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Control of color in dyed paper Bond, Tracy 1988

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C O N T R O L OF 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 FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF APPLIED SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF E L E C T R I C A L ENGINEERING We accept this thesis as conforming to the required standard T H E UNIVERSITY OF 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 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 con-trol. 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 perfor-mance, 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 M c G i l l 8 2.1.6 Alderson, Atherton & Derbyshire 9 2.1.7 Conclusions 10 2.2 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 M c G i l l 13 2.2.5 Alderson, Atherton & Derbyshire 13 iii 2.2.6 Conclusions 14 2.3 Industrial Control System Vendors 15 2.3.1 AccuRay C o . L t d 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 Co . L t d . 17 2.3.5 Conclusions 18 3 M o d e l o f D y e T r a n s p o r t t h r o u g h a P a p e r M a c h i n e 19 3.1 Continuous Model 19 3.2 Discrete Model 21 4 T h e D y e C o n c e n t r a t i o n / C o l o r R e l a t i o n s h i p 25 4.1 Color Measurement 25 4.2 Color Sensors 27 4.3 T h e D y e / C o l o r Matrix A 29 4.3.1 Calculating A at a Color Setpoint 31 5 T h r e e D y e I n p u t / T h r e e C o l o r O u t p u t M o d e l 34 5.1 Complete Model 34 5.2 Simplified Model 37 5.2.1 Simplified Model in Predictor Form 38 6 D i s c u s s i o n o f C o n t r o l M e t h o d s 40 6.1 Closed-Loop Control 40 6.1.1 Gain Scheduling 44 6.2 Open-Loop Control 45 iv 7 C o n t r o l l e r s d e r i v e d f r o m t he P r e d i c t o r 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 Extended Prediction Horizon 48 8 D a h l i n C o n t r o l l e r s 50 8.1 Decoupling Using Complete A - 1 50 8.1.1 Adaptive Controller 52 8.2 Decoupling Using Diagonalized A - 1 53 9 A d a p t i v e P a r a m e t e r I d e n t i f i c a t i o n 55 9.1 Mult i -Output R L S Algorithm 55 9.1.1 R L S Applied with the Predictor-based Controllers 58 9.1.2 R L S Applied with the Dahlin Controllers 61 9.2 Constraint of Identified Parameters 64 10 S i m u l a t i o n s 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 73 11.1 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 B i b l i o g r a p h y 96 A p p e n d i c e s 99 v K K A -j^ and of Simulated Dyes and Undyed Paper 100 B A C S L Simulation of Dye Transport Process 102 vi 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 Id 77 11.5 Simulation Subseries le 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 Summary 90 vii List of Figures 3.1 Block Diagram of Dye Transport 20 3.2 Approximate Discrete Block Diagram of Dye Transport 23 8.3 Application of the Dahlin Controller 52 10.4 Block Diagram of ACSL-Simula ted Dye Transport Model 67 10.5 Dye Plots for a Typical Simulation R u n 71 10.6 Color Plots for a Typical Simulation R u n 72 viii Acknowledgement I thank Dr . G u y Dumont for his advice and guidance in my research. I thank Christos Zervos for his assistance in operating the Pulp & Paper Centre's / u V A X computer. I also thank my wife, Jane, and my daughter, Katie, for their patience during the long process of completing this thesis. ix Nomenclature S Y M B O L DEFINITION O L Open-Loop C L Closed-Loop T F Transfer Function R L S Recursive Least Squares R M L Recursive M a x i m u m Likelihood Z O H Zero Order Hold M I M O Multi-Input Mult i -Output G M V C Generalized M i n i m u m Variance Controller E H C Extended Horizon Controller Din The column vector [Din,i DiN<2 Din^Y containing the ratios of injected Dye 1, Dye 2 and Dye 3, respectively, to dry fibre in the fresh pulp [gm. dye/gm. fibre] D The column vector \D\ D2 D3]T containing the concentrations of Dye 1, Dye 2 and Dye 3, respectively, in the dried paper product [gm. dye/gm. fibre] C The column vector \C\ C 2 Cz]T containing the color of the dried paper product in the L* a* b* scale A The incremental dye/color 3 x 3 matrix \ij A n element of the matrix A 1Z Retention of dye on the fourdrinier x T'di Transport delay from dye addition point to fourdrinier [min.] T j 2 Transport delay in fibre recovery process [min.] T'd Transport delay in dryers [min.] Td2 Accumulated time delay in fibre recovery process = T'd2 + Tdl [min.] Td Overall time delay in dye transport process T'd + T'di [min.] Ta Sampling interval [min.] d Discrete dye transport delay = Td/Ta + 1 dh Discrete prediction horizon p Desired C L discrete time constant for a Dahlin controller = e T* Tc Desired C L continuous time constant [min.] p Weighting factor in the G M V C Vii V21 V3 Constant offset vectors Wi, W2, W3 White noise vectors g _ J The .7th backward shift operator: shifts sampled data j sampling periods backwards X(q^1) The backward shift operator polynomial X ^ f o ^ x j ( l ~ i deg[X] Highest negative power in the polynomial X(q~1) term[X] Number of terms (including the q° term) in the polynomial X(q~x) = deg[X] + 1 Ai(q~1) Denominator polynomial of discrete dye transport model for Dye i Bi(q_1) Numerator polynomial of discrete dye transport model for Dye i A(q~l) Denominator polynomial 1 + Y^i^ aj9.^ ° f common discrete dye xi transport model for all dyes B(q~1) Numerator polynomial Y f f l l ^ bjQ* of common discrete dye transport model for all dyes 6X Incremental change in the scalar or vector X A X X — q'1 X where X is a scalar or vector containing sampled data rrace[X] The sum of the diagonal elements of the matrix X F(q~1), G(q~1) Solutions of differential Diophantine E q . 5.45 FB(q-') F{q-*)B{q-*) FE'(q-l) F(q-L)E'(q-*) A or \j Wavelength in the visible spectrum [nm.] i?(A) The reflectance distribution (function of wavelength) of the dyed paper K(X) The absorption coefficient (function of wavelength) of the dyed paper 5(A) The scattering coefficient of the dyed paper ^ ( A ) The absorption/scattering coefficient = 7^  (A) The absorption/scattering coefficient of the undyed paper as defined by E q . 4.26 ^ i ( A ) The absorption/scattering rate coefficient of Dye i as defined by E q . 4.26 xn Chapter 1 Introduction. This thesis will examine and evaluate by computer simulations different methods for the feedback control of color in dyed paper. These methods will include adaptive and non-adaptive techniques. The 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). Another objective is to achieve a good response to changing color setpoints. The paper industry's interest in 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). The thesis is organized in the following way: Chapter 2. The literature on color and its control in the paperrnaking industry is reviewed. This includes: • Dye input/color 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 in the dried paper product) is developed. Chapters 4. The nonlinear relationship between dye concentration in 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. The results of Chapters 3 and 4 are combined to make a three dye input/three color output model. Chapter 6. The problems of controlling this process are discussed. The two closed-loop control methodologies (predictor-based and Dahlin) to be used in this thesis are introduced. Open-loop control is also discussed. Chapter 7. The predictor-based controllers (based on a process model in predictor form) are developed. These are direct/implicit adaptive controllers. Chapter 8. The Dahlin controllers are developed. These are indirect/explicit adaptive controllers. Chapter 9. T h e identification algorithms used by the predictor-based and Dahlin adaptive controllers are described. Chapter 10. The computer-simulated paper dyeing process is described and its pa-rameters given. Chapter 11. The results of applying the controllers developed in Chapters 7 & 8 to the simulated process are given. Chapter 12. The 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 in the literature will be described. Generally, the models are similar to that shown in F ig . 3.1. As well, color control methods used in the literature will be described. 2.1.1 Belanger In [2], Belanger uses a dye transport model similar to that used in this thesis (see Fig . 3.1), except for the absence of a time delay in the fibre recovery/white water process. The 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. Adaptive 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. The model is thus reduced from a second order model to: D e~TdS Din rxa + 1 where: • Td = 3.14 min. • Ti = 2.10 min. (2.1) 3 Chapter 2. Review of Color Control Literature 4 • T, = T d / 4 = 0.80 min. It is assumed that these parameters are those of the paper machine on which the controller was tested. This is a small-scale paper machine used by the Dyestuffs Division of Imperial Chemical Industries of Blackley, England. A discrete integrating L Q R which minimizes the performance index: oo oo YXD - D'«Y + / > £ ( A D i n ) 2 0 0 is designed for the 1Z = 1 model of E q . 2.1. The control signal weighting factor 0 < p < 1 in the performance index is left as an unspecified tuning parameter. Assuming that all retentions between 0.4 and 1.0 are equally likely, the p (i.e. p") is selected which minimizes: S(p) = S'(p,0A)+ S'(p,0.5)+ --- + S'(p,1.0) (2.2) w here: I*{K) I{p,1Z) = r { D - D ' e t ) 7 Jo r(Tl) d= mm I(p,K) dt (2.3) when the L Q R is applied to a simulated continuous dye transport process. I(p,7V} is calculated from the output D of this process with retention calR, when it is controlled by an L Q R with weighting factor p. The problems with this method appear to be: 1. Using the L Q R based on the simplified process model shown in E q . 2.1, the selected p gives the "best" (using the above criterion) performance, but that performance is not very "good" . The value of I(p*,lZ) at 7c = 0.4 is more than 4 times that at 7Z = 1.0. Chapter 2. Review of Color Control Literature 5 2. It seems that seeking to minimize I(p, 0A) + I(p, 0.5)H \~I(p, 1.0) instead of the normalized S'(p,0A) + S'(p,0.5) H (- 5"(,o,1.0) would give better performance in 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 in 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], Chao 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. The process is modelled as in F ig . 3.1 in Chapter 3 (except that there is no time delay in the fibre recovery process), but the authors show that a first order model with time delay: provides a good approximation to the full model at values at 7Z from 0.8 to 1.0. They use dye/fibre units. The parameter values used in the model above were: • Td = 1.4 min. • T\ = 0.6 min. D in (2.4) Chapter 2. Review of Color Control Literature 6 • T„ — 1.0 min. 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 min. time delay in the white water process. The authors feel that an integrating controller will compensate for this. Their method decouples the three dye transport processes with A - 1 and controls the dye addition rates with three identical Dahlin controllers. They 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. They 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 min. Perhaps if the estimated open-loop time constant had been larger to model the time delay in 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. The 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 e~Td" A n {rlS + l){r2s + 1) V " ; The parameter values used in the model above were: Chapter 2. Review of Color Control Literature 7 • Td = 2.1 min. • Tj = 1.25 min. m T2 = 0.33 min. • T„ = 0.75 min. 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. This 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. Dahlin controllers). 2. Multivariable Controller Using the value of A and three identical dye transport models this paper con-structs a 3-input, 3-output state-space model. Augmenting the states to provide integral action, a Linear Quadrature Regulator L Q R is designed. This controller was actually suggested and developed by Sandraz [20]. T h e paper concludes that both these controllers give similar performance, in part be-cause the state space model assumes identical dye transport models as do the three monovariable controllers. 