Recent Developments in Spatially Distributed Control Systems on the Paper Machine
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Transcript of 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
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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
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The Paper Machine
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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.
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Press Section
suctionpresses
• Newly-formed paper sheet is pressed and further de-watered.
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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.
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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.
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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.
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Paper Machine Process
Measurement gauges
MDCD
weight moisture caliper
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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
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Scanning Sensor
• Paper properties are measured by a sensor traversing the full sheet width.
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Cross-Directional Control
Measured profile response, y(t)
Actuator setpoint array, u(t)
CD
MD
Sensor measurements
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Profile Control Loop
LAN connection
LAN connection
INPUT SIGNAL, u(t)
OUTPUT SIGNAL, y(t)
CONTROLLER, K(z)
PROCESS, G(z)
TARGET, r(t)
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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
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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
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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
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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
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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
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Objective function of CD MPC
22
nom
1
1
2
1
2
43
21
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
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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
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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
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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
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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
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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
)|
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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
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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.
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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
spatial frequency [cycles/metre]
dynamical frequency [cycles/second]
|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
dynamical frequency [cycles/second]
sp
ati
al
fre
qu
en
cy
[
cy
cle
s/m
etr
e]
Two-dimensional frequency bandwidthcontour
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Control sensitivity function for single array systems
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• 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)
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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
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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
dynamical frequency [cycles/second]
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
spatial frequency [cycles/metre]
dynamical frequency [cycles/second]
|tyd
(,ei2
)|
the surface for |tyd
(,ei2)|=0.7071
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Tuning procedure
Scaling Model preparationInput plant info and knob positions
Horizon calculation
Spatial tuning Dynamical tuning
Results display Output tuning parameters
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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
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Spatial tuning knobs in the tool
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Tune the controller using spatial gain functions
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Dynamical tuning knobs in the tool
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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.
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Example 1: linerboard paper machine (2)
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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.
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Example 2: Supercalendars(2)
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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%)
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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
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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.
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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)
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Current and Proposed Solutions
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Solutions for Identification of Alignment
Current Industrial Solutions:- Open-Loop Bumptest
- Closed-Loop Probing
Proposed Solution:
- Closed-loop bumptest
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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
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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.
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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.
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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.
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Overview of Background Theory
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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.
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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.
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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
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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
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Example
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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
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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.
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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
+ +
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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
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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
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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]
Frequency [radians/eng unit]
-0.02 -0.01 0 0.01 0.020.8
0.85
0.9
0.95
1
1.05
Ma
gnitu
de
Frequency [radians/eng unit]
HIGH SIDE
-0.02 -0.01 0 0.01 0.02-1
-0.5
0
0.5
1
Pha
se [
radi
ans]
Frequency [radians/eng unit]
High side phase has a slope of 50.9mm at zero frequency.
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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)
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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.
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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.