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State estimation of a solid-state polymerization reactor for PET based on improved SR-UKF.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
Published online 19 August 2009 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.306
Research Article
State estimation of a solid-state polymerization reactor for
PET based on improved SR-UKF
Ji Liu and Xing Sheng Gu*
Research Institute of Automation, East China University of Science and Technology, Shanghai 200237, China
Received 7 September 2008; Revised 18 February 2009; Accepted 23 February 2009
ABSTRACT: A state estimator for the continuous solid-state polymerization (SSP) reactor of polyethylene terephthalate
(PET) is designed in this study. Because of its invalidity in the application to some of the practical examples such
as SSP processes, the square-root unscented Kalman filter (SR-UKF) algorithm is improved for the state estimation
of arbitrary nonlinear systems with linear measurements. Discussions are given on how to avoid the filter invalidation
and accumulating additional error. Orthogonal collocation method has been used to spatially discretize the reactor
model described by nonlinear partial differential equations. The reactant concentrations on chosen collocation points
are reconstructed from the outlet measurements corrupted with a large noise. Furthermore, the error performance of
the developed ISR-UKF is investigated under the influence of various initial parameters, inaccurate measurement noise
parameters and model mismatch. Simulation results show that this technique can produce fast convergence and good
approximations for the state estimation of SSP reactor. ? 2009 Curtin University of Technology and John Wiley &
Sons, Ltd.
KEYWORDS: state estimation; solid-state polymerization reactor; improved square-root unscented Kalman filter; error
performance
INTRODUCTION
The widespread application of polyethylene terephthalate (PET) in the production of automotive tire cords
and soft drink bottles has led to a multibillion dollar
market for the resin.[1] The PET resin obtained by the
traditional synthetic routes (melt process) generally has
an average molecular weight of 15 000?20 000 kg/kmol
and an intrinsic viscosity (IV) of 0.55?0.65 dl/g, which
is unsuitable for bottle production (bottle grade IV
0.72?0.85 dl/g) or high-strength fibers (tire-cord resin
IV 0.95?1.05 dl/g). The continuous solid-state polymerization (SSP) process is the most feasible way to
increase the molecular weight of PET and other stepgrowth polymers, such as poly(butylene terephthalate)
(PBT), poly(ethylene naphthalate) (PEN) and nylon
6.6.[2]
The PET melt is extruded to form about 2-mmlong solid pellets. As shown in Fig. 1, these pellets are
fed continuously from the top of the reactor, which is
essentially an empty cylinder, and an inert gas enters
*Correspondence to: Xing Sheng Gu, Research Institute of Automation, East China University of Science and Technology, Box. 303,
No. 130, Meilong Road, Shanghai 200237, China.
E-mail: xsgu@ecust.edu.cn
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
from the opposite end. This inert gas removes the
reaction byproducts so that high molecular weights can
be obtained from the reaction. Meanwhile, it also acts
to partly regulate the reactor temperature. It is assumed
that both the solid flow and the gas flow are close to
ideal plug-flow conditions.[3]
The principal polycondensation chain-building reaction within the solid PET pellets is of the form
Pn + Pm ? Pm+n + C
(1)
where C is a byproduct, such as ethylene glycol (EG)
or water. Pn , Pm and Pn+m represent the polymer with
various polymerization degrees. The mechanism of the
SSP can be summarized in the following three steps[3] :
1. The reaction step involving the reaction of the
corresponding end groups;
2. The diffusion of the byproducts (water and EG)
through the polymer matrix (internal diffusion step)
to the surface;
3. The diffusion of these byproducts from the surface
to the inert gas (interphase diffusion step).
The reaction process involves both radial mass transfer in the solid particles and axial mass transfer in
Asia-Pacific Journal of Chemical Engineering
PET pellet
N2 and byproducts
pure N2
PET pellet
Figure 1. Diagram of an SSP
reactor.
the reactor. This is a multidimensional and multiphase process. The underlying continuous dynamics
of the reactor is described by nonlinear partial differential equations (PDEs) taking into account the
reactant concentrations, comprising the reactions of
the corresponding end groups, reactant convection,
and internal and interphase diffusion of byproducts.
Hence, the overall dynamics of the SSP reactor can
be described by a nonlinear distributed parameter
system.
The quality indices of polymer, for example intrinsic viscosity, average molecular weight and polymerization degree, are highly dependent on the operating conditions such as the reaction residence time
and the reaction temperature. The knowledge of the
time-varying spatial concentration profiles in the reactor is mandatory for the production quality control, monitoring and diagnosis of the SSP process
for PET. However, it is technically difficult to measure the time-varying concentration profiles in the
whole reaction tube. Instead, most of PET resin manufacture will use the above indices for quality control because they can be analyzed by on-line instrumentation. The concentration profiles of the reactants
have to be estimated indirectly from the few outlet
measurements.
