CONTROL ENGINEERING LABORATORY

One- and two-dimensional control of

paper machine: A literature review

Markku Ohenoja

Report A No 38, October 2009

ii

University of Oulu

Control Engineering Laboratory

Report A No 38, October 2009

One- and two-dimensional control of paper machine: A literature review

Markku Ohenoja

University of Oulu, Control Engineering Laboratory

Abstract: This literature review introduces the controls of paper machine, basis weight being the

example case. As control problem is highly affected by the sensing system and the data processing, also

different sensor arrangements, with their shortcomings, and the most common data filtering techniques are

explained. Both the cross-directional (CD) control and the machine-directional (MD) control are presented.

Also, a brief description of two-dimensional control theory is given.

MD and CD control loops require different measurement information. However, the data coming from the

most traditional scanning sensing systems is mixed. The common data filtering techniques assume the

separability of the MD and CD, where large part of the variation is accounted in residual variation and only

part of the variations can be removed. Model predictive control (MPC) has gained a lot of attention in paper

machine control and it has been applied to MD control some time ago. MPC provides good tools for

controlling the more complex CD process also, but the size of the problem is huge so certain

approximations are inevitable. Besides the input-output model of the control algorithm, CD system requires

often models or on-line identification to compensate the sheet shrinking phenomena.

Focus of the research has naturally changed with time. First, the reduction techniques to make the CD

control problem feasible were investigated. Later, as MPC has become more common, also the robustness

issues, i.e. model accuracy and uncertainty, have become more important. Also the alignment (shrinkage)

problem has gained a lot of attention lately, since it is an essential part of the model accuracy. Also, the

research in the area of sensing systems has been continuous.

Keywords: paper machine, web forming, cross direction control, machine direction control, two-

dimensional control

ISBN 978-951-42-9316-0 University of Oulu

ISSN 1238-9390 Control Engineering Laboratory

PL 4300

FIN-90014 University of Oulu

iii

Contents

1 INTRODUCTION ...................................................................................................... 1 1.1 Scanning sensor ...................................................................................................... 2 1.2 Optional sensing systems ........................................................................................ 3

2 PROFILE ESTIMATION ........................................................................................... 5 2.1 CD/MD separation .................................................................................................. 5 2.2 Reducing the computational load ............................................................................ 7 2.3 Discussion ............................................................................................................... 9

3 MACHINE-DIRECTIONAL CONTROL ................................................................ 11 3.1 Sources of the variations ....................................................................................... 11 3.2 Control strategies in MD ....................................................................................... 12

3.3 Discussion ............................................................................................................. 13

4 CROSS-DIRECTIONAL CONTROL ...................................................................... 14 4.1 Sources of the variations ....................................................................................... 14 4.2 Mapping the actuators and the measurements ...................................................... 15

4.3 Response models ................................................................................................... 17 Spatial response of a slice lip actuator ............................................................. 18

Spatial response of a dilution actuator ............................................................. 20

Dynamic response ............................................................................................ 21

Two-dimensional response ............................................................................... 21 Disturbance models .......................................................................................... 21

4.4 Control strategies in CD........................................................................................ 22 Linear control ................................................................................................... 22 Model predictive control .................................................................................. 23

4.5 Discussion ............................................................................................................. 26

5 2D-POLYNOMIALS ................................................................................................ 28

5.1 2D linear control ................................................................................................... 29 Prediction.......................................................................................................... 29

Identification .................................................................................................... 30 Control .............................................................................................................. 31

State-space representation ................................................................................ 32 5.2 2D model predictive control ................................................................................. 33 5.3 Discussion ............................................................................................................. 34

6 SUMMARY AND CONCLUSIONS ....................................................................... 36

7 REFERENCES ......................................................................................................... 39

1

1 INTRODUCTION

Introduction to the controllable parameters of paper machine and the conventional

scanning sensor system are explained first. Additional measurement systems are also

introduced briefly. For a quick overview of this literature review, see Chapter 6.

There are four main quality parameters to be controlled in paper manufacturing: basis

weight (grammage), moisture, ash content (filler content) and caliper (thickness). The

control problem is two-dimensional, since variations occur both across the paper web and

along the production line. However, separate control systems are used for the machine-

direction (MD, along the line) and cross-direction (CD, across the web). Basis weight in

the MD is controlled by adjusting the amount of pulp flowing into the headbox and in the

CD by controlling the pulp flow distributed to the wire (slice lip) or the consistency of

pulp flow distributed to the wire (dilution) in different locations. The MD control takes

care of the setpoint value of the basis weight and the CD control tries to distribute the

pulp evenly to the wire. The control problem is illustrated in Figure 1.1.

Figure 1.1. Estimation and control of basis weight variations [11].

This literature review concentrates on basis weight, but the nature of the problem is

similar for all properties mentioned above. Variations in moisture are controlled with

steam boxes and water sprays. Caliper is mostly affected by the pressure of the calender

2

nips. Other paper properties, such as formation, gloss and opacity are not considered

here. It should be noted, that there are strong interactions between variables.

1.1 Scanning sensor

Control actions are based on the measurements which are usually provided by the

scanning sensor. These measurements can be taken only from sufficiently dry paper web

at the end of the production line. This causes long time delays between the actuators and

measurements. Additionally, since the sensor travels slowly back and forth across the

web, while the paper web moves fast in other direction, the forming measurement path is

zig-zag-pattern (see Figure 1.2). The horizontal parts in the path correspond to the fact

that it takes some seconds to change the direction of the scan.

Figure 1.2. Measurement path caused by the movement of the paper web and the

traversing scanner.

The producing path can be an angle less than one degree and only small amount of the

paper web, down to 0.001 %, is actually measured [5]. The measurement resolution

calculated from the experimental setup in [48] is roughly 1.7 centimetres in the CD and

1.2 metres in the MD assuming that the MD value is estimated at every sample. Based on

the same setup, the scanning angle is 0.85 degrees. Usually a bit higher resolution values

are reported [10]. The specifications of the calculations above are given in Table 1.1.

Table 1.1. Calculation of the measurement resolution.

Width of the

sheet (CD), W

Number of data-

points in CD, N

Speed of the

machine, v

Time of a

single scan, t

10 m 560 1600 m/min

≈ 27 m/s

25 s

Sampling

frequency, F

CD

resolution

MD resolution Scanning

angle

tN = 22.4 1/s

NW = 0.017 m

FW = 1.205 m

tvW

1sin = 0.848

3

Another problem related to the traversing scanner is the separability of the measurements

in MD and in CD. The measurements contain mixed information from which the CD and

the MD variation need to be estimated. At simplest, an exponential filtering is used to

capture the CD variations i.e. the measurements in each CD position is a weighted

average of several scans. Correspondingly, the MD variations are estimated from the

average of a single scan. This however results in slow detection of upsets and slow

control response. [11] Also, the variations are not totally separable. Large part (70%) of

the short term variation is accounted in residual variation which is uncorrelated either

with CD or MD. Almost all variations on MD scales shorter than 10 metres are random.

[39, p.35] Additionally, some MD variations will be aliased into the CD measurement

due to the periodic nature of the variations and the sensor path.

1.2 Optional sensing systems

There have been efforts to increase the measurement resolution. This would facilitate the

CD and MD measurement separation and therefore profile estimation. Most convenient

way would be using several adjacent scanning sensors in the same frame or add a fixed

sensor array to the process, but the sensor technology is very expensive. Inspection of the

measurement patterns in different sweeping directions and ranges with more than one

scanning sensor demonstrates that the same pattern can be achieved with different

arrangements [56]. The best estimation results in spatio-temporal domain were achieved

with a row of scanning sensors, each of them sweeping only part of the paper web.

However, considering also the engineering point of view of the sensing system, an

arrangement with full web sweeping in opposite directions was more preferable (i.e. less

frequent accelerations and decelerations) [56]. Also, adding a single stationary sensor to

detect variations more efficiently has been considered [17], but the improvement to the

CD closed-loop system performance may be quite moderate in some cases [22].

One recent development to the scanning sensor is the possibility to control the movement

of the sensor. The scanning speed and width is changed to suit the control situation and

the process state i.e. the scanning takes place where the faster measurements and better

estimates are required. [76] The sensor paths of the above mentioned sensing systems are

illustrated in Figure 1.3.

4

Figure 1.3. Different sensing system arrangements. Multiple scanning sensors (a),

multiple scanning sensors with different scanning directions and phases (b), array of

stationary sensors (c) and situation-based scanning (d).

