Design of Filter-Based Controllers for Cross-Direction ... - TSpace

Design of Filter-Based Controllers for Cross-Direction Control on Paper Machines

Andrew John Thake

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Department of Chernical Engineering and Applied Chemistry

University of Toronto

O Copyright by Andrew John Thake (1997)

1*1

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Design of Filter-Based Controllen for Cross-Direction Control on Paper Machines Master of Applied Science. 1997 Andrew John Thake School of Graduate Studies. Department of Chernical Engineering and Applied Chemistry University of Toronto

Abstract Cross-Directional control has been studied extensively by both academia and industry resulting in a large body of literature. However, there are few instances of industrial implementation of these advanced CD control designs due to limitations in the existing control hardware platforms. In order to address this issue, a set of methods for approximating the behaviour of advanced CD controllers with an F R filter has been developed. These controller approximation methods use either Least Squares or Semidefinite Programming. In addition to open-loop controller approximations, a closedloop approximation method has been developed that uses Successive Semidefinite Programming. A set of performance measures for assessing the performance degradation due to the filter approximation has been formulated. Resdts indicate that there is linle performance or robustness degradation due to the filter approximation. These controller approximation techniques are formulated such that these methods have wider applicability to a range of controller approximation problems.

Acknowledgments 1 woufd like to thank Dr. Cluett and his group for adopting me following my supervisor's migration to Edmonton. 1 wouid like to thank the other original member of the group, Aseema.

1 would like to thank my supervisors: Dr. Fraser Forbes (University of Alberta) and Dr.

lim McLeiIan (Queen's University). The quality of this thesis is a direct reflection of the ceaseless interest exhibited by them. 1 woufd like to acknowledge the support of NSERC during my time at the University of

Toronto. Although the 'guys' fkom Queen's (Boyd, Doug, Eric, Ryan and Sean) were not directly involved in my thesis work, I would like to acknowledge their hospitality during my fiequent trips to Kingston. Throughout my years in school, my family (Murn, Dad and James) has aiways pushed me to put forth an honest effort. This thesis is an example of the results of their influence.

Last but not least, I would like to thank my fiancee. Sarah. for her constant encouragement, understanding, love and. above ail. patience.

iii

Table of Contents Abstract Acknowledgments Table of Contents List of Figures Nomenclature 1

Introduction 1.1 Motivation 1.2 Thesis Outline 1.3 Thesis Contributions

2

Cross Directional Control Design The Papemaking Process Profile Measurement CD Profile Modeling 2.3.1 Empirical Identification Methods 2.3.1.1 Steady-State CD Models 2.3.1-2 MDICD Models 2.3.2 Advanced Modeling Methods 2.3 2.1 Advanced CD Models 2.3.2.2 M D K D Coupled Models Existing CD Contro 1 Designs 2.4.1 Off-Line Methods 2.4.2 On-Line Methods 2.4.3 Sumrnary of CD Control Designs Benchmark CD Controller Design 2.5.1 CD Model 2.5.2 One-Step-Ahead Optimal CD Control 2.5.2.1 No input Penaiization 3.5.3 An Example: MI vs. IP Contml 2.5 -3.1 Tuning Considerations

3

Filter Approximation Methods Controller Approximation 3.1.1 Analogy to Open-Loop Model Reduction The Filter Approximation Method 3.2.1 Input-Output Interpretation 3.2.2 Interpretation as Controller Output Matching Least Squares Filter Approximation 3.3.1 Development of the LS Method 3.3.2 Limitations of the LS Method Semidefinite Programming Filter Approximation 3.4.1 Semidefinite Programming

ii iii iv vii viii

3.5 3-6

3.7 4

3.4.2 Filter Approximation via SP 3-4.3 Stability Constraint 3.4.3.1 LM1 Regions 3.4.3.2 Symmetrization of HX 3.4.3-3 Constrained SP Filter Approximation Direct Filter Design 3 S. 1 Direct Filter Developrnent Closed-Loop Filter Approximation 3 -6.1 Development of the Sensitivity Function 3.6.2 Problem Formulation 3.6.3 Linearization Procedure 3.6.4 SP Formulation 3-6.5 Successive Semidefinite Prograrnming Summary

CD Control System Analysis Asyrnptotic Behaviour of the Closed-Loop Systern Performance Criteria for Filter Approximations 4.2.1 Existing CD Controller Performance Cntena 4.2.2 r measure 4.2.2.1 Benchmark Values of r 4.2.3 Frequency Response of the Controller-Filter Mismatch 4.3 Robust Stability of the Closed-Loop System 4.3.1 Types of Uncertainty 4.3.2 Robust Stability Analysis via Monte Carlo Methods

4.1 4.2

5

Case Study

5.1 5.2 5.3

5.4 5.5

6

Physical System Comparison of Open-Loop Filter Approximation Methods Andysis of Filter Approximation Performance 5.3.1 Vaiidity of the Syrnmetrization Approximation 5.3.2 Effect of Filter Length 5.3.3 Cornparison of Filter Approximation and CD Controller 5.3-3.1 Performance Cornparison 5.3.3.2 Robust Stability Comparison Comparison of Open and Closed-Loop Approximation Methods Summq

Conclusions and Recommendations 6.1 Summary and Conclusions 6.2 Recomrnendations References

Appendix A: Mathematical Formulations SP Formulation for Minirnization of a Matrix N o m A. 1 A.2 SSP Linearization A.3 Frequency Response Development

103 103 104

Appendix B: MATLAB Program Synopsis

107

I 06

List of Figures A Simple Industrial Paper Machine Sheet Property Measurement Using a Scanning Gauge Sensor Classification Scheme for CD Profile Modeling Methods Classification Scheme for CD Controller Review CD Control of Basis Weight via Deformation of the Slice Lip CD impulse Response Nomenclature Definition CD Impulse Response and Initiai CD Basis Weight Profile Basis Weight and CD Actuator Profiles After 100 ControI Intervals Basis Weight Variance Trajectory for MI and IP Controller Effect of Tuning Constants on CD Basis Weight Variance Anderson and Liu's Classification Scheme for Controller Approximation Anderson and Liu's Additive Representation of Controller Approximation Problem Comparison of Mode1 Reduction and Controller Approximation for both Open-Loop and Closed-Loop Methods The General Idea Rehind SP The Successive Semidefinite Programming Method The Approximate Sensitivity Matrix Frequency Response of the Filter/Controller Mismatch Plant-Mode1 Mismatch Uncerrainty Description Robust Stability Analysis Procedure Spatial Filters Generated by the Various Filter Approximation Methods Error in the Elements of the Closed-Loop State Matrix Approximation Cornparison of Filters of Different Lengths SP Objective Function vs. Filter Length Cornparison of the Basis Weight and CD Actuator Profiles M e r 100 Control intervals Comparison of the Output Variance Trajectories for the Original Controller and its Filter Approximation Cornparison of the Objective Function Trajectories for the Original Controller and its Filter Approximation Asymptotic Behaviour of the Original CD Controller and its Filter Approximation Performance Degradation due to the Filter Approximation Measured by r Singular Value Frequency Response of the Controller/Filter Mismatch Singular Value Frequency Response of the Original Closed-Loop System and its Approximations Initial Estimates and the Closed-Loop Filter Returned via SSP Closed-Loop Objective Function vs. Iteration Number for Both Solutions Comparison of Objective Function for the Open -Loop and Closed-Loop Approximations Singular Value Frequency Response of the Controiler/Filter Mismatch for Open and Closed-Loop Approximations vii

7 9

11 17 17 25 31 31 32 33 35 37 39 49

59 68 70

71 73 76 78 79 80 81 81

82 83 84 85 86

88 89 89 90

Nomenclature Note that matrices will be denoted by upper-case bold and vectors by lower-case bold. is a The following convention will be used in dimensioning of vecton and matrices: real matrix of dimension wxs while ji,.. i l is a real vector of dimension w x 1. matrix containing current error profile in its columns

first-order difference representation disturbance entenng systern controller/filter mismatch number of zones subject to edge effects CD e m r profile hi&-order process transfer hinction approximated low-order process transfer fùnction CD mode1 rnatrix feedback gain matrix approximated feedback gain matrix number of CD actuators mismatch in sensitivity matrices number of CD measurements factored matrix from the one-step-ahead objective function factored matrix fiom the one-step-ahead objective function value of the objective function for the approximated control system value of the objective function for the original control system positive definite weighting matrix state in a corresponding state-space mode1 positive definite weighting rnatrix performance degradation measure sensitivity matrix for the approximated system sensitivity matrix for the original system dummy variable in SP formulation positive definite weighting matrix CD actuator adjustment plant/model mismatch rnatrix positive definite weighting matrix filter length positive definite weighting matrix filter feedback gain matnx containing the filter weights feedback gain matrix about which linearization made desired CD profile at next interval current CD profile initial offset fiom the uniform CD profile shift operator

viii

Greek Symbols uncertainty in a gain element parameter for the "size" of the linearization region in SSP approximation error vector scalar tuning constant in the one-step-ahead optimal control problem percentage uncertainty in a gain element weighting term in the LS filter approximation step size in RS analysis scaiar h g constant in the one-step-ahead optimal control problem proportionality constant spectral radius of a matrix singular value of a matrix standard deviation of CD profile CD impulse response coefficient eigenvalue of a matrix randorn number matrix of random nurnbers

Ab breviations ARIMA

CD CL DF EE FA FVT

GU

IP LMI

LQ LQG

LS MD MI haMo

PTV QP QPF

RE RS

SIS0

SP

Auto-Regressive htegrating Mo Cross-Direction Closed-Loop Direct FiIter Edge Effects Filter Approximation Final Value Theorem Gain Uncertainty Input Penalization Linear Matrix Inequality Linear Quadratic Linear Quadratic Gaussian Least Squares Machine Direction Matrix Inversion Multi Input Multi Output Periodic Time Varying Quadratic Prognunming Quadratic Penalty Function Robust Ellipsoid Robust Stability Single Input Single Output Semidefinite Programrning

SPWS

SSP

Semidefinite Programming Without Stability Constraint Successive Semidefinite Programrning

Chapter 1 Introduction The field of cross-direction (CD) control has received a good deal of attention from both the industrial and academic çornmunities over the past decades: in particular. the pulp and paper community has been very active in this research [Heavenet al. (1994)l. CD control is typicdy encountered in sheet-forming processes. In general. a sheet-forming process involves the formation of a continuou mat of materiai: exarnples of this type of process include the manufacture of paper, strip steel and the extrusion of plastic [Boyle (1977)l.Although the work presented in this thesis is applicable to the general class of sheet-forming processes, the control of CD profiles on a paper machine will be the particular exarnple examined.