2.1.4 Sandraz This P h . D thesis [20] is the source of the state space model and multivariable controller used in Lebeau et al. [12]. It identifies the dye transport model in 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 both 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 = 1 (2.6) Din rlS + l V ' T h e parameter values used in the model above were: • Td = 2.25 min. • T i = 1.7 min. • T. = 0.75 min. 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 M c G i l l The chapter " T h e Measurement of Optical Variables" in Measurement & Control in Papermaking [16] discusses the advantages of the continuous (versus batch) dyeing of paper and also describes some important variables in the coloring process. These variables together with those given by Clapperton [6] are listed in Section 6.1. This reference discusses the benefits of controlling the ratio of the dye addition rate to the flowrate of incoming fibre. This method results in less color deviation due to paper breaks and basis weight changes. Chapter 2. Review of Color Control Literature 9 2.1.6 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 Some dyers and colorists use the Instrumental Match Prediction (IMP) method to achieve a particular color (see Alderson et al. [1] and Quinn [19]). The method of estimating the incremental dye/color matrix A suggested by Alderson et al. is that used in this thesis and is described in 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 , the first step change in dye concentration Din, calculated using: An = A-lC"e\ (2.7) is applied to the undyed sheet. After the color has reached the steady state color C (Quinn [19] estimates an 8 min. wait), D — L>in and A (and thence A - 1 ) is recalculated —* —> using D. The next step change in dye concentration ADin is calculated using: ADin = A.-\C"*-C) (2.8) and then: 4 <- 4 + A 4 (2.9) This process is repeated until the desired color is achieved. This 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 in order to design feedback controller with a much shorter sampling interval. However, the variable A - 1 gain aspect of the above method is retained by using G a i n Scheduling. Chapter 2. Review of Color Control Literature 10 2.1.7 Conclusions Chao 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. By 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 in 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 in the paper. Chao et al. and Lebeau et al. both use Dahlin controllers. Lebeau et al. find that 3 decoupled Dahlin controllers perform as well as a multi-input multi-output L Q R . 2.2 Selection of Dyes and Identification of A The selection of a "good" set of dyes to achieve a particular color range is complex. It requires experience and advice from a dye manufacturer. There are two objectives in selecting a set of dyes: • To achieve the desired color. • To have good controllability at the desired color. The 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. to control L*, a*, and b* in the 3-dimensional V a* b" color scale requires 3 dyes, but to control only a* and b" requires only 2). 2. The 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 in 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 in achieving a desired color by itself, should not be used. The 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 in a dye set to span larger volumes of the color space, since large counteracting flowrates which darken the paper may be required in that space. Controllability once color is achieved will be poor. Therefore, orthogonality (if that term may be used for a nonlinear dye/color relation-ship) in 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 Chao et al. [5] and is described in Subsection 2.2.2. To cover the whole color space, 24 dyes or more may be required. The 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 pro-duction 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 in Subsection 2.2.5 and Section 4.3. 2.2.1 Belanger To estimate the dye/color matrix A, Belanger suggests the theoretical calculation de-scribed in Subsection 2.2.5 and Section 4.3. Chapter 2. Review of Color Control Literature 12 2.2.2 Chao & Wickstrom This 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 con-trollability. 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. This means that only a thin plane in the color three space is achievable. Assuming that a particular color in that plane has been achieved, control about that point will be difficult, especially if a correction is required perpendicular to the thin plane. In that case large increments in the red dye & blue dye are cancelled by a large decrement in the green dye in 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 = J I ( 2 . 1 0 ) and where is an element of A. T h e value of E lies between 0 (i.e. singular A) and 1 (i.e. orthogonal A) 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 scale was used, but it was not necessary to control the lightness L of the near-white paper made at Consolidated Papers Inc., Wisconsin Rapids, W I . The process is thus a two dye input / two color (a and 6) output process. The authors suggest that, if necessary, the lightness could be controlled separately using the addition of a white filler. The 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). The color matrix A is identified online by making separate step changes in each of the three dye flowrates. Chapter 2. Review of Color Control Literature 13 2.2.3 L e b e a u , V i n c e n t & R a m a z This 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 in large errors in a A - 1 calculated from it. Therefore, this paper suggests that A - 1 be identified directly instead of A. W i t h the process under closed-loop control, step changes were made in each of the three color setpoints separately and the resulting changes in dye addition rates were measured. This calibration would have to be performed at every color setpoint. 2.2.4 M c G i l l The chapter " T h e Measurement of Optical Variables" in Measurement & Control in Papermaking [16] states the necessity of having a range of dyes with known and stable properties in order to produce a wide range of colors (i.e. several sets of orthogonal dyes which are capable of achieving colors in 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 Match Prediction (IMP) color matching method, Alderson et al. [1] suggest that A can be determined, at a particular D and C , using numerical differentiation. First, they develop a theoretical relationship between dye and color (as described by Eqs. 4.20, 4.22, 4.24, 4.26 in Chapter 4). T h e n , these equations are differentiated numerically as described in Section 4.3 to determine A. In E q . 4.26 the value of the coefficient ^ ( A ) for each of the dyes in use (i.e. ^^- (A) for Dye 1, ^^- (A) for Dye 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 in the visible spectrum. 5^(A) for one dye only and ^(X) can be determined in the lab using a spectrophotometer and small sheets as follows. Several sheets are made up with varying concentrations of the dye D. At n wave-lengths Xj (j = 1 , . . . ,n), the reflectance of each sheet is measured with the spectropho-tometer. If an abridged spectrophotometer is used, n is limited by the sensor. Using the Kubelka-Munk Equation (Eq. 4.24) the value of ' j (Aj) is calculated. This 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) and zero intercept ^(Xj) where: = ^i)D + §f(Xj) (2.11) 2.2.6 C o n c l u s i o n s Chao et al. identifies the dye/color matrix A online by perturbing the three dye flowrates to the paper machine. Alternatively, Lebeau et al. identifies the color/dye matrix A - 1 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. The theoretical dye/color model suggested by Alderson et al., together with specific optical properties of the dyes, has proved successful in color matching. The 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. This 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 15 2.3 Industrial Control System Vendors The color control systems described below are described and compared in a short article by the P I T A Engineering Technology Working Group [17]. The emphasis is on the color sensors used by each system. A comparison of spectrophotometers and colorimeters as color sensors is made in Section 4.2. 2.3.1 AccuRay Co. Ltd. R. Marclii [15], an AccuRay systems engineer, briefly describes this color monitoring and control system. The color sensor is located between the calendar stack and the paper reel. The sensor is an abridged spectrophotometer manufactured by the Macbeth Color Communications C o . 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 at the 16 wavelengths. It then calculates the XYZ (and thence Lab) color value for any of the C I E standard illuminants. Either the Lab or XYZ scale can be used for feedback control of the dye input. The control algorithm uses an incremental color/dye matrix to convert color errors to dye errors. The incremental dye/color matrix is calculated using a proprietary method based on the Kubelka-Munk equation. Since details in 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 in the lab and stored in 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 in 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 in past production runs to achieve a particular color are also recorded in memory for use in repeat runs. 2.3.2 Babcock-Bristol Co. Ltd. K . L . T o d d et al. [22], Babcock-Bristol engineers, briefly describe the "Color Eye" color monitoring and control system. The color sensor is an abridged spectrophotometer also manufactured by the Macbeth Color Communications C o . Babcock-Bristol holds the exclusive European distribution contract for the Macbeth sensor in the paper indus-try. The Lab scale is used for feedback control. The control algorithm is described as a proprietary multi-variable nonlinear dead-time controller. Few other details of this proprietary algorithm are given. The control algorithm uses an incremental color/dye matrix to convert color errors to dye errors. The 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. This identification must be performed for every new paper color and is therefore less general than that used by AccuRay which, by identify-ing 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. The flowrates of dyes used in past production runs to achieve a particular color are also recorded in the computer's memory for use in repeat runs. Chapter 2. Review of Color Control Literature 17 2.3.3 M e a s u r e x I n t e r n a t i o n a l S y s t e m s L t d . R. P. Shead [21], a Measurex engineer, briefly describes this color monitoring and control system. The color sensor is a abridged spectrophotometer manufactured by Measurex. This sensor is able to scan the paper web in the cross-machine direction. It uses two continuous illumination sources: a tungsten-halide lamp and a mercury vapor U V source. The 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. The 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 C o n t r o l Co. L t d . M . Lehtoviita [13], an A l t i m engineer, briefly describes the "Colourkeeper" color mon-itoring and control system. The color sensor is a Hunter D43 colorimeter (not a spec-trophotometer). 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. The control algorithm is stated to be a multivariable P.I .D. one with a 3 x 3 decoupling (i.e. in-cremental color/dye) matrix. The 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 concentra-tions desired, process model, and "controller's calibration values" (the 3 x 3 decoupling matrix, it is assumed). 2.3.5 Conclusions The Measurex abridged spectrophotometer measures the reflected light at 32 wave-lengths, twice the number measured by the Macbeth spectrophotometer. AccuRay 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 in Section 4.3. AccuRay'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 in 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. At 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. The dyed web moves from the fourdrinier through the presses to the dryers. The dried paper is finally wound onto a roll at the reel. 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 in F i g . 3.1. It agrees with the model used by Belanger [2] and Chao et al. [5]. In Fig . 