# Model parameterization_v4 (doc, 447 KiB) - Infoscience ```18th European Symposium on Computer Aided Process Engineering – ESCAPE 18
Bertrand Braunschweig and Xavier Joulia (Editors)
Model Parameterization Tailored to
Real-time Optimization
Beno&icirc;t Chachuat,a Bala Srinivasan,b Dominique Bonvina
a
Laboratoire d’Automatique, Ecole Polytechnique F&eacute;d&eacute;rale de Lausanne (EPFL),
Station 9, CH-1015 Lausanne, Switzerland
b
D&eacute;partement de G&eacute;nie Chimique, Ecole Polytechnique de Montr&eacute;al,
C.P. 6079 Succ. centre ville, Montr&eacute;al (QC), H3C 3A7, Canada
Abstract
Challenges in real-time process optimization mainly arise from the inability to build and
adapt accurate models for complex physico-chemical processes. This paper surveys
different ways of using measurements to compensate for model uncertainty in the
context of process optimization. Three approaches can be distinguished according to the
process model and repeats the optimization, modifier adaptation modifies the
constraints and gradients of the optimization problem and repeats the optimization,
while direct input adaptation turns the optimization problem into a feedback control
problem and implements optimality via tracking of the necessary conditions of
optimality. This paper argues in favor of modifier adaptation, since it uses a model
parameterization, measurements, and an update criterion that are well tailored to
meeting the KKT conditions of optimality.
Keywords: Measurement-based optimization; Real-time optimization; Plant-model
1. Introduction
Optimization of process performance has received attention recently because, in the face
of growing competition, it represents the natural choice for reducing production costs,
improving product quality, and meeting safety requirements and environmental
regulations. Process optimization is typically based on a process model that is used by a
numerical procedure for computing the optimal solution. In practical situations,
however, an accurate process model can rarely be found with affordable effort.
Uncertainty results primarily from trying to fit a model of limited complexity to a
complex process. The model-fitting task is further complicated by the fact that process
data are usually noisy and signals often do not carry sufficient information for efficient
process identification. Therefore, optimization using an inaccurate model might result in
suboptimal operation or, worse, infeasible operation when constraints are present .
Two main classes of optimization methods are available for handling uncertainty. The
main difference relates to whether or not measurements are used in the calculation of
the optimal strategy. In the absence of measurements, robust optimization is typically
used, whereby conservatism is introduced to guarantee feasibility for the entire range of
expected variations . When measurements are available, adaptive optimization can
help adjust to process changes and disturbances, thereby reducing conservatism . It is
interesting to note that the above classification is similar to that found in control
problems with the robust and adaptive techniques.
An optimal solution has to be feasible and, of course, optimal. In practice, feasibility is
2
B. Chachuat et al.
often of greater importance than optimality. In the presence of model uncertainty,
feasibility is usually enforced by the introduction of backoffs from the constraints. The
availability of measurements helps reduce these backoffs and thus improve performance
. Generally, it is easier to measure or infer constrained quantities (e.g. temperature or
pressure) than estimate gradients of the cost and constrained quantities. These elements
clearly set a priority of actions in the framework of adaptive optimization.
This paper discusses three adaptive optimization approaches that differ in the way
measurements are used to refine the process model, and the updated model is used
subsequently for optimization [7,17]; (ii) modifier adaptation, where modifier terms are
added to the cost and constraints of the optimization problem, and measurements are
used to update these terms [8,10,20]; and (iii) direct input adaptation, where the inputs
are adjusted by feedback controllers, hence not requiring numerical optimization but a
considerable amount of prior information for control design [9,21,25].
These approaches are surveyed and compared in the first part of the paper. A critical
discussion follows, which argues in favor of modifier-adaptation methods that share
many advantages of the other methods. An important issue that is not addressed in this
paper concerns the availability of appropriate measurements.
2. Static Optimization Problem
For continuous processes operating at steady state, optimization typically consists in
determining the operating point that optimize process performance such as minimization
of operating cost or maximization of production rate, while satisfying a number of
constraints (such as bounds on process variables or product specifications). In
mathematical terms, this optimization problem can be stated as follows:
 
