C LOSED - LOOP I DENTIFICATION WITH MPC
FOR AN I NDUSTRIAL
S CALE CD C ONTROL P ROBLEM
BY
DANIEL R. S AFFER II
F RANCIS J. D OYLE , III
U NIVERSITY
OF
D ELAWARE
D EPARTMENT OF C HEMICAL E NGINEERING
N EWARK , DE 19716 USA
Abstract
This paper details an approach to implement closed-loop identification in a model predictive control framework for
the cross-direction control of basis weight. The closed-loop identification technique uses the concepts of Markov parameters to determine a step response model that can then be used within a model predictive controller. The technique
is applied to an industrial scale paper machine simulation benchmark problem for cases of varying degrees of shrinkage. Performance of the wet and dry end full array sensors are compared. Also the performance of the closed-loop
identification is compared for a nominal case and two cases in which shrinkage occurs in the drying process.
1
1 Introduction
For a number of years, researchers have been exploring methods to determine control relevant models for CrossDirectional (CD) control of sheet and film forming processes. Most identification techniques have considered open-loop
bump tests of some or all of the actuators. One technique that has been studied by a number of researchers is to reduce
the order of the model by transforming the spatial response of the actuators. These techniques have included the use
of Gram Polynomials [1], pseudo-singular value decomposition [2], and principle components analysis [3, 4]. Upon
transformation, the parameters within the new model space have been identified using open-loop bump tests. A number
of studies [5, 6, 7, 8] have used a least-squares technique to fit step or impulse response data to a given shaping function.
While open-loop identification is theoretically possible, the techniques outlined above are not without their disadvantages. First, the process under investigation would need to be operated in open-loop for an extended period of time.
Second, the required excitations may cause the paper that is produced to be of low quality. Since the fastest of today’s
paper machines runs at speeds of over 1700 meters per minute [9], the loss of product could be quite costly for paper
manufacturers.
An alternative approach involves closed-loop identification of a model, which calls for persistent excitation while
collecting data and a method to separate the dynamics of the process from the known dynamics of the controller.
A number of surveys have been published on the topic of closed-loop identification, e.g. [10, 11]. There are three
main approaches to identification in closed-loop: direct, indirect, and joint input-output. The direct approach refers
to a technique that ignores control interaction and attempts to identify the open-loop system from input and output
signals. Direct approaches yield consistent and optimally accurate models if a correct noise model is assumed [11].
The indirect approach identifies a closed-loop model and uses knowledge of the controller to determine the open-loop
model. While knowledge of the regulator is necessary for indirect methods, fixed noise models can be used while
guaranteeing consistency. Finally, the joint input-output approach uses an extra input or set-point signal and noise to
identify a system whose outputs are the outputs and the actual inputs of the process [11]. The estimates of the open-loop
system are consistent regardless of the assumed noise model so long as the feedback has a certain linear structure [11].
Some of the above mentioned approaches have been applied to paper machine control. One researcher [12] has
treated the closed-loop identification problem for CD control with a method demonstrating the joint input-output approach and proved that one could determine the open-loop sensitivity function from closed-loop data due to the inherent
2
time delay between actuation and the sensor location in sheet and film processes. The author demonstrated the estimation of steady-state response gain and location in a framework of parallel SISO estimators. Earlier, a number of
researchers applied various closed-loop identification techniques to Machine Direction (MD) models for control of a
paper machine [13, 14, 15]. While [13, 14] use only direct techniques to identify a difference equation model and a
frequency response model, respectively, [15] compares direct and indirect methods to identify a difference equation
model.
In this paper, an indirect closed-loop identification technique originally developed by Phan et al. [16] has been
extended to develop an unconstrained model predictive controller for a paper machine. While this technique is an
indirect method, it is unique in that an observer is explicitly used within the identification equations [16]. Markov
parameters were identified and converted into a step response model. A demonstration of closed-loop identification
using the model predictive controller was carried out on an industrial scale benchmark paper machine simulation that
considers shrinkage of the sheet during drying. Finally, the newly identified model was compared to the original model
that did not account for shrinkage within the MPC framework.