3.1, the following definitions apply: • Din =f The ratio of injected dye to undyed fibre in the incoming pulp [gm. of 19 Chapter 3. Model of Dye Transport through a Paper Machine 20 W E T E N D P R O C E S S D R Y E R S D, - © - e di T l » + 1 F O U R . -I) F I B R E R E C O V E R Y P R O C E S S e "2 1 T2 3+1 n D • T i Figure 3.1: Block Diagram of Dye Transport dye/gm. of fibre] D =f T h e concentration of the dye in the dried paper [gm. of dye/gm. of fibre] def Volume of wet end reservoirs Total liquid flowrate [min.] def Volume of fibre recovery reservoirs r . i * T 2 — Total hquid flowrate [min.j • 7c =f Retention of dye on the fourdrinier Td = f Transport time delay from dye addition point to fourdrinier [min.] def r | Transport time delay in fibre recovery process [min.] a T'd == Transport time delay in dryers [min.]. The block diagram can be re-arranged so that the T'd transport delay is combined with both the Td and the T'd transport delays. Defining new symbols: Chapter 3. Model of Dye Transport through a Paper Machine 21 . Td2 d^ Td2 + T'di • Td d=f rd + Vdl If we assume that all transport delay occurs in the dryers (i.e. Td2 — 0), the transfer function of the dye transport model is: D xKe-Tdt (3.12) D i n 1 (n«+i)(^ f+i) ' T 2 S + 1 xHe-T*> (3.13) (TIS + 1){T2S + 1)-{1-K) T2S + 1 T i r 2 s2 + (TI + r 2 ) s + Tl (r2s + l)e-T*B + n ± a B + 1 TZe-TdS (3.14) (3.15) This model has unity gain. If dye or fibre is lost, the gain would be changed. The unit of D is the ratio of dye to dry fibre. The color of the paper at the reel is largely determined by this ratio and this unit allows for ratio control of dye addition rate (i.e. dye addition rate equals the desired D{n multiplied by the flowrate of undyed pulp). Ratio control causes the paper color to be less sensitive to changes in undyed fibre flowrate due to changes in basis weight or paper breaks. 3.2 Discrete Model If the transport time delay Td2 is not zero, a discrete T F : I - ^? cannot be calculated directly from 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 22 • T„ = Sampling interval 4eg[A] j = l -j • B(q-*) def ^deglB] — l^j=0 J In this thesis, the empirical method of determining the T F was to select Ts, deg[A] and deg[B] and then carry out a discrete R L S identification of the continuous A C S L -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 Ts should be selected so 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 Td evenly. In order to get estimates of rfe^[^4] and deg[B], the approximate discrete model of F ig . 3.2 will be used. In F ig . 3.2, the following definitions apply: • n 2 =f truncated 1 a def o def 1 - e def 1 — e T2 The discrete transfer function of the dye transport model is: Chapter 3. Model of Dye Transport through a Paper Machine 23 Z O H ( r . ) -n2 W E T E N D P R O C E S S b' l - a j g - 1 F O U R . 4 F I B R E R E C O V E R Y P R O C E S S l-ai'g- i-n D R Y E R S D Figure 3.2: Approximate Discrete Block Diagram of Dye Transport If T„ = 0.66 min. and Td2 < 2.0 min. (as it is in these simulations), then n2 < 3 and E q . 3.18 indicates that reasonable estimates for depfA] and deg[B] would be 3 and 2 respectively. For example, if the continuous A C S L model with the parameters: • T i = 0.94 min. • r 2 = 0.75 min. • Td2 = 1.0 min. • Td = 2.0 min. • 1Z = 0.8 was identified with T, = 0.66 min. , 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 24 • a a , a 2 , a 3 = - 0.905,0.159,-0.017 » 60,61,62 = 0.406,-0.167,-0.003 The output of this discrete model simulated the output of the continuous model with only 0.02% error, when the model was simulated for 80 min. (after the coefficients had been identified) with a known zero-order-held random input. The zeros of A(q~1) are .716, .094 ± .122* and the zeros of B(q~l) are .429,-.017. The discrete model of the process is therefore stable and minimum phase. W i t h Ts = 0.66 min. , </e-gr[^ 4] = 1 and deg[B] = 0, the R L S algorithm identified the following coefficients: • 01 = - 0 . 5 2 3 • 60 - 0.407 The 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'Q b'0' in the denominator is small (eg. it equals 0.02 for the model parameters above), then E q . 3.18 can be approximated by: D _ g-d TZb'0 An ~ ( 1 - a i r 1 ) ( 3 " 1 9 ) This is the reason that a discrete model with de#[vl] = 1 and deg[B] = 0 is quite accurate in simulating the continuous model. Chapter 4 The Dye Concentration/Color Relationship 4.1 Color Measurement A n object's color in the XY Z color space is determined by: X = /•700nm J400nm /•700nm 1(A) R{X) x{X) dX Y = JiOOnm r700nm I(X)R(X)y(X)dX Z = JiOOnm 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 . 2 / n m . ] • R(X) = T h e reflect ance distribution of the object illuminated • a:(A), y(A), z(X) = f The sensitivity distributions of the three retinal pigments of the Standard Observer as defined by the Commission Internationale de l'Eclairage (CIE) in 1931 and updated in 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 , 67,63) 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 in 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\\ = yj6C? + SCI + SCi (4.21) the color space/scale (C\,C2iCz) can be defined (as a function of the C I E 1964 XYZ scale) in several ways: none of which exactly duplicates perceived color difference, due to the non-linearity of human vision. This metric is especially useful in L Q control in which the performance index has the same form. The two color scales ( C I E L A B and C I E L U V ) recommended by the C I E in 1976 (see Billmeyer et al. [3]) will be briefly described. The C I E L A B 1976 L*a*b* scale: L* = 1 1 6 ( ^ ) 3 - 1 6 * n a = 5 0 0 [ ( ^ ) i - ( £ ) * ] (4.22) b* = 2 0 0 [ ( £ ) * - ( y - ) i ] where: • Xn,Yn,Zn = f The X,Y,Z values of the illuminant. is a "uniform" scale in that it approximates the uniformity of spacing of the Munsell Color Space. The C I E L A B L*a*b* scale is an opponent-type system (like the Hunter 1958 Lab scale) in which L* measures light(lOO) to dark(O), a* measures red (positive) to green (negative), and b* measures yellow (positive) to blue (negative). This scale has been widely used in the paper industry and was recently recommended by Jordan et al. [10] in a P P R I C Report, although they only studied near-white paper colors. The C I E L U V 1976 L*u*v* scale: L* = calculated as above. Chapter 4. The Dye Concentration/Color Relationship 27 u 13L > ' - u'J (4.23) v 13L*(v' - v'n) where: 4.X u X + 15Y + 3Z 9Y v X + 15Y + 3Z u'niv'n ~ u'iv' calculated using Xn, Yn, Zn. measures just perceptible color differences where || 8C ||= 1 is a just perceptible differ-ence. Although not recommended by the C I E in 1976, the F M C - 1 and 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 in some industries although their calculation from the XYZ scale is complicated. In the simulations of this thesis, the L*a*b* scale will be used and C i , C 2 , C 3 will refer to the L*,a*,b* values on that scale. As well, the results of the simulation runs will be given as the sum of the average squared deviations of the colors from their setpoints (i.e. the average value of || SC ||2) during different intervals of the run. 4.2 C o l o r Sensors There are two types of color measuring devices: colorimeters and spectrophotometer. T h e colorimeter uses three colored filters (ie. the x, y and z filters) to separate the light reflected from an object. Behind 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 Dye Concentration/Color Relationship 28 The spectrophotometer measures the continuous 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 over the entire visible spec-trum. It can then calculate the XY Z color value for any of the C I E standard illuminants using E q . 4.20. Metamerism is the phenomenon in which two differently dyed objects show the same color under a particular illuminant. By 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 in the cross direction (i.e. from 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. The Dye Concentration/Color Relationship 29 4.3 T h e D y e / C o l o r M a t r i x A For scattering absorbing mixtures such as dyed paper, the Kubelka-Munk equation is used to calculate the reflectance R(X) from a thick non-fluorescent sheet (derived by Billmeyer [3], Mac A d a m [14]): * ( A ) = l + f ( A ) - v / f ( A ) 2 + 2 f ( A ) (4.24) where: • R(X) ==f The reflectance distribution (function of wavelength) of the dyed pa-per. • K(X) = f The absorption coefficient (function of wavelength) of the dyed paper. • 5(A) =f The scattering coefficient of the dyed paper. • ^ ( A ) == = The 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], Bonham [4], M a c A d a m [14]) can be made: -sW = ^ A ) (4-25) or: f W = 7 f ( A ) + E ^ ( A ) A (4-26) where: • Kf(X) =f The absorption coefficient of the undyed paper • Sf(X) =f T h e scattering coefficient of the undyed paper Chapter 4. The Dye Concentration/Color Relationship 30 • KT<i(X) = T h e absorption rate coefficient of dye i • ~sj(X) is the absorption/scattering coefficient of the undyed paper • ^^-(A) = f The absorption/scattering rate coefficient of dye i • D{ =f The concentration of dye i. The absorption/scattering coefficients ^ (A ) , ^MA) , ^^(A), ^^(A) can be determined experimentally with a spectrophotometer as described in Subsection 2.2.5. J .S. Bonham [4] has derived an extended Kubelka-Munk equation which includes fluorescent dyes. To solve the extended equation, the value of |^(A) of each dye in its absorption band and the quantum efficiency of each fluorescent dye in its emission band must be known. Obviously, the function Fc relating dye concentration and color on the L*a*b* scale (combining Eqs. 4.20, 4.22, 4.24, 4.26) is a nonlinear one: C = FC(D) (4.27) where: . C = [ d C 2 C 3 ] T . D d=r [D, D2 D3f -*' B y differentiating Fc at a particular dye concentration D , a 3 x 3 matrix A which relates incremental changes in dye concentration to incremental changes in color can be found: 8C = A5D (4.28) where: Chapter 4. The Dye Concentration/Color Relationship 31 • 6C = [8d 8C2 8C3}T • 3D d= [8Dt 8D2 SD3]T Since 8C = C — C , E q . 4.28 can be written: C = A8D + C. (4.29) Letting 8D = D - D, E q . 4.29 becomes: 6 = AD + VX (4.30) where Vx = C - A D . Calculation of A: A can be determined, at a particular D and C , using numerical differentiation as follows. C is calculated for the dye concentrations D using Eqs. 4.20, 4.22, 4.24, 4.26. Adding a small change 8Di to Dye 1, C is calculated for the dye concentrations D + [8DX 0 0 ] r . Now, 8& is - C". Adding a small change 8D2 to Dye 2, & is calculated for the dye concentrations D + [0 8D2 0 ] r . Now, 6C is C — C . Similarly, 8C is calculated for a small change 8D3 to Dye 3. Finally, A is calculated using: A = sdl sd2 sd3 SDi SD2 SD3 (4.31) 4.3.1 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 . This calculation can be done offline for each desired paper color before a production run begins. Because of the integration in E q . 4.20, it is very difficult to calculate A directly from C (i.e. to differentiate the inverse function Fcl(C)). Similarly, it is very difficult to calculate directly from C the corresponding Chapter 4. The Dye Concentration/Color Relationship 32 -•set D . Therefore, denoting the desired A by Aset, both A°et and D~" must be calculated iteratively using the algorithm: 1. Initiallize D with an estimate of D 2. C = FC{D) (using Eqs. 4.20, 4.22, 4.24, 4.26) 3. A = F A ( D ) (using numerical differentiation of Eqs. 4.20, 4.22, 4.24, 4.26 described above) 4. D <— D + A - 1 (C — C°et) ^tset 5. IF (C - C ) > M A X _ E R R O R , T H E N go to step 2. 6. Finally, A set set A and D = D E x a m p l e : In reality, this algorithm uses approximate (i.e. measured), not exact, absorption/scattering coefficients ^ ( A ) , ^ - ( A ) , ^ ^ ( A ) , ^ - ( A ) in E q . 