  : g  , y  0
minimize:  p   :  p  , y p

subject to: G p
where  
n
respectively;
gp :
n

ny
and y p 
p :

ng
n

ny
ny
p
(1)
p
stand for the process input (set point) and output vectors,

is
the
plant
performance
index;
and
is the vector of constraints imposed on the input and output
variables.
In contrast to continuous processes, the optimization of batch and semi-batch processes
consists in determining time-varying control profiles, u(t), t0  t  tf . This typically
involves solving a dynamic optimization problem, possibly with path and terminal
constraints. A practical way of solving such problems is by parameterizing the control
profiles using a finite number of parameters  , e.g., a polynomial approximation of
u(t ) on finite elements. Although the process is dynamic in nature, a static map can be
used to describe the relationship between the process inputs  and the outcome of the
batch y p (t f ) . Hence, the problem can be regarded as a finite-dimensional static
optimization problem similar to (1), and the optimization approaches discussed in the
following sections can also be used in the framework of run-to-run optimization of
batch and semi-batch processes (see, e.g., ).
In practice, the mapping relating the process inputs and outputs is typically unknown,
and only an approximate model is available,
Model parameterization tailored to real-time optimization
3
y  f  ,  
with y 
ny
(2)
representing the model outputs, and  
n
the model parameters, and
the input-output mapping. Accordingly, an approximate solution of
f:


Problem (1) is obtained by solving the following model-based optimization problem:
n
ny
n
minimize:   ,   :   , y,  

subject to: y  f  ,  
G  ,   : g  , y,    0
(3)
If the objective and constraint functions in (1) and (3) are continuous and the feasible
domains of these problems are nonempty and bounded, optimal solution points  p and
  are guaranteed to exist for (1) and (3), respectively . Note that such optimal
points may not be unique due to nonconvexity. Provided that the active constraints
satisfy a regularity condition at the optimal solution, the KKT conditions – also called
necessary conditions of optimality (NCO) – must hold at the solution point . For
Problem (3), the KKT conditions read:


G   ,   0,    0 ,
 
G 
 ,    
 ,   0,


  G   ,   0



where   



(4)
ng
is the vector of Lagrange multipliers. The KKT conditions involve the

G
quantities G,
, which will be denoted collectively by  subsequently.
and


3. A Classification of Real-time Optimization Schemes
Real-time optimization (RTO) improves process performance iteratively by adjusting
selected optimization variables using available measurements. The goal of this closedloop adaptation is to drive the operating point towards the true plant optimum in spite of
inevitable structural and parametric errors. RTO methods can be classified in different
ways. This section presents one such classification based on the quantities that can be
adapted, as illustrated in Fig. 1; note that the methods of columns 1 and 2 use repeated
numerical optimization, whilst those of column 3 do not.
The standard way of implementing RTO is via the so-called two-step approach , also
referred to as repeated identification and optimization in the literature. In the first step,
the values of (a subset of) the adjustable model parameters  are estimated using the
process output measurements. This is typically done by minimizing the lack of closure
in the steady-state model equations (2), such as the weighted sum of squared errors
between the measured outputs y p and the predicted outputs y .
4
B. Chachuat et al.
Figure 1: Optimization scenarios that use measurements to adapt for feasibility and optimality.
A key, yet difficult, decision in model-parameter adaptation is to select the parameters
to be adapted. These parameters should be identifiable, represent actual changes in the
process, and contribute to approach the process optimum; also, model adequacy proves
to be a useful criterion to select candidate parameters for adaptation . Clearly, the
smaller the subset of parameters, the better the confidence in the parameter estimates,
and the lower the required excitation (also the better the regularization effect). But too
few adjustable parameters can lead to completely erroneous models, and thereby to a
false optimum.
In the second step, the updated model is used to determine a new operating point by
solving an optimization problem similar to (3). Model-parameter adaptation schemes
can be written generically using the following two equations (see Fig. 2):
 k   k1   upd y p ( k1 )  y( k1  k1 ),
 k   opt  k 
(5)
(6)
where  upd is the map describing model-parameter adaptation, such that  upd (0)  0 ;
 opt , the map describing the optimization step. Note that the handles for correction, i.e.
the adjustable model parameters  are a subset of the model parameters. The use of
auxiliary measurements (the process outputs y p ) presents the advantage that any
available measurement can be used.
Optimizer (6)
Model-Parameter
Static Plant
Model parameterization tailored to real-time optimization
5
Figure 2. Optimization via model-parameter adaptation: Two-step approach
It is well known that the interaction between the model-parameter adaptation and
reoptimization steps must be considered carefully for the two-step approach to achieve
optimality. If the model is structurally correct and the parameters practically
identifiable, convergence to the plant optimum can be achieved in one iteration.
However, in the presence of plant-model mismatch, whether the scheme converges, or
to which point it converges, becomes anybody's guess. This is due to the fact that the
adaptation objective might be unrelated to the cost and constraints in the optimization
problem; hence, minimizing the mean-square error in y may not help in our quest for
feasibility and optimality. To alleviate this difficulty, Srinivasan and Bonvin 
presented an approach in which the criterion for model-parameter adaptation is
modified to account for the subsequent optimization objective.
Convergence under plant-model mismatch has been addressed by several authors [3,8];
it has been shown that optimal operation is reached if model adaptation leads to
matched KKT conditions for the model and the plant.
Theorem 1. Let the model-parameter adaptation (5) be such that the quantities 
predicted by the model match the plant measurements  p . Then, upon convergence,
the model-parameter adaptation scheme (5-6) reaches an (local) optimum of the plant.
A proof of this result is readily obtained from the assumption that the KKT conditions
predicted by the model equal those achieved by the plant. With such a matching, the
converged solution corresponds to a (local) plant optimum.
Although Theorem 1 is straightforward, the KKT-matching assumption is difficult to
implement in practice. KKT matching requires (i) an adequate model parameterization
so that all the components of the KKT conditions can be matched, (ii) adequate
measurements so that the  p quantities can be estimated, and (iii) an adaptation
criterion that enforces KKT matching such as    p 2 .
In order to overcome plant-model mismatch, several variants of the two-step approach
have been presented in the literature. Generically, they consist in modifying the cost and
constraints of the optimization problem such that the KKT conditions of the model and
the plant can match. The optimization problem with modifiers can be written as follows:
minimize:   ,  :   ,   T



