preprint version of published journal: N. Hashemian and A. Armaou, Simulation, model-reduction and state estimation of a two-component coagulation process, AIChE J., 2016, 62, pp 1557-1567, DOI:10.1002/aic.15146 Simulation, Model-Reduction and State Estimation of a Two-Component Coagulation Process Negar Hashemian and Antonios Armaou Dept. of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802 Abstract We address the issue of state estimation of an aggregation process through (i) using model reduction to obtain a tractable approximation of the governing dynamics, and (ii) designing a fast moving-horizon estimator for the reduced-order model. We first use the method of moments to reduce the governing integro-differential equation down to a nonlinear ordinary differential equation (ODE). This reduced-order model is then simulated for both batch and continuous processes and the results are shown to agree with constant Number Monte Carlo (cNMC) sim- ulation results of the original model. Next, the states of the reduced order model are estimated in a Moving Horizon Estimation (MHE) approach. For this purpose we first employ Carleman linearization and represent the nonlinear system in a bilinear form. This representation lessens the computation burden of the estimation problem by allowing for analytical solution of the state variables as well as sensitivities with respect to decision variables. INTRODUCTION In chemical engineering, material science and biology there are a multitude of processes character- ized by dispersed phenomena. Systems involving such processes merit a particle population study Correspondence concerning this article should be addressed to A. Armaou at [email protected]
Transcript
preprint version of published journal: N. Hashemian and A. Armaou, Simulation, model-reduction and
state estimation of a two-component coagulation process, AIChE J., 2016, 62, pp 1557-1567,
DOI:10.1002/aic.15146
Simulation, Model-Reduction and State Estimation of
a Two-Component Coagulation Process
Negar Hashemian and Antonios Armaou
Dept. of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802
Abstract
We address the issue of state estimation of an aggregation process through (i) using model
reduction to obtain a tractable approximation of the governing dynamics, and (ii) designing a
fast moving-horizon estimator for the reduced-order model. We first use the method of moments
to reduce the governing integro-differential equation down to a nonlinear ordinary differential
equation (ODE). This reduced-order model is then simulated for both batch and continuous
processes and the results are shown to agree with constant Number Monte Carlo (cNMC) sim-
ulation results of the original model. Next, the states of the reduced order model are estimated
in a Moving Horizon Estimation (MHE) approach. For this purpose we first employ Carleman
linearization and represent the nonlinear system in a bilinear form. This representation lessens
the computation burden of the estimation problem by allowing for analytical solution of the
state variables as well as sensitivities with respect to decision variables.
INTRODUCTION
In chemical engineering, material science and biology there are a multitude of processes character-
ized by dispersed phenomena. Systems involving such processes merit a particle population study
Correspondence concerning this article should be addressed to A. Armaou at [email protected]
rather than the traditional mass balance performed for continuous media. There are numerous
examples of these processes ranging from industrial applications to biological reactions. Crys-
tallization, polymerization, viral infections, and collocalization of enzymes in cells are just a few
instances of these processes that occur in our everyday life and motivate the work in this article.
The common characteristic of all these processes are the individual members of the population or
the particles. These particles are distinguished by their type, size and/or composition. Mostly, a
population balance governs the dynamic behavior of particulate systems. This results in complex
mathematical models, specifically for systems in which the particles are characterized by two or
more properties. The simplest system, in this area, is a two-component aggregation system with no
chemical reactions. This system is mostly used in pharmaceutical applications where through use
of a solvent called excipient the particles in a drug powder adhere together and form granules. In an
ideal granulation process the components, solvent and solute, are well mixed and the distribution of
the solute mass is uniform. The deviation from the mean of the solute mass in aggregates quantifies
the granulation quality which is referred to as blending degree. However, this output property is
not easily measurable during evolution of the process. This article focuses on these bi-component
granulation processes and estimation of the distribution of components in the product.
The population balance equation for granulation systems determines the dynamics of a bivariate
distribution function in form of an integro-differential equation. The variables of this distribution
function are size and composition of the aggregates. However, the weighting function in calculating
the integrals called coagulation kernel is mostly a function of particle size and independent of
composition. Solving this equation has been subject of many research efforts in the literature .1–4
For a general kernel, there is no analytical solution to the population balance equation. Matsoukas
et al. proposed the constant-Number Monte Carlo (cNMC) algorithm to simulate these dynamics
in both batch processes and semi-batch processes where there is a continuous inflow of binder.5,6
In the cNMC method, there should always be a constant number of particles in the “simulation
box”. Therefore, the box expands/contracts as the population decreases/increases to maintain the
same number of particles. Smith and Matsoukas showed this method is more accurate and faster
than the traditional Monte Carlo algorithms where a finite number of particles are considered in
2
a fixed-volume simulation box.7 More specifically, cNMC method reduces the decay of accuracy
in comparison to the traditional Monte Carlo simulation method, from square-root dependence to
logarithmic dependence on the number of coagulation events. In this article, we extend the use of
cNMC algorithm to continuous processes and utilize it for simulation of a bi-component coagulation
system.