4.26 and it is worthwhile to see how, —*set _ 1 for specific values of C , the estimated A determined by the algorithm compares with the exact A - 1 . For C = [74.2 3.57 9.88]T, a yellow -orange color: - l -13.5 3.23 - .656 -16.6 -4.89 7.82 -3.11 4.71 3.74 x 10 - 2 (4.32) T h e difference between A 1 and the corresponding A was 20.3% using: 100 II A" A w here: X def 3 3 i = l j = l (4.33) Chapter 4. The Dye Concentration/Color Relationship 33 +set For C = [74.2 16.0 10.0] r, a red-orange color: A " 1 = -16.7 4.58 .255 -13.2 -3.06 6.66 -6.47 2.95 4.87 x 10" (4.34) The difference between A 1 and the corresponding A * was 22.9%. These differ-ences are less than the 25% difference between the estimated 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 Appendix A for the coefficient values). Convergence of the above algorithm occurred within 8 iterations. The 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 ) . Even though the inverse function FQ1(C) 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 con-centration in the dried paper and the paper's color are combined to make a complete three dye input/three color output model. Taking three discrete dye transport models 9 A^CT1) ^ ( w ^ h 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: Aiiq-1) 0 0 . 0 A2{q'1) 0 0 0 Az{q~l) EM-1) + 0 0 £> = g-' 0 E2(q~') B1(q~1) 0 0 0 0 0 0 0 EM'1) B2(q-1) 0 0 B3{q-X) Di. (5.35) wh ere: D d ^ [ D i D2 D3]T . Din d^ [Din<1 Din<2 Din,3]T Adding measurement noise to E q . 4.30, an expression for the color vector is obtained: C AD + Vi + W2 (5.36) where: 34 Chapter 5. Three Dye Input/Three Color Output Model 35 C d=f [ d C2 C3f 9 Vi *= Constant offset vector • W2 — White measurement noise vector Rearranging we obtain: D = A " 1 C - A - 1 Vi - A - 1 W2 (5.37) Substituting E q . 5.37 into E q . 5.35 we obtain: Axiq-1) 0 0 0 A2{q-X) 0 0 0 A^q-1) E1(q-1) 0 0 + 0 E2(q~1) 0 0 0 Esiq-1) where: A " 1 C = q'd Biiq-1) 0 0 0 B2(q-1) 0 0 0 B^q-1) Axiq-1) 0 0 0 A^q-1) 0 0 0 A3{q^) D{ A - 1 W2 + V2 d e f • Vjs = Constant offset vector = To make Mr1) o o 0 A2{q~1) 0 o o A 3(<r a) ^ ( T 1 ) o o 0 A^q-1) 0 0 0 A^q-1) A 1 monic, we multiply by A : A _ 1 V i Ai{q-1) 0 0 0 A2{q-1) 0 0 0 A^q-1) A - 1 C = q~dA S i f ? - 1 ) o 0 0 ^ ( g - 1 ) 0 o o ^(r 1 ) A , Chapter 5. Three Dye Input/Three Color Output Model 36 + £r(? _ 1) o o 0 E^q-1) 0 0 0 ^3 * ( g _ 1 ) W4 + Vg or: (5.38) A A ( g - 1 ) A - 1 C = g - d A B ( 9 - 1 ) A n + E * * ( g - 1 ) P F 4 + V3 (5.39) where: AGr 1) d e f • Biq-1) = E * ^ - 1 ) D ^ F E * * ( g - i ) d e f A ^ 1 ) 0 0 0 A2(q~1) 0 o o A3(<r]) B1(q-') 0 0 0 B2(q-1) 0 0 0 Bsiq'1) E{(q-1) 0 0 0 ^ ( g - 1 ) 0 0 0 E^q-1) Efiq-1) 0 0 o £2**(?-1) 0 0 0 £ 3 * * ( g _ 1 ) E * ( Q 1 ) W > 3 = f The spectral factorization of ^ ( g - 1 ) 0 0 0 E2(q^) 0 0 0 E3(q~1) Chapter 5. Three Dye Input/Three Color Output Model 37 A ^ 1 ) 0 0 0 A2(q-1) 0 A - 1 ^ 0 0 A 3 ( g - 1 ) E** ( 9 - ! ) WA = The spectral factorization of A E * ( g - ] ) W3. = r Constant offset vector = AVo 5.2 S i m p l i f i e d M o d e l It is reasonable to assume that the transport dynamics of the dyes are similar. This 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. The color control literature consistently makes this assumption. Letting Aj(q~1) — A(q~x) and Bj(q~1) = B(q~1) for j — 1,2,3, E q . 5.39 simplifies to: Aiq-^C = q-'ABiq-^Din + W'iq-^Wt + Va (5.40) By subtracting E q . 5.40 at two successive samplings, the constant disturbance vector is eliminated and the simplified differential model is obtained: Aiq-^AC = q-dAB(q-l)ADin + E"{q-1)W4 (5.41) where: d e f A ( 1 - g " 1 ) . E ' V 1 ) = E - ' O r ^ l - i r 1 ) The simplified differential model is used explicitly to identify the color/dye process and this identified model is then used to derive the Dahlin controller as described in Chapter 8. Chapter 5. Three Dye Input/Three Color Output Model 38 5.2.1 Simplified Model in Predictor Form To generate the e^-step-ahead predictor, the scalar polynomials F(q~1) and G'(g_1) are calculated by solving the scalar differential Diophantine Equation: 1 - F { q - 1 ) A ( q - 1 ) { l - q - 1 ) + q - l > G ' ( q - 1 ) (5.42) or: F ( g - , M ( « - 1 ) ( l - r 1 ) = l - f 4 G V ) (5-43) W h e n q~l = 1, G'(l) = 1. Therefore G'(g_1) can be expressed as: G V 1 ) = l + G ^ X l - g - 1 ) (5.44) Substituting this expression into E q . 5.43, we obtain the differential Diophantine E q . : F(q~l) Aiq-W - q-*) = (1 - q~^ ) - q~d" G{q^){\ - q~l) (5.45) where: • dh = Discrete prediction horizon • deg[F) = 4 - 1 • term[F] = dh • deg[G] = deg[A] - 1 • term[G] = deg[A] Multiplying E q . 5.41 by F(q'1) and then eliminating F(q~l) A(q~1)(l - q'1) using E q . 5.45, we obtain the form of the 4-step-ahead predictor: (1 - q-d* ) C - q~d« G(q^)AC = q-d F(q~1)AB(q~1) ADin + F f a - ^ E ' V 1 ) ^ 4 (5.46) Chapter 5. Three Dye Input/Three Color Output Model 39 Moving the matrix A to the left of the term containing it, we obtain: ( l _ f 4 ) c - f 4 G{q~1)AC = q~d A F(q~1)B(q~1) ADin + F(q~1)E"(q^1) W4 (5.47) Multiplying by A - 1 and taking the ADin term to the left hand side, we obtain the predictor in linear control form (see Goodwin et al. [9]): q~d FBiq-1) ADin = A _ 1 ( l - q-dh)C -q~d* A - 1 G ( g - 1 ) A C - F E ' ( g - 1 ) F F 3 (5.48) where: FB(q^) F{q-*)B{q-i) FE'(q->) * f F(q-')E'(q^) . E'iq-1)^ = X-xE"{q^)WA = (1 - g - ^ E ^ g " 1 ) W3 The predictor form is used to derive the predictor-based controllers as described in 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 counter-acting disturbances (e.g. the addition of colored broke to the pulp). Another objective is to achieve a good response to changing color setpoints. Both 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 in Chapter 4 ) are available. These estimates are used to ini-tialize 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 from the output dye concentration D can be determined). The 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 in 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 in dye concentration to color: C = A6D + C. The identification of the process is in 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 in the color setpoint and changes in other process variables such as (from Clapperton [6]): • T y p e (i.e. Sulphite, Kraft , T M P ) and color of undyed pulp • Amount 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) • Amount of active bleach left in the undyed pulp • Strength and quality of the dyes • Moisture in finished sheet. As well the color is subject to step disturbances caused by: Chapter 6. Discussion of Control Methods 42 • Addit ion of dyed or undyed broke • Change in the offset vector C in E q . 4.29 due to changes in the process variables listed above. These disturbances require an integrating controller to counteract them. The unconstrained adaptive controllers (both explicit and implicit) are found to be unstable as the identification procedure attempts to converge to the process parame-ters. To prevent this, the parameters are constrained using Parameter Projection as suggested by Praly [18] and described in Chapter 9. The identified parameters are kept within a "projection sphere" with radius /3 about the a priori estimated parameters (i.e. A , B(q~1), A(q~1) or parameters calculated from them). If the constraint radius (3 is zero, a non-adaptive controller is obtained since the controller parameters are con-strained to be those calculated from the a priori estimated parameters. The following types of controllers are derived and tested in this thesis. P r e d i c t o r - b a s e d C o n t r o l l e r s : 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 dn is equal to d, the J-step-ahead predictor predicts the expected value of the process output d sampling intervals (i.e. steps) in 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. Allowing dn to exceed d, the tf^-step-ahead predictor gives the expected value of the process output dh sampling intervals in the future based on present and past values of the process output and present, past, and some future values of the process input. 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 in this thesis), the future values of the process input are set to the present value of the process input algebraically. The 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. The form of the process model to be identified is: q~dADin = -X>i<ri-dAAn + A- 1 (C-<r i (7) i=l - A - 1 G(q~1)q~d AC (6.49) By simply shifting to data taken d sampling intervals later and setting q+d C to C , the controller is obtained: A An = -£A<r'AAn + A " 1 ( ( ? " ' - ( ? ) - A - 1 G ( g - 1 ) A C (6.50) These are implicit/direct adaptive controllers. The process is identified "implicitly" in the form of a predictor model and the controller is derived "directly" from the identified process model by a time shift. The details are given in Chapter 7. D a h l i n C o n t r o l l e r s : T h e Dahlin controllers are members of the Smith family of time-delay-compensation controllers. The Smith controllers cancel both the poles and zeroes of the process in order to achieve desired C L responses. The Dahlin controllers are designed to achieve a first-order C L response <f1_^q~f^ • If P equals zero, the C L response becomes the zero-order response q~d and the controller is called a Deadbeat controller. This is the fastest setpoint change for a process with a discrete time delay of d sampling intervals. Chapter 6. Discussion of Control Methods 44 Since the Dahlin controllers cancel both the poles and 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. As well, Dahlin controllers are integrating controllers. In this thesis, the Dahlin controllers are used as explicit/indirect adaptive con-trollers. The process is identified "explicitly" in its simplest form (by identifying the parameters A , B(q~1) and A(q~1) of E q . 5.41) and the controller is derived "indi-rectly" in that the non-adaptive method of calculating a Dahlin controller is used after the process has been identified. The details are given in Chapter 8. For good convergence properties of the R L S algorithm, Elliot & Wolovich [8] rec-ommend that the number of parameters estimated should not exceed 10-12. Praly [18] feels that explicit/indirect controllers are more robust. 6.1.1 G a i n S c h e d u l i n g 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 m r — _ 1 C using the method of Subsection 4.3.1, then A can be used as the controller gain (i.e. G a i n Scheduling). During a setpoint change, an estimate of the mean gain A m is obtained by calcu-lating A at the color halfway between the two color setpoints . This mean gain is used in the C L controller in the time delay (ie. the d sampling intervals) when the critical changes in 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 and adaptive) simulated in this thesis, this Chapter 6. Discussion of Control Methods 45 method of G a i n Scheduling is used. 6.2 O p e n - L o o p C o n t r o l ~*8et,b From past production runs of a certain color (i.e. C ) of dyed paper, the steady state input concentrations (i.e. D{n ) of the corresponding dyes may be well known. If 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 dye output concentrations to the new dye output concentrations) can be achieved with O L control. For example, if the fastest setpoint change T F (i.e. Deadbeat) is desired: = ° ~ d 6.51 and the transport model is: 3 - « ~ J B ^ (6.52) Mr') ' —f then, removing D by substitution, the following O L controller determines the transition of Din from Din to Din : ° " - A { " ^ (6.53) For this application, it is essential that the steady state gain of ? A(^-1) ^ ^ E UIU^V> -* -* set since, at steady state, D{n should equal Din . As before, the minimum phase property of B(q~1) is essential. O L control could be applied for at least the duration of the transport delay time (i.e. d sampling intervals) , after which C L control would be resumed. 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 in Chapter 6. This 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 in predictor form (i.e. E q . 5.48), one obtains: deg[FB] q-d £ fliq-iADin = i=0 A - 1 (C - q~dh C) - q~dh A " 1 G(q~1) AC - F E ' ( g _ 1 ) W3 (7.54) 7.1 With Minimum Prediction Horizon Setting dh to its minimum useful value of d and partitioning the sum produces: deg[FB] q~d ADin = [- £ fl>iq~iq~d ADin + A - 1 (C - q~d C) i=l - q~d A - 1 G(q~1)AC - FE'(q^)vV3} I fb0 (7.55) or: deg[FB] q-dADin = - £ (fbi/fb0)q-iq-d ADin + (A-l/fb0)(C-q-dC) 46 Chapter 7. Controllers derived from the Predictor Model 47 - q~d (A" 1 / fb 0 ) G(q~l)AC - (FE ' (g - 1 ) / fb 0 ) W3 (7.56) E q . 