LOW DENSITY POLYETHYLENE GRADE TRANSITION CONTROL USING NEURAL WIENER MODEL PREDICTIVE CONTROL WITH SOFT SENSOR

PI controller presented a sluggish behavior whereas a shorter settling time and minimal
overshoot were observed from the QDMC performance.

Pladis and Kiparissides (1998) implemented QDMC for controlling a two-zone
LDPE autoclave reactor. A multilayer control hierarchy approach was applied in their
study, where two control layers were applied to the autoclave reactor control. The first
layer was the regulatory control managed by the PI controller. The purpose of the
regulatory control layer was to control the reactor temperature by manipulating the
mixture’s initiator flow rate and CTA flow rate. The second layer was the supervisory
control, managed by QDMC to control polymer production and quality. Here, the
QDMC was utilized to provide optimal setpoint values for the PI controllers of the first
zone temperature and CTA concertation. Based on the reactor start-up and feed
disturbance tests, the QDMC managed to control the reactor within the specified
conversion and quality set point.

A type of model-based controller (MBC), which is called Extended Predictive
Self Adaptive Control (EPSAC), was used by Anghelea and De Keyser (2001) for
multi-zone LDPE tubular reactor control. The EPSAC is a control strategy that utilizes
an adaptive long-range prediction of the process output to calculate the controller’s
action. The prediction is made using a black-box model of the process dynamics. The
EPSAC controller was applied to control peak temperature in the first zone and weight
average degree of polymerization at the third zone outlet. The controller’s objective
function was formulated into two approaches: solidary (combined control vectors) and
selfish (separate control vectors). Based on the performance tests, both control
approaches produced comparable results and can outperform the PI controllers.

25

Naidoo et al. (2007) discussed several technical and practical issues regarding
the nonlinear model predictive control (NMPC) application in industrial continuous
polymer manufacturing processes, including the LDPE tubular reactor plant. In the
plant, a proprietary NMPC was implemented in a cascade structure control scheme as
the master quality controller. The slave composition controller was manipulated to
regulate the reactor’s temperature and CTA flow rate. The NMPC process model was
identified using the bounded derivative network (BDN) model; an extended neural
network model with an integral analytical function (Turner and Guiver, 2005). The
implementation of NMPC in the plant demonstrated encouraging results in minimizing
grade transition time, minimizing product quality variability, maximizing production
capacity, and reducing raw material consumptions.

Jacob and Dhib (2012) developed the NMPC algorithm with unscented Kalman
Filter (UKF) for a multi-zone multi-feed LDPE autoclave reactor. Here, the UKF was
used for online estimation of the plant states and disturbances. The NMPC algorithm
used in this work consisted of two subcomponents, namely, the target calculator and
regulator. First, the target calculator was used to obtain a feasible new state and input
targets based on the desired set point. The regulator then calculated an optimal open-
loop control trajectory to achieve the new targets, which was obtained from the previous
target calculation. In their study, the performance of the UKF-NMPC controller was
faster in rejecting unmeasured disturbances than the KF-LMPC.

The current industrial control application of the LDPE tubular reactor in the
literature was investigated by Skålén et al. (2016). In their research, a proprietary
NMPC framework was developed to control product quality during grade transitions.
The NMPC was equipped with a detailed physical model of the process, which included

26

recycling streams, delays, compressor dynamics, and uncertainties in reaction kinetics.
A simple decentralized update scheme using a correction factor was used to update the
model. The NMPC was used to control polymer MFI, functional groups content, and
co-monomer contents by manipulating the CTA and co-monomer feed flow rates. The
controller algorithm was based on the SQP method with constraints on the input MVs
and the rate of change of MVs. The controller was programmed in FORTRAN and
installed on a dedicated server and communicated with the distributed control system
(DCS) through a process historian database. The application of NMPC in the LDPE
plant managed to minimize the grade transition time and off-spec products via faster
and improved control compared to manual transition.

A summary of the control schemes mentioned earlier is presented in Table 2.1.
Based on Table 2.1, most of the control studies for the LDPE tubular and autoclave
reactor implement the MPC control scheme. The MPC controller is preferable in the
industry due to its inherent multivariable control, explicit constraints handling, and
availability of extensive commercial tools (in modeling and controller implementation)
(Piche et al., 2000). Based on a recent study in the LDPE reactor control schemes, the
application of NMPC is more common than LMPC. This trend also coincides with
NMPC applications for polymer reactor control elsewhere (Nogueira et al., 2020; Qin
and Badgwell, 2003). The ability to handle a strong nonlinearity process is one of the
advantages of NMPC over LMPC (Seki et al., 2001). In the NMPC, the objective
function is commonly solved using Sequential Quadratic Programming (SQP). Since
the LDPE polymerization process exhibits a nonlinear behavior, a nonlinear
programming method such as SQP is more appropriate than Quadratic programming
(QP), which is typically used by LMPC.

