Recent Developments in Spatially Distributed Control Systems on the Paper Machine

Recent Developments in Spatially Distributed Control

Systems on the Paper Machine

Greg Stewart and James FanHoneywell, North Vancouver

Presented by Guy DumontUniversity of British Columbia

2 HONEYWELL - CONFIDENTIAL CDC-ECC'05 Seville, Spain

Outline

• Industrial Paper Machine Operation

• Selected recent developments:- Automatic Tuning for Multiple Array Spatially Distributed

Processes

- Closed-Loop Identification of CD Controller Alignment

Industrial Paper Machine

Operation


The Paper Machine


Headbox and Table

sheettravel

• Pulp stock is extruded on to a wire screen up to 11 metres wide and may travel faster than 100kph.

Initially, the pulp stock is composed of about 99.5% water and 0.5% fibres.


Press Section

suctionpresses

• Newly-formed paper sheet is pressed and further de-watered.


Dryer Section

finished reel

• The pressed sheet is then dried to moisture specifications

The paper machine picturedis 200 metres long and the paper sheet travels over 400 metres.


Dry End

scanner

• The finished paper sheet is wound up on the reel.

The moisture content at the dry end is about 5%. It began as pulp stock composed of about 99.5% water.


Control Objectives

• Properties of interest:- weight

- moisture content

- caliper (thickness of sheet)

- coating & misc.

• Regulation problem: to maintain paper properties as close to targets as possible.

• Variance is a measure of the product quality.


Paper Machine Process

Measurement gauges

MDCD

weight moisture caliper


Cross-Directional Profile Control

• control objective: flat profiles in the cross-direction (CD)

• a distributed array of actuators is used to access the cross-direction

CD

MD


Scanning Sensor

• Paper properties are measured by a sensor traversing the full sheet width.


Cross-Directional Control

Measured profile response, y(t)

Actuator setpoint array, u(t)

CD

MD

Sensor measurements


Profile Control Loop

LAN connection

LAN connection

INPUT SIGNAL, u(t)

OUTPUT SIGNAL, y(t)

CONTROLLER, K(z)

PROCESS, G(z)

TARGET, r(t)


Supercalendering process

• Supercalendering is often an off-machine process used in the production of high quality printing papers

• The supercalendering objectives are to enhance paper surface properties such as gloss, caliper and smoothness

• Typical end products are magazine paper, high end newsprint and label paper


Supercalenders

Off Machine Supercalender

• Gloss, caliper and smoothness are all affected by:- The lineal nip load

- The sheet temperature

- The sheet moisture content

• With the induction heating actuators we can change the sheet temperature and the local nipload

• With the steam showers we can change the sheet temperature and the sheet moisture content

Automatic Tuning for Multiple Array Spatially Distributed Processes


Automated Tuning Overview

• Control problem-Multi-array cross-directional process models

- Industrial model predictive controller configuration

• Objectives of automated tuning• Two-dimensional frequency domain• Tuning procedure• Industrial software and examples• Conclusions


Multiple-array CD process models

• Multiple-array process model:

ly.respective arraysactuator and

arrayst measuremen theof numbers the and

,, ,1,, ,1 with , ,, where

)()(

),()()(

)(

)(

)(

)(

)()(

)()(

)(

)(

)(

11

11

1

1111

uy

uyn

jm

ii

ijijij

NNNNN

N

N

NN

NjNiCuCdy

zhBzG

zDzUzG

zd

zd

zu

zu

zGzG

zGzG

zy

zy

zY

j

yuuyy

u

y


Trial and error,Closed-loop simulations

CD-MPC weights andclosed-loop prediction

LAN (local area

network)