While the current status of the linear estimation theory is a mature topic, challenges exist for the estimation of more complex systems (nonlinear, distributed,
hybrid, large-scale physical models), in particular goaloriented modeling issues (estimation for control, diagnosis, detection, monitoring, etc).[4] The SSP reaction
process is such a complex system with the characteristic of nonlinear distributed parameter, which is hybrid
and time varying.
The extended Kalman filter (EKF), which used
to be the first choice for state estimation of nonlinear models,[5,6] uses a local slope linearization
approach for calculating the mean and covariance of
the random variables. Such approximations are valid
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
STATE ESTIMATION OF A SSP REACTOR FOR PET
only for ?small? perturbations around the mean. Currently, unscented Kalman filter (UKF) and squareroot UKF (SR-UKF) are the widely used nonlinear filtering strategies, in which the state distribution is approximated by a small set of carefully chosen support points called sigma points.[7]
It is accurate to the second (extendable to third)order Taylor series expansion for arbitrary nonlinearity, while the EKF can only achieve first-order
accuracy. Furthermore, it is not necessary to compute the Jacobian matrices in the UKF. As a result,
the UKF can avoid substantial computational demand
and may become a powerful candidate for real-time
implementation.
Applications of UKF and SR-UKF can be found in
open literature, mainly for navigation purposes, e.g.
the tracking of a vehicle entering the atmosphere,[8,9]
navigation of an unmanned helicopter[10] and the estimation of spacecraft attitude.[11] However UKF has
been barely applied to solve chemical engineering
problems especially for nonlinear distributed parameter systems.[12] One of the successful applications
has been the state estimation of a lumped parameter
CSTR system by Romanenko and Castro.[13] Another
one is for the stratified domestic hot water storage tank described by nonlinear PDEs.[12] The estimation used both a distributed parameter observer
and the standard UKF posterior to the model
discretization.
A major drawback of the UKF is its instability
due to the numerical calculation error.[4] The covariance matrices may lose their positive semidefiniteness during the filtering procedure. To improve the
numerical properties, the SR-UKF was proposed by
Wan and van der Merwe.[14] In their implementation, the Cholesky factors of the covariance matrices will be propagated directly, similar to the standard square-root Kalman filter (SRKF), avoiding the
requirement of re-factorizing at each time step. The
SR-UKF algorithm, making use of powerful linear algebra techniques, orthogonal matrix triangularization (QR
decomposition) and Cholesky factor update, improves
numerical stability and can usually guarantee positive semidefiniteness of covariance matrices. Although
more robust than the UKF, it is found that the SRUKF cannot be applied for the state estimation of
SSP processes and some other nonlinear stochastic
systems[15,16] because the updated matrix of Cholesky
factor is still negative definite, which is caused by two
problems. One is the negative zeroth weight of sigma
points, probably causing Cholesky factor update of
the predicted state and measurement covariance matrices failing.[17] The other is the numerical computing
error probably causing the Cholesky factor update of
the updated covariance matrix failing. This paper contributes to the improvement of the existing SR-UKF
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
379
380
J. LIU AND X. GU
Asia-Pacific Journal of Chemical Engineering
algorithm so that it can be applied to arbitrary nonlinear systems with linear measurements such as the SSP
process.
The rest of the paper is organized as follows: First of
all, a simplified one-dimensional dynamic model of the
SSP reactor, described by nonlinear PDEs, is introduced
as well as the orthogonal collocation method (OCM)[18]
used for spatial discretization. The improved SR-UKF
(ISR-UKF) algorithm is then developed with theoretical explanation on how to avoid filter instability and
bringing in additional error. Next, the state estimator
of SSP reactor for PET is implemented on the basis of
the ISR-UKF. The time-varying concentrations within
the reactor are reconstructed from the outlet measurements. Finally, conclusions and future work will be
addressed.
MODELING
Dynamic model of SSP reactor
Although there are extensive publications on modeling of melt reactors for PET, few have been found for
SSP reactors. Most of them are particle models,[1,19,20]
describing the dynamics within micro particles. Modeling on SSP macro reactor can be found in a few
publications.[3,21] It was recognized that these SSP models are very complex with many uncertainties, consequently leading to calculation and solution problems.
The model selected is for the estimation of concentration profiles used as quality soft-sensor for quality
control. This type of model only deals with the reactant concentrations. Operating conditions, such as flow
velocity and reaction temperature, will be chosen as
input variables. A simplified macro reactor model based
on the mass balance, proposed earlier by the authors,[22]
will be adopted. This distributed parameter model consists of a set of nonlinear PDEs with initial and boundary conditions.
Boundary conditions:
g(t, 0) = g0 , e(t, 0) = e0
?e(t, z )
?g(t, z )
|z =1 =
|z =1 = 0
?z
?z
e(0, z ) = g(0, z ) = 0
(3)
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
(5)
where g, e and Z are, respectively, the average concentrations (in the unit of mol/ ) of EG, hydroxyl end
groups and diester end groups; t is the dimensionless
time with respect to reaction residence time ? ; and z
is the dimensionless axial coordinate with respect to
reactor length L. Dz is the average diffusion coefficient
of EG. ? is the effective factor of the interphase diffusion of EG. ? and K are, respectively, the reaction
rate constant and equilibrium constant. It is evident that
model parameters Dz , ? and ? are related to operating
conditions.
