simulation studies of paper machine basis weight control. august 2010

CONTROL ENGINEERING LABORATORY

Simulation studies of paper machine basis

weight control

Markku Ohenoja, Ari Isokangas & Kauko Leiviskä

Report A No 43, August 2010

ii

University of Oulu

Control Engineering Laboratory

Report A No 43, August 2010

Simulation studies of paper machine basis weight control

Markku Ohenoja, Ari Isokangas & Kauko Leiviskä

University of Oulu, Control Engineering Laboratory

Abstract: This report presents some simulation results of basis weight control in paper machine. In

particular, the dilution actuator width, actuator response width and the effect of scanner measurement

uncertainty are studied. Also different controller configurations in machine direction control are tested.

Basis weight is one of the most important quality parameters in paper manufacturing. The control problem

is two-dimensional, since variations occur both across the paper web (cross-direction, CD) and along the

production line (machine direction, MD). In addition, the basis weight is measured from the end of the line

and the actuators are at the beginning of line. This means long measurement delay, which makes the

problem more difficult.

The simulations are performed using the simulator originally developed in Tampere University of

Technology. The control performance is evaluated using the criteria of 2σ standard deviation, spectral

properties, and variance reduction percentages of disturbances.

CD control performance decreases with wider CD actuator widths. Simulations suggested that better CD

control performance was achieved using wider actuator response widths than it was identified earlier in a

mill. Increased scanner measurement uncertainty decreased the CD control performance, since the

increased measurement uncertainty provides worse CD estimates. MD control performance is related to the

control interval applied and, of course, the tuning of the controller. Predictive controller simulated here was

found to be practical if different control intervals are applied.

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

ISBN 978-951-42-6271-5 University of Oulu

ISSN 1238-9390 Control Engineering Laboratory

PL 4300

FIN-90014 University of Oulu

iii

CONTENTS

1 INTRODUCTION 1

2 BASIS WEIGHT CONTROL 2 2.1 Variations in paper machine 2

2.2 Control 2 2.3 Measurements 3

3 PERFORMANCE CRITERIA 5

4 SIMULATION STUDIES 8 4.1 Simulated disturbances 8 4.2 Definitions in simulations 8

5 RESULTS 10 5.1 Effect of CD actuator width 10

5.2 Effect of CD actuator response width 11 5.3 Effect of scanner measurement uncertainty 12 5.4 Simulation of wire section problem 13

5.5 Simulation of unstable CD disturbance 13 5.6 MD simulations 14

6 SUMMARY AND CONCLUSIONS 16

7 REFERENCES 17

APPENDIX

1

1 INTRODUCTION

The studies reported here are a part of QVision project, which aims to measure paper web

properties of paper machine in high resolution, for example to enable better elimination

of paper web disturbances. Simulations in this report show how the actuator settings of

the headbox effect on paper web quality in cross direction. The effect of different control

strategies on paper quality is studied in machine direction. First factor is the effect of CD

actuator width on simulated paper quality. Then the effect of the actuator response width,

which depends on machine speed and head box construction, on paper quality is studied

by keeping the actuator width constant and studying the effect of the actuator response

width on paper quality. Also the performance of the CD control with dynamically

changing CD profile disturbances is evaluated. In addition, the control performance is

studied in different cases (varying scanner measurement uncertainty and a simulated wire

section problem). The MD control simulations include testing different controllers and

control intervals.

The simulation studies utilise the simulator originally developed in Tampere University

of Technology [8,9]. Authors have made following additions:

- Smart distribution of the actuators when the actuator width (i.e. actuator spacing) is

changed.

- A possibility to change the actuator response according to the actuator width.

- The usage of dilution actuator control model with the base, which differs from the

simulated response model, i.e. mismatch for more realistic simulation.

- Measurement delays from CD actuators to scanner measurement.

- MD control development. PI controller tuning and development of novel model

predictive controller (MPC).

- Profile estimations based on exponential filtering and scan averages.

Chapter 2 gives a brief introduction to the basis weight control problem in paper

machine. For more details, see e.g. [11]. In Chapter 3, the criteria for the evaluation of the

control performance are given. The simulated cases are shortly introduced in Chapter 4

and the simulation results can be found from Chapter 5. Finally, Chapter 6 summarizes

the main findings.