Another effort is to get the measurements closer to the start of the production line i.e.

improving the dynamic bandwidth of the control system. Additional measurements could

be available from the existing camera-based systems in paper machines, which are

currently used detecting the dry-line in wire section (of a fourdrinier paper machine i.e.

single-wire) or defect detection from the finished base paper. Dry-line camera option is

examined in [5] and [9], where the information from the camera is converted into digital

form. However, the information was not yet applied to an on-line automated process

control. The restrictions on data processing have not made it feasible to use the

information coming from the defect detection system, which uses an array of high speed

CCD cameras. However, recent development of data storage capabilities and data

processing techniques may help to overcome this problem. One example of using CCD

cameras to estimate sheet properties and variations has been given in [19]. It is expected,

that these camera-based systems can offer the kind of information needed in control and

can fulfil the requirements of the real-time data processing in near future. Also other

optical measurements have been proposed. One of them used light transmissisivity signal

from the stationary array of sensors and from the scanning optical sensor at the wet end

of the paper machine, where the latter one calibrates the first one. Additionally, the

traditional scanning sensor in the dry end was also required for the calibration. The

control signal was generated for each slice lip positions. [2] Some measurement

applications related to other paper properties than basis weight can be found for example

from [39, p.50].

Improving the sensing system on paper machine, as well as the actuator speed, could lead

to the need of two-dimensional signal processing and control systems. Hence, this paper

gives an introduction both to the one-dimensional control of basis weight, which assumes

the separability of the CD and MD, and to the two-dimensional control. It is notable, that

methods used in the academic field may differ significantly from the applied methods in

industry.

5

2 PROFILE ESTIMATION

CD control loops make control actions to different CD positions at the same time instant.

That is why there have to be a measurement available at each time instant in each CD

actuator position. As we know, traditional traversing scanning sensor can’t provide this.

Sometimes this problem is handled by assuming that measurements are taken at the same

time instant since the variation in CD is much slower than in MD. More sophisticated

option is to calculate estimations for the measurement. Chapter 2.2 concerns the need of

reducing the dimensions of the cross-directional input-output model, where the number of

measurement locations has the most significant contribution. The methods used have a

tendency to smooth and filter profiles in a preferable way.

2.1 CD/MD separation

MD profile is traditionally calculated simply from the average of each scan. Improving

the MD measurement bandwidth by several orders of magnitude is possible by providing

estimates of the MD variations as the scanner moves across the web. Other methods are

concerning the improvements made to the measurement hardware. [5]

Sometimes also CD profile is averaged as the mean of the last 4 to 10 scans. Besides

producing slow control action, this method poses several other problems in profile

estimation [1, p.209]:

Mixing of CD and MD variation in estimation (aliasing), when MD variation has

the period close to the scan time

MD variation that has twice larger period (i.e. 8-20 scans) than the observation

period that CD estimate uses can lead to “flying profiles”, where estimates

become unstable close to the edges (i.e. MD trend tilts the profiles and the

direction of tilting changes from profile to profile due to the fact that every other

profile measurement is obtained in reverse time order)

Diagonal waves and excessive residual variation make the estimate noisy

More detailed description of this aliasing problem can be found from [39, p.32]. Aliasing

may also occur with an array of sensors due to insufficient spatial filtering within

measurement process [14].

More efficient estimation methods for CD/MD separation use models to provide

estimates of the variations at any point of the sheet. Usually these are based on the

Kalman filtering and least-squares optimal estimation theory. Some examples are:

periodic, time varying Kalman filter, e.g. [17,22]

two-dimensional transfer function, e.g. [68,69]

recursive least-squares approach and extended Kalman filter, e.g. [11]

wavelets, e.g. [10]

principal components analysis, e.g. [35]

6

Expectation-maximization algorithms, e.g. [38]

Extended generalised sampling theorem, e.g. [7]

Both the moisture estimation model and the basis weight estimation model have been

derived in [71]. The nonlinear moisture model assumes known MD dynamics, but the

linear basis weight model allows estimating the MD dynamics with Kalman filtering. The

moisture model was improved in [72]. The related works can be found from [11] and

[73].

The full two-dimensional variation can be constructed from the scanned data in Fourier

domain. The analysis in [7] is based on generalised sampling theorem and it considers

scanning as a “one-dimensional” sample of a process that is varying in two dimensions.

Variation can be constructed at any point by taking a linear combination of the

measurements at this CD location on each scan. The analysis shows that the angle of the

scan is a design parameter.

More straightforward solution to the full two-dimensional variation estimation is given in

[17], where stationary sensor is used with scanning one. Additional sensors may improve

the estimation error, but due to the significant system delay, robustness considerations

and model and measurement accuracy, obtaining a better CD estimate may not

significantly enhance closed-loop system performance [22]. Another possibility is to use

fixed array of sensors, which takes measurements across the full width of the sheet at

each sample time i.e. profiles are measured continuously. This way the process

disturbances can be modelled [17,36].

Figure 2.1 shows the need of a good estimation technique for CD profile. Unprocessed

signal (dotted line) is very far from the actual CD variation (solid line) due to the MD

variation mixed to it. Exponential averaging (dash-dotted line) filters some of the MD

variation, but the mean level of the CD variation is incorrect due to the lag caused by

filtering. [7] Reconstruction with the estimation method (linear combination in frequency

domain) used in [7] shows a good performance and would be an obvious choice to use in

control system (dashed line).

It should be noted, that CD profile or CD dimension may refer either to the actuator

position or to the measurement position. There is a difference between the CD

dimensions of these two due to the shrinking phenomenon and other web alignment

problems. These are addressed in chapter 4.2.

7

Figure 2.1. CD variation at one MD location [7]. Solid line is the true variation, dotted

line is the unprocessed signal, dash-dotted line is the signal from exponentially averaged

scans and dashed line is the reconstruction with the generalised sampling approach.

2.2 Reducing the computational load

The scanning measurement provides multiple measurements for each actuator in CD. We

will find out later, that it is advisable to describe the CD process in reduced resolution.

Usually basis function expansions are used for this purpose. Basis function methods both

reduce the dimensions of the data and, if correctly applied, filter out most of the

uncontrollable high frequency variations. Possible basis function expansions are [6,14]:

Fourier (sinusoidal) methods [24,26]

Chebyshev polynomials [12,16,24,25]

wavelets [4,27]

the eigenvectors of the correlation matrix of the CD variations (principal

components) [28,33,36,44]

splines

square impulse i.e. top hat functions

These bases suit well for smoothing the data and orthogonal ones (first three) present

some analytic and computational benefits. The orthogonal basis functions bk capture

patterns of the variation from the original data. They form a design matrix B, which can

be used to project part of the original variation data Y in reduced space as in Equation 2.1.

The bases are orthogonal if BTB = I. [39, p.79]

8

(2.1)

where W is the weighting matrix and Z2 contains the residual variation or uncontrollable

variations, which are not captured by the basis functions.

For example, the general form for the discrete Chebyshev polynomials (Gram

polynomials) is [25]:

(2.2)

where m is the polynomial order and cmN is the normalization parameter (equals to 1 in

[25]). There are N measurement points, which are denoted by i running from zero to N-1.

For the discrete Chebyshev polynomials the weighting matrix W=I. Figure 2.2 shows a

measured CD profile and some schematic approximations with different polynomial

orders.

Figure 2.2. CD profile approximations with Gram polynomials [25].

The order of the polynomials must be less than the number of the actuators in order to

facilitate some cooperation between adjacent actuators [25]. In [25], 420 measurements

9

were reduced to seventh order Gram polynomial and it took 59 simulated scans to

complete the identification. Hence control design included seven controllers, one for each

order of polynomial.

The CD profile was estimated with discrete Chebyshev polynomials also in [24]. The

problem was solved only at the sampling points and interpolated between them. The

simulations contained 70 actuators and 500 data points in CD, which were described by

the first 40 Chebyshev coefficients. Hence, the dimensionality of the problem was

reduced from 500 to 40.

In [3], different wavelet filters were tested for process data. For the industrial data,

compression ratios of roughly 27:1 were obtained. The algorithm can be adjusted to

provide either a minimum error (loss of information) or a constant compression ratio.

Another work based on wavelet transformation is [4]. There, the dimensions of the

response model, describing the relationship between the setting of the slice lip actuators

and the basis weight profile, were reduced from 420x88 to 53x22 using discrete wavelet

transform domain.