The model assumes constant feed concentrations
g0 and e0 : that is, the reactant concentrations in the
PET pellet from the melt process are assumed to be
constants. For a complete study, the model must be
able to simulate the start-up procedure when the reactor
is gradually filled with solid particles. Hence, an empty
reactor is presumed as the initial condition. The pellets
are continuously fed into the reactor with a flow velocity
v which is a control variable determining the reaction
residence time, defined as ? = vL.
In an industrial application, the reaction residence
time ? is a discrete variable controlling the polymerization degree. The larger the ? , the higher the molecular
weights of the PET obtained. In the state estimation of
the SSP reactor, ? is chosen as a control variable, the
concentrations g(t, z ) and e(t, z ) are the state variables
and the outlet concentration measurements g(t, 1) and
e(t, 1) are the observation data.
Model discretization
The state estimator is subsequently designed on the
basis of the discretized model. For chemical processes,
the OCM has been widely adopted. One of the advan-
Dz ? ? 2 g(t, z )
4g(t, z )Z (t, z )
?g(t, z ) ?g(t, z )
2
+
= 2
+ ??? e (t, z ) ?
?t
?z
K
L
?z 2
?e(t, z ) ?e(t, z )
4g(t, z )Z (t, z )
+
= ?2?? e 2 (t, z ) ?
?t
?z
K
e(t, z )
Z (t, z ) = 1 ?
2
Initial conditions:
(4)
(2)
tages of this method is that the collocation points are
chosen automatically, which tend to converge much
faster than the method of line.[23] The less number of
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STATE ESTIMATION OF A SSP REACTOR FOR PET
collocation points required reduces the computational
effort in terms of calculation time and data storage.
In the OCM, concentration is approximated by a
nonsymmetrical polynomial of the form
C (t, z ) = C0 (t) + zC1 (t) + z (1 ? z )
N
ai (t)Pi ?1 (z )
i =1
(6)
where C (t, z ) represents g(t, z ) or e(t, z ); C0 (t) =
C (t, 0), the feed concentration; C (t, 1) is the outlet concentration. {Pn (z )} are a set of polynomials
such that Pn (z ) is orthogonal to Pm (z ) when m = n.
The collocation points are the roots of Pn (z ) : z1 =
0, z2 , z3 , . . . , zN +1 , zN +2 = 1, of which z1 and zN +2 are
the boundary points.
Here, the transformation of Eqn (2) to ordinary
differential equations (ODEs) can be expressed as
N +2
N +2
Dz ? dgi (t) +
Aij gj (t) = 2
Bij gj (t)
dt
L j =1
j =1
4gi (t)(1 ? ei (t)/2)
2
+ ??? ei (t) ?
K
(7)
N +2
dei (t) Aij ej (t)
+
dt
j =1
4gi (t)(1 ? ei (t)/2)
= ?2?? ei2 (t) ?
(8)
K
Aij and Bij are obtained from the roots of the
orthogonal polynomials; gi and ei are the timevarying concentration at the i th collocation point zi .
i goes from 2 to N + 1. For the boundaries, we
have:
g1 (t) = g0 , e1 (t) = e0
N
+2
j =1
A(N +2)j gj (t) =
N
+2
As discussed in last section, the measurement equation is determined by
y(t) = H x(t) + v(t)
where H is a constant 2 О (2N + 2) matrix. y(t) is the
two-dimensional observation vector, which is equal to
the outlet concentration measurements of EG and the
hydroxyl end groups, contaminated by an uncorrelated
zero-mean Gaussian white noise v(t).
THE IMPROVED SQUARE-ROOT UNSCENTED
KALMAN FILTER
Since its proposal by Julier and Ulhman,[7] the UKF is
mainly used for an arbitrary nonlinear system. However, numerical instability often causes the covariance matrix P losing its positive definiteness during the filtering procedure.
? Consequently, the sigma
points ?i ,k ?1 = x?k ?1 ▒ ( (n + ?)P)i cannot be correctly calculated. Furthermore, the most computationally expensive operation in the UKF is the calculation of the square root of the covariance matrices at each time step for the new set of sigma
points.
Although proposed by Wan and van der Merwe as
a countermeasure,[14] the SR-UKF algorithm seems to
encounter the same problem as the UKF. It will be
discussed further in subsequent sections. The SR-UKF
cannot always be successfully implemented because the
updated matrices are probably negative definite while
updating the Cholesky factors (available as ?cholupdate?
in Matlab) such as
(c)
Sk? = cholupdate{Sk? , ?0,k |k ?1 ? x??
k , W0 } (14)
(c)
Syk = cholupdate{Syk , Y0,k |k ?1 ? y??
k , W0 } (15)
(9)
or
A(N +2)j ej (t) = 0
(10)
j =1
So the distributed parameter model described by
Eqns (2)?(5) is discretized to the 2N + 2 ODEs, which
can be rewritten as in state space as
x?(t) = F (x(t), u(t))
(11)
where F is the known nonlinear functions defined by
Eqns (7),(8) and (10). X is the unknown time-varying
state vector defined as
x = [x1 , x2 , . . . x2N +2 ]
= [g2 , g3 , . . . , gN +2 , e2 , e3 , . . . , eN +2 ]T
Sk = cholupdate{Sk? , Kk Syk , ?1}
(12)
u represent the control variables.