LAN

Directconnection

LAN connected when needed

Sensor measurements Actuator setpoints CDProcesses

CD-MPCController

Real timeQP solver

Model identification

Industrial MPC Configuration

Efficient and robust tuning

Automated MV Tuning


Objective function of CD MPC

22

nom

1

1

2

1

2

43

21

QQ

H

jQ

H

jQ

sp

jkUUjkU

jkUYjkYkVcp

)()(

)()(ˆ)(

• The objective function

is minimized subject to actuator constraints for optimal control solution

Aggressiveness penalty

Energy penaltyPicketing penalty

Measurement weightPrediction horizon Control horizon


Objectives of automated tuning

• The tuning problem is to set the parameters of the MPC:- Prediction and control horizons (Hp, Hc)

- Optimization weights (Q1, Q2, Q3, Q4)

To provide good closed-loop performance with respect to model uncertainty (balance between performance and robustness)

• Software tool requirements:- Computationally efficient implementation required for use in

the field

- Easy to use by the expected users


Circulant matrices and rectangular circulant matrices

246810

13579

102468

91357

810246

79135

681024

57913

468102

35791

ccccc

ccccc

ccccc

ccccc

ccccc

ccccc

ccccc

ccccc

ccccc

ccccc

R

1

2

3

4

5

4

3

2

1

0

510

~0000

0~

000

00~

00

000~

0

0000~

0000

0000

0000

0000

0000

b

b

b

b

b

b

b

b

b

b

FRF H

A 10-by-5 rectangular circulant matrices

12345

51234

45123

34512

23451

ccccc

ccccc

ccccc

ccccc

ccccc

C

1

2

2

1

0

5

~0000

0~000

0000

0000

0000

5

a

a

a

a

a

FCF H

A 5-by-5 circulant matrices


Two-dimensional frequency

• Based on the novel rectangular circulant matrices (RCMs) theory for CD processes,

),(~0

),(~0

),(~0

00

00

00

0),(

0),(

0),(

)(

1

1

1

0

zg

zg

zg

zg

zg

zg

FzGF

ij

qij

qij

pij

ij

ij

Hnijm j

,),(~,),(,,),(~,),(,),())((

sfrequencie spatail theacross modelplant array -single theof aluesSingular v

110 zgzgzgzgzgzG kijkijijijijij


Single-array plant model in the 2-D frequency domain

01

23

45

10-3

10-2

10-10

0.01

0.02

0.03

0.04

0.05

0.06

spatial Nyquist frequency dynamical Nyquist frequency

spatial frequency [cycles/metre]

dynamical frequency [cycles/second]

|g(

,ei2

)|


Multiple-array plant model in the 2-D frequency domain

• The model can be considered as rectangular circulant matrix blocks; and its 2-D frequency representation is

000000

000000

),(

),(

),(~),(

),(

)(

1

1

0

zg

zg

zg

zg

zg

PFzGFPk

kTu

Huyy

)),,(~()),,((,)),,(~()),,(()),,(())((

sfrequencie spatail theacross modelplant array -multiple theof aluesSingular v

110 zgzgzgzgzgzG kk


Closed-loop transfer function matrices

1)()()( zKzGIzTyd

• Performance defined by sensitivity function

Kr K(z)Ysp U(z) Y(z)

G(z)+

D(z)

_

++

1 )()()()( zKzGIzKzTud

• Robust Stability depended on control sensitivity function

• Derive the closed-loop transfer functions of the system with unconstrained MPC.


Sensitivity function for single array systems

01

23

45

10-810

-610-410

-21000

0.20.40.60.8

11.21.4

1.61.8



|tyd

(,ei2

)|

the surface for |tyd

(,ei2)|=0.7071

1 2 3 4 5 6

x 10-3

0.5

1

1.5

2

2.5

3

3.5

4

4.5

|tyd

(,ei2)<0.7071

|tyd

(,ei2)|=0.7071


sp

ati

al

fre

qu

en

cy

[

cy

cle

s/m

etr

e]

Two-dimensional frequency bandwidthcontour


Control sensitivity function for single array systems


• For additive unstructured uncertainty

where is the representation of Tud(z) in the two -dimensional frequency domain.