In [10] it is mentioned, that the benefit of the DWT (discrete wavelet transform) is the

ability to completely reconstruct the original signal with the inverse DWT. Wavelet

decomposes the data to details and approximations. Four orthogonal functions (i.e. filters)

are required: First of all, the two decomposition filters are used to determine the wavelet

coefficients for the details and for the approximations. Next, the data can be

reconstructed with these coefficients and two reconstruction filters. In practice, the

decomposition filters B and the original data Y forms the wavelet coefficients C, the

matrices are truncated and the reconstruction is made (see Equation 2.3). [39, p.82]

(2.3)

Principal component analysis can be considered as an empirical orthogonal functions

method, since it does not use a priori chosen basis functions, but instead uses the data set

itself to determine the natural set of coordinates to analyse the data [36]. A particular

principal components analysis, known as Karhunen-Loeve expansion, to identify process

disturbances and input modes in lower dimensions has been used in several papers

[28,33,36,44]. Precise explanation can be found in [36]. In one example, the dimensions

of the control problem were reduced from 125 to 2 [28].

2.3 Discussion

Shakespeare [39] warns that the data may be misrepresented if the filtering is not done

correctly. The control systems are able to move only part of the variations. That is why

the performance is usually expressed as variance of the error e.g. standard deviation. The

mismatched sampling and filtering can, however, lead to artificially low values of

standard deviation. For example, by using the response to smooth a profile and sampling

10

it at the actuator Nyquist frequency will lead to this situation. A lower sampling

frequency based on the highest significant frequency in the response is more preferable.

[39, p.54]

11

3 MACHINE-DIRECTIONAL CONTROL

The MD control is quite straightforward and the dynamics of the actuator responses can

be described with first-order transfer function with dead time. Hence, the problem may

be solved with simple PI-controllers. However, the interactions between different quality

parameters can be easily taken account with multivariate model predictive control.

The machine-directional control is less problematic in paper machines than the cross-

directional control. Yet, there is a trade-off between regulatory control and setpoint

following of the MD control [40]. In practice, one should probably use a different

controller for setpoint steps than for disturbance rejection [41]. MD variation is usually

characterized as a multivariate process with substantial dead times and greatly differing

time constants and many identification and control techniques developed in other process

industries may be readily adapted for the paper industry [39, p. 1]. Dead times vary from

a few seconds to some minutes and time constants from few tens of seconds up to tens of

minutes for different quality parameters [1, p.214].

The process can be described more precisely with the nonlinear process model introduced

in [40]. It describes the white water system, where thick stock flow and filler flow are

controlled by corresponding valve positions and where dry weight and ash content are the

observed outputs. Moisture is assumed to be constant. Disturbances come from the thick

stock consistency which affects both to the fibre and the filler consistencies in the

headbox. Also the varying filler retention level was used as an additional disturbance,

although nowadays it is a controllable disturbance [42]. In commercial paper machine,

the retention is controlled by the flow of a retention aid polymer [45]. This process model

has been used as a benchmark plant in [41,42,43].

Another nonlinear model and a linearised version for the forming zone of the paper

machine are introduced thoroughly in [45]. Here, the basis weight and the retention are

selected as outputs. The input of the linear model is the thick stock flow and disturbances

are caused by the consistency and the fines content. The linear model is used for defining

controllability within pre-defined limits on input and output variables. The nonlinear

model was used to confirm the results of the linear model.

3.1 Sources of the variations

Most of the variations in MD can be attributed to disturbances and pulsations occurring

prior to the headbox [11]. For example, the fluctuations of the consistency, the fines

content and the dissolved solids cause variations in first-pass retention and basis weight

[45]. Shakespeare [39, p.32] has listed following causes:

poorly maintained process equipment (i.e. valves)

improper tuning of control loops for materials supplied to the machine (flows,

concentrations, levels in the wet end, headbox pressure)

suboptimal dimensioning of machine services (air, vacuum and shower systems)

12

variation in headbox pressure

reciprocating devices on the machine (doctor blades, cleaning showers)

nonuniform wearing of the wire during operation.

3.2 Control strategies in MD

The performance of different control strategies have been compared in [40,41] and it

seems that modern control schemes have more to offer for setpoint changes than for

disturbance rejection [41]. Performance evaluations can also be found from [42,43].

The most traditional way to control the machine directional variation is the Smith

predictor and its discrete time version, the Dahlin controller. The Smith predictor

includes an internal model of the process with and without dead time in addition to some

controller structure. [40] Usually a PI- or a PID-controller is used with a low-order model

(see for example [42]). An example of the Dahlin controller is presented in [41]. Wang et

al. [11] describe that both the Dahlin controller and the Smith predictor can be derived

using pole-placement. The Dahlin controller cancels all process zeros, while the Smith

predictor preserves process zeros but cancels all process poles [11].

A modification of the Smith predictor makes it possible to vary the control signal

between the samples. It computes output predictions also in the intersample region while

the true process output is used as a correction signal. Hence, the controller can be tuned

for the process with no delay. [42] Similar control structure was introduced also in [46]

(see Figure 3.1).

Figure 3.1. The Smith predictor for time-delay systems [46]. K(s) is the controller, G(s)

is the delay free part of the actual plant and Go(s) is the delay free part of the model.

Some other alternatives for the control strategies are loop-shaping designs (i.e. H∞) and

predictive controllers, such as dynamic matrix control (DMC) and generalised predictive

control (GPC). The difference between the two latter ones is that DMC uses a step

coefficient model whereas GPC is based on a discrete-time transfer function model. [40]

H∞ control has been used by Whidborne et al. (see [40]), where the aim is to minimise the

13

highest peak in the Bode plot of a weighted closed-loop system to control the error.

Makkonen et al. [42] also introduced an internal model control (IMC) which is a

feedforward type of controller.

Some GPC approaches can be found from [42] and [43], but more precise work is

presented in [11]. Wang et al. [11] have used GPC by extending a 2nd

-order ARMA

model to an ARMAX model with an assumption of an additional stationary sensor. This

way a measurement in MD would be available at each sample time. The ARMAX model

was identified with an extended Kalman filter. Additionally, in order to apply the GPC

for adaptive control, the ARMAX model was transformed into following CARIMA

(controlled auto-regression and integrated moving average) model:

(3.1)

The time delay was placed explicitly in the model. Alternatively, the CARIMA could be

identified also by using the increments of the measured output and control input with the

extended Karman filter. The GPC controller calculates the predicted control increments

over a certain horizon, but updates only the next control action (receding horizon

principle). If the minimum variance predictions through the prediction horizon are given

as in Equation 3.2., then the optimization criterion is a linear quadratic function J1

(Equation 3.3) where control moves ũ can be solved from Equation 3.4. [11]

(3.2)

(3.3)

(3.4)

3.3 Discussion

Besides the control structure, also the tuning the controllers need attention. For example,

Baki et al. [46] pointed out the need of more robust control algorithms for basis weight

control and implemented a PID controller with derivative filtering to a pilot paper

machine. With the new tuning algorithm, both the dynamic and steady-state responses

were better than provided with PI- or Smith predictive control [46].

14

4 CROSS-DIRECTIONAL CONTROL

CD control is very challenging even due to the problems explained so far. There are also

additional problem of sheet shrinkage and wandering which requires actuator-

measurement mapping. This is viewed in Chapter 4.2. Another problem comes from the

fact that actuator responses are not known exactly; hence response models are somewhat

inaccurate. Response modelling approaches and identification methods for the slice lip

actuators (continuum device) and dilution actuators (discrete device) are introduced in

Chapter 4.3. The control methods presented in the literature (see Chapter 4.4) are often

based on linear control or model predictive control, where solving methods can be

divided further to quadratic and linear programming.

There are numerous things to consider in cross-directional control of some property in

paper machine. First of all, the scanning measurement is not able to provide information

at each time instant at each CD location. Also, the control problem is large due to the

high amount of measurement and actuator positions. Secondly, there are different

numbers of actuators in different actuator arrays, which are all used to control the same

property or they are affecting to different properties due to the cross-variations. CD

process is quite unique, although there is some similarity with polymer film extrusion

process (for the description of the process, see [6]). However, solutions from other

industries may not be applicable due to the critical limitations in papermaking. For

example, the solid content can vary across the sheet and process involves as loss of

moisture and dry matter. Since exact control is not possible, usually the control target is

to minimize the deviation of sheet properties in CD (a minimum CD variance control)

[39, p.2]. Also, overlapping of the effecting zones of the actuators prevents removing

some disturbances. The rule of thumb is that only disturbances with wavelengths twice

wider than the actuator spacing can be removed [14,15,26]. Including these

uncontrollable modes of variation to the response model leads to rank deficiencies and

numerical problems in optimization [39, p.3].