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
(16)
where Sk? , Syk and Sk , respectively, are the Colesky
factors of the predicted state and measurement covariance matrices and the updated covariance matrix. Kk
is the filter gain matrix. ?0,k |k ?1 and Y0,k |k ?1 represent,
respectively, the time update of the zeroth sigma point
?
of the state vector and its observation. x??
k and y?k are,
respectively, the mean estimate of state vector and the
mean observation. The zeroth weight of sigma points,
W0(c) , is defined as
W0(c) =
T
(13)
?
+ 1 ? ?2 + ?
n +?
(17)
where n is the state dimension and ?, ? and ? are the
adjustable parameters of the filter, of which ? and ? are
very small.
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
381
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J. LIU AND X. GU
Asia-Pacific Journal of Chemical Engineering
It has been found from our investigation that
if designed on the basis of the standard UKF or
SR-UKF, the state estimator for the SSP reactor
would encounter the above problems. The algorithm
broke down at second iteration. This work intends
to improve the SR-UKF performance. The state-space
model after spatial discretization for a SSP reactor
is
x(k + 1) = F (x(k ), u(k )) + w(k )
y(k ) = H (k )x(k ) + v(k )
(18)
where x(k ) is the n-dimensional state vector of the
system at time step k , u(k ) is the input vector, y(k ) is
the measurement vector, w(k ) and v(k ), respectively,
represent the process noise and measurement noise, F
Pk? =
2n
and the updated matrix from Eqn (15) is
?
P ▒ v uu T = Syk Syk T + sgn(W0(c) ) |W0(c) |(Y0,k |k ?1
? T
? y??
k )(Y0,k |k ?1 ? y?k )
(21)
If the zeroth weight W0(c) is negative to ensure a fine
tuning of the weights so that the overall prediction
error can be reduced,[17] these two matrices are most
possibly negative definite. According to Julier,[7] a
useful heuristic is to select n + ? = 3 in Eqn (17)
when the state is assumed Gaussian. Therefore, when
the algorithm approximates higher order systems or
probability density distributions, for a fine tuning of
the weights, ? is probably much smaller than zero
and the above updated matrices will become negative
definite.
Improvement 1: A modification of the covariance
matrix used by Julier and Uhlmann[17] is as follows:
Wi(c) [?i ,k |k ?1 ? ?0,k |k ?1 ][?i ,k |k ?1 ? ?0,k |k ?1 ]T + Qk
i =1
Pyk =
2n
Wi(c) [Yi ,k |k ?1 ? Y0,k |k ?1 ][Yi ,k |k ?1 ? Y0,k |k ?1 ]T + Rk
(22)
i =1
is the nonlinear state update function and H is the
measurement matrix. Moreover, w(k ) and v(k ) are the
uncorrelated zero-mean Gaussian white noise sequences
and their covariance matrices are, respectively, Qk and
Rk .
Calculation of Colesky factors of predicted
state and measurement covariance matrices
Problem 1: According to linear algebra theory, if S is
the original Cholesky factor of P , that is P = SS T , rank1 update matrix of the Cholesky factor S is available
in MATLAB as
S ? = cholupdate{S , u, ▒v }
(19)
The process is actually
the Colesky factorization of
?
the matrix P ▒ v ?
uu T ,[14] which requires that the
updated matrix P ▒ v uu T must be positive semidefinite. But in the SR-UKF, the updated matrix inferred
from Eqn (14) is
?
(c)
? ?T
T
P ▒ v uu = Sk Sk + sgn(W0 ) |W0(c) |(?0,k |k ?1
? T
? x??
k )(?0,k |k ?1 ? x?k )
(20)
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
?Covariance? is evaluated on ?0,k |k ?1 or Y0,k |k ?1 .
1
, i = 1,
Since all of the weights Wi(c) =
2(n + ?)
и и и , 2n are positive, the covariance matrices must be
positive semidefinite. So we modify the Cholesky factorization of the above covariance matrices to
?
Sk = qr [ W1(c) (?1,k |k ?1 ? ?0,k |k ?1 ) и и и
(c)
W2n
(?2n,k |k ?1
? ?0,k |k ?1 )
T
Qk ]
Syk = qr [ W1(c) (Y1,k |k ?1 ? Y0,k |k ?1 )
иии
T
(c)
W2n (Y2n,k |k ?1 ? Y0,k |k ?1 )
Rk ] (23)
This improvement avoids the negative definite covariance matrices caused by negative zeroth weight and
substitutes the Cholesky factor rank-1 update (denoted
as cholupdate) by the QR decomposition.