Although extremely useful, stochastic simulations are computationally cumbersome. Subse-
quently, stochastic modeling is sometimes impractical, specially in the applications demanding high
computational costs such as state estimation, optimization, and process control. One solution is to
derive and use deterministic models that instead of microscopic behavior of the individual elements
focus on bulk statistics of the stochastic process under study. The method of moments is a powerful
technique in this matter. It can be used to transform the integro-differential governing coagula-
tion processes into a set of ODE. In this method, the closure of the obtained ODE set is critical.
Although, this closure issue is nonexistent in the examples with constant or additive coagulation
kernels, it introduces a challenge in more complicated examples such as the case of a Brownian
kernel. Direct quadrature method of moments is an efficient technique for single-component coag-
ulation simulations.8,9 However, Marshal el al. showed this method is not as accurate as cNMC
simulation for bi-component coagulation processes.10 Yu el al. employed Taylor expansions to de-
rive a closed finite dimensional ODE system that models a single-component coagulation process.11
The accuracy of the system improves with increasing the Taylor expansion’s order, which is equal
to the number of moments considered in the model. In this manuscript, we exploit the idea in Yu
el al.’s work but for bi-component coagulation processes. This results in a tractable ODE model
of the process. The accuracy of this reduced-order model is verified in a simulation study, when it
is compared against stochastic simulation results of the process. The verified model is finally used
for online estimation of the internal dynamics of the process, which is the second objective of this
work.
In the reduced-order model obtained from application of the method of moments, the states are
statistical moments of the particles’ distribution. Knowledge of these states yields comprehensive
information on different output properties, such as the blending degree. This information is also
3
very critical for process control purposes. The scientific literature is abundant with numerous
observer design techniques for estimation of the unknown state variables of a dynamic system using
measurements of the input and output trajectories. Moving Horizon Estimation (MHE) is one
of these design techniques that is specifically powerful in handling unknown disturbance signals
acting on the system or unknown noise signals corrupting the output measurements.14–18 MHE
uses the past and current output measurements and defines an error cost function over a moving
time window. It then finds the most probable realization of the state variables at the current time
by minimizing a cost function. Although this method has many advantages over other alternative
methods, it is computationally demanding. In this article, in order to deal with this issue, we use
a modified MHE design proposed in a previous work by the authors.23 In this modified design,
first, Carleman linearization is used to represent nonlinear systems in a bilinear form. Not only
we can reduce computational cost by analytically solving for the response of the bilinear system,
but also the KKT matrix of the optimization problem is found analytically which decreases the
computational complexity even further. This design framework is employed for the coagulation
process. Lastly, the designed estimator is shown to closely track the evolution of the statistical
moments of the particles using only noisy measurements of the particle population in the exit flow.
It is worth mentioning that there are many observer and controller design methods in the
literature for particulate processes.24,26–29 For example Kalani and Cristofides modeled a Titania
aerosol reactor in which a combination of a chemical reaction, coagulation and convective transport
happens. In their model, the collision kernel includes a Brownian and a turbulent shear collision
frequency function. Then, they used method of moments to control this process. However, because
of the Brownian kernel, the fractional moments appear in the ODEs. To address this issue, they
assumed the aerosol particles have a lognormal distribution.24 Hulburt and Katz, in another
work, used Laguerre polynomial expansions to obtain closed ODEs from the infinite set of moment
equations.25 Chiu and Christofide also employed this idea to approximate a crystallizer’s PDE
model by a finite ODE set. They next employed the reduced model to maximize the mean size
of the particles.26 Mesbah et al. used method of moments in a batch polymorphic crystallization
process and reduced the integro-differential model to a nonlinear differential algebraic equation
4
set.29 Then, they designed a stochastic model predictive controller to maximize the desired product
while meeting some probabilistic constraints. However, to the best knowledge of the authors there
has been no work on the estimation and control of bi-component coagulation processes.
In this manuscript, we describe and mathematically represent the coagulation process. Next, the
cNMC algorithm and the method of moments are explained. The article reviews and implements the
modified MHE design technique on the coagulation process. The last section starts with comparing
the simulation results obtained by cNMC and method of moments in both batch and continuous
processes. This section concludes with estimating the critical moments in the continuous process
in the presence of noise in the feed flow concentration and the output measurement.