This thesis considers the problem of designing control systerns for the regdation of CD sheet properties. This work is motivated by the fact that. in generd, industry hcts not kept pace with developments in the CD control literature in terms of implementation [Dumont (1986)). Heaven et al. (1994) state that the low incidence of applications of advanced CD control methods in

the pulp and paper industry is due to system constraints (limitations in hardware and software). The aim of this thesis is not to provide a new CD control design; rather. the aim is to develop

a systematic method by which advanced CD control schernes can be approximated by a simple fixed controI structure available in the installecl hardware. As a consequence of the previous goal, a more general aim of this thesis is to establish a rigorous method by which high-dimensional

multivariable controilers can be approxirnated in some simpler form which can then be more readily implemented. The first reference to CD profile control in the scientific literature was made by Burkhard and Wrist (1954) who stated that there are three components to basis weight (massi per unit area)

-

Introduction

9

variation in the manufacture of paper: niachine-direction (MD), crossdirect ion (CD)and random components of variation. The machine-direction

(MD)is

the direction of the movernent of the

sheet of material on the machine, tvhile the cross-direction (CD)is the direction perpendicular to the motion of the sheet. It was not until the introduction of the scanning gauge sensor in the 1960s that the importance of CD variation was recognized [Cutshd (1991)!. The first attempt at closed-loop regdation of CD sheet properties was made by Beecher and Bareiss (1970). The

CD control problem has received a good deal of attention kom that day forward. There are a number of key properties of the CD control problem that make it both an interesting

and challenging problern. The primary challenge is due to the physical nature of the process under consideration: a sheet-forming process. Simply put. the challenge of modelling and controlling a sheet-forming process resdts from the inherently coupled mdtivariable nature of the process

[Heaven et al. (1994)l. The common simpiification of dividing the CD into zones results in a system wit h significant coupling between zones [Dumont (l99O)j. Secondly. the development of methods for the rneasurement of CD profiles is still an on-going activity [Rigopoulos et ai. (1996)l.

In general, a CD profile is inferreci (not directly measured) by a scanning sensor which traverses the moving sheet of product. The scanning speed relative to sheet speed is generdy very slow leading to the problem of extracting the CD profile fiom the measured data [Dumont (1990)l.The

final challenge associatd with the CD control problem is the design of controllers that efficiently regulate the CD property profile while requiring acceptable actuation. Over the past several decades, quality constraints on paper products have continually grown more stringent [Heaven et al. ( l994)I. Paper properties such as t hickness. moisture content, smoothness and strength must be regulated more tightly to give a more uniform product. The underlying motivation responsible for the interest in the problern of sheet property regdation is econornics. By reducing the sheet variations in the MD, significant reductions in raw material and energy consumption, increased production levels and improvements in product quality have been realized [Wallace (1981)l. Reduction of sheet variations in the CD has also been advantageous economically. In particular. by reducing these CD profile variations. the paper machine efficiency is improved due to the impact of profile variations on sheet breaks. final paper quality and product

rejects [Wallace ( 1981)j .

3

Introduction

The remainder of this chapter is devoted to the underlying motivation behind this work. an ovewiew of the contents of this thesis and a summary of the contributions of this thesis.

Motivation

1.1

As stated earlier. the p r i m q reason for the implementation lag in new CD control methods is due to limitations in the installed base of CD control hardware [Heaven et al. (1994)l. In kt.

a significant portion of today's hardware consists of older. or "legacy", distributed control

systems with programrning capabilities that do not readily support the implementation of complex multivariable control designs (e.9.. [&IeasurexDevron (1990)j). Faced with this challenge. the control engineer has two options: purchase a new control system or attempt to incorporate sorne of the advanced control designs using the functions that are available within the current hardware platform. It is the latter of these two options which will be addressed in this thesis. This incorporation of some advanced high-dimensional control design

using functions available within the current hardware platform can be considered to be a controiier approximation pro blem.

The problem of controller approximation is encountered and addressed in an ad hoc manner on a daily basis in many industriai settingç. The development of a systematic. tractable approach for

the approximation of high-dimensional control designs in some readily implemented form wodd be a useful tool in many industrial applications. The specific problem to be addressed in this thesis is the approximation of some advanced CD control design by a Finite Impulse Response

(FIR)filter. A filter

is a control structure that is

supported by the control platforms under consideration in this thesis. This technique of CD control is termed *'spatial filtering" or "convolution decoupling" [Measurex Devron ( 1990)]. The solution

to the previow problem involves the development of a systematic method of approximating the behaviour of a high-dimensional controiler (advanced CD controller) in some lower-order form

(filter).

Introduction

1.2

Thesis Outline

-As discussed in the previous section. t his thesis concentrates on the development of a met hod by which advanced CD controllers may be approximated as a flter. r\t the beginning of Chapter 2. a brief description of the operation of a paper machine is given. The remainder of the chapter is devoted to a detailed review of the CD control literature with separate sections on CD profile measurement. modeiling and control. The chapter concludes with the developrnent of a o n e s t e p ahead optimal CD controller whïch is the particular CD controller to be used as a benchmark controller for the fiiter approximations.

In Chapter 3. the curent technique of spatial filtering is discussed. Based on this. the filter approximation problem is posed which aims to incorporate the behaviour of the advanced CD control design in a filter. X number of different techniques are then proposed to solve the Filter Approximation problem using either Least Squares or Semidefinite P r o g r a d n g . Chapter 4 is concerned with analyzing the resulting filter-based CD control system. The asymptotic behaviour of the closed-loop system and its inherent directionaiity are analyzed. Performance criteria for the approximation process are also developed in this chapter. Findy, a

means of comparing the robust stability to plant-mode1 mismatch of the two control systerns is developed.

A hil case study contrasting the various approximation schemes is given in Chapter 5. Following the selection of the most effective approximation scherne. the effect of filter length on the approximation quality is investigated. and a cornparison is made between the performance of the original CD controller and its filter approximation. Finaily, Chapter 6 provides the conclusions h m this work and recornmendations for W h e r investigation in this area.

1.3

Thesis Contributions

The primary contribution in this work is the development of a systematic method by which an advanced CD control design can be approximated by a spatial filter. This approximation method

Introduction

5

provides a "bridge" between the developments in the CD control literature and the CD control schemes that are readily implemented within the existing platforms.

In particuiar. the general problem for approximating an advanced CD controller as a filter is forrnulated. -4 variety of rnethods to solve the filter approximation problem are proposed. The different approximation methods are cornparecf in terms of preservation of closed-loop stabiiity, ease of tuning and performance degradation. The application of this approximation scheme is not necessariiy limited to CD control. A tool

of this kind allows an a d w c e d controller synthesis method to be ernployed for the tuning of simple and readily available controllers. For example. in Thake et al. (1997),a Linear Quadratic regulator was designed for a standard 2 x 2 distillation control problem. and the performance of this LQ regulator was then approximated by a multi-loop PI control design. In many cases, a multi-Ioop PI control system is more readily implemented than the original L Q regulator design.

The secondary contribution from this work is a detailed review of the current CD control literature. The CD profile modeiling techniques are classified as being either an Empirical Identification Method or an Advanced Modelling hlethod foliowing sirnilar Iogic t o that of Kjaer et al. (1994). Existing CD control a l g o r i t h are classified into two categories: Off-Line Design

Methods or On-Line Optimization blethods.

Chapter 2

Cross Direct ional Controller Design As stated previo~sly~ the aim of this work is to provide a means by which the advanced methods proposed in the CD controI literature c a n be readily impiemented in existing CD control hardwaxe.

This chapter begins with a brief explanation of the operation of a paper machine leading to a review of the work that has been reported in the CD control iiterature. The literature review is divided into three sections: CD profile measurement, CD profile modelling and CD controiler design methods. The remainder of the chapter is devoted to the deveiopment of a one-stepahead optimal CD controller. This controller is subsequently used in a case study illustrating the various Filter Approximation methods in Chapter 5.

2.1

The Papermaking Process

The process of manufacturing a final paper product Çom papermaking stock may be divided into six main steps (see Figure 2.1) [Smook (1982)l. A pressurized headbox discharges a jet of paper-rnaking stock under a slice lip on to the moving forming fabric. The amount of stock being discharged at a aven position across the width of the sheet may be varied by adjusting the siice lip opening, using rnotorized actuators which deform the slice lip. The forrning fabric is desigmd to form the fibres contained in the papermaking stock into a continuous matted web, while water present in the stock is drained by suction. The continuous web is then passed through a series of presses where additional water is removeci, and the fibre web is consolidateci prior to the sheet entering the dryer section of the paper machine. The dryer section removes most of the remaining water by evaporation using a series of steam-heated cylinders. Following the dryer section. the

Cross Directional Controller Design

7

sheet is fed to the calender stack where a series of presses reduce sheet thickness and smooth the sheet surface. Finally. the dried and calendered sheet is wound into a reel of paper. Between the calender stack and reel. the paper properties are meâsured by a scanning bauge sensor.

Figure 2.1: -4 Simple Industrial Paper ),lachine

The examples in this thesis wiIl concentrate on the wet-end of a paper machine. The wet-end of the machine involves the distribution of papemaking stock under the slice lip (flexible beam) onto

the moving forxning fabric and subsequent fibre web formation. If either the fiow under the slice Lip or stock consistency is not uniform, the basis weight (mass per unit area) wiil vary. This variation in basis weight directly affects the product quality. Sheet properties such as web strength, opacity, Bexibility, density, and thickness are directly related to basis weight uniformity. Non-uniform basis weight results in uneven drying and uneven calendering of the sheet which yields a final paper product of non-uniform appearance. uneven absorption of coat ings and uneven printing quality [Wallace (1981)l. It is for these reasons that the control of bais weight on a paper machine has received considerable attention over the past t hree decades. The control of basis weight and other sheet properties on a paper machine belong to a general class of problerns termed control of web or sheet-forming processes. Control of a web-fonning

process involves two dimensions: the machine direction (MD) and cross direction (CD). Commonly, the assumption that CD variations are sufliciently slow relative to the MD variations is

made [Dumont (lggO)!. This assumption leads to the CD and MD problems being treated independently. These MD variations are also assurneci to influence al1 CD positions simuitaneously [Dumont et al. (1993)j.

Cross Direct ional Controiler Design

8

Ai1 CD control systems require three common components: a reliable measurernent (estimate) of the CD property profile. a mode1 of the CD process and a CD controller design. The following sections review the CD measurement . rnodelling and control techniques currently available in the

CD control Iiterature.

2.2

Profile Measurement

The majority of installed paper sheet control systems rely on the above assumption regarding the independence of the MD and CD control problerns [Dumont (1990), Kjaer et al. (1994)l. Unfortunately. the method of measurement used to collect the data required for any CD control scheme is not as easily decoupled into the MD and CD directions.

Typicdy, t h e control of a sheet property in the *LID is based on the average of the current measured profile, termed the scan average.

Due to the significant time delay between the MD ac-

t uator(s) and the scanning gauge. dead-time compensation techniques such as the Smith Predictor

are standard MD control strategies [Bergh and MacGregor (l987)].

CD control is based upon an estimate of the current CD profile. It should be noted that the CD profile is not directly measured by a scanning sensor. Instead. an estimate of the CD profile is inferred fiom the scanning sensor measurements. The paper properties are measured between the calender stack and the reel with a sensor that traverses the paper machine in the

CD [Kristinsson and Dumont (1996)l. Figure 2.2 shows the "zig-zag" path taken by a typical scanning sensor measurement device on a paper machine.

Cross Direct ional Cont roller Design

f sheet rnovement

Figure 2.2: Sheet Property Measurernent Using a Scanning Gauge Sensor

The path of the scanning sensor on the sheet is at a s h d o w angle relative to the MD due to the large difference in the speed of the sheet on the machine and the scanning sensor [Dumont (1990)l.