2 Model Identification and Model Predictive Control
2.1 Background
Model Predictive Control (MPC) has attracted considerable attention as an effective method for CD control [5, 17, 18,
19, 20, 21]. The key challenges for CD control include time delays between the headbox and the measurement area,
production rate and grade transition changes, plant-model mismatch due to paper shrinkage or shifting, and the need to
compute control moves for a large number of actuators in a short period of time. The advantage to using MPC for CD
control include the ability to handle constraints on the rate, magnitude, and bending moment of the actuators, and the
ability to compensate for the interactions between the different actuator positions in the CD as well as interactions of
multiple headboxes in the Machine Direction (MD).
A number of methods have been developed to make MPC more efficient for use in large scale CD control. One
approach [5, 22] employs an unconstrained model predictive controller to solve an off-line Quadratic Programming
(QP) problem that results in a linear time invariant (LTI) controller. With this method, constraint handling is applied by
3
scaling the actuator moves to preserve the direction of the input. This is the same technique that has been used for other
LTI CD control strategies such as
H1 or Dahlin controllers.
For these controllers, however, the constraint handling
compensation often results in sub-optimal performance as compared with algorithms that account explicitly for actuator
limitations within the optimization framework.
A number of on-line optimization implementations of MPC have been suggested as well. Rao et al. [20] propose
the use of deadbeat control when actuator velocity (U ) constraints are not necessary, or a modified constrained QP
MPC by exploiting the sparse model structure to reduce computational time. Those authors point out that deadbeat
controllers are implementable in certain situations, however, systems that require actuator velocity constraints would
not benefit from this technique. The authors also note that constrained QP techniques are still computationally infeasible
for current computing platforms. VanAntwerp [23] showed that a reformulated constraint set, using ellipsoids to bound
the constraint set, results in an efficient sub-optimal MPC scheme for CD control. Model reduction techniques, such as
the adaptive principle component analysis [17], have been coupled with receding horizon control to produce results of
similar quality to those of the full MPC problem.
Some researchers have attempted Linear Programming (LP) based formulations of MPC. Doyle and co-workers
[8, 19] have demonstrated a number of different Linear Programming techniques to further improve the computational
efficiency within the on-line optimization. Others have used an L1 norm for both a maximum deviation from setpoint
and as a method of minimizing the range of deviations across the CD [24].
2.2 Model Identification
Prior to building a model predictive controller that could manage sheet shrinkage, an identification study was performed.
The goal was to maintain control of the CD profile using a simple CD controller while determining a model to map
the headbox actuator movements to each of the possible full-array sensor banks: one at the end of the fourdrinier table
to collect wet-end measurements, and one at the end of the machine to collect dry-end measurements (Figure 1). The
identification was performed for three conditions: no shrinkage in the cross direction, 2% shrinkage in the CD, and 5%
shrinkage in the CD.
Although other methods of model identification have been described, e.g. [5, 17], the identification procedure
used in this work was adapted from Phan et al. [16] and is merely summarized below. The details of the study can
4
be found in the original work which included using pure gain and dynamic feedback controllers. As this paper will
extend the technique to the MPC framework, a summary of the closed-loop identification technique for a dynamic
feedback controller will be shown. If a pure gain controller is used, such as an LQR design with full state feedback, the
generalized procedure reduces back to the pure gain controller closed-loop identification case.
In this study, the open-loop process is assumed to be linear and can be expressed in discrete state-space form:
x(i + 1)
=
Ax(i) + Bu(i)
y(i)
=
Cx(i) + Du(i)
(1)
x is the state vector of size nx, u is the vector of inputs of size nu, and y is the vector of outputs from the system
of size ny . The overall objective is to determine the discrete impulse, H, (or step, Su ) response coefficients for the
model that can be represented as combinations of the state-space matrices, A, B, C, and D:
where
H(0)
=
D;
H(i) = CAi,1 B; i = 1 : : : p
Su(0)
=
D;
Su (i) = D +
i
X
=1
CA ,1B; i = 1 : : : p
The controller is also in state-space form:
where
s(i + 1)
=
Ps(i) + Qy(i)
u(i)
=
Rs(i) + Sy(i)
(2)
s is the controller state vector of size ns. An excitation signal v of size nu used in the identification procedure
is added to the input u. The excitation signal is assumed to be persistently exciting and uncorrelated with measurement
noise.
2.3 State-Space MPC
In order to implement the closed-loop identification routine, a state-space formulation of MPC is utilized in this work.