Multi-Component Mass Balance
Each particle in a binary-component aggregation process is composed of two components: the
solute (drug) and the solvent (excipient). The total mass of a particle is denoted by p which is
also representative of the particle’s size. Moreover, s denotes the mass of the solute content in the
particle. The ratio of s to p is indicative of the particle’s composition. The probability distribution
function f(p, s) describes the population distribution of the particles as a function of their mass and
their solute content. In other words, the number of particles with a total mass within the differential
range [p− dp2 , p+ dp
2 ] and a solute mass in the range [s− ds2 , s+ ds
2 ] is given by f(p, s)dpds. For
simplicity, we combine the total and solute mass variables in an augmented vector, r = (p, s). Also,
hereafter, we refer to this augmented vector as characteristic size, or size in short.
During the coagulation process, the particles may collide and form a larger particle. The
aggregation kernel, k12 = k(r1, r2) is a function reflecting the dynamic behavior of particles in
the system. More specifically, it determines the rate of formation of particles of size r1 + r2 from
collision of particles of size r1 and r2. Conservation of particle and solute mass leads the mass
balance formulation in.5 This formulation considers an infinitesimal range of characteristic particle
size characterized by [r1 − dr12 , r1 + dr1
2 ]. As Fig. 1 suggests particles may appear in or disappear
from this range because of the following two reasons:
5
1. The particle population in the range increases when smaller particles stick together and form
a particle in this range. Say one of the particles has a size r2 < r1, then if it collides into a
particle of size r1 − r2 > 0 the result would be a new particle in the range considered. This
particle formation happens with a rate equal to k(r1 − r2, r2).
2. The particles within the range may form bigger particles when colliding with an arbitrary
particle of size r2 with a rate of k(r1, r2).
Taking the above two phenomena into account yields the net rate of change of the aggregate
population in the range as follows.
∂f(r1)
∂t=
1
2
r1∫0
k(r1 − r2, r2)f(r1 − r2) f(r2)dr2 −∞∫
0
k(r1, r2)f(r1)f(r2)dr2
(1)
It is important to note that for any particle the total mass is always greater than the mass of solute.
Therefore,
f(s, p) = 0 for s > p
If dependence of the distribution function f(.) on the composition of particles is ignored, then
the integro-differential equation obtained above simplifies to the so called Smoluchowski coagulation
equation. When the coagulation kernel is either constant or additive, the integro-differential equa-
tion can be analytically solved. However, the nature of the particle agglomeration process is more
complicated and the real life examples are modeled by more complex kernels such as Brownian.
For these complicated kernels, there is no explicit analytical solution for the distribution function.
This article next addresses this issue following two different approaches. Specifically, we employ a
cNMC algorithm and method of moments to simulate bi-component coagulation processes in batch
and continuous mode.
6
Constant-Number Monte Carlo Algorithm
This section utilizes a cNMC algorithm to numerically simulate the evolutions in a bi-component
agglomeration process. Constant-Number Monte Carlo shows two advantages over the traditional
Monte Carlo approaches: First, it is faster; second, the simulation accuracy decays at a much lower
rate.7 Figure 2, schematically shows how this algorithm works. The simulation box always includes
a fixed number of particles. As a result, when dilution or thickening occurs, the volume of the box
increases or decreases, respectively. This section, first explains cNMC as was originally presented
in5 for a batch process. Then, we modify this algorithm and extend its use to continuous processes.
Batch Process
As explained before, the simulation box considers a constant number of particles denoted by N.
Each of these particles is characterized by a size vector r. For any pair of particles, i and j,
there exists a kernel, ki,j , expressing their chance of collision and forming a larger particle. The
maximum and average of these kernel functions over all possible pairs of particles are denoted by
kmax and kave, respectively. At every sampling time we randomly pick two particles. The ratio
of the corresponding kernel to kave shows the agglomeration probability of the selected pair of
particles. We then generate a random number ρ ∈ (0, 1) as a metric for acceptance or rejection
of the agglomeration of the candidate pair. When ρ >ki,jkave
, the pair is rejected and the random
pair selection is repeated until the acceptance criterion is met. In the case, ρ ≤ ki,jkave
, the pair is
accepted. In order to systematically update the population with the accepted pair, we reassign the
properties of the ith particle as follows.
ri + rj → ri
Because after an agglomeration event the number of particles per unit volume decreases, next we
need to introduce a new particle into the simulation box. To do so, we randomly select an existing
particle in the box and copy its properties to redefine the jth particle. This addition of a new
7
particle affects the total volume of the box in the following manner:
ptotalptotal + pj
V → V (2)
where ptotal is the total mass of the particles before the update, and pj is the total mass of the new
jth particle after the update.
