three dye system are multivariable and nonlinear with a significant transport time delay: thus the incentive for adaptive control. Several predictor-based and D a h l i n controllers with gain scheduling are designed, tested i n simulation, and compared.  Adaptive versions using parameters identified  with Recursive Least Squares ( R L S ) are also tested.  For practical applications, the  non-adaptive D a h l i n algorithm with gain scheduling is shown to offer the best performance, together with relative ease of use.  ii  Table of Contents  Abstract  ii  Acknowledgement  vi  Nomenclature  vii  1  Introduction.  1  2  Review of Color Control Literature  3  2.1  Process Models & Control Methods  3  2.1.1  Belanger  3  2.1.2  Chao & Wickstrom  5  2.1.3  Lebeau, Vincent & Ramaz  6  2.1.4  Sandraz  7  2.1.5  McGill  8  2.1.6  Alderson, Atherton & Derbyshire  9  2.1.7  Conclusions  2.2  10  Selection of Dyes and Identification of A  10  2.2.1  Belanger  11  2.2.2  Chao & Wickstrom  12  2.2.3  Lebeau, Vincent & Ramaz  13  2.2.4  McGill  13  2.2.5  Alderson, Atherton & Derbyshire  13  iii  2.2.6 2.3  3  4  5  6  Conclusions  14  Industrial Control System Vendors  15  2.3.1  AccuRay Co. Ltd  15  2.3.2  Babcock-Bristol C o . L t d  16  2.3.3  Measurex International Systems L t d  17  2.3.4  A l t i m Control C o . L t d .  17  2.3.5  Conclusions  18  M o d e l of D y e Transport  through a Paper Machine  19  3.1  Continuous M o d e l  19  3.2  Discrete M o d e l  21  The Dye Concentration/Color  Relationship  25  4.1  Color Measurement  25  4.2  Color Sensors  27  4.3  The Dye/Color Matrix A  29  4.3.1  31  Calculating A at a Color Setpoint  Three Dye Input/Three  Color Output Model  34  5.1  Complete M o d e l  34  5.2  Simplified M o d e l  37  5.2.1  38  Simplified M o d e l i n Predictor F o r m  Discussion of C o n t r o l M e t h o d s  40  6.1  Closed-Loop Control  40  6.1.1  44  6.2  G a i n Scheduling  O p e n - L o o p Control  45  iv  7  8  Controllers derived from the Predictor M o d e l  46  7.1  W i t h M i n i m u m Prediction Horizon  46  7.2  W i t h E x t e n d e d Prediction Horizon  48  Dahlin Controllers 8.1  Decoupling Using Complete A 8.1.1  8.2  9  9.2  50  - 1  Adaptive Controller  Decoupling Using Diagonalized A  Adaptive 9.1  50  52 - 1  Parameter Identification  53  55  Multi-Output R L S Algorithm  55  9.1.1  R L S A p p l i e d with the Predictor-based Controllers  58  9.1.2  R L S A p p l i e d with the D a h l i n Controllers  61  Constraint of Identified Parameters  10 S i m u l a t i o n s  64  66  10.1  Process & Controller Parameters  66  10.2  Description of the Simulation R u n  69  11 R e s u l t s 11.1  73  Performance Summary  90  12 D i s c u s s i o n  91  13 C o n c l u s i o n s  94  Bibliography  96  Appendices  99  v  K  A -j^ and B  K  of Simulated Dyes and Undyed Paper  A C S L Simulation of Dye Transport Process  vi  100 102  List of Tables  11.1  Simulation Subseries l a  74  11.2  Simulation Subseries l b  75  11.3  Simulation Subseries l c  76  11.4  Simulation Subseries I d  77  11.5  Simulation Subseries l e  78  11.6  Simulation Subseries If  79  11.7  Simulation Subseries l g  80  11.8  Simulation Subseries l h  81  11.9  Simulation Subseries 2a  82  11.10Simulation Subseries 2b  83  11.11 Simulation Subseries 2c  84  11.12Simulation Subseries 2d  85  11.13Simulation Subseries 2e  86  11.14Simulation Subseries 2f  87  11.15Simulation Subseries 2g  88  11.16Simulation Subseries 2h  89  11.17Performance S u m m a r y  90  vii  List of Figures  3.1  Block D i a g r a m of Dye Transport  20  3.2  Approximate Discrete Block D i a g r a m of D y e Transport  23  8.3  A p p l i c a t i o n of the D a h l i n Controller  52  10.4 Block D i a g r a m of A C S L - S i m u l a t e d Dye Transport M o d e l  67  10.5 D y e Plots for a T y p i c a l Simulation R u n  71  10.6 Color Plots for a T y p i c a l Simulation R u n  72  viii  Acknowledgement  I thank D r . G u y D u m o n t for his advice and guidance i n my research. I thank Christos Zervos for his assistance in operating the P u l p & Paper Centre's / u V A X computer.  I  also thank my wife, Jane, and my daughter, K a t i e , for their patience during the long process of completing this thesis.  ix  Nomenclature  SYMBOL  DEFINITION  OL  Open-Loop  CL  Closed-Loop  TF  Transfer Function  RLS  Recursive Least Squares  RML  Recursive M a x i m u m Likelihood  ZOH  Zero Order H o l d  MIMO  Multi-Input Multi-Output  GMVC  Generalized M i n i m u m Variance Controller  EHC  E x t e n d e d Horizon Controller  Di  T h e column vector [Di ,i Di 2  n  Di ^Y  N<  n  n  containing the ratios of  injected Dye 1, D y e 2 and D y e 3, respectively, to dry fibre i n the fresh pulp [gm. d y e / g m . fibre] D  T h e column vector \D\ D  D]  T  2  3  containing the concentrations of  Dye 1, D y e 2 and Dye 3, respectively, i n the dried paper product [gm. d y e / g m . fibre] C  T h e column vector \C\ C  2  Cz]  T  containing the color of the dried  paper product i n the L* a* b* scale A  T h e incremental dye/color 3 x 3  matrix  \ij  A n element of the matrix  1Z  Retention of dye on the fourdrinier  A  x  T'  Transport delay from dye addition point to fourdrinier [min.]  Tj  Transport delay i n fibre recovery process [min.]  di  2  T'  Transport delay i n dryers [min.]  Td  A c c u m u l a t e d time delay i n fibre recovery process  d  2  = T' + T d2  dl  [min.]  Overall time delay in dye transport process  Td  T' + T' d  di  [min.]  T  Sampling interval [min.]  d  Discrete dye transport delay = T /T + 1  dh  Discrete prediction horizon  p  Desired C L discrete time constant for a D a h l i n controller  a  d  a  = e* T  T  Desired C L continuous time constant [min.]  c  p  Weighting factor i n the  Vii V21 V3  Constant offset vectors  Wi, W , 2  g  W  3  GMVC  W h i t e noise vectors T h e .7 backward shift operator: shifts sampled data th  _ J  j sampling periods backwards X(q^ )  T h e backward shift operator polynomial X ^ f o ^ j l ~  deg[X]  Highest negative power i n the polynomial  term[X]  N u m b e r of terms (including the q° term) i n the polynomial  1  x  (  i  X(q~ ) 1  X(q~ ) x  = deg[X] + 1 Ai(q~ )  Denominator polynomial of discrete dye transport model for Dye i  Bi(q )  Numerator polynomial of discrete dye transport model for Dye i  A(q~ )  Denominator polynomial 1 + Y^i^  1  _1  l  xi  j9.^ ° f common discrete dye  a  transport model for all dyes Numerator polynomial Y f f l l ^ bjQ* of common discrete dye  B(q~ ) 1  transport model for all dyes 6X  Incremental change i n the scalar or vector X  AX  X — q' X where X is a scalar or vector containing sampled data  rrace[X]  T h e sum of the diagonal elements of the matrix  1  F(q~ ), 1  X  G(q~ ) Solutions of differential Diophantine E q . 5.45 1  FB(q-')  F{q-*)B{q-*)  FE'(q-l)  F(q-L)E'(q-*)  A or \j  Wavelength i n the visible spectrum [nm.]  i?(A)  T h e reflectance distribution (function of wavelength) of the dyed paper  K(X)  T h e absorption coefficient (function of wavelength) of the dyed paper  5(A)  T h e scattering coefficient of the dyed paper  ^(A)  T h e absorption/scattering coefficient =  7^ (A)  T h e absorption/scattering coefficient of the undyed paper as defined by E q . 4.26  ^ (A) i  T h e absorption/scattering rate coefficient of Dye i as defined by E q . 4.26  xn  Chapter 1  Introduction.  T h i s thesis will examine and evaluate by computer simulations different methods for the feedback control of color i n dyed paper. These methods will include adaptive and non-adaptive techniques. T h e object of the control is to maintain the paper color at a desired setpoint by counteracting disturbances (e.g. the addition of colored broke to the pulp). A n o t h e r objective is to achieve a good response to changing color setpoints. T h e paper industry's interest i n continuous color monitoring and control is high, due to the demand for higher paper quality, for better, more uniform optical characteristics, and for less off-standard product (ie. broke). T h e thesis is organized i n the following way:  Chapter 2. T h e literature on color and its control i n the paperrnaking industry is reviewed. T h i s includes: • Dye i n p u t / c o l o r output models and control methods used. • T h e selection of dyes and methods of identifying the dye/color matrix. • Color control systems developed by industrial control system vendors.  Chapters  3 . A model for the transport of dye (from its addition to the pulp to its  concentration i n the dried paper product) is developed. Chapters 4. T h e nonlinear relationship between dye concentration i n the dried paper and the paper's color is examined. Color measurement scales and color sensors  1  Chapter 1. Introduction.  2  are also discussed.  Chapters 5. T h e results of Chapters 3 and 4 are combined to make a three dye i n p u t / t h r e e color output model.  Chapter 6. T h e problems of controlling this process are discussed. T h e two closedloop control methodologies (predictor-based and Dahlin) to be used i n this thesis are introduced. Open-loop control is also discussed.  Chapter 7. T h e predictor-based controllers (based on a process model i n predictor form) are developed. These are direct/implicit adaptive controllers.  Chapter 8. T h e D a h l i n controllers are developed. These are indirect/explicit adaptive controllers.  Chapter 9. T h e identification algorithms used by the predictor-based a n d D a h l i n adaptive controllers are described.  Chapter 10. T h e computer-simulated paper dyeing process is described and its parameters given.  Chapter 11. T h e results of applying the controllers developed i n Chapters 7 & 8 to the simulated process are given.  Chapter 12. T h e results are discussed. Chapter 13. Conclusions are made.  Chapter 2  Review of Color Control Literature  2.1  Process Models & Control Methods  In this section, models of the dye transport  process used i n the literature  described. Generally, the models are similar to that shown i n F i g . 3.1.  will be  A s well, color  control methods used i n the literature will be described.  2.1.1  Belanger  In [2], Belanger uses a dye transport F i g . 3.1),  model similar to that used i n this thesis  except for the absence of a time delay i n the fibre recovery/white  (see  water  process. T h e retention of the dye 7Z (i.e. the fraction of the dye retained on the wire) is considered a key unknown model parameter and is known to vary from 0.4 to  1.0  depending on the dye. A d a p t i v e control is suggested as potentially applicable to the problem, but instead a sensitivity minimization analysis is used to tune a non-adaptive Linear Quadrature Regulator L Q R . Initially, 1Z is assumed to be 1.0.  T h e model is  thus reduced from a second order model to:  D D  e~ ra + 1 TdS  in  x  where:  • T  =  • Ti  =  d  3.14 m i n . 2.10 m i n . 3  (2.1)  Chapter 2. Review of Color Control Literature  • T, =  T /4 d  =  4  0.80 m i n .  It is assumed that these parameters are those of the paper machine on which the controller was tested. T h i s is a small-scale paper machine used by the Dyestuffs Division of Imperial C h e m i c a l Industries of Blackley, E n g l a n d . A discrete integrating L Q R which minimizes the performance index: oo  oo  YXD  -  D'«Y  0  +  />£(AD  i n  )  2  0  is designed for the 1Z = 1 model of E q . 2.1. T h e control signal weighting factor 0 < p < 1 in the performance index is left as an unspecified tuning parameter. all retentions between 0.4 and 1.0 are equally likely, the p (i.e.  Assuming that  p") is selected which  minimizes:  S(p)  =  S'(p,0A)+  S'(p,0.5)+  (2.2)  --- + S'(p,1.0)  w here:  I*{K) r(Tl) I{p,1Z)  =  mm I(p,K)  =  r{D-D' Jo  d  e t  )  7  dt (2.3)  when the L Q R is applied to a simulated continuous dye transport process. calculated from the output  D  of this process with retention  calR, when  I(p,7V} is  it is controlled  by an L Q R with weighting factor p. T h e problems with this method appear to be: 1. Using the L Q R based on the simplified process model shown i n E q . 2.1, selected p gives the "best"  (using the above criterion) performance, but that  performance is not very " g o o d " . T h e value of 4 times that at 7Z =  1.0.  the  I(p*,lZ) at 7c = 0.4 is more than  Chapter 2. Review of Color Control Literature  2. It seems that seeking to minimize I(p, normalized  0A) + I(p, 0.5)H  S'(p,0A) + S'(p,0.5) H  5  \~I(p, 1.0) instead of the  (- 5"(,o,1.0) would give better performance  i n terms of reducing the output error under the full range of 7Z. 3. In order to decouple the three dye transport processes (by calculating D from C, the color sensor output), the dye/color matrix A is required. Belanger assumes that A is estimated using the theoretical calculation suggested by Alderson et al. and described i n detail in Section 4.3.  However, he does not discuss the effect  of the difference between this estimated A and the exact A on the  controller's  performance.  2.1.2  Chao & Wickstrom  In [5], C h a o et al. emphasize the importance of having a dye/color matrix A which is almost orthogonal so that a color range can be achieved efficiently and with good controllability (see Subsection 2.2.2 for details). It is conjectured that the dye passing through the wire is attached to fine fibres and that 80-90% of fibres remain on the mesh.  T h e process is modelled as i n F i g . 3.1 in  Chapter 3 (except that there is no time delay i n the fibre recovery process), but the authors show that a first order model with time delay:  D  (2.4)  in  provides a good approximation to the full model at values at 7Z from 0.8 to 1.0. T h e y use dye/fibre units. T h e parameter values used i n the model above were:  • T  d  •  T\  =  =  1.4 m i n . 0.6 m i n .  Chapter 2. Review of Color Control Literature  • T„ —  6  1.0 m i n .  It is assumed that these parameters were those of a paper machine at Consolidated Papers Inc., Wisconsin Rapids, W I , on which the authors tested their controller. A l though the authors include no time delay in their fibre recovery/white water process model, a plot showing the step response of a paper machine indicates that there ia a 2.7 m i n . time delay i n the white water process.  T h e authors feel that an integrating  controller will compensate for this. T h e i r method decouples the three dye transport processes with A  - 1  and controls the  dye addition rates with three identical D a h l i n controllers. T h e y found this controller worked well, although during setpoint changes there was significant overshoot.  They  attribute the overshoot to the unmodelled time delay in the white water process. T h e y found that reducing the closed-loop time constant (from 4.8 to 1.0 min.) reduced the overshoot, although the open-loop time constant of the transport model was 0.6 m i n . Perhaps if the estimated open-loop time constant h a d been larger to model the time delay i n the white water process there would have been less overshoot.  2.1.3  Lebeau, Vincent & Ramaz  In [12], Lebeau et al. assume that the dyes fix to the fibres quickly and that, therefore, the dynamics of dye transport are the same as those of fibre transport. T h e dynamics of fibre transport correspond to the hydraulic properties (including time delays) of the pulp preparation process and the paper machine.  It also assumes that all three dye  transport processes are identical. For the dye transport model, the authors use a second order model with time delay:  D An  e~ " Td  {r  lS  + l){r s + 1) 2  T h e parameter values used in the model above were:  V  "  ;  Chapter 2. Review of Color Control Literature  • T  d  =  • Tj =  m  2.1 m i n . 1.25 m i n .  =  0.33 m i n .  • T„ =  0.75 m i n .  T  2  7  It is assumed that these parameters were those of an experimental paper machine at the Centre Technique du Papier, Grenoble, France on which the authors tested their controller. T h i s paper uses two methods:  1. Three. Mono variable Controllers: Decouple the three dye transport processes with A  - 1  and control the three dye  addition rates with three identical pole-zero placement controllers with process cancellation (i.e. D a h l i n controllers).  