2

2 BASIS WEIGHT CONTROL

2.1 Variations in paper machine

Basis weight is probably the most important quality parameter in paper manufacturing.

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

(cross-direction, CD) and along the production line (machine direction, MD).

Additionally, there are diagonal waves i.e. variations that cannot be classified to CD or

MD variations. In addition, a large part of the variation takes place at too high

frequencies for control. This part of the total variations is called residual variation. Figure

1 demonstrates the variations in basis weight.

Figure 1. Variations in a paper machine.

2.2 Control

Separate control systems are used for the machine-direction and cross-direction. Basis

weight variation in MD is controlled by adjusting the amount of pulp flowing into the

headbox. Variation in CD is controlled with numerous actuators in different CD locations

either by adjusting the pulp flow (slice lip headbox) or by adjusting the consistency of

pulp flow (dilution headbox) distributed to the wire. The MD control takes care of the

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

3

the wire while preserving the setpoint i.e. the mean value in CD. The control problem is

illustrated in Figure 2.

Figure 2. Estimation and control of basis weight variations [1].

CD actuators response is wider than the actuator hence the response affects also the

adjacent actuators. These interactions are taken into account by defining the control

actions that minimise the observed CD variation for the whole width of the paper i.e.

setpoint values for each actuator are calculated simultaneously. Usually only a part of the

calculated control action is performed to ensure stability. MD control is performed

usually with PI-controllers or model predictive controller (MPC).

2.3 Measurements

Control actions are based on the profile estimates which are calculated from the

measurements 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 a long

delay between the actuators and measurements. Additionally, since the sensor travels

back and forth across the paper web at speed from 0 to 1 m/s, while the paper web moves

fast in other direction, the measurement path forms a zig-zag-pattern (see Figure 3). The

horizontal parts in the path correspond to the fact that the scanner stays some seconds at

the paper edge before changing the direction.

4

Figure 3. Measurement path caused by the movement of the paper web and the traversing scanner.

Only a small amount of the paper web is actually measured, e.g. 0.2 % [2] or 0.001 % [3].

The movement of the sensor and the paper web means that the measurements contain

mixed information of CD and MD variations. Some MD variation may be aliased into the

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

The measurement frequency of the scanner can be up to 4000 Hz. Scanner measurements

are usually averaged in e.g. 1 cm data boxes, which are then used in calculating the

profiles. The CD profile estimates are conventionally generated using exponential

filtering i.e. the estimate in each CD position is a weighted average of data boxes. The

MD profile is estimated from the average of a single scan. It means that the new MD

profile information is available when a single scan is completed. Depending on the speed

of the scanner and the width of the paper web, it takes up to 50 seconds to perform a

scan. Due to exponential filtering, changes in CD profile are detected even slower than

MD changes. More sophisticated methods use Kalman filtering to give profile estimates

at each time instant. This improves the dynamic bandwidth of the measurement

significantly.

5

3 PERFORMANCE CRITERIA

The most typical criterion for evaluating the control performance is 2σ i.e. twice the

standard deviation of the measurements. The criterion describes the average deviation

from the mean value. The 2σ value means that 95 % of measured values deviate less than

2σ value from the average value. It is also the criterion constantly observed in paper

manufacturing process. In these simulation studies, the 2σ, and other criteria, are

calculated from the actual data, not from the profile estimates.

The control performance is evaluated separately in machine direction and cross direction.

The following criteria Ci are used (the acronym in parentheses):

- 2σ standard deviation in CD (2σ CD):

1

)(

*2 1

2

1

M

xx

C

M

i

i

(1)

where x is the average of data points xi. M is the number of data points in CD, xi

are calculated from:

N

x

x

N

k

k

i

1 (2)

where N is the number of data points in MD.

- 2σ standard deviation in MD (2σ MD)

1

)(

*2 1

2

2

N

xx

C

N

i

i

(3)

where x is the average of data points xi. N is the number of data points in MD, xi

are calculated:

M

x

x

M

k

k

i

1 (4)

where M is the number of data points in CD.

6

With simulated data, it is possible to compare the controlled and the uncontrolled paper

webs. Hence, the reduction of variance is also used as a performance criterion. The

criterion gives the percentage of the removed disturbances in simulation. The whole finite

amount of data in simulation at its full resolution is used to perform the calculations.