Furthermore, the development will be limited to an unconstrained controller, which leads ultimately to an equivalent LTI
formulation. State-space formulations have been considered by a number of researchers [25, 26, 27] and the particular
step-response formulation by Li et al. [25] is employed in this study.
As with most predictive controllers, the algorithm can be decomposed into three discrete elements: prediction,
correction, and compensation. In the case of unconstrained MPC, the compensation step takes the form of a fixed state
5
feedback law with a suitably defined state:
U(k) =
KmpcE(kjk)
Kmpc [Y(kjk) , R(k)]
=
where
(3)
E(kjk) is the predicted future error of the process, Y(kjk) is the predicted future outputs, and R(k) is the
predicted desired trajectory. Since CD control is a regulatory problem, the reference trajectory will be set to zero at all
times in this formulation. The prediction and correction equations for a step response model are given by:
Y(kjk , 1)
=
MY(k , 1jk , 1) + Su U(k , 1)
(4)
Y(kjk)
=
Y(kjk , 1) + Kf (y(k) , NY(kjk , 1))
(5)
where
2
M
N
=
=
0 Iny 0 0
0 . . . Iny . . . 0
6
6
6
6
6
6
6 .
6 .
6 .
6
6
6
6
6
6
6
6
6
4
..
.
0 0
0 0
0 0
..
.
..
.
..
.
0
0
..
.
0 . . . Iny 0
0 . . . 0 Iny
0 0 Iny
Iny 0 0
3
2
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
S=
Su (2)
Su (3)
..
.
Su(p , 1)
Su (p)
Su(p + 1)
3
2
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
6
6
6
6
6
6
6
6
6
= 66
6
6
6
6
6
6
6
4
Kf
Iny
0ny
..
.
0ny
0ny
0ny
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
Iny is an identity matrix the size of the number of outputs, ny. The state-space form of MPC results from combining
Equation (4) with a single shifted version of itself [25]:
2
Y(k + 1jk + 1)
=
PmpcY(kjk) + Qmpc 664
2
U(k) =
RmpcY(kjk) + Smpc 664
3
y(k) 77
5
U(k , 1)
3
y(k) 77
5
U(k , 1)
where
2
6
6
4
Pmpc Qmpc
Rmpc Smpc
3
2
7
7
5
= 64
6
[I , Kf N] M
Kmpc
6
Kf [I , Kf N] Su
0
0
3
7
7
5
(6)
Y(kjk), become the states, s(i), of Equation (2). Similarly, the plant outputs, y(k), and
the previous control moves, U(k , 1), become the inputs, y(i), to the controller and the current control moves,
U(k), are the outputs, u(i), from the controller of Equation (2).
The predicted future outputs,
To complete the problem specification, the traditional quadratic objective function [26] is used:
2
p
X
3
m
2 + X k,u U(j )k2 5
4
min
k
,
Y
(
k
+
`
)
k
y
U `=1
j =1
(7)
This optimization problem can be recast as the following least-squares problem [26]:
2
6
6
4
,y Su
,u
3
2
7
7
5
U = 664
,y (,MY(kjk))
0
3
7
7
5
(8)
with the following solution:
U =
STu ,Ty ,y Su + ,Tu ,u ,1 STu ,Ty ,y (,MY(kjk))
(9)
The controller gain can thus be realized as:
Kmpc = STu ,Ty ,y Su + ,Tu ,u,1 STu ,Ty ,y
(10)
Equation (6) will be applied in the closed-loop identification routine of the following section as the controller
S
of Equation (2). The goal of the identification will be to determine the step response model, u , within the MPC
framework.
2.4 Identification Algorithm
The identification scheme of from Phan et al. [16] uses the state-space form of the system, Equation (1), and the
controller, Equation (2), in a combined form in a two step process. First, the proposed state-space open-loop model and
the state-space controller are combined to form a closed-loop state-space representation (Equations 30-32 of [16]):
xa (i + 1)
=
Aacxa (i) + Bacua (i)
ya (i)
=
Ca xa(i) + Daua(i)
ua (i)
=
Fa ya(i) + va(i)
with a feedback controller:
7
(11)
The addition and subtraction of an observer term,
Maya, to the state equation in Equation (11) yields the closed-loop
observer form of the state-space system (Equation 33 of [16]):
xa (i + 1)
=
A acxa (i) + B acza (i)
ya (i)
=
Caxa (i) + Daua (i)
(12)
Ma is determined by a deadbeat condition, (Aac + MaCa)p = 0. This allows one to follow the dynamics related to the
controller and the process independent of the observer dynamics. The deadbeat condition, however, will impose certain
limitations on minimum data sets and model orders that are allowable for closed-loop identification. The observer plays
a key role in the identification of the system parameters. It is the observer gain that allows one to proceed from the
closed-loop observer state-space model to the open-loop state-space model.