The rneasured profile fiom a scaming sensor will contain MD variation. CD variation and measurement noise [Heaven et al. (l994)I. Heaven et al. (1994) state there are four main steps in the c~tractionof a CD profile from scanned data. The rneasured data may be initially corrected for uneven sampling periods due to the nature of the scanning path. The sampling periods rnay not be equal since the sensor may move off the sheet at eit her end and then stop to change direction. The measured data is then decoupled into MD and CD variation using various techniques. Since there are typically more measurernents than actuators on today's paper machines, the CD profile is usually mapped from the higher rneasurement resolution to the actuator resolution.

This mapping algorithm must also

account for nonlinear sheet shrinkage and web wander which cause misaiignment between the measurements and actuators. Finally, the CD profile is filtered (typically with an exponential filter) to address any rapid MD variation that may be present to give the persistent CD profile,

which can be used for CD control of the corresponding sheet property [Heaven et al. (1994)l.

Cross Directional Controuer Design

10

Slost scanning measurernent techniques in industry are based upon the Evponential %Idtiple

Scan Trending approach [Dahlin (1970)l. The MD variation is characterized by the variation in the scan average. while the CD profile is based on the output of an exponential filter of current

and previous CD measurements. This method has b e n fomd to be very slow in detecting process upsets and results in very sluggish control. Due to these limitations. much research has focussed on improving the methods of treating the data provided by a scanning gauge. Taylor (1991) describes two approaches for improving CD profde estimation which involve varying the speed of the scanning gauge. One approach advocates decreasing the scan speed

which increaseç the averaging time for each CD rneasurement zone and Ieads to a decrease in the cut-off bequency. The other approach involves increasing the scanning speed to remove MD wiability from the CD profile more quickly by coliecting data faster. Chen (1992) has proposed a dual Kalman filter to exploit the fact that each individual rneasurement is a portion of a fui1

scan. This approach uses a gradual updating of the CD profile estimate and frequent updating of

the MD variation. Resdts show that the scheme allows for better grade change tracking due to more responsive MD control action. Wang et al. (1993) have developed an on-line algorithm that employs an Extended Kalrnan Filter to update the model parameters of a linear transfer function describing the hID dynamics. while an exponential forgetting and resetting algorithm is used for

CD profile estimation. This algorithm may be suitable for implementation in conjunction with an adaptive control scheme (Wang et al. (1993)l. Tyler et al. (1995) considered the implications

of CD dynarnics which are not negligible. and modelled the sheet forming process and sensor as a Periodic Time Varying (PTV) system. The performance of scanning sensors was compared to

stationaxy sensors. and a performance index to identlfy the value of additional scanning sensors was proposed. Recent work by Rigopoulos et al. (1996) employs an on-line Prïnciple Component

Anaiysis to identify the significant modes of the underlying profile. By identibng the significant modes of the underlying profile, a controller can be designed to respond oniy to the most relevant

disturbance profile at any give time [Rigopoulos et al. (1996)l. The major implication of the limitations inherent in current industrial scanning sensor technology is that the control of sheet properties is restricted by the inability to fuUy observe the proces

at any given time. As a result. most CD control systems are designed to reject low frequency

Cross Directional Cont rouer Design disturbances only.

2.3

CD Profile Modelling

There is a wide range of e'cisting modelling work in the literature ranging fiom impulse response representations through to b i t e element analysis of the slice lip. This review of CD profile modelling approaches will be divided using a classification sirnilar to that of Kjaer e t al. (1994) and wi11 classify the modeiling approaches into two general categories: Empirical Identification methods and Advanced Modelling met hods. Figure 2.3 shows the classification scheme to be used.

Modclling

tD Empncal Identification

Figure 2.3: Classification Scheme for CD Profiie Modelling Methods

2.3.1

Empirical Identification Methods

Empirical Identification Methods involve the construction of a process mode1 using experimen-


22

tally obtained input/output data and do not use any mechanistic laws concerning the physical nature of the proces. These methods are further divided into steady-state CD models (no temporal djnamics) and MD/CD models which contain temporal dynamia but no tw~dimensional coupling.

2.3.1-1

Steady-StateCD Models

The first atternpt at describing the effect of slice-lip opening on the CD basis weight profile using a linear model. assuming steady-state in the MD. was made by Beecher and Bareiss (1970).The

sheet was divided along the CD into n zones with the basis weight in each zone ( y i ) being related to the flow (u,) under the slice lip at a given CD position by a proportionality constant ( f i ] ) .

For a square control problem (the number of CD actuators=n), this mode1 form requires the estimation of n2 coefficients. fi,. The number of parameters is greatly reduced by assuming that the responses are symrnetric about a given actuator and identical across the machine with edge effects ignored [Boyle ( l977)j. The distributed parameter nature of the sheet property contro1 problem was recognized by Wilhelm and Fjeld (1983). A number of assumptions were made to sirnplify the problem. The temporal (MD) process response was assumed to be independent of position across the sheet, and the spatial coupling (CD) was assumed to be static. The assumption of static coupiing in the CD irnpiies that, following an actuator adjustrnent. steab-state is achieved in the CD at the next

scanning p a s . The previous two assumptions were termed to be the separability property which allowed the MD and CD problerns to be treated independently. The assumption that the CD response is independent of position across the sheet, except at the edges (boundary of the sheet

in the CD), is referred to as the hornogeneity property. The final assumption waç that the CD response is symrnetric about a given actuator except at the edges; this assumption is termed the symmetry property. Based on these assumptions. a model relating a CD sheet property to a set of manipulated

variables (typically CD actuators) was formulateci in temis of convolutions for both the continuous

Cross D irectionai Cont roller Design

13

and discrete cases. This model is also expressed in matrk form where the open-loop CD (spatial) impulse response for each actuator is contained in the steady-state gain matrix (H):

where yk is the CD profile and Auk is vector of CD actuator adjustments at tirne intend "k".

2.3.1.2

MD/CD Models

The above model formulation does not include any temporal dynamics associated with an adjustment of the CD actuators or time deiay between an actuator adjustment and the corresponding measurement. These temporal dynarnics. when combined with the CD coupling. provide a more complete picture of the property dynamics on the sheet and must be considered in the design of an M D control system.

If the dynamics of the actuators and dead-time are similar across the sheet, the MD dynamics may be formulateci in terrns of a scalar transfer function which is then combined with the matrix formulation representing the CD interactions [Laughlin e t al. (1993). Halouskova et al. (1993), Duncan (1995)l. Haiouskova et al. (1993) represent the relationship between the slice lip and the resultant CD profile as a curve describecl in terms of derivatives.

Tr adit ional parame t er estimation techniques, including l e s t squares and maximum likelihood rnethods. have been applied to the problem of estirnating a model which includes both MD dynamics and the CD response [Heaven et al. (1996)l. The MD dynamics are represented a s being first-order plus time delay, and the steady-state CD response was expressed in terms of a secondorder critically darnped response shape. The description of the CD reçponse as a second-order critically darnped response is motivated by the characteristic centrai pulse with smailer negative side lobes ( s e Figure 2.6) observeci on many paper machines [Wilhelm and Fjeld (1983)l. Related work has addresseci the concerns regarding the collection of process data via bump tests and the accompanying disruption to the normal operation of the paper machine [Duncan(1996)l. By representing the MD dynamics as a pure tirne delay, an estimate of the open-loop CD impulse response can be evtracted from an estimate of the closed loop system response. The necessary data is collected under normal operating conditions with a perturbation signal added to the CD

Cross Directionai Controller Design actuator setpoints.

2.3.2

Advanced Modelling Methods

For the purposes of this work. an advanceci modelling method is considered to be any form of mode1 that mes information about the fundamental nature of the process. These advanced modeilhg methods are further classified into two categories: Advanced CD Modeis and MD/CD Coupled Models.

2.3.2.1

Advanced CD Models

Since CD control of sheet properties on the wet-end of a paper machine usually involves the deformation of the slice lip by motorized actuators. a number of researchers have chosen to model the slice lip as a beam under multiple loads. Balakrishnan and McFarlin (1985) approximated the slice lip as a metd bar supported by actuators (rnodelIed as springs). This model accounted for backlashing observed in the slice vpening in zones adjacent to an adjusted actuator. This model form produced a cornmonly encountered steady-state CD response similar to a Gaussian-shaped curve. A similar approach employed a finite eiement solution of the partial differential equation governing the slice lip behaviour [Kjaer et al. (1994)l. Recently, Knstinsson and Dumont (1996) have proposed the use of a finite number of Gram polynomials to approximate a CD profile. The slice lip was modelied as a flexible beam under multiple loads corresponding to the CD actuators, and its profile was then approximated in terrns of Gram polynomials. The advantage of this approach is that the Erequency content of the approximated profiles can be controlled by the number of Gram polynornials used. Related work has shown that using orthogonal basis functions to represent a CD profile ailows the profile to be separated into controllable and uncontrollable

components [Heath ( 1996)l.

2.3.2.2

MD/CD Coupled ModeIs

Recent work [Chen and Adler (1990),Kjaer et al. (1994)l has suggested that the assumed independence of the MD and CD behaviour was not suitable for modelling profile behaviour on a paper machine. The following modelling approaches attempt to address the coupling between

the MD ana CD' and these methods are divided into two categories. The first category includes

15

Cross Directional Cont roiler Design tw*dimensiond

empirical identification methods which address MD/CD coupling. The second

category contains modelling techniques that are based on the fundamental nature of the process

and also address hID/CD coupling. 2D Empirical Identification Met hods Bergh and MacGregror ( 1987) developed linear t ransfer function models using r ime-series analysis to describe the process. disturbances and the MD/CD interaction. The MD model was formulated as a linear transfer function relating the

MD actuator to the scan average

of the output, wMe

a rnuitivariate ARLPvIA or state-space time series model was proposed to relate the vector of spatial responses to the CD actuators. The motivation for this work rested on the fact that the independence of the CD and !vID problems relied on the assumption that CD disturbances were slow or infrequent. Recent work has employed a two-dimensional ciifference equation or

a partial differential equation to describe the mutualiy dependent nature of the MD and CD

systerns [Wellstead and Heath ( l992)]. Wellstead et al. (1996) use this two-dimensional linear transfer function representation iz a recursive on-line estimation of the model parameters. This approach requires the estimation of many more parameters in cornparison to the non-coupled

MD/CD representations discussed in 52.3.1.2. Fundamental MD/CD Modelling Methods None of the -4dvanced Modelling approaches mentioned. thus far. have been based on the funda-

mental nature of the web-forrning process. The most fundamentai approach involved modeling the Bow from the headbos on to the Fourdrinier wire and the resultant effects on CD basis weight profiles using basic fluid mechanics [Westermeyer (1987)l.Spatial currents under the slice lip and on the moving wire were shown to affect the rneasured CD basis weight profiles significantly. Chen and Adler (1990) propose that profile response behaviour be described in terms of the superposition of wave propagations.

~ ( i . t=) f f(i

+ ut) + f b ( i - ut)

(2-3)

where p(i, t) is the profile at CD position i and time t . The time is measured as the time from the

exit from the slice lip. The terms f f ( 2 + u t ) and f b ( i

- ut) are the wave cornponents propagating

Cross Directional Controiler Design

16

in the CD towards the kont and back edges of the machine respectively. The speed in the CD is v: this implies that ut is the distance in the CD from position i. The profile response was observed to be very dependent on the type of paper being produced. More specifically, the response is generaily narrow for lighter grades of paper due to higher machine speeds and broader for heaw grades due to a slower machi~espeed. Their findings indicate that the profile response mode1 may change in both order and shape for different production grades.