Calculation of Colesky factor of the updated
covariance matrix
Problem 2: It has been mentioned that the updated
matrix must be positive semidefinite while updating the
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STATE ESTIMATION OF A SSP REACTOR FOR PET
Cholesky factor. In the SR-UKF technique, the updated
matrix inferred from Eqn (16) is as follows:
?
(24)
P ▒ v uu T = Sk? Sk?T ? Kk Syk (Kk Syk )T
The subtraction of two positive semidefinite matrices easily results in a negative definite matrix. It is
easy for the updated covariance matrix in Eqn (24)
to lose its positive semidefiniteness owing to the
accumulated computing error. It is well known that
UKF-based algorithms are sensitive to the computing
error accumulation,[4] similar to the standard Kalman
filter.
Improvement 2: As the measurement equation
for a nonlinear system is linear as described in
Eqn (18), the updated covariance matrix can be derived
with the same equations as in classical Kalman
filtering
?1 T T
) Pxy
Pk = Pk? ? Kk Pyy KTk = Pk? ? Pxy (Pyy
(25)
Meanwhile, from the UKF, it can be known that the
matrices Pyy and Pk? are symmetrical. So the above
equation can be rewritten as
Pk = Pk? ? Pk? HkT (Hk Pk? HkT + Rk )?1 Hk Pk?
(26)
In order to improve the numerical stability, the standard SRKF algorithm[24] modifies Eqn (26) by using the
Cholesky factors Sk? and Sk instead of Pk? and Pk , which
are propagated directly at each iteration. The modification comes from the square-root covariance filtering
(SRCF).[25]
Sk = Sk? ? Sk? Fk (UkT )?1 (Uk + Rk )?1 FkT
(27)
where Fk and Uk are subject to
Fk =
and
(Sk? )T HkT
1
Uk = [FkT Fk + Rk ] 2
Step 1: Initialization
x?0 = E [x0 ],
S0 = chol{E [(x0 ? x?0 )(x0 ? x?0 )T ]}
Step 2: Sigma point calculation and time updating
?k ?1 = [x?k ?1 x?k ?1 + (n + ?)Sk
x?k ?1 ? (n + ?)Sk ]T
?k |k ?1 = F [?k ?1 , uk ?1 ]
x??
k
=
(29)
Hence, originated from the SRKF, Eqns (27)?(29)
are used to calculate the Cholesky factor Sk to improve
the numerical stability of the SR-UKF. It is easy to
prove that the nonzero matrix FkT Fk is positive definite.
As a result, the square root of the matrix FkT Fk + Rk in
Eqn (29) can be solved by Cholesky factorization. Of
course, the matrix FkT Fk + Rk may also lose positive
definiteness during the filtering process. In this case the
QR decomposition is again imposed.
Rk ]T )
(30)
Uk = qr([FkT
Therefore, the ISR-UKF algorithm for state estimation of arbitrary nonlinear system with linear measurements is summarized as follows:
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
Wi(m) ?i ,k |k ?1
i =0
Sk? = qr{[ W1(c) (?1,k |k ?1 ? ?0,k |k ?1 )
T
(c)
W2n
(?2n,k |k ?1 ? ?0,k |k ?1 )
Qk ] }
иии
Yk |k ?1 = Hk ?k |k ?1
y??
k =
2n
Wi(m) Yi ,k |k ?1
i =0
Step 3: Measurement updating
Syk = qr{[ W1(c) (Y1,k |k ?1 ? Y0,k |k ?1 )
(c)
W2n
(Y2n,k |k ?1 ? Y0,k |k ?1 )
Rk ]T }
иии
Pxk yk =
(28)
2n
2n
? T
Wi(c) [?i ,k |k ?1 ? x??
k ][Yi ,k |k ?1 ? y?k ]
i =0
Kk = (Pxk yk /SyTk )/Syk
?
x?k = x??
k + Kk yk ? y?k
Fk = (Sk? )T HkT
1
Uk = [FkT Fk + Rk ] 2
Sk = Sk? [I ? Fk (UkT )?1 (Uk +
Rk )?1 FkT ]
Error of the improved filter
In Improvement 1, the modified covariance matrix, taking Pyy for example, is equivalent to calculating the
covariance using the original form plus a term [y??
k ?
T
[17]
Y0,k |k ?1 ][y??
k ? Y0,k |k ?1 ] . According to Julier et al .,
the calculated covariance is
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
383
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J. LIU AND X. GU
Pyk =
?fx Pk? ?fx T
+E
Asia-Pacific Journal of Chemical Engineering
3
2
D 2 f (Dx
D 3 f (Dx f )T
Dx f (Dx
f )T
f )T
+ x
+ x
3!
2 О 2!
3!
where ?fx is the Jacobian of F [?] with respect to x(k ),
and the operator Dx f evaluates the total differential
of F [?] when x = x ? x. The modified covariance
ensures positive semidefiniteness and does not truncate
any order. Thus the new filter is still correct up to
the second order with error in the fourth or higher
order.