4.1 Sources of the variations

Process variation often takes a sinusoidal shape, especially in heavier grades [21]. It is

commonly assumed that fluctuations in CD occur on a much slower time scale than in

MD [22]. This is also the reason why the first problem described above is not considered

to be crucial; due to the slow variation of the CD properties, it can be assumed that

measurements are taken at the same time instant. Naturally some MD operating

conditions (headbox consistency level, slice opening, stream valve control, pulp pH) have

some impact to CD profiles [21].

One problem for controlling CD profile comes from the edge profile variations on the

wire. Especially for heavy paper grades the slurry coming out from the slice lip can fall

off the wire very easily. This may be partly resolved by curling the wire toward the centre

or installing deckle boards to the edges. However, these methods always create several

15

edge reflection wave boundaries. Additionally, the junctions between the deckle boards

and headbox are not smooth, which causes turbulence. [21]

4.2 Mapping the actuators and the measurements

In conventional CD control systems, mapping is used to correctly align actuator zones

with measurement zones. This may cause some loss in profile resolution or poor

controller performance. Misalignment of a few centimetres can result in sawtooth

profiles, since actuators are adjusted to compensate for errors outside their control zone

[25]. The allowable mapping error that causes system to diverge has been examined for

plastic film extrusion, where a mis-mapping of the full spacing between actuators were

required before system became unstable [53]. However, the widely accepted mis-map

threshold of one third of the spacing between CD actuators was exceeded with the

expense of the spatial bandwidth and dynamic response of the control system [53] i.e. the

tuning of the control affects to the mis-mapping stability. Also, the width of the CD

response has an effect on the mis-map threshold. For narrow actuators, even small

changes in alignment would lead to instability. However, these small mapping errors may

be hidden by the small signal to noise ratio. [65]

Besides the difficulties caused by the traversing scanning sensor, also other phenomena

make it necessary to adjust the measurements for the right actuators or in other words, to

find the center of the actuator response. Mapping algorithm usually compensates for

nonlinear sheet shrinkage, sheet wandering and actuator spacing [15]. Another problem

comes from the different resolutions of the different actuator arrays. Arrays include

different number of actuators; hence they may not be capable to remove the unwanted

effects caused by the others [20]. As a whole, the factors affecting the mapping between

measurements and actuators in CD process are [48]:

geometric alignment of the CD actuators and measurement devices

position of the individual actuator within the array

wandering of the paper web

paper shrinkage through the drying process

flow pattern of the extracted liquid paper stock in the wire

Figure 4.1 illustrates the shrinkage phenomenon and the mapping problem between

actuator locations and their responses.

The shrinkage occurs in the dryer section and it varies across the web, gaining its

maximum values at the edges. The phenomenon can be described with a function s=f(s’)

that relates a particle with the wet cross coordinate s to a particle with the dry cross

coordinate s’. This mapping function can be used to calculate the shrinkage curve B(s)

(Equation 4.1) and vice versa. It is usually necessary to update the mapping function f for

example when certain 2σ-value exceeds, since shrinkage varies both with time and

process state. This can be done with mini-bumb tests (non-intrusive), which does not

affect the quality (amplitude is approximately equal to the standard deviation) and using

statistical analysis methods or fuzzy logic. [37]

16

(4.1)

Figure 4.1. Shrinkage profile for a high-speed newsprint machine [66] and alignment

problem [65].

Misalignment due to shrinkage is magnified in faster machines. Misalignment larger than

one-third of the spacing between actuators can cause instability of the CD control system.

Many shrinkage models assume that the shrinkage phenomenon occur symmetrically

which is not the real case. Mapping problem is a non-linear and slowly time varying

phenomenon. [48] Hence, current approaches try to detect the mapping directly from the

closed-loop data instead of using a model. One approach is the CD tensor model in [48].

Alignment of actuators with their response centres has been examined also in [64] and

[65]. The latter one used a procedure where it was detected whether mapping is incorrect

and re-mapping was performed if necessary. It is said that the updating the actuator

response model is rarely necessary, but the updating the alignment need to be done more

often [65]. The online method of mapping in [64] is based on the correlation cancellation

of the source signal and noise. In [61], a closed-loop identification algorithm was

developed to update the step-response model used by a MPC control to compensate the

effects of shrinkage.

Actual shrinkage can also be measured. The forming fabric leaves periodic imprints to

the paper which can be recorded either measuring the intensity of the light transmitted

through the paper using a CCD camera [66] or alternatively measuring the amount of

fluorescence light from the paper [67]. The image coming from the CCD camera or the

time series coming from the fluorescence sensor is then examined in frequency spectrum

to detect the wire marks. Once the location of these marks is known, the shrinkage profile

can be calculated.

17

4.3 Response models

The basis weight is controlled either with slice lip actuators or dilution actuators. Slice lip

actuators control the amount of fiber suspension exiting the headbox by locally changing

the height of the slice opening. The spacing of the actuators are typically between 75 to

100 mm. Dilution actuators change the local concentration of fibres by diluting the

suspension with a flow of low consistency water. The dilution zones are typically 40 to

60 mm wide. [75, p.11]

Despite the type of the actuator, some common assumptions are shared. Common

assumptions for response models are [21]:

the response shape is symmetric

the response is identical for all actuators in an actuator array

the total profile response is equal to the sum of the individual actuator responses

Additionally, the input-output models usually assume that the dynamic and spatial

responses of the actuator can be separated. The dynamic response actually comprises two

dynamic components; dynamic response of the actuator to setpoint changes and the sheet

response to the actuator action [35]. The dynamic response is usually described as a first-

order plus delay transfer function [25]:

(4.2)

In most cases, the dynamics may be just a dead time [25].

The spatial response of the system is usually represented by copying single actuator

response to form an interaction matrix. Hence, the input-output relationship becomes

[25]:

(4.3)

The problem with this kind of model may arise from the interaction matrix G, which

cannot be approximated without uncertainty and may approach singularity. When the

number of measurements (number of rows) is not an integer multiple of the number of

actuators (number of columns), each column in matrix will have different entries. [25]

Also, the calculation of the optimal control output by inversion is not straightforward for

non-square matrix. The matrix may be squared for example by reducing the number of

measurement locations to match the number of actuators. The main problem, however,

comes from the size of the interaction matrix. Due to the high amount of measurement

locations and actuator numbers, the interaction matrix is very large and inverting even a

square matrix may be problematic [28]. Hence, the inversion has to be approximated or

the matrix must be truncated to the lower dimension. There are several ways to truncate

18

the model, as well there are several ways to model (identify) the spatial response of single

actuator.

Spatial response of a slice lip actuator

The spatial response can be identified by a parameterisation of single actuator move (e.g.

[21,27]), defining the full dimensional model and reducing it (e.g. [25]) or identifying it

directly from the data, which is the most popular alternative (e.g. [12,21,24]). The typical

steady-state response of a single slice lip actuator is shown in Figure 4.2. One should

notice, that the width of the response is much wider than the actuator width i.e.

adjustment of a slice lip actuator affects also to the adjacent actuator positions.

Additionally, it is clear that this effect is in opposite direction than to the adjusted

actuator.

Figure 4.2. Typical response of a slice lip actuator [30].

The profile response of single actuator can be interpreted as a phenomenon of wave

propagations. Machine speed and dewatering process determine the profile response

width and the response shape. For lighter weight paper products, the coefficients of the

response model are not too difficult to determine, a simple bump test or on-line

identification method can be applied. For heavier grades, the model size (response width)

and response shape may change dramatically and the actual numerical values of the

model coefficients are much smaller than those for lighter weight paper. Wave

propagation can be described by a following function: [21]

(4.4)

19

where w(x,t) is the wave magnitude, ff(x+vt) is the wave component propagating toward

the front edge of the machine, fb(x-vt) is the wave component propagating toward the

back edge of the machine and v is the speed of the wave propagation in CD [21].

Another function for the shape of the CD response at the dry-line is given in [27] and the

equation contains amplitude, damping factor, spatial frequency and phase shift:

(4.5)

For light grades, the response is quite typical bump response, phase shift is zero and

problem is hence one-dimensional. For heavier grades, the response is M-shape and also

the phase shift must be included [27]. The spatial coupling can vary from 1 to 5 adjacent

zones on each side of an actuator uniformly across the sheet [34].