.4 more fundamental approach. in which the CD basis weight response and the accompanying

hlD/CD coupling were considered simultaneously. was motivated by the observation that the flow exiting the headbox exhibits two-dimensional behaviour [Kjaer et al. (1994)l. The overall tw-dimensional

impulse response is then d e h e d as the convolution of the actuator dynarnics,

slice lip bending, and the flow field function. The flow field function is intended to describe the changes in the CD response as it evolves dong the machine in the MD. This type of mode1 which accounts for the evolution of CD profiles in the MD is intended to give a mode1 capable of making

CD control Iess dependent on MD operating levels. It is apparent fiom the preceding discussion that there is a wide range of modelling approaches that may be used to characterize the CD response. The limiting factor is not the degree of t~~tino1og-y in CD profile rnodeiling, but rather the ability to adequately observe the process beyond steady-state in the CD. This is the reason why most implemented

CD control designs

draw their CD models from 52.3.1 [Heaven et al. (1994)j.

2.4

Exist ing CD Controller Designs

The comrnon theme among the existing CD control designs is that the resulting controller is of high-dimension. The designs will be classified into two categories: Off-Line methods and On-Line methods. Off-Line methods are CD control methods which may include optimization techniques but no optimization calculations are performed on-line. On the other hand, the On-Line methods require some form of on-line optimization. parameter updating or constraint checking (refer to Figure 2.4).


(

CD Control Methods

) on-linc opttmizarion paramncr aumiaion

Figure 2.4: Classification Scheme for CD Controller Review

Since the control of CD basis weight profiles on industrial paper machines is typicdy achieved by adjustment of actuators which physically deform the slice Iip, the control systerns must be

designeci to prevent excessive strain and W e a r on the slice Lip (refer to Figure 2.5).

Figure 2.5: CD Control of Basis Weight via Deformation of Slice Lip

Excessive deformation has been typically defineci as large differences in adjacent actuator lev-

els. Various terms have been used in the Literature to describe this problem including ;profile oscillation!' [Boyle(1977)], "jack-t*jack lirnit" [Siler (l984)], "bending stress lirnits" [Chen and

Wilhelrn (l986)]! "sawtoothing" [Brewster (1989)] and "picketing" [Heaven et al. (l996)]. From

Cross Directional Controller Design this point on. the problem of excessive differences in adjacent actuator levels will be referred to as "excessive bending stress" .

2.4.1

Off-Line Methods

The 6rst attempt at automatic regulation of CD sheet properties was made by Beecher and Bareiss (1970). Using a linear mode1 to represent the behaviour between basis weight and stock flow under the slice lip, a simple reglatory control systern was designed. Cdculation of a set of control moves required t hat a matrix containing the proport ionality constants ( refer to Equation (3.1)) be inverted. This work also alludes to the physical constraints on the actuation; more

specifically. it is stated that the control moves must be small relative to the absolute actuator

level.

Based on the work of Beecher and Bareiss (L9'70), the weil known CD control approach known as M a t r k Inversion

(MI)was developed [Boyle (1977)l. Let the CD process be modelled as

in Equation (2.2). For the special case where the CD profile is estimated at the same spatial

Çequency as the actuator spacing and provided that H is invertible, the control actions can be

directly calculateci as:

where

is the desired profile at the next sarnpling instant. This approach is based simply

on inversion of the steady-state gain model (N). D~leto the excessive actuation required by this

approach, the MI method is not a feasible CD control design [Boyle (1977), Dumont (1990)l. One of the first atternpts at CD sheet property control was made by Boyle (1977). As stated above, he cieveloped the kiI rnethod of CD control and discussed the excessive controi action that results from such a control design for a five actuator system. Boyle used a Quadratic Programming (QP) approach with a second-order ciifference representation of slice lip bending to

address the problem of excessive bending stress and a constraint giving bounded actuator levels. Simulation results in&cated that the QP method gave feasible CD actuation: however, the on-fine implementation of QP on the hardware of the 1970s was not achievable according to Boyle. In order to address the problem of bending stress limits with the M I approach, a quadratic objective


19

function was proposed which penalizes deviations fiom the desired profile dong with a penalty term to suppress excessive actuator moves [Boyle (l978)]:

where J is the quadratic objective hmction to be minimized. v is a vector containing the CD profile at the next interval. b is a scalar tuning constant and u contains the CD actuator setpoints. Simulation results for a system of six actuators and eleven CD measurements indicated that the problem of infeasible actuation could be resolved by appropriate adjustment of 6 .

-4 similar one-stepahead optimal CD controller was proposed by Wilhelm and Fjeld (1983) who noted the problems of ill-conditioning of the matrix inverse and constraint handling. The

neeci to consider the spatial interaction in a CD basis weight control scheme, dong with the resultant instability if the interaction is not accounted for? is identified by Karllson et al. (1985).

An optimal controi stratew similar to that of Boyle (1978) is discussed. Implementation of this scheme using high resolution CD profile measurements on an industriai paper machine having approximately 50 CD actirators and the favourable results are discussed by Norberg (1989). Most CD control systems tend to focus on reducing the variance in the profile of a single sheet property without regard for the interactions between different CD actuator systerns. It is well known that there is significant interaction between independent CD actuator systems on a paper machine (e.g. adjusting the slice-lip to regulate the CD basis weight profile affects the moisture profile which is typically controlled by another set of CD actuators) [Mustonen and Ritala (1992)]. This problem is addressed by the coordinated control of multiple sheet properties. Wilhelm and Fjeld (1983) treated the problem of coordinated control with a modified onestepahead controller t hat considered the interaction arnong sheet properties. A st ate-space approach using eigenstructure assignment to decouple the responses of sheet properties with respect to different

CD profiles such as basis wight and moisture is proposed by Manness et al. (1992). Another approach uses a state equation dong with a quadratic objective hnction giving various weights to mutually dependent profile properties [Mustonen and Ritala ( l992)].

Balakrishnan and McFarlin (1985) used a different approach for detennining the optimal CD


20

actuator moves. Instead of direct ly calculating the act uator adjustments based on the observed error profile. a vector of desired bending moments for the slice lip is determined based on the observed error profile. Using their elastic beam model of the slice lip. the set of actuator adjustments is then calculated to give the desired deformation of the slice lip. Since a matrix containhg the

bending moment coefficients matrix m u t be inverted. this method is sirnilar to the MI method of CD control. Using the twedimensiond linear transfer function representation of a sheet forming process, a regulator which is the two-dimensional analog of generalized Minimuni Variance Control has been proposed [Wellstead and Heath (1992)l. However. there is no direct attempt to satisfy specifxc conçtraints such as slice lip bending stress lirnits. An example with 50 CD actuators and 100 control intervais is presented. The results show quick convergence in the estimates of the model parameters and promising cont roi performance.

Based on new developments in the design of headboxes. a CD basis weight control technique termed "consistency profilings' has been proposed [Vyse et al. (1995)l. Instead of physicaily ad-

justing the slice lip with actuators to vary the flow of stock at a given position across the sheet, the stock is diluted at each CD position as required to infiuence the basis weight. This technique is intended to overcome some of the curent difficulties in controlling narrow spatial streaks in bais weight .

Laughlin e t al. (1993) examineci the CD basis weight control problem in terrns of robust performance in the presence of modelling uncertainty. The goal of this work was to design a

CD controller that maintained stability and satisfied a bound on the maximum singuiar value of the closed-loop sensit ivity function despit e modeiling error. The issue of controller robustness was addressed by detuning the scalar controller dynarnics. A simulation example having 20 CD

actuators was used to discuss the issues of disturbance direction. Their results indicated that the disturbance that was most ciosely aligneci with the plant matrix (the maximum input singular vector) was most easily rejected which agrees with previous work [Skogestad et al. (l988)I. An

alternative approach for designing a robust CD controlier has been proposed by Duncan (1995).

Robustness is again achieved by detuning of the controiier; in this case, a scdar term in the precompensator rnatrix is adjusted to give a control system that will adjust actuators in the


21

region of an error only. rather than across the entire width of the sheet . Uncertainties in the steady state CD response and the actuator dynamics are considered. Aithough most of the Off-Line methods revieweà in this section acknowledge limits on the

CD actuation required. there is no guarantee that the actuation will not become unfavourable or damage the slice lip. This is because the penalization terms included in the objective function formulations (refer to Equation (2.5)) simply suppress unfavourabIe actuation: these terrns do not guarantee that unfavourable actuation uill not occur. In order to guarantee that actuation lirnits are respected. it is necessary to consider on-line optimization subject to constraints or a constraint checking algorithm. 2.4.2

On-Line Methods

.4 CD control method is classified as an On-Line method if some form of on-line optimization,

parameter updating or constraint checking is required. As stated earlier. Boyle (1977) proposed a QP approach to address the excessive actuation of the MI approach but found that the compu-

tations were prohibitive on the hardware of that era.

The QP approach used by Boyle (1977) is revisited by Chen and Wilhelrn (1986). Although the QP solution is optimal. Impiementation again proved to be impractical due to computation requirements. A Quadratic Penalty Function approach (QPF), dthough suboptimal, was found to be readily implementabIe with good results. The solution involved the recursive solution of the one-step-ahead optimal control problem with a constraint checking algorithm and corresponding adjustment of the control weightings to achieve feasibility. Simulation results are inchded for a 19 actuator case, and the QPF approach was found to perforrn similarly to the QP control

scherne. The QPF approach has been irnplemented in a commercial package with successful results reported in over 40 mills. Recently. an alternative approach to solving the QPF formulation of the CD basis weight problem has been proposed to give decreased computation requirements [Braatz and VanAntwerp (l996)I. The new solution method, termed the Robust Ellipsoid (RE) algorithm. involves the approximation of the plant mode1 matrix as a circulant matrix and a p proximation of the constraints as an ellipsoid. The RE algorithm was shown to require much less computation than the QPF formulation and gave similar performance for the sarne example given

Cross Directional Controller Design in Chen and Wilhelm (1986) [Braatz and V a n h t w x p (1996)l.