2. Multivariable  Controller  Using the value of A and three identical dye transport models this paper constructs a 3-input, 3-output state-space model. A u g m e n t i n g the states to provide integral action, a Linear Quadrature Regulator L Q R is designed. T h i s controller was actually suggested and developed by Sandraz [20]. T h e paper concludes that b o t h these controllers give similar performance, i n part because the state space model assumes identical dye transport models as do the three monovariable controllers.  2.1.4  Sandraz  T h i s P h . D thesis [20] is the source of the state space model and multivariable controller used i n Lebeau et al. [12]. It identifies the dye transport model i n open loop using the  Chapter 2. Review of Color Control Literature  8  cross-correlation between an input containing a pseudo-random binary sequence and the output.  It compares the fit of the impulse responses of b o t h a first and second  order T F ' s to the cross-correlation data and found they fit equally well. For designing the controllers, the simpler first-order model with time delay is used:  ±L D  in  =  1 r  lS  .)  (2 6  + l  V  '  T h e parameter values used in the model above were:  • T  d  •  Ti  = =  • T. =  2.25 m i n . 1.7 m i n . 0.75 m i n .  It is assumed that these parameters were those of the same paper machine modelled by Lebeau et al. [12] which is confirmed by the similarity of the parameter values. In the state-space model of the color/dye process Sandraz uses the dye transport model shown above for each of the three dyes.  2.1.5  McGill  T h e chapter " T h e Measurement of O p t i c a l Variables" in  Measurement & Control in  Papermaking [16] discusses the advantages of the continuous (versus batch) dyeing of paper and also describes some important variables i n the coloring process. variables together with those given by C l a p p e r t o n [6] are listed i n Section 6.1.  These This  reference discusses the benefits of controlling the ratio of the dye addition rate to the flowrate of incoming fibre. T h i s method results in less color deviation due to paper breaks and basis weight changes.  Chapter 2. Review of Color Control Literature  2.1.6  9  Alderson, Atherton & Derbyshire  Some dyers and colorists use the Instrumental M a t c h Prediction ( I M P ) method to achieve a particular color (see Alderson et al. [1] a n d Q u i n n [19]).  T h e method of  estimating the incremental dye/color matrix A suggested by Alderson et a l . is that used i n this thesis and is described i n Section 4.3. -*8et  In order to achieve the color setpoint C  , the I M P method first estimates A as-  suming the dye concentration D = 0. W i t h the inverse A dye concentration Di , n  , the first step change i n  calculated using:  An  (2.7)  = A- C" \ l  e  is applied to the undyed sheet. After the color has reached the steady state color C ( Q u i n n [19] estimates an 8 m i n . wait),  D — L>i and A (and thence A  ) is recalculated  —>  —*  using  - 1  n  D. T h e next step change i n dye concentration ADi  n  AD  in  is calculated using:  (2.8)  = A.-\C"*-C)  and then:  4  <- 4  + A 4  (2.9)  T h i s process is repeated until the desired color is achieved. T h i s type of control can be described as discrete integral feedback control with a variable gain A  -  1  and a sampling  interval which is longer than any dye transport dynamics . However, the long sampling interval means more off-standard paper is produced before a correction can be made. In this thesis, the dye transport dynamics have been modelled i n order to design feedback controller with a much shorter sampling interval.  However, the variable A  aspect of the above method is retained by using G a i n Scheduling.  -  1  gain  Chapter 2. Review of Color Control Literature  2.1.7  10  Conclusions  C h a o et al. and Sandraz conclude that a simple first-order dye transport model provides a good model with which to design a C L controller. B y assuming a retention 7Z equal to 1.0 Belanger also uses a first-order model and then designs a L Q R controller which performs well if 0.4 < K < 1.0. Dye input and dye output i n the dye transport model are generally measured as concentrations (i.e. the ratio of dye to fibre), since color is best expressed as a function of dye concentration i n the paper. C h a o et al. and Lebeau et al. b o t h use D a h l i n controllers. Lebeau et al. find that 3 decoupled D a h l i n controllers perform as well as a multi-input multi-output L Q R .  2.2  Selection of Dyes and Identification of A  T h e selection of a " g o o d " set of dyes to achieve a particular color range is complex. It requires experience and advice from a dye manufacturer.  There are two objectives i n  selecting a set of dyes: • T o achieve the desired color. • T o have good controllability at the desired color. T h e following rules should be followed: 1. T h e number of dyes used to achieve a particular color should equal the number of color space dimensions to be controlled (i.e. 3-dimensional V  to control L*, a*, and b* in the  a* b" color scale requires 3 dyes, but to control only a* and b"  requires only 2). 2. T h e desired color should be on the inside of the color space volume spanned by a set of dyes. In other words, all dyes should be added i n substantial amounts so the  Chapter 2. Review of Color Control Literature  11  amount of each dye can be increased or decreased to compensate for disturbances. A s an example, one dye, which seems to be adequate i n achieving a desired color by itself, should not be used. T h e desired color would be on the edge of any color space volume spanned by a dye set that includes that dye. 3. Similar dyes should not be used in a dye set, since they span a small volume of the color space and disturbances could cause that volume to drift away from the desired color. Controllability once color is achieved will be poor. 4. Near-opposite dyes should not be used i n a dye set to span larger volumes of the color space, since large counteracting flowrates which darken the paper may be required i n that space. Controllability once color is achieved will be poor. Therefore, orthogonality (if that term may be used for a nonlinear dye/color relationship) i n the dye set causes those dyes to span a reasonably large color space volume and orthogonality of the incremental dye/color matrix A provides optimal controllability. A measure of orthogonality or dye efficiency of A is suggested by C h a o et al. [5] and is described in Subsection 2.2.2. T o cover the whole color space, 24 dyes or more may be required. T h e estimation of an incremental dye/color matrix A at a particular color setpoint is critical for good closed-loop control. A may be identified online during paper production or measured offline in a lab using small sheets dyed to or near the desired color. Alternatively, it may be calculated using a theoretical dye/color model (with certain optical properties of the dyes) as described i n Subsection 2.2.5 and Section 4.3.  2.2.1  Belanger  T o estimate the dye/color matrix A, Belanger suggests the theoretical calculation described in Subsection 2.2.5 and Section 4.3.  Chapter 2. Review of Color Control Literature  2.2.2  12  Chao & Wickstrom  T h i s paper emphasizes the importance of having a color matrix A with a high degree of orthogonality so that a color range can be achieved efficiently and with good controllability.  It illustrates this point with a selection of red, blue & green dyes where  the red dye & blue dye additively have a similar color effect to the green dye. means that only a t h i n plane i n the color three space is achievable.  This  Assuming that  a particular color i n that plane has been achieved, control about that point will be difficult, especially if a correction is required perpendicular to the t h i n plane. In that case large increments i n the red dye & blue dye are cancelled by a large decrement in the green dye i n order to achieve a small color change.  In response to this problem,  the authors define the normalized determinant of A or degree of orthogonality of A as  the dye efficiency E where:  E = and where  is an element of A.  1 (i.e. orthogonal A)  J  I  ( 2  T h e value of E lies between 0 (i.e. singular A)  .  1 0 )  and  where larger values are desirable.  In the example above, the determinant of A is very large (i.e. 438,000) due to the large conversion factors from dye units to color units, but the dye efficiency E is less than .03 !  In this paper, the 3 dimensional Hunter Lab  not necessary to control the lightness  scale was used, but it was  L of the near-white paper made at Consolidated  Papers Inc., Wisconsin R a p i d s , W I . T h e process is thus a two dye i n p u t / two color (a and 6) output process.  T h e authors suggest that, if necessary, the lightness could be  controlled separately using the addition of a white filler. T h e authors state that the color matrix A does not change significantly for the same grades but does depend on the  furnish (i.e. the type and composition of the pulp stock). T h e color matrix A is  identified online by making separate step changes i n each of the three dye  flowrates.  Chapter 2. Review of Color Control  2.2.3  Lebeau, Vincent &  13  Literature  Ramaz  T h i s paper points out that it is the inverse color matrix A  -  1  that is used to decouple the  three dye transport models and that small errors in an identified A can result i n large errors in a A  -  calculated from it. Therefore, this paper suggests that A  1  -  1  be identified  directly instead of A. W i t h the process under closed-loop control, step changes were made i n each of the three color setpoints separately and the resulting changes in dye addition rates were measured. T h i s calibration would have to be performed at every color setpoint.  2.2.4 The  McGill chapter " T h e Measurement of Optical Variables" in Measurement  Papermaking  & Control in  [16] states the necessity of having a range of dyes with known and stable  properties i n order to produce a wide range of colors (i.e. several sets of orthogonal dyes which are capable of achieving colors i n the 3-dimensional color space).  It mentions  that some dye manufacturers can provide dye formulae to produce a color specified by the papermaker.  2.2.5  A l d e r s o n , A t h e r t o n &: D e r b y s h i r e  In their Instrumental M a t c h Prediction ( I M P ) color matching method, Alderson et al. [1] suggest that A can be determined, at a particular D differentiation.  and C , using numerical  First, they develop a theoretical relationship between dye and color  (as described by E q s . 4.20, 4.22, 4.24, 4.26 in Chapter 4).  T h e n , these equations are  differentiated numerically as described i n Section 4.3 to determine A. In E q . 4.26 the value of the coefficient ^ ( A ) for each of the dyes i n use (i.e. ^ ^ - ( A ) for D y e 1, ^ ^ - ( A ) for D y e 2, ^ ^ - ( A ) for Dye 3) and the value of the coefficient if^(A) for  Chapter 2. Review of Color Control Literature  14  the undyed paper are required, where A is wavelength i n the visible spectrum. for one dye only and ^(X)  can be determined i n the lab using a  5^(A)  spectrophotometer  and small sheets as follows. Several sheets are made up with varying concentrations of the dye D. lengths Xj (j = 1 , . . . ,n),  A t n wave-  the reflectance of each sheet is measured with the spectropho-  tometer. If an abridged spectrophotometer is used, n is limited by the sensor. the K u b e l k a - M u n k E q u a t i o n ( E q . 4.24)  Using  the value of ' j ( A j ) is calculated. T h i s data is  collected for all the sheets. Finally, at each wavelength Xj, the best linear fit through the ^(Xj)  points of varying dye concentration determines the slope ^(Xj)  intercept ^(Xj)  where:  =  2.2.6  and zero  ^i)D  + §f(Xj)  (2.11)  Conclusions  C h a o et al.  identifies the dye/color matrix A  flowrates to the paper machine. matrix A  - 1  online by perturbing the three dye  Alternatively, Lebeau et al. identifies the color/dye  online by perturbing the three color setpoints to the feedback controller.  These online identifications may be required for each different color of paper produced and they may generate considerable off-spec paper. T h e theoretical dye/color model suggested by Alderson et al., together with specific optical properties of the dyes, has proved successful i n color matching. T h e value of A corresponding to any color witliin the range a set of dyes can be  calculated, once the  optical properties of the dyes have been measured. T h i s method is used by Belanger to calculate A and it seems a better method of determining A than online identification, which must be performed at each new color setpoint.  Chapter 2. Review of Color Control Literature  2.3  15  Industrial Control System Vendors  T h e color control systems described below are described and compared in a short article by the P I T A Engineering Technology W o r k i n g G r o u p [17]. T h e emphasis is on the color sensors used by each system. A comparison of spectrophotometers and colorimeters as color sensors is made i n Section 4.2.  2.3.1  AccuRay Co. Ltd.  R. Marclii [15], an A c c u R a y systems engineer, briefly describes this color monitoring and control system.  T h e color sensor is located between the calendar stack and the  paper reel. T h e sensor is an abridged spectrophotometer Color Communications C o .  manufactured by the M a c b e t h  and it uses a high-intensity pulsed xenon flashtube for  illumination. T h e U V content of this source excites optical brightening agents ( O B A ) (also known as fluorescent whitening agents ( F W A ) ) which are fluorescent dyes used for whitening yellowish pulp stock. It has a diffraction grating to separate by wavelength the light reflected from the dyed paper. Sensors measure the intensity of the reflected light at 16 wavelengths of the visible spectrum. Having the illumination intensity vs. wavelength relationship, the control computer can calculate the reflectance wavelengths. It then calculates the XYZ CIE  at the 16  (and thence Lab) color value for any of the  standard illuminants. Either the Lab or XYZ  scale can be used for feedback  control of the dye input. T h e control algorithm uses an incremental color/dye matrix to convert color errors to dye errors.  T h e incremental dye/color matrix is calculated  using a proprietary method based on the K u b e l k a - M u n k equation. Since details i n this paper are so scanty, it can only be guessed that the absorption/scattering  coefficients  (which are functions of wavelength) for upto 48 dyes are measured i n the lab and stored i n the control computer's memory. Further, it is guessed that a method similar to that  Chapter 2. Review of Color Control Literature  16  suggested by Alderson et al. [1] and described i n Section 4.3 is used to calculate the incremental dye/color matrix.  Inversion provides the incremental color/dye matrix.  T h e control algorithm is a proprietary "self-tuning" one which "remembers previous control actions and makes modifications, based on current process conditions, to the control equations".  It is assumed that the flowrates of dyes used i n past production  runs to achieve a particular color are also recorded i n memory for use i n repeat runs.  2.3.2  Babcock-Bristol Co. Ltd.  K . L . T o d d et al. [22], Babcock-Bristol engineers, briefly describe the " C o l o r E y e " color monitoring and control system. T h e color sensor is an  abridged spectrophotometer also  manufactured by the M a c b e t h Color Communications C o . B a b c o c k - B r i s t o l holds the exclusive E u r o p e a n distribution contract for the M a c b e t h sensor in the paper industry. T h e Lab scale is used for feedback control. T h e control algorithm is described as a proprietary multi-variable nonlinear dead-time controller.  Few other details of this  proprietary algorithm are given. T h e control algorithm uses an incremental color/dye matrix to convert color errors to dye errors.  T h e incremental dye/color matrix is de-  termined by making small color changes about a desired paper color either during production runs or in the lab.  T h i s identification must be performed for every new  paper color and is therefore less general than that used by A c c u R a y which, by identifying the absorption/scattering coefficient (as a function of wavelength) for each dye, can  estimate the dye/color matrix when dyes are used in any combination or concentration. T h e flowrates of dyes used i n past production runs to achieve a particular color are also recorded i n the computer's memory for use in repeat runs.  