Full dimensional square interaction matrix was identified in [25]. This was then used to

define a reduced matrix in terms of Gram polynomials. Another solution, without

knowledge of full dimensional matrix, was to estimate the reduced matrix directly from

the orthogonal parameters of the measured profile and the slice lip polynomial.

The response of the slice lip can be examined more precisely by modelling the actuator

bending with full dynamic partial differential equation [31]. The model concerns the

forces applied to each screw in slice lip and the positions of the screws. The results in

[31] show that the assumption of the separability of the dynamic and spatial component is

valid in most cases. Another first principle model and its simplification are presented in

[70]. The CD response is a shifted and amplified version of the slice deflection curve thus

the model can be identified directly using cross correlation approach.

Partial differential equations (PDE) approach is not convenient for control purposes. That

is why PDEs are approximated using an above mentioned interaction matrix approach,

that most of the currently installed systems use. In this approach, infinite system is

truncated to finite-dimensional by some basis function expansion. Another approach is to

discretise PDEs to give a 2D polynomial transfer function description. Usually finite

difference procedures are applied. [6]

Basis functions (i.e. rational polynomials or arbitrary orthogonal functions) are used to

express the spatial response, and often similarly the measured CD profile, as referred in

chapter 2.2. Basis function expansion decouples the system, so that each spatial mode can

be considered separately and it restricts the response to those spatial components that can

be regulated. Hence, the reduction of the system dimensions follows. [14] Also the

actuator profile and control action may be given in same reduced space. The

representation with orthogonal polynomials provides accuracy and smoothness only for

either responses near the centre of the sheet or those near the edge, but not both [39,

p.92].

The response of an actuator array can be analysed also in spatial-frequency domain using

Fourier transform. This leads to an expression for the spatial bandwidth of the array and a

20

methodology for the design of an actuator array and signal processing. The cut-off

frequency of a spatial low-pass filter can be determined in cases where the actuator

response is close to the “ideal” response. One example happens to be the basis weight

response in slice lip adjustment. [26] One has to notice, that Fourier methods require the

spatial response to be periodic [14].

Spatial response of a dilution actuator

The spatial response of a dilution actuator differs from the spatial response of a slice lip

actuator. A typical response of a dilution actuator is illustrated in Figure 4.3. The most

notable difference comes from the fact, that the response does not have the negative

effect on the adjacent actuators. Also, the width of the response is not as wide as for the

slice lip actuator. This means that the dilution actuator requires more accurate mapping

information [75, p.11].

Figure 4.3. Spatial responses of basis weight to a slice lip actuator and to a dilution

actuator [75,p.8].

The response of a dilution actuator can be well approximated with Gaussian [39,p.21]:

(4.6)

where a is the amplitude and s is the dispersion parameter. The offset b may be neglected

if the CD control action satisfied the zero mean over the sheet [39, p.73]. The parameter

xk is the mapping position.

21

Dynamic response

As mentioned earlier, the dynamic response of an actuator is usually described by first-

order plus delay transfer function. While it is assumed that the dynamic response is

similar for all actuators in an actuator array, it may differ significantly between different

types of actuators.

The process dynamics can be dominated by the sensor filtering and the transport delay, if

the dynamic response of an actuator is very fast comparing to the scan time. An example

is the dynamic response of moisture to a rewet shower, where the rise time could be less

than one scan time. An opposite example is the rise time of the dynamic response of

caliper to an induction heating actuator. The response could be more than 10 times the

scan time while the dead time due to the transport delay is small since the actuator is

located near the scanning sensor. Thus this rise time is the main factor of this process

dynamics. [75,p.6]

In [39, p.60], it is said that it took several scan times (i.e. 75-100 seconds) to the dilution

actuator response to reach the steady-state in open-loop. However, the adjustment made

to the actuator was significantly higher than the adjustments in normal operation (1%

dilution ratio change compared to less than 0.1% changes in normal control actions). In

[60], the time constant of the dynamic response of a slice lip actuator was found to be 3

seconds while the dead-time delay was 35 seconds.

Two-dimensional response

In practice, the actuator response models have to be identified from bump-tests, which is

an inefficient way. One solution is to use two-dimensional sheet variation data, which is

available with the new measurement technology. Both the dynamic and spatial parts of

the response model are identified with random probing actions made to the actuators and

using KL expansions to separate MD and CD variations. This method is able to identify

individual responses for each CD actuator including both dynamic components. [35,60]

Another method to identify the CD response from industrial data is to use wavelet

transform [27]. Some identification methods for two-dimensional process models are

introduced in [6]. The identification of polynomial model can be done with modified

conventional recursive estimators when full array data is available. Scanning sensor data

need sparse data methods, e.g. EM (expectation maximization) algorithms. [6]

Disturbance models

There have been attempts to include the process disturbances in the process model as

well. While some approaches have assumed some structure of the disturbance model in

order to estimate the full profile variations [16,17,30], have others derived identified the

disturbance model from the data [28,33,36,44].

A correlated disturbance model for the use with polynomial models was derived in [16].

Disturbance model considers CD correlation in process noise and distinguishes between

22

two types of disturbances; infrequent, spatially correlated, low-spatial-frequency

disturbances and random disturbances, which may be spatially correlated [16].

Disturbance model can be identified by capturing most significant modes of spatial

disturbances entering the process with a PCA method, more precisely the Karhunen-

Loeve expansion [28,33,36,44]. Temporal changes were modelled with autoregressive

time series modelling.

4.4 Control strategies in CD

The control strategy is based on minimising some measure of the cross-directional error,

since it is not possible to generate cross-direction control actions that will produce the

desired basis weight at all measurement positions. The most common approach is to

minimise the expected cross-directional variance of the variations (residual error).

Another option is to minimise the expected range of future values. [30]

Additional things to consider in CD control are the constraints of the actuators.

Traditionally, the basis weight is controlled with a machine wide steel beam, which is

bended for example by screws forming openings (i.e. slice lip). This slice lip setup has

the following limits [33]:

The actuator has a maximum (and minimum) allowable opening.

The actuator has a maximum (and minimum) bending moment i.e. adjacent

actuators have maximum difference in their openings, which cannot be exceeded.

The change rate of the actuator opening has a maximum (and minimum) value in

order to avoid wear and tear of actuators. The allowable rate can depend also on

the openings of two adjacent actuators [4].

The sum of current actuator positions must provide zero mean (in order to keep

MD value at its setpoint value).

Linear control

Linear control usually assumes that the dynamics are equal to the dead time i.e. control

actions are made only for steady-states. Linear control cannot take constraints into

account very effectively, since it can only penalize the control action in the objective

function. Clipping or scaling leads to decreased closed-loop performance of the

controller. Anti-windup controllers, e.g. observer-based compensation or internal model

control based compensation may be used to modify the linear control system. [13]

The optimal control output can be calculated as the least-squares solution (Equation 4.7)

by inverting the steady-state input-output model. This may require a square structure of

the interaction matrix and usually a Toeplitz symmetric structure is used, where the same

element is repeated along each diagonal of the matrix i.e. adjustment of any actuator in an

array affect symmetrically to the nearest neighbours of this actuator [13]. For the non-

square matrices, the inversion can be approximated with pseudo-inverse (see Equation

4.8) if the columns of the response matrix are assumed to be rank deficient. If the rank of

23

the response matrix is unknown or it is known not to have full rank, it is recommended to

compute the inverse via the singular value decomposition. [39, p.125,171]

(4.7)

(4.8)

One example of linear quadratic optimal control has been given in [15], where the

performance was compared to mapped PI-controller. A minimum variance controller for

scalar disturbance dynamics and for multivariable disturbance model was used in [16].

Actuator response and disturbances were represented by Chebyshev polynomials.

Kristinsson and Dumont [25] used a Dahlin controller, which is capable to compensate

the long time delay. Control limits were applied to the control signal by an anti-windup

design. The interaction matrix dimensions were reduced with Gram polynomials. The

simulations showed, that with this approach, the total bending of the slice can be reduced

significantly comparing to a traditional control and the method is more robust. The basis

weight profile is not as smooth for the controller with Gram polynomial, since it does not

try to remove the high-frequency components, which a traditional controller removes as

long as the model mismatch is not too big. [25]

Arkun and Kayihan [28] used a reduced-order internal mode controller (IMC), since

model dynamics are known to be non-minimum phase, and hence cannot be inversed.