7.56 is the form in 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+d (inserting data taken d sampling intervals later), the d-step ahead predictor of q+dC is obtained: deg[FB] ADin = - £ {fbi/jb0)q-iADin + {lL-i/jb0){q+dC-C) 1=1 - (A-1 /fb^Giq-1) AC (7.57) where: • q+dC =f The ef-step ahead prediction of the color vector. • C — q° C = f The present color vector. • q~x C for i > 0 = f Past color vectors. • A =f (1 — q_i) == Change between two successive samples. • D{n = q° D{n —' The imminent control signal vector (to be calculated by controller). • q~l for i > 0 =f Past control signal vectors. e X = f Estimate of corresponding X provided by the R L S algorithm . , -* _ t -*set T h e controller is obtained by setting q+ C to the desired color setpoint C : deg[FB] ^ ^ ^ ^ , • v ADin = - £ fbilfb0q-iADin + A.-llfb0{Cn-C) - A ^ V ^ G ^ 7 7 ) A C (7.58) Chapter 7. Controllers derived from the Predictor Model 48 This is the M i n i m u m Variance Controller M V C in Control Configuration. It is a dead-beat controller in 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 Koivo [11]) —* the change in control signal ADin is attenuated by dividing E q . 7.57 by (1 + p) where p is a weighting factor. We obtain: deg[FB] , « ^ ADin = [- £ fbJfbvq-'AD^ + A-i/fbviC8* -C) i=l - A ~ ^ G [ q r T ) AC] I(1 + P) (7.59) 7.2 With 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 ). Taking E q . 7.54: deg[FB] q~d E fliq-i*Din = A " 1 (1 - q-d» )C - g - d " A " 1 G(q~1) AC - FE'(q-1)Wz and substituting the partition: deg[FB] dh-d-l deg[FB] E + fi>dh-dq-{dh-d) + E (7-60) i=0 i=zdh-d+\ we obtain the form in which the predictor is identified: dh-d-l deg[FB] q-d»ADin = [(- E A ? - ' - E fbiq-')q-d ADm Chapter 7. Controllers derived from the Predictor Model 49 + A " 1 (1 - q-d* )C - q~dh A " 1 G(q~1) AC - F E ' ( g _ 1 ) W3] / fbdh_d (7.61) By multiplying by q+dh (inserting data taken dn sampling intervals later), the dy,-step ahead predictor of q+d,lC is obtained. B y setting q+d,,C = C , the controller is ob-tained: deg[FB] AAn = (- E fbjfb^q-1- E fbJfb^q-^-'AD, i=0 i=dh-d+l + A-*/fbdh_d (C'et - C) - A - 1 / ^ ^ G(q-') AC (7.62) If we assume the future unknown ADin values in Y^=od 1 foil ft>dh-d<ldh d 1 ADin are equal to the imminent control signal change ADin, then the controller becomes: deg[FB] f t . s ADin = [- E fbi/fbdh-dqdh-d~iADin + A~x I fbdh-d{Cee -C) i=dfL—d-\-l - A~rifbd~Td GZF) AC] I (1 + d E 1 lWbd~Z) (7.63) i=0 Chapter 8 Dahlin Controllers 8.1 Decoupling Using Complete A 1 Dahlin controllers are discrete, integrating controllers that are simple to apply to minimum-phase, stable processes with a time delay (such as the dye transport pro-cess). A n introduction to this controller is given in Chapter 6. This chapter will do detailed derivations of this type of controller. If the discrete dye transport model (ignoring noise) for Dye i is: Din,i Mq-1) and the desired C L response is: (8.64) (l-pq-^Di = q-d(l-p)D?t (8.65) then the corresponding Dahlin controller is: Hiq-^Biiq-^D^i = (l-p)Ai(q-1){Drt-Di) (8.66) where: • p =f The desired C L discrete time constant = e - ^ where Tc is the desired continuous time constant. . Il(q^) * 1 - pq^ - (1 - p)q~d 50 Chapter 8. Dahlin Controllers 51 Note that if p = 0, a Deadbeat controller is obtained. The Dahlin 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 in the simulations. In order to determine D\et - D{ for i = 1,2,3 (i.e. D - D), let: D —D = A _ 1 ( C -C) (8.67) The Dahlin controller is an integrating controller since H(q~1) includes the factor (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. The block diagram of Fig . 8.3 illustrates the application of one Dahlin controller to the flow of Dye i and the use of A - 1 to decouple the dyes. The contribution of color C\ only is detailed. The remaining dye controllers are similar. In Fig . 8.3, <f>ij is an element of 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-X)D = q-d Biq-^Din (8.68) and, if the desired C L responses are identical, the decoupled M I M O Dahlin controller is: H(q-')B(q-')Din = (l-p)A(q-')(Dset -D) (8.69) Substituting E q . 8.67 in the above produces: H{q-i)B{q-*)Din = (l-p)A(q-')A-1(CSet - C) (8.70) Chapter 8. Dahlin Controllers 52 tset Di) Dahlin Dinti Controller Dye i Transport Di other dye other dye Dye -to-Color Transform C2 © - <f>il (C°2et -<f>i3 9; Color Sensor Figure 8.3: Application of the Dahlin Controller When A(q 1), B(q 1) and A 1 are substituted, the non-adaptive controller is obtained. 8.1.1 A d a p t i v e C o n t r o l l e r Applying R M L to the differential simplified model of the color/dye process (from E q . 5.41): A{q-*)AC = q-dAB{q-1)ADin + E"{q-1)W4 (8.71) the parameters of this model can be explicitly identified. A (actually A 6 0 ) can be determined from the first terms of AB(q~1). From the other terms and the previously identified value of Ab0, the coefficients of B(q~1) (actually B(q~1) / b0) can also be determined. See Chapter 9 for details of the identification. The determination of the coefficients of A(q~1) is straightforward. Using the explicitly identified parameters, Chapter 8. Dahlin Controllers 53 E q 8.70 becomes: iset H{q-1)B{q-')lb0Din = (l-p)A(q-1)Ab0 (CT" - C) 8.2 Decoupling Using Diagonalized A - 1 (8.72) If the off-diagonal elements of 0jj in Fig . 8.3 are set to zero then Bristol's method is being applied: H{q-*)B(q-1)Din = 0n 0 0 0 02 2 0 0 0 <f>33 This method allows that, in 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. (l-p)A{q-*) (cset-c) (8.73) To apply it, find: (A"1) 011 </»21 031 012 022 032 013 023 033 (8.74) where: A = T h e relative gain matrix T where: An A 1 2 A A 2 1 A 2 2 A A 3 1 A 3 2 A 13 (8.75) 7 n 7i2 7i3 721 722 723 731 732 733 (8.76) Chapter 8. Dahlin Controllers 54 is calculated by letting: Hj = xij<l>ji for i,j = 1,2,3. (8.77) 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 in row i (if jij is not too large). According to Deshpande [7] if some values of relative gain in 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] r the calculated relative gain matrix is: 0.88 0.43 -0.31 0.15 0.37 0.48 -0.02 0.20 0.82 This would indicate that G\ should be controlled with Dye 1 (i.e. the dye represented by the first column). Similarly, it indicates that C 3 should be controlled with Dye 3. This leaves C2 to be controlled with Dye 2 even though two elements in the second row are close in 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] r. This can 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 C2 and Dye 3: an interaction which was ignored by the previous pairing of dyes and colors. C h a p t e r 9 A d a p t i v e P a r a m e t e r I d e n t i f i c a t i o n 9.1 M u l t i - O u t p u t R L S A l g o r i t h m If the initial estimates of the process parameters are in error or if the parameters change in time, a Recursive Least Squares (RLS) algorithm may be useful in identifying more accurate values of the parameters. At 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 in Chapter 5 are 3-dimensional, the R L S algorithm in this application will use the 3-output form: Y = M T 6 + W (9.78) where: ~* d e f • Y = 3 x 1 column vector containing model output d e f • . • M = UQ x 3 matrix containing measured data • 0 =f 7i© X 1 column vector containing model parameters -* d e f • W = 3 x 1 column vector containing the identification/prediction error. The actual contents of these vectors and matrix depend on which process model is used. T h e contents for two alternative models are given in the Subsections 9.1.1 and 9.1.2. The R L S algorithm used in 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 O c where W is the a priori prediction error. 3. G = P M ( M T P M + A ) - 1 —* 4. Update the unconstrained parameter vector ®u using: 0V = ®c + GW 5. A Kalman-type covariance matrix update is used to maintain an almost-constant tra.ce[P] (i.e. 200 <trace[P] < 320). First, calculate trace[P]. Update the scalar a (where 0.1 < a < 1.0) as follows. If trace[P] > 320, then: a * - a - 0 . 2 5 ( a - 0 . 1 1 ) . If trace[P] < 200, then: a <- a + 0.25(1.0 - a). Finally, update the covariance matrix P using the Kalman-type update (i.e. adding the incremental covariance matrix o:P'): P <- ( P - G M T P ) + a P ' 6. Update the constrained parameter vector Qc using Parameter Projection (see Section 9.2): 0 c = 0 g + m i n [ l , B } {QU-QE) || 0C/" — 0£' || Chapter 9. Adaptive Parameter Identification 57 T ~* -* . . . . 7. W = Y — M © 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 an overbrace) used in the adaptive controller (i.e. in the predictor-based controllers, the coefficients of G(q *) and FB(q x)/fb0 and the elements of A 1 / fbQ; in the Dahlin controllers, the coefficients of A(q *) and B(q 1)/&o, and the elements of Ab0). • ®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. in the predictor-based controllers, the coefficients of G(q 1) and FB(q 1)/fb0 and the elements of A 1 /fb0; in the Dahlin controllers, the coefficients of A(q"1) and B(q~1)/b0, and the elements of A 6 0 ) -• I ^ f The identity matrix • G = f n© x 3 gain matrix • P = f n © x 71© covariance matrix • A = Constant 3 x 3 covariance matrix representing an estimate of cSfW'PFr] = I in the simulations. • P ' = Constant n© x n© diagonal matrix = 0.41 in the simulations. Chapter 9. Adaptive Parameter Identification 58 This R L S algorithm uses a Kalman-type covariance matrix update, which is an alternative to the forgetting factor update. The Kalman-type update is useful if cer-tain parameters (for example, the ith parameter) are known to have more variability (i.e. higher variances as represented in 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 ith diagonal element) are increased. In the simulations, P ' contains ng identical elements since it is assumed that all parameters have equal variance. In comparison to the Kalman-type update with its ng degrees of freedom, the forgetting factor update has only one. The variable scalar a keeps P within a reasonable range. If P was too large, then the gain G would be too large and 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 A p p l i e d w i t h the P r e d i c t o r - b a s e d C o n t r o l l e r s As given by E q . 7.56 in Chapter 7, the form of the predictor with minimum prediction horizon to be identified is: deg[FB] q~dADin = - E (fbi/fb0)q-iq-d ADin + (A-l/fb0)(C-q-dC) i=l -q'd(A-l/Jb0)G(q-')AC -(FE'(q-l)/fb0)W3 (9.79) Substituting E ( g _ 1 ) for —(FE'(q~1)/fb0) and W for W3, we obtain: deg[FB] q-dADin = - E (fbi/fb0)q-iq-d ADin + (A-l/fb0)(C-q-dC) i=l -q-d(A-l/ft,0)G(q-')AC + E(q-1)W (9.80) where: Chapter 9. Adaptive Parameter Identification 59 E^q-1) 0 0 • E ( g - 1 ) d ^ f 0 ^ ( g " 1 ) 0 0 0 E3{q~l) ^ ( g - 1 ) f o r i = 1,2,3 = f 1 + £ g f ] e^-g^ Fitting 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 . This 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-dAD2 q~dAD3]T (9.81) Chapter 9. Adaptive Parameter Identification 60 and q~d-1AD2 q~d-1AD3 q-d~2AD1 q-d-2AD2 q'd-2AD3 -fb2/fb0 q-d~3AD1 q-d'3AD2 q-d~3AD3 -fb3/fb0 q - d - A A D i q-d-*AD2 q-d'4AD3 q-d'5ADi q-d~sAD2 q-d~5AD3 -fbjfb0 q'C, - q~dCx 0 0 <f>n/fl>o q°C2 - q'dC2 0 0 $12/fl>0 q°C3 - q-dC3 0 0 013/'ft>0 0 q°C1 - q~dC1 0 02i//&o 0 q°C2 - q~dC2 0 $22/fl>0 0 q°C3 - q~dC3 0 023/A) 0 0 q°Cl - q-dd 031 / fl>0 0 0 q°C2 - q~dC2 032 /flo 0 0 q°C3 - q~dC3 033/ foo Lio L20 L30 -go i n L2\ L31 • -9i L12 I>22 L32 -92 q-lWt 0 0 el,l q-2W, 0 0 e l , 2 0 q~lW2 0 e 2 , l 0 q~2W2 0 e2,2 0 0 q-HV3 e3,i 0 0 q-2W3 e3,2 where, in this example: Chapter 9. Adaptive Parameter Identification 61 • d = 4 • deg[A] = 3 • deg[B] = 2 • deg[F] = d - 1 = 3 • deg[G] = deg[A] - 1 = 2 • deg[FB] = deg[F] + deg[B] = 5 • deg[E) = 2 • ng = 23 and: • q~l X =f Value of X taken I sampling intervals in the past. • (f>ij/fb0 (with i,j = 1,2,3) =f Elements of A - 1 /fb0 • Lij = Zl=1(<t>ik/fboyq-d-j&Ck • {(j>ijlfb0)* = f Previous value of (/>ij/fb0 The prediction error vector used in the R M L is the a posteriori prediction error. 9.1.2 RLS Applied with the Dahlin Controllers As given by E q . 