Robust Stability (RS) Condition

))((

1),(~supmax1)()(

22

ii

judj

ud eetzzT

),(~ 2 jud et

)( 2je

K(z) G(z)+

+

(z)


Impact of MPC weights on Sensitivity Function1

• Interesting result:- MPC weight Q2 on u does not impact the spatial bandwidth- MPC weight Q4 does not impact the dynamical bandwidth

• Encourages a separable approach to the tuning problem:

1 “Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes”, Automatica, vol 40, no. 11, p. 1891-1903, 2004.

1 2 3 4 5 6 x 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

|tyd(,ei2

)<0.7071


spat

ial f

req

uen

cy

[

cycl

es/

met

re]

-3

Q4

Q2

01

23

45

10-810

-610-410

-21000

0.20.40.60.8

11.21.4

1.61.8



|tyd

(,ei2

)|

the surface for |tyd

(,ei2)|=0.7071


Tuning procedure

Scaling Model preparationInput plant info and knob positions

Horizon calculation

Spatial tuning Dynamical tuning

Results display Output tuning parameters


Spatial tuning knobs in the tool


Tune the controller using spatial gain functions


Dynamical tuning knobs in the tool


Example 1: linerboard paper machine (1)

Four CD actuator arrays:

u1 = Secondary slice lip;

u2 = Primary slice lip;

u3 = Steambox;

u4 = Rewet shower;

Two controlled sheet properties:

y1 = Dry weight;

y2 = Moisture;

Overall model G(z) is a 984-by-220 transfer matrix.

Performance comparison between traditional decentralized control and auto-tuned MPC.


Example 1: linerboard paper machine (2)


Example 2: Supercalendars (1)

Four CD actuator arrays:

u1 = top steambox;

u2 = top induction heating;

u3 = bottom steambox;

u4 = bottom induction

heating;

Three controlled sheet properties:

y1 = caliper;

y2 = top gloss;

y3 = bottom gloss;

Overall model G(z) is a 2880-by-190 transfer matrix.

Performance comparison between traditional decentralized control, manually tuned MPC, and auto-tuned MPC.


Example 2: Supercalendars(2)


Example 2: Performance Comparison

Before control(2sigma)

Traditional control(2sigma)

ManualTuning(2sigma)

AutomatedTuning(2sigma)

Caliper 0.0882 0.0758(-14.06%)

0.0565(-35.94%)

0.0408(-53.74%)

TopsideGloss

2.8711 4.0326(+40.45%)

2.8137(-2%)

1.5450 (-46.19%)

WiresideGloss

3.5333 2.7613(-21.85%)

2.6060(-26.24%)

2.3109(-34.60%)


Conclusions

• A technique was presented for solving an industrial controller tuning problem – multi-array cross-directional model predictive control.

• To be tractable the technique leverages spatially-invariant properties of the system.

• Implemented in an industrial software tool.

• Controller performance was demonstrated for two different processes.

Closed-Loop Identification of CD Controller Alignment


Motivation

Actuator profile

MeasuredBump response

CD position [space]

• Uncertainty in alignment grows over time and can lead to degraded product and closed-loop unstable cross-directional control.

• Typically due to sheet wander and/or shrinkage.


Motivation

• In many practical papermaking applications the alignment is sufficiently modeled by a simple function.

• We assume it to be linear throughout this presentation.(Although the proposed technique is not restricted to linear alignment.)

0 5 10 15 20 25 30 35 40 450

100

200

300

400

500

600

700

800

CROSS-DIRECTIONAL ACTUATOR NUMBER

PO

SIT

ION

OF

RE

SP

ON

SE

CE

NT

ER

xj = f(j)


Current and Proposed Solutions


Solutions for Identification of Alignment

Current Industrial Solutions:- Open-Loop Bumptest

- Closed-Loop Probing

Proposed Solution:

- Closed-loop bumptest


Feedback diagram

• The standard closed-loop control diagram.- r = target (bias target)- u = actuator setpoint profile- y = scanner measurement profile

Gu y

dy

++++

du

K-

+r


Open-Loop Bumptest

• Procedure- Open-loop insert perturbation at du

- Then record the response in y, to extract model G.

