Recent Developments in
Spatially Distributed Control
Systems on the Paper Machine
Greg Stewart and James Fan
Honeywell, North Vancouver
Presented by Guy Dumont
University of British Columbia
Outline
• Industrial Paper Machine Operation
• Selected recent developments:
- Automatic Tuning for Multiple Array Spatially Distributed
Processes
- Closed-Loop Identification of CD Controller Alignment
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Industrial Paper
Machine
Operation
The Paper Machine
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Headbox and Table
• Pulp stock is
extruded on to a
wire screen up to
11 metres wide
and may travel
faster than
100kph.
sheet
travel
Initially, the pulp stock is composed of
about 99.5% water and 0.5% fibres.
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Press Section
• Newly-formed paper
sheet is pressed and
further de-watered.
suction
presses
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Dryer Section
finished reel
• The pressed sheet
is then dried to
moisture
specifications
The paper machine pictured
is 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
weight
moisture
MD
caliper
CD
Measurement gauges
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Cross-Directional Profile Control
• control objective:
flat profiles in the
cross-direction (CD)
CD
• a distributed array of
actuators is used to
access the
cross-direction
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
CD
Actuator setpoint
array, u(t)
Sensor
measurements
MD
Measured profile
response, y(t)
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Profile Control Loop
INPUT SIGNAL, u(t)
LAN connection
CONTROLLER,
K(z)
PROCESS,
G(z)
TARGET, r(t)
LAN connection
OUTPUT SIGNAL, y(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
• Gloss, caliper and
smoothness are all affected
by:
- The lineal nip load
- The sheet temperature
- The sheet moisture content
Off Machine Supercalender
• 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
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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
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Multiple-array CD process models
• Multiple-array process model:
 y1 ( z )   G11 ( z )  G1N ( z )   u1 ( z )   d1 ( z ) 
u

 


 
Y ( z)       


      
 y ( z ) G ( z )  G
 u ( z ) d ( z )
(
z
)
N
N
1
N
N
y u
 y   y
  Nu   N y 
 G ( z )U ( z )  D( z ),
Gij ( z )  Bij  hij ( z )
where yi , d i  C
m1
, u j C
n j 1
, with i  1,  , N y , j  1,  , N u ,
N y and N u the numbers of the measuremen t arrays
and actuator arrays respective ly.
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Industrial MPC Configuration
Actuator setpoints
LAN
Direct
connection
CD
Sensor measurements
Processes
CD-MPC
Controller
Real time
QP solver
Trial and
Efficient
anderror,
robust
Closed-loop
simulations
tuning
20
LAN
(local area network)
LAN connected
when needed
Model identification
CD-MPC weights and
Automated MV Tuning
closed-loop prediction
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Objective function of CD MPC
Prediction horizon Measurement weight Control horizon
• The objective function
Aggressiveness penalty
Hp
2
j 1
Q1
V (k )   Yˆ (k  j )  Ysp

 U (k  j )  Unom
  U (k  j )
H c 1
j 1
2
Q3
2
 U (k  j ) Q
4
2
Q2

is minimized subject to actuator constraints
for optimal control solution
Picketing penalty
Energy penalty
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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
A 5-by-5 circulant matrices
 c1
c
 2
C  c3

c4
c5
c5
c1
c2
c3
c4
c4
c5
c1
c2
c3
c3
c4
c5
c1
c2
c2 
c3 
c4 

c5 
c1 
a0
0

H
F5  C  F5   0

0
 0
0 0
a1 0
0 a2
0 0
0 0
0
0 
0

0
a~1 
0
0
0
a~2
0
A 10-by-5 rectangular circulant matrices
 c1
c
 2
 c3

 c4
c
R 5
 c6
c
 7
 c8
c
 9
c10
24
c9
c7
c5
c10
c8
c6
c1
c9
c7
c2
c10
c8
c3
c4
c1
c2
c9
c10
c5
c3
c1
c6
c4
c2
c7
c8
c5
c6
c3
c4
b0
0

0

0
0
F10  R  F5H   ~
b5
0

0
0

 0
c3 
c4 
c5 

c6 
c7 

c8 
c9 

c10 
c1 

c2 
HONEYWELL - CONFIDENTIAL
0
b1
0
0
0
0
0
b2
0
0
0
b3
0
0
0
0
~
b4
0
0
0
0
0
~
b3
0
0
0
~
b2
0
0
0
0
0 
0

0
b4 

0
0

0
0
~
b1 
CDC-ECC'05 Seville, Spain
Two-dimensional frequency
•
Based on the novel rectangular circulant
matrices (RCMs) theory for CD processes,
Fm Gij ( z )FnHj
 g ij ( 0 , z )






0






0






0
g ij ( 1 , z )

g ij ( p , z )
0
0

0
g~ij ( q , z )
0

0


0




0



0




0



~

g ij ( q 1 , z )