T
k
subject to: G  ,  : G  ,    G  G     0
where  G &deg;
G &deg;
ng
is a constraint bias term,   &deg;
n
(7)
n ng
the constraint-gradient modifier; these modifiers will be denoted
collectively by  subsequently.
 The constraint bias  G represents the difference between the measured and predicted
constraints,  G  G p ( )  G( ,  ) , evaluated at the previous operating point  k1 .
scheme is rather straightforward and corresponds to common industrial practice .
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B. Chachuat et al.
 The cost-gradient modifier   represents the difference between the estimated and
predicted values of the cost gradient,  T    p    , evaluated at the previous

 
 

operating point  k1 . The pertinent idea of adding a gradient modifier to the cost
function of the optimization problem dates back to the work of Roberts  in the
late 1970s. Note that it was originally proposed in the framework of two-step
methods to better integrate the model update and optimization subproblems and has
led to the so-called ISOPE approach .
 The constraint-gradient modifier  G , finally, represents the difference between the
estimated and predicted values of the constraint gradients,  T   G p  G  ,
G
 
 

evaluated at the previous operating point  k1 . The idea of adding such a first-order
modifier term to the process-dependent constraints, in addition to the constraint bias
 G , was proposed recently by Gao and Engell . This modification allows
matching, not only the values of the constraints, but also their gradients.
Overall, the update laws in modifier-adaptation schemes can be written as (see Fig. 3):

% k1 ,  )
k  k1   upd  p ( k1 )  (
 k   opt k 

(8)
(9)
(Beware of notation: compare 7 and 8; the meaning of the (.) is not the same) where
% G
%
 % 
, with , G as defined in Problem (7); and the modifier-adaptation
   G,
,
   