- reduction of variance in CD [%] (CD var reduction)

100*

1

)(

1

)(

100

1

2

1

2

3

M

zz

M

xx

CM

i

i

M

i

i

(5)

where x are the controlled paper web points values and z the values of uncontrolled paper

web.

- reduction of variance in MD [%] (MD var reduction)

100*

1

)(

1

)(

100

1

2

1

2

4

N

zz

N

xx

CN

i

i

N

i

i

(6)

Power spectrum gives information on the nature of the variations. The data is transformed

into spectrum data with Matlab function ―fft‖, fast Fourier transform. Here the spectrum

has been used as a visual inspection tool showing how different components of simulated

variations have been attenuated by the control actions. Figure 4 illustrates the typical

power spectra of uncontrolled paper web and controlled paper web in machine direction

and in cross direction. Also, the sum of spectrum values and variance of spectrum values

are used as performance criteria (CD spectrum sum, CD spectrum variance).

7

0 0.005 0.01 0.015 0.020

100

200

300

400

Hz

MD power spectral density

Simulated disturbances

Controlled web

0 0.05 0.1 0.150

0.1

0.2

0.3

0.4

0.5

1/cm

CD power spectral density

Simulated disturbances

Controlled web

Figure 4. MD and CD power spectra for input disturbances and controlled paper web. The simulated

disturbances in CD spectrum reach the peak values of around five.

From the MD spectrum, it can be seen that the input disturbances have frequencies

around 0.001 and 0.002 Hz, where the higher frequency component has also higher

amplitude. The controller has attenuated these components quite nicely. From the CD

spectrum, it is visible that the main disturbance components with wavelengths around

0.01 and 0.02 1/cm have been efficiently damped by the control actions.

8

4 SIMULATION STUDIES

4.1 Simulated disturbances

The input disturbances are sine waves of form:

xfAd 2sin (7)

where A is the amplitude of the variation and f is its frequency. x is the length of the

simulation (MD) or width of the simulated paper web (CD). The amplitudes and

frequencies of simulated variations are listed in Table 1. In addition, simulator generates

also zero-mean residual variation with a variance of 0.1. In CD simulation studies in

Chapters 5.1, 5.2 and 5.3 MD, disturbances are not simulated and MD control is turned

off. Likewise, the CD components and CD control are not included in MD simulations in

Chapter 5.6. The simulation in Chapter 5.4 and 5.5 contains different disturbance

scenarios.

Table 1. Simulated disturbances.

CD component 1 CD component 2 CD component 3 MD component 1 MD component 2

A 0.8 0.7 0.3 0.7 0.5

f 0.1 0.04 0.0025 0.0017 0.001

The simulated disturbances demonstrate both the fast variations that most likely cannot

be removed by the control systems and slower variations that are somewhat controllable.

4.2 Definitions in simulations

CD actuator width (act width [cm]) is the ratio of paper web width and the number of

actuators. The response width (resp width) describes the width of CD actuator response

in cross direction (CD). The actuator response width is 100 mm at 50 % amplitude of

response [7]. The response width parameter 4 cm results in similar response as in [7].

In result tables, the following properties are presented to display settings used in

simulations:

- First centre means the place of the first CD actuator response peak at the paper web

(at left side). The place of the first centre is calculated on the basis of the actuator

width.

- CD control on means whether the CD control is enabled (=1) or disabled (=0).

- MD control means which type of control is used in simulation. 1 means the pre-

computed GPC and 2 the PI controller.

9

- Saato means which measurements are the bases of estimates in simulations. 1 means

the scanner measurement based on Kalman filter and 4 the scanner measurement

based on exponential filtering.

- Delta u MD means how often control actions in CD are performed.

- Delta u CD means how often control actions in MD are performed.

- Alfa describes how much of calculated CD control action is adopted.

- N means the length of the paper web in MD (s).

- M the width of the paper web in CD (cm).

10

5 RESULTS

5.1 Effect of CD actuator width

This simulation case illustrates the effect of CD actuator spacing (act width [cm]) in the

dilution headbox. The simulation results using Kalman filter-based estimation are

presented in Table 2. The results using exponential filtering are presented in Table 3. The

actuator response width (resp width) is constant 4 cm, which results the width of 10 cm at

50 % amplitude of response, which was earlier identified in a mill [7].