A model predictive controller has been designed for a coating process that considers actuator constraints explicitly [Braatz et al. (1992)l.The constraints are addressed in three distinct fashions: explicitly via QP. inclusion in the quadratic objective function or by scaling of control moves to achieve feasibility. Scaling is achieved by modif@ng the magnitude of the wctor of actuator moves. not its direction, and was found to perform almost as well a s the explicit solution via QP for a coating example with 12 CD actuators with limited zone-tozone coupling. Bergh and MacGregor (1987) have proposed an Nstep-ahead Linear Quadratic Gaussian

(LQG) controller which is capable of simultaneously regulating both the M D and CD variations. Due to the scanning path. the output vector of the state-space representation (regular pattern of 2n vectors), the Kalman filter gain m a t r k and the resulting optimal controller mat& change at each control interval. However. for t h e infinitestep-ahead control problem, the Kalman filter

and optimal controIler matrices each converge to a repeated pattern of 272 matrices corresponding to the pattern of the output vector of the state-space model. This allows these matrices to be calcuiated off-line and. if possible. implemented depending on the current scanning gauge position. The cont rol is implemented using bot h lumped and spatiaily distributeci actuators wit h examples having 4 CD actuators. Alt hough this research gave a very interesting approach to CD control, it requires programming capabilities far beyond those of many installed hardware platforms. Due to variations in machine speed. environmental conditions and changing production grades, the CD profile response may vary over time [Chen et al. (1986)l.In order to address this process variability, an adaptive control scheme, in which the profile response is identified on-line using

a generalized least squares approach, was proposed by Chen and Wilhelm (1986). A simulation example with 30 CD actuators and piant-mode1 mismatch is used to demonstrate the value of this adaptive control scheme. In particular, it was found that the estimation algorithm performed weil for s m d model perturbations but tended to fail for the drastic changes encountered following a grade change. The problem of large variations in CD profile response due to changing paper grades and operating leveis was addressed via a gain scheduling approach [Adler and Marcotte (1994)l. The gain scheduling approach aTasfound to work well for the large profile variations experienced during paper grade changes. nhile the iterative least squares estimation allows the process model

Cross Direct ional Controller Design

23

to be updated as required. -An alternative adaptive CD controller for regdation of basis weight is proposed by Halouskova et al. (1993). -\n LQG controller. .n-hich includes a penalty terrn on the successive moves of a given actuator. with a recursive identification algorithm is described. Long horizon d-mmic prograrnming is used to address the process coupling and process time delay.

This approach could be useful if the delay between an actuator adjustment and the scanning gauge is greater than one interval which is generally not the case [Wilhelm and Fjeld (1983)l. Using the Gram polynomial approximations of the CD profile and slice lip profile. Kristinsçon

and Dumont (1996) have designeci a CD controller which provides for upper lirnits on the allowable bending stress on the slice lip. This controlfer is contrasted with the control availabie via the MI method. For the case of a badly conditioned mode[ matrix. the Gram polynomial-based controller performs much better t han the MI case wit h no high frequency deflection of the siice lip. Of course. the actuation required by the MI met hod is grossly infeasible with slice lip deflections on the order of kilometres.

The major limitation of these On-Line methods is that the on-line computation required exceeds the capabilities of a large portion of installeci CD control hardware [Heaven et al. (l994)].

2.4.3

Summary of CD Control Designs

It is evident from the preceding discussion that CD control designs generally result in controller gain matrices of dimension equai to the nuniber of CD actuators and CD rneasurements. On modern paper machines having over LOO CD actuators and hundreds of CD measurements. the

CD controller can be of very high dimension.

2.5

Benchmark CD Controller Design

In general, most of the previous CD control designs are variants of the one-stepahead optimal controller proposed by Boyle (1978) and Wilhelm and Fjeld (1983). For this reason, this CD control design wilI be adopted as the benchmark design for this thesis. This CD controiler will then be approximated as a spatial filter in tater chapters. A crucial distinction must be made a t this point. The aim of this work is not to propose a new control design for the CD control problem,


24

but rather to approximate available control methods in a form that is readily implementable within existing hardware. The foliowing CD control design method is one of many possible approaches.

and the onestepahead optimal CD controller has been chosen because it is one of the most cornmon CD controller approaches. It should be emphasized that any CD control design that cari be expressed as a feedback gain rnatriw can be approximated using the filter approximation methods presented in Chapter 3.

The use of a steady state model to describe CD profile variations is a widely used assurnption due to the slow nature of the variations relative to the MD variations !Dumont(l990), Wihelrn and Fjeld (1983)l. Frorn a practical standpoint. the frequency range over which CD control of basis weight using slice lip adjustment is limited by the scanning sensor. In fact. CD control is limiteci to steady-state control in most cases. For these reasons. the use of a steady-state CD mode1 is reasonable. For the purposes of this work. the changes in the curent profile and actuator profiles are assumed to be linearly related by the following steady-state model [Wilhelm and Fjeld (1983))

where ykinxllis a vector of offsets from the nominal profile (usually a Rat profile) at time "k",

Hbxmlis a m a t r k cont aining the procw model and Auk(,,

11 is a

vector containing the individual

actuator moves at time interval "k". There is no guarantee that the model matrix H is either invertible or well-conditioned. The problems of mat rix invertibility and iü-conditioning have been noted by various researchers [Wilhelrn and Fjeld (1983), Laughiin et al. (1993)l. Thus, considerable care m u t be taken when formdating the model matrix to ensure that it is both weii-conditioned and invertible. In this thesis, we assume the model matrix H is formed as follows:


where &,,, are the CD impulse weights observed at the scanning gauge For CD position 3" at "j" positions to the right of CD position "i". -4s shown in Figure 3.6. a positive value represents a

shift of **j"positions to the right of the position of interest. whereas a negative value represents a shift to the left.

Zone

Figure 2.6: CD Irnpuise Response Nomenclature Definition

This is a general formulation for the mode1 matrix (H) which reflects the possibility. that the estimated CD profile and actuator spacings may occur at different spatial fiequencies. For the purposes of this work, we will assume that the crossdirection profile is measured at a spatial

frequency at Ieast a s high as that of the actuator spacing (i.e., n 2 m). This is generally the case

in CD control applications [Heaven et al. (lgW)]. It is worth noting at this point that, in the most general case, H is a non-syrnmetric real

matriu: however, should the same symmetric CD impulse response be used for each actuator with no edge effects, H becornes a Toeplitz matrix. In such a situation there are some advantageous properties of H such as convenience in robust CD controller design [Laughlin et al. (1993)]or

Cross Directional Cont roller Design transformation to circulant form [Featherstone and Braatz (1995)l. Since any candidate design for the control of CD profiIe properties on a paper machine must provide a balance between sheet property regulation and the required actuation. an optimal control design is an obvious choice. The following section provides a detailed development of the one-stepahead optimal CD controller.

2.5.2

One-Step-Ahead Optimal CD Control

The one-stepahead optimal CD control designs reported in the Iiterature are formulated to remove CD offset while penalizing unfavourable actuation [Boyle (1978). Wilhelrn and Fjeld (19831. Consider the case where bot h excessive bending stress and "excessive actuation" are considered tO

be undesirable. "Excessive actuat ion" refers to excessive adjust rnents to any given actuator at

successive control intervais: excessive actuation is undesirable due to actuator Wear. The problem formulation for this situation requires that the difference between adjacent actuator levels be penalized (for excessive bending stress) dong with differences between successive moves of a given actuator. This leadç to the following one-stepahead optimal control probiem for regdation of

CD property profiles:

subject to, ?k+ 1 = Yk -k H

where

a~k

is the desired property profile at time "k

time *'k + 1"' QCxnl,T[(m-l) x ( m - 11, and

+ 1' gk+1 is the predicted property profile at !

are suitably chosen positive definite weighting

matrices, yk is the CD profile provided by the scanning sensor and Dl(m-i)xml is d e h e d as:


The term DAuk produces a vector of differences in actuator moves between adjacent actuators representing bending stress. A first-order bending penalty has b e n proposed by a number of researchers [Siler (1984). Chen and Wilhelm (1986)] and is chosen over a second-order f o m for several reasons. Most irnportantly the first-order formulation results in a rank deficiency of one. while the second-order gives a rank deficiency of two (discussed in detail in 52.5.3). Due to the lesser rank deficiency of the first-order bending constraint. iess nunierical problems are encountered with t his formulation. Xote that. in this forrnulation of the optimal control Problem (2.7), differences in adjacent actuator changes (DAuk)are suppresed and not the levels of adjacent actuators (Duk).Direct suppression of excessive bending stress would require acting on the difference in adjacent actuator ievels (Duk). The (Duk) formulation would work to eliminate any excessive bending stress initiaily present in the actuator profile when the controler was activateci. Suppression of the difference in adjacent actuator changes (DAuk)cannot eliminate an undesirable actuator profile that exists prier to actikxtion of the control system. but it will serve to suppress any increase in bending stress due to automatic controi. This difficulty can be easily addressed by ensuring that the actuator profile is desirable prior to engaging the cont roller.

The optimization probiem given by Problem (2.7) can be converted into an unconstrained optimization problem by substituting for the predicted property profile function. yielding:

Stationarity of the first derivative with respect to Auk requires that:

yk+I in

the objective


2(yi+, - y k

- HAU~)'Q(-H) + ~ A U : R + ~( D A U ~TD ) ~= O

which can be sirnplified to yield:

AU^

=

(H'Q~H + RT+DTTTD)-~HT QT (y*+L- yt)

(2.11)

Thus. the control law can be writ ten in terms of a feedback controller gain matriu (K) as:

au&= K(Y;+I - yk) where the feedback gain matrk is defined as:

The portion of the feedback gain matrix that must be inverted is invertible when it has rank m. Although the last terrn D ~ T ~ is singular D ( r a d of rn - l), the sum of the three terms is invertible provided T is sufficiently small relative to R and H is reasonably conditioned. The r d deficiency of the bending stress suppression term has implications with respect to tuning of the controller: this is discussed in 52.5.3.1.

If there is no reason to suppress profile deviations at one sheet position more than at any other position. the weighting matrix Q can be chosen as Q = I. Similarly, if there is no reason to suppress excessive actuation or bending stress at one sheet position in preference to any other

R and T can be chosen as R =y1 and T = M where 7 and X are some suitably chosen. positive scalars. Such choices for Q , R and T result in the following

sheet position, the weighting matrices

feedback controller gain matrix:

Zlotice that this control problem formulation has the advantage of only a single tuning parameter for suppressing each of ac t uator "bending stress" and "excessive actuation" . thus greatly

Cross Directional Controller Design simpli&ing controller tuning. In this case. the feedback gain mat ri^ K can be considered an approximate left pseudminverse of the model matriu H. whose conditioning is controlled by the tuning parameters 7 and A. .A strong anaiogy can be drawn with the statistical method of ridge regression in which the conditioning of the regessor is improved by the addition of a non-singular rnatriu component [Mason et al. (1989)l.

2.5.2.1

No Input Penalization

The original MI approaches to CD control did not contain actuator suppression terms in their formulation [Boyle (l977)],and t hese designs are still used as a benchmark against which to measure the performance of new designs [Kristinsson and Dumont (l996)].If the CD actuation is not penalized in any form (7 and X = 0): the feedback gain mat*

defined by Equation (2.14j

becomes:

and can be considered as t h e left pseudo-inverse for the mode1 rnatrix H. This solution draws a strong parailel with the mat ri^ Inversion (MI) methods to CD profile control (recall that the MI methods of CD control simply attempt to invert the plant model matrtv [Boyle (1977),Dumont

(1990)l). Consider the situation where the CD profile measurernent and actuator spacing are identicai, and the CD impulse response model is symmetric and identicai for each actuator. This is a common case in the literature [Boyle (1978).Wilhelm (1983)l.In this situation. H is a symmetric n x n matrix hrther. if H is non-singular. Equation (2.15) reduces to:

which gives the MI method of CD control. Thus, the MI approach to CD profile control for a square system is a syecial case of the onestepahead optimal control problem with no input move penalization. Various authors have noted that rnatrix inverse approaches have exhibiteci considerable actuator "bending stress" problerns [Boyle (1978),Wilhelrn and Fjeld (1983),Kristinsson and Dumont


30

(1996)j.This optimal control formulation provides a framework in which the source of such exces sive actuation can be understood. Yotice that in the optimal control problem formulation of this section. actuator moves were not penalized: any solution of a problem formulated in this rnanner

will not consider actuat ion restrictions. Since the optimization problem used to design the controller in Equation (2.7) penalizes the actuator changes (Auc) rather than actuator leveis (uk), the resulting positional form of the CD controller will contain integral action. This follows from the standard result for discrete systems [Marchetti et al. (1983)l.