The key steps of the method are:

conditioning of the original input-output space using singular value

decomposition (SVD) to get around the problem of model ill-conditioning

identification of the dominant disturbance modes that should be rejected

construction of the lower-dimensional subspace for the controller followed by

reduced order IMC design in that space

“lifting” the computed control action from the design subspace to the original

input space and implementing it on the actual process

The identification of the disturbance modes was done with PCA. It should be noted, that

here PCA (KL-expansion) was used in the context of closed-loop control, whereas for

example in [36] it was used only for identification of disturbance profiles. This means,

that the modelled effect of the control inputs must be first subtracted from the

measurements before PCA is applied [28].

Model predictive control

CD process is truly multivariable and runs suboptimally with traditional CD-controllers,

which are used for controlling only one sheet property. That is why multivariable

controllers, such as MPC, should be used. In MPC, coupling of the actuator and

measurement arrays is done internally, setpoints are calculated with respect to the

physical limitations of the actuators and spatial responses are not assumed to be uniform

and symmetric. [23]

24

The role of predictive control has a long history in the consideration of CD control.

Already in 1977, it was observed that the control problem could be posed as finding the

minimum of a quadratic cost function subject to linear inequality constraints. Basis

functions allow to reconsider this problem with a possibility to satisfy all the

requirements of CD control; good dynamic behaviour, satisfaction of input constraints

and parsimonious representation. [24]

The optimization can be performed by means of linear programming (LP) or quadratic

programming (QP) [4]. LP gives a solution to the 1-norm or ∞-norm minimization

problem while in QP the problem is to minimize the variance of the sheet profile [13].

For more detailed description of these solution methods, see [47] and [34], respectively.

Additionally, solution methods based on elliptic approximation of quadratic MPC

problem were presented in [52,55]. Here the focus is to introduce how different authors

have approached the use of MPC in CD control.

For example, Backström et al. [23] introduced a model predictive controller to a

linerboard machine. Each of the four actuator array-measurement array pairs are

represented in discrete-time. Measurement resolution is down-sampled to the highest

resolution of the all actuator arrays. The model was identified with an automated

identification tool. Process disturbances were assumed as integrating moving average

processes. Next, the model was converted to the state-space form and objective function

(minimize the sampled variance of the CD error profiles) was formulated as MPC

problem. Current state was estimated with Kalman filter. Optimization problem and the

constraints provide a quadratic program, so a QP solver is used at each scan (i.e. every 23

seconds). [23] The objective function is presented in Equation 4.9.

(4.9)

where R contains the future target profile, Ŷ contains the predicted future profiles over a

prediction horizon p, ΔU contains the future actuator setpoint profiles over the control

horizon c. R, Y, ΔU and U are calculated for each scan, but the time index k has been

omitted from the equation. Q1 is the weighting matrix for weighting different sheet

properties and CD locations and Q3 is for energy maintenance (i.e. to penalize actuator

fighting etc.). Q2 is the main weighting matrix used for the penalization of actuator moves

i.e. aggressiveness of control. Also, different actuator arrays and CD locations can be

weighted with Q2. [23]

Heath [24] uses an orthogonal function, discrete Chebyshev polynomial with uniform

weighting to describe the process. The linear distributed parameter model includes also

terms for residual variation (ξ) and sensor noise (e). By representing above mentioned

parts in terms of orthonormal basis functions, the controllable part of the process may be

represented by the multivariable model:

25

(4.10)

where N(z-1

) and D(z-1

) are finite-order polynomials and B is the interaction matrix.

The control criterion was based on a GPC type criterion and it was solved subject to

inequality constraints on the inputs:

(4.11)

where Ei,j(z-1

) and Fi,j(z-1

) comes from the spectral factorisation and partitioning of the

noise i.e.

(4.12)

Problem can be solved either via a quadratic program or iterative algorithms (e.g. mixed-

weights least-squares) [24].

In [4], an adaptive predictive control strategy was applied in discrete wavelet domain.

The output prediction included model-based output prediction and a term for

compensation of predictive errors. It was not stated if the optimization problem was

solved using linear programming or quadratic programming.

A broad analysis of linear programming in MPC was made in [47]. LP requires linear

objective function. It was stated that LP-MPC can outperform QP-MPC being

computationally efficient and can be modified easily. However, LP can suffer from

undesirable idle or deadbeat behaviour.

The PCA based work by Arkun et al. [28] mentioned before has been reported with MPC

algorithm in [33,44]. Again, this approach assumes that the measurements (or

estimations) are available at each CD location at each sample time. In [33], the

disturbance model is based on the lower dimensional subspace projection. The time

dependence of the disturbance profile can be captured with autoregressive (AR) time

series modelling. Vector autoregressive (VAR) method should be used if the correlation

between the temporal modes is found to be significant. Also the original input-output

relationships, as well as the constraints need to be reduced to the same subspace. Finally,

a reduced order augmented state space model is used in calculating the optimum control

action (see Table 4.1). The steady state gain matrix is not inversed here, so the problem

with ill-conditioning will not arise. [33]

26

Table 4.1. Model equations in [33].

Input-output relationship in reduced subspace:

Superscript L denotes lower order subspace, Φ is the projection vector

in KL expansion and t is the time dependence of the disturbance

profile. Approximated disturbance profile in original space can be

retained by:

Reduced order augmented state-space model:

Subscript p refers to process model and d to disturbance model. e is the

noise of the autoregressive model.

The control criterion is:

Where, Q, R, S are weighting matrices. The solution was given by a

quadratic program.

The controller synthesis has been made also in two-dimensional frequency domain

(Fourier transform). In [62], the process model with square Toeplitz interaction matrix

was approximated with symmetric circulant matrices. This way the stability analysis

became easier and bounds for robust control could be derived. The system uses a family

of SISO controllers, one for each spatial frequency. The amount of the controllers was

halved. Since squaring the process model is difficult and infeasible, the system was

interpreted with rectangular circulant matrices in [63]. The MPC algorithm was based on

[23] and the solver was adopted from [34].

4.5 Discussion

Above mentioned methods concern only single property control. Multiple property

control i.e. multiple properties are controlled with the same actuators has been discussed

for example in [33]. Each property is described with its own gain and disturbance

matrices. Model becomes non-square, so a third intermediate space need to be introduced

where the calculations can be performed and the optimal solution can be projected back

to the original space. Also the constraints need to be projected again. Other examples of

multiple property control are given in [23,51].

27

There has also been an attempt to assist MD control performance with CD actuators. This

kind of coordinated MD control improves the MD response at the expense of

degradations of the CD profile. For the simulated case of moisture, MD response was

indeed faster. Besides the increased closed loop speed of the response, the CD actuator is

required to handle the MD transients only and should return to its original position after

the transient passes so that it will have full control capability in CD profiling. Another

requirement is to determine how much CD profile degradation is allowable. [20]

Although there have been a lot of effort reducing the computational cost of model

predictive control in CD (see for example [34]), the results may not always be sufficient

for on-line applications. Especially, if one is expecting to increase the resolution of the

measurements, the problem will be more emphasized and the use of MPC may become

impossible.

28

5 2D-POLYNOMIALS

The interaction matrix approach introduced above is based on the assumption of

separating the spatial and the dynamic terms of the actuator response. PCA and wavelet

methods allowed model to adapt with time and such an assumption was not required. The

improvement of sensing and actuating systems may lead to a situation, where this kind of

assumption may no longer be allowable. The 2D-methods presented in this chapter give

some theoretical tools to be used with two-dimensional measurements in control

engineering.

As referred earlier, Duncan and Corscadden [31] have examined the validity of the

assumption of separability of the dynamic and spatial responses. They pointed out, that

the assumption may not hold when the dynamic bandwidth of the system increases. This

kind of increase is achievable by improving the speed of the actuators and measurement

resolution. As a result, it may be necessary to design control algorithms that do not

require the assumption of separability or can accommodate deviations from separability.

[31] One suitable approach may be to use techniques based upon two-dimensional

polynomial methods developed for the optimal control [6,14,68,69]. This study has taken

place in University of Manchester Institute of Science and Technology Control Systems

Centre (UMIST) and the earliest works are from the mid-80s. Another example which is

classified here as two-dimensional control is taken from reference [44], where process

disturbances were first modelled and a simultaneous model predictive control of CD and

MD variations was applied. The coupled control approach was first introduced in [77]

where it is said to allow improved control when the CD profile variations are slow or

infrequent in nature.