In Improvement 2, the Cholesky factor Sk of covariance matrix Pk is strictly derived from the SRCF. The
improvement should not bring in further error with the
improved numerical properties similar to those of the
SRKF.
Therefore, it can be concluded that the modification
on the SR-UKF algorithm introduces no additional
error.
+ ...
(31)
Thus, the state-space equations will consist of Eqn (11)
with N = 5. According to Eqn (12), the state vector is
12 dimensional and defined as
x(t) = [g(t, z2 ), и и и g(t, z7 ), e(t, z2 ) и и и , e(t, z7 )]T
(34)
The feed concentrations[22] on z1 = 0 are g(t, z1 ) =
10?5 mol/ and e(t, z1 ) = 0.0187 mol/ .
For integration, the sampling interval T = 0.002 is
selected. So the model can be approximated in discrete
time.
(35)
xk +1 = xk + TF (xk , uk )
and restricted by the measurement equation
yk = H xk + vk
(36)
0 0 0 0 0 1 0 0 0 0 0 0
with H =
0 0 0 0 0 0 0 0 0 0 0 1
where vk is the white noise sequence and its covariance
matrix is assumed as Rk = diag[10?10 , 10?4 ].
For simulation, the initial concentration profile in the
SSP reactor is assumed to be unknown. The initial
state vector and its covariance matrix for the filter are
arbitrarily chosen as follows:
ESTIMATOR IMPLEMENTATION FOR THE SSP
REACTOR
In this section an earlier lumping state estimation
method combining OCM with ISR-UKF is applied
to the SSP reactor. The SR-UKF is explained to
encounter the second problem above, which causes the
updated covariance matrix unable to be updated. Next
an ISR-UKF-based estimator is implemented and its
performance will be evaluated. In the following, we will
present the detailed design, especially on how to choose
the number of collocation points, initial profiles and the
filter parameters, etc.
Design and simulation
The state estimation of the concentration profiles in SSP
reactor is based on the dynamic model described by the
Eqns (2)?(5). First of all, the model is transformed to
a set of ODEs with the OCM. Here the time?spatial
concentration profile C (t, z ) is approximated by the
time-varying concentrations Ci (t, zi ) on five internal
collocation points and two boundary points, that is N =
5. The internal points are generated from orthogonal
polynomials? roots, which are
z2 = 0.0469,
z3 = 0.231,
z5 = 0.769,
z6 = 0.953
z4 = 0.5,
(32)
and the boundary points are
z1 = 0,
z7 = 1.
(33)
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
x?0 = [10?4
10?4
10?4
10?4
10?4
10?4
10?4
10?4
10?4
10?4
P0 = block ? diag{10?10 I6 , 10?4 I6 }
10?4
10?4 ]T
(37)
Using the standard SR-UKF algorithm to the state
estimate model, the program ceases while carrying out
the Cholesky factor update in Eqn (16) at the second
time step. The error information is that the updated
matrix is not positive definite. It is evident that the
pause is caused by the second problem described in the
Section Calculation of Colesky Factors of the Updated
Covariance Matrix. Besides, if ? = ?9, in order to
ensure n + ? = 3, the zeroth weight W0(c) becomes negative. So the first problem described in the Section Calculation of Colesky Factors of Predicted State and Measurement Covariance Matrices also probably happens.
However, the ISR-UKF avoids the filtering invalidation
while applied to the SSP process. The ISR-UKF-based
state estimator of SSP reactor achieves success.
The start-up process states are first estimated for the
reactor from initial empty to gradually filled with solid
particles. The observation data corrupted by the noise
are shown in Fig. 2, representing the concentration measurements of EG and hydroxyl at the outlet. The states
are estimated from the observation serials based on
the ISR-UKF. Figure 3 depicts the state and estimation
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STATE ESTIMATION OF A SSP REACTOR FOR PET
x 10-5(mol/O)
(a)
(b)
Observation of hydroxyl concentration
Observation of EG concentration
7
6
5
4
3
2
1
0
-1
-2
-3
0
0.5
1
1.5
2
2.5
3
3.5
4
(mol/O)
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
0
0.5
1
1.5
t
2
2.5
3
3.5
4
t
Figure 2. Plots of the available (simulated) observations of (a) EG and (b) hydroxyl in start-up
process.
of hydroxyl concentrations on all internal collocation
points during this start-up process. It shows the fact
that solid particles gradually fill the reactor from the
inlet.
Although large at beginning due to the arbitrary
initial parameter settings of the filter, the estimation
error rapidly converges. Even with the observation data
highly corrupted by noise (shown in Fig. 2), the filter
can overcome the disturbance and produce very good
results.
Next, states of the dynamic process will be estimated when the operating conditions are changed. In
the SSP process, the reaction residence time is one
of the operating variables determined by the flow
velocity of the pellets. Figure 4 displays the observation data from on-line measurement when the residence time decreases from 30 to 10 h. The estimation results at certain collocation point are shown
in Fig. 5. The filter parameters are kept the same.