For a general Poisson process, the mean time elapsed before occurrence of the next random
event is given by:30
∆t =1
V∑ki,j
(3)
where V is the volume of the simulation box. Then, specifically for this aggregation process, the
elapsed time for a large enough N is approximately equal to:
∆t ≈ 1
V(n2
)kave
≈ 2
N2V kave(4)
Continuous Process
This article modifies the simulation technique described above to be used in a continuous process.
As opposed to the batch process, here we consider three possible events: (i) introduction of new
particles to the system as a feed, (ii) coagulation of particles as before, and (iii) particles exiting
the system as an output flow. At every simulation step, the stochastic simulation algorithm first
needs to determine which of these events occurs and how long it takes.
The events occur with different probabilities:
Pevent =Revent
Rin +Rcoag +Rout, event ∈ {in, coag, out} (5)
where P and R denote the probability and the rate of a particular event, respectively. The occur-
rence rates of all the events are listed in Table 1.
A random variable ρ with a uniform distribution over [0, 1] is used to determine which event
happens at any simulation step. Also, similar to the batch simulation, the average time elapsed
8
before the occurrence of the next event is obtained from Eq. (3):
∆t =1
Rin +Rcoag +Rout(6)
After the random selection of the event, the algorithm needs to simulate occurrence of the
selected event and update the particle population accordingly. Since simulation of the coagulation
event is the same as before, we focus on explaining the input and output flows and how to simulate
their effects on the particle population. For this purpose, let n represent the number of particles
that enter/leave the reactor during any simulation step.
The input flow has a particle distribution that is potentially different from the mixture in the
reactor. Define Cf to be the number of particles in the input flow and α to be the input/output
flow rate, where both the quantities are normalized by the reactor’s volume. Now, when n particles
enter the reactor, to maintain the number of particles constant, we randomly remove n particles
from the simulation box and replace them by the new entering particles. As a result, the volume
of the simulation box shrinks according to the following expression:
ptotal + pin − pdptotal + pin
(V +n
Cf)→ V (7)
where ptotal, pin and pd are the total mass of particles before the update, the total mass entered to
the box, and the mass of the particles removed from the box, respectively.
Now consider the output flow of the reactor. Since we assume that the reactor is well-mixed,
properties of output flow is the same as the reactor’s content. Hence, there is no change in the
particle distribution of particles in the box in this event. Consequently, the volume of the simulation
does not change.
Table 1 gives the occurrence rate and the corresponding volume change for each event.
9
Model Reduction by Method of Moments
As discussed before, the original integro-differential equation does not have an explicit analytical
solution. Monte Carlo simulation is a reliable approach to study the dynamical behavior of the
particles, but it is computationally demanding. This section seeks for an alternative and more
efficient solution by relying on model reduction techniques. More specifically, applying method
of moments to Eq. (1), we derive an ODE system that describes the dynamics the population’s
probabilistic moments.
To accomplish this goal, we first define mixed moment variables as follows:
Mij =
∫ ∞0
∫ p
0pisjf(p, s) dpds (8)
Therefore M00,M10, and M01 represent the total number of particles, the total mass, and the total
solute mass in a unit of volume, respectively. In a batch process, M10 and M01 have constant
values. However, in a continuous process because of the input and output streams these moments
may change in time.
To obtain rate of change of Mij with i, j ∈ Z+, Eq. (1) is multiplied by pisj and then integrated
over the region used in the definition of moments in Eq. (8). Matsoukas et al.5 show that after
some algebra and simplifications this results in the following equation:
d[Mij ]
dt=
1
2
∞∫0
∞∫0
drdr′[A(r + r′)−A(r)−A(r′)]f(r)f(r′)k(r, r′) (9)
where the function A(r) = pisj .
Here, we assume the aggregation kernel between particles is Brownian and calculated as:
k12 = k0 (2 + (p1
p2)13 + (
p2
p1)13 ) (10)
where k0 is a constant coefficient. Incorporating the Brownian kernel and using the moments’ def-
inition, Eq. (9) determines the dynamics of the population’s probabilistic moments. Theoretically,
10
all the moments are needed to recover the exact particle distribution. But, for practicality reasons,
we restrict i, j to the index set I = {0, 1, ...,m − 1}. This results in a total of m2 states which
construct the state vector x of the reduced-order model.
x = {Mij |i, j ∈ I}
In calculation of the time derivative of Mij , the left-hand side of Eq. (9) depends on some mixed
moments Mk,j with fractional indices.
x = F (x,Mkj) (11)
where k ∈ Q+ and j ∈ I. This expression of the system, however, is an approximation of the actual
dynamics and becomes more accurate as more probabilistic moments are considered. We deal with
the fractional moment Mkj , by utilizing a truncated Taylor expansion of pk about a desired point
p = p:
pk =
∞∑l=0
(k
l
)pk−l(p− p)l (12)
where
(k
l
)=
k∏n=1
l − n+ 1
n. It is more convenient to expand the system around the average mass
of the particles i.e. p = M10M00
.