The basis of the 2D systems theory comes from filtering applications in image

processing. The basic idea is to represent a process output variable at a particular point on

a surface as an explicit function of two Cartesian coordinate variables which jointly

define the surface. [8] Again, it is not possible to describe the process efficiently with

physical models and process identification methods are needed to be used. The two

approaches introduced in [6] were 2D-ARMAX model and two-dimensional state-space

model. Also state-space models with mixed structure of discrete in one direction and

continuous in the other direction has been studied by Kaczorek (see [57]).

More universal description on 2D systems theory is given in [57], which gives a review

on whole area including controllability, observability, causality, construction of state

space models (realisation theory), stability and stabilisation, feedback control and finally,

filtering. It was mentioned in [57] in 1996 that the area of applicable control theory and

controller design for multidimensional systems has gained very small attention. The

development has been focused more on multidimensional filtering, although there have

been some progress in control algorithms for special cases of 2D linear systems, referring

for example to the work in UMIST.

29

5.1 2D linear control

Prediction

There are some key elements when considering two-dimensional polynomials:

Second shift operator

Support of the polynomials

The solution polynomials are of infinite degree (Diophantine equations)

Polynomial condition requirements

Truncation of the supports near the edges

The one-dimensional polynomial model can be expanded to two-dimensional polynomial

model by introducing a second shift operator and a support for the polynomials. Equation

5.1 shows an example of 2D-ARMAX model. Possible supports are non-symmetric half-

plane (NSHP), quarter-plane (QP) and symmetric half-plane (SHP). SHP requires the

whole current row to be measured before it can be recursively updated (raster scanning).

[6] This is not a problem, if fast measurements from an array of sensors are available.

(5.1)

The delay v is in vertical direction n (corresponding to the MD). The shift operators w

and z (in CD and MD, respectively) and the two-dimensional polynomials A, B and C can

be defined in a following way [6]:

(5.2)

(5.3)

The nonzero coefficients in polynomial A forms a support, which is actually a region in

available past data S(n,m). The support is analogous to the degree of a one-dimensional

autoregressive polynomial. The coefficients of polynomial C are associated with the

moving average part of the model. The size and the shape of the supports determine

which past values of signal and noise contribute to the current value of the signal. [49

p.474]

The theory is certainly applicable for infinite webs. For the finite webs of the true

processes, model is only an approximate since the support for the model must be

truncated for the data near the edges. [6] The assumption of the semi-infinite global past

corresponds to the assumption of a semi-infinite past in one-dimensional prediction and

control. Appropriate algorithms can be achieved by simply truncating the local supports

30

at the edges, as was shown in [68]. The simulations showed that the truncation affects

only at the edges.

The book covering the two-dimensional self-tuning algorithms [49] uses 2D-models to

predict only the future values of pixels along the row i.e. the cross-direction. However, if

the predictions are considered away from the row (machine direction), a different

approach is required. The two-dimensional Diophantine equation in the latter case

contains infinite polynomials (F and G). The resulting infinite-degree polynomials can be

truncated, but the producing result of the generalized predictors will be sub-optimal.

Optimal predictor can be derived if these polynomials are expressed as a ratio of two

finite-degree polynomials i.e. they are expressed as rational transfer functions of finite

order (causal polynomials Φ and Γ). [68]

Identification

Optimal least-squares predictor for known two-dimensional CARMA process was

derived in [68]. The algorithm was extended to self-tuning (recursive) form in [69]. The

presence of finite edges requires estimating the noise explicitly. In infinite case as well as

in one-dimensional prediction, the noise is implicit. [68] For the first case, the estimation

of the noise process must be done by using the model itself [69]. For the self-tuning case,

the identification of the 2D-polynomial model produces an estimate of the noise process

and it can be applied to minimum variance controller [6]. This was demonstrated in [69],

where parameters were estimated with 2D-AML (two-dimensional approximate

maximum likelihood). The noise estimate comes from the residuals of the 2D-AML. The

procedure is following:

Form the data vector using past values of output and input and the noise estimate

Form the prediction error

Update the covariance matrix

Update the parameter estimate with the covariance matrix

Form new residual (noise estimate) with the current parameter estimates

Use the parameter estimates to obtain polynomials X, Y and Z

Apply the controller

The method of identification of the parameters for these models depends on the sensing

system. Recursive methods for both the stationary array sensors and scanning sensor

were introduced in [6]. Conventional estimators can be applied with some modifications.

Some mentioned methods are 2D forgetting factor method, regularized forgetting and

sparse matrix methods. The applicable method depends also on the type of the support

(for example, SHP cannot use point-to-point recursion). [6] The modifications include

housekeeping changes in how the data are stacked and how the polynomial estimates are

removed or changes concerning how the information matrix is updated in the means of

radial distance from the current data point [18].

Also the support needs to be estimated. Standard 1D-methods for model order testing

may be appropriate for 2D support estimation. Some examples are product moment

methods, Akiake’s information criterion and final prediction error criterion. [6]

31

Control

After the model parameters have been obtained, model can be used for control purposes.

The minimum variance control law can be constructed directly from the predictor. The

generalized minimum variance controller incorporates a set point and weights to the input

and to the output. [68] The data in [68] is from a raster scan, where it propagates from left

to right and starts from the beginning at each row. Hence, it does not correspond to any

measurement method in paper machine. The algorithm was extended to self-tuning

(recursive) environment with generalised minimum variance controller in [69]. Also the

set-point tracking and offset handling of the system was considered. The paper makes no

claims for the convergence properties of the algorithms. [69]

As mentioned above, parameter identification generates automatically a residual process

which can be used as a noise estimate [18]. Generalised minimum variance control has

been extended to two-dimensions and an example is given in [8]. The cost function V has

the form:

(5.4)

The pseudo-output governing the controller must be given as:

(5.5)

P, Q and R are (truncated) 2D-polynomials, which filter the original input/output

processes u and y and the setpoint r to produce required control behaviour. The control

signal can be calculated from Equation 5.6, where Φ’ and Γ’ are the results of an

operation (Equations 5.7-5.9) corresponding the partitioning in one-dimensional GMV.

[8]

(5.6)

(5.7)

(5.8)

(5.9)

Two methods are presented in [69] to achieve a given set-point; tracking column-by-

column (weighting each column separately) and integral control. The first one is based on

the multivariable analysis in [68] and requires manipulation of scalar matrices

corresponding to the multivariable problem. The disadvantage of this method is the

32

necessity of inverting large matrices. The integral method is simple, but takes longer time

to settle and the stability (choosing polynomials P and Q) is not guaranteed. [69]

There are two different approaches also for the offset descriptions; the incremental model

(zero mean noise process) and the static model (unique disturbance in every vertical

position). Incremental model allows a neat solution, but e.g. the case where the offset is

constant or slowly varying and the noise is purely sensor noise, the model may be

unsuitable. For the static model, the control is simple, since it is closely related to the set-

point problem. However, it introduces severe difficulties to the parameter estimation.

Control law does not need an estimation of disturbances, but the estimation of the other

parameters requires it. This increases the dimensions of the problem hence increasing the

computational time and also the convergence rate of the estimation. It may be possible to

approximate smooth offsets with some low-order polynomials etc. However, optimal

control cannot be achieved. It can be concluded that incremental model should be used

when possible. [69]

State-space representation

2D-ARMAX model can be easily transformed into state-space forms, which are more

versatile when considering controller synthesis. For example Roesser’s model and

Fornasini-Marchesini’s model may be applied (for more information see [58,59]). Also

other model forms may be found from the field of two-dimensional signal processing.

Equation 5.10 shows a state-space model which is based on 2D-ARMAX model with

quarter plane support and Roesser’s formation. [6]

(5.10)

The vertical and horizontal state vectors are comprised in local state vector x and the

arrow notation denotes propagation of these state vectors as follows:

(5.11)

(5.12)

The state-space model allows making state estimates for k1 and k2 steps ahead in CD and

MD, respectively. The estimates can naturally be applied to predictive control.

The state-space model can be used with generalised predictive controller. The

representation for the use with scanning sensor (missing data problem) can be found from

[38] and [6]. The procedure without equations is following:

1. Initialize the predictor using the one-step ahead prediction along a row i.e. use the

available a one-step-ahead prediction in both directions subject to the current

measuring position.

33

2. Replace missing data points with their prediction from the previous location in a

row.

3. Use these predictions to give k1 columns ahead horizontal state estimates for the

current row and vertical estimates for the next row in these estimated positions.

4. Calculate k2 rows ahead state estimates subject to the current measurement row.

As explained above, the first row comes from the previous step using one-step-

ahead predictions. Next rows are based on the other estimates.