map,  upd , is such that  upd (0)  0 . The handles for correction are the modifier terms
 instead of  used in the context of model-parameter adaptation. Also, the
measurements/estimates  p required for adaptation are directly related to the KKT
conditions. Auxiliary measurements are not used directly in this framework, although
they might be used to estimate  p . Observe the one-to-one correspondence between
the number of measurements/estimates and the number of adjustable parameters. In
particular, identifiability is automatically satisfied, and so are the KKT-matching
conditions upon convergence.
Optimizer (9)
Static Plant
Modifier
Model parameterization tailored to real-time optimization
7
Figure 3. Optimization via modifier adaptation: Matching the KKT conditions
Modifier-adaptation schemes possess nice theoretical properties, as summarized by the
following theorem.
Theorem 2. Let the cost and constraint functions be parameterized as in Problem (7).
Also, let the information on the values of  p be available and used to adapt the
modifiers  . Then, upon convergence, the modifier-adaptation scheme (8-9) reaches
an (local) optimum of the plant.
A proof of this result is easily obtained by noting that, upon convergence, the modified
constraints G in (7) match the plant constraints G p , and the gradients of the modified
cost and constraint functions match those of the plant (see also ). It follows that the
active set is correctly determined and the converged solution satisfies the KKT
conditions.
that the parameterization and the update procedure are both intended to match the KKT
conditions. However, provided  p can be measured or estimated, the modifieradaptation scheme is better tailored to the task than model-parameter adaptation. Note
that modifier adaptation corresponds to model-predictive control with a one-step
prediction horizon; such a short horizon is justified because the system is static.
However, since the computed modifiers are valid only locally, modifier-adaptation
schemes require some amount of filtering/regularization (either in the modifiers or in
the inputs) to avoid too aggressive corrections that may prevent convergence.
This last class of methods provides a way of avoiding the repeated numerical
optimization by turning the optimization problem into a feedback control problem that
directly manipulates the input variables. This is motivated by the fact that practitioners
like to use feedback control of selected variables as a way to counteract plant-model
mismatch and plant disturbances, due to its simplicity and reliability compared to online optimization. The challenge is to find functions of the measured variables which,
when held constant by adjusting the input variables, enforce optimal plant performance
[19,21]. Said differently, the goal of the control structure is to achieve a similar steadystate performance as would be realized by an (fictitious) on-line optimizing controller.
In the presence of uncertainty, the inputs determined from off-line solution of Problem
(3) for nominal parameter values satisfy the NCO (4) but typically violate the NCO
related to the plant itself. Hence, a rather natural idea is to correct the input variables 
so as to enforce the NCO for the plant [1,9,14]; in other words, the controlled variables
are chosen as the NCO elements, with the corresponding set points equal to zero.
Tracking the NCO of the plant consists of three steps: (i) determining the active set
(positivity condition on Lagrange multipliers), (ii) enforcing the active constraints, and
(iii) pushing the sensitivity to zero. Determining the active set requires a switching
strategy, whereby a constraint is included in the active set when it is attained, and
deactivated when its Lagrange multiplier becomes negative . Since this switching
logic renders the scheme more complex, it is often assumed, in the interest of simplicity,
that the active constraints do not change. Note that such an assumption is always
verified in the neighborhood of an optimal solution and is observed in many practical
situations.
Once the active set is known, the inputs are split into : (i) constraint-seeking directions
that are used to track the active constraints, and (ii) sensitivity-seeking directions that
8
B. Chachuat et al.
are adapted to force the reduced gradients to zero. The active constraints Gap and the
G ap
 I  P P  , with P :
, need to be evaluated.


Since, in general, the constrained quantities are measured or can be reliably estimated,
adjusting the inputs in the constraint-seeking directions to track the active constraints is
rather straightforward [4,25,27]. Adjusting the sensitivity-seeking directions is more
involved, mainly due to the difficulty of evaluating the gradient terms. Fran&ccedil;ois et al.
 proposed a two-time-scale adaptation strategy, wherein adaptation in the sensitivityseeking directions takes place at a much slower rate than in the constraint-seeking
directions.
Input-adaptation schemes obey the following equations (see Fig. 4):
reduced cost gradient  r  p :
G
a
p
 p


( k ),  r  p ( k )   swi  p ( k )