~
g ij ( 1 , z )
Singular v alues of the single - array plant model across the spatail frequencie s
 (Gij ( z ))  g ij ( 0 , z ) , g ij ( 1 , z ) , g~ij ( 1 , z ) , , g ij ( k , z ) , g~ij ( k , z ) , 

25

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Single-array plant model in the 2-D frequency domain
0.06
|g(,e
i2 
)|
0.05
0.04
dynamical Nyquist frequency
spatial Nyquist frequency
0.03
0.02
0.01
0
10
-1
5
10
dynamical frequency
 [cycles/second]
26
4
-2
3
2
10
-3
1
0
HONEYWELL - CONFIDENTIAL
spatial frequency
 [cycles/metre]
CDC-ECC'05 Seville, Spain
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
 g ( 0 , z )

g ( 1 , z )


g~ ( 1 , z )



H T
Py Fy G ( z ) Fu Pu  



0
0
 0
 



 0
0
0











g ( k , z )

g ( k , z )



0
0
0




0 
0
0
Singular v alues of the multiple - array plant model across the spatail frequencie s
 (G ( z ))   ( g ( 0 , z )),  ( g ( 1 , z )),  ( g~ ( 1 , z )), ,  ( g ( k , z )),  ( g~ ( k , z )), 
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Closed-loop transfer function matrices
• Derive the closed-loop transfer functions of the
system with unconstrained MPC.
D(z)
Ysp
Kr
+
_
K(z)
U(z)
G(z)
+
+
Y(z)
• Performance defined by sensitivity function
1


Tyd ( z )  I  G( z ) K ( z )
• Robust Stability depended on control sensitivity function
Tud (z)  K (z)I  G(z)K (z)
1
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|tyd (,e
i2
)|
i2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
10
the surface for |tyd(,e
10
29
)|=0.7071
-2
-4
10
-6
10
dynamical frequency
 [cycles/second]
10
-8
0
spatial frequency  [cycles/metre]
Sensitivity function for single array systems
1
2
3
4
5
4.5
4
3.5
|t ( ,e
i2
yd
3
)|=0.7071
2.5
2
1.5
|t ( ,e
yd
1
i2
)<0.7071
0.5
1
2
3
4
5
6
-3
dynamical frequency  [cycles/second]
x 10
Two-dimensional
frequency bandwidth
contour
spatial frequency
 [cycles/metre]
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Control sensitivity function for single array systems
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Robust Stability (RS) Condition
(z)
K(z)
G(z)
+
+
• For additive unstructured uncertainty (e j 2 )
Tud ( z )( z ) 