Chapter 2. Review of Color Control  2.3.3 R.  17  Literature  Measurex International Systems L t d .  P. Shead [21],  a Measurex engineer, briefly describes this color monitoring and  control system. Measurex.  T h e color sensor is a abridged spectrophotometer  manufactured by  T h i s sensor is able to scan the paper web i n the cross-machine direction.  It uses two continuous illumination sources: vapor U V source.  a tungsten-halide l a m p and a mercury  T h e U V source excites the O B A ' s .  It has a diffraction grating to  separate by wavelength the light reflected from the dyed paper. Sensors measure the intensity of the reflected light at 32 wavelengths of the visible spectrum. Having the illumination intensity vs. wavelength relationship, the control computer can calculate the reflectance  at the 32 wavelengths. It then calculates the XYZ  color value for any  of the C I E standard illuminants. Either the Lab or C I E L A B 1976 L*a*b* scale can be used for feedback control and for the operator display. T h e control algorithm is not described, except to say that a 3 x 3 "decoupled" matrix (the incremental color/dye matrix) is used and that there is feedforward from drystock flow (to ratio the dye flowrate, it is supposed), clay flow, and machine speed (to calculate the dye transport delay time, it is supposed).  2.3.4  A l t i m Control Co. L t d .  M . Lehtoviita [13], an A l t i m engineer, briefly describes the "Colourkeeper" color monitoring and control system. T h e color sensor is a Hunter D43 colorimeter trophotometer).  (not a spec-  It uses two n o n - U V containing quartz halogen light sources for illu-  mination and 6 colored filters to calculate directly the XYZ  color values. T h e control  algorithm is stated to be a multivariable P . I . D . one with a 3 x 3 decoupling (i.e. i n cremental color/dye) matrix.  T h e controller, using an Intel 8085 with only 8 K b . of  memory, seems to be less sophisticated than that of the previous systems. Before use,  Chapter 2. Review of Color Control Literature  18  this system must be provided with, among other data, the following: dye concentrations desired, process model, and "controller's calibration values" (the 3 x 3 decoupling matrix, it is assumed).  2.3.5  Conclusions  T h e Measurex abridged spectrophotometer  measures the reflected light at 32 wave-  lengths, twice the number measured by the M a c b e t h spectrophotometer. A c c u R a y seems to use the most sophisticated method of determining the dye/color matrix A . A is  calculated using a theoretical dye/color model (together with specific  optical properties of the dyes) similar to that given i n Section 4.3.  A c c u R a y ' s system  can store in memory the optical properties of upto 48 dyes. A l l the control systems record the steady-state dye flowrates used i n past production runs to achieve particular paper colors. W i t h this a priori knowledge of dye flowrates, feed-forward open-loop control of the dye input can be used during a setpoint change to a previously produced color.  Chapter 3  Model of Dye Transport through a Paper Machine  3.1  Continuous Model  In paper machines that add dye to the pulp continuously (as opposed to machines that dye the pulp in batches), the dye is added to the wet end process as close as possible to the headbox (i.e.  the reservoir from which the pulp flows onto the fourdrinier) to  minimize the transport time delay from the addition point to the fourdrinier. O n the other hand, the addition point must be far enough from the headbox to ensure that the dye has adequate time to fix (i.e. bond) to the pulp fibres. A t the fourdrinier (i.e. a moving screen) the sheet or web is formed as much of the pulp water drains through the screen.  Some fibre and dye also escapes from the web with the water, but they  are recovered and returned to the beginning of the wet end process through the fibre recovery/whitewater process. presses to the dryers.  T h e dyed web moves from the fourdrinier through  T h e dried paper is finally wound onto a roll at the reel.  the The  color sensor is located just before the reel so that the color of the finished paper can be measured. A block diagram for the transport of a single dye through a paper machine (if the flowrate of paper fibre is constant) is shown i n F i g . 3.1. It agrees with the model used by Belanger [2] and C h a o et al. [5]. In F i g . 3.1, the following definitions apply: • Di  n  =  f  T h e ratio of injected dye to undyed fibre i n the incoming pulp [gm. of  19  Chapter 3. Model of Dye Transport through a Paper Machine  WET  END  PROCESS D,  DRYERS  FOUR.  -I)  -©-  e  d  i  T l » +  20  n  1  D  FIBRE RECOVERY PROCESS 1  e  "2  T2  3+1  Figure 3.1: Block D i a g r a m of Dye Transport d y e / g m . of fibre] =  D  •  *  Ti  T 2  T h e concentration of the dye i n the dried paper [gm. of d y e / g m . of fibre]  f  def  Volume of wet end reservoirs Total liquid flowrate [min.]  def  Volume of fibre recovery reservoirs r . i T o t a l hquid flowrate [min.j  —  • 7c  =  =  T  d  Retention of dye on the fourdrinier  f  f  Transport time delay from dye addition point to fourdrinier [min.]  def Transport time delay i n fibre recovery process [min.] r|  a T'  ==  d  Transport time delay i n dryers [min.].  T h e block diagram can be re-arranged so that the T' both the T  d  and the T'  d  d  transport delay is combined with  transport delays. Defining new symbols:  Chapter 3. Model of Dye Transport through a Paper M a c h i n e  . T  d2  • T  d  ^  d  T  d2  = r  d f  d  21  + T'  di  + V  dl  If we assume that all transport delay occurs i n the dryers (i.e. T  d2  — 0), the transfer  function of the dye transport model is:  D D i n  xKe(n«+i)(^f+i) '  1  T 2 S  (S  + 1){T S  Tir  s  TI  TS  +  2  2  2  +  (r s + 2  +  1  (TI +  xHe- *> T  +  2  (3.12)  Tdt  1)-{1-K)  1  TZe-  TdS  r ) s + 2  l)e- * T  Tl  (3.13) (3.14)  B  (3.15)  + n ± a +1 B  T h i s model has unity gain. If dye or fibre is lost, the gain would be changed. T h e unit of D is the ratio of dye to dry fibre. T h e color of the paper at the reel is largely determined by this ratio and this unit allows for dye addition rate equals the desired  D{  n  ratio control of dye addition rate (i.e.  multiplied by the flowrate of  undyed pulp).  R a t i o control causes the paper color to be less sensitive to changes in undyed fibre flowrate due to changes i n basis weight or paper breaks.  3.2  Discrete Model  If the transport time delay T  d2  is not zero, a discrete T F :  I - ^? cannot be calculated directly f r o m the Laplace T F of the continuous dye transport model with zero-order-held ( Z O H ) input. In that discrete T F :  •  d  =  £ +1  Chapter 3. Model of Dye Transport through a Paper Machine  • T„  =  Sampling interval 4eg[A]  -j  j=l  •  B(q-*)  22  ^deglB] l^j=0  def —  J  In this thesis, the empirical method of determining the T F was to select T , deg[A] s  and deg[B] and then carry out a discrete R L S identification of the continuous  ACSL-  simulated model while it was excited by a known random input (with Z O H ) . In order to insure that the discrete model of the dye transport process has no nonminimum-phase zeros due to a fractional sampling interval the sampling interval T should be selected so s  it divides the process time delay Td evenly. If the interval between color measurements is limited by the color sensor then interpolation could be used to generate a new color measurement at a sampling interval which does divide T  d  evenly.  In order to get estimates of rfe^[^4] and deg[B], the approximate F i g . 3.2 will be used. In F i g . 3.2, the following definitions apply: • n  2  =  f  1a  truncated  def  o  def  1 - e  def  1 —e  T  2  T h e discrete transfer function of the dye transport model is:  discrete model of  Chapter 3. Model of Dye Transport through a Paper Machine  WET END PROCESS ZOH (r.)  DRYERS  4  b'  l-ajg-  FOUR.  23  D  1  FIBRE RECOVERY PROCESS  -n  l-ai'g-  2  Figure 3.2:  i-n  Approximate Discrete Block D i a g r a m of Dye Transport  If T„ = 0.66 m i n . and T  d2  < 2.0 m i n . (as it is i n these simulations), then n  E q . 3.18 indicates that reasonable estimates for depfA] and  2  < 3 and  deg[B] would be 3 and 2  respectively. For example, if the continuous A C S L model with the parameters: •  Ti  • r  2  • T  d2  • T  d  =  0.94 m i n .  =  0.75 m i n . =  =  • 1Z =  1.0 m i n . 2.0 m i n . 0.8  was identified with  T, = 0.66 m i n . , rfe(/[^l] = 3 and deg[B] = 2, the R L S algorithm  identified the following coefficients:  Chapter 3. Model of Dye Transport through a Paper Machine  • a ,a ,a a  2  » 60,61,62  3  = =  24  - 0.905,0.159,-0.017 0.406,-0.167,-0.003  T h e output of this discrete model simulated the output of the continuous model with only 0.02% error, when the model was simulated for 80 m i n . (after the coefficients h a d been identified) with a known zero-order-held r a n d o m input.  T h e zeros of A(q~ ) are .716, .094 ± .122* and the zeros of B(q~ ) 1  l  are .429,-.017.  T h e discrete model of the process is therefore stable a n d m i n i m u m phase. With  T = 0.66 m i n . , </e-gr[^4] = 1 and deg[B] = 0, the R L S algorithm identified the s  following coefficients: • 01 = • 60 -  -0.523 0.407  T h e output of this discrete model simulated the output of the continuous model with 7.0% error, and it was observed that the gain (i.e. 0.85) was close to the dye retention 7c, even though the steady-state gain of the continuous model is unity. Referring to E q . 3.18 one can see that, if the factor (1 — 7c) b' b' ' i n the denominator Q  0  is small (eg. it equals 0.02 for the model parameters above), then E q . 3.18 can be approximated by: D  _  g- TZb' d  0  An ~ ( 1 - a i r ) 1  T h i s is the reason that a discrete model with de#[vl] = accurate i n simulating the continuous model.  (3  1 and  "  19)  deg[B] = 0 is quite  Chapter 4  The  Dye Concentration/Color Relationship  Color Measurement  4.1  A n object's color i n the XY  Z color space is determined b y : /•700nm  X  =  Y  =  Z  =  J400nm /•700nm JiOOnm r700nm JiOOnm  1(A) R{X) x{X) dX  I(X)R(X)y(X)dX 1(A) R(X) z(X) dX  where: • I(X)  =  f  T h e intensity distribution (function of wavelength of visible light) of  the illumination source [ W . / m . / n m . ] 2  •  R(X)  =  T h e reflect ance distribution of the object illuminated  • a:(A), y(A), z(X)  =  f  T h e sensitivity distributions of the three retinal pigments of  the Standard Observer as defined by the Commission Internationale de l'Eclairage ( C I E ) i n 1931 and updated i n 1964. Colors can be specified by the C I E 1964 XYZ  scale, but, for control purposes, it  is useful to have a color space/scale ( C i , 6 7 , 6 3 ) and a color difference metric (|| 6C ||) which corresponds closely to color difference  as perceived by the average observer. De-  viations from the desired color setpoint can then be measured i n units that correspond  25  Chapter 4.  The Dye Concentration/Color  Relationship  26  to the apparent color differences between the actual product color and the desired color. Defining the color difference metric as:  \\SC\\ = the color space/scale (C\,C iCz) 2  yj6C? + SCI + SCi  (4.21)  can be defined (as a function of the C I E 1964  XYZ  scale) i n several ways: none of which exactly duplicates perceived color difference, due to the non-linearity of human vision. T h i s metric is especially useful i n L Q control i n which the performance index has the same form. T h e two color scales ( C I E L A B and C I E L U V ) recommended by the C I E i n 1976 (see Billmeyer et al. [3]) will be briefly described. T h e C I E L A B 1976 L*a*b* scale:  116(^)3-16 *n  L*  =  a  = 500[(^)i - ( £ ) * ]  (4.22)  200[(£)*-(y-)i]  b* = where:  • X ,Y ,Z n  n  n  =  f  T h e X,Y,Z  values of the illuminant.  is a " u n i f o r m " scale i n that it approximates the uniformity of spacing of the Munsell Color Space. T h e C I E L A B L*a*b* scale is an opponent-type system (like the Hunter 1958 Lab scale) i n which L* measures light(lOO) to dark(O), a* measures red (positive) to green (negative), a n d b* measures yellow (positive) to blue (negative).  T h i s scale  has been widely used i n the paper industry and was recently recommended by Jordan et a l . [10] i n a P P R I C Report, although they only studied near-white paper colors. T h e C I E L U V 1976 L*u*v* scale:  L*  =  calculated as above.  Chapter 4.  The Dye Concentration/Color  Relationship  u  13L > '  v  13L*(v' - v' )  27  - u'J  (4.23)  n  where:  4.X  u  X + 15Y + 3Z  9Y  v u  measures  'ni 'n v  X + 15Y + 3Z ~  u  'i ' v  calculated using X , Y , Z . n  n  n  just perceptible color differences where || 8C ||= 1 is a just perceptible differ-  ence. A l t h o u g h not recommended by the C I E i n 1976, the F M C - 1 a n d F M C - 2 color scales generate metrics which approximate the " M a c A d a m Ellipses" (within which an observer considers the colors to be matched).  These scales have been used i n some  industries although their calculation f r o m the XYZ In the simulations of this thesis, the refer to the  scale is complicated.  L*a*b* scale will be used a n d C i , C , C 2  3  will  L*,a*,b* values on that scale. A s well, the results of the simulation runs  will be given as the s u m of the average squared deviations of the colors from their setpoints (i.e. the average value of || SC ||) during different intervals of the r u n . 2  4.2  C o l o r Sensors  There are two types of color measuring devices: colorimeters a n d spectrophotometer. T h e colorimeter uses three colored filters (ie. the x, y and z filters) to separate the light reflected from an object.  B e h i n d each filter there is a corresponding detector which  measures the intensity of the light passed. Therefore, these detectors provide directly the XYZ  color values of the object, but these values are only valid for the particular  standard illuminant being used.  Chapter 4.  The  The Dye Concentration/Color  Relationship  spectrophotometer measures the continuous  28  reflectance vs.  wavelength rela-  tionship over the entire visible spectrum. A diffraction grating is used to separate by wavelength the light reflected from an object. A detector scans the separated light to measure its intensity. Having the illumination intensity vs.  wavelength relationship of the illumination  source, the sensor computer can calculate the reflectance t r u m . It can then calculate the XY using E q . 4.20.  over the entire visible spec-  Z color value for any of the C I E standard illuminants  M e t a m e r i s m is the phenomenon i n which two differently dyed objects  show the same color under a particular illuminant. B y calculating XYZ  values for two  different illurninants (eg. C I E Illuminant A for incandescent light and Illuminant D65 for sunlight), a spectrophotometer can detect metamerism between two objects.  For  on-line color measurement, an abridged spectrophotometer provides a faster reading. As before, a diffraction grating is used to separate by wavelength the reflected light, but the intensity of separated light is measured at only a discrete number (16 or 32) of wavelengths by the same number of detectors. O n a paper machine, the color sensor is located just before the reel so that the color of the finished paper can be measured. Some sensors scan i n the cross  direction  (i.e. f r o m one side of the sheet to the other) so that the average color of the sheet is obtained.  A backing plate on the side of the sheet opposite the sensor is sometimes  used to prevent the background color from distorting the measurement. If the color of the backing plate is similar to the desired paper color (i.e. simulating a thick sheet), a more accurate color measurement, one independent of sheet thickness, is obtained.  Chapter 4.  4.3  For  The Dye Concentration/Color  Relationship  29  T h eDye/Color Matrix A  scattering absorbing mixtures such as dyed paper, the K u b e l k a - M u n k equation is  used to calculate the reflectance R(X)  from a thick non-fluorescent sheet (derived by  Billmeyer [3], M a c A d a m [14]): *(A)  =  l + f(A)-v f(A) /  2  + 2f(A)  (4.24)  where: • R(X)  == T h e reflectance distribution (function of wavelength) of the dyed paf  per. • K(X)  =  • 5(A)  =  T h e absorption coefficient (function of wavelength) of the dyed paper.  