With the above state-space framework in mind, data of the closed-loop system in question is collected. From this
data, particularly the
v(i), y(i), u(i), and Y(iji) one can determine the closed-loop observer Markov parameters, H ac,
according to:
ya = H acVa
(13)
where
2
ya (i)
2
Va
=
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
=
6
6
6
6
6
6
4
3
y (i) 77
7
U (i , 1) 77 i = 1 : : : N
7
5
Y (iji)
ua(1) ua (1) ua (2) 0
ua (p + 1) ua (N , 1)
za(N , 2)
za(0) za (p , 2) za (p , 1) za(N , 3)
ua (p)
za (0) za(1) za (p , 1)
..
.
0
..
.
..
.
..
.
..
0
0
0
z a (p )
..
.
..
.
za (0)
za (1)
.
8
..
.
..
.
za (N , p , 1)
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
2
2
ua(i)
=
6
6
4
u (i )
Y (i + 1ji + 1)
3
7
7
5
za (i) =
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
3
v1 (i) 77
7
v2 (i) 777
7
y (i) 777
7
U (i , 1) 77
7
5
Y (iji)
In this case, p is the model memory and determines the order of the model to be identified. Also, N is the number of
time samples that are used to identify the model. The value of p must be chosen such that p
The number of samples N necessary for the identification to proceed is bounded by N
(nx + ns)=(ny + ns).
p(nu + ny)=nu and may need
to be significantly larger if a significant amount of noise is present. These limitations are necessary for the open-loop
CAk B, where k p, to tend towards zero after a finite time as well as the need to have
T
as many independent rows of V as possible [28]. Also, each element of the excitation signal, v=[v1 v2 ] , where v1 acts
on the process input to directly excite the process and v2 on the setpoint to identify low frequency behavior, should be
impulse response coefficients,
persistently exciting and uncorrelated. In the examples below, the excitation signals were selected to be pseudo-random
binary sequences having a sample time greater than that of the sensor sample rate, each of which had different seeds for
the random number generator.
The least-squares solution to the equation above for the closed-loop observer Markov parameters is:
H ac = yaVay
where
(14)
Vay is the pseudo-inverse of the matrix Va. The accuracy and efficiency of this algorithm is directly related to
the size of the pseudo-inverse problem, p(nu + ns) N , and the number of closed-loop observer Markov parameters
that are to be identified. The closed-loop observer Markov parameters can be interpreted as the output and observer
responses to an impulse input:
H ac(0) = Da; H ac(i) = CaA iac,1 B ac; i = 1 : : : p
From the closed-loop observer Markov parameters, one can uniquely determine the so-called closed-loop system
Markov parameters,
Hac, by partitioning the closed-loop observer Markov parameters as:
H ac(i)
=
H ac(0) H (1)
ac (1)
H (2)
ac (1) 9
H (1)
ac (p)
H (2)
ac (p)
and solving for the partitioned parameters:
Hac(i)
=
i
X
,
ac(0)Fa ,1 Hac(i , )
H (1)
H (2)
ac (i) +
ac ( ) I , H
=1
F
where a is a matrix of the controller parameters:
Fa
=
2
6
6
4
Smpc Rmpc
Qmpc Pmpc
(15)
3
7
7
5
The closed-loop Markov parameters are equal to the closed-loop system’s impulse response:
Hac(0) = Da ; Hac(i) = CaAiac,1 Bac; i = 1 : : : p
Similarly, the open-loop system Markov parameters,
Ha(i)
=
Ha, can be determined through:
Hac(i) (I , Fa Hac(0)) +
i,1
X
=1
Hac( )Fa Ha(i , )
(16)
The open-loop Markov parameters are used to determine the series of outputs of the system, given the excitation and
the previous inputs to the system. From these Markov parameters, the open-loop impulse response,
H(0) = D; H(i) = CAi,1B; i = 1 : : : p
can be determined from the upper ny nu portion of the
Ha(i) matrix. From the impulse response coefficients, H(i),
S
the step response coefficients, u (i) for use within the model predictive controller can be determined by:
Su (i)
=
i
X
=1
Ha( )
(17)
3 Example and Results
The benchmark problem studied in this paper is a proprietary Matlab and Simulink model of a paper machine [8].