By applying the first m terms of Eq. (12) in (8), the moments with fractional order can be
approximated by a function of x. Here, we use the second order Taylor expansion:
Mkj ≈M0j pk(1 +
k2
2− 3k
2) +M1j p
k−1(2k − k2) +M2j pk−2k
2 − k2
(13)
By increasing the order of Taylor expansion, the approximation in (13) becomes more accurate.
However, it requires the geometric increase of the number of states in the dynamic system (11)
which makes the computations more complex.
Moreover, note that for the continuous process the input and output streams should be consid-
11
ered in the population balance equation:
∂f(r1)
∂t=
1
2
r1∫0
k(r1 − r2, r2)f(r1 − r2) f(r2)dr2 −∞∫
0
k(r1, r2)f(r1)f(r2)dr2
+ α[fin(r1)− f(r1)]
(14)
where function fin gives the distribution of particles in the input stream.
We can close these ODE sets at any desired approximation level of the original infinite dimen-
sional ODEs, denoted by m. Appendix gives a closed ODE system for the batch and continuous
system employing the approximation. This approximation considers the first nine moments, Mi,j
where i, j ∈ {0, 1, 2}. However, since M10 and M01 moments are constant in the batch process,
the states reduce to 7. The simulation results in the next section, indicate the number of moments
with integer indices employed to reduce the system suffice for the underlying coagulation process.
In what follows, we will discuss how to use this derived ODE system to design an estimator for the
output properties. Also, the next section discusses about the moments estimation in the presence
of noise in the feed flow stream.
System State Estimation
This section implements the MHE approach to estimate the unmeasurable output properties. In
particle processes, one of the important output properties is the total variance of excess solute. Let
the average solute per particle mass be denoted by φ = M01M10
. So, the squared excess solute in a
particle is:
ε(r) = (p− φs)2 (15)
and the sum squared of excess is:
ε =
∫ ∞0
ε(r)dr = M02 − 2φM11 +M20 (16)
12
Therefore ε can be obtained by state estimation of the dynamic system. The system derived in the
previous section is presented in the compact form as follows:
x = F (x) +Dw
y = x1 + ν(17)
where w ∈ Rn and ν ∈ Rp are noise signals on the system dynamics and output measurements.
Moreover, D has an appropriate dimension. To simplify the computations, we transform the states
so that the desired nominal operating point denoted by x0 and the corresponding noise-free output
is at the origin.
MHE uses the history of the output trajectory, to update the states estimated in the past
and also obtain the most probable values for the current states. To bound the dimension of the
optimization problem, MHE defines an objective function over a moving window:
J(x0, {wi}Ni=1) = ‖x0 − x0‖2Π +
N∑i=1
(‖νi‖2Q + ‖wi‖2R)
Subject to:
wi ∈W, xi ∈ R+ and νi ∈ V
(18)
where νi = yi − yi, ‖z‖2M = zTMz; Q,R, and Π are tuning parameters with a symmetric positive
definite structure; and W and V are compact sets. The moving time window includes the N recent
unknown state vectors. This window moves one step forward at each sampling time. We assume the
noise wi and νi remain constant over the time interval (ti, ti+1]. Also, an uncertain prior knowledge
about the initial point x0 is available, which we denote by x0. We use the estimated state x1 at
t− 1 as the prior knowledge, namely x0. This estimated state would be out of the moving window
at the next sampling time, t.
The nonlinear dynamic optimization in (18) makes MHE slow. In oder to accelerate the opti-
mization computations, the following section employs the Carleman linearization and approximates
the nonlinear system given by Eq. (17) in a bilinear form.
13
Carleman Linearization
Using Taylor expansion the nonlinear system in Eq. (17) can be represented in a polynomial form:
x =
∞∑k=0
Akx[k] +Nw
y = x1 + ν
(19)
where Ak = 1k!∂kf∂xk|x=0, and x[k] denotes the kth Kronecker product calculated recursively from
x[k] = x ⊗ x[k−1] and x[0] = 1. Carleman linearization method truncates this expansion at the ηth
term and also defines an augmented state vector as follows:
x⊗ =
[xT x[2]T · · · x[η]T
]T(20)
This notation helps us present the approximated system in a compact bilinear form:23
x⊗ 'Ax⊗ +Ar +n∑l=1
[Dlx⊗wl +Dl0wl]
y 'Cx⊗ + ν
(21)
where A,Ar, Dl and Dl,0 matrices have the following structure:
A =
a1,1 a1,2 · · · a1,η
a2,0 a2,1 · · · a2,η−1
0 a3,0 · · · a3,η−2
......