Table 2. Simulation results as a function of CD actuator width using Kalman filter-based estimation.

2σ CD 0.45 0.53 1.15 1.17 1.19 1.21 1.26 1.31 1.35 1.37 1.41 1.41

2σ MD 0.20 0.20 0.20 0.21 0.20 0.21 0.20 0.19 0.19 0.19 0.20 0.19

CD spectrum sum 3.70 12.23 84.65 84.15 86.38 90.64 97.16 105.77 114.38 118.16 124.01 125.28

CD spectrum variance 0.00 0.25 21.52 24.42 25.85 25.65 25.62 26.57 27.36 26.82 27.80 28.54

CD var reduction 91.94 88.71 44.81 44.76 43.51 40.51 35.91 31.52 26.05 23.29 20.09 18.15

act width 4 5 6 7 8 9 10 11 12 13 14 15

resp width 4 4 4 4 4 4 4 4 4 4 4 4

first centre 2 3 3 3 4 4 5 5 6 6 7 7

CD control on 1 1 1 1 1 1 1 1 1 1 1 1

saato 1 1 1 1 1 1 1 1 1 1 1 1

delta u CD 11 11 11 11 11 11 11 11 11 11 11 11

alfa 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9

N 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

M 500 500 500 500 500 500 500 500 500 500 500 500

Table 3. Simulation results as a function of CD actuator width using exponential filtering.

2σ CD 0.35 0.46 1.14 1.16 1.16 1.20 1.25 1.27 1.36 1.36 1.41 1.42

2σ MD 0.20 0.19 0.19 0.19 0.20 0.20 0.20 0.20 0.20 0.21 0.19 0.20

CD spectrum sum 4.36 13.45 83.15 84.31 85.09 90.93 97.61 101.13 115.07 116.13 125.39 126.03


CD var reduction 95.18 91.52 45.39 44.52 44.84 41.09 37.20 32.62 26.95 22.82 19.27 18.60

act width 4 5 6 7 8 9 10 11 12 13 14 15

resp width 4 4 4 4 4 4 4 4 4 4 4 4

first centre 2 3 3 3 4 4 5 5 6 6 7 7

CD_control_on 1 1 1 1 1 1 1 1 1 1 1 1

saato 4 4 4 4 4 4 4 4 4 4 4 4

delta_u_CD 11 11 11 11 11 11 11 11 11 11 11 11

alfa 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9

N 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

M 500 500 500 500 500 500 500 500 500 500 500 500

All CD performance criteria suggest that CD control performance decreases with wider

CD actuator widths. The simulated variations have the wavelengths of 10, 25 and 400

cm. Only disturbances wider than two times of the actuator width can be controlled

[4,5,6]. The results support the theory, since only the actuator widths 4 and 5 cm can

reduce the most frequent disturbance with 10 cm wavelength. According to theory [4, 5,

11

6], the actuator widths 13 – 15 cm can only reduce the disturbance of 400 cm wavelength

disturbance. The results in Table 3 using exponential filtering as an estimation method

are slightly better than in Table 2 using Kalman filter.

Narrower actuator widths allow the elimination of disturbances with shorter wavelengths.

On the other hand, paper web shrinking or wandering cause problems and the problem is

emphasized with narrower actuator widths. In simulations, shrinking and wandering were

not considered.

5.2 Effect of CD actuator response width

In the dilution headbox, the width of the actuator response is constant. The following

simulations are performed to study whether the response width is optimal or could

additional improvement be achievable if the actuator response would be narrower or

wider. The effect of the CD actuator response width was studied by keeping the actuator

width constant at 10 cm and changing the width of actuator response (resp_width). To

study only the effect of the actuator response width, the mismatch between the actuator

response and control response was removed i.e. the simulated response and control

response model are congruent. The results are presented in Table 4.

Table 4. Simulation results as a function of the CD actuator response width using exponential

filtering.