27

Table 2.1 Review of LDPE high-p

No Authors Reactor Control Model MV se
(Year) Type strategy type

1 Singstad et AC MVNLC - F_ mono
al. (1992)

2 Ham and AC FSF - F_i
Rhee (1996) AC
AC
28 TR

Berber and LMPC LEM F_mono
3 Coşkun

(1996)

4 Pladis et al. LMPC LEM F_CTA
(1998)

Anghelea

5 and De LMPC LEM F_ mono
Keyser

(2001)

pressure polymerization control scheme

election CV selection App. NL MFI
analysis estimation

o; F_init T_reactor I - -

init T_reactor S BD -

o; F_cal T_reactor I MSS -

A; F_init MFI; Conv S - Y

o; F_CTA DP; T_reactor S - -

6 Naidoo et al. TR NMPC NLEM F_CTA
(2007)

7 Jacob and AC NMPC FPM F_mono
Dhib (2012)

8 Skålén et al. TR NMPC FPM F_CM;
(2016)

29 9 This work TR NMPC NLEM F_CTA

*Key: App. = Application; AC = Autoclave; TR = Tubular; MVNLC = m
Linear MPC; NMPC = Nonlinear MPC; FPM = First principle model; LE
F_mono = Monomer flowrate; F_init = initiator flowrate; F_CTA = CTA
temperature; MFI = Melt flow index; DP = Degree of polymerization; GI
concentration co-monomer; C_FG = concentration functional group; Con
MSS = Multiple Steady state; DNL = degree of nonlinearity; NLI = Nonl

A; F_init MFI; GI I -Y

o; F_init T_reactor; MWW S - -

F_CTA MFI; C_CM; I -Y
C_FG

A; F_init MFI; Conv S DNL; NLI; Y

multivariable nonlinear control; FSF = Full state feedback; LMPC =
EM = Linear empirical model; NLEM = Nonlinear empirical model;
A flow rate; F_CM = Co-monomer flowrate; T_reactor = Reactor
I = Gloss index; MWW = Weight average molecular weight; C_CM =
nv = conversion; I = industry; S = simulation; BD = Bifurcation diagram;
linearity index; Y = Yes;

The application of nonlinear empirical modeling for the NMPC process model
shown in Table 2.1 is still limited. Although the FPM can provide accurate estimation,
the development of such a model requires extensive cost, maintenance (i.e., coping with
the process uncertainties), and specific for a single process only (Piche et al., 2000).
An appealing alternative for this dilemma is the application of an empirical model using
the nonlinear model identification technique (Schoukens and Ljung, 2019). One of the
available nonlinear model identification techniques is block-oriented modeling
(Schoukens and Tiels, 2017).

Unlike typical black-box modeling, the block-oriented modeling approach is
more transparent due to its straightforward physical interpretation based on its
combined block gains (Lawryńczuk, 2019). In addition, the block-oriented model
identification technique is straightforward, requires low computational effort, and easy
to incorporate a priori process knowledge (Shafiee et al., 2008). The block-oriented
model class comprises a wide range of model configurations, which involve linear
dynamic systems and nonlinear static elements. One of the widely implemented block-
oriented models is the Wiener model. This model has been applied in many modeling
case studies and has displayed the capability of describing a broad class of nonlinear
systems and processes (Janczak, 2005).

Most of the earlier control schemes for the LDPE polymerization process, as
shown in Table 2.1, focus on tight temperature control by following a predetermined
product recipe to achieve the desired end-use properties. The reactor control target has
recently shifted from controlling reactor temperature to controlling polymer quality
such as MFI. One of the reasons for this transformation is due to the improving
development of online analytical equipment especially the rheometer, which is used to

30

measure MFI. The ability of current online rheometer such as the capillary rheometer
(Kamykowski, 2018; Whelan and Brydson, 2002) to produce faster and reliable MFI
measurements is an advantage for LDPE reactor control. Moreover, due to the current
polymer production trend, i.e., reducing off-spec polymer and maximizing production,
polymer quality is chosen as a direct control objective rather than tight control of
operating parameters (Ohshima and Tanigaki, 2000).

In order to measure the polymer MFI in the plant, an online rheometer and
laboratory testing are required, which are prone to measurement delays (Skålén et al.,
2016). Thus, the application of soft sensors to predict the polymer MFI (Sharmin et al.,
2006) can improve the MPC performance. Nonetheless, most researchers have not
explicitly considered the polymer conversion variable, which can influence the
production throughput. Current control practices emphasize achieving optimal
economic performance, which includes product quality and quantity (Ellis et al., 2014).

As shown in Table 2.1, several researchers have conducted the nonlinearity
analysis of the LDPE polymerization process by using a bifurcation diagram and
multiple steady state analyses. This procedure is essential to identify the nonlinear
behavior of the process. However, a more straightforward method can be performed
using the degree of nonlinearity analysis, which is based on process observations
(Pearson, 2003). In addition, the nonlinearity index (NLI) measurement can also be used
to evaluate the process behavior based on a statistical approach (Uddin et al., 2018).