Measurements of cross direction variations can be achieved with either a full array wet end sensor at the end of the
fourdrinier table or a full array dry end sensor located at the end of the machine (Figure 1). Each of these sensor banks
were modeled to contain roughly 360 independent sensor locations distributed evenly across the machine while the
headbox has 45 evenly spaced actuator locations. The objective in each case is to control the CD variation of basis
weight in the presence of sustained disturbances.
10
Like many other sheet and film processes, the simulation model employs first-order-plus-time-delay dynamics between the actuator and sensor with a Toeplitz symmetric input-output interaction matrix. Prior to dry-end sensing, the
outputs of the model can be nonlinearly remapped, based on the percentage of shrinkage. The time delays between
the headbox and the sensor locations are included in Figure 1. The model assumes that no significant shrinkage has
occurred prior to the wet end measurements.
While the theory of closed-loop identification allows one to identify each input-output relationship, the magnitude
of inputs and outputs in an industrial paper machine prohibit one from identifying the model of every actuator position at
every sensor location. As the number of inputs and outputs increases for a particular identification problem, it becomes
increasingly difficult to excite all of the necessary directions, as well as save and perform the calculations on the large
data matrices even on current computer hardware. For the present study, 5 actuator locations were selected across the
span of the sheet and all were simultaneously identified using the Markov parameter approach outlined in Section 2.4.
The remaining actuator to sensor location responses will be interpolated from the 5 identified responses.
The excitation signals chosen for the examples were a set of random binary sequences, each with a different seed
value and each with a five second hold time. The prediction horizon within the MPC routine was chosen to be 4 due to
the low order of dynamic character and the need to minimize the number of states for the controller in Equation (6). The
move horizon was chosen to be 1 as it was found that more control moves made the controller too sluggish. The original
model in the MPC routine was assumed to be a first-order plus time delay model with a diagonal gain matrix with a
gain that is only 80% of the actual process’s diagonal gain elements. Once the model is identified, the newly identified
model will be tested against this original controller to determine the improvement was achieved with the identification
procedure.
An example of the type of data that was collected for analysis is shown in Figure 2. The figure shows the data
for analysis of a dry end measured system that incurs only a minimal nonlinear shrinkage effect, which in turn results
in an uneven actuator response distribution. Noise was simulated with a zero mean random number generator with
variance of 1 (gm=cm2 )2 and 20:1 signal to noise ratio, which is significantly more than values reported for industrial
scanning sensors. While only the first 300 significant (when a change was noticed) sample times were used, 350 samples
were collected to compensate for the significant deadtime between the headbox and the sensor locations. Deadtime is
not included as part of the model as it is assumed to be known and is added to the full model after identification of
11
the actuator response. The number of samples were chosen to accommodate the condition N
N 4(5 + 360)=5 = 292.
p(nu + ny)=nu, or
Figure 3 shows the identified actuator responses using only the first 300 significant time
points from the data set. From the peaks of the waves from each response shape, the centers of the other actuator
responses can be extrapolated. This center identification is shown for both dry end trials in Figure 4. Notice that the
extrapolation routine is able to detect the 2% edge shrinkage between the two cases as well as the uneven distribution
of centers in both cases.
The lower right plot of Figure 3 shows the steady state response of the third identified actuator as compared to the
expected profile. The mean squared error for this actuator response profile was found to be 1:01 10,6 (gm=cm2 )2 ,
roughly 1% of the maximum response gain. The error is mainly concentrated in areas of the cross direction that the
response actually has no real effect. This is potentially due to the attempt to identify many actuators at the same time
and could possibly be eliminated if each estimation was only performed on a window of the CD instead of the entire
CD.
Each identified model was tested for its disturbance rejection capability within the MPC framework. The disturbance
introduced at the headbox to which each trial was subjected is shown in Figure 5. A typical result from one of the trials
can be seen in Figure 6. The results of these simulations are summarized in Table 1 as a scaled value of the CD
variance calculation as outlined in [29]. The MPC controller formed with the identified model proved to reduce the
cross-directional variance by roughly 76 , 78% if a wet end full array sensor were used. Also, for trials in which the
sensor was assumed to be at the dry end, a reduction of 74 , 82% of the CD variance was found after implementation
of the identified model. It was also found that the identified model had a reduction in the amount of CD variance in the
cases when a significant amount of edge shrinkage occurs.