2σ CD 1.32 1.19 1.23 1.24 1.20 1.18 1.16 1.13 1.14 1.14 1.15 1.15

2σ MD 0.23 0.36 0.38 0.23 0.21 0.21 0.20 0.19 0.20 0.20 0.20 0.20

CD spectrum sum 109.73 86.28 88.32 96.20 90.18 87.16 83.64 80.41 80.76 81.25 82.53 81.96


CD var reduction 28.34 42.38 38.89 37.70 41.44 43.38 45.53 46.93 47.21 47.01 47.39 46.94

act width 10 10 10 10 10 10 10 10 10 10 10 10

resp width 1 2 3 4 5 6 7 8 9 10 11 12

first center 5 5 5 5 5 5 5 5 5 5 5 5

CD control on 1 1 1 1 1 1 1 1 1 1 1 1

saato 4 4 4 4 4 4 4 4 4 4 4 4

delta u CD 11 11 11 11 11 11 11 11 11 11 11 11

alfa 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9

N 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

M 500 500 500 500 500 500 500 500 500 500 500 500

According to Table 4, the results are mainly at the same magnitude for all actuator

response widths, except the narrowest ones. Actuator response width 8 cm results in the

smallest 2σ CD standard deviation and the smallest CD spectrum sum, though the wider

actuator response widths result in better CD variance reduction. Figure 4 illustrates the

CD actuator response width 8 cm (red line). Blue line illustrates the actuator response

width of 10 cm at 50 % response amplitude, which is identified in a mill [7]. The blue

line corresponds the actuator response width (resp_width) of 4 cm. Simulations therefore

suggest that, in this case, better CD control performance is achieved using wider actuator

response widths than it was identified in the mill.

12

Figure 4. Blue line describes the response width of 10 cm at 50 % response amplitude, which was

identified in a mill [7]. According to simulation results, better results are obtained using wider CD

actuator response widths. Red line describes the actuator response width 8 cm.

5.3 Effect of scanner measurement uncertainty

The effect of scanner measurement uncertainty (sigma_mitt_0) was studied on paper

web quality, Table 5.

Table 5. Simulation results as a function of scanner measurement uncertainty. sigma_mitt_0 2 4 6 8 10 12 14 16 18 20 22 24

2σ CD 1.18 1.21 1.17 1.25 1.26 1.28 1.31 1.40 1.30 1.39 1.34 1.42

2σ MD 0.20 0.19 0.20 0.19 0.20 0.20 0.21 0.20 0.20 0.20 0.20 0.20

CD spectrum sum 94.39 99.32 103.79 118.31 126.72 139.05 143.51 156.13 142.68 143.44 166.85 184.89


CD var reduction 42.76 39.95 43.27 35.91 36.69 32.47 31.24 20.68 31.55 21.39 27.36 19.98

act width 6 6 6 6 6 6 6 6 6 6 6 6

resp width 4 4 4 4 4 4 4 4 4 4 4 4

first centre 3 3 3 3 3 3 3 3 3 3 3 3

CD_control_on 1 1 1 1 1 1 1 1 1 1 1 1

saato 4 4 4 4 4 4 4 4 4 4 4 4

delta_u_CD 11 11 11 11 11 11 11 11 11 11 11 11

alfa 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9

N 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

M 500 500 500 500 500 500 500 500 500 500 500 500

Increase in the scanner measurement uncertainty naturally decreases the control

performance, since the uncertainty reflects to the estimates. CD control has managed to

13

reduce variation in CD in all cases even with high sigma_mitt_0 values. The CD profile

is estimated using exponential filtering, which seems to eliminate the effect of increased

measurement uncertainty.

5.4 Simulation of wire section problem

Figure 5 illustrates simulation of disturbance in the wire section, where the basis weight

at CD locations 200 - 230 cm is decreased by 1 unit. The actuator width is 10 cm and

only residual noise and simulated wire section disturbance are used in cross direction

(CD).

Figure 5. The simulation of wire section problem, where the basis weight is decreased by one unit in

CD locations 200 – 230 cm. The picture on right illustrates the effect of CD control at MD row 1000.

The CD control has managed to eliminate the wire section disturbance. In this case, the

disturbance width (30 cm) is three times wider than actuator width (10 cm). According to

earlier studies, the wavelengths twice wider than the actuator spacing can be removed

[4,5].

5.5 Simulation of unstable CD disturbance

In earlier simulations, the incoming CD variations were normal waves with constant

wavelengths and amplitudes. Here the control performance of the linear steady-state

optimal controller was tested against dynamically changing, or unstable, CD variations.