Based on the earlier discussion, the MPC performance can be influenced by its
optimizer, process model, and soft sensor (for delayed process) selection. In the
industry, the application of NMPC with gradient-based optimization is favored over the
complex MPC, such as stochastic or robust MPC (Mayne, 2016). Such MPC needs to

31

solve complex optimal control problems online, requiring more processing time and
computing effort (Lee, 2011). A comparable situation can be observed in Table 2.1, as
the NMPC implemented in the LDPE plant uses the SQP technique as the optimizer.
The discussion on the MPC process model and soft sensor selection is continued in
section 2.4.2 and section 2.4.3.

2.4.2 Process model

The success of any model-based controller such as MPC depends on the
accuracy of its process model. For a nonlinear case such as the polymerization reactor,
the process model must adequately capture the dynamics of the process to provide an
accurate estimation of the actual process during online control. One of the aspects that
can affect the sustainability of the MPC performance in the industry is the ease of
identifying (or developing) the model (Forbes et al., 2015). Based on the authors, the
poor workflow and high complexity of the model identification (or development)
process can overwhelm the occasional inexpert industrial personnel, which can create
opportunities for mistakes to be made. These challenges often occur during the MPC
maintenance in the plant, where it needs to be re-identified, usually by non-expert
personal. Moreover, a complex identification (or development) procedure may produce
a poorly developed process model that can affect the sustainability of the MPC
performance. The non-intuitive model identification (or development) process can also
hamper the user’s confidence and motivation to update the process model as often as
needed (Forbes et al., 2015).

Thus, to overcome some of these issues, the application of nonlinear model
identification techniques is considered to be more practical for industrial applications

32

(Schoukens and Ljung, 2019). A nonlinear model identification technique refers to
developing a parametric-based input-output model using a nonlinear identification
algorithm. One of the well-known nonlinear model identification techniques is the
block-oriented model (Schoukens and Tiels, 2017). Unlike black-box modeling, the
block-oriented modeling approach is more transparent due to its straightforward
physical interpretation, based on a combination of block gains (Lawryńczuk, 2019). In
addition to that, the identification method for the block-oriented models is
straightforward, requires low computational effort, and is able to incorporate priori
process knowledge (Shafiee et al., 2008).

The block-oriented model class comprises a wide range of model
configurations, which involve a linear dynamic system and a nonlinear static element.
The most widely implemented block-oriented models are the Wiener and the
Hammerstein model. Both models have been applied in many case studies modeling
and have displayed the capability of describing a broad class of nonlinear systems, e.g.,
biology and medicine (Janczak, 2005). Nonetheless, the Wiener model is more
acceptable than the Hammerstein model due to its transient response and the dynamic
modeling capabilities of its nonlinear part (Pearson and Pottmann, 2000). Previously,
the application of the Wiener model based NMPC has been addressed by several
researchers in the literature (Sudibyo, 2018). However, its application in polymerization
reactor control is still new.

Jeong et al. (2001) and Song and Rhee (2004) tested the application of Wiener
NMPC for continuous methyl methacrylate (MMA) polymer conversion and weight-
average molecular weight control. The Wiener model was developed using state space
and polynomial models for the dynamic linear and nonlinear static block, respectively.

33

The developed Wiener model NMPC showed a satisfactory control performance of the
polymerization reactor compared to a linear MPC during simulation and experiment
tests.

Shafiee et al. (2008) utilized piecewise linear functions (PWL) for the Wiener
model’s nonlinear part. Similarly, the dynamic linear element was identified using state
space modeling technique. The researcher mentioned that PWL is a powerful tool for
modeling and can be inverted to generate an inverse model. The piecewise linear
Wiener model was embedded with the NMPC control scheme to control the solution
polymerization of MAA in a jacketed CSTR. The simulation results presented a notable
performance of the NMPC with fast response in setpoint tracking and unmeasured
disturbance rejection compared to the LMPC.

Arefi et al. (2008) implemented a Wiener model-based NMPC using state space
(SS) and neural network (NN) model for controlling a plug-flow reactor during the
chlorination of propylene process. This process involves the reaction between the
propylene and chorine gas inside the reactor to produce allyl chloride. In their work, the
model identification procedures were performed using the SLICOT toolbox in Matlab.
Based on the test results, the Wiener NMPC exhibited excellent responses in setpoint
tracking and disturbance rejection compared to the LMPC and PI controllers. This
showed the advantage of applying a nonlinear controller over a linear controller.
However, their work only covered reactor temperature control, which is still insufficient
in polymerization quality control.

Lawryńczuk (2013) developed the Wiener model using a polynomial and NN
model for the NMPC control scheme. The NMPC control algorithm developed with
online linearization was used to control the free radical polymerization of the MMA

34

polymer inside a jacketed CSTR. The significant nonlinearity of the polymerization
process had made the linear model inaccurate and, in consequence, ruined the
performance of the LMPC. Instead, the Wiener model managed to model the
polymerization process properly. The NMPC was able to track the polymer molecular
weight set point with good accuracy and computational efficiency.