. . ....
0 0 · · · aη,1
, Ar =
a1,0
0
0
...
0
14
Dl =
0 0 · · · 0 0
dl,2 0 · · · 0 0
0 dl,3 · · · 0 0
......
. . ....
...
0 0 ... dl,η 0
, Dl0 =
dl,1
0
0
...
0
ai,k =i−1∑α=0
I [α]n ⊗Ak ⊗ I [i−1−α]
n
dl,i =
i−1∑α=0
I [α]n ⊗Nl ⊗ I [i−1−α]
n
and Nl is the lth row of matrix N .
Note that with the current structure of x⊗, some states are repetitive. This causes an unneces-
sary increase in the system’s dimension. To avoid the excessive computation efforts, we eliminate
the repetitious states and modify the corresponding linearization matrices. Therefore, the dimen-
sion of the bilinear system reduces to:
r =
η∑i=1
(n− 1 + i
n− 1
)
Also, in general, the relation between the states in definition (20) may be lost during the system’s
time evolution. As a result, it is required to reset the augmented state vector x⊗ using the original
state variables.31 However, MHE updates x0 at every sampling time and calculates the extended
state using the updated original state variables.
Simulation
In this section, to implement the proposed MHE design, we first evaluate the performance of the
method of moments. To perform this evaluation, we use the result obtained by cNMC simulation
as a benchmark and compare it with reduced-order system derived by method of moments for
both batch and continuous processes. For the batch and continuous processes we consider 5000
15
particles in the Monte Carlo simulation. This number was seen to be sufficient for obtaining the
expected values in these stochastic phenomena, as when the simulations are repeated the results
do not change. We assumed k0 = 0.01 in all the simulations. However, at every sampling time it
is required to obtain kave, which makes the simulation very slow for this population of particles.
To deal with this issue, we estimate kave using only the random particles considered during the
selection of the reacting particles for the next event. The simulation assumes that initially half of
the particles are only composed of solute, and the other half is only solvent. Additionally, at the
beginning, all the particles have a mass equal to 1 and the number of particles in the unit volume
is 1. For the continuous process, α = 0.5, Cf = 2 and a uniform feed flow is assumed with particles
of characteristic size r = (1, 0.1). Lastly, the moments for the cNMC simulation at any sampling
time are calculated according to the following expression.
Mij =1
V
N∑l=0
pilsjl (22)
where (pl, sl) are the characteristic size of the ith particle.
Figure 3 and Figure 4 show the moments obtained by both the methods in the batch process and
the continuous process, respectively. The reduced model predicts the system’s behavior properly.
Also, the results show that only the first three terms of Taylor expansion suffice to approximate
the fractional moments as a function of the state variables of the system.
In Figure 3, we observe that the zeroth order moment monotonically decreases and finally
converges to 1 particle per unit volume. This is expected, as the batch process does not reach
an equilibrium and the particles continue merging and growing until only one particle remains.
Furthermore, M10 and M10 are seen to remain constant in Fig. 3. This results from the conservation
of solvent and salute mass in the reactor.
Figure 4 demonstrates the simulation results for the continuous process. There are clearly two
phases in the response plots of all the moments in this figure. First, there is a transient phase
which lasts an interval proportional to the residence time of the reactor. Next, all the moments
reach their corresponding steady state level after all the transient effects have died out.
16
Switching our focus to the estimation problem, we only consider continuous processes. The
number of the particles in the output flow, M00 can be measured. This is because in practice
online measurement of the other output properties moments is not as easy. During the simulation
of our estimator’s performance, we assume the feed flow concentration, Cf , is corrupted by a white
noise signal uniformly distributed over the interval [−0.5, 0.5]. Also, there is another zero-mean
uniformly distributed white noise signal affecting the output measurements with a maximum value
of 0.025. The MHE design parameters are chosen to be:
Q = 100, R = 10−4,Π0 = 0.1I
Moreover, the sampling time and the estimation horizon are chosen to be ∆T = 0.25 and N = 3,
respectively.
As discussed before, the measured output of the system is M00 (with noise degradation) shown
in Fig. 5.A. Additionally Fig. 5.B shows that the estimation error is very small in comparison with
the measurement and process noise. Also, Fig. 6 presents the estimated and true blending degrees
using Eq. (16). This figure shows that small changes in concentration may have an important effect
on ε, which necessitates design of a controller in parallel with this estimator.