f  T h e scattering coefficient of the dyed paper.  f  • ^ ( A ) ==  =  T h e absorption/scattering coefficient of the dyed paper.  For a dyed material (i.e. dyed paper) where most of the scattering is done by the substrate (i.e. paper fibres), the following approximation (Billmeyer [3], B o n h a m [4], M a c A d a m [14]) can be made:  -sW  ^ A )  =  (4-25)  or:  fW  =  7f(A) +  E^(A)A  where: • Kf(X)  =  • Sf(X)  =  f  f  T h e absorption coefficient of the undyed paper T h e scattering coefficient of the undyed paper  (4-26)  Chapter 4.  The Dye Concentration/Color  • K i(X)  =  T<  T h e absorption  Relationship  30  rate coefficient of dye i  • ~sj(X) is the absorption/scattering coefficient of the undyed paper  • ^^-(A) =  • D{  =  f  f  rate coefficient of dye i  T h e absorption/scattering  T h e concentration of dye i.  T h e absorption/scattering coefficients  ^ ( A ) , ^ M A ) , ^^(A), ^^(A) can  be determined  experimentally with a spectrophotometer as described i n Subsection 2.2.5. J.S. B o n h a m [4] has derived an extended K u b e l k a - M u n k equation which includes fluorescent dyes.  T o solve the extended equation, the value of |^(A) of each dye i n  its absorption b a n d and the q u a n t u m efficiency of each fluorescent dye i n its emission band must be known. Obviously, the function Fc relating dye concentration and color on the L*a*b* scale (combining E q s . 4.20, 4.22, 4.24, 4.26) is a nonlinear one:  C  =  (4.27)  F (D) C  where:  . C =  [dC C ] 2  . D = [D, D d  r  2  3  T  Df 3  -*' B y differentiating Fc at a particular dye concentration D , a 3 x 3 matrix A which relates incremental changes i n dye concentration to incremental changes i n color can be found: 8C  where:  =  A5D  (4.28)  Chapter 4.  • 6C =  The Dye Concentration/Color  [8d 8C  8C }  • 3D = [8D 8D  SD ]  2  d  t  2  Relationship  31  T  3  3  T  Since 8C = C — C , E q . 4.28 can be written:  C  Letting 8D = D - D,  =  x  A8D  + C.  (4.29)  AD  + V  (4.30)  E q . 4.29 becomes:  6  where V  =  =  X  C - A D .  Calculation of A: can be determined, at a particular D  A  follows. 4.26.  C  is calculated for the dye concentrations  A d d i n g a small change 8Di to Dye 1, C  + [8D 0 0 ] . Now, 8&  D  a n d C , using numerical differentiation as  r  X  is  - C". A d d i n g a small change 8D  2  is calculated for a small change 8D  =  3  is C  to D y e 2, & is — C . Similarly,  to Dye 3. Finally, A is calculated using:  sd SDi  4.3.1  0 ] . Now, 6C r  2  A  using E q s . 4.20, 4.22, 4.24,  is calculated for the dye concentrations  calculated for the dye concentrations D + [0 8D 8C  D  l  sd  SD  2  2  sd  (4.31)  3  SD  3  Calculating A at a Color Setpoint  For G a i n Scheduling, it is necessary to calculate an estimate of A (actually A  1  ) at a  -*set  particular color setpoint C  . T h i s calculation can be done offline for each desired paper  color before a production r u n begins. Because of the integration i n E q . 4.20, it is very difficult to calculate A  Fc (C)). l  directly from C  (i.e. to differentiate the inverse function  Similarly, it is very difficult to calculate directly f r o m  C  the corresponding  Chapter 4.  -•set D .  The Dye Concentration/Color  Therefore, denoting the desired  Relationship  A by A ,  both  set  A°  32  and  et  D~"  must be calculated  iteratively using the algorithm: 1. Initiallize D with an estimate of D 2. C = F {D) (using E q s . 4.20, 4.22, 4.24, 4.26) C  3. A = F A ( D ) (using numerical differentiation of E q s . 4.20, 4.22, 4.24, 4.26 described above)  4. D <— D + A 5. I F (C -  ^tset  C  6. Finally, A  - 1  (C — C° ) et  ) > M A X _ E R R O R , T H E N go to step 2.  set  A and  set  D  =  D  Example: In reality, this algorithm uses approximate (i.e. measured), not exact,  absorption/scattering  coefficients ^ ( A ) , ^ - ( A ) , ^ ^ ( A ) , ^ - ( A ) i n E q . 4.26 and it is worthwhile to see how, _ 1  —*set  for specific values of C with the exact A  - 1  .  , the estimated A  For C  = [74.2 3.57 9.88] , a yellow -orange color: T  -l  T h e difference between A  determined by the algorithm compares  1  -13.5  3.23  -.656  -16.6  -4.89  7.82  -3.11  4.71  3.74  and the corresponding A 100  II  A"  x 10  -2  (4.32)  was 20.3% using:  A  w here:  X  def  3  3  (4.33) i=lj=l  Chapter 4.  +set  For C  The Dye Concentration/Color  Relationship  33  = [74.2 16.0 10.0] , a red-orange color: r  -16.7 A"  =  1  -13.2 -6.47  T h e difference between A  1  4.58 .255 -3.06  (4.34)  6.66 x 10"  2.95 4.87  and the corresponding A * was 22.9%.  ences are less than the 25% difference between the estimated  These differ-  absorption/scattering  coefficients -$j{X), ~^{^) (where i = 1,2,3), and the the exact absorption/scattering coefficients yjr(A), -jf(X)  (where i = 1,2,3) (see A p p e n d i x A for the coefficient values).  Convergence of the above algorithm occurred within 8 iterations.  T h e algorithm,  when tested, seemed robust and, for a wide range of initial values of D , it converged on identical values of  D  (and therefore A ) . E v e n though the inverse function  FQ (C) 1  is  difficult to define explicitly, this behaviour indicates a one-to-one relationship between  D  and C  .  Chapter 5  Three Dye Input/Three Color Output Model  5.1  Complete Model  In this chapter, three dye transport models and the relationship between the dye concentration i n the dried paper and the paper's color are combined to make a complete three dye input/three color output model. T a k i n g three discrete dye transport models  9  A^CT )  ^  1  ( ^h w  i — 1,2,3) and adding  the white noise vector W\, we obtain the following 3-input/3-output A R M A X model:  0  Aiiq- ) 1  . 0  0 1  2  0  0  £> = g-'  0  0  0  E (q~')  0  0  0  EM- ) 1  2  1  2  0  l  z  0  B (q- )  0  A {q~ )  +  1  1  0  A {q' )  0  B (q~ )  0  Di.  0 B {q- ) 3  X  (5.35)  EM' ) 1  wh ere:  D . D  d  in  ^ d  [Di  D  ^ [D  2  in<1  D] 3  D  T  in<2  D ,] in 3  T  A d d i n g measurement noise to E q . 4.30, an expression for the color vector is obtained:  C  AD + Vi + W  2  where: 34  (5.36)  Chapter 5. Three Dye Input/Three Color Output Model  =  C  d  *=  9 Vi • W  [d C  f  Cf  2  3  Constant offset vector W h i t e measurement noise vector  —  2  35  Rearranging we obtain:  D  =  A "  1  C - A  -  1  Vi - A  -  1  W  (5.37)  2  Substituting E q . 5.37 into E q . 5.35 we obtain:  0  Axiq- ) 1  0  0 X  2  0  0  1  C = q'  0  d  0  0  0  E (q~ )  0  0  0  0 1  1  0 0  A^q- ) 1  0  Esiq- )  {  B^q- )  0  1  D  0  1  2  Axiq- )  0  1  2  0  B (q- )  0  1  1  +  A"  A^q- )  E (q- ) 1  1  0  A {q- )  0  Biiq- )  0  A-  1  W + V 2  A {q^) 3  where:  ^(T )  o  1  • Vjs  =  def  Constant offset vector  =  0  o  0  A {q~ )  1  T o make  1  o  o 0  Ai{q- ) 1  0 0  1  2  0  0  A  1  _ 1  Vi  A^q- ) 1  monic, we multiply by A :  A (<r ) a  3  0  A {q- )  0  A  o 1  2  0  A^q- )  0  Mr )  o  o  0  ^(g- )  0  o  o  ^(r )  1  A-  0 A^q- ) 1  1  C = q~ A d  0  Sif?- )  1  A, 1  2  Chapter 5.  Three Dye Input/Three  £r(? )  o  _1  +  0  36  Color Output Model  o 0  E^q- ) 1  0  0  + Vg  W  4  ^3*(g  _ 1  (5.38)  )  or: AA(g- )A- C 1  1  =  g  -  d  A B ( - ) A n + E**(g- )PF 1  9  1  4  + V  (5.39)  3  where: A ^ def  AGr ) 1  1  0  )  0  1  2  B (q-') 1  0  =  1  ^  F  0  0  g  Bsiq' ) 1  0  0  0  ^(g- )  0  0  0  1  E^q- ) 1  0  1  E**( -i)  0  0  Efiq- )  def  ]  3  1  1  D  A (<r )  2  E{(q- ) 1  o  B (q- )  0  E*^- )  0  A (q~ )  o  • Biq- )  0  0  o  £ **(? )  0  0  0  -1  2  £ **(g 3  _ 1  )  ^(g- )  0  1  E*(Q  1  )W  > 3  =  f  T h e spectral factorization of  0 0  E (q^) 2  0  0 0 E (q~ ) 1  3  Chapter 5. Three Dye Input/Three Color Output Model  A^ )  0  1  0 0 9  A  5.2  r  0  1  0  E**( -!) W =  0  A (q- ) 2  37  =  A -  ^  1  A (g- ) 1  3  T h e spectral factorization of A E * ( g - ) ]  Constant offset vector  =  W. 3  AVo  Simplified M o d e l  It is reasonable to assume that the transport dynamics of the dyes are similar. T h i s is the case when the three dyes are of the same type (i.e.  acid-type or base-type) with  similar fixing dynamics or when the dyes fix to the pulp fibres so rapidly that the dye dynamics are the same as that of the pulp. makes this assumption. Letting  T h e color control literature consistently  Aj(q~ ) — A(q~ ) and Bj(q~ ) = B(q~ ) for j — 1,2,3, 1  x  1  1  E q . 5.39 simplifies to:  Aiq-^C  =  q-'ABiq-^Din  + W'iq-^Wt  + Va  (5.40)  B y subtracting E q . 5.40 at two successive samplings, the constant disturbance vector is eliminated and the simplified  Aiq-^AC  differential model is obtained: =  q- AB(q- )AD d  l  in  + E"{q- )W 1  4  (5.41)  where: A  def  (1-g" )  . E'V ) 1  The  simplified  1  =  E-'Or^l-ir ) 1  differential model is used explicitly to identify the color/dye process  and this identified model is then used to derive the D a h l i n controller as described in Chapter 8.  Chapter 5.  Three Dye Input/Three  38  Color Output Model  Simplified M o d e l in Predictor Form  5.2.1  T o generate the e^-step-ahead predictor, the scalar polynomials F(q~ ) 1  and  G'(g ) _1  are calculated by solving the scalar differential Diophantine E q u a t i o n : 1  -  (5.42)  F{q- )A(q- ){l-q- )+q- >G'(q- ) 1  1  1  l  1  or:  F( - M(«- )(l-r ) ,  g  1  = l - f  1  G'(g )  W h e n q~ = 1, G'(l) = 1. Therefore  GV )  G V )  (5-43)  can be expressed as:  _1  l  4  = l+G^Xl-g- )  1  (5.44)  1  Substituting this expression into E q . 5.43, we obtain the differential F(q~ ) l  Aiq-W  - q-*)  Diophantine E q . :  (1 - q~^ ) - q~ " G{q^){\ - q~ )  =  d  (5.45)  l  where: • dh  =  • deg[F)  Discrete prediction horizon  =  • term[F] • deg[G] • term[G]  4 - 1  =  dh deg[A] - 1  = =  deg[A]  M u l t i p l y i n g E q . 5.41 by F(q' ) 1  a n d then eliminating F(q~ )  A(q~ )(l  l  1  - q' ) using 1  E q . 5.45, we obtain the form of the 4-step-ahead predictor: (1 - q- * ) C - q~ « G(q^)AC d  d  q- F(q~ )AB(q~ ) d  1  1  AD  in  = + Ffa-^E'V ) 1  ^ 4  (5.46)  39  Chapter 5. Three Dye Input/Three Color Output Model  M o v i n g the matrix A to the left of the term containing it, we obtain:  ( l _  f  4 ) c -  f  4 G{q~ )AC  q~ A F(q~ )B(q~ ) d  M u l t i p l y i n g by A predictor i n  -  1  =  1  1  AD  1  and taking the  AD  in  in  + F(q~ )E"(q^ ) 1  W  1  (5.47)  4  term to the left hand side, we obtain the  linear control form (see G o o d w i n et al. [9]): q~ FBiq- ) d  AD  1  A  _ 1  in  =  ( l - q- )C  A- G( - )AC-FE'( - )FF  -q~ *  dh  1  d  g  1  g  1  3  (5.48)  where:  FB(q^) FE'(q->) . E'iq- )^ 1  F{q-*)B{q-i) *f =  F(q-')E'(q^) X- E"{q^)W x  A  =  (1 - g - ^ E ^ g " ) 1  W  3  T h e predictor form is used to derive the predictor-based controllers as described i n Chapter 7.  Chapter 6  Discussion of Control Methods  T h e objective of control is to maintain the paper color at a desired setpoint by counteracting disturbances (e.g. the addition of colored broke to the pulp). A n o t h e r objective is to achieve a good response to changing color setpoints.  B o t h adaptive and non-  adaptive techniques will be considered. In this thesis, it is assumed that estimated dye transport parameters (from which an estimated discrete model of the transport process can be calculated) and  estimated  optical properties of the dyes (from which the dye/color matrix A can be calculated theoretically as described i n Chapter 4 ) are available. These estimates are used to initialize and constrain the adaptive controllers and to design the non-adaptive controllers. These discrete controllers (with Z O H ) are applied to a simulated continuous process with the actual dye transport parameters and actual optical properties of the dyes (with which the color resulting f r o m the output dye concentration D can be determined). T h e performances of the non-adaptive controllers are compared with each other.  Further-  more, the performances of the non-adaptive controllers are used as standards against which to evaluate the performances of the corresponding adaptive controllers.  6.1  Closed-Loop Control  There is a long time delay from the dye addition point to the color sensor. In order to ensure that the discrete model of the dye transport process has no nonminimum-pha.se zeros due to a fractional time delay, the sampling interval should be selected so that it 40  Chapter 6. Discussion of Control Methods  41  divides the process time delay evenly. If this is done, the discrete model of the process is stable, minimum-phase, and is quite simple to control, although the long time delay adds some difficulty. T h e dye/color relationship is nonlinear and can be described by a constant matrix (i.e. A ) only for incremental  color changes about a particular color.  Generally, however, the relationship is monotonic i n that an increase in the amount of a particular dye will consistently increase or decrease a particular color component. Repeating E q . 4.29, we see how the incremental  dye/color matrix A relates incre-  mental changes i n dye concentration to color:  C  =  A6D + C.  T h e identification of the process is i n large part the identification of the  incremental  dye/color matrix A which is required by any controller. Initially, only an estimate A of A is available. A can also change with changes i n the color setpoint and changes in other process variables such as (from C l a p p e r t o n [6]): • T y p e (i.e. Sulphite, K r a f t , T M P ) and color of undyed pulp • A m o u n t of sizing (sizing helps fix the dye to the fibre) • p H of pulp (affects acid-type dyes especially) • Temperature of drying cylinders (affects acid-type dyes especially) • A m o u n t of active bleach left i n the undyed pulp • Strength and quality of the dyes • Moisture i n finished sheet. As well the color is subject to step disturbances caused by:  Chapter 6. Discussion of Control Methods  42  • A d d i t i o n of dyed or undyed broke • Change i n the offset vector C i n E q . 4.29 due to changes i n the process variables listed above. These disturbances require an integrating controller to counteract them. T h e unconstrained adaptive controllers (both explicit and implicit) are found to be unstable as the identification procedure attempts to converge to the process parameters.  T o prevent this, the parameters are constrained using Parameter Projection as  suggested by P r a l y [18] and described i n Chapter 9. T h e identified parameters are kept within a "projection sphere" with radius /3 about the a priori estimated (i.e. A ,  parameters  B(q~ ), A(q~ ) or parameters calculated f r o m them). If the constraint radius 1  1  (3 is zero, a non-adaptive controller is obtained since the controller parameters are constrained to be those calculated from the a priori estimated parameters.  T h e following  types of controllers are derived and tested i n this thesis. Predictor-based Controllers: In processes with considerable time delay, the d^-step-ahead predictor can be used for feedback control if the predicted process output is set to the desired output setpoint. First, assuming that  d is equal to d, the J-step-ahead predictor predicts the expected n  value of the process output d sampling intervals (i.e. steps) i n the future based on present and past values of the process output and present and past values of the process input. If there is a stochastic or other disturbance to the process, the accuracy of the prediction deteriorates. the  Allowing d  n  to exceed d, the tf^-step-ahead predictor gives  expected value of the process output dh sampling intervals i n the future based on  present and past values of the process output and present, past, and some  future values  of the process i n p u t . These future values of the process input (which must be set to some values for feedback control) provide the controller based on the (/^-step-ahead  Chapter 6. Discussion of Control Methods  43  predictor with the flexibility to achieve specific control objectives.  For example, if a  robust controller is wanted (as it is i n this thesis), the future values of the process input are set to the present value of the process input algebraically. T h e resulting present value of the process input calculated by the predictor/controller is then applied as the control input. Let us illustrate the derivation of this controller with a d-step-ahead predictor. T h e form of the process model to be identified is:  = -X> <r - AAn + A - ( C - < r ( 7 ) i d  q~ AD d  in  i=l  - A  1  i  -  1  i  (6.49)  G(q~ )q~ AC 1  d  B y simply shifting to data taken d sampling intervals later a n d setting q  +d  C to C  ,  the controller is obtained:  A  An =  -£A<r'AAn + - A -  1  G (  g  -  1  A"  1  ((?"'