From this brief review, it can be noted that several Wiener-based NMPC have
been successfully applied in polymerization reactors. Moreover, from the results of the
controller performance, the NMPC with Wiener model has shown excellent results in
controlling the reactors compared to the LMPC. However, there has been no application
of the Wiener-based NMPC in the LDPE tubular reactor.

2.4.3 Soft sensor

In the polymer manufacturing process, the LDPE quality is determined by its
melt flow index (MFI). However, due to the challenging engineering activity and the
complexity of the process, MFI is commonly evaluated offline using laboratory testing,
which consumes almost two hours to complete (Bafna and Beall, 1997). Thus, this will
cause a time delay in the quality control system since there is no new quality
measurement during this period. As an alternative, the installation of an online analyzer,
such as near-infrared spectroscopy or ultrasound for measuring the melt index, is
performed (Coates et al., 2003). However, the high installation cost, maintenance effort,
and sensor reliability have limited its application in real plants (Cheng and Liu, 2015).
Moreover, the existence of a physical delay in the online analyzer measurement due to
its location in the production line has produced an inevitable delay in the readings
(Sharmin et al., 2006). The wide application of distributed control system (DCS) in the

35

industrial process has made a large amount of process data to be routinely recorded
(Kadlec et al., 2009). In such cases, developing a soft sensor based on the measured
data using data-driven techniques is a feasible option. This recorded data, which is also
referred to as historical data, can serve as the input signals for the soft sensor.

The term “soft sensor” is a combination of two words; “software” which refers
to the computer program that is used to develop the model, and “sensors” due to its
similar function with typical hardware sensors. In general, a soft sensor, virtual sensor,
or inferential estimator is a technique of estimating process parameters such as quality
measurement, process properties, and faults in various applications when a hardware
sensor is unavailable or unsuitable for making direct measurements (Abeykoon, 2018).
Moreover, soft sensors are capable of estimating process states that are challenging to
measure using hardware sensors due to considerable measurement delays, technological
restrictions, or high instrument expenses (Ahmad et al., 2020).

Based on Kano and Fujiwara (2013), the significant benefits of using soft
sensors are; to stabilize the product quality using online estimation, reduce energy and
material consumption through optimal operation, and cross-check the hardware sensor
measurement comparing it with soft sensors. On the other hand, the application of soft
sensors can also be quite challenging due to the need for expert knowledge, effort, and
time required to develop it. Besides, the quality of the training/validation data and the
limitation of the soft sensor applications due to its specificity to the collected data has
become an issue for soft sensor development (Abeykoon, 2018).

Soft sensors are commonly applied for real-time process monitoring, modeling
and control, fault detection and estimation, and process diagnostics. These sensors have
been used in various industrial applications such as polymerization reactors, distillation

36

columns, cement kilns, food processing, and paper/pulp production industry to estimate
the product quality parameters (Fortuna et al., 2007). Bremner et al. (1990) studied the
empirical relationship between the MFI and the weight average molecular of HDPE and
LLDPE. From their work, such a relationship can be established for linear polymers
with a similar molecular polydispersity index (PI) and processing history. They
concluded that for branched polymers (i.e., LDPE), the condition is more complex, and
no general relationship can be obtained unless the rheology state of the polymer can be
determined. A similar observation is also reported by Azmi et al. (2019) for their quality
model development of the LDPE tubular reactor using commercial MFI information
from the literature.

Based on the situation mentioned earlier, the selection of soft sensors for the
LDPE polymerization process must consider the complicated relationship of the MFI
with other process parameters. In the past, the LDPE quality has been inferred using a
multivariate statistical method, which is the partial least square (PLS) model by
MacGregor et al. (1992). The PLS model provides a systematic approach for developing
an inferential model from a larger number of correlated variables. A nonlinear
transformation of the variables is made to adapt the linear structure of the PLS model
with the nonlinear LDPE process. Their final PLS model had used six latent vectors that
explained 93% of the variations in polymer quality, with a measurement error of 10%
based on the residual variance.

Kumar et al. (2003) performed a similar approach on the multivariate statistical
technique using the principal component analysis (PCA) for a commercial LDPE/EVA
(ethylene-vinyl acetate copolymer) process monitoring. The PCA can obtain a new
relationship model, which can be explained based on fewer variables by using the

37

variable transformation method. To monitor the process, multivariate-monitoring
charts, namely Squared Prediction Error (SPE) and Hotelling’s statistic, which
condense most of the process variables into a single plot, were applied. The PCA based
monitoring scheme was reported to be able to predict multiple faults successfully.