 k   k1   con G ap ( k1 ), r  p ( k1 )
(10)
(11)
where  swi is the map describing the switching logic for determination of the active set;
 con , the map describing the controller, such that  con (0, 0)  0 . The handles for
correction are the process inputs  , i.e., no specific parameterization is required here.
Both the active constraints and the reduced cost gradient are forced to zero, e.g., with a
discrete integral-type controller.
-
Controller
(11)
Static Plant
Switching
Logic (10)
Figure 4. Optimization via input adaptation: Tracking the NCO using feedback control
Direct input adaptation also possesses nice theoretical properties, as summarized by the
following theorem.
Theorem 3. Let the information on the values of  p be available and used to adapt the
active set and the inputs as given by (10-11). Then, upon convergence, the inputadaptation scheme (10-11) reaches an (local) optimum of the plant.
Note that the active process constraints and reduced gradients are both zero upon
convergence. Moreover, since the positivity of the Lagrange multipliers is guaranteed
by the switching logic, the active set is correctly identified and the NCO are satisfied.
The key question lies in the design of the controller. Unlike optimization-based
schemes, the required smoothening is provided naturally via appropriate controller
tuning.
3.4. Performance evaluation
A systematic approach for evaluating the performance of adaptive optimization
schemes, labeled the extended cost design approach, has been presented in . It
incorporates measures of both the convergence rate and the effect of measurement
noise. Interestingly, it is shown that, in the presence of noise, a standard two-step
Model parameterization tailored to real-time optimization
9
algorithm may perform better, in terms of the proposed metric, than modified
algorithms compensating for plant-model mismatch such as ISOPE. Another approach
to performance characterization for adaptive optimization has been proposed in ,
which considers the backoff from active inequality constraints required to ensure
feasibility. Therein, better adaptive optimization approaches produce smaller backoffs.
(Also Welz et al. ?)
4. Use of Measurements for Feasible and Optimal Operation
This section discusses the two main rows in Fig.1. The feasibility issue is addressed
first, and various gradient estimation techniques are considered next.
4.1. Feasible Operation
In practical applications, guaranteeing feasible operation is often more important than
achieving the best possible performance. Hence, first priority is given to meeting the
process constraints (such as safety requirements and product specifications) and only
second priority to improving process performance in terms of the objective function.
Interestingly, the results of a variational analysis in the presence of small parametric
error support the priority given to constraint satisfaction over the sensitivity part of the
NCO . More specifically, it has been shown that failure to adapt the process inputs in
the constraint-seeking directions results in cost variations in the order of the parameter
variations  ; in contrast, failure to adapt the inputs in the sensitivity-seeking
directions gives cost variations in the order of  2 only.
The ability to guarantee feasible operation is addressed next for the three classes of
methods presented above. With model-parameter adaptation, since the plant constraints
are predicted by the process model, constraint matching may be difficult to meet, e.g.,
when the model is updated by matching a set of outputs not directly related to the active
constraints. With modifier adaptation, feasibility is guaranteed upon convergence,
provided that all the constraint terms are measured ; yet, ensuring feasibility does not
necessarily imply that the correct active set has been determined due to the use of
possibly inaccurate cost and constraint gradients, e.g., when gradient modifiers are not
considered. Finally, in input adaptation, feasibility is trivially established when the
active set is known and does not change with the prevailing uncertainty. However, as
soon as the active set changes, tracking the previous set of active constraints may lead to
infeasibility. A switching logic can be used to remove this limitation, but it requires
experimental gradient information to be available. The use of a barrier-penalty function
approach, which removes the need to know the active set, has also been proposed .
When feasibility cannot be guaranteed, conservatism is typically introduced in the form
of constraint backoffs. Backoffs are also used to enforce feasibility when some of the
constraints are difficult to measure.
Taking a system from a feasible operating point to optimality requires accurate gradient
information. With model-parameter adaptation, convergence is relatively fast since the
updated model is used to estimate the gradient. In the other two schemes, the gradient
information is estimated experimentally, thereby slowing down convergence
significantly.
lies in the estimation of this gradient information. The finite-difference scheme used in
the original ISOPE paper  is known to be inefficient for large-scale, slow and noisy
processes. Hence, alternative techniques have been developed, which can be classified
10
B. Chachuat et al.
as either model-based approaches or perturbation-based approaches.
Model-based approaches allow fast derivative computation by relying on a process
model, yet only approximate derivatives are obtained. In self-optimizing control
[12,21], the idea is to use a plant model to select linear combinations of the outputs, the
tracking of which results in good performance, also in the presence of uncertainty; one
can say that these linear combinations of outputs approximate the process derivatives.
Also, a way of calculating the gradient based on the theory of neighbouring extremals
has been presented in ; however, an important limitation of this approach is that it
provides only a first-order approximation and the accuracy of the derivatives depends
strongly on the reliability of the plant model.
The idea behind perturbation methods is to estimate process derivatives using variations
around the operating point. However, this may be very time consuming if one has to
wait for a new steady state to be reached. Extremum-seeking control [1,14] attempts to
obtain the cost sensitivity by superposing a dither signal to the plant inputs. In dynamic
model identification, the plant is approximated by a dynamic model during the transient
phase between two successive steady states [16,31,11]; since the derivatives are
calculated from the identified dynamic model, the waiting time needed for reaching a
new steady state is avoided (reference to Isermann). Other perturbation-based
use current and past (steady-state) measurements to compute a gradient estimate based
on Broyden’s formula . For the case of multiple identical units operating in parallel,
Srinivasan considered perturbations along the unit dimension rather than the time
dimension, thereby allowing faster derivative estimates . In principle, the smaller
the difference between the operating points, the more accurate the derivative
approximation, but conditioning issues might arise due to measurement noise and plant
disturbances. A way of avoiding this latter deficiency is presented in .
5. Comparison of the approaches
In this section, we take a critical look at the three aforementioned classes of adaptive
optimization schemes in terms of various criteria. We also argue in favor of modifieradaptation methods, in the sense that they provide a parameterization that is well
tailored to KKT matching.
The analysis presented in Table 1 shows many facets of the problem. It is interesting to
input adaptation; several attractive features are shared between the first and second
columns, while other features are shared between the second and third columns.
The methods differ mainly in the handles and the measurements that are used for
correction. The major drawback of model-parameter adaptation is that KKT matching
can be very difficult to achieve with the (arbitrary) parameterization  and (arbitrary)
auxiliary measurements y p . In comparison, modifier adaptation resolves the
challenging task of selecting the adjustable parameters by introducing the modifiers 
as handles. Also, the measurements  p are directly related to the KKT conditions, and
their number equals that of the handles  , i.e., there results a square adaptation
problem. In other words, since the handles  are essentially decoupled, no
sophisticated technique is required for their update. Moreover, since KKT matching is
enforced, reaching a (local) plant optimum is guaranteed upon convergence. This leads
and use the “adequate” measurements” for solving optimization problems on-line.
Model parameterization tailored to real-time optimization
11
Input adaptation differs from the other two adaptation schemes in that a process model
is not used on-line, thus removing much of the on-line numerical burden and
complexity.
Another important element of comparison is the need of experimental gradient
gradients to guarantee (local) optimality. However, obtaining this information is usually
time consuming and slows down the entire adaptation process.
Note that the use of an updated process model allows determining changes in the active
set and provides faster convergence. Yet, in practice, the convergence of modelparameter adaptation and modifier adaptation is typically slowed down by the
introduction of filtering that is required to avoid unstable behavior that would result
from plant-model mismatch and measurement noise.
Model-parameter
Modifier
Input