-(?)  ) A C  (6.50)  These are implicit/direct adaptive controllers. T h e process is identified " i m p l i c i t l y " i n the f o r m of a predictor model and the controller is derived "directly" from the identified process model by a time shift. T h e details are given i n Chapter 7. Dahlin Controllers: T h e D a h l i n controllers are members of the Smith family of time-delay-compensation controllers. T h e S m i t h controllers cancel both the poles a n d zeroes of the process i n order to achieve desired C L responses. T h e D a h l i n controllers are designed to achieve a first-order C L response  _^ ~f^ • If P equals zero, the C L response becomes the zero-  <f 1  q  order response q~ and the controller is called a Deadbeat controller. T h i s is the fastest d  setpoint change for a process with a discrete time delay of d sampling intervals.  Chapter 6. Discussion of Control Methods  44  Since the D a h l i n controllers cancel both the poles a n d zeroes of the process, the process must be minimum-phase, stable processes for stable C L performance. If this is the case, these controllers are simple to apply. A s well, D a h l i n controllers are integrating controllers. In this thesis, the D a h l i n controllers are used as explicit/indirect adaptive controllers. T h e process is identified "explicitly" i n its simplest f o r m (by identifying the  parameters A , B(q~ ) and A(q~ ) of E q . 5.41) a n d the controller is derived "indi1  1  rectly" i n that the non-adaptive method of calculating a D a h l i n controller is used after the process has been identified. T h e details are given i n Chapter 8. For good convergence properties of the R L S algorithm, Elliot & Wolovich [8] reco m m e n d that the number of parameters estimated should not exceed 10-12. Praly [18] feels that explicit/indirect controllers are more robust.  Gain Scheduling  6.1.1  Because of the nonlinearity of the dye/color relationship, the dye/color matrix or  pro-  cess gain A varies with color. If the process gain A can be calculated for the desired -*set  C  m  r  —_1  using the method of Subsection 4.3.1, then A  can be used as the controller gain  (i.e. G a i n Scheduling). D u r i n g a setpoint change, an estimate of the mean gain A lating A at the color  m  is obtained by calcu-  halfway between the two color setpoints . T h i s mean gain is used  in the C L controller i n the time delay (ie. the d sampling intervals) when the critical changes i n dye flowrate are made using the difference between the two setpoints (i.e. actually the difference the new setpoint and the current color). After the time delay, the color should be close to the new setpoint and A , calculated using the new setpoint, can be used. In the controllers (both non-adaptive a n d adaptive) simulated i n this thesis, this  Chapter 6. Discussion of Control Methods  45  method of G a i n Scheduling is used.  6.2  Open-Loop Control ~*8et,b  F r o m past production runs of a certain color (i.e. C input concentrations (i.e. D{  ) of dyed paper, the steady state  ) of the corresponding dyes may be well known. If  n  an accurate dye transport model is available, any desired setpoint change T F from the current color C  to the new color C  (actually the desired T F from the current  output concentrations to the new dye output concentrations) can be achieved with  dye  O L control. F o r example, if the fastest setpoint change T F (i.e. Deadbeat) is desired: =  °~  d  J  B  6.51  and the transport model is:  -  3  «  ~  ^  (6.52)  Mr') '  —f  then, removing D by substitution, the following O L controller determines the transition of D  in  from D  in  to D  in  : ° "  -  A  {  " ^  For this application, it is essential that the steady state gain of -*  -*  since, at steady state, D{  n  of B(q~ ) 1  is essential.  ?  A(^- ) 1  ^ ^  E  ^>  UIU  V  set  should equal Di  n  . A s before, the m i n i m u m phase property  O L control could be applied for at least the duration of the  transport delay time (i.e. d sampling intervals) resumed.  (6.53)  , after which C L control would be  Chapter 7  Controllers derived from the Predictor Model  In processes with considerable time delay, the 4-step-ahead predictor can be used for feedback control if the predicted process output is set to the desired output setpoint. A n introduction to this controller is given i n Chapter 6. T h i s chapter will do detailed derivations of this type of controller. Introducing the definitions:  • deg[FB]  =  deg[F] + deg[B]  and applying them to the process model i n predictor form (i.e. E q . 5.48), one obtains: deg[FB]  q-  £  d  fl q- AD i  i  =  in  i=0  A-  7.1  (C - q~  1  dh  C) - q~  dh  A"  G(q~ ) AC  1  1  - FE'(g  _ 1  ) W  3  (7.54)  W i t h Minimum Prediction Horizon  Setting dh to its m i n i m u m useful value of d and partitioning the sum produces: deg[FB]  q~ AD d  =  in  [-  £  fl> q~ q~ i  i  d  AD  in  + A  - 1  (C - q~ C) d  i=l - q~ A d  1  G(q~ )AC 1  I fb  - FE'(q^)vV } 3  0  or: deg[FB]  q- AD d  in  = -  £  (fb /fb )q- qi  0  i  d  AD  in  46  +  (A- /fb )(C-q- C) l  0  d  (7.55)  47  Chapter 7. Controllers derived from the Predictor Model  - q~ ( A " / f b ) G(q~ )AC 1  d  - (FE'(g-1)/fb )  l  0  0  W  (7.56)  3  E q . 7.56 is the f o r m i n which the multi-output R L S algorithm (or the R M L algorithm, if the last term is desired) identifies the predictor.  Chapter 9 gives details of this  algorithm. B y multiplying by q  (inserting data taken d sampling intervals later), the d-step  +d  ahead predictor of q+dC  is obtained: deg[FB]  ADin  =  -  £  {fb /jb )q- AD i  i  0  +  in  {lL- /jb ){ + C-C) i  0  q  d  1=1  - (A-  1  /fb^Giq- )  (7.57)  AC  1  where: • q C  =  • C  q° C  +d  —  T h e ef-step ahead prediction of the color vector.  f  =  • q~ C for i > 0 x  • A  =  • D{  =  (1 — q )  f  Past color vectors.  f  ==  _i  =  n  T h e present color vector.  f  Change between two successive samples.  —'  q° D{  n  T h e imminent control signal vector (to be calculated by  controller). • q~ e  X  for i > 0  l  =  f  =  Past control signal vectors.  f  Estimate of corresponding X provided by the R L S algorithm . , -*  T h e controller is obtained by setting q  +  _  deg[FB] ^ ^ ^ ^  AD  in  =  -  £  ,  fb lfb q- AD i  0  A^V^G^  t  -*set  C to the desired color setpoint C  i  7 7  in  ) AC  +  •  :  v  A.- lfb {C -C) l  0  n  (7.58)  Chapter 7. Controllers derived from the Predictor Model  48  T h i s is the M i n i m u m Variance Controller M V C i n Control Configuration. It is a deadbeat controller i n that the control signal is such that the desired C  is expected to be  achieved ci-steps ahead. In the Generalized M i n i m u m Variance Controller G M V C (developed by K o i v o [11]) —*  ADi  the change i n control signal  n  is attenuated by dividing E q . 7.57 by (1 +  p) where  p is a weighting factor. W e obtain: deg[FB]  AD  =  in  [-  £  ,  fbJfbvq-'AD^  «  ^  + A-i/fbviC * 8  -C)  i=l  - A~^G[  7.2  q  r  T  AC] I(1 + )  )  (7.59)  P  W i t h Extended Prediction Horizon  A generalization of the M V C is the Extended Horizon Controller E H C developed by Ydstie [23].  Allowing the prediction horizon dh to exceed d makes the M V C more  robust and is useful when the delay time d is uncertain (i.e. the selected dh is greater than the estimate of d ). T a k i n g E q . 7.54: deg[FB]  E  q~  d  A"  1  fl q- *Din  =  i  i  (1 - q- » )C - g - " A " d  d  G(q~ ) AC -  1  FE'(q- )W  1  1  z  and substituting the partition: deg[FB]  d -d-l  deg[FB]  h  E  + fi>d -dq- -  {dh d)  h  +  i=0  (7-60)  E i=zd -d+\ h  we obtain the form i n which the predictor is identified: d -d-l  deg[FB]  h  q- »AD d  in  =  [(- E A ? ' - E -  fb -')q-  d  iq  AD  m  Chapter 7. Controllers derived from the Predictor Model  + A"  (1 - q- * )C - q~ A "  1  d  - FE'(g B y multiplying by q+dh  _ 1  dh  ) W ] / fb _ 3  dh  (inserting data taken  ahead predictor of q C  G(q~ ) AC  1  1  (7.61)  d  d sampling intervals later), the dy,-step n  is obtained. B y setting q C  +d,l  49  , the controller is ob-  = C  +d,,  tained: deg[FB]  AAn  = (- E fbjfb^q- E i=0 i=d -d+l + A-*/fb _ (C' - C) - A - / ^ ^  fbJfb^q-^-'AD,  1  h  dh  et  d  If we assume the future unknown  AD  values i n  in  equal to the imminent control signal change deg[FB]  AD  in  = [-  E  G(q-')  1  ADi , n  Y^=o  d  1  foil ft>d -d<l h  .  t  i  i=df —d-\-l  dh d  dh d  i  dh d  1  ADi are n  then the controller becomes:  f  fb /fb - q - ~ ADin  (7.62)  AC  s  + A~ I fb -d{C x  dh  ee  -C)  L  - A~ ifbd~Td r  GZF) AC] I (1 + E1 lWbd~Z) d  i=0  (7.63)  Chapter 8  Dahlin Controllers  8.1  Decoupling Using Complete A  1  D a h l i n controllers are discrete, integrating controllers that are simple to apply to minimum-phase, stable processes with a time delay (such as the dye transport process). A n introduction to this controller is given i n Chapter 6. T h i s chapter will do detailed derivations of this type of controller. If the discrete dye transport model (ignoring noise) for Dye i is:  D ,i  (8.64)  Mq- ) 1  in  and the desired C L response is:  (l-pq-^Di  = q- (l-p)D? d  (8.65)  t  then the corresponding D a h l i n controller is:  Hiq-^Biiq-^D^i  =  (l-p)A (q- ){Dr -D ) i  1  t  (8.66)  i  where:  • p =  f  T h e desired C L discrete time constant  continuous time constant.  . Il(q^)  *  1 - pq^ - (1 - p)q~  d  50  =  e ^ where -  T is the desired c  Chapter 8. Dahlin Controllers  51  Note that if p = 0, a Deadbeat controller is obtained. T h e D a h l i n controller can be seen to cancel both the poles and zeros of the process and therefore can only be applied to stable, minimum-phase processes. It can be applied to the dyeing process since B(q~ ) 1  is minimum-phase for the sampling intervals used i n the simulations. In order to determine D\  et  - D for i = 1,2,3 (i.e. D {  D  =  —D  A  _  1  - D), let:  ( C -C)  (8.67)  T h e D a h l i n controller is an integrating controller since H(q~ ) includes the factor (1 — 1  g ). - 1  Hence we can expect any step disturbance (eg. colored broke addition) to the  desired paper color to be completely counteracted and any color setpoint change to be accomplished. E q . 8.67 allows the three decoupled dye controllers to be implemented. T h e block diagram of F i g . 8.3 illustrates the application of one D a h l i n controller to the flow of Dye  i a n d the use of A  -  1  to decouple the dyes. T h e contribution of color  C\ only is  <f>ij is an element of  detailed. T h e remaining dye controllers are similar. In F i g . 8.3, A- . 1  Using the Simplified Dye Transport Model: If the transport dynamics of the three dyes are similar (i.e.  Q  A^}- ) 1  ^) then they can  be combined:  A(q- )D  =  X  q- Biq-^Din  (8.68)  d  and, if the desired C L responses are identical, the decoupled M I M O D a h l i n controller is:  H(q-')B(q-')D  = (l-p)A(q-')(D  set  in  -D)  (8.69)  Substituting E q . 8.67 i n the above produces:  H{q-i)B{q-*)D  in  =  (l-p)A(q-')A- (C 1  Set  - C)  (8.70)  Chapter 8. Dahlin Controllers tset  Di)  52  Din i  Dahlin Controller  Di  Dye i Transport  t  other dye other dye  ©-  9;  <f>il (C°  et 2  -  Dye -toColor Transform  C  2  Color Sensor  <f>i3  Figure 8.3: A p p l i c a t i o n of the D a h l i n Controller W h e n A(q  8.1.1  1  ), B(q  ) and A  1  1  are substituted, the non-adaptive controller is obtained.  Adaptive Controller  A p p l y i n g R M L to the differential simplified model of the color/dye process  (from  E q . 5.41):  A{q-*)AC  =  q- AB{q- )AD d  1  in  +  E"{q- )W  the parameters of this model can be explicitly identified. determined from the first terms of AB(q~1). identified value of Ab0,  1  A  (8.71)  4  (actually A 6 ) can be 0  F r o m the other terms and the previously  the coefficients of B(q~1)  (actually B(q~1) / b0) can also be  determined. See Chapter 9 for details of the identification. T h e determination of the coefficients of A(q~ ) is straightforward. 1  Using the explicitly identified parameters,  Chapter 8. Dahlin  Controllers  53  E q 8.70 becomes:  H{q- )B{q-')lb D 1  8.2  0  iset  = (l-p)A(q- )Ab 1  in  Decoupling Using Diagonalized A  0jj  If the off-diagonal elements of  (CT" - C)  0  (8.72)  - 1  i n F i g . 8.3 are set to zero then  Bristol's method is  being applied:  H{q-*)B(q- )D 1  =  in  0n (l-p)A{q-*)  0  0  0  0  0  0  22  0  (c -c) set  (8.73)  <f>  33  T h i s method allows that, i n some coupled multi-input, multi-output processes, each control input be determined by feedback from only one corresponding output as if the  process were a decoupled one. T o apply it, find:  (A" ) 1  011  </»21 031  012  022 032  013  023 033  (8.74)  where:  A  The  =  A13  An  A 1 2  A 2 1  A 2 2  A  A 3 1  A 3 2  A  7n  7i2 7 i 3  721  722 723  731  732 733  (8.75)  relative gain matrix T where:  (8.76)  Chapter 8. Dahlin Controllers  54  is calculated by letting:  Hj  =  ij<l>ji for i,j = 1,2,3.  (8.77)  x  Bristol's method states that generally color C{ can be controlled using dye Dj only (using feedback of C ; ) if 7y is the largest positive element i n row i (if jij large).  is not too  A c c o r d i n g to Deshpande [7] if some values of relative gain i n a row are close  then interaction between controlled outputs is likely, especially if the response times of the loops are similar. In these cases, Bristol's method cannot be used. Because of the highly interactive nature of dyes and color, color control does not seem a good candidate for this simplified form of C L control.  In the simulations,  Bristol's method did not perform well which can be explained as follows. At C  = [74.2 16.0 10.0] the calculated r  relative gain matrix is:  0.88  0.43  -0.31  0.15  0.37  0.48  -0.02  0.20  0.82  T h i s would indicate that G\ should be controlled with Dye 1 (i.e. the dye represented by the first column). Similarly, it indicates that C should be controlled with Dye 3. T h i s 3  leaves C  2  to be controlled with Dye 2 even though two elements i n the second row are  close i n magnitude. W h e n this control scheme was implemented using a diagonalized A  -  1  control was poor, especially after C  was changed to [74.2 3.57 9.88] . T h i s can r  be explained by examining the relative gain matrix at the new setpoint: 0.80  0.27  -0.07  0.13  0.28  0.59  0.07  0.45  0.48  In it, the second row shows that there is now a strong interaction between C  2  and Dye  3: an interaction which was ignored by the previous pairing of dyes and colors.  Chapter 9  Adaptive Parameter  9.1  Identification  Multi-Output R L SAlgorithm  If the initial estimates of the process parameters are i n error or if the parameters change in time, a Recursive Least Squares ( R L S ) algorithm may be useful i n identifying more accurate values of the parameters. A t each sampling, the R L S algorithm updates the last set of identified parameters to obtain a new set of identified parameters using the newly sampled data.  Since all the models developed i n Chapter 5 are 3-dimensional,  the R L S algorithm i n this application will use the 3-output form:  Y  =  M 6 T  + W  (9.78)  where: ~*  • Y • M • 0 -*  • W  def  = 3 x 1 def  =  =  column vector containing model output •  .  UQ x 3 matrix containing measured data 7i© X 1 column vector containing model parameters  f  def  = 3 x 1  column vector containing the identification/prediction error.  T h e actual contents of these vectors and matrix depend on which process model is used. T h e contents for two alternative models are given i n the Subsections 9.1.1 and 9.1.2. T h e R L S algorithm used i n this thesis is:  55  Chapter 9. Adaptive Parameter Identification  56  1. Initialize Y, M with available data; initiallize P with 1001; initiallize 0 ^ with  the estimated parameter vector. 2. W  — Y — M  3. G = P M ( M  Oc  T  PM  where W + A)-  is the a priori prediction error.  1  —*  4. U p d a t e the unconstrained parameter vector ®u using:  0V = ® + GW c  5. A K a l m a n - t y p e covariance matrix update is used to maintain an almost-constant tra.ce[P] (i.e. 200 <trace[P] < 320). First, calculate trace[P]. U p d a t e the scalar a (where 0.1 < a < 1.0) as follows. If trace[P] > 320, then:  a  *-  a-0.25(a-0.11).  