~
 1  max sup t ud ( j , ei 2 ) 
j

1
 ((ei 2 ))
~
j 2
) is the representation of Tud(z) in the two where tud ( , e
dimensional frequency domain.
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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
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
10
the surface for |tyd(,ei2)|=0.7071
-2
10
-4
10
-6
10
dynamical frequency
 [cycles/second]
3
10
-8
0
1
4
5
2
spatial frequency
 [cycles/metre]
spatial frequency  [cycles/metre]
|tyd (,e
i2
)|
• Encourages a separable approach to the tuning problem:
4.5
4
3.5
3
Q4
2.5
2
1.5
|tyd( ,e
1
i2
)<0.7071
Q2
0.5
1
2
3
4
5
-3
6 x 10
dynamical frequency  [cycles/second]
“Two-dimensional frequency analysis for unconstrained model predictive control of cross-directional
processes”, Automatica, vol 40, no. 11, p. 1891-1903, 2004.
1
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Tuning procedure
Input plant info and
knob positions
Scaling
Model preparation
Horizon calculation
34
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
35
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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)
43
Traditional Manual
control
Tuning
(2sigma)
(2sigma)
0.0758
0.0565
(-14.06%)
(-35.94%)
Automated
Tuning
(2sigma)
0.0408
(-53.74%)
1.5450
(-46.19%)
Caliper
0.0882
Topside
Gloss
2.8711
4.0326
(+40.45%)
2.8137
(-2%)
Wireside
Gloss
3.5333
2.7613
(-21.85%)
2.6060
2.3109
(-26.24%) (-34.60%)
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Conclusions
• A technique was presented for solving an industrial
controller tuning problem – multi-array crossdirectional model predictive control.
• To be tractable the technique leverages spatiallyinvariant properties of the system.
• Implemented in an industrial software tool.
• Controller performance was demonstrated for two
different processes.
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Closed-Loop Identification
of CD Controller Alignment
Motivation
• 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.
Measured
Bump response
Actuator profile
CD position [space]
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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.)
800
POSITION OF RESPONSE CENTER
700
600
xj = f(j)
500
400
300
200
100
0
47
0
5
10
15
20
25
30
35
CROSS-DIRECTIONAL ACTUATOR NUMBER
HONEYWELL - CONFIDENTIAL
40
45
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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
du
r
+
-
50
K
+
+ u
G
HONEYWELL - CONFIDENTIAL
dy
+
+ y
CDC-ECC'05 Seville, Spain
Open-Loop Bumptest
• Procedure
- Open-loop insert perturbation at du
- Then record the response in y, to extract model G.
du
r
+
-
K
+
+ u
G
dy
+
+ y
• 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.
du
r
+
-
K
+
+ u
G
dy
+
+ y
• 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.
dr
r
+
+
dy
+
K
u
G
+
+ y
• 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
Spatial domain
Spatial Frequency domain
g (x )
g ( )
g p ( x)  g ( x   )
g p ( )  e j g ( )
• 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
r
+
-
K
u
G
y
• At steady-state (temporal frequency =0) the closedloop input and output can be written in spatial
frequency:
y( )  1gg(() k) k(() ) r ( ),
u( )  1 gk(() k) ( ) r ( )
• For those spatial frequencies where the control has
integral action we find the steady-state expressions:
y ( )  r ( )
57
u( )  g ( ) 1 r ( )
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CDC-ECC'05 Seville, Spain
Practical Consequence
• Combining these results we see that the change in alignment
is contained in the phase of the actuator array:
u( )  g p ( ) 1 r ( )  e j g ( )1 r ( )
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
58
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CDC-ECC'05 Seville, Spain
Example
59
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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
60
60mm
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CDC-ECC'05 Seville, Spain
Regular steady-state closed-loop operation
r
+
u
y
G
MEASUREMENT
-
K
• CD controller tuned ‘as usual’
with integral action at low
spatial frequencies.
CLOSED-LOOP STEADY-STATE PROFILES UNDER NORMAL OPERATION
193
192
191
50
100
150
200
250
ACTUATOR
20
0
BIAS TARGET
-20
61
0
5
10
15
20
25
30
35
40
45
193
192
191
50
100
150
CROSS-DIRECTION
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200
250
CDC-ECC'05 Seville, Spain
Closed-loop response of profiles
dr
r
+
+
+
K
u
G
• Bumps inserted into the bias
target profile while CD control is
in closed-loop.
y
MEASUREMENT
CLOSED-LOOP STEADY-STATE PROFILES WITH BIAS TARGET BUMPS
193
192
191
50
100
150
200
250
ACTUATOR
20
0
BIAS TARGET
-20
62
0
5
10
15
20
25
30
35
40
45
193
192
191
50
100
150
CROSS-DIRECTION
HONEYWELL - CONFIDENTIAL
200
250
CDC-ECC'05 Seville, Spain
ACTUATOR
MEASUREMENT
Response relative to baseline profiles
DIFFERENCE BETWEEN BUMPED AND NORMAL CLOSED-LOOP PROFILES
1
0
-1
50
BIAS TARGET
150
200
250
2
0
-2
0
63
100
5
10
15
20
25
30
35
40
45
1
0
-1
50
100
150
CROSS-DIRECTION
HONEYWELL - CONFIDENTIAL
200
250
CDC-ECC'05 Seville, Spain
ACTUATOR
MEASUREMENT
Profile partitioning
DIFFERENCE BETWEEN BUMPED AND NORMAL CLOSED-LOOP PROFILES
1
0
-1
50
200
250
0
-2
5
10
15
20
25
30
35
40
45
1
0
-1
50
100
150
CROSS-DIRECTION
DFT
gain phase
64
150
2
0
BIAS TARGET
100
200
250
DFT
gain
HONEYWELL - CONFIDENTIAL
phase
CDC-ECC'05 Seville, Spain
Frequency domain analysis of actuator profile
HIGH SIDE
1.05
1
1
Magnitude
Magnitude
LOW SIDE
1.02
0.98
0.96
0.94
0.8
-0.02 -0.01
0
0.01
0.02
Frequency [radians/eng unit]
1
Phase [radians]
0.5
Phase [radians]
0.9
0.85
0.92
-0.02 -0.01
0
0.01
0.02
Frequency [radians/eng unit]
0
-0.5
-0.02 -0.01
0
0.01
0.02
Frequency [radians/eng unit]
Low side phase has a slope
of 29.5mm at zero
frequency.
65
0.95
0.5
0
-0.5
-1
-0.02 -0.01
0
0.01
0.02
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.
65
29.5mm
ALIGNMENT SHIFT [eng units]
60
55
xj = f(j)
50
50.9mm
45
40
35
30
25
20
66
0
5
10
15
20
25
30
35
CROSS-DIRECTIONAL ACTUATOR NUMBER
HONEYWELL - CONFIDENTIAL
40
45
CDC-ECC'05 Seville, Spain
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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CDC-ECC'05 Seville, Spain
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. 37173723, Hawaii, USA, 2003.
•
G.E. Stewart, “Reverse Bumptest for Closed-Loop Identification of CD Controller Alignment”,
US Patent filed Aug. 22, 2005.
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