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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Control of color in dyed paper Bond, Tracy 1988
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Title | Control of color in dyed paper |
Creator |
Bond, Tracy |
Publisher | University of British Columbia |
Date Issued | 1988 |
Description | This 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. The 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 Dahlin controllers with gain scheduling are designed, tested in simulation, and compared. Adaptive versions using parameters identified with Recursive Least Squares (RLS) are also tested. For practical applications, the non-adaptive Dahlin algorithm with gain scheduling is shown to offer the best performance, together with relative ease of use. |
Subject |
Paper Dyes and dyeing Color |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-09-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0302146 |
URI | http://hdl.handle.net/2429/28366 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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