Rallo et al. (2002) evaluated three different neural network (NN) models, which
were Fuzzy ARTMAP, Radial Basis Function (RBF) with Clustering Average, and
Dynamic RBF for estimating the MFI of an LDPE tubular reactor process. The complete
development of the hybrid models was covered in their work. A sensitivity analysis was
conducted using self-organizing maps (SOM) to select relevant process features and
reduce input variables. The initial input variables consisted of 25 process variables from
the reactor, compressor, and extruder sections. The NN model was used to estimate six
LDPE product grades clustered into three groups based on their average MFI values and
polymer densities. From the simulation results, the Dynamic RBF yielded a slightly
more accurate prediction of the MFI than NN fuzzy ARTMAP and RBF clustering
average model for single grade sensors. The results indicated that hybrid NN model-
based sensor implementation provided satisfactory reliability and accuracy in
estimating the LDPE MFI grade.

Sharmin et al. (2006) applied data-based multivariate regression methods as an
alternative solution to the limited quality measurement in the industrial LDPE autoclave
reactor. In their work, the partial least square (PLS) method was used to develop a soft
sensor for estimating the MFI properties during grade transition operations. From their
experience, the online rheometer only provided accurate measurements during steady
state conditions. However, during grade transitions, online measurement was erroneous.
To develop a composite model (i.e. estimating several polymer grades in a single

38

model), a dataset of a month was collected with the usual time length for producing any
grade of polymer that varied from two to seven days at the plant. A bias update scheme
was used to update the soft sensor prediction with current online MFI measurement.
The developed composite model was divided into high MFI and low MFI (between 2
to 7). During the grade transition simulation, the composite models can predict the MFI
appropriately. However, if the model focused on each grade separately, the performance
was poor. Thus, the composite model was only used during grade transition, while a
single grade model was used during the steady state operation.

The review earlier shows that the number of soft sensor applications in the
LDPE polymerization process is still limited. Based on a study by Kadlec et al. (2009),
neural network models such as multilayer perceptron (MLP) and recurrent network
(RN) are commonly applied techniques in the soft sensor computational learning
method after the multivariate statistical technique. Moreover, the application of the
neural network model as soft sensor in polymer monitoring and processing is also well
accepted (Abeykoon, 2018). Thus, the development of a neural network model in
estimating polymer quality parameters in the LDPE polymerization reactor can be
further explored. In addition, the input selection and pre-processing technique for soft
sensor development is a task that needs careful consideration (May et al., 2011). Thus,
an input selection methodology is needed to approach this task systematically.

2.5 Summary of review
The industrial significance of the LDPE high-pressure tubular reactor

technology has steered a number of research and development over the past 30 years
(Kiparissides et al., 2010). Most of the research focuses on developing a comprehensive
LDPE model that can predict the polymer molecular and morphology properties

39

(Dietrich et al., 2019; Peikert et al., 2019; Pladis et al., 2015). Moreover, other areas
such as process simulation (Bokis et al., 2002), control (Naidoo et al., 2007), and soft
sensor development (Rallo et al., 2002) have also shown promising results in improving
the productivity and sustainability of the LDPE process.

Polymer process simulators such as Aspen Plus have managed to provide a
quick and reliable solution for developing an accurate FPM for the LDPE
polymerization process. Additionally, the Aspen Plus simulation model has been
reported to be successfully validated with LDPE plant data (Bokis et al., 2002). The
developed simulation model can be used in many areas, such as control, optimization,
and safety analysis. Moreover, the application of the Wiener model in the NMPC
control scheme has presented a convincing motivation for it to be implemented in the
LDPE tubular reactor control. The advantage of using a nonlinear empirical model such
as the Wiener model over the FPM has been elaborated in section 2.4.2 earlier.

Furthermore, the development of a soft sensor model can assist the NMPC in
controlling the LDPE process by estimating the delayed quality measurements. The
application of soft sensors in the LDPE tubular reactor control has been mentioned in
Skålén et al. (2016) and Naidoo et al. (2007) work. However, only minimal explanation
on this topic is provided. Thus, a detailed study on the soft sensor model development
and the effect on control performance in the LDPE tubular reactor is necessary.

40

CHAPTER 3
METHODOLOGY
3.1 Research Outline
The flow chart of the overall research methodology of this present work is
shown in Figure 3.1. The details of each step are elaborated in subsequent sections.

Research
Objective 1

Research
Objective 2

Research
Objective 3

Research
Objective 4

Figure 3.1 Research methodology flow chart

41

3.2 Development of steady state model using Aspen Plus
The initial procedure in developing a steady state model in Aspen Plus is to

complete the Properties section, which includes specifying the components used in the
simulation and selecting the simulation property method. For the polymer simulation,
the polymer segment definition and polymerization attribute have to be specified. For a
more accurate simulation, the heat of polymerization can be specified in the pure
component parameters. After the Properties section is completed, the next section is the
Simulation section.