If trace[P] < 200, then:  a  <-  Finally, update the covariance adding the  a + 0.25(1.0 -  a).  matrix P using the K a l m a n - t y p e update (i.e.  incremental covariance matrix o:P'): P  6. U p d a t e the  <-  ( P - G M  T  P )  +  aP'  constrained parameter vector Qc using Parameter Projection (see  Section 9.2):  0c  =  0 +min[l, g  B  }  || 0C/" — 0£' ||  {QU-QE)  Chapter 9. Adaptive Parameter Identification  ~*  T  7. W = Y — M  57  .  -*  ©c  where  .  .  .  is the a posteriori prediction error.  8. Wait until the next sampling of the process and update Y and M with that sampled data 9. G o to step 2. In this algorithm, let: • 0 c contain the identified parameters (represented by a n overbrace) used i n the adaptive controller (i.e. i n the predictor-based controllers, the coefficients of G(q *) and FB(q )/fb x  0  and the elements of A  the coefficients of A(q *) and B(q  1  1  / fb ; Q  i n the D a h l i n controllers,  )/&o, and the elements of  Ab ). 0  • ®E contain the estimated parameters used to constrain the identified parameters. It contains the parameters (represented by an overline) derived from the  estimated  process model (i.e. i n the predictor-based controllers, the coefficients of G(q and FB(q  )/fb  1  coefficients of • I ^  =  • P  =  • A  =  • P'  and the elements of A  A(q" ) and B(q~ )/b , 1  1  0  1  )  /fb ; i n the D a h l i n controllers, the 0  and the elements of A 6 ) 0  T h e identity matrix  f  • G  =  0  1  f  f  n© x 3  gain matrix  n © x 71© covariance matrix Constant 3 x 3 covariance matrix representing an estimate of cSfW'PFr]  I i n the simulations. =  Constant n© x n© diagonal matrix  =  0.41 i n the simulations.  Chapter 9. Adaptive Parameter  Identification  58  T h i s R L S algorithm uses a K a l m a n - t y p e covariance matrix update, which is an alternative to the forgetting factor update. T h e K a l m a n - t y p e update is useful if certain parameters (for example, the i  parameter) are known to have more variability  th  (i.e. higher variances as represented i n the corresponding elements of P ) . In order to maintain these higher values the corresponding diagonal elements of the incremental matrix P ' (in the example, the i  diagonal element) are increased. In the simulations,  th  P ' contains ng identical elements since it is assumed that all parameters have equal variance.  In comparison to the K a l m a n - t y p e update with its ng degrees of freedom,  the forgetting factor update has only one. T h e variable scalar a keeps P within a reasonable range. If P was too large, then the gain G would be too large a n d the algorithm would adapt too quickly. If P was too small, then the gain G would be too small and the algorithm would not adapt.  9.1.1  R L S Applied with the Predictor-based Controllers  A s given by E q . 7.56 i n Chapter 7, the f o r m of the predictor with m i n i m u m prediction horizon to be identified is: deg[FB]  q~ AD d  =  in  -  E  (fb /fb )q- qi  i  0  d  AD  in  +  (A- /fb )(C-q- C) l  0  d  i=l  -q' (A- /Jb )G(q-')AC d  l  0  -(FE'(q- )/fb )W l  Substituting E ( g  _ 1  ) for  0  (9.79)  3  —(FE'(q~ )/fb ) 1  and  0  W for W , we obtain: 3  deg[FB]  q- AD d  in  =  -  E  (fb /fb )q- qi  0  i  d  AD  in  +  (A- /fb )(C-q- C) l  0  d  i=l  -q- (A- /ft, )G(q-')AC d  l  0  + E(q- )W 1  where:  (9.80)  Chapter 9. Adaptive Parameter Identification  0  E^q- ) 1  • E(g  - 1  )  d  ^  f  0  0  ^(g" )  0  0  1  0  1  ^ ( g - ) f o r i = 1,2,3  =  f  59  E {q~ )  1+ £gf  l  3  ]  e^-g^  F i t t i n g the model of E q . 9.80 into the R L S form of E q . 9.78 by using the following definitions of Y,  M , 0.  T h i s is a Recursive M a x i m u m Likelihood ( R M L ) structure  that minimizes the number of parameters to be identified (since the terms of A  -  1  are  identified only once).  Y  =  [q^ADr  q- AD d  2  q~ AD ] d  3  T  (9.81)  60  Chapter 9. Adaptive Parameter Identification  and  q~ - AD  q~ - AD  q- ~ AD  q- - AD  2  q' - AD  q- ~ AD  q- ' AD  2  q- ~ AD  d 1  d  d  q  2  d 2  1  3  d 3  1  - d - A  A  D  q- ' ADi d 5  3  d 2  d  3  3  3  2  q- ' AD  q- ~ AD  2  q- ~ AD  d  s  d 4  d  -fb /fb  0  -fb /fb  0  2  q- -*AD d  i  d 1  2  3  3  5  -fbjfb  3  0  q'C, - q~ C  x  0  0  <f>n/fl>o  q°C -  q' C  2  0  0  $12/fl>0  q°C -  q- C  0  0  013/'ft>0  0  02i//&o  0  $22/fl>0  0  023/A)  d  d  2  d  3  3  0  q°C - q~ C  0  q°C - q~ C  0  q°C - q~ C  0  0  q°C -  0  0  q°C - q~ C  0  0  q°C - q~ C  d  1  d  2  d  3  1  2  3  q- d d  l  d  2  d  3  Lio  L20  L30  in  L2\  L31  L12  I>22  L  031 / fl>0  032 /flo  2  033/ foo  3  -go •  -9i  32  -92 e  q- W  0  0  q- W,  0  0  0  q~ W  0  q~ W  0  0  q-HV  e ,i  0  0  q- W  e,  l  t  2  where, i n this example:  l  2  2  2  e  l,2  0  e  0  e,  2,l 2 2  3  2  l,l  3  3  3 2  61  Chapter 9. Adaptive Parameter Identification  • d  =  4  • deg[A] = 3 • deg[B] = 2 • deg[F] = d - 1 = 3 • deg[G] = deg[A] - 1  = 2  = deg[F] + deg[B] = 5  • deg[FB]  • deg[E) = 2 • ng =  23  and: =  • q~ X l  f  Value of X taken I sampling intervals i n the past.  • (f>ij/fb (with i,j  = 1,2,3)  0  • Lij  =  =  f  Zl (<t>ik/fb yq- - &C =1  • {(j>ijlfb )* = 0  f  o  d j  Elements of A  1  /fb  0  k  Previous value of  (/>ij/fb  0  T h e prediction error vector used i n the R M L is the a  9.1.2  -  posteriori prediction error.  R L S Applied with the Dahlin Controllers  A s given by E q . 5.41 i n Chapter 5, the form of the incremental process model to be identified is:  Aiq-^AC  =  g  - A S ( g - ) A A n + E"(g- )#4 d  1  1  (9.82)  Chapter 9. Adaptive Parameter Identification  Substituting E ( g ~ ) for E " ( g 1  _ 1  62  ) a n d W for W , we obtain:  Aiq-^AC  4  =  q- AB(q- )AD d  1  +  in  E{q- )W l  (9.83)  where:  = l+ur^  d  A{ -') q  5( - ) 9  X  f  =  deg[A]  E% i b 9 B)  jq  j  0  Eiiq- ) 1  E(<T )  def  X  .  0  0  ^((T )  0  0  J ^ , - ) fori = 1,2,3 1  0  1  d  =  1+ ^  f  E {q- ) 3  -.deg\E]  1  ^  F i t t i n g the model of E q . 9.83 into the R L S form of E q . 9.78 by using the following definitions of Y,  M , 0.  T h i s is a Recursive M a x i m u m Likelihood ( R M L ) structure  that minimizes the number of parameters to be identified (since the terms of A i n  AB(q~ ) 1  are identified only once).  Y  =  [ A d A C  2  AC ] 3  T  (9.84)  Chapter 9. Adaptive Parameter Identification  Ad  q- AC  2  q-'ACs  a  g- Ad  q' AC  2  q~ AC  a  g- Ad  q~ AC  q- AC  03  if"  1  2  3  1  2  3  A 6  <T AAn,2  0  0  A1260  - AD  in>3  0  0  A1360  0  <r AAn,i  0  A 21K  0  q' AD  0  A 22 bo  0  q- AD  0  A 23 ^0  0  0  q- AD  0  0  q- AD  0  0  q~ AD  Ln  L21  Li  61/60  L22  L32  62/60  0  0  0  0  d  d  d  d  q- W 2  where, i n this example:  =  2  3  0  q  •  Z  2  2  3  0  d  d  • deg[B] = 2  2  x  - AAn,i  g  • deg[A] = 3  63  y  in<2  in>3  d  d  u  in>1  1  in<2  d  3  q~ W  2  0  0  q~ W  0  0  0  q~'W  0  0  q~ W  2  2  3  A32&0  e  i,i  e  l,2  e  2,l  e2,2 3  2  A i bo  A 33 60  in<3  0  l  —* 0 — KJ  0  3  e ,i 3  63,2  Chapter 9. Adaptive Parameter Identification  • n  64  = 20  g  and: =  • q~ X l  Value of X taken / sampling intervals i n the past.  f  • Xijb (with i,j = 1,2,3) 0  • Li,  = EU^ikboTq- -  d  f  d j  • (Xijbu)*  =  f  =  f  Elements of A&o  AD  Previous value of  inik  \ijb  0  A g a i n , the prediction error vector used i n the R M L is the  9.2  a posteriori prediction error.  Constraint of Identified Parameters  If the parameters identified by R L S are unconstrained, the controller is unstable. In order to provide robustness, the method of constraint suggested by P r a l y [18] (i.e. —*  Parameter Projection) was used. T h e constrained parameter vector 0 c  1 S  kept within  a "projection sphere" (defined with respect to the norm || * ||) with radius (3\\ ®E \\ about the estimated parameter vector ®E using: 0c  =  0 f i + min[l,  ..}{eu-e ) II  (9.85)  E  II 0 y  —0 £  where: • Bu  =  • 0  =  C  T h e unconstrained parameter vector provided by R L S ,  f  The  constrained parameter vector containing the parameters (repre-  sented by an overbrace) used i n the adaptive controller (i.e. i n the predictorbased controllers, the coefficients of G(q~1) and  FB(q~ )/fb 1  and the elements of  0  A / f b ; i n the D a h l i n controllers, the coefficients of A(q~ ) and B(q~ )/b , and - 1  l  0  the elements of A f e ) ; 0  1  0  Chapter 9. Adaptive Parameter Identification  ==  • QE  65  T h e estimated parameter vector containing the parameters (represented  by an overline) derived f r o m the apriori  estimated process model (i.e. i n the  predictor-based controllers, the coefficients of G(q elements of A  1  /fb ; 0  x  ) and FB(q  1  )/fb  0  i n the D a h l i n controllers, the coefficients of A(q  and the ) and  1  B(q )/b , and the elements of A b ), 1  • j3 =  f  0  0  A n adjustable parameter (0 < (3 < 1) which controls the radius of con-  straint. Notice that, if /3 is set to zero, 0 c is forced to equal 0 # i n E q . 9.85 a n d the resulting controller is non-adaptive. In the simulations of this thesis: —*  dim[X] II^H  =  £ 1 * 1 t=i  (9.86)  Chapter 10  Simulations  10.1  Process & Controller Parameters  T h e simulations were carried out on a fiYAX  computer at the P u l p & Paper Center  at U . B . C . T h e continuous three dye transport process as modelled i n Section 3 and the conversion to the three color output were simulated with an A d v a n c e d Continuous Simulation Language ( A C S L ) program. T h e R L S identification and control algorithm (both discrete) were performed by F O R T R A N subroutines called by the A C S L program after each sampling interval T . s  For each dye, the dye transport  model shown i n F i g . 10.4  is simulated pseudo-  continuously with an A C S L language program. T h e " D E R I V A T I V E " section of the A C S L program which simulates the transport of Dye 1, Dye 2, Dye 3 and a disturbing dye is given i n A p p e n d i x B . A l l the transport models share the following parameters:  •  T\  • r  2  =  1.25 m i n .  =  1.0 m i n .  a Td  =  1.333 m i n .  • T  =  1.6 or 2.0 or 2.4 m i n .  • K  =  0.65 or 0.80 or  2  d  0.95  66  Chapter 10.  Simulations  67  W E T E N D  Din  ZOH (r.)  DRYERS  PROCESS  Noise  FOUR.  l  V  1  l  Tl.+  D  71  e  -T s d  r = 0.33  FIBRE  R E C O V E R Y  PROCESS  1-11  T28+1  Figure 10.4:  Block D i a g r a m of A C S L - S i m u l a t e d Dye Transport M o d e l —*  —*  In simulation the color C is calculated from D C h a p t e r 4 at 31 points of frequency (i.e.  Aj with  using E q s . 4.20,  4.22,  4.24,  4.26 in  j — 1 , . . . , 3 1 ) . T h e actual absorp-  tion/scattering coefficients used i n the process simulation ( ^ j ( A ) , ^jf-(A), ^ ^ ( A ) for i  the controlled dyes,  (A) for the unknown disturbing dye, and -j^-(A) for the undyed  fibre) are given i n A p p e n d i x A . D y e i is a magenta (ie. purple) dye which absorbs i n the yellow/green wavelengths. D y e yellow/green wavelengths. D y e  3  2  is a yellow/green dye which absorbs at all but the  is a fluorescent red/yellow dye which emits (simulated  using negative absorption coefficient values) in the red/yellow wavelengths and absorbs elsewhere. These dyes can generate colors i n the red to yellow range. T h e fluorescence of the red/yellow dye allows for better contol of lightness L*, since non-fluorescent dyes can only decrease L*. Controllers: T h e controllers were designed using the  estimated discrete T F  corresponds to the continuous dye transport process with  9  •5i''^? n  This T F  estimated parameters:  Chapter 10.  • r  •  v  =  Simulations  0.94 m i n . 0.75 m i n .  —  f~2~  •  =  • Td  =2.0  • K  =  68  1.0 m i n . min.  0.8  T h e discrete T F used had coefficients which provided the least squares fit (for specified degrees rfe^r[A] and  deg[B]) to the continuous process at the specified sampling interval  T.. In the non-adaptive case (3 = 0.0; i n the adaptive case f3 = 0.2. for each of the controllers were: Dahlin:  • p ( C L discrete time constant) =  where T  c  Deadbeat:  • p ( C L discrete time constant) = G M V C :  • p (Weighting factor)  =3.0  • dh = d = 7 for T = 0.33 m i n . s  • dh = d — 4 for T — 0.66 m i n . s  0.0  = 0.66 m i n .  Other parameters  Chapter 10.  Simulations  69  EHC:  • 4  — 10, d — 7 for T, = 0.33 m i n .  • 4  = 6,  d = 4 for T = 0.66 m i n . t  T h e number of terms containing past prediction errors i n the R L S f o r m (i.e.  deg[E(q~ )] 1  defined i n Section 9.1) was set at zero. It was found that, if deg[E(q~ )] was greater 1  than zero, the improvement i n C L performance was small. A l l the controllers used G a i n Scheduling of A as described i n Subsection 6.1.1. A was calculated using the method described i n Subsection 4.3.1. and the estimated coefficients n ^ - ( A ) , ^ ^ ( A ) , ^ ^ " ( A ) for the dyes and ^ ( A ) for the undyed fibre (given i n A p p e n d i x A ) . These coefficients are 25% more than the corresponding  actual coeffi-  cients. A l l dye/color calculations are done at 31 points of frequency A.  10.2  Description of the Simulation Run  E a c h controller was tested using a standard 70 m i n . simulation r u n . D u r i n g four of the 10 m i n . intervals of the simulation r u n , the squared deviations of the three colors from their respective setpoints were measured and summed. A t the end of the interval, the total was averaged by dividing by the number of sampling intervals to produce, what will be called, the  combined color variance for that interval. These combined color  variances are recorded i n the tables of Chapter 11. Initially, when the controller is inserted, the process is i n steady state with the color output and setpoint unequal. In the 0 to 10 m i n . interval, the measured combined color variance indicates the controller's ability to eliminate this initial error. A t the 20 m i n . mark the addition of colored broke is simulated by the addition of a disturbing dye  Chapter 10.  (with ^ ^ ( A )  Simulations  coefficient).  70  In the 20 to 30 m i n . interval, the measured combined  color variance indicates the controller's ability to counteract a step disturbance. the 40 m i n . mark, a color setpoint change is made.  At  In the 40 to 50 m i n . interval,  the combined color variance (measuring the color deviations from the new setpoints) indicates the setpoint response speed of the controller. Finally, the measured combined color variance i n the 60 to 70 m i n . interval indicates the performance long after a setpoint change. T h e dye input and color plots for a typical simulation run are shown i n Figure 10.5 and Figure 10.6, respectively. and 1Z = 0.80.  In this r u n , the process parameters are T  d  = 2.0 m i n .  T h e controller is a non-adaptive D a h l i n controller with T„ = 0.66 m i n . ,  de^fvl] = 1, and a C L time constant T of 0.66 m i n . c  Chapter 10.  71  Simulations  I 3KQ  Figure 10.5:  Z 3KQ  6 3KQ  D y e Plots for a T y p i c a l Simulation R u n  T anoioo Figure 10.6:  z anoioo  e anoioo  C o l o r Plots for a T y p i c a l Simulation R u n  Chapter 11  Results  Twenty-four controllers are tested: 4 controller types (i.e. D a h l i n , Deadbeat, G M V C , E H C ) used non-adaptively (/? = 0.0) and adaptively (/? = 0.2) with three combinations of T„, degfA] and  deg[B]. A l l of the 24 controllers are tested i n two series of simulation  runs (one series uses a particular color setpoint sequence; the second series uses the reversed sequence). E a c h series contains 8 different subseries i n which certain parameters (i.e. Td, 7c, and noise) of the simulated process are varied. T h e tables i n this section report the measured color variances of the 24 controllers for each subseries. A l l of the 24 controllers are therefore tested i n 16 simulation runs. In Section 10.2 the color variances from the 16 runs are summed and averaged to produce the Average Performance Summary (Table 11.17). T h e process parameters used i n each series of simulation runs are:  • T= d  1.6 or 2.0 or 2.4 m i n . ,  • K = 0.65 or 0.80 or 0.95, • Measurement a n d / o r process noise added.  73  Chapter 11. Results  74  SIMULATION SERIES  1  Simulation runs with setpoint sequence C  = [74.2,3.57,9.88]  T  = [74.2,16.0,10.0]  C  T  Simulation Subseries l a • N o Noise  • T = 2.0 m i n . d  • K = 0.80 Dahlin  13 = 0.0 Min.  