2.5.3

An Example: Matrix Inversion (MI) vs. OneStepAhead with Input

Penalization (IP) The following example is inciuded to contrast the controI possible using a controller based on the general onestepahead optimal CD control formulation in Problem (2.7) and the -MI approach. Recall that the MI approach is the optimal control problem with y = A = O and wiil be contrasted with the IP controller (A = 5 and y = 7.5).

The CD impulse response chosen for this example is shown in Figure 2.7; this type of CD impulse response model is termed a '=flat-top" model and is generdly encountered on a heavy grade paper machine [Measurex Devron (1990)l. As shown in the upper plot in Figure 2.7, the "flat-top" model has non-zero CD impulse weights at the centre zone and the four adjacent zones on either side. The initial b a i s weight profile shows a unity offset at the centre zone.

Cross Directiond Controuer Design

-

-6

-4

O

-2

2

4

6

CD Zone

O

40

20

60

100

80

CD Zone

Figure 2.7: CD Impulse Response and Initial CD Basis Weight Profile

After 100 control intervais. the basis weight and actuator profiles for both the MI controller and the IP controller (y = 7.5 and A = 5) are shown in Figure 2.8. Matrix Inversion

Input Penalization k 5 and ~ 7 . 5

A A and y=û

O

20

60 CD Zone

40

CD Zone

80

100

O

20

60 CD Zone

40

80

CD Zone

Figure 2.8: Basis Weight and CD Actuator Profiles After 100 Control Intervals

100


,As expected. the disturbance has been eliminated by the MI controller (only 1 control move was required. see Figure 2.9). The output profile corresponding to the IP controller shows more

sluggish disturbance rejection. The rnost important distinction between the two control designs is the required actuation. The actuator profile corresponding to the MI xnethod exhibits large bending stress with large offset from the nominal actuator level of zero. The actuator profile corresponding to the IP controller is much smoother and the magnitude of the offset from the nominal actuator level is smaller than that of the MI profile. Of course, this is to be expected because the MI approach is andogous to Minimum Variance Control.

O

20

40

60

80

100

fime Step k

Figure 2.9: Basis Weight CD Variance Trajectory for the MI and IP Controllers As can be seen in Figure 2.9. the MI controller reduces the output variance ta zero after only

one control move. while the IP controller brings the output variance down in a more gradua1 fashion. Due to the integral action inherent in the IP controller, the CD basis weight profile should be returned to the desired flat profile eventually. The asymptotic behaviour of the CD controller systerns mil1 be considered in detail in s4.1. 2.5.3.1

Tuning Considerations

The compromise between profile regdation and required actuation is governeci by the choice of the weightings given to the penalty terms for excessive actuation and excessive bending stress in


33

Problern (2.7): however. there is one more consideration that must be accounted for in tuning of the optimal controller. The bending stress suppression term X (hutlTD ~ (Auk) D is a singular rnatrkx In direct contrast. the penalty term corresponding to excessive actuation. 71, is perfectly condition&. For this reason. it is advisable to tune the optimal controller with y > X (for the onestepahead optimal CD controller design) t O lessen the effects of the singularity introduced through the bending stress term on the feedback gain matrix (K). Figure 2.10 shows the effect of the tuning constants. 7 and A, on the CD basis weight variance profile for the previous example.

Figure 2.10: Effect of the Tuning Constants on the CD Basis Weight Variance

In this section. the onestepahead optimal CD controller has been developed. This will be the CD controller to be approximated in a filter forrn using the approximation methods discussed

in the next chapter.

Chapter 3

Filter Approximation Methods In s2.4. many of the existing methods. available in the literature. for the design of CD controllers were discussed. One of the common themes among these methods was that the resuitant controller was of high dimensionality. However. a large nurnber of existing CD control systems are not able

to support many of these advanced control designs due to their lirnited programrning capabilities ( e-g.. [Measurex Denon (199O)l). The problem presented in this chapter is the incorporation

of some of these advanced control designs using functions that are available within an existing hardware platform.

This chapter will consider the design of filter-based controlleis for CD control problems. A filter has been chosen as the lower-order controller structure since it is a function available in many existing CD control hardware platforms ( e.g., [Yeasurex Devron (lggO)]). However, the following sections will present a generd approach by which high-dimensional controllers can be approximated in some reduced form. These approaches can be extended to other applications as shown by Thake et al. (1997).

The incorporation of an advanced CD controller design into a filter belongs to the generd class of problems termed controller approximation (reduction). Prior to developing the methods for Filter Approximation of advanced CD controllers. the generd problem of controiler approximation is discussed.

3.1

Controller Approximation

In generai. the approximation of a given controller design with some lower (reduced) order form

35

Filter Approximation Methods

is termed a controller approximation (reduction) problem. The lower-order controller may be of a reduced-order and/or of a specified structure. The approximation should strive to match

the performance of the original control design as close- as possible while not causing instability. Furthermore, the controlier approximation should strive to preserve the robustness properties of the original design [Anderson and Liu (1989)l. Anderson and Liu (1989) classify controller approximation into two categories: Direct and Indirect met hods. They use the following diagram to summarize t heir ckssification scheme.

Reduaion

Controiier Approximatioa

Figure 3.1: Anderson and Liu's ( 1989) Classification Scheme for Controller Approximation

The Direct methods consider the design of a low-order controller directly from the high-order plant. The parameters of the low-order controller are usually determined by some optirnization method. The common approach is to minimize a quadratic performance index subject t o the constraints that the controuer be of h e d degree. stable and time-invariant [Anderson and Liu (l989)].

A typical formulation of the Direct Design method involves h d i n g the parameters of a dynarnic stabilizing compensator (Ka) of dimension r < p (where p is the number of states) to minimize the following quadratic performance index [Vasil'eva and Dombrovskii (1995)l:

Typically, a set of algebraic equations can be generated that constitute necessary conditions on the controller parameters to rninimize the performance index. The numerical solution of this set of equations tends to be cumbersome [Vasil'eva and Dombrovskii (l995)]. In addition to these difficulties, the multiplicity of the solutions may be very great [Anderson and Liu (1989)).

36

Filter Approximation Met hods

Even if a satisfactory solution to a Direct design method can be determineci. there are still other limitations to these approaches. The major problem with a Direct method is that there is no benchmark against which to measure the low-order controller performance. More specifically, there is no convenient means of asswing the performance loss due to synthesis of the low-order controller. Secondly, it may not be ciear how to tune a low-order controller for a high-order process to achieve closed-loop goals such as bandwidth [Anderson and Liu (1989)l. Indirect methods are classified into two subcategories: model reduction and controller a p proximation. Mode1 reduction t ethniques approximate the high-order plant as a low-order plant with the approximate plant used for the design of a low-order controller. Anderson and Liu ( 1989) make two criticisms of t his approach. First. the approximation is performed at an early

stage in the method (plant approximation). The approximation error then propagates through the synthesis of the low-order controller. The second criticism stems from the fact that it is the closed-loop system that m u t be approximated for satisfactory closed-bop performance.

This im-

plies that one needs advance knowledge of the controller prior to model reduction. This critickm is &O made by Aguirre (1994) who makes the distinction between open-loop model reduction and

clos&-loop model reduction. Open-loop model reduction attempts to match the process transfer functions while closed-loop model reduction attempts to match the closed-loop system for a giuen controller. Aguirre (1994) clairns that an excelIent open-loop approximation will not necessarily guarantee a reduced-order rnodel which produces an accurate simplified closed-loop system.

The second type of Indirect method is controller approximation. This approach considers the approximation of a high-order controller (designecl using the high-order plant model) by a low-order controller. As noted eariier. Anderson and Liu (1989) state that any controller approximation method must preserve closed-loop stability and strive to maintain the closed-loop performance and transfer function. The approximated closed-loop system is modeled as shown in Figure 3.2.

Filter .4pproximat ion Methods

I

Figure 3.2: Anderson and Liu's (1989) Additive Representation of the Controller Approximation Problem

The presemtion of closed-loop stability is addressed by the following optimization prob is detennined such that the left hand side of Problem lem. The lower-order controller (Ka)

(3.2) is minimized, and it is a stabilizing compensator if the inequality constra.int is satisfied [Anderson and Liu (1989)l.

min Ka

~I [ K ( S )

- K.(s)] G ( s )[If K ( S ) G ( S ) ] - ' < ~ ~1~

(3.2)

Anderson and Liu (1989) pose the probiem of approximating the closed-loop performance of the high-order controller (K) as:

min II [ W s ) Ka

- K&)l

V(s),lI

(3.3)

where V(s) is an appropriately chosen weighting matrix to d o w frequency weighting of the approximation. Finally, the presemtion of the closed-loop transfer hinction is formulated using a 6 r s t order approximation of the complementary sensitivity matrices [Anderson and Liu (1989)l:

Further, Anderson and Liu (1989) state t hat convenient algorithms to solve the kequency-weighted approximation problems above are not available. The use of "brute force?' optimization is generally precluded due to the high dimensionality of the problems, and, in practice, state-space mode1


38

reduction methods. such as the optimal Hankel norm approximation. are used to give suboptimai solutions [Anderson and Liu ( 1989)1. The case of discrete controller approximation was recently addressed by hfadievski and Anderson (1995). A lifting technique is used to convert the Periodic Time Varying system (continuous plant and discrete controller) to a time-invariant system. The second step of their re duction algorithm involves minimization of an objective function similar to that of Problem (3.4) [Anderson and Liu (1989)l. It should be noted that there are no explicit constraints on the closedloop stability of the approximated closed-loop system. Secondly, it is not the closed-loop systems

which are being matched: it is equivalent to a weighted minimization of the mismatch in the open-loop controller responses. This problem is essentially a model reduction problem.

3.1.1

Analogy to Open-Loop Model Reduction

-4nderson and Liu (1989) claim that the problems of open-loop mode1 reduction and controuer approximation are fundarnental1y different . They argue t hat open-Ioop model reduction is based

on open-loop considerst ions. while any cont rolIer approxirnat ion scheme should account for the existence of the plant. preserve closed-loop stability and preserve (as far as possible) closed-loop performance. Figure 3.3 summarizes the similarities and differences between model reciuction and controller approximation [or both the open and cioseci-loop cases. The process t r a d e r function matrk is denoteà by G . while the sensitivity matrix is denoted by S (the approximations are indicated by a subscripted a).