Here, the process flowsheet is developed using predefined unit operation blocks
that are available in Aspen Plus. For a simulation that requires heat transfer, especially
for the reactor jacket, the heat transfer coefficient information is necessary. In order to
simulate the polymerization process, the polymer reaction mechanisms and kinetic
information are required. After the simulation model is finalized, it needs to be
simulated and validated. These procedures are illustrated in Figure 3.2, and these steps
are further explained in the following subsections.

Specify Specify EOS Specify Heat of
components Polymerization

Develop process Determine Heat Determine kinetic
flowsheet Transfer mechanisms

Coefficient, U

Determine kinetic Model Completed
parameters Validation

Figure 3.2 Steps for developing steady state model using Aspen Plus
42

3.2.1 Tubular reactor model
The reactor model used in this study was based on Asteasuain and Brandolin

(2008). The reactor model was initially developed by the researchers using the
gPROMS simulation software. It was later expanded to include the molecular weight
distribution (MWD) (Asteasuain and Brandolin, 2009) and bivariate distribution of
MWD with long-chain branching (LCB) and short-chain branching (SCB) (Dietrich et
al., 2019). The reactor model was validated based on industrial operation data and
product samples (Asteasuain et al., 2001a; Brandolin et al., 1996). A schematic diagram
of the tubular reactor model is shown in Figure 3.3.

Figure 3.3 A schematic diagram of the tubular reactor (Agrawal et al., 2006)
Based on the figure, the tubular reactor has a similar configuration to a typical

industrial reactor with a length to diameter ratio of over 20000 (Bokis et al., 2002). The
reactor operates at a temperature ranging from 70°C at a reactor inlet and can achieve
320°C at the peak (Asteasuain and Brandolin, 2008). The pressure inside the reactor is
2200 bar, and the axial stream velocity is 11 m/s (Asteasuain and Brandolin, 2008). The
elevated temperature and pressure conditions are necessary for the polymerization to
occur inside the reactor. In addition, the high velocity stream is vital to ensure the gel

43

effect phenomenon can be safely avoided (Cioffi et al., 2001). The feed to the reactor
contains ethylene gas (monomer), oxygen, and traces of hydrocarbons.

Moreover, a chain transfer agent (CTA) (also known as telogen) is also added
at the feed stage. The addition of the CTA will control the excessive production of large
polymer molecules, which can escalate the polymer mixture viscosity. Thus, CTA is
typically used to control the polymer’s final grade, which relates to the Melt Flow Index
(MFI). Here, propane is used as the CTA with n-butane as inert. Initiators are used to
initiate the polymerization process by decomposing into free radicals that bond with
ethylene molecules to form active polymer chain molecules. Tert-butyl peroxypivalate
(TBPPI) is selected as the first initiator, and tert-butyl 3,5,5 trimethyl-peroxyhexaonate
(TBPIN) is selected as the second initiator (Agrawal et al., 2006; Brandolin et al., 1996).

Furthermore, the reactor is divided into five zones. The first two zones are used
for preheating the reactor mixture to the optimal temperature for the first initiator
injection. An initiator is used to initiate the polymerization process. The initiator is
usually selected from the peroxide or azo compound based on its activation temperature
in order to achieve the desired polymer conversion within the operating temperature
range (Brandolin et al., 1996). In the third zone, the first initiator is injected to kick-
start the polymerization process by forming free radicals. At the same time, oxygen is
also decomposed to produce similar free radicals. The polymer’s chain initiation and
propagation reaction will stop after the initiator is depleted.

The fourth zone is regarded as the cooling zone. Since ethylene polymerization
is an exothermic process, the excess heat needs to be removed to match the second
initiator activation temperature. In the fifth and final zone, the second initiator is
introduced, and the polymerization reaction is resumed. The amount of the initiator

44

corresponds to the polyethylene final conversion rate. However, the uncontrolled
amount of initiator injection can lead to a dangerous reactor runaway situation.

The tubular reactor’s operating and feed conditions are tabulated in Table 3.1.
All the components used in this study are available and selected from the Aspen Plus
database. The current tubular reactor operating conditions are considered as the steady
state values inside the Aspen Plus model. The reactor jacket flow rate is determined by
the using heat transfer rate equation, :

= ∆ (3.1)

3.2.2 Equation of State (EOS)

One of the most critical aspects of process simulation is the accurate description
of the process's thermodynamic properties and phase behavior (Carlson, 1996). A
variety of thermodynamic models can be found in the open literature. These models
belong to the activity-coefficient category or the equation-of-state (EOS) category. In
the case of LDPE, the nature of the species involved (hydrocarbons, light gases) and the
high-pressure conditions inside and outside the reactor suggest the use of EOS.
Guerrieri et al. (2012) have presented a good review of EOS applications in polymer
modeling. They have concluded that EOS development for polymers remains a very
active area of research, and it is difficult to recommend a specific EOS for certain
applications.

Among popular EOS in the polymeric system are Sanchez-Lacombe (SL),
Statistical Associating Fluid Theory (SAFT), and Perturbed-chain statistical fluid

45

theory (PC-SAFT). In this study, the PC-SAFT EOS is selected due to its capability to
estimate the ethylene state and thermodynamic properties more accurately than other
EOS (Bokis et al., 2002).