Gu y

dy

++++

du

K-

+r

• Whenever possible, closed-loop techniques are preferred in a quality-conscious industry.


Closed-Loop Probing

• Procedure- Keep controller in closed-loop- Insert probing perturbation du on top of the actuator profile- Then record the response in y, to extract model G.

Gu y

dy

++++

du

K-

+r

• Technique relies on transient response of y. In practice a noisy process and scanning sensor make dynamics difficult to extract reliably.


Proposed Solution: Closed-Loop Bumptest

• Procedure- Leave loop in closed-loop control- Insert perturbation on target dr as shown- Record the response in the actuator profile u.

Gu y

dy

++K+r

dr

+ +

• The control loop is exploited to extract alignment information. No need of addressing (exciting and modeling) dynamics to extract alignment information.


Overview of Background Theory


Spatially Invariant Systems

• The theory of spatially invariant systems allows the convolution to be converted to multiplication in the frequency domain- Allows the spatial

frequency response of the entire array to be written as the Fourier transform of the response to a single actuator1

1S.R. Duncan, "The Cross-Directional Control of Web Forming Processes", PhD thesis, University of London, 1989.


Appearance of Alignment in Frequency Domain

)(xg

)()( xgxg p

)(g

)()( geg jp

Spatial domain Spatial Frequency domain

• A shift in x will appear as a linear term in the phase of its Fourier transform.


Closed-loop spatial frequency response

)()( ),()( )()(1)(

)()(1)()(

rury kgk

kgkg

)()( ry

Gu yK

-+r

• At steady-state (temporal frequency =0) the closed-loop input and output can be written in spatial frequency:

• For those spatial frequencies where the control has integral action we find the steady-state expressions:

)()()( 1 rgu


Practical Consequence

• Combining these results we see that the change in alignment is contained in the phase of the actuator array:

)()()()()( 11 rgergu jp

Practical consequence: We can identify changes in the alignment of the CD process by inserting perturbations into the setpoint to the CD controller.

Advantages:• Straightforward execution• CD control can remain in closed-loop – no need to work against the control action• Size of disruption in paper is more predictable than with actuator bumps


Example


Simulation Setup

• We introduce a combined sheet wander and shrinkage into the simulation by artificially moving the low side and high side sheet edges by 20mm and 60mm respectively.

20mm 60mm


Regular steady-state closed-loop operation

Gu yK

-+r

50 100 150 200 250191

192

193

ME

AS

UR

EM

EN

T

CLOSED-LOOP STEADY-STATE PROFILES UNDER NORMAL OPERATION

0 5 10 15 20 25 30 35 40 45-20

0

20

AC

TU

AT

OR

50 100 150 200 250191

192

193

BIA

S T

AR

GE

T

CROSS-DIRECTION

• CD controller tuned ‘as usual’ with integral action at low spatial frequencies.


Closed-loop response of profiles

• Bumps inserted into the bias target profile while CD control is in closed-loop.

50 100 150 200 250191

192

193

ME

AS

UR

EM

EN

T

CLOSED-LOOP STEADY-STATE PROFILES WITH BIAS TARGET BUMPS

0 5 10 15 20 25 30 35 40 45-20

0

20

AC

TU

AT

OR

50 100 150 200 250191

192

193

BIA

S T

AR

GE

T

CROSS-DIRECTION

Gu y

K+r

dr

+ +G

u yK+r

dr

+ +


Response relative to baseline profiles

50 100 150 200 250-1

0

1

ME

AS

UR

EM

EN

T

DIFFERENCE BETWEEN BUMPED AND NORMAL CLOSED-LOOP PROFILES

0 5 10 15 20 25 30 35 40 45

-2

0

2

AC

TU

AT

OR

50 100 150 200 250-1

0

1

BIA

S T

AR

GE

T

CROSS-DIRECTION


Profile partitioning

50 100 150 200 250-1

0

1

ME

AS

UR

EM

EN

T

DIFFERENCE BETWEEN BUMPED AND NORMAL CLOSED-LOOP PROFILES

0 5 10 15 20 25 30 35 40 45

-2

0

2A

CT

UA

TO

R

50 100 150 200 250-1

0

1

BIA

S T

AR

GE

T

CROSS-DIRECTION

DFT DFT

gain gain phasephase


Frequency domain analysis of actuator profile

Low side phase has a slope of 29.5mm at zero frequency.