5.41 in Chapter 5, the form of the incremental process model to be identified is: Aiq-^AC = g - d A S ( g - 1 ) A A n + E " ( g - 1 ) # 4 (9.82) Chapter 9. Adaptive Parameter Identification 62 Substituting E (g~ 1 ) for E " ( g _ 1 ) and W for W4, we obtain: Aiq-^AC = q-dAB(q-1)ADin + E{q-l)W (9.83) where: A{q-') d=f l + u r ^ 5 ( 9 - X ) = E%9iB)bjq-j deg[A] E(<T X ) d e f Eiiq-1) 0 0 0 ^ ( ( T 1 ) 0 0 0 E3{q-1) . J ^ , - 1 ) fori = 1,2,3 d=f 1 + ^ ^ Fitting 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 . This 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 in AB(q~1) are identified only once). -.deg\E] Y = [ A d A C 2 A C 3 ] T (9.84) Chapter 9. Adaptive Parameter Identification 63 if"1 A d q-1AC2 q-'ACs ax g - 2 A d q'2AC2 q~2AC3 a2 g - 3 A d q~3AC2 q-ZAC3 03 g - d A A n , i 0 0 A u 6 0 <TdAAn,2 0 0 A1260 q-dADin>3 0 0 A1360 0 <r d AAn,i 0 A 21K 0 q'dADin<2 0 A 22 bo 0 q-dADin>3 0 A 23 ^ 0 0 0 q-dADin>1 —* 0 — A 3 i bo 0 0 q-dADin<2 1 KJ A32&0 0 0 q~dADin<3 A 33 60 Ln L21 L3i 61/60 L22 L32 62/60 0 0 e i , i q-2Wy 0 0 e l ,2 0 q~lW2 0 e 2 , l 0 q~2W2 0 e2,2 0 0 q~'W3 e 3 , i 0 0 q~2W3 63,2 where, in this example: • deg[A] = 3 • deg[B] = 2 • = 2 Chapter 9. Adaptive Parameter Identification 64 • ng = 20 and: • q~l X =f Value of X taken / sampling intervals in the past. • Xijb0 (with i,j = 1,2,3) =f Elements of A&o • Li, d=f EU^ikboTq-d-j ADinik • (Xijbu)* = f Previous value of \ijb0 Again, the prediction error vector used in the R M L is the a posteriori prediction error. 9.2 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 Praly [18] (i.e. —* Parameter Projection) was used. The 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: 0 c = 0 f i + min[ l , ..}{eu-eE) (9.85) II 0 y — 0 £ II where: • Bu = T h e unconstrained parameter vector provided by R L S , • 0 C = f The constrained parameter vector containing the parameters (repre-sented by an overbrace) used in the adaptive controller (i.e. in the predictor-based controllers, the coefficients of G(q~1) and FB(q~1)/fb0 and the elements of A - 1 / f b 0 ; in the Dahlin controllers, the coefficients of A(q~l) and B(q~1)/b0, and the elements of Afe 0 ) ; Chapter 9. Adaptive Parameter Identification 65 • QE == The estimated parameter vector containing the parameters (represented by an overline) derived from the apriori estimated process model (i.e. in the predictor-based controllers, the coefficients of G(q x) and FB(q 1)/fb0 and the elements of A 1 /fb0; in the Dahlin controllers, the coefficients of A(q 1) and B(q 1)/b0, and the elements of A b0), • j3 = f 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# in E q . 9.85 and the resulting controller is non-adaptive. In the simulations of this thesis: —* dim[X] I I ^ H = £ 1 * 1 - (9.86) t=i Chapter 10 Simulations 10.1 Process & Controller Parameters The simulations were carried out on a fiYAX computer at the Pulp & Paper Center at U . B . C . The continuous three dye transport process as modelled in Section 3 and the conversion to the three color output were simulated with an Advanced Continuous Simulation Language ( A C S L ) program. The 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 Ts. For each dye, the dye transport model shown in F i g . 10.4 is simulated pseudo-continuously with an A C S L language program. The " 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 in Appendix B . A l l the transport models share the following parameters: • T\ = 1.25 min. • r 2 = 1.0 min. a Td2 = 1.333 min. • Td = 1.6 or 2.0 or 2.4 min. • K = 0.65 or 0.80 or 0.95 66 Chapter 10. Simulations 67 Din ZOH (r.) Noise l T28+1 W E T E N D P R O C E S S F O U R . F I B R E R E C O V E R Y P R O C E S S 1-11 D R Y E R S 1 V 71 e-Tds T l . + l D r = 0.33 Figure 10.4: Block Diagram of A C S L - S i m u l a t e d Dye Transport Model —* —* In simulation the color C is calculated from D using Eqs. 4.20, 4.22, 4.24, 4.26 in Chapter 4 at 31 points of frequency (i.e. Aj with j — 1 , . . . , 31 ) . The actual absorp-tion/scattering coefficients used in the process simulation ( ^ j i ( A ) , ^jf-(A), ^ ^ ( A ) for the controlled dyes, (A) for the unknown disturbing dye, and -j^-(A) for the undyed fibre) are given in Appendix A . Dyei is a magenta (ie. purple) dye which absorbs in the yellow/green wavelengths. D y e 2 is a yellow/green dye which absorbs at all but the yellow/green wavelengths. D y e 3 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 in the red to yellow range. The fluorescence of the red/yellow dye allows for better contol of lightness L*, since non-fluorescent dyes can only decrease L*. C o n t r o l l e r s : T h e controllers were designed using the estimated discrete T F 9 •5i'?n'^ - This T F corresponds to the continuous dye transport process with estimated parameters: Chapter 10. Simulations 68 • rv = 0.94 min. • f~2~ — 0.75 min. • = 1.0 min. • Td = 2 . 0 min. • K = 0.8 The 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; in the adaptive case f3 = 0.2. Other parameters for each of the controllers were: D a h l i n : • p ( C L discrete time constant) = where Tc = 0.66 min. D e a d b e a t : • p ( C L discrete time constant) = 0.0 G M V C : • p (Weighting factor) =3 .0 • dh = d = 7 for Ts = 0.33 min. • dh = d — 4 for Ts — 0.66 min. Chapter 10. Simulations 69 E H C : • 4 — 10, d — 7 for T, = 0.33 min. • 4 = 6, d = 4 for Tt = 0.66 min. The number of terms containing past prediction errors in the R L S form (i.e. deg[E(q~1)] defined in Section 9.1) was set at zero. It was found that, if deg[E(q~1)] was greater than zero, the improvement in C L performance was small. A l l the controllers used G a i n Scheduling of A as described in Subsection 6.1.1. A was calculated using the method described in Subsection 4.3.1. and the estimated co-efficients n^- (A) , ^ ^ ( A ) , ^^"(A) for the dyes and ^ ( A ) for the undyed fibre (given in Appendix 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 Each controller was tested using a standard 70 min. simulation run. During four of the 10 min. intervals of the simulation run, the squared deviations of the three colors from their respective setpoints were measured and summed. At 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 in the tables of Chapter 11. Initially, when the controller is inserted, the process is in steady state with the color output and setpoint unequal. In the 0 to 10 min. interval, the measured combined color variance indicates the controller's ability to eliminate this initial error. At the 20 min. mark the addition of colored broke is simulated by the addition of a disturbing dye Chapter 10. Simulations 70 (with ^ ^ ( A ) coefficient). In the 20 to 30 min. interval, the measured combined color variance indicates the controller's ability to counteract a step disturbance. At the 40 min. mark, a color setpoint change is made. In the 40 to 50 min. 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 in the 60 to 70 min. interval indicates the performance long after a setpoint change. The dye input and color plots for a typical simulation run are shown in Figure 10.5 and Figure 10.6, respectively. In this run, the process parameters are Td = 2.0 min. and 1Z = 0.80. The controller is a non-adaptive Dahlin controller with T„ = 0.66 min. , de^fvl] = 1, and a C L time constant Tc of 0.66 min. Chapter 10. Simulations 71 I 3KQ Z 3KQ 6 3KQ Figure 10.5: Dye Plots for a Typical Simulation R u n T anoioo z anoioo e anoioo Figure 10.6: Color Plots for a Typical Simulation R u n Chapter 11 Results Twenty-four controllers are tested: 4 controller types (i.e. Dahlin, 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 in two series of simulation runs (one series uses a particular color setpoint sequence; the second series uses the re-versed sequence). Each series contains 8 different subseries in which certain parameters (i.e. Td, 7c, and noise) of the simulated process are varied. The tables in this section report the measured color variances of the 24 controllers for each subseries. A l l of the 24 controllers are therefore tested in 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). The process parameters used in each series of simulation runs are: • Td= 1.6 or 2.0 or 2.4 min. , • K = 0.65 or 0.80 or 0.95, • Measurement and/or process noise added. 73 Chapter 11. Results 74 S I M U L A T I O N S E R I E S 1 Simulation runs with setpoint sequence C = [74.2,3.57,9.88]T C = [74.2,16.0,10.0]T Simulation Subseries l a • No Noise • Td = 2.0 min. • K = 0.80 D a h l i n D e a d b e a t G M V C E H C 13 = 0.0 (3 = 0.2 (3 = 0.0 (3 = 0.2 (3 = 0.0 (3 = 0.2 (3 = 0.0 (3 = 0.2 M i n . W i t h T. = 0.33 min. , deg[A] = 1, deg\B) = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.380 1.380 1.240 1.250 1.420 1.420 1.660 1.660 .3470 .3910 .2730 .2920 .2590 .3780 .3550 .6040 44.90 45.70 41.50 41.10 47.70 62.60 54.60 57.80 .0030 < .001 < .001 < .001 .0473 < .001 .0043 < .001 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.400 1.390 1.290 1.290 1.690 1.690 1.880 1.880 .3860 .4070 .3320 .3340 .4340 .7160 .5400 .8270 45.50 45.90 42.90 42.70 54.90 57.50 61.00 58.70 .0042 < .001 .0020 < .001 .0081 .0026 .0138 .0049 M i n . W i t h Ta = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 1.390 1.640 1.280 1.490 1.650 1.650 1.720 1.720 .4190 .4810 .3550 .4080 .3400 .5820 .3510 .5660 44.50 51.40 42.30 55.10 54.50 73.90 56.20 60.30 < .001 .3010 < .001 .6690 .0012 .0150 < .001 < .001 Table 11.1: Simulation Subseries l a Chapter 11. Results Simulation Subseries l b 75 • No Noise • Td = 2.0 min. • 11 = 0.65 D a h l i n D e a d b e a t G M V C E H C 3 = 0.0 3 = 0.2 3 = 0.0 3 = 0.2 3 = 0.0 3 = 0.2 3 = 0.0 3 = 0.2 M i n . W i t h Ts = 0.33 min. , deg[A] = 1, deg[B] = 0 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 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 -- 10 1.540 1.530 1.420 1.420 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 M i n . W i t h Ta = 0.66 min. , 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 Simulation Subseries l c 76 • No Noise • Td = 2.0 min. • K = 0.95 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h Ta = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.300 1.300 1.190 1.190 1.330 1.330 1.550 1.550 .3980 .4160 .3140 .3170 .3010 .4350 .4040 .6430 41.60 44.30 39.30 42.60 44.60 115.0 50.50 64.00 < .001 < .001 < .001 < .001 .0017 < .001 < .001 < .001 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.310 1.310 1.230 1.230 1.570 1.570 1.750 1.750 .4450 .4560 .3850 .3850 .4880 .7400 .5780 .8620 42.30 45.10 40.50 45.40 50.60 63.10 56.50 53.00 < .001 < .001 < .001 < .001 < .001 < .001 < .001 < .001 M i n . W i t h Ts = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 1.310 1.580 1.230 1.450 1.550 1.550 1.610 1.610 .4620 .5350 .3980 .4650 .4040 .5740 .4020 .6360 41.90 52.70 40.50 57.30 50.90 67.50 52.10 55.70 < .001 .4330 < .001 .7030 < .001 < .001 < .001 < .001 Table 11.3: Simulation Subseries l c Chapter 11. Results Simulation Subseries Id 77 • No Noise • Td = 1.6 min. • 11 = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h Ts = 0.33 min. , deg[A] = 1, deg[B) = 0 0 - 1 0 20 - 30 4 0 - 5 0 60 - 70 1.170 1.180 1.230 1.240 1.620 1.490 1.480 1.480 .2930 .3000 14.30 5.390 23.50 .3200 .3060 .5630 38.00 38.30 126.0 53.80 121.0 50.60 48.20 44.20 < .001 < .001 64.00 25.30 247.0 1.350 .0013 < .001 M i n . W i t h Ta = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.190 1.190 1.170 1.160 1.510 1.510 1.720 1.720 .3250 .3300 2.4.80 4.210 .3780 .6590 .4780 .7830 38.50 38.60 54.90 49.60 49.20 46.00 55.90 51.80 .0012 .0047 62.60 95.90 .0024 < .001 .0053 .0019 M i n . W i t h T, = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 40 - 50 60 - 70 1.220 1.510 1.170 1.410 1.470 1.470 1.570 1.570 .3620 .4430 1.480 1.250 .2740 .4910 .2940 .5090 39.50 43.50 73.00 72.70 47.90 45.30 51.00 45.40 < .001 .4770 78.80 176.0 < .001 < .001 .0041 < .001 Table 11.4: Simulation Subseries Id Chapter 11. Results Simulation Subseries le 78 • No Noise • Td = 2.4 min. • K = 0.80 D a h l i n D e a d b e a t G M V C E H C 3 = 0.0 j3 = 0.2 (3 = 0.0 (3 = 0.2 3 = 0.0 (3 = 0.2 3 = 0.0 3 = 0.2 M i n . W i t h T, = 0.33 min. , deg[A] = 1, deg[B) = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.650 1.630 1.700 1.750 1.800 1.770 1.880 1.880 .4140 .4420 10.50 4.090 22.80 .8170 .4080 .6180 54.20 50.80 110.0 98.50 189.0 79.80 63.60 96.90 .0267 .0486 98.20 518.0 670.0 15.60 .0270 .0493 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.650 1.640 1.600 1.570 1.900 1.900 2.060 2.060 .4380 .4630 .4340 .4800 .4730 .7210 .5880 .8330 54.90 59.30 55.60 79.00 61.60 97.50 66.80 82.00 .0327 .0221 .4400 1.090 .0359 .2150 .0504 .0292 M i n . W i t h T . = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 1.590 1.780 1.520 1.660 1.890 1.890 1.920 1.920 .4700 .5020 .4370 .5430 .4060 .5880 .4040 .5930 50.80 54.30 48.90 54.00 63.70 101.0 63.90 99.20 < .001 .4840 .4280 1.160 .0232 .8060 .0049 < .001 Table 11.5: Simulation Subseries le Chapter 11. Results Simulation Subseries If 79 • Measurement Noise Only • Td = 2.0 min. • K = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 8 = 0.2 8 = 0.0 0 = 0.2 0 = 0.0 8 = 0.2 0 = 0.0 3 = 0.2 M i n . W i t h T, = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 10 20 - 30 40 - 50 60 - 70 2.270 2.360 2.430 2.460 2.770 2.860 2.560 2.570 1.250 1.300 1.710 1.760 2.110 1.700 1.270 1.450 45.90 57.10 43.10 51.10 49.00 71.40 55.90 60.20 .9200 1.290 1.350 1.490 2.090 1.560 .9250 1.010 M i n . W i t h Ts = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.300 2.320 2.320 2.350 2.590 2.590 2.770 2.770 1.270 1.330 1.390 1.450 1.290 1.430 1.350 1.480 46.90 57.30 44.50 53.60 56.20 73.40 62.10 63.70 1.020 1.110 1.230 1.290 .9670 1.190 .9080 .9660 M i n . W i t h Ta = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 40 - 50 60 - 70 2.290 2.540 2.280 2.520 2.660 2.660 2.740 2.740 1.290 1.340 1.390 1.490 1.290 1.350 1.320 1.400 45.80 48.50 44.20 49.90 56.10 68.30 57.60 58.00 .9740 .9890 1.170 1.180 1.130 1.140 1.120 1.460 Table 11.6: Simulation Subseries If Chapter 11. Results Simulation Subseries l g 80 • Process Noise Only • Td = 2.0 min. • 11 = 0.80 D a h l i n D e a d b e a t G M V C E H C a = o.o a = 0.2 a = o.o 3 = 0.2 3 = 0.0 3 = 0.2 3 = 0.0 3 = 0.2 M i n . W i t h TB = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.190 2.410 2.040 2.200 2.380 2.460 2.520 2.510 1.110 1.330 1.130 1.250 1.400 1.250 1.110 1.270 47.00 66.30 43.20 63.30 51.00 87.30 57.20 68.80 1.640 2.310 1.720 2.320 2.100 2.280 1.590 1.580 M i n . W i t h T. = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 10 20 - 30 40 - 50 6 0 - 7 0 2.200 2.200 2.090 2.090 2.530 2.530 2.710 2.710 1.130 1.220 1.120 1.190 1.160 1.490 1.280 1.620 47.50 56.70 44.60 50.20 57.00 75.20 63.20 70.30 1.650 1.850 1.680 1.900 1.610 1.750 1.520 1.700 M i n . W i t h T, = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 2.140 2.290 2.020 2.150 2.540 2.540 2.560 2.560 1.150 1.180 1.120 1.170 1.110 1.300 1.110 1.370 45.80 48.60 43.30 48.90 56.80 70.60 58.40 62.30 1.440 1.360 - 1.480 1.470 1.700 1.730 1.560 1.360 Table 11.7: Simulation Subseries l g Chapter 11. Results 81 Simulation Subseries l h • Measurement Noise & Process Noise • Td = 2.0 min. • K = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h Tg = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 10 2 0 - 3 0 40 - 50 60 - 70 3.080 3.340 3.190 3.420 3.670 3.690 3.400 3.410 2.050 2.430 2.610 2.850 2.890 2.690 2.050 2.110 47.90 61.60 44.50 50.20 51.10 79.60 58.60 68.50 2.470 4.020 2.960 3.740 3.850 3.560 2.420 2.330 M i n . W i t h Ts = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 3.100 3.100 3.090 3.110 3.430 3.430 3.590 3.590 2.020 2.050 2.190 2.260 2.020 2.180 2.090 2.200 49.00 65.00 46.10 60.20 58.40 80.90 64.40 69.70 2.480 3.020 2.670 3.150 2.400 2.750 2.260 2.290 M i n . W i t h T8 = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 4 0 - 5 0 60 - 70 3.040 3.200 3.000 3.170 3.520 3.520 3.570 3.570 2.020 2.060 2.160 2.220 2.080 2.160 2.090 2.160 47.00 48.40 44.90 49.00 58.60 68.00 59.80 61.10 2.240 2.060 2.430 2.240 2.600 2.880 2.460 2.320 Table 11.8: Simulation Subseries l h Chapter 11. Results 82 S I M U L A T I O N S E R I E S 2 Simulation runs with setpoint sequence C^ = [74.2,16.0,10.0] r =4> C"^ = [74.2,3.57,9.88]T Simulation Subseries 2a • No Noise • Td = 2.0 min. • U = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h Te = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 4 0 - 5 0 60 - 70 2.320 2.300 2.070 2.070 2.390 2.390 2.810 2.810 .9440 1.020 .7400 .7720 .6980 1.290 .9730 1.910 43.00 45.40 40.30 41.60 43.90 42.80 50.90 49.60 .0034 .0013 < .001 .0017 .0789 < .001 .0033 < .001 M i n . W i t h T„ = 0.66 min. , deg[A\ = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.340 2.340 2.160 2.160 2.850 2.850 3.170 3.170 1.050 1.090 .9000 .9250 1.190 2.280 1.470 2.510 43.30 45.30 40.50 41.30 51.90 52.00 57.70 56.60 .0044 < .001 .0021 < .001 .0047 .0080 .0103 .0055 M i n . W i t h T„ == 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 40 - 50 60 - 70 2.320 2.770 2.120 2.490 2.790 2.790 2.900 2.900 1.140 1.500 .9630 1.260 .9220 1.950 .9550 2.170 43.60 49.20 40.50 45.00 50.70 51.00 52.90 49.30 < .001 .2750 < .001 .1720 < .001 < .001 < .001 < .001 Table 11.9: Simulation Subseries 2a Chapter 11. Results Simulation Subseries 2b 83 • No Noise • Td = 2.0 min. • 11 = 0.65 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h Ta = 0.33 min. , deg[A) = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.580 2.520 2.300 2.280 2.650 2.640 3.080 3.080 .8420 .9960 .6340 .7080 .6090 1.050 .9120 1.620 46.60 49.40 43.70 46.90 47.30 46.90 55.30 55.90 .0558 .0086 .0285 .0136 .7340 .0168 .0541 .0387 M i n . W i t h Ts = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 10 20 - 30 40 - 50 60 - 70 2.600 2.590 2.410 2.410 3.130 3.130 3.480 3.480 .9510 1.010 .7780 .8030 1.160 1.930 1.440 2.190 47.00 48.10 43.90 44.00 56.80 57.60 63.60 63.10 .0675 .0178 .0454 .0101 .0861 .1800 .1310 .0975 M i n . W i t h T. = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 4 0 - 5 0 60 - 70 2.570 3.030 2.350 2.710 3.060 3.060 3.180 3.180 1.070 1.330 .8890 1.080 .8000 1.620 .8760 1.730 47.70 53.00 44.10 47.90 54.80 54.40 57.40 53.00 .0142 .1700 .0051 .2450 .0291 .0040 .0116 < .001 Table 11.10: Simulation Subseries 2b Chapter 11. Results Simulation Subseries 2c 84 • No Noise • Td = 2.0 min. • 1Z = 0.95 Dahlin Deadbeat G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h T, = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 2 0 - 3 0 40 - 50 60 - 70 2.160 2.150 1.960 1.960 2.220 2.220 2.600 2.600 1.070 1.100 .8480 .8560 .8130 1.640 1.090 2.260 40.70 42.50 38.20 39.90 41.60 41.40 47.80 46.00 < .001 < .001 < .001 < .001 .0033 < .001 < .001 < .001 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.170 2.170 2.030 2.030 2.630 2.630 2.940 2.940 1.200 1.210 1.040 1.040 1.310 2.730 1.550 2.910 41.00 42.70 38.60 39.60 48.50 51.40 53.70 54.60 < .001 < .001 < .001 < .001 < .001 < .001 < .001 < .001 M i n . W i t h TB = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 40 - 50 60 - 70 2.170 2.640 2.010 2.420 2.600 2.600 2.700 2.700 1.240 1.660 1.070 1.420 1.080 1.910 1.080 2.390 41.30 47.00 38.60 43.80 47.80 55.60 49.70 52.30 < .001 .4690 < .001 .2900 < .001 < .001 < .001 < .001 Table 11.11: Simulation Subseries 2c Chapter 11. Results Simulation Subseries 2d 85 • No Noise • Td = 1.6 min. • K = 0.80 Dahlin Deadbeat G M V C E H C 3 = 0.0 3 = 0.2 3 = 0.0 3 = 0.2 3 = 0.0 3 = 0.2 3 = 0.0 8 = 0.2 M i n . W i t h Ts = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.970 2.000 2.070 2.070 2.770 2.550 2.500 2.500 .8000 .7080 44.50 16.50 122.0 .9340 .8370 1.810 36.30 33.60 227.0 60.90 134.0 40.10 45.40 45.00 < .001 < .001 106.0 33.50 53.80 .0209 < .001 < .001 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 1.990 2.010 1.970 1.940 2.550 2.550 2.910 2.910 .8870 .7510 4.200 7.670 1.030 2.200 1.290 2.470 36.80 33.70 116.0 116.0 46.40 47.70 52.60 54.20 < .001 < .001 8.860 8.010 .0015 < .001 .0058 .0015 M i n . W i t h Tt = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 2.050 2.560 1.960 2.370 2.490 2.490 2.650 2.650 .9800 1.320 2.720 2.720 .7450 1.870 .7970 1.920 38.00 41.90 199.0 166.0 45.20 45.80 48.20 47.20 < .001 .4400 174.0 217.0 < .001 < .001 .0037 < .001 Table 11.12: Simulation Subseries 2d Chapter 11. Results Simulation Subseries 2e 86 • No Noise • Td= 2.4 min. • n = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 3 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h T, = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 2 0 - 3 0 40 - 50 60 - 70 2.790 2.760 3.090 3.280 3.120 3.040 3.190 3.190 1.100 .9530 25.50 31.50 95.10 2.950 1.110 1.960 50.40 47.20 235.0 418.0 339.0 61.30 57.00 59.40 .0238 .0034 136.0 251.0 425.0 .9960 .0200 .0183 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.800 2.770 2.710 2.090 3.210 3.210 3.470 3.470 1.180 1.130 1.120 1.130 1.310 2.140 1.620 2.400 50.70 47.60 47.40 53.00 58.10 58.90 63.30 62.20 .0311 .0075 .8790 1.630 .0254 .0101 .0280 .0219 M i n . W i t h Ta = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 4 0 - 5 0 60 - 70 2.670 2.980 2.540 2.780 3.210 3.210 3.230 3.230 1.270 1.550 1.150 1.650 1.080 1.910 1.090 2.190 50.00 51.70 46.90 58.50 57.60 58.50 58.50 55.90 < .001 .5030 1.030 6.140 .0115 .0021 .0025 < .001 Table 11.13: Simulation Subseries 2e Chapter 11. Results Simulation Subseries 2f 87 • Measurement Noise Only • Td = 2.0 min. • K = 0.80 D a h l i n D e a d b e a t G M V C E H C 3 = 0.0 3 = 0.2 8 = 0.0 8 = 0.2 3 = 0.0 3 = 0.2 8 = 0.0 8 = 0.2 M i n . W i t h TB = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 6 0 - 7 0 3.220 3.350 3.240 3.330 3.720 3.830 3.700 3.710 1.790 1.800 2.140 2.180 2.850 2.290 1.840 2.490 44.20 46.30 42.50 43.20 45.20 48.70 52.30 54.40 .8900 1.070 1.220 1.320 1.790 1.340 .8960 .9450 M i n . W i t h Ts = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 10 20 - 30 40 - 50 60 - 70 3.230 3.250 3.180 3.220 3.740 3.740 4.060 4.060 1.810 1.800 1.860 1.890 1.940 2.560 2.190 2.720 44.20 45.20 41.90 42.70 52.70 56.70 58.60 59.40 .9700 1.050 1.150 1.250 .9230 .9310 .8840 .8780 M i n . W i t h T„ = 0.66 min. , deg[A] = 3, deg[B) = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 3.220 3.700 3.140 3.560 3.780 3.780 3.920 3.920 1.910 2.040 1.930 2.130 1.760 1.990 1.830 2.040 44.60 47.00 41.90 44.10 51.60 55.20 53.90 51.30 .9340 .9530 1.100 1.140 1.060 1.030 1.050 1.300 Table 11.14: Simulation Subseries 2f Chapter 11. Results Simulation Subseries 2g 88 • Process Noise Only • Td = 2.0 min. • % = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h T, = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 10 20 - 30 40 - 50 60 - 70 3.460 3.730 3.260 3.480 3.730 3.830 3.870 3.850 2.060 2.480 2.050 2.230 2.410 2.500 2.050 2.680 43.60 47.00 40.90 42.80 44.40 51.40 51.20 55.70 1.090 1.410 1.130 1.440 1.370 1.480 1.040 1.040 M i n . W i t h Ts = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 4 0 - 5 0 60 - 70 3.450 3.470 3.300 3.320 3.870 3.870 4.060 4.060 2.130 2.350 2.080 2.300 2.200 3.060 2.390 3.220 44.30 45.00 41.60 42.00 52.50 60.20 58.10 63.70 1.110 1.260 1.120 1.200 1.030 1.090 .9070 .8950 M i n . W i t h T, = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 40 - 50 60 - 70 3.280 3.400 3.120 3.220 3.900 3.900 3.890 3.890 2.130 2.250 2.050 2.190 2.090 2.670 2.050 2.850 44.50 48.00 41.50 44.40 51.60 58.60 53.70 56.40 .9290 .8170 .9620 .8810 1.120 1.230 1.010 .9980 Table 11.15: Simulation Subseries 2g Chapter 11. Results 89 Simulation Subseries 2h • Measurement Noise & Process Noise • Td = 2.0 min. • U = 0.80 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h Ts = 0.33 mm. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 4.380 4.780 4.400 4.700 5.010 4.910 4.800 4.810 2.910 3.570 3.390 3.600 3.870 3.830 2.920 3.150 44.70 46.60 43.10 43.30 46.20 52.40 52.60 53.90 1.950 2.400 2.220 2.480 3.030 3.060 1.910 1.860 M i n . W i t h T, = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 4 0 - 5 0 60 - 70 4.370 4.390 4.340 4.330 4.790 4.790 4.970 4.970 2.820 2.830 2.950 3.000 2.890 3.470 3.070 3.390 45.20 46.30 42.70 43.40 53.30 57.30 58.90 62.20 1.990 2.280 2.110 2.390 1.890 2.060 1.730 1.640 M i n . W i t h Tg = 0.66 min. , deg[A] = 3, deg[B] = 2 0 - 10 20 - 30 4 0 - 5 0 60 - 70 4.210 4.390 4.160 4.320 4.910 4.910 4.940 4.940 2.850 2.900 2.940 3.070 2.840 3.170 2.850 3.010 45.40 48.60 42.70 45.80 52.50 56.60 54.60 54.70 1.770 1.650 1.920 1.840 2.040 2.090 1.940 1.970 Table 11.16: Simulation Subseries 2h Chapter 11. Results 11.1 P e r f o r m a n c e S u m m a r y 90 D a h l i n D e a d b e a t G M V C E H C 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 0 = 0.0 0 = 0.2 M i n . W i t h T3 = 0.33 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.340 2.418 2.298 2.377 2.635 2.624 2.714 2.714 1.104 1.227 6.929 4.661 17.62 1.525 1.122 1.603 44.93 48.24 75.28 73.96 84.27 61.97 53.79 59.01 .5715 .7863 25.93 52.54 88.24 1.955 .5620 .5561 M i n . W i t h T„ = 0.66 min. , deg[A] = 1, deg[B] = 0 0 - 1 0 20 - 30 40 - 50 60 - 70 2.352 2.354 2.269 2.270 2.740 2.740 2.975 2.975 1.148 1.175 1.471 1.835 1.230 1.810 1.403 1.950 45.46 48.17 49.34 53.00 54.26 62.65 60.18 61.86 .5910 .6653 5.177 7.365 .5698 .6401 .5377 .5413 M i n . W i t h T, = 0.66 min. , deg[A] = 3, deg[B) = 2 0 - 1 0 20 - 30 40 - 50 60 - 70 2.313 2.612 2.206 2.457 2.739 2.739 2.811 2.811 1.197 1.345 1.336 1.464 1.094 1.539 1.113 1.629 44.99 49.26 54.94 58.57 53.74 62.51 55.59 57.87 .5198 .7238 16.46 25.72 .6100 .6849 .5747 .5884 Table 11.17: Performance Summary C h a p t e r 12 Discussion As expected for all the controllers, the largest color variances occur in the 40-50 min. interval after the color setpoint change and the smallest occur in the 60-70 min. steady-state interval . The Summary Table (Table 11.17) shows that adaptive controllers perform worse on average than their non-adaptive counterparts. The exception is the improvement that the adaptive Deadbeat controller with Ts = 0.33 min. , ete(7[A] = 1 shows over its non-adaptive counterpart. The reason can be seen in Tables 11.4 and 11.12 where both controllers were destabilized by a low ^=1 .6 , but the adaptive controllers had lower color variances. In general, the lack of improvement shown by the non-adaptive controller in the 40-50 min. 