The simulated variations were the two controllable wavelengths in Table 1 with the

amplitude changing in time. The amplitudes in Table 1 define the boundaries for the

changing amplitude. The time behaviour follows also a normal wave, but the phase is

different for the both variation components. The actuator settings were act_width=6,

resp_width=4.

Figure 6 shows that the disturbances are efficiently damped with both estimation

methods. Performance criteria show that the results are now better with Kalman filter.

Reduction in CD variation for the exponential filtering and Kalman filter was 72% and

14

84%, respectively. Unstable CD disturbance does not seem to have a significant effect on

control behaviour. This was expected with the current robust controller settings (only 10

% of control action is performed at each control interval) and the control interval is, of

course, longer than the dead-time between actuators and measurements. Most likely,

more variation could be damped with more aggressive tuning or shorter control interval.

0 0.05 0.1 0.150

10

20

1/cm

CD power spectral density

Exponential filtering , =7.673 , =0.10816

Kalman filtering , =7.4021 , =0.085071

Input disturbances , =39.9188 , =3.1537

Figure 6. CD power spectrums for input disturbances and controlled paper, when control actions are

based on different estimation methods.

5.6 MD simulations

In MD simulations, the results for the two different controllers as a function of MD

control interval are presented. The results for the PI controller are presented in Table 6

and the results for the pre-computed Generalised Predictive Controller (GPC) in Table 7.

GPC is constructed as explained in [10] using a first order plus delay model, see also

Appendix. Notice that the simulated MD response follows a second order model. The

simulated paper web width is 400 cm i.e. the time consumed by a single scan is 8

seconds. Hence, control intervals of the scan time and multiple of scan times are studied

with a traditional estimating mode. The smaller control intervals are performed with the

estimates provided by Kalman filter. The control performance is only evaluated

according to disturbance rejection properties. Setpoint following and stability in setpoint

changes are not taken into account when tuning the controllers.

15

Table 6. Simulated results as a function of MD control interval using PI controller.

2σ CD 0.20 0.20 0.20 0.20 0.20 0.19 0.20 0.21 0.21 0.20 0.20 0.20

2σ MD 2.E+10 43.74 0.56 0.70 0.75 0.78 0.77 0.79 0.80 0.82 0.84 0.87

MD spectrum sum 7.E+22 5.E+05 79.61 121.27 140.42 151.53 146.54 155.19 161.92 168.89 176.10 190.82

MD spectrum variance 2.E+41 4.E+07 1.01 3.79 5.52 6.59 5.99 6.83 7.41 8.07 8.86 9.98

MD var reduction -2.E+22 -1.E+05 80.40 69.84 64.59 62.27 63.23 61.21 59.19 57.22 55.68 51.96

MD control 2 2 2 2 2 2 2 2 2 2 2 2

saato 1 1 4 4 4 4 4 4 4 4 4 4

delta u MD 2 4 8 16 24 32 40 48 56 64 72 80

N 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

M 400 400 400 400 400 400 400 400 400 400 400 400

The PI controller was tuned using a gain term P equal to one and the integration term five

times the control interval. This tuning gives great results with the control interval of 8

seconds. However, the MD control performance decreases with increased MD control

interval. It is obvious that the controller should be tuned more carefully when changing

the control interval. Simulations with the constant tuning parameters produced unstable

control performance. For the shorter control intervals, the PI controller fails in control,

because all simulations have negative MD variance reduction value meaning that control

has actually increased variation.

Table 7. Simulated results as a function of MD control interval using GPC controller.

2σ CD 0.19 0.21 0.19 0.21 0.20 0.20 0.21 0.20 0.20 0.20 0.20 0.20

2σ MD 0.60 0.69 0.69 0.70 0.71 0.69 0.71 0.69 0.72 0.74 0.73 0.78

MD spectrum sum 90.70 117.60 120.49 122.71 124.62 118.08 124.23 120.34 130.76 136.09 133.59 153.13

MD spectrum variance 2.05 3.84 3.97 4.33 3.58 3.60 3.86 3.05 4.41 4.85 4.71 6.12

MD var reduction 77.10 70.50 69.96 69.30 68.50 70.67 69.03 69.32 66.82 66.26 66.72 61.94

MD control 1 1 1 1 1 1 1 1 1 1 1 1

saato 1 1 4 4 4 4 4 4 4 4 4 4

delta u MD 2 4 8 16 24 32 40 48 56 64 72 80

N 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

M 400 400 400 400 400 400 400 400 400 400 400 400

The pre-computed GPC can be tuned with a single tuning parameter λ, which is set here

to the constant value of one. Since the model in the controller is based on sampling time

one, the best control performance is achieved with the smallest control interval. The MD

control performance decreases with the increased MD control interval, but the

degradation is not as clear as with PI controller. Another option to use the GPC is to

identify the process model with the sampling time determined by the control interval. The

tests not presented here, however, showed that the tuning parameters must also be altered

when the control interval changes.