Filter -4pproximation Met hods

OPEN-LOOP

CLOSED-LOOP

MODEL

REDUCTION

CONTROLLER

APPROXIMATION

Figure 3.3: Cornparison of Mode1 Reduction and Controuer Approximation for both Open-Loop and Closed-Loop Met hods

As shown in Figure 3.3. the problem of open-loop model reduction considers the problem of determining the pararneters of an optimum reduced-order model, Ga, such that some approximation criterion is rninirnized. In the past. model reduction has been used as a technique to simpIifv control design5 by decreasing the dimensionality of the dynamic system. Typically, an open-loop model reduction algorithm will compare the output responses of the original and reduced-mode1 for some input signal (usually a step or an impulse) [Edgar (1975)j. Closed-loop model reduction is quite similar; however. for a regdatory control problem, the pararneters of the low-order plant models (for a given controller) are determined to minirnize the misrnatch in the sensitivity matrices corresponding to the high and low-order plant models [Aguirre (lgM)]. The controller approximation schemes are also shown in Figure 3.3. The open-loop controuer approximation approach may be considered to be @valent

to the open-loop model reduction

prcblem. A particular error profile (input signal) is passed through the fuil controller (original model) and the controller approximation (reduced-order model) with the pararneters of the controller approximation (parameters of the reduced order model) being determineci to minimize some objective hnction [Aguirre ( l9W)1. Similarly, the closed-loop controller approximation problem


40

can be considered to be equivalent to the closed-Ioop rnodel reduction problem. The only ciiffer-

ence. in this case. is that the parameters of the low-order controller are to be deterrnined rather

than those of the low-order plant model. and the plant model is k e d in this case.

3.2

The Filter Approximation Problem

One approach to CD profile control. currently being used in some industriai CD control systems, is filtering of the measured CD error profile [Measurex Dewon (1990)].Curent practice involves

detennining the filter weights in a such a marner as to bbdeconvolve5the process effects from the

CD error profile. leaving the underlyîng actuator moves which could produce the error profile. In order to remove the observed CD error profile. the negative of these actuator moves is implemented to compensate for the CD error. This approach to CD profile control is referred to as ''spatial filtering or "convolution decoupling" [Neasurex Denon (1990)]. For the common case where the actuator spacing fiequency is identical to the CD zone spacing kequency used by the controller [Heaven et al. (l994)], a square control problem results (n = m). Consider the following error profile:

where et(,,

11

is the error profile. y;+ll,,

is the desired profile at the next sampling instant and

is the current property profile. The technique of spatial filtering is summarized by the

following equation:

where * denotes convolution. x is the spatial fiker of length W . and the filter length (w) is an odd integer (typically w

« the

number of CD actuators (n)). The filter length is odd due to

the physicd nature of the system. If an actuator is adjusted. the response will be typically observed at the scanning gauge in both the corresponding zone and an equal number of adjacent zones on either side. The filter length can be considered to be a "tuning" parameter of the CD control design (the effects of filter length will be discussed in 55.2)- The pre-multiplier matrix

Filter Approximation Slethods elements of e& * x. InO [n ] t runcates the first and last The ph-mica1 justification for the truncation of the convolution is due to the nature of the physical s-ystem under consideration (the wet-end of a paper machine). Adjustment of a CD actuator flexes the slice lip which generates water waves on the wire. This results in the basis weight being iduenced at several zones adjacent to the actuator [Kristinsson and Dumont (1996)l.For

"-

example. it is reasonabie that one would want to calculate the 6rst actuator move. Aul, baseci

on the observed response in the first Only the first

zones rather than just the corresponding first zone.

9 zones are considered because the center element of the filter, x&,

2

should

coincide with the zone in which the C D actuator is located. The interactions bekveen zones on

a paper machine m u t be taken into account in any CD control design. The resultant instability of CD control systerns that fail to acknowledge this inherent zone-to-zone coupling has been well documented [Karlsson et al. ( 1985)1. The convolution operation in Equation (3.6) can be conveniently reformulated in terrns of standard rnatrix operations. Then. design of a spatial filter involves finding the elements of a

6lter x~~~ such that the set of actuator moves. Auklnx11 is given by:

where A[nxwihas the forrn:

Filter Approximation 'ulethods

42

where the ei is the ich element of the error profile ee. The pre-multiplier matnx above is as defined in Equation (3.6).

The problem of determining the elements of the 6iter may be considered as the solution to the over-determined set of linear equations in Equation (3.7). The problem defined in Equation (3.7)

can be equivalently writ ten:

where Xi,,,i

has the form:

where xz are the unknown filter weights and the pre-multiplier matrix takes the same form as in

Equation (3.7). There are two interesting interpretations of the problem posed in Equation (3.9): input-out put matching and controller out put matching.

3.2.1

Input-Output Matching Interpretation

In the past, the design of a spatial filter for CD control problems has been tackied in temx of matching input-output data [Measurex Denon (1990)l.The problern posed in Equation (3.9) may be interpreted in terms of determining a matrut X such that post-multiplying by the observed profile response, Ay, from an identification test reproduces the test input signai, Autest. Exact matcbng of the input-output data can only occur with a perfect process model, Le., X = H-'.

Filter Approximation Met hods

43

Thus. attempting to determine the filter elements based on matching input-output data by methods such as Ieast squares is equil-alent to approxirnating the inverse of the process mode1 with a

filter. This can be seen by comparing Equations (2.4) and (3.9). Such approaches would sutfer similar difficulties to the Matrix Inversion approach (see 52.4.1).

3.2.2

Interpretation as Controller Output Matching

The second. and more relevant. interpretation of Equation (3.9) is manifesteci in the similarity between the feedback control iaw given in Equat ion (3.12) and the reforrnulated Nter caiculation in Equation (3.9). Llore specifically. both equations airn to determine a set of actuator adjustments,

Auki to compensate for an obsemed error profile. ek. In this interpretation. X takes on the role of a feedback gain matriu. and we can write:

where K represents the high-order controller to be approximated. The above problem amounts to attempting to match the controlier outputs in some manner. It is crucial ta note that any

feedback gain matrü. K, resulting from an advanced CD contml design may be inserted into Equation (3.11). Since the CD contro1 problem is generally a regdation problem with the desired profüe being uniform. the setpoint profile is y ; ,

= O. Equation (3.5) becomes et = -y&, and

Equation (3.11) becomes:

or:

The previous equation c m be interpreted as determining the filter elements to approximate a

given feedback gain matrix: K. in some manner. There are a number of possible approaches that can be used to approximate the feedback gain matrix by a filter. It is worth noting that the filter approximation problem posed above involves the approximation

44


of a feedback gain matris of large dimension i typically having over 10000 independent elements) by a spatial filter-based fë-edback gain niatrix

(X) which typically has 15-35 independent elements

(the filter weights). The procedure of filter approximation is summarised below.

K(adz7anced

CD)

-

The following sections will propose severai different methods for spatial filter design. The first. and most simple. method involves a least squares approach to match the outputs of the original and filter-based controllers for a given disturbance direction. The idea of matching the outputs of the controllers is extended to a design that is independent of disturbance direction using Semidefinite Prograrnming. These two approaches. the Least Squares and Semidefinite Programming approximation methods. are based upon matching the outputs of the full and filterbased controllers. The controller approximation techniques are exampies of open-loop techniques.

A Direct method for filter design is also proposed in 53.5.1 w h c h does not rely on the design of

an original controller to be subsequently approximated. Finally. the design of a fiiter based on an approximation of the original closed-loop system is proposed.

3.3

Least Squares Filter Approximation

Since X contains only a small number of parameters ( w fiiter weights where w

let hods

60

the scope of controller approximation problems that can be addresseci with SP. 2.e.. non-linear problem formulations. The SSP niethod. dthough very basic. may provide motivation for other research into solving SP problems induding non-linearities.

3.7

Summary

This chapter has classified the filter approximation problem as one of controller approximation. The similarities/differences with standard mode1 reduction have been given. In order to solve the filter approximation problern. a number of methods have been proposed. -4 constraint on the location of the ciosed-loop poles has been formulateci as an L'c,II. Finally. the SSP method was

developed to address the closed-Ioop approximation problem. These methods will be compared using performance measures. developed in the next chapter. via a case study in Chapter 5.

Chapter 4

CD Control System Analysis If an advanced CD controller is to be approximated as a filter using the methods in the previous chapter. one must possess a means of assessing the performance and robustness losses due to the approximation. The anaiysis of a filter-bas4 CD control system is the focus of this chapter. In particular, the asymptotic behaviour and robust stability of the resultant closed-loop system are investigated. -41s0, a set of performance measures are developed to quantifi the performance loss

due to the filter approximation. measured relative to the original CD controller.

4.1

Asymptotic Behaviour of the Closed-Loop System

Once a filter-based CD control system has been designed. it is necessary to investigate the ability of this controller to compensate for some CD error profile. In order to assess the asymptotic

behaviour of a CD control system. it is convenient to express the closed-loop system in a statespace form. In this way, the asymptotic behaviour of the CD control systern in compensating for an existing error profile c m be interpreted as the state response to initial conditions. The system

may be written in terms of a state-space mode1 as:

where q+,

11

feedback law:

is a vector of the States. The set of actuator moves is calculated using the standard

CD Control System Analysis

The feedback gain matrix (X) is arbitrary for this discussion but must give a stable closedloop system.

This proviso aliows the standard Final Value Theorem (FVT) to be applied

[Ogunnaike and Ray (1994)l. Sote that Equation (4.2) is a feedback control law in a velocity forrn. hlarchetti et al. (1983) show that a velocity form of a control law contains integral action.

Using Equation (4.2), the state equation from Equation (4.1), with yi,, = 0. becomes:

The z transform of

qr;+i

is given below [Vande Vegte (1990)l:

where z is the shift operator. The difference equation in Equation (4.3), with initial conditions qo, can be expressed in z transforrns using the above equation. The initiai state condition (q*)is the initial offset from the desired profile when the CD controller is engaged.

Equation (4.5) can be rearranged to give the state response t o the initial conditions. The output response also follows easily since yk = Iqk and y0 = 1%.


The FVT can be used to determine the asymptotic CD output profile. yk as k -. oo, in response to an initiai offset profile [Ogumaike and Ray (1994)l.

lim yk = lim(1 - Z - ' ) ~ ( Z )

k-a3

z-

1

The previous result indicates t hat the output profile will be returned to the desired unifoini profile in the limit as k

-

m. According to Equation (4.6),this result is true for any initial offset

profile. yo. Even with the filter-based controller. integral action is contained in the resulting controller, and zero steady-state offset is obtained. Although the above result shows that the CD profile is eventudy returned to its setpoint. there is no guarantee regarding the time required for this to occur. This asymptotic behaviour of a filter-based CD control system wiU be confirmeci

through a simulation example in the next chapter.

4.2

Performance Criteria for Filter Approximations

This section considers the development of performance criteria to quanti& the quality of a flter approximation. When considering CD controller performance criteria, it is important to malce the distinction between a measure of CD controller performance and a measure of the performance

l o s due to approximation. It is the second of these measures which is the most relevant to this work.

As stated earlier. it is assumed that the CD controller to be approximated possesses adequate performance which. ideally. would be matched by the filter approximation. A performance criterion specific to optimal CD controllers is developed first: the r measure. Then, a means of assessing approximation quality applicable to the general class of CD controllers is developed: frequency response of the controller/filt er mismatch. Before discussing t hese two criteria, existing

CD controller performance measures are reviewed.