Table 3.1 LDPE tubular reactor operating and feed conditions (Asteasuain et al.,
2001)

Reactor Parameters Values Unit
Length/Diameter (L/D) 27800
Internal Diameter 0.05 m
Number of Zones 5
Zone length (Lzi, i = 1…5) 60; 100; 180; 510; 540; m
Inlet temperature 76 ºC
Inlet pressure 2250 atm
Global Heat transfer coefficient
(Ui, i = 1…5) 1088.6; 1088.6; 837.4; 628.0; 196.8; W/m2K
Mean Jacket Temperature
(Tji, i = 1…5) 168; 225; 168; 168; 168; ºC
Reacting mixture density kg/m3
Monomer flow rate 530 kg/s
Oxygen flow rate 11 kg/s
CTA flow rate 6.8 x 10-5 kg/s
Inert flow rate 7.4 x 10-2 kg/s
Initiator 1 flow rate 2.2 x 10-1 kg/s
Initiator 2 flow rate 1.0 x 10-3 kg/s
1.6 x 10-4

46

3.2.3 Heat of Polymerization

The heat of polymerization is the heat required to convert one mole of monomer
to one mole of segments of a polymer chain:

∆ = − (3.4)

where H POL is the heat of polymerization, H s is the enthalpy of the segment, and

Hm is the enthalpy of the monomer. In Aspen Plus, the polymer molar enthalpy is equal

to the enthalpy of the segment for a homopolymer system:

= (3.5)

where H p is the polymer molar enthalpy. The enthalpy of the monomer, Hm , is

calculated using available standard methods in Aspen Plus. There are two categories of

methods for calculating H p in Aspen Plus, which are activity coefficient-based models

and equation-of-state based models. In the activity coefficient-based models, the

polymer molar enthalpy is calculated as follows:

(3.6)

( ) = ( ) + ∫ ( )



( ) = ( ) + ∆ ( ) (3.7)

where C l is the heat capacity of the polymer, H ig (Tref ) is the heat of formation of the
p

polymer at the ideal-gas state and at a reference temperature, Tref , (usually set to 298.15

K), and conH (Tref ) is the heat of condensation of the polymer at Tref . The model

parameters for C l (CPLVK) are acquired by data fitting or can be estimated using the
p

group contribution method in Aspen Plus. H ig (Tref ) (DHFVK) and conH (Tref )

47

(DHCON) by default is estimated by the group contribution method as used in this
study.

Typically, the DHFVK of polymer segments is fine-tuned using the heat of
polymerization data. By performing Analysis or Design Spec in Aspen Plus, the heat of
formation (DHFVK) of the polymer segment is back-calculated so that the estimated
heat of polymerization matches the literature data. Data regarding the heat of
polymerization can also be found in Brandrup et al. (1999) polymer handbook. In this
case, the heat of polymerization from Asteasuain et al. (2001b) work is 21500 cal/mol.
Thus, from the fine-tuning process, the DHFVK value obtained is -2.669 x 107 J/kmol.
The complete procedures for fine-tuning the LDPE heat of polymerization in Aspen
Plus is presented in Appendix A.

3.2.4 Process flowsheet

Based on Figure 3.3, five reactor models are used to represent the entire tubular
reactor model. These reactors represent the reactor zones, as mentioned earlier. The
RPLUG model in Aspen Plus is selected as the reactor block model, which is a rigorous
model block for plug flow reactors. It assumes that perfect mixing occurs in the radial
direction and that no mixing occurs in the axial direction. The RPLUG block can
simulate multiple phase conditions, applying thermal fluid stream and handling
chemical reactions. Thus, it is suitable for simulating commercial reactors with high
mass velocity, high Reynolds number, and catalyst. Here, the RPLUG reactor block
with counter-current thermal fluid is employed. The initiator (INIT) injection point is
placed at the beginning of Zone 3 (Z3) and Zone 5 (Z5) using the MIXER block. The
CTA feed is separated from the primary feed stream to control it as a manipulated
variable. The whole reactor's pressure drop is estimated to be around 10% from the

48

reactor inlet pressure, as commonly observed in industrial tubular reactor operations
(Bokis et al., 2002).

3.2.5 Heat Transfer Coefficient (HTC)

The equation for the overall heat transfer coefficient, , considers four
resistances in series: those of the film at the reaction mixture side, the reactor wall, the
film at the cooling/heating jacket side and the fouling build-up, , as shown in the
equation below. Note that the U coefficient depends on the variables that are calculated
as functions of the axial position x:

1 = 1 + 2.3 10 ( 0) + ℎ ( ) 1 + ( ) (3.8)
( ) ℎ ( ) 2 ( 0⁄ )

where 0 and are the jacket internal diameter and reactor internal diameter,
respectively, with as the thermal conductivity of the wall. A review of heat transfer

coefficient used in ethylene polymerization is reported by Lacunza et al. (1998).