16

6 SUMMARY AND CONCLUSIONS

CD control performance decreases with wider CD actuator widths. The rule of thumb is

that only disturbances with wavelengths twice wider than the actuator spacing can be

removed [4, 5, 6]. The simulated results support this theory. Narrower actuator widths

allow the elimination of disturbances with shorter wavelengths. On the other hand, paper

web shrinking or wandering cause problems and the problem is emphasized with

narrower actuator widths. In simulations, shrinking or wandering was not considered.

Simulations suggest that better CD control performance is achieved using wider actuator

response widths than it was identified earlier in a mill [7]. Although the analysis is

incomplete, the results suggest that there would be additional improvement achievable

with different design of the dilution actuators and headbox construction. Naturally, wider

response is also achieved by decreasing the speed of the machine, which, however, is not

an acceptable option in most cases.

Increased uncertainty in the scanner measurement decreases the CD control performance.

The result is expected, since in some point the capability of the estimation method to

filter out the noise is exceeded and the increased measurement uncertainty provides

worse CD estimates. With the controller settings used in simulations, unstable CD

variations do not harm the control performance significantly. However, most likely a

more aggressive control strategy would remove more variations, but the robustness may

become an issue.

MD control performance with different control intervals decreases as the interval

increases. The degradation is more profound with PI controller than with pre-defined

Generalised Predictive Controller. With careful tuning the performance of the PI

controller may be better than with the GPC, but the stability is easily lost. GPC can be

applied with different control intervals without re-tuning, still achieving a good control

performance in each case.

The performance criteria used in these simulation studies contain some basic measures of

control performance. The simulated data allows also evaluating the reduction of the

overall variation. An effort to calculate some numerical values from spectrum data is also

given. Since all measures give similar behaviour, it cannot be concluded if one is suited

better than another. The behaviour of performance criteria should also be studied with the

data from profile estimates.

Finally, the authors want to point out, that the results of the control performance are, of

course, case sensitive and related to the simulated disturbance scenarios. Solid

conclusions based on these results should not be made. For a more comprehensive control

studies, different methods should be applied. The simulations here are more or less

performed also to demonstrate the usability of the simulator as a research tool. The

modular structure of the simulator allows also using mill data as an input.

17

7 REFERENCES

1. Wang X. G., Dumont G. A. & Davies M. S. (1993) Adaptive Basis Weight Control in

Paper Machines. Second IEEE Conference on Control Applications, Vancouver,

B.C., September 13-16, p. 209-216.

2. Ratilainen M. (2004) Paperikoneen laatusäätöjärjestelmän uusinnan vaikutukset

laatusäätöjen suorituskykyyn. Diplomityö. Lappeenrannan teknillinen korkeakoulu.

3. Kjaer A. P., Wellstead P. E. & Heath W. P. (1997) On-Line Sensing of Paper

Machine Wet-End Properties: Dry-Line Detector. IEEE Transactions on Control

Systems Technology, 5(6), p. 571-585.

4. Duncan S. R. (2001) Two-dimensional control systems: Learning from

implementations. Journal of Electronic Imaging, 10(3), p. 661-668.

5. Heaven E. M., Jonsson I. M., Kean T. M., Manness M. A. & Vyse R. N. (1994)

Recent Advances in Cross Machine Profile Control. IEEE Control Systems

Magazine, 14(5), p. 35-46.

6. Duncan S. R. & Bryant G. F. (1997) The Spatial Bandwidth of Cross-directional

Control Systems for Web Processes. Automatica, 33(2), p. 139-153.

7. Shakespeare J. (2001) Identification and Control of Cross-machine Profiles in Paper

Machines: A Functional Penalty Approach. Ph.D. thesis, Tampere University of

Technology, Finland.