CD Control System h a l y s i s

4.2.1

Existing CD Controller Performance Criteria

In the literature. the benchmark performance measure of a CD controlIer is the 20, or

cri trajectory

(over successive control intervais) where a,(P)is the standard deviation of the output CD profle at interval *'F. The main problem wit h this approach is that it does not include a measure of the required actuation which has been identified by many authors as a concern [Boyle (1977), Dumont (1990)l. Karlsson et al. (1982) make the point that the information provided by 2ay or

0; trajectories

should be interprered with care. More specifically. two profiles rnay have the

same variance while possessing vastly different CD profiles (one of which rnay be acceptable while

the other rnay not be acceptable). i n e n comparing CD controllers. the approach taken in the literature has been to run a simulation and compare the controllers in terms of their 2uy or

O$

trajectories and actuation [Kristinsson and Dumont (1996),Chen et al. (1986)l. The design of robust CD controllers, in the cont inuous time domain. has used the Structured Singular Value to compare various design schernes [Laughlin et al. (1993)). Since the main thrust of this work is to propose methods by which CD controllers of hi& dimension rnay be approximated as a filter. a performance measure must naturally indicate the

quality of a filter approximation. 4.2.2

r measure

Since the "standard" CD controIler rnay be considered a variant of the one-çtepahead optimal controller originally proposed by Boyle (1977),the trajectory of the respective objective functions is a usefui tool for comparison of the original controiler and its filter approximation. The comparison of the objective function v a l u s gives a direct indication of the approximation quality and considers the actuation required in addition to the behavior of the output profiles. As discussed in 33.3.2. the control system rnay exhibit very different performance for two different disturbances due to the high-directionality of the system (as evidenced by the ill-conditioning of the plant mode1 matrix. H). Thus. it would be more accurate to consider the relative performance of the two control systems given the same error profile, rather than comparing objective function trajectories. This is because the two control systems rnay not be encountering a common profile after the first control move which does not provide a fair basis for comparison of performance.


65

For the general optimal CD control design. the following mesure is proposed as a means of quanti-ng

where

P'

the performance degradation due to the approximation.

and

Pa, are the values

of the objective function for the original controller and its

approximation respect ively. For the specific case of one-stepahead optimal control. the original objective function kom

which the LQ controuer is designeci is shown below:

where:

K =( H ~ H + 71 t A D ~ D ) - ~ H ~

(4.10)

At the optimum. the value of the scalar objective function is found by substituting the optimal value of the decision variable (Auk) into (4.8):

Popt = (Y;+, - Y where

~ ~ N K ( Y L +-~ YL)

(4.12)-

N K = (1- HK)'(I - HK) +TK~K+X(DK)*(DK).Following the approximation of the

controller by some method. i-e. Least Squares or Semidefinite Programming, the value of the objective function for this approximate controller. X. is given by:

66

CD Control System .4nalysis

where Nx = (1 - H X ) ~ ( I HX)+?X~X+A(DX)~(DX).

As defmed in Equation (4.7). a measure of the performance degradation c m be defined as:

where a perfect approximation corresponds to r = O. Then, the bourids are O 5

T

5 cm. This

measure is only meaningful nhen the value of the objective Function. Pqt is non-zero for the given error profile.

In mathematical t e m . the rnatrix N K = (1 - H K ) ~ (-I HK)+K*K+A(DK)*(DK)

must

be positive definite. The following explanation holds only for the case where y and X are non-zero

(not the Mi approach). In order to show that the matrix N K is positive debite, the controller,

K, must be of full rank. Recall t hat for an invertible plant mode1 matrix (H), the optimal CD controller is generated via an inverse (refer to equation (4.10)), and if K exists. it will be invertible

which implies Ml rank. Since the sum of positive definite (positive semidefinite) matrices is a positive definite (positive semidefinite) matrix [Hom and Johnson (l985)],showing NK to be positive d e f i t e arnounts to showing one of the three te-

to be positive definite. Since the

control problem has been assumed to be square and K has been assumed to be of full r d , the t e m K ~ isKpositive definite [Ortega (1985)l. Thus, NK is positive definite. The same result is true for the filter-based controller, X. It is of full rank due to its structure (refer to Equation

(3.10)). 4.2.2.1

Benchmark Values of r

It shodd be noted that the performance ratio given by Equation (4.14) is directly affect4 by the direction of the error profile (y;+l - y k ) . The following discussion will propose a rnethod by


67

which a set of benchrnarks on the performance ratio may be determineci. The numerator of the performance measure in Equation (4.14) is the difference in the d u e s of the objective functions:

Papp - Popt = (Y;+' - ytlT (N.Y- N K ) The matrix

- Y*)

(4.15)

- N K governs the difference in the objective function values for a given

error profile. Assurning that the approximation is imperfect.

Nx - NK will be a square. posi-

tive definite m a t e . Further. the eigenvalues and singular values of such a matrix are identical

[Hom and Johnson (1985)l.B y perfoming an SVD on this matriu. it is possible to determine the srnailest and largest d u e s of the performance measure. The upper and lower benchmarks on r are defined as the ratios of the following singular values:

Given an error profile, these benchmarks rnay be used to assess the performance degradation to be expected due to the fiiter approximation. During operation, the value of r can be generated to assess the performance loss at each control interval relative to the original CD controller. However. it should be noted that this measure of performance degradation is only applicable to

CD controllers designed from the rninimization of some objective function. A means of comparing the original CD controller and its filter approximation, that is more widely applicable. is required.

4.2.3

F'requency Response of the ControIler-Filter Mismatch

Recail that the control of CD profiles is. in almost d l cases, a regulation problem. Based on t h , it is reasonable to conclude that one would like the sensitivity matrices of the original and filter approximation controllers to be similar. As in Chapter 3, let:

CD Control Systern Analysis

68

be the sensitivity matrices of the original CD controller and filter approximation respectively. If the mismatch between the control systerns is modeiled as being additive:

X=K+E where E contains the controller mismatch. the following holds:

or, using Equation (4.20): z

[I - (1- H(K i E))2-11 S = 1

The previous equation can be rearranged to give an expression for the approximate sensitivity matrix in t e m of the original controtler sensitivity matrix and the mismatch matrix

+ SKHE!-'. Please refer to Appendix A.3 for the fdi development of Equation

where M = [1

(4.22). The expression in Equation (4.22) has a physical interpretation (refer to Figure 4.1). The output response of the approximate sensitivity mat*

(yTt) is the output response of the original

systern (yqt) passed through the mismatch contained in M.

Figure 4.1: The Approximate Sensitivity Matrix

Using the Cauchy Schwarz Inequality [Skogest ad and Post lethwaite (l996)],the foUowing holds:

CD Control System Anaiysis

The previous result gives an upper bound on the norm of IISII. Since the original controller is

-

assumed to possess satisfactory performance which the filter attempts to match. it is desired that

/(SI1 llSKll which corresponds to llMl( 5 1 over the frequency range of interest. Using the above resuit . the quality of the filter approximation can be examined by generating the kequency response of the mismatch matrix. M. where:

The mismatch between controliers can be examined as a Eunction of Çequency. Although these controllers are designed to compensate for steady-state disturbances. the closed-loop system is a d p a m i c system as evidenced by Equation (4.18). Figure 4.2 shows an exmple of a singular value frequency response plot of M. which is the analog of a Bode plot for Multi Input Multi Output (MIMO) systems [Ogunnaike and Ray (l994)].Any system response is contained in the band containing the curves describing the frequency response of the maximum and minimum singular values of the rnismatch. The highest amplification is given by the curve correspond-

ing to 8(M), while the smallest amplification is denoted by the curve corresponding to g(M) [Skogestad and Postlethwaite (1996)j.If a portion of the band containing a system response Lies above OdB. the rnismatch in the controllers can result in amplification of error for a given distur-

bance for the filter-approximat ion cont roi system. Whet her or not amplification is encountered depends on how the disturbance is alignecl with the mismatch [Skogestad et al. (198811.Inspection of this plot yields information regarding the performance degradation to be expected over the frequency range of interest.

CD Conti01 System halysis

Figure 4.2: Frequency Response of the Filter/Controiler Mismatch

4.3

Robust Stability of the Closed-Loop System

Any controuer approximation should preserve the robustness to modelling uncertainty. The CD impulse response models used to design one-stepahead controilers can vary significantly in dayt d a y operation [Chen et ai. (1986)l.In fact. a paper machine may experience changes in machine speed, headbox consistency and other wet-end conditions that may affect the CD response in both magnitude and shape [Laughlin et al. ( 1993)l. For this reason, a satisfactory filter approx-

imat ion should exhibit similar robust ness to plant-mode1 mismatch as the original CD controiler.

4.3.1

Types of Uncertainty

Two types of uncertainty will be considered in this work: gain uncertainty and edge effects. Gain uncertainty is considered to be uncertainty in the individual elements of the CD impulse response for a given actuator. The second major form of uncertainty to be treated in this andysis is edge effects. Edge effects are common on most paper machines and invoIve the different behaviour observeci at the edges of the sheet in the CD. These effects are due to slurry falling off of the machine or waves hitting the guards (deckieboards) [Adler and Marcotte (1994)l.

CD Control S-ystem Analysis

--

Figure 4.3: Plant-Mode1 11ismatch Uncert ainty Description The uncertainty due to edge effects and gain uncertainty wiil be considered to be additive as shown in Figure 4.3. The uncertainty in the elements of the mode1 matrix (H) is contained in the uncertainty rnatriu. U. For t h e case of simple gain uncertainty. the true plant Ml1 take the

following form:

where b i , is the uncertainty in the CD impulse parameter

wi,j- Gain

uncertainty will be modelled

exactly as shown in the above equation.

Edge effects involve uncertainty in the CD impulse response near the edges of the sheet in the

CD. Let e be the nurnber of zones subject to edge effects at each edge of the machine. In this case, the uncertainty matrix becornes:

As shown in Equation (4.26). the case of edge uncertainty is a special case of the general gain

CD Control System Andysis

72

uncertainty. The significance of gain uncertainty and edge effects will contrasteci via simulation in the next chapter. The following discussion gives an overview of the methods used.

4.3.2

Robust Stability Analysis via Monte Carlo Methods

A number of advanceci methods exkt for Robust Stability (RS) anaiysis such as Structured Singular Value analysis [Skogestad and Postlethwaite (1996)l and Rç of Interval Matrices

[Keel and Bhatt acharyya ( l995)!. However. these methods require considerable cornputation.

Due to the reiative simplicity of Monte Carlo simulation methods. this method was chosen for performing the RS analysis. More specifically. a "crude" Monte Carlo method will be uçed. Crude Monte CarIo methods involve the generation of some random observations from a distribution [Hillier and Lieberman (lggo)]. In this case. a matrix. ZfnXnI, of random numbers is generated. Each element. cZJ,is a random number â o m a uniform distribution where -1 5

Gj

5 1. By generating n2 random numbers,

the uncertainty in a given element wiIl be independent of the uncertainty in other elements of the model matrix, Also. this description allows the variation in the CD impulse response hom actuator to actuator to be addressed. .4 uniform distribution was chosen because a level of uncertainty is to be tested. More specifically, one would like to test combinations of uncertainty across the entire range to assess whether closed-loop stability is attained for the given percentage uncertainty. Each element of the uncertainty matrix d l be defined as follows:

where q is the % uncertainty in an impulse model coefficient estimate and wu is the nominal

rnodel coefficient. The actual model coefficient, h, = d i j

+ r)wij

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