The estimation was performed in MATLAB using a 3.4 GHz Intel Core i7-3770 processor. The
simulation used the fmincon function to find the optimal estimation variables at each sampling step.
Four different variations of the estimation problem were considered, with all of them giving the same
results as those in Fig. 5. In the most basic version, the nonlinear reduced-order model derived in
was directly employed. In another version, this nonlinear model was replaced with its second order
Carleman linearization. The other two versions also used the Carleman linearized model. However,
instead of relying on fmincon to numerically approximate the gradient and Hessian of the problem’s
objective function, these quantities were accurately calculated using analytical expressions. The
two first cases were solved using the ‘trust-region-reflective’ algorithm, whereas, the‘interior-point’
algorithm was used in the last two cases. The latter method was more efficient than the former
as it benefits from knowledge of the gradient of the cost function. Table 2 reports the simulation
17
runtimes for all these four cases.
According to this table, use of the Carleman linerized model and analytical calculation of the
derivatives of the optimization problem’s objective function, together, accelerates the simulation
more than 8 times. This is while the optimal estimates are very minimally affected and the method
has almost the same performance level as the nonlinear MHE.
Conclusions
This article addresses the problem of state estimation in a coagulation process. The distribution
of particles in this process is described as a function of the particles’ mass and composition. The
particle balance for this system results in an integro-differential equation which does not have a
closed form analytical solution in general.
Using the method of moments, the original model is represented as an ODE set, where the states
are distribution moments of the particles. This representation enables the application of estimation
methods to the coagulation process. This work also exploited Taylor expansions to approximate
some moments in terms of other moments to guarantee the closure of the reduced-order model.
The simulation results of the ODE set are seen to be in agreement with the results of Monte Carlo
simulations.
Blending degree is one of the important characteristics of the process, which can be directly
expressed as a function of the reduced order model of the process. However, there are noise signals
acting on the process dynamics and affecting the measurements of the system’s output. Therefore,
MHE is a proper approach to estimate these states, while taking into account the constraints of
the system. However, this method needs significant online numerical computational power to solve
the associated dynamic optimization in real time. This manuscript used Carleman linearization to
approximate the nonlinear continuous system in a bilinear form. This form admits an analytical
solution, which allows for the analytic calculation of the gradient and Hessian matrix of the cost
function with respect to the decision variables. The results showed MHE combined with Carleman
linearization reduces the running time by more than 8 times in comparison with the nonlinear
18
MHE.
Acknowledgments
This work was supported by the National Science Foundation, CBET Award 12-634902. Also, the
authors would like to thank Themis Matsoukas for helpful discussions on Monte Carlo simulations
of bi-component coagulation processes.
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21
Appendix
This Appendix, derives the two ODE sets approximating the batch and continuous coagulation
processes, by incorporating Eq. (13) in Eq. (1) and (14), respectively.
Batch Process
Here, to simplify the equations, we scale M10 to unity and therefore M01 = φ. Also, we define the
state vector of the reduced model in the following form:
x =
[M00 M11 M20 M02 M12 M21 M22
]T
Consequently, the expression for evolution of the moments is given by the following ODE set:
We assume the noise vector in a small enough time interval is constant. Hence, the bilinear system
derived by Carleman linearization method has an analytical solution in the interval [ti, ti+1]:
xi,⊗ = exp(∆TAi)xi−1,⊗ + A−1i (exp(∆TAi)− I)(Ar +
m∑j=1
Dj0wi,j) (26)
where ∆T = ti − ti−1, Ai = A+∑m
j=1Djwi,j and wi =
[wi,1 wi,2 ... wi,m
]T.
Consequently, using chain rule the first derivatives of the estimation error, vi with respect to
the elements of x0 and wi vectors are given by:
∂vi∂x0,k
= −CEiEi−1...E1∂x0,⊗∂x0,k
(27)
and,
∂vi∂wp,k
= −CEiEi−1...Ep+1
(∂Ep∂wp,k
xp−1
+ [A−1p
∂Ep∂wp,k
+∂A−1
p
∂wp,k(Ep − I)](Ar +
m∑j=1
Dj0wi,j) + A−1p (Ep − I)Dk0
) (28)
where Ei = exp(∆TAi) and p ≤ i. Also, the estimation error is independent to the noise signal in
future and therefore, for p > i,∂vi∂wp,k
= 0.