8. Ylisaari J., Konkarikoski K., Ritala R., Kokko T., & Mäntylä M. (2009) Simulator for

Testing Measurement, Estimation and Control Methods in 2D Processes, Proceedings

of the MATHMOD Vienna - Vienna International Conference on Mathematical

Modelling, February 11-13, Vienna, Austria, 2009.

9. Ohenoja M., Ylisaari J., Leiviskä K. & Ritala R. (2010) Analyzing 2-Dimensional

Variation Based on Scanning and Imaging Measurements. Accepted to be presented

in TAPPI PaperCon 2010, May 2-5, Atlanta, GA, USA.

10. Bordons C. & Camacho E. F. (1998) A Generalized Predictive Controller for a Wide

Class of Industrial Processes. IEEE Transactions on Control Systems Technology,

6(3), p. 372-387.

11. Ohenoja M. (2009) One- and two-dimensional control of paper machine: a literature

review. Report A No 38, Control Engineering Laboratory, October 2009.

18

APPENDIX

A Generalized Predictive Controller, GPC [10]

Model Predictive Control (MPC) does not refer to a single control algorithm, but a family

of algorithms that differ in terms of model applied and the in the way the control actions

are calculated. They have the following step-wise procedure in common

- An explicit process model is used in predicting the future plant output over the

prediction horizon.

- A sequence of the future manipulated variable is calculated optimally so that the

predicted output is as close as possible to the desired future output.

- The first value in the sequence of manipulated variables is applied to the plant.

The following presentation is based on Reference [10] that introduces a procedure

starting from the first order model and ending up with a predictive controller having pre-

calculated control parameters that leave only one tuning parameter. The basic GPC

algorithm is given e.g. in [A1].

GPC algorithm minimizes the following cost function

2

1

2 2

1

ˆ ( ) 1uNN

j N j

J E y t j t w t j j u t j

(A1)

and calculates the future control sequence so that the future plant output is forced close to

the reference trajectory. Following denotations are in use; E{-} is the expectation

operator, y t j t is the optimum j-step prediction of the system output until time t, N1

and N2 are the minimum and maximum costing horizons, Nu is the control horizon, (j) is

the sequence of control weights, w(t+j) is the future reference trajectory and u(t+j-1) is

the control signal trajectory.

GPC in [10] uses a first order plus delay model

( )1

d sKG s e

s

(A2)

K is the process gain, is the time constant and d is the time delay. This model can be

transformed into [10]

( 1) (1 ) ( ) ( 1) ( ) ( 1)y t a y t ay t b u t d t (A3)

In (A3), a=e-T/

, b=K(1-a), d=d/T and is zero-mean white noise. Notice that the dead

time is assumed to be an integer multiple of the sampling time. The minimum costing

horizon must be greater that the dead time. In the following, N1=d+1 and N2=d+Nu.

19

According to [10] the best expected value for the output is

ˆ ˆ ˆ( ) (1 ) ( 1 ) ( 2 ) ( 1)y t d j t a y t d j t ay t d j t b u t j (A4)

assuming that two output estimates on the right-hand side are known. Minimizing (A1)

gives the formula for the control increment

1 2 1ˆ ˆ( ) ( ) ( 1 ( ) ( )y y ru t l y t d t l y t d t l r t (A5)

where r(t) is the reference signal. The coefficients li are functions of a, b and (t).

Reference [10] gives detailed information on how to calculate control parameters in

advance and how the algorithm is working.

References

A1 D. W. Clarke, C. Mohtadi, and P. S. Tuffs, ―Generalized predictive control—Part

I: The basic algorithm,‖ Automatica, 23, pp. 137–148, 1987.

20

ISBN 978-951-42-6271-5

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27. Baroth R. Literature review of the latest development of wood debarking. August

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28. Mattina V & Yliniemi L, Process control across network, 39 p. October 2005.

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37. Mäyrä O & Leiviskä K, Modelling in methanol synthesis. December 2008.

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39. Paavola M & Leiviskä K, ESNA – European Sensor Network Architecture. Final

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40. Virtanen V & Leiviskä K, Process Optimization for Hydrogen Production using

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41. Keskitalo J & Leiviskä K, Mechanistic modelling of pulp and paper mill wastewater

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43. Ohenoja M, Isokangas A & Leiviskä K, Simulation studies of paper machine basis

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