5. Obtain the final state estimates.

6. Calculate the predicted output.

Additionally, step 1 requires the initial conditions for the states and step 4 requires

appropriate boundary conditions.

5.2 2D model predictive control

In [44], two-dimensional disturbance profile was measured with an array of stationary

sensors, or alternatively, the estimates of the full profile based on the measurements from

the scanning sensor were used. The most significant features of this profile were captured

with principal components analysis (PCA). PCA describes the stochastic evolution of the

CD profile with temporal modes an. The more correlated the CD lanes are, the less modes

are needed to describe them [36]. The method filters and compresses the data very

efficiently. The dimensions of the data were reduced from 20x400 to 2x400. The

evolution of the temporal modes (MD profile) was modelled with linear autoregressive

sequences (AR). This provides a predictor for the control purposes. It was mentioned in

[33], that AR should only be used if there is no significant correlation among the modes

of retained temporal vectors. If significant correlation exists, vector autoregressive

(VAR) process model is preferable. Additionally, the selection of the AR or the VAR

model order should not be made with traditional methods (AIC, BIC), since they do not

take into account the effect of time series modelling to the other operations in the

algorithm. [33]

The approach in [44] is non-recursive and identification needs to be done in open-loop.

The derived disturbance model ND(k) was simulated with a MPC algorithm, which

accounts both the CD and MD actuator input-output behaviour i.e. the algorithm couples

two one-dimensional models into a two-dimensional control problem (see Equations

5.13-5.15). The interaction matrix GCD was assumed to be square, as can be seen from

Equation 5.15.

(5.13)

34

(5.14)

(5.15)

The actuator transfer function coefficients in [44] were taken from [77] and they are

dMD=7, dCD=1, g11=2.0 and g12=0.8. The MD actuator transfer function gMD was modelled

as a first order system with a gain of -0.024 and a pole at 0.88. Finally, the problem was

casted into a state-space form.

The next paper [36] of the research group explains comprehensively the identification of

the full profile disturbances. The contribution of this later work comparing to the earlier

one, is that the approach is adaptive and can be implemented on-line. The control

application was not considered in [36], neither were the mapping issues.

5.3 Discussion

Clearly, the methods mentioned in Chapter 5.1 could be implemented to the systems,

which are truly two-dimensional i.e. variations are controlled in two directions at the

same time. The actuators currently in paper machines are used to control strictly only one

direction. So, is there any use for the two-dimensional methods in current processes? The

following may give some insight of the potential of two-dimensional methods (based on

the 2D-theory).

The total variations can be decomposed to mean level, MD variation and CD variation,

where only CD variation depends on both directions. Additionally, it is usual to

decompose the CD variation further to a true CD variation, depending only upon CD

position, and to an uncontrollable residual variation. However, by using two-dimensional

approach, CD variation is considered in both directions, which means that some part of

the residual variation is controlled. [14]

The problem of two-dimensional methods in controlling a paper machine comes from the

edge effects. For finite width webs the 2D model is only approximate, because the

support of the model must be truncated for the data near the edges [6].

35

Additionally, there are some condition requirements for the polynomials used to solve the

Diophantine equation and used by the controllers. Some of the polynomials need to be

inverse-stable in order to the prediction order to be bounded and the closed-loop output to

be stable. However, there are not always solutions, which will satisfy these conditions. In

general, the applicability of the two-dimensional controllers is questionable since the

relation between poles and zeros of 2D transfer functions and system response is not fully

understood and the relation between continuous processes and their discrete

representations is not as clear as in one-dimensional case. [68]

36

6 SUMMARY AND CONCLUSIONS

Paper machine uses different actuators and control loops for the cross-directional (CD)

and machine-directional (MD) control. Whereas MD problem is quite straightforward,

CD control is a quite complex system. Model predictive control (MPC) provides good

tools for controlling the CD, since the limits of the actuators can be efficiently taken into

account. Besides the input-output model of the control algorithm, CD system requires

often models or on-line identification to compensate the sheet shrinking phenomenon.

Sometimes also process disturbances are modelled in order to make better predictions.

Another concern is the data processing of the measurements coming from the scanning

sensor. Data need to be filtered in a way, that the mixed information between MD and

CD variations could be decomposed. The high amount of data points and the vast number

of actuators make the dimensions of the input-output model huge, so usually some basis

function expansion is applied to reduce the dimensions. The common techniques assume

the separability of the MD and CD, which does not remove all the variation as possible

with the CD control system. Especially the development of the actuators and sensing

systems may lead to a situation, where truly two-dimensional methods should be applied

instead of current ones.

The literature cited in this review mostly concerns the papermaking process, although

some works prefer to handle the sheet, or web, forming processes in general. Few cited

works use the plastic film extrusion as an example process. The classification can be

found from table 6.1.

Table 6.1. Cited literature by the process.

Sheet forming in general Papermaking Plastic film machines

6, 7, 13, 14, 16, 17, 18, 22,

26, 30, 33, 36, 38, 50, 51,

77

3, 4, 5, 8, 9, 10, 15, 19, 20,

21, 23, 25, 27, 28, 31, 32,

34, 35, 37, 44, 47, 48, 52, 55

12, 24, 53, 54

Table 6.2 shows how the cited literature is divided between the most important topics of

this review. Naturally, most extensive works cover several of these topics. It seems that

the aim of the research in this area has changed with time. Earlier, the focus has been on

finding efficient techniques to reduce the size of the CD control problem (basis functions)

in order to make different control methods feasible. As model predictive control has

gained more attention, also the robustness issues have become more important and they

are involved in many recent studies e.g. [62,63]. Robust control is related to the model

accuracy and uncertainty, which means that the model reduction techniques may need to

be reconsidered. Also the alignment and shrinkage problems have gained a lot of

attention lately, since they are essential parts of the model accuracy. Especially, the non-

intrusive closed-loop methods have been under investigation e.g. [61,64,65]. Research in

the area of sensing systems has been quite continuous, but without any remarkable

breakthroughs.

37

Table 6.2. Cited literature by the topic.

Sensing CD control MD

control

Two-

dimensional

methods

Basis

function

expansion

Profile

estimation

Response

identification

Disturbance

identification

2, 5, 6, 7,

9, 10, 17,

19, 22,

56, 76

3, 4, 6, 10,

12, 16, 24,

25, 26, 27,

28, 35, 36,

44

3, 4, 5, 10,

17, 22, 24,

25, 27, 35,

71, 72, 73

4, 6, 12, 15,

16, 17, 21,

23, 24, 25,

26, 27, 31,

35, 60

16, 17, 24,

28, 30, 33,

36, 44

11, 40,

41, 42,

43, 45,

46

6, 7, 8, 14, 18,

38, 44, 57, 68,

69, 77

Finally, table 6.3 lists some of the authors, whose contributions have been in more than

one paper cited in this review. The subject column indicates roughly where the main

focus of the author has been. Notice, that this list is only an opinion of the reviewer.

Some certain observations from this list, however, can be made. Depending on one’s

interests, for example robust control in sheet forming processes, the works of

VanAntwerp and Stewart may be a good starting point for further reading. The work with

two-dimensional control has been in the hands of Duncan, Heath, Wellstead and Zarrop.

For another example, the alignment problem has been studied by Loewen and co-

workers.

38

Table 6.3. Cited literature by the author.

Author References Subjects

Arkun 28, 33, 36, 44 identification, control, PCA

Backström J. U. 23, 34 MPC

Corscadden 16, 30, 31 minimum variance control

Chen 35, 60 identification

Dumont 3, 10, 11, 25, 27, 43, 48, 63-65, 70-73 CD control, MD control

Duncan 6, 7, 14, 16, 17, 18, 26, 30, 31, 50, 53 2D methods, controllability

Ghofraniha 27, 70 identification, wavelets

Heath 5, 8, 12, 24, 31, 68, 69 CD control, basis functions

Isaksson 40, 41 MD control

Kayihan 28, 36 identification, PCA

Kjaer 5, 12 CD control

Loewen 48, 64, 65 alignment

Nesic 3, 10 wavelets, filtering

Rigopoulos 36, 44 PCA, MPC

Stewart 23, 62, 63 MPC, robust control

Taylor 17, 53 controllability

VanAntwerp 13, 32, 52, 55 MPC, robust control

Wellstead 5, 6, 7, 8, 12, 18, 68, 69 2D methods

Wang X. G. 11, 71, 73 identification, control

Vyse 15, 23 multiple actuators/variables

Zarrop 6, 18, 38 2D methods

39

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