Deadbeat  G M V C  (3 = 0.2 (3 = 0.0 (3 = 0.2  E H C  (3 = 0.0 (3 = 0.2 (3 = 0.0  T. = 0.33 m i n . , deg[A] = 1, deg\B) = 0  With  0-10  1.380  1.380  1.240  1.250  1.420  1.420  1.660  1.660  20 - 30  .3470  .3910  .2730  .2920  .2590  .3780  .3550  .6040  40 - 50  44.90  45.70  41.50  41.10  47.70  62.60  54.60  57.80  60 - 70  .0030  < .001  < .001  < .001  .0473  < .001  .0043  < .001  0-10  1.400  1.390  1.290  1.290  1.690  1.690  1.880  1.880  20 - 30  .3860  .4070  .3320  .3340  .4340  .7160  .5400  .8270  40 - 50  45.50  45.90  42.90  42.70  54.90  57.50  61.00  58.70  60 - 70  .0042  < .001  .0020  < .001  .0081  .0026  .0138  .0049  Min.  With  Min. 0-10  With 1.390  1.640  T, = 0.66 m i n . , deg[A] = 1, deg[B] = 0  T = 0.66 m i n . , deg[A] = 3, deg[B] = 2 a  1.280  1.490  1.650  1.650  1.720  1.720  20 - 30  .4190  .4810  .3550  .4080  .3400  .5820  .3510  .5660  40 - 50  44.50  51.40  42.30  55.10  54.50  73.90  56.20  60.30  60 - 70  < .001  .3010  < .001  .6690  .0012  .0150  < .001  < .001  Table 11.1: Simulation Subseries l a  (3 = 0.2  Chapter 11. Results  75  Simulation Subseries l b • N o Noise  • T = 2.0 m i n . d  • 11 = 0.65 Dahlin  Deadbeat  G M V C  EHC  3 = 0.0  3 = 0.2  0 -- 10  1.520  1.500  1.360  1.350  1.560  1.550  1.820  1.820  20 - 30  .2920  .4010  .2280  .2830  .2500  .3200  .3140  .5150  40 - 50  49.90  49.70  46.20  46.20  53.30  60.20  59.50  63.80  60 - 70  .0706  .0178  .0270  .0103  .9160  .0218  .0998  .0235  Min.  With T  Min. 0 -- 10  3 = 0.0 s  3 = 0.2  = 0.33 m i n . ,  W i t h T, = 0.66 m i n . , 1.540  1.530  1.420  1.420  3 = 0.0  3 = 0.2  3 = 0.0  3 = 0.2  deg[A] = 1, deg[B] = 0  deg[A] = 1, deg[B] = 0 1.850  1.850  2.060  2.060  20 - 30  .3270  .3720  .2740  .2990  .4020  .6510  .5320  .7830  40 - 50  50.30  49.00  47.80  45.30  60.00  67.00  66.50  64.50  60 - 70  .0934  .0202  .0489  .0231  .1320  .0539  .1490  .1300  Min.  With T  a  = 0.66 m i n . , de<jr[A] =  3, deg[B] = 2  0 -- 10  1.530  1.790  1.400  1.600  1.810  1.810  1.880  1.880  20 - 30  .3860  .4290  .3200  .3650  .2810  .4920  .3120  .5280  40 - 50  49.40  54.30  46.70  54.70  59.60  69.90  61.50  63.80  60 - 70  .0141  .2000  .0053  .3520  .0436  .0306  .0284  .0044  Table 11.2: Simulation Subseries l b  Chapter 11. Results  76  Simulation Subseries l c • N o Noise  • Td = 2.0 m i n . • K = 0.95 Dahlin  0  = 0.0  Min.  0  Deadbeat  = 0.2  0 = 0.0  With  0 = 0.2  G M V C  0 = 0.0  0 = 0.2  E H C  0 = 0.0  0 = 0.2  T = 0.33 m i n . , deg[A] = 1, deg[B] = 0 a  0-10  1.300  1.300  1.190  1.190  1.330  1.330  1.550  1.550  20 - 30  .3980  .4160  .3140  .3170  .3010  .4350  .4040  .6430  40 - 50  41.60  44.30  39.30  42.60  44.60  115.0  50.50  64.00  60 - 70  < .001  < .001  < .001  < .001  .0017  < .001  < .001  < .001  Min.  With  T, = 0.66 m i n . , deg[A] = 1, deg[B] = 0  0-10  1.310  1.310  1.230  1.230  1.570  1.570  1.750  1.750  20 - 30  .4450  .4560  .3850  .3850  .4880  .7400  .5780  .8620  40 - 50  42.30  45.10  40.50  45.40  50.60  63.10  56.50  53.00  60 - 70  < .001  < .001  < .001  < .001  < .001  < .001  < .001  < .001  Min. 0-10  With 1.310  1.580  T = 0.66 m i n . , deg[A] = 3, deg[B] = 2 s  1.230  1.450  1.550  1.550  1.610  1.610  20 - 30  .4620  .5350  .3980  .4650  .4040  .5740  .4020  .6360  40 - 50  41.90  52.70  40.50  57.30  50.90  67.50  52.10  55.70  60 - 70  < .001  .4330  < .001  .7030  < .001  < .001  < .001  < .001  Table 11.3: Simulation Subseries l c  Chapter 11. Results  77  Simulation Subseries I d • N o Noise  • Td = 1.6 m i n . • 11 = 0.80 Dahlin  0 = 0.0 Min. 0-10  Deadbeat  0 = 0.2  0 = 0.0  With 1.170  1.180  0 = 0.2  G M V C  0 = 0.0  0 = 0.2  E H C  0 = 0.0  0 = 0.2  T = 0.33 m i n . , deg[A] = 1, deg[B) = 0 s  1.230  1.240  1.620  1.490  1.480  1.480  20 - 30  .2930  .3000  14.30  5.390  23.50  .3200  .3060  .5630  40-50  38.00  38.30  126.0  53.80  121.0  50.60  48.20  44.20  60 - 70  < .001  < .001  64.00  25.30  247.0  1.350  .0013  < .001  Min. 0-10  With 1.190  T = 0.66 m i n . , deg[A] = 1, deg[B] = 0 a  1.190  1.170  1.160  1.510  1.510  1.720  1.720  20 - 30  .3250  .3300  2.4.80  4.210  .3780  .6590  .4780  .7830  40 - 50  38.50  38.60  54.90  49.60  49.20  46.00  55.90  51.80  60 - 70  .0012  .0047  62.60  95.90  .0024  < .001  .0053  .0019  Min.  With  T, = 0.66 m i n . , deg[A] = 3, deg[B] = 2  0 - 10  1.220  1.510  1.170  1.410  1.470  1.470  1.570  1.570  20 - 30  .3620  .4430  1.480  1.250  .2740  .4910  .2940  .5090  40 - 50  39.50  43.50  73.00  72.70  47.90  45.30  51.00  45.40  60 - 70  < .001  .4770  78.80  176.0  < .001  < .001  .0041  < .001  Table 11.4: Simulation Subseries I d  Chapter 11. Results  78  Simulation Subseries l e • N o Noise  • T = 2.4 m i n . d  • K = 0.80 Dahlin  3 = 0.0 Min. 0-10  Deadbeat  j3 = 0.2  (3 = 0.0  With 1.650  (3 = 0.2  G M V C  3 = 0.0  E H C  (3 = 0.2  3 = 0.0  3 = 0.2  T, = 0.33 m i n . , deg[A] = 1, deg[B) = 0  1.630  1.700  1.750  1.800  1.770  1.880  1.880  20 - 30  .4140  .4420  10.50  4.090  22.80  .8170  .4080  .6180  40 - 50  54.20  50.80  110.0  98.50  189.0  79.80  63.60  96.90  60 - 70  .0267  .0486  98.20  518.0  670.0  15.60  .0270  .0493  Min.  With  T, = 0.66 m i n . , deg[A] = 1, deg[B] = 0  0-10  1.650  1.640  1.600  1.570  1.900  1.900  2.060  2.060  20 - 30  .4380  .4630  .4340  .4800  .4730  .7210  .5880  .8330  40 - 50  54.90  59.30  55.60  79.00  61.60  97.50  66.80  82.00  60 - 70  .0327  .0221  .4400  1.090  .0359  .2150  .0504  .0292  0-10  1.590  1.780  1.520  1.660  1.890  1.890  1.920  1.920  20 - 30  .5930  Min.  With T.  = 0.66 m i n . , deg[A] = 3, deg[B] = 2  .4700  .5020  .4370  .5430  .4060  .5880  .4040  40 - 50  50.80  54.30  48.90  54.00  63.70  101.0  63.90  99.20  60 - 70  < .001  .4840  .4280  1.160  .0232  .8060  .0049  < .001  Table 11.5: Simulation Subseries l e  Chapter 11. Results  79  Simulation Subseries If • Measurement Noise O n l y  • T = 2.0 m i n . d  • K = 0.80 Dahlin  0 = 0.0 Min. 0 - 10  Deadbeat  8 = 0.2 8 = 0.0 0 = 0.2 With  2.270  2.360  G M V C  E H C  0 = 0.0 8 = 0.2 0 = 0.0  3 = 0.2  T, = 0.33 m i n . , deg[A] = 1, deg[B] = 0 2.430  2.460  2.770  2.860  2.560  2.570  20 - 30  1.250  1.300  1.710  1.760  2.110  1.700  1.270  1.450  40 - 50  45.90  57.10  43.10  51.10  49.00  71.40  55.90  60.20  60 - 70  .9200  1.290  1.350  1.490  2.090  1.560  .9250  1.010  Min. 0-10  With 2.300  2.320  T = 0.66 m i n . , deg[A] = 1, deg[B] = 0 s  2.320  2.350  2.590  2.590  2.770  2.770  20 - 30  1.270  1.330  1.390  1.450  1.290  1.430  1.350  1.480  40 - 50  46.90  57.30  44.50  53.60  56.20  73.40  62.10  63.70  60 - 70  1.020  1.110  1.230  1.290  .9670  1.190  .9080  .9660  Min.  With  T = 0.66 m i n . , deg[A] = 3, deg[B] = 2 a  0 - 10  2.290  2.540  2.280  2.520  2.660  2.660  2.740  2.740  20 - 30  1.290  1.340  1.390  1.490  1.290  1.350  1.320  1.400  40 - 50  45.80  48.50  44.20  49.90  56.10  68.30  57.60  58.00  60 - 70  .9740  .9890  1.170  1.180  1.130  1.140  1.120  1.460  Table 11.6: Simulation Subseries If  Chapter 11. Results  80  Simulation Subseries l g • Process Noise O n l y  • T = 2.0 m i n . d  • 11 = 0.80 Dahlin  a =  o.o  Min. 0-10  Deadbeat  a = 0.2  a=  With 2.190  2.410  o.o  3 = 0.2  G M V C  3 = 0.0  3 = 0.2  E H C  3 = 0.0  3 = 0.2  T = 0.33 m i n . , deg[A] = 1, deg[B] = 0 B  2.040  2.200  2.380  2.460  2.520  2.510  20 - 30  1.110  1.330  1.130  1.250  1.400  1.250  1.110  1.270  40 - 50  47.00  66.30  43.20  63.30  51.00  87.30  57.20  68.80  60 - 70  1.640  2.310  1.720  2.320  2.100  2.280  1.590  1.580  Min.  With  T. = 0.66 m i n . , deg[A] = 1, deg[B] = 0  0 - 10  2.200  2.200  2.090  2.090  2.530  2.530  2.710  2.710  20 - 30  1.130  1.220  1.120  1.190  1.160  1.490  1.280  1.620  40 - 50  47.50  56.70  44.60  50.20  57.00  75.20  63.20  70.30  60-70  1.650  1.850  1.680  1.900  1.610  1.750  1.520  1.700  0-10  2.140  2.290  2.020  2.150  2.540  2.540  2.560  2.560  20 - 30  1.150  1.180  1.120  1.170  1.110  1.300  1.110  1.370  40 - 50  45.80  48.60  43.30  48.90  56.80  70.60  58.40  62.30  60 - 70  1.440  1.360 -  1.480  1.470  1.700  1.730  1.560  1.360  Min.  With  T, = 0.66 m i n . , deg[A] = 3, deg[B] = 2  Table 11.7: Simulation Subseries l g  81  Chapter 11. Results  Simulation Subseries l h • Measurement Noise & Process Noise  • T = 2.0 m i n . d  • K = 0.80 Dahlin  Deadbeat  0 = 0.2  0 - 10  3.080  3.340  3.190  3.420  3.670  3.690  3.400  3.410  20-30  2.050  2.430  2.610  2.850  2.890  2.690  2.050  2.110  40 - 50  47.90  61.60  44.50  50.20  51.10  79.60  58.60  68.50  60 - 70  2.470  4.020  2.960  3.740  3.850  3.560  2.420  2.330  0-10  3.100  3.100  With  Min.  With  0 = 0.2  0 = 0.0  0 = 0.2  E H C  0 = 0.0 Min.  0 = 0.0  G M V C  0 = 0.0  0 = 0.2  T = 0.33 m i n . , deg[A] = 1, deg[B] = 0 g  T = 0.66 m i n . , deg[A] = 1, deg[B] = 0 s  3.090  3.110  3.430  3.430  3.590  3.590  20 - 30  2.020  2.050  2.190  2.260  2.020  2.180  2.090  2.200  40 - 50  49.00  65.00  46.10  60.20  58.40  80.90  64.40  69.70  60 - 70  2.480  3.020  2.670  3.150  2.400  2.750  2.260  2.290  Min.  With  T = 0.66 m i n . , deg[A] = 3, deg[B] = 2 8  0-10  3.040  3.200  3.000  3.170  3.520  3.520  3.570  3.570  20 - 30  2.020  2.060  2.160  2.220  2.080  2.160  2.090  2.160  40-50  47.00  48.40  44.90  49.00  58.60  68.00  59.80  61.10  60 - 70  2.240  2.060  2.430  2.240  2.600  2.880  2.460  2.320  Table 11.8: Simulation Subseries l h  82  Chapter 11. Results  SIMULATION SERIES 2 Simulation runs with setpoint sequence C^  = [74.2,16.0,10.0] =4> C"^ = [74.2,3.57,9.88] r  Simulation Subseries 2a • N o Noise  • T = 2.0 m i n . d  • U = 0.80 Dahlin  0  = 0.0  Min.  0  Deadbeat  = 0.2  0=  With  0.0  0  = 0.2  G M V C  0=  0.0  0=  E H C 0.2  0  = 0.0  0=  0.2  T = 0.33 m i n . , deg[A] = 1, deg[B] = 0 e  0-10  2.320  2.300  2.070  2.070  2.390  2.390  2.810  2.810  20 - 30  .9440  1.020  .7400  .7720  .6980  1.290  .9730  1.910  40-50  43.00  45.40  40.30  41.60  43.90  42.80  50.90  49.60  60 - 70  .0034  .0013  < .001  .0017  .0789  < .001  .0033  < .001  2.340  2.340  Min. 0-10  With  T„ = 0.66 m i n . , deg[A\ = 1, deg[B] = 0 2.160  2.160  2.850  2.850  3.170  3.170  20 - 30  1.050  1.090  .9000  .9250  1.190  2.280  1.470  2.510  40 - 50  43.30  45.30  40.50  41.30  51.90  52.00  57.70  56.60  60 - 70  .0044  < .001  .0021  < .001  .0047  .0080  .0103  .0055  Min. 0 - 10  With 2.320  2.770  T„ == 0.66 m i n . , deg[A] = 3, deg[B] = 2 2.120  2.490  2.790  2.790  2.900  2.900  20 - 30  1.140  1.500  .9630  1.260  .9220  1.950  .9550  2.170  40 - 50  43.60  49.20  40.50  45.00  50.70  51.00  52.90  49.30  60 - 70  < .001  .2750  < .001  .1720  < .001  < .001  < .001  < .001  Table 11.9: Simulation Subseries 2a  T  Chapter 11. Results  83  Simulation Subseries 2b • N o Noise  • T = 2.0 m i n . d  • 11 = 0.65 Dahlin  0  = 0.0  Min. 0-10  0=  Deadbeat 0.2  0  With 2.580  2.520  = 0.0  0=  G M V C 0.2  0=  0.0  0  E H C = 0.2  0  = 0.0  0=  0.2  T = 0.33 m i n . , deg[A) = 1, deg[B] = 0 a  2.300  2.280  2.650  2.640  3.080  3.080  20 - 30  .8420  .9960  .6340  .7080  .6090  1.050  .9120  1.620  40 - 50  46.60  49.40  43.70  46.90  47.30  46.90  55.30  55.90  60 - 70  .0558  .0086  .0285  .0136  .7340  .0168  .0541  .0387  Min.  With  T = 0.66 m i n . , deg[A] = 1, deg[B] = 0 s  0 - 10  2.600  2.590  2.410  2.410  3.130  3.130  3.480  3.480  20 - 30  .9510  1.010  .7780  .8030  1.160  1.930  1.440  2.190  40 - 50  47.00  48.10  43.90  44.00  56.80  57.60  63.60  63.10  60 - 70  .0675  .0178  .0454  .0101  .0861  .1800  .1310  .0975  Min.  With  T. = 0.66 m i n . , deg[A] = 3, deg[B] = 2  0-10  2.570  3.030  2.350  2.710  3.060  3.060  3.180  3.180  20 - 30  1.070  1.330  .8890  1.080  .8000  1.620  .8760  1.730  40-50  47.70  53.00  44.10  47.90  54.80  54.40  57.40  53.00  60 - 70  .0142  .1700  .0051  .2450  .0291  .0040  .0116  < .001  Table 11.10: Simulation Subseries 2b  Chapter 11. Results  84  Simulation Subseries 2c • N o Noise  • T = 2.0 m i n . d  • 1Z = 0.95  Dahlin 0 = 0.0 0 = 0.2 Min. 0-10  Deadbeat 0 = 0.0 0 = 0.2  With 2.160  2.150  GMVC 0 = 0.0 0 = 0.2  EHC 0 = 0.0 0 = 0.2  T, = 0.33 m i n . , deg[A] = 1, deg[B] = 0 1.960  1.960  2.220  2.220  2.600  2.600  20-30  1.070  1.100  .8480  .8560  .8130  1.640  1.090  2.260  40 - 50  40.70  42.50  38.20  39.90  41.60  41.40  47.80  46.00  60 - 70  < .001  < .001  < .001  < .001  .0033  < .001  < .001  < .001  Min.  With  T, = 0.66 m i n . , deg[A] = 1, deg[B] = 0  0-10  2.170  2.170  2.030  2.030  2.630  2.630  2.940  2.940  20 - 30  1.200  1.210  1.040  1.040  1.310  2.730  1.550  2.910  40 - 50  41.00  42.70  38.60  39.60  48.50  51.40  53.70  54.60  60 - 70  < .001  < .001  < .001  < .001  < .001  < .001  < .001  < .001  0 - 10  2.170  2.640  2.010  2.420  2.600  2.600  2.700  2.700  20 - 30  1.240  1.660  1.070  1.420  1.080  1.910  1.080  2.390  40 - 50  41.30  47.00  38.60  43.80  47.80  55.60  49.70  52.30  60 - 70  < .001  .4690  < .001  .2900  < .001  < .001  < .001  < .001  Min.  With  T = 0.66 m i n . , deg[A] = 3, deg[B] = 2 B  Table 11.11: Simulation Subseries 2c  Chapter 11. Results  85  Simulation Subseries 2d • N o Noise  • Td = 1.6 m i n .  • K = 0.80 Dahlin 3 = 0.0 3 = 0.2  Deadbeat 3 = 0.0 3 = 0.2  GMVC 3 = 0.0 3 = 0.2  W i t h T = 0.33 m i n . , deg[A]  Min.  s  EHC 3 = 0.0 8 = 0.2  = 1, deg[B] = 0  0-10  1.970  2.000  2.070  2.070  2.770  2.550  2.500  2.500  20 - 30  .8000  .7080  44.50  16.50  122.0  .9340  .8370  1.810  40 - 50  36.30  33.60  227.0  60.90  134.0  40.10  45.40  45.00  60 - 70  < .001  < .001  106.0  33.50  53.80  .0209  < .001  < .001  1.990  2.010  1.970  1.940  W i t h T, = 0.66 m i n . , deg[A]  Min. 0-10  = 1, deg[B] = 0  2.550  2.550  2.910  2.910  20 - 30  .8870  .7510  4.200  7.670  1.030  2.200  1.290  2.470  40 - 50  36.80  33.70  116.0  116.0  46.40  47.70  52.60  54.20  60 - 70  < .001  < .001  8.860  8.010  .0015  < .001  .0058  .0015  W i t h T = 0.66 m i n . , deg[A] = 3,  Min.  t  deg[B] = 2  0-10  2.050  2.560  1.960  2.370  2.490  2.490  2.650  2.650  20 - 30  .9800  1.320  2.720  2.720  .7450  1.870  .7970  1.920  40 - 50  38.00  41.90  199.0  166.0  45.20  45.80  48.20  47.20  60 - 70  < .001  .4400  174.0  217.0  < .001  < .001  .0037  < .001  Table 11.12: Simulation Subseries 2d  Chapter 11. Results  86  Simulation Subseries 2e • N o Noise  • T=  2.4 m i n .  • n=  0.80  d  Dahlin  0 = 0.0 Min.  Deadbeat  0 = 0.2 3 = 0.0 With  0 = 0.2  G M V C  0 = 0.0 0 = 0.2  E H C  0 = 0.0  0 = 0.2  T, = 0.33 m i n . , deg[A] = 1, deg[B] = 0  0-10  2.790  2.760  3.090  3.280  3.120  3.040  3.190  3.190  95.10  2.950  1.110  1.960  20-30  1.100  .9530  25.50  31.50  40 - 50  50.40  47.20  235.0  418.0  339.0  61.30  57.00  59.40  60 - 70  .0238  .0034  136.0  251.0  425.0  .9960  .0200  .0183  Min. 0-10  With 2.800  2.770  T, = 0.66 m i n . , deg[A] = 1, deg[B] = 0 2.710  2.090  3.210  3.210  1.130  1.310  2.140  1.620  2.400  53.00  58.10  58.90  63.30  62.20  .0254  .0101  .0280  .0219  20 - 30  1.180  1.130  1.120  40 - 50  50.70  47.60  47.40  60 - 70  .0311  .0075  .8790  1.630  Min.  With  3.470  3.470  T = 0.66 m i n . , deg[A] = 3, deg[B] = 2 a  0 - 10  2.670  2.980  2.540  2.780  3.210  3.210  3.230  3.230  20 - 30  1.270  1.550  1.150  1.650  1.080  1.910  1.090  2.190  40-50  50.00  51.70  46.90  58.50  57.60  58.50  58.50  55.90  60 - 70  < .001  .5030  1.030  6.140  .0115  .0021  .0025  < .001  Table 11.13: Simulation Subseries 2e  Chapter 11. Results  87  Simulation Subseries 2f • Measurement Noise O n l y  • T = 2.0 m i n . d  • K = 0.80 Dahlin  3 = 0.0 Min. 0-10  Deadbeat  3 = 0.2 8 = 0.0 With  3.220  3.350  8 = 0.2  G M V C  E H C  3 = 0.0 3 = 0.2 8 = 0.0  8 = 0.2  T = 0.33 m i n . , deg[A] = 1, deg[B] = 0 B  3.240  3.330  3.720  3.830  3.700  3.710  20 - 30  1.790  1.800  2.140  2.180  2.850  2.290  1.840  2.490  40 - 50  44.20  46.30  42.50  43.20  45.20  48.70  52.30  54.40  60-70  .8900  1.070  1.220  1.320  1.790  1.340  .8960  .9450  Min. 0 - 10  With 3.230  3.250  T = 0.66 m i n . , deg[A] = 1, deg[B] = 0 s  3.180  3.220  3.740  3.740  4.060  4.060  20 - 30  1.810  1.800  1.860  1.890  1.940  2.560  2.190  2.720  40 - 50  44.20  45.20  41.90  42.70  52.70  56.70  58.60  59.40  60 - 70  .9700  1.050  1.150  1.250  .9230  .9310  .8840  .8780  Min.  