-0.02 -0.01 0 0.01 0.020.92

0.94

0.96

0.98

1

1.02

Ma

gnitu

de

Frequency [radians/eng unit]

LOW SIDE

-0.02 -0.01 0 0.01 0.02-0.5

0

0.5

Ph

ase

[rad

ian

s]


-0.02 -0.01 0 0.01 0.020.8

0.85

0.9

0.95

1

1.05

Ma

gnitu

de


HIGH SIDE

-0.02 -0.01 0 0.01 0.02-1

-0.5

0

0.5

1

Pha

se [

radi

ans]


High side phase has a slope of 50.9mm at zero frequency.


Derivation of global alignment

• Here we make an assumption of linear alignment shift and thus need only two points to define a straight line.

• Confirm that the ends of the straight line correspond to the 20mm and 60mm alignment change.

0 5 10 15 20 25 30 35 40 4520

25

30

35

40

45

50

55

60

65

CROSS-DIRECTIONAL ACTUATOR NUMBER

ALI

GN

ME

NT

SH

IFT

[en

g un

its]

29.5mm

50.9mmxj = f(j)


Conclusions

• The proposed closed-loop bumptest uses a perturbation in the setpoint profile and identifies the response of the actuator array.

• Technique is sensitive to changes in alignment of the paper sheet – a practically important model uncertainty.

• Technique can be implemented with minor changes to existing installed base of CD control systems.

• Initial experiments have begun on industrial paper machines.

• While not necessary to date, more complex shrinkage models would simply require more than two bumps for identification.


References

CDC-ECC 2005 - TuB09, Process Control II• J. Fan and G.A. Dumont, “Structured uncertainty in paper machine cross-directional control”,

to appear in TuB09, Process Control II , Seville, Spain, 2005.• Borrelli, Keviczky, Stewart, “Decentralized Constrained Optimal Control Approach to

Distributed Paper Machine Control” TuB09, Process Control II , Seville, Spain, 2005

Other• J. Fan and G.E. Stewart, “Automatic tuning of large-scale multivariable model predictive

controllers for spatially-distributed processes”, US Patent (No.:11/260,809) filed 2005.• J. Fan, G.E. Stewart, G.A. Dumont, J. Backström, and P. He, “Approximate steady-state

performance prediction of large-scale constrained model predictive control systems”, IEEE Transactions on Control Systems Technology, vol 13, no. 6, p. 884-895, 2005.

• J. Fan, G.E. Stewart, and G.A. Dumont, “Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional processes”, Automatica, vol 40, no. 11, p. 1891-1903, 2004.

• J. Fan, “Model Predictive Control for Multiple Cross-Directional Processes: Analysis, Tuning, and Implementation”, PhD thesis, The University of British Columbia, Vancouver, Canada, 2003.

• J. Fan and G.E. Stewart, “Fundamental spatial performance limitation analysis of multiple array paper machine cross-directional processes”, ACC 2005, p. 3643-3649 Portland, Oregon, 2005.

• J. Fan, G.E. Stewart, and G.A. Dumont, “Two-dimensional frequency response analysis and insights for weight selection of cross-directional model predictive control”, CDC’03, p. 3717-3723, Hawaii, USA, 2003.

• G.E. Stewart, “Reverse Bumptest for Closed-Loop Identification of CD Controller Alignment”, US Patent filed Aug. 22, 2005.

Recent Developments in Spatially Distributed Control Systems on the Paper Machine

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