There were some drawbacks to this identification technique, however. While the amount of time necessary to
collect the data needed for identification is small, 300 significant samples in this case, the amount of time necessary
for identification would make this technique implausible for on-line implementation. While the procedure consumed
only 400 MB of memory, the computational demand limits the on-line capabilities of this numerical technique. For
the problem above, the identification routine took nearly 6 minutes on a Sun Ultra-4 400 MHz machine with 1 GB of
RAM and 2.5 GB of swap space. Using a Linux based PC with AMD Thunderbird 800 MHz processor and 256 MB
of memory and 500 MB of swap space, the same identification routine took
12
3 minutes to produce the same results.
Finally, using a WindowsXP based PC with an AMD Athalon 4 processor running at 1.2 GHz with 516 MB of memory
and dynamic swap space, the routine took only 1:5 minutes. While a faster processor will improve the efficiency of the
algorithm, the times are still far too long to be implemented on-line. Implementation of this type of technique in its
current form could be performed on demand in response to changes in operation that may affect the parameters of the
model. Potential improvements to the current numerical technique will be outlined in the next section.
Finally, varying noise levels were tested to determine the amount of necessary data, N, and the corresponding
increase in computation time to identify the actuator responses. Table 2 summarizes this study. While the mean squared
error is an order of magnitude higher for the levels of noise above 5%, the computational demand and the necessary
amount of data increases linearly as a function of the noise while the computation time increases exponentially as a
function of the data required.
4 Summary
A closed-loop identification routine has been developed in the model predictive control framework. Simulation studies
on an industrial benchmark simulator point to the efficiency of the approach. Starting with a simple model approximation, the closed-loop identification technique yielded a controller that outperformed the approximation-based controller
in the cases of plant-model mismatch in the center locations of the actuator responses.
While the identification technique only needed the minimum amount of data to identify the responses, a large amount
of computation time was needed to manipulate the large data arrays that result from the low number of data points. This
can be attributed to a number of issues regarding the implementation of this technique. First, the dimensionality of the
states in the MPC routine, even for small move and prediction horizons, can become quite large. A reduction in the
size of the state-space model used in MPC, such as that described in [27] could allow the identification routine to be
performed much faster.
Some improvements could also be made to the algorithm implementation to increase the efficiency of the results,
yielding a more tractable on-line technique. First, sparse matrix mathematics could be employed to reduce the computational load of the algorithm. A majority of the computational demand is the recombination of the large Markov
parameter matrices from closed-loop observer, to closed-loop to open-loop. Similar sparsity techniques have been
employed in recent constrained MPC code for CD control [8, 30]. Also, as mentioned earlier in attempting to reduce
13
inaccuracies in response identification at areas in the CD far away from the actuator location, windowing the CD to identify specific sections for each actuator would also decrease the number of samples needed, and thus the computational
and memory demand.
While the technique described above has been formulated to only allow for unconstrained controller techniques,
it may be desirable, especially for CD control, to implement closed-loop identification with constrained controllers.
A possible extension to the method described in this paper could be to use soft constraints, implementing system
constraints as additional terms in the objective function. This technique would allow the process to react optimally
while accounting for the constraint set without the necessity for hard constraints that preclude the application of the
described technique.
Acknowledgments
The authors would like to acknowledge the financial support of the University of Delaware and the technical and
financial support of Weyerhaeuser through the University of Delaware’s Process Control and Monitoring Consortium.
Also, a thanks to Dr. Jay Lee of the Georgia Institute of Technology School of Chemical Engineering for his valuable
comments.
14
References
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17
List of Figures
1
Paper machine with wet and dry end full array sensors. The times represent approximate time delays at
various points in the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
21
A representative data set that was used in the identification analysis. The set was taken from a simulation
of a machine that incurs only a small amount of nonlinear shrinkage during drying and is sensed using
a dry-end sensor. The top figures represent the excitation signals,
v1 and v2 , the center plots are the
controlled input u and the dry end basis weight response to the excitation and the control action, y with
noise. The lower plot is the dry end basis weight response, y , without noise. A number of extra samples
are collected to compensate for the deadtime between the actuator and sensor locations.