Number of parameters
n
ng  n (ng  1)
n
Measurements
yp
p
p
Number of measurements
ny
ng  n (ng  1)
ng  n (ng  1)
  p
None
y  yp
2
2
No
Yes
Yes
Repeated optimization
Yes
Yes
No
On-line use of process model
Yes
Yes
No
Model predictive
Model predictive
Any
Smoothening
External filter
External filter
Controller tuning
Choice of active sets
Optimization
Optimization
Switching logic
Requirement for feasibility
Constraint
matching
None
Correct
active set
Requirement for optimality
KKT
matching
None
None
Controller type
Table 1. Comparison of various real-time optimization schemes (Regularization?)
6. Conclusions
This paper provides a classification of real-time optimization schemes and analyzes
their ability to use measurements to enforce optimality of the plant. The similarities and
differences between the various schemes are highlighted, and it is shown that modifieradaptation schemes use a parameterization, measurements, and an update criterion that
are tailored to the matching of KKT conditions.
To improve the performance of adaptive optimization, it may be useful to combine
specific features of the various methods. For example, the combination of modelparameter adaptation (which ensures fast convergence for the first few iterations and
detects changes in the active set) with input adaptation (which provides the necessary
12
B. Chachuat et al.
gradients in the neighborhood of the plant optimum) has been demonstrated in .
Another interesting combination would be to use modifier adaptation at one time scale
and perform model-parameter adaptation at a slower rate, thus giving rise to a two-timescale adaptation strategy.
Acknowledgments
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```
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