The same conclusion can be drawn on the filter
performance as the start-up simulation. The selection of initial state of the filter will be discussed
later.
x 10-3 (mol/O)
z1
Concentration of hydroxyl
18
z2
16
z3
14
z4
z5
12
10
z6
8
z7
6
4
2
0
-2
0
0.5
1
1.5
2
t
2.5
3
3.5
4
The state (solid) and its estimation (dash)
of hydroxyl on collocation points during the start-up
process. This figure is available in colour online at
www.apjChemEng.com.
Figure 3.
x 10-5 (mol/O)
(a)
(b)
Observation of hydroxyl concentration
Observation of EG concentration
3.2
3
2.8
2.6
2.4
2.2
2
0
0.5
1
1.5
2
2.5
3
t
(mol/O)
0.018
0.017
0.016
0.015
0.014
0.013
0.012
0.011
0.01
0.009
0
0.5
1
1.5
2
2.5
3
t
Figure 4. Observations of (a) EG and (b) hydroxyl (b) when ? decreases to 10 h from 30 h.
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
385
J. LIU AND X. GU
Asia-Pacific Journal of Chemical Engineering
(a)
(b)
x 10-5 (mol/O)
3.5
Concentration of EG on z5
Concentration of hydroxyl on z5
state
estimate
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
x 10-3 (mol/O)
16
3
0
0.5
1
1.5
2
2.5
14
10
8
6
4
2
0
-2
3
state
estimate
12
0
0.5
1
1.5
t
(c)
(d)
Outlet concentration of hydroxyl
3.1
3
2.9
state
estimate
2.8
2.7
2.6
2.5
2.4
2.3
2.2
0
0.5
1
1.5
2
2.5
3
t
x 10-5 (mol/O)
3.2
Outlet concentration of EG
2
2.5
3
(mol/O)
0.015
0.0145
simulate
estimate
0.014
0.0135
0.013
0.0125
0.012
0.0115
0.011
0.0105
0
0.5
1
1.5
2
2.5
3
t
t
Figure 5. The states and estimations of EG and hydroxyl concentrations on the fifth collocation
points (a) and (b), and the right boundary in (c) and (d) when ? decreases to 10 h from 30 h.
Performance evaluation
x 10-5 (mol/O)
3
It is well known that the performance of the Kalman
filter is largely determined by such factors as initial
parameters of the filter, process and measurement noise
parameters and uncertain model parameters. Here the
performance of the designed estimator is evaluated
under the impact of various initial parameters, inaccurate measurement noise parameters and model mismatch.
Initial parameters
The initial parameters of a filter largely determine the
estimation error at the initial stage. The influence will
become asymptotically negligible as more and more
data are processed. These initial parameters mainly
include initial state vector and its covariance matrix.
In reality, the initial state should be a zero-vector, not
as defined in Eqn (37).
x?0 = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]T
(38)
Its covariance matrix decreases to P0 = block ? diag
{10?10 I6 , 10?6 I6 }. The estimation results at the third
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
2.5
Concentration of EG on z4
386
state
estimate
2
1.5
1
0.5
0
-0.5
0
0.5
1
1.5
2
t
2.5
3
3.5
4
Figure 6. The EG concentration and its estimate when
X?0 is accurately set.
internal points are shown in Fig. 6, from which we can
see that it is much better estimated.
When the operating variables are changed, the
unknown initial profile of the state estimator is approximated by incorporating the known inlet concentrations
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
STATE ESTIMATION OF A SSP REACTOR FOR PET
x 10-5 (mol/O)
(a)
Concentration of EG on z5
Concentration of hydroxyl on z5
16
3
simulate
estimate
2.5
2
1.5
1
0.5
0
-0.5
-1
x 10-3 (mol/O)
(b)
3.5
0
0.5
1
1.5
2
2.5
3
14
state
estimate
12
10
8
6
4
2
0
-2
0
0.5
1
1.5
t
2
2.5
3
t
Figure 7. The state and its estimate of EG (a) and hydroxyl (b) when X?0 is properly chosen.
with the available measurement information. This can
be done by a linear correlation of inlet and outlet concentrations. So the initial state vector of the filter in the
dynamic simulation is chosen as follows:
?5
?5
?5
x?0 = [1.09 О 10 , 1.46 О 10 , 2 О 10 , 2.54 О 10 ,
2.91 О 10?5 , 3 О 10?5 , 0.0184, 0.0172, 0.0154,
0.0135, 0.0123, 0.012]T
(39)
For the same conditions (the residence time decreases
from 30 to 10 h), the simulation results are shown in
Fig. 7. Compared to Fig. 5 (a) and (b), the justification
in the initial parameter setting of the filter can remarkably improve the estimation performance.