3.2.6 Polymerization Mechanisms

The polymerization reaction mechanisms presented by Asteasuain et al. (2001b)
are used in this work. The complete polymerization mechanisms are shown from
equations 3.9 to 3.18. The polymerization mechanisms used here are similar to those
used to model the industrial LDPE process (Brandolin et al., 1996).

49

Initiator-Decomposition , → 2 (0) (3.9)
Oxygen-Decomposition 2 + → 2 (0) (3.10)
Chain Initiation (0) + → ( ) (3.11)
(3.12)
Propagation ( ) + → ( + 1) (3.13)
(3.14)
Chain transfer to polymer ( ) + ( ) → ( ) + ( ) (3.15)
(3.16)
Chain transfer to transfer agent ( ) + → ( ) + (0) (3.17)
(3.18)
Chain transfer to solvent ( ) + → ( ) + (0)

Termination by combination ( ) + ( ) → ( + )

Thermal degradation ( + 1) → ℎ ( ) + (0)
(Beta-scission) ( ) → ( )

Short Chain Branching
(Back-biting)

3.2.7 Kinetic Parameters

The polymerization kinetic parameters, which are pre-exponential factors ( ),
activation energy ( ), and activation volumes (Δ ), are taken from Agrawal et al.
(2006). The kinetic parameters follow the Arrhenius law for temperature and pressure
dependency as follows:

= o − + Δ (3.19)


50

The initiators and beta scission activation energy is fine-tuned to achieve the desired
reactor temperature profile, polymer conversion, and molecular weight (Bokis et al.,
2002). The initial values are based on the original data as in Agrawal et al. (2006) work.
These properties need to be adjusted to accommodate the differences in the modeling
technique (i.e., between Aspen Plus and Agrawal model) and to suit the validation data
(Bokis et al., 2002).

Here, the Aspen Data Fitting technique is used to retune the initiator's kinetic
activation energy to meet the reactor temperature profile, while the Aspen Design Spec
is used to achieve the preferred number average of molecular weight (MWN) by
modifying the beta scission kinetic activation energy. In addition, the initiator efficiency
is also fine-tuned to fit the desired temperature profile. Initiator efficiency represents
the fraction of generated radicals that are effective in initiating polymer chain growth
(Bokis et al., 2002). These initiator efficiency values are obtained from Brandolin et al.
(1996).

The thermal degradation kinetic from Agrawal et al. (2006) is replaced by beta-
scission kinetic since both are similar (Asteasuain et al., 2001a). The chain initiation
reaction rate is faster than the propagation reaction rate. Thus, it can be set the same or
faster than the propagation reaction rate (Bokis et al., 2002). In this work, the chain
initiation reaction rate is set to be similar to the propagation reaction rate as done by
Bokis et al. (2002). The complete kinetic parameters used in this work are tabulated in
Table 3.2.

51

Table 3.2 List of kinetic parameters used in this study (Agrawal et al., 2006)

Kinetic Parameters

Kinetic Mechanisms 0 Ea ∆ Efficiency
(1/sec) (J/kmol) (m3/kmol)
Initiator-Decomposition 0.95
(Initiator 1) 1.00 x 1014 1.32414 x 108 0.0140 0.95
Initiator-Decomposition 0.25
(Initiator 2) 1.00 x 1012 1.23931 x 108 0.0116
Initiator-Decomposition
(Oxygen) 1.60 x 1011 1.33605 x 108 -0.0121

Chain-Initiation 4.00 x 105 1.7431 x 107 -0.0168

Propagation 4.00 x 105 1.7431 x 107 -0.0168

Chain Transfer-Polymer 5.20 x 104 3.6844 x 107 -0.0190

Chain Transfer-CTA 7.00 x 104 1.8406 x 107 0.0000

Chain Transfer -Solvent 7.00 x 104 1.8406 x 107 0.0000

Beta-Scission 7.70 x 109 8.74296 x 107 -0.0100

Termination-Combination 8.70 x 108 1.5282 x 107 0.0092

Short Chain-Branch 1.20 x 1010 6.0537 x 107 0.0000

3.3 Dynamic model
A dynamic model can be obtained by exporting the current Aspen Plus steady

state model into Aspen Dynamic using the flow-driven simulation option. The flow-
driven simulation option is selected since ethylene gas at high pressure behaves

52

similarly to liquids (Stoiljkovic and Jovanović, 2019). The dynamic model would
function as a real plant in the control scheme by receiving process inputs from the
controller and producing process output online. The benefits of the dynamic model are
presented in Mcmillan (2006) work.

3.4 MFI model
Since the referred literature (Asteasuain et al., 2001b) does not provide the MFI

values, these values are correlated based on weight average molecular weight (MWW).
The relationship between MFI and MWW has been studied by several researchers
(Kazakov et al., 1974; Seavey et al., 2003