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. The Summary Table shows that, in general, the predictor-based controllers do worse than the Dahlin/Deadbeat controllers. In the 20-30 min. and 60-70 min. inter-vals , the predictor-based controllers sometimes show the same variances as the best Dahlin/Deadbeat controllers in the same T„ and ete(jr[yl] category (eg. the non-adaptive E H C controllers in the 60-70 min. interval with Ta = 0.66 min. and cZe(?[/4] = 1). Inspecting the Summary Table, we see that the performance of the non-adaptive 91 Chapter 12. Discussion 92 G M V C and E H C for T, = 0.66 min. are similar. This makes sense if we note the sim-ilarities in the structures of the two controllers (i.e. Eqs. 7.59 and 7.63) where the de-nominator {1+p) of the G M V C corresponds to the denominator (1 +5Zf=o d _ 1 foil f°dh-d) 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 from a setpoint (i.e. makes quick setpoint changes). This may be the reason the predictor-based controllers do not compare well with the Dahlin/Deadbeat controllers. The predictor-based were made integrating controllers by calculating F(q~1) and G(q~1) using the <itjfferera<ia/Diopha.ntine E q . 5.45. The resulting controllers are sensitive to perturbation unless their denominators are quite large (in fact, p > 0 is necessary for stability for the G M V C ) . Therefore, the added complexity of the predictor-based controllers (i.e. solving the Diophantine Eq.) is not warranted for a process in which the dynamics are simple to control. Except when Td = 1.6 or 2.4 min. , the Deadbeat controllers give lower variances than the Dahlin controllers , as expected. But the Deadbeat controllers are less robust in the face of deviations in Td- Noting that the table entries with large variances indicate an unstable controller, we see that a low Td (i.e. Tables 11.4 and 11.12) causes instability in all the Deadbeat controllers, while all the Dahlin controllers continue to do well. For a high Td (i.e. Tables 11.5 and 11.13) it's interesting to note that only the Deadbeat controller with T„ = 0.33 min. is destabilized. Not only do the Deadbeat controllers with T„ = 0.66 min. remain stable but the Deadbeat controller with TB = 0.66 min. and rfe^[yl] = 3 does better than its Dahlin counterpart. Both low and high Td have little effect on the stability of the Dahlin controllers which do better than their Deadbeat counterparts except for the case just mentioned. At steady state (60-70 min. Chapter 12. Discussion 93 variance) the Deadbeat controllers also show their sensitivity by giving higher variances than the Dahlin controllers when noise is added (Tables 11.6-8 and 11.14-16). In general, the Summary Table shows that, for T„ = 0.66 min. , the model degree deg[A] has little effect on performance, especially with the Dahlin controllers. In other words, the added complexity of the desr[.A] = 3 transport model brings little improvement over the <ie(/[.4] = 1 model. The reason may be that, even if the dec/fA] is increased, the model A(q~l) and B(q~l) is still generated from estimated (i.e. T[, etc.), not exact, parameters . That model accuracy is not critical for good control is illustrated in Tables 11.2 and 11.10 (where 1Z = 0.65) and in Tables 11.3 and 11.11 (where K = 0.95). Even with this large change in 1Z both the Dahlin and Deadbeat controllers provide good control. Because of the easy controllability of the stable, minimum-phase dye transport pro-cess, it seems that any reasonable non-adaptive Dahlin controller with G a i n Scheduling —*8et (i.e. A calculated using the optical properties of the dyes and C ) gives good C L per-formance. C h a p t e r 13 C o n c l u s i o n s The adaptive controllers perform worse on average than their non-adaptive counter-parts. 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. As well, the Summary Table shows that in general the predictor-based controllers do worse than the Dahlin/Deadbeat controllers. Therefore, the added complexity of the predictor-based controllers (i.e. solving the Diophantine Eq.) is not warranted. Increasing the order of the dye transport model from rfeg[A] = 1 to 3 brings little improvement in C L performance since both models are generated from 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 in its possible range. Because of the easy controllability of the stable, minimum-phase dye transport process, it seems that anon-adaptive Dahlin/Deadbeat controller with Gain Scheduling gives good performance. The non-adaptive Deadbeat performs best if Td is known accurately . Otherwise, the non-adaptive Dahlin controllers with C L time constant Tc = 0.66 min. give good robust control and that with Ts = 0.66 min. and tZe^[A] =1 gives good control with 94 Chapter 13. Conclusions 95 simple implementation. A good estimate of the process gain A (from which the controller gain A 1 is calcu-lated) is important for good control. If the gain A is calculated at each desired paper color C 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. Even 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. Din ) of the corresponding dyes are well known, then open-loop control, as described in Section 6.2, should be considered for making efficient color setpoint changes. Bibliography [1] Alderson, J . V . , E . Atherton, A . N . Derbyshire. Modern Physical Techniques in Colour Formulation. Journal of the Society of Dyers & Colourists, 77:p.657 (1961) [2] Belanger, P.R. A Paper Machine Color Control System Design Using Modern Techniques. IEEE Transactions on Automatic Control, AC-14:p.610 (1969) [3] Billmeyer, F . W . , M . Saltzman. Principles of Color Technology, John Wiley & Sons, New York (1981) [4] Bonham, J.S. Fluorescence & Kubelka-Munk Theory. COLOR, research & appli-cation, 11 (1986) [5] Chao, H . , W . Wickstrom. The Development of Dynamic Color Control on a Paper Machine. Automatica, 6:p.5 (1970) [6] Clapperton, R . H . Modern Paper-Making. Basil Blackwell L t d . , Oxford (1952) [7] Deshpende P .B . , R . H . A s h , Elements of Computer Process Control. ISA, Research Triangle Park, N C (1981) [8] Elliot , H . , W . A . Wolovich. Parameterization Issues in Multivariable Adaptive Con-trol. Automatica, 20:p.533 (1984) [9] Goodwin, G . C . , K . S . Sin. Adaptive Filtering, Prediction, & Control. Prentice-Hall, Englewood Cliffs (1984) [10] Jordan, B . D . , M . O'Neil l . The Case for Switching to the C I E L* a* b* Colour Description for Paper. Miscellaneous Report of PPRIC, M R 106 (1987) 96 [11] Koivo, H . N . A Multivariable Self-Tuning Controller. Automation, 16:p.351 (1980) [12] Lebeau, B . , J .P. Vincent, A . Ramaz. On-line Color Control System: A Case Study on an Experimental Paper Machine. The 3rd I F A C / P R P Conference (PRP3) , Brussels, Belgium (1976) [13] Lehtoviita, M . New On-line Colour and Brightness Measurement and Control Sys-tem. The 5th I F A C / P R P Conference (PRP5) , Antwerp, Belgium (1983) [14] MacAdarn , D . L . Color Measurement, V o l . 27, Springer-Verlag Series in Optics, Springer-Verlag, (1985) [15] Marchi , R. Latest Developments Allow On-line Color and Shade Measurement and Control. Pulp & Paper, 60:p.l24 (1986) [16] M c G i l l , R . J . Chapter 18 in Measurement and Control in Papermaking. A d a m Hilger L t d . , Bristol (1980) [17] P I T A Engineering Technology Working Group, On-line Colour Control. Paper Technology and Industry, 24:p.l7 (1983) [18] Praly, L . Robustness of Indirect Adaptive Control based on Pole-Placement De-sign. The 1st I F A C Workshop on Adaptive Systems in Control and Signal Pro-cessing, San Francisco (1983) [19] Quinn, M . Automatic Colour Control on the Paper Machine. Paper Technology, 9:p.317 (1968) [20] Sandraz, J .P. Identification et Commande 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 and Control: The Present State of the Ar t . Conference of the Swedish Association of Pulp & Paper Engineers, Stockholm (1984) [22] T o d d , K . L . , V . Holding, C . Freel, The Control of Colour on a Paper Machine. Conference of the Swedish Association of Pulp & Paper Engineers, Stockholm (1984) [23] Ydstie, B . E . Theory and Application of an Extended Horizon Self-Tuning Con-troller. AIChE Journal, 31:p.l771 (1985) 98 A p p e n d i c e s 99 Appendix A -g* and of Simulated Dyes and Undyed Paper 100 ndix A. -j* and of Simulated Dyes and Undyed Paper K A [nm 1 Kr,2 •Kt,2 Kj_ [iiiii.j St St sf St St St St St s, 400 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 410 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 420 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 430 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 440 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 450 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 460 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 470 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 480 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 490 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 500 0.00 0.00 0.20 0.25 0.20 0.25 0.19 0.24 0.00 510 0.20 0.25 0.00 0.00 0.20 0.25 0.19 0.24 0.10 520 0.20 0.25 0.00 0.00 0.20 0.25 0.19 0.24 0.10 530 0.20 0.25 0.00 0.00 0.20 0.25 0.19 0.24 0.10 540 0.20 0.25 0.00 0.00 0.20 0.25 0.19 0.24 0.10 550 0.20 0.25 0.00 0.00 0.20 0.25 0.19 0.24 0.10 560 0.20 0.25 0.00 0.00 -.20 -.15 0.19 0.24 0.10 570 0.20 0.25 0.00 0.00 -.20 -.15 0.19 0.24 0.10 580 0.20 0.25 0.00 0.00 -.20 -.15 0.19 0.24 0.10 590 0.20 0.25 0.00 0.00 -.20 -.15 0.19 0.24 0.10 600 0.20 0.25 0.00 0.00 -.20 -.15 0.19 0.24 0.10 610 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 620 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 630 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 640 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 650 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 660 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 670 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 680 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 690 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 0.20 700 0.00 0.00 0.20 0.25 -.20 -.15 0.19 0.24 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 " D 0 1 = I N T E G ( ( D I ( 1 ) + D W 1 - D 0 1 +NO(7)*GAUSS(0.0,1 .0)) / T 1 , D 0 1 I C ) D 0 2 = I N T E G ( ( D I ( 2 ) + D W 2 - D 0 2 +NO(8)*GAUSS(0.0 ,1 .0)) / T 1 , D 0 2 I C ) D 0 3 = I N T E G ( ( D I ( 3 ) + D W 3 - D 0 3 +NO(9)*GAUSS(0.0 ,1 .0)) / T 1 , D 0 3 I C ) D 0 4 = I N T E G ( ( D 1 S T + D W 4 - D 0 4 ) / T 1 , D 0 4 I C ) " T i m e Delay in Dryers" D D 0 1 = D E L A Y ( R E T ( 1 ) * D 0 1 , D D 0 1 I C , T D 1 1 , 3 0 0 ) D D 0 2 = D E L A Y ( R E T ( 2 ) * D O 2 , D D O 2 I C , T D 1 2 , 3 0 0 ) D D 0 3 = D E L A Y ( R E T ( 3 ) * D O 3 , D D O 3 I C , T D 1 3 , 3 0 0 ) D D 0 4 = D E L A Y ( R E T ( 3 ) * D O 4 , D D O 4 I C , T D 1 4 , 3 0 0 ) " T i m e Delay in Fibre Recovery Process" D D W 1 = D E L A Y ( ( l . - R E T ( l ) ) * D O l , D D W l I C , T D 2 1 , 5 0 0 ) D D W 2 = D E L A Y ( ( l . - R E T ( 2 ) ) * D O 2 , D D W 2 I C , T D 2 2 , 5 0 0 ) D D W 3 = D E L A Y ( ( l . - R E T ( 3 ) ) * D O 3 , D D W 3 I C , T D 2 3 , 5 0 0 ) D D W 4 = D E L A Y ( ( l . - R E T ( 3 ) ) * D O 4 , D D W 4 I C , T D 2 4 , 5 0 0 ) "Fibre Recovery Process" D W 1 = I N T E G ( ( D D W 1 - D W 1 ) / T 2 , D W 1 I C ) D W 2 = I N T E G ( ( D D W 2 - D W 2 ) / T 2 , D W 2 I C ) D W 3 = I N T E G ( ( D D W 3 - D W 3 ) / T 2 , D W 3 I C ) D W 4 = I N T E G ( ( D D W 4 - D W 4 ) / T 2 , D W 4 I C ) 102 

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