Measurement noise
The performance of a filter largely relies on the process
and measurement noise parameters of the system. The
performance improves with the increasing of signal-tonoise ratio. In the previous section, we have demonstrated that ISR-UKF can handle the signal with heavy
disturbance. However, the use of Kalman filters requires
accurate information on the noise distribution. This is
rather difficult since disturbance is unpredictable and
uncertain. In the EKF estimation, error can be easily
accumulated and even diverge owing to the inaccurate
noise parameters presumed by the user.
In order to evaluate the performance of the ISRUKF under the impact of inaccurate measurement
noise parameters, we carried out another trial for Rk =
diag[10?12 , 10?6 ] different from the actual value Rk =
diag[10?10 , 10?4 ]. Simulation results are depicted in
Fig. 8, where ?estimate 1? represents the estimation
when the approximate value of noise covariance is taken
as Rk = diag[10?10 , 10?4 ] and ?estimate 2? represents
the estimation when Rk = diag[10?12 , 10?6 ]. It can be
seen that, in the case of a wrong noise parameter,
although the estimation error increases at the beginning, it converges faster and tracks the states well. It
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
state
estimate1
estimate2
6
Concentration of EG on z5
?5
x 10-5 (mol/O)
8
4
2
0
-2
-4
0
0.5
1
1.5
2
2.5
3
t
Figure 8. The state and its estimates of EG concentration
under different noise covariances Rk = diag[10?10 , 10?4 ]
(estimate1) and Rk = diag[10?12 , 10?6 ](estimate2).
proves that the ISR-UKF works well under inaccurate
measurement noise estimation.
Model mismatch
The above simulations are based on an ideal situation
where the exact system model is available to the estimator. This is often unrealistic in industrial practice. So,
investigation has been conducted on the performance
of the ISR-UKF when the filter model is mismatched
to the system. It can happen in reality when uncertainty comes from diffusion coefficient Dz and effective
factor ?.
First, the effective factor ? is reduced 5%. From
Fig. 9 it can be seen that the mismatched effective factor
would significantly lower the capability of the estimator.
Besides, the estimation error increases along the axial
Asia-Pac. J. Chem. Eng. 2010; 5: 378?389
DOI: 10.1002/apj
387
J. LIU AND X. GU
Asia-Pacific Journal of Chemical Engineering
reaction output and interphase diffusion. The sensitivity of Dz to the observation is smaller than that
of ?. Many researchers have carried out sensitivity
analysis.[26] This simulation result is in agreement with
the fact that uncertain model parameters with various sensitivities affect the Kalman filtering performance
differently.[27]
Hence, the factors for the model mismatch play
different roles on the estimation performance. The
effective factor ? lowers significantly the capability of
the estimator, whereas the diffusion coefficient Dz has
only slight impact. A future work could develop an
adaptive filter algorithm or a joint estimation algorithm
for the state and parameter ? based on the ISRUKF.
x 10-5 (mol/O)
Concentrations of EG
3
2.5
z7
2
z4
1.5
z2
1
0
0.5
1
1.5
t
2
2.5
3
CONCLUSIONS
Figure 9. The state (solid) and its estimate (dash) of EG
concentration when ? reduces 5% at the second, fourth
and right boundary points. This figure is available in colour
online at www.apjChemEng.com.
x 10-5 (mol/O)
3.1
state
estimate1
estimate2
3
Outlet concentration of EG
388
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
0
0.5
1
1.5
t
2
2.5
3
Figure 10. The state and its estimate of EG concentration
when ? reduces 5% (estimate1) and 10% (estimate2).
coordinate of the reactor because the influence of the
reduction of ? on the reaction and interphase diffusion
gradually increases when the reaction proceeds. If ?
reduced further, the estimation error will become larger,
as shown in Fig. 10 where ? is reduced by 10%.
However, the estimation of hydroxyl is only slightly
affected since ? does not directly affect its mass
balance.
The change of diffusion coefficient shows that the
estimator can work well with the mismatch of Dz to
the extent of 10 times the actual value because the
internal diffusion of EG is much smaller than the
? 2009 Curtin University of Technology and John Wiley & Sons, Ltd.
This paper presents the design of a novel state estimator
for a typical distributed chemical reactor, the continuous
SSP reactor for PET. An improved ISR-UKF is developed on the basis of the standard SR-UKF algorithm.
Simulation results in both the start-up and dynamic
process show that the ISR-UKF used for the state estimation of SSP reactor converges fast and reconstructs
the concentration profiles well even with highly contaminated observation data and inaccurate filter parameters.
It can be further concluded from the study that
the shape of initial profile and accuracy of measurement noise covariance do not affect the long-term
convergence behavior of the ISR-UKF, although they
have some influence at the initial stage. It is also
found that the factors for the model mismatch play
different roles in the estimation performance. The
effective factor lowers significantly the capability of
the estimator, while the diffusion coefficient affects
slightly.
The successful estimation of the reactant concentration in SSP reactor will be useful as a quality index
soft-sensor for production quality control.
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No.60774078), Shanghai Commission of Science and Technology (Grant
No.08JC1408200) and Shanghai Leading Academic
Discipline Project (Project No. B504).
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389
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