3
. . . . . . . .
22
A step response identification set from the data collected in Figure 2. Each response is labeled for the
actuator, nu, that it represents, y (nu). The lower right plot shows the identified steady state response of
the third identified actuator as compared to the expected response. The mean squared error between the
two is 1:01 10,6 (gm=cm2 )2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
23
Center identification from the data collected in Figure 2 (x and solid line) and from data representing a
machine that has 2% edge shrinkage (o and dashed line). . . . . . . . . . . . . . . . . . . . . . . . . .
24
5
Disturbance profile used in each of the MPC case studies. . . . . . . . . . . . . . . . . . . . . . . . . .
25
6
A representative closed-loop trial showing the resulting output (y ) and input (u) profiles for the identified
model on the machine corresponding to Figures 2, 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . .
26
List of Tables
1
Performance of MPC controllers before and after closed-loop identification. The 2 , CD test was used
as the performance criteria and all values are scaled against the worst case within the set. . . . . . . . .
2
19
Computation time and identification error in the steady state profiles for varying levels of noise. The
computation times are for a WindowsXP based PC with an AMD Athalon 4 processor running at 1.2
GHz with 516 MB of memory and dynamic swap space. . . . . . . . . . . . . . . . . . . . . . . . . .
18
20
sensor
shrinkage
before ID
after ID
percent improvement
location
(%)
CD
CD
after identification
wet end
0
0.980
0.216
76
wet end
2
0.994
0.214
78
wet end
5
1.000
0.217
78
dry end
0
0.767
0.027
74
dry end
2
0.842
0.094
75
dry end
5
0.849
0.032
82
Table 1: Performance of MPC controllers before and after closed-loop identification. The 2 , CD test was used as the
performance criteria and all values are scaled against the worst case within the set.
19
Signal to
Min data
computation
Mean squared error
noise ratio
required (N)
time (sec)
in steady state profile
gm=cm2 )2
(
20:1
300
86.273
1.010
10,6
4:1
400
96.739
2.314
10,5
2:1
475
124.059
1.679
10,5
4:3
600
178.777
4.437
10,5
Table 2: Computation time and identification error in the steady state profiles for varying levels of noise. The computation times are for a WindowsXP based PC with an AMD Athalon 4 processor running at 1.2 GHz with 516 MB of
memory and dynamic swap space.
20
Sensor Banks
1
0
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Actuator / Sensor
0
1
0
1
Lane
0
1
0
1
0
1
0
1
0
1
0
1
0
1
Cross Direction
11
00
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
00
11
Actuators
Machine Direction
Headbox
Slice Lip
1
0
0
1
11
00
00
11
11
00
00
11
00
11
00
11
00
11
1s.
~30 s.
Figure 1: Paper machine with wet and dry end full array sensors. The times represent approximate time delays at
various points in the model.
21
Figure 2: A representative data set that was used in the identification analysis. The set was taken from a simulation of
a machine that incurs only a small amount of nonlinear shrinkage during drying and is sensed using a dry-end sensor.
The top figures represent the excitation signals,
v1 and v2 , the center plots are the controlled input u and the dry end
basis weight response to the excitation and the control action, y with noise. The lower plot is the dry end basis weight
response, y , without noise. A number of extra samples are collected to compensate for the deadtime between the
actuator and sensor locations.
22
Figure 3: A step response identification set from the data collected in Figure 2. Each response is labeled for the
actuator,
nu, that it represents, y(nu).
The lower right plot shows the identified steady state response of the third
identified actuator as compared to the expected response. The mean squared error between the two is
(gm=cm2)2 .
23
1:01 10,6
350
300
250
sensor
200
150
100
Identified centers, no edge shrinkage
Interpolated centers, no edge shrinkage
Identified centers, 10% edge shrinkage
Interpolated centers, 10% edge shrinkage
50
0
0
5
10
15
20
25
30
35
40
45
actuator
Figure 4: Center identification from the data collected in Figure 2 (x and solid line) and from data representing a
machine that has 2% edge shrinkage (o and dashed line).
24
Figure 5: Disturbance profile used in each of the MPC case studies.
25
Figure 6: A representative closed-loop trial showing the resulting output (y ) and input (u) profiles for the identified
model on the machine corresponding to Figures 2, 3 and 4.
26