Similarly, the second order derivatives of vi are available analytically:
∂2vi∂x0,l∂x0,k
= −CEiEi−1...E1∂2x0,⊗
∂x0,l∂x0,k(29)
∂2vi∂x0,l∂wp,k
= −CEiEi−1...Ep+1∂Ep∂wp,k
Ep−1...E1∂x0,⊗∂x0,k
(30)
25
∂2vi∂wp,l∂wp,k
= −CEiEi−1...Ep+1
(∂2Ep
∂wp,l∂wp,kxp−1 +
[∂A−1p
∂wp,l
∂Ep∂wp,k
+∂A−1
p
∂wp,k
∂Ep∂wp,l
+ A−1p
∂2Ei∂wp,l∂wp,k
+∂2A−1
p
∂wp,l∂wp,k(Ep − 1)
](Ar +
m∑j=1
Dj0wi,j)
+[A−1p
∂Ep∂wp,k
+∂A−1
h
∂wp,k(Ep − I)
]Dl0 +
[A−1p
∂Ep∂wp,l
+∂A−1
p
∂wp,l(Ep − I)
]Dk0
),
(31)
and
∂2vi∂wp,l∂wq,k
= −CEiEi−1...Ep+1∂Ep∂wp,l
Ep−1...Eq+1
(∂Eq∂wq,k
xq−1
+ [A−1q
∂Eq∂wq,k
+∂A−1
q
∂wq,k(Eq − I)](Ar +
m∑j=1
Dj0wi,j)
) (32)
where q < p ≤ i.
Using information provided by Eq. (27-32) in form of gradient and Hessian accelerated the
dynamic optimization in (18).
26
Table 1: Possible events in a continuous process
Event Rate
Coagulation Rcoag = N2/(2V kave)
Inlet stream Rin = αCfV/n
Outlet stream Rout = αN/n
27
Table 2: Run Time in the Simulations
gradient Hessian 2ndorder nonlinear
No No 21.0(s) 47.7(s)Yes No 6.4(s) -Yes Yes 5.7(s) -
28
Figure 1: Two possible scenarios in coagulation process for particles in a specific range
29
Figure 2: cNMC algorithm
30
0 5 10 15 200.7
0.8
0.9
1
time
M00
Monte CarloMethod of Moments
0 5 10 15 200
0.5
1
1.5
2
time
M12
0 5 10 15 200.5
1
1.5
2
2.5
time
M21
0 5 10 15 200.4
0.5
0.6
0.7
0.8
time
M02
0 5 10 15 200.5
1
1.5
2
time
M20
0 5 10 15 200.25
0.5
0.75
1
time
M11
0 5 10 15 200
1
2
3
4
5
time
M22
Figure 3: Comparison of method of moments and cNMC simulation results for a batch process; solidlines and circles represent the moments obtained by method of moments and cNMC simulation,respectively. 31
0 5 10 15 201
1.2
1.4
1.6
1.8
2
time
M00
Monte CarloMethod of Moments
0 5 10 15 20
1.2
1.6
2
time
M10
0 5 10 15 20
0.25
0.5
time
M01
0 5 10 15 200.2
0.3
0.4
0.5
time
M11
0 5 10 15 20
1.2
1.8
2.4
time
M20
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
time
M02
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
time
M12
0 5 10 15 200.25
0.3
0.35
0.4
0.45
0.5
time
M21
Figure 4: Comparison of method of moments and cNMC simulation for the continuous process; solidlines and circles represent the moments obtained by method of moments and cNMC simulation,respectively. 32
A.
0 1 2 3 4 5
time
1.8
1.9
2M
00
TrueMeasuredEstimated
B.
0 1 2 3 4 5
1.9
2
2.1
time
M10
True Moment
Estimated Moment
0 1 2 3 4 5
0.19
0.2
0.21
time
M01
0 1 2 3 4 50.22
0.23
0.24
time
M11
0 1 2 3 4 52.2
2.3
2.4
time
M20
0 1 2 3 4 50.022
0.023
0.024
0.025
time
M02
0 1 2 3 4 50.29
0.31
0.33
time
M21
0 1 2 3 4 5
0.03
0.032
time
M12
0 1 2 3 4 50.05
0.052
0.054
time
M22
Figure 5: (A) True, measured and estimated first moment in the continuous process; solid black,dashed black and gray lines represent the measured, estimated and true outputs, respectively. (B)Moments estimation using Carleman linearization in the presence of noise on the feed concentration(Cf ) and the output measurements (M00); black and gray lines represent the estimated and truemoments, respectively.
33
0 1 2 3 4 52.2
2.3
2.4
2.5
time
ε
Figure 6: True and estimated blending degree in the continuous process; black and gray linesrepresent the estimated and true ε, respectively.