With  T„ = 0.66 m i n . , deg[A] = 3, deg[B) = 2  0-10  3.220  3.700  3.140  3.560  3.780  3.780  3.920  3.920  20 - 30  1.910  2.040  1.930  2.130  1.760  1.990  1.830  2.040  40 - 50  44.60  47.00  41.90  44.10  51.60  55.20  53.90  51.30  60 - 70  .9340  .9530  1.100  1.140  1.060  1.030  1.050  1.300  Table 11.14: Simulation Subseries 2f  Chapter 11. Results  88  Simulation Subseries 2g • Process Noise O n l y  • T = 2.0 m i n . d  • % = 0.80 Dahlin  0 = 0.0 Min. 0 - 10  Deadbeat  0 = 0.2 0 = 0.0 With  3.460  3.730  0 = 0.2  G M V C  0 = 0.0 0 = 0.2  E H C  0 = 0.0  0 = 0.2  T, = 0.33 m i n . , deg[A] = 1, deg[B] = 0 3.260  3.480  3.730  3.830  3.870  3.850  20 - 30  2.060  2.480  2.050  2.230  2.410  2.500  2.050  2.680  40 - 50  43.60  47.00  40.90  42.80  44.40  51.40  51.20  55.70  60 - 70  1.090  1.410  1.130  1.440  1.370  1.480  1.040  1.040  Min. 0-10  With 3.450  3.470  T = 0.66 m i n . , deg[A] = 1, deg[B] = 0 s  3.300  3.320  3.870  3.870  4.060  4.060  20 - 30  2.130  2.350  2.080  2.300  2.200  3.060  2.390  3.220  40-50  44.30  45.00  41.60  42.00  52.50  60.20  58.10  63.70  60 - 70  1.110  1.260  1.120  1.200  1.030  1.090  .9070  .8950  Min.  With  T, = 0.66 m i n . , deg[A] = 3, deg[B] = 2  0 - 10  3.280  3.400  3.120  3.220  3.900  3.900  3.890  3.890  20 - 30  2.130  2.250  2.050  2.190  2.090  2.670  2.050  2.850  40 - 50  44.50  48.00  41.50  44.40  51.60  58.60  53.70  56.40  60 - 70  .9290  .8170  .9620  .8810  1.120  1.230  1.010  .9980  Table 11.15: Simulation Subseries 2g  89  Chapter 11. Results  Simulation Subseries 2h • Measurement Noise & Process Noise  • T = 2.0 m i n . d  • U = 0.80 Dahlin  0  = 0.0  Min.  0  Deadbeat = 0.2  0=  0.0  0=  G M V C 0.2  W i t h T = 0.33 m m . , s  0=  0.0  E H C  0=  0.2  0=  0.0  0=  0.2  deg[A] = 1, deg[B] = 0  0-10  4.380  4.780  4.400  4.700  5.010  4.910  4.800  4.810  20 - 30  2.910  3.570  3.390  3.600  3.870  3.830  2.920  3.150  40 - 50  44.70  46.60  43.10  43.30  46.20  52.40  52.60  53.90  60 - 70  1.950  2.400  2.220  2.480  3.030  3.060  1.910  1.860  W i t h T, = 0.66 m i n . , deg[A] = 1,  Min. 0-10  4.370  4.390  4.340  4.330  deg[B] = 0  4.790  4.790  4.970  4.970  20 - 30  2.820  2.830  2.950  3.000  2.890  3.470  3.070  3.390  40-50  45.20  46.30  42.70  43.40  53.30  57.30  58.90  62.20  60 - 70  1.990  2.280  2.110  2.390  1.890  2.060  1.730  1.640  W i t h T = 0.66 m i n . , deg[A] = 3,  Min.  g  deg[B] = 2  0 - 10  4.210  4.390  4.160  4.320  4.910  4.910  4.940  4.940  20 - 30  2.850  2.900  2.940  3.070  2.840  3.170  2.850  3.010  40-50  45.40  48.60  42.70  45.80  52.50  56.60  54.60  54.70  60 - 70  1.770  1.650  1.920  1.840  2.040  2.090  1.940  1.970  Table 11.16: Simulation Subseries 2h  90  Chapter 11. Results  11.1  Performance Summary  Dahlin  0  = 0.0  Deadbeat 0.2  0  With  Min. 0-10  0=  2.340  = 0.0  0  = 0.2  G M V C  0=  0.0  0=  E H C 0.2  0=  0.0  0=  0.2  T = 0.33 m i n . , deg[A] = 1, deg[B] = 0 3  2.418  2.298  2.377  2.635  2.624  2.714  2.714  20 - 30  1.104  1.227  6.929  4.661  17.62  1.525  1.122  1.603  40 - 50  44.93  48.24  75.28  73.96  84.27  61.97  53.79  59.01  60 - 70  .5715  .7863  25.93  52.54  88.24  1.955  .5620  .5561  With  Min. 0-10  2.352  2.354  T„ = 0.66 m i n . , deg[A] = 1, deg[B] = 0 2.269  2.270  2.740  2.740  2.975  2.975  20 - 30  1.148  1.175  1.471  1.835  1.230  1.810  1.403  1.950  40 - 50  45.46  48.17  49.34  53.00  54.26  62.65  60.18  61.86  60 - 70  .5910  .6653  5.177  7.365  .5698  .6401  .5377  .5413  With  Min.  T, = 0.66 m i n . , deg[A] = 3, deg[B) = 2  0-10  2.313  2.612  2.206  2.457  2.739  2.739  2.811  2.811  20 - 30  1.197  1.345  1.336  1.464  1.094  1.539  1.113  1.629  40 - 50  44.99  49.26  54.94  58.57  53.74  62.51  55.59  57.87  60 - 70  .5198  .7238  16.46  25.72  .6100  .6849  .5747  .5884  Table 11.17: Performance S u m m a r y  C h a p t e r 12  Discussion  A s expected for all the controllers, the largest color variances occur i n the 40-50 m i n . interval after the color setpoint change and the smallest occur i n the 60-70 m i n . steadystate interval . T h e S u m m a r y Table (Table 11.17) shows that adaptive controllers perform worse on average than their non-adaptive counterparts. that the adaptive Deadbeat controller with T its non-adaptive counterpart.  s  T h e exception is the improvement  = 0.33 m i n . , ete(7[A] =  1 shows over  T h e reason can be seen i n Tables 11.4 and 11.12 where  both controllers were destabilized by a low ^ = 1 . 6 , but the adaptive controllers h a d lower color variances. In general, the lack of improvement shown by the non-adaptive controller i n the 40-50 m i n .  interval is not surprising since we cannot expect R L S  to identify the rapidly changing A . Before and after the setpoint change, we might expect R L S to give an improved estimate of A but only if the process receives adequate perturbation/excitation.  In the adaptive controllers, the degree of "adaptiveness" 3  was set at 0.2. It was found that C L performance degraded at higher values. T h e S u m m a r y Table shows that, i n general, the predictor-based controllers do worse than the D a h l i n / D e a d b e a t controllers. In the 20-30 m i n . and 60-70 m i n . intervals , the predictor-based controllers sometimes show the same variances as the best D a h l i n / D e a d b e a t controllers i n the same T„ and ete(jr[yl] category (eg. the non-adaptive E H C controllers i n the 60-70 m i n . interval with T  a  = 0.66 m i n . and cZe(?[/4] = 1).  Inspecting the S u m m a r y Table, we see that the performance of the non-adaptive  91  92  Chapter 12. Discussion  G M V C a n d E H C for T, = 0.66 m i n . are similar. T h i s makes sense if we note the similarities i n the structures of the two controllers (i.e. E q s . 7.59 a n d 7.63) where the denominator  {1+p) of the G M V C corresponds to the denominator (1 +5Zf=o  d _ 1  foil f°d -d) h  of the E H C . There is a trade-off between designing a controller which is an integrating controllers (i.e. counteracts step disturbances) and one which minimizes deviation f r o m a setpoint (i.e.  makes quick setpoint changes).  T h i s may be the reason the predictor-based  controllers do not compare well with the D a h l i n / D e a d b e a t controllers. T h e predictorbased were made integrating controllers by calculating  F(q~ ) a n d G(q~ ) using the 1  1  <itjfferera<ia/Diopha.ntine E q . 5.45. T h e resulting controllers are sensitive to perturbation unless their denominators are quite large (in fact, p > 0 is necessary for stability for the  GMVC). Therefore, the added complexity of the predictor-based controllers (i.e. solving the  Diophantine E q . ) is not warranted for a process i n which the dynamics are simple to control. Except when T  = 1.6 or 2.4 m i n . , the Deadbeat controllers give lower variances  d  than the D a h l i n controllers , as expected. B u t the Deadbeat controllers are less robust in the face of deviations i n T d  Noting that the table entries with large variances  indicate an unstable controller, we see that a low T  d  (i.e. Tables 11.4 and 11.12) causes  instability i n all the Deadbeat controllers, while all the D a h l i n controllers continue to do well. For a high T  d  (i.e. Tables 11.5 a n d 11.13) it's interesting to note that only the  Deadbeat controller with T„ = 0.33 m i n . is destabilized. Not only do the Deadbeat controllers with T„ = 0.66 m i n . remain stable but the Deadbeat controller with T  B  =  0.66 m i n . a n d rfe^[yl] = 3 does better than its D a h l i n counterpart. B o t h low and high T  d  have little effect on the stability of the D a h l i n controllers which do better than their  Deadbeat counterparts except for the case just mentioned. A t steady state (60-70 m i n .  93  Chapter 12. Discussion  variance) the Deadbeat controllers also show their sensitivity by giving higher variances than the D a h l i n controllers when noise is added (Tables 11.6-8 and 11.14-16). In general, the Summary Table shows that, for T„ = 0.66 m i n . , the model degree deg[A] has little effect on performance, especially with the D a h l i n controllers. In other words, the added complexity of the desr[.A] = 3 transport model brings little improvement over the <ie(/[.4] = 1 model. T h e reason may be that, even if the dec/fA] is increased, the model  A(q~ ) and B(q~ ) is still generated from estimated (i.e. l  l  T[, etc.), not exact, parameters . T h a t model accuracy is not critical for good control is illustrated i n Tables 11.2 and 11.10 (where 1Z = 0.65) and i n Tables 11.3 and 11.11 (where K = 0.95). E v e n with this large change i n 1Z b o t h the D a h l i n and Deadbeat controllers provide good control. Because of the easy controllability of the stable, minimum-phase dye transport process, it seems that any reasonable non-adaptive D a h l i n controller with G a i n Scheduling —*8et  (i.e. A calculated using the optical properties of the dyes and C formance.  ) gives good C L per-  Chapter  13  Conclusions  T h e adaptive controllers perform worse on average than their non-adaptive counterparts. Because of the easy controllability of the stable, minimum-phase dye transport process, it is not clear that, even if an improved estimate of the process were obtained, the benefit (in terms of improved C L performance) would warrant the added complexity of the R L S identification. A s well, the S u m m a r y Table shows that i n general the predictor-based controllers do worse than the D a h l i n / D e a d b e a t controllers.  Therefore, the added complexity of  the predictor-based controllers (i.e. solving the Diophantine E q . ) is not warranted. Increasing the order of the dye transport model f r o m rfeg[A] = 1 to 3 brings little improvement i n C L performance since b o t h models are generated f r o m  estimated , not  exact, process parameters . Variations in the actual dye retention 1Z have little effect on good C L performance if the  estimate of 7Z used to generate the transport model is the mean value i n its possible  range. Because of the easy controllability of the stable, minimum-phase dye transport process, it seems that anon-adaptive D a h l i n / D e a d b e a t controller with G a i n Scheduling gives good performance. T h e non-adaptive Deadbeat performs best if Td is known accurately . Otherwise, the non-adaptive D a h l i n controllers with C L time constant T robust control and that with T  s  = 0.66 m i n . and tZe^[A] =1  94  c  = 0.66 m i n . give good gives good control with  Chapter 13. Conclusions  95  simple implementation. A good estimate of the process gain A (from which the controller gain A lated) is important for good control. If the gain A is color C  1  is calcu-  calculated at each desired paper  using a theoretical dye/color model and certain measured optical properties  of the dyes (i.e.  G a i n Scheduling), this compensates for the nonlinear nature of the  dye/color relationship. E v e n for paper colors that have not been produced previously the gain A can be calculated, as long as the optical properties of the particular dyes are known. If from past production runs of a certain color of dyed paper, the steady state input concentrations (i.e. Di  n  ) of the corresponding dyes are well known, then open-loop  control, as described i n Section 6.2, setpoint changes.  should be considered for making efficient color  Bibliography  [1] Alderson, J . V . , E . Atherton, A . N . Derbyshire. M o d e r n Physical Techniques i n Colour Formulation. Journal  of the Society of Dyers & Colourists,  77:p.657 (1961)  [2] Belanger, P . R . A Paper Machine Color Control System Design U s i n g M o d e r n Techniques. IEEE  Transactions  on Automatic  [3] Billmeyer, F . W . , M . Saltzman. Principles New  Control, AC-14:p.610 (1969)  of Color Technology, John W i l e y & Sons,  York (1981)  [4] B o n h a m , J . S . Fluorescence & K u b e l k a - M u n k Theory. COLOR, research & application, 11 (1986) [5] C h a o , H . , W . W i c k s t r o m . T h e Development of D y n a m i c Color Control o n a Paper Machine. Automatica,  6:p.5 (1970)  [6] C l a p p e r t o n , R . H . Modern  Paper-Making.  [7] Deshpende P . B . , R . H . A s h , Elements  Basil Blackwell L t d . , O x f o r d (1952)  of Computer Process Control. I S A , Research  Triangle Park, N C (1981) [8] E l l i o t , H . , W . A . Wolovich. Parameterization Issues i n Multivariable Adaptive C o n trol. Automatica,  20:p.533 (1984)  [9] G o o d w i n , G . C . , K . S . Sin. Adaptive  Filtering,  Prediction,  & Control. Prentice-Hall,  Englewood Cliffs (1984) [10] Jordan,  B . D . , M . O ' N e i l l . T h e Case for Switching to the C I E L* a* b* Colour  Description for Paper. Miscellaneous  Report of PPRIC, 96  M R 106 (1987)  [11] K o i v o , H . N . A Multivariable Self-Tuning Controller.  Automation, 16:p.351 (1980)  [12] Lebeau, B . , J . P . Vincent, A . R a m a z . On-line Color Control System: A Case Study on an Experimental  Paper Machine. T h e 3rd I F A C / P R P Conference  (PRP3),  Brussels, B e l g i u m (1976) [13] Lehtoviita, M . New On-line Colour and Brightness Measurement and Control System. T h e 5th I F A C / P R P Conference ( P R P 5 ) , Antwerp, B e l g i u m (1983) [14] M a c A d a r n , D . L .  Color Measurement, V o l . 27, Springer-Verlag Series i n Optics,  Springer-Verlag, (1985) [15] M a r c h i , R . Latest Developments Allow On-line Color and Shade Measurement a n d Control.  Pulp & Paper, 60:p.l24 (1986)  [16] M c G i l l , R . J . Chapter  18 i n  Measurement and Control in Papermaking. A d a m  Hilger L t d . , Bristol (1980) [17] P I T A Engineering Technology Working G r o u p , On-line Colour Control.  Paper  Technology and Industry, 24:p.l7 (1983) [18] Praly, L . Robustness of Indirect Adaptive Control based o n Pole-Placement D e sign. T h e 1st I F A C Workshop o n Adaptive Systems i n Control a n d Signal Processing, San Francisco (1983) [19] Q u i n n , M . A u t o m a t i c Colour Control on the Paper Machine.  Paper Technology,  9:p.317 (1968) [20] Sandraz, J . P . Identification et C o m m a n d e Multidimensionnelle d'une Unite P i lote de Fabrication de Papier. P h . D . Thesis, L'Universite Scientifique de Grenoble (1973)  [21] Shead, R . P . Colour Measurement  a n d Control:  T h e Present State of the A r t .  Conference of the Swedish Association of P u l p & Paper Engineers,  Stockholm  (1984) [22] T o d d , K . L . , V . Holding, C . Freel, T h e Control of Colour on a Paper Machine. Conference of the Swedish Association of P u l p & Paper Engineers,  Stockholm  (1984) [23] Ydstie, B . E . T h e o r y a n d A p p l i c a t i o n of an Extended Horizon Self-Tuning C o n troller.  AIChE Journal, 31:p.l771 (1985)  98  Appendices  99  Appendix A  -g* and  of Simulated Dyes and Undyed Paper  100  ndix A. -j* and K of Simulated Dyes and Undyed Paper  A [nm 1 [iiiii.j 400 410 420 430 440 450 460 470 480 490 500 510 520 530 540 550 560 570 580 590 600 610 620 630 640 650 660 670 680 690 700  St  0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  St  0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  K, s  r 2  0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 f  •Kt,2  St  0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25  St  0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20 -.20  St  0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15 -.15  St  0.19 0.19 0.19 0.19 0.19  0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19  Kj_ St  0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24  s,  0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20  Appendix B  A C S L Simulation of Dye Transport Process  "Wet E n d Process with Gaussian Noise A d d e d " D01=  INTEG((DI(1) + D W 1 - D 0 1 +NO(7)*GAUSS(0.0,1.0))  /T1,D01IC)  D 0 2 = INTEG((DI(2) + D W 2 - D 0 2 +NO(8)*GAUSS(0.0,1.0))  /T1,D02IC)  D 0 3 = INTEG((DI(3) + D W 3 - D 0 3 +NO(9)*GAUSS(0.0,1.0))  /T1,D03IC)  D 0 4 = INTEG((D1ST + D W 4 -D04)  /T1,D04IC)  " T i m e Delay i n D r y e r s " DD01=  DELAY(RET(1)*D01,DD01IC,TD11,300)  DD02=  DELAY(RET(2)*DO2,DDO2IC,TD12,300)  DD03=  DELAY(RET(3)*DO3,DDO3IC,TD13,300)  DD04=  DELAY(RET(3)*DO4,DDO4IC,TD14,300)  " T i m e Delay i n F i b r e Recovery Process" DDW1= DELAY((l.-RET(l))*DOl,DDWlIC,TD21,500) DDW2=  DELAY((l.-RET(2))*DO2,DDW2IC,TD22,500)  DDW3=  DELAY((l.-RET(3))*DO3,DDW3IC,TD23,500)  DDW4=  DELAY((l.-RET(3))*DO4,DDW4IC,TD24,500)  " F i b r e Recovery Process" D W 1 = I N T E G ( ( D D W 1 -DW1)/T2, DW1IC) D W 2 = I N T E G ( ( D D W 2 -DW2)/T2, DW2IC) D W 3 = I N T E G ( ( D D W 3 -DW3)/T2, DW3IC) DW4= I N T E G ( ( D D W 4 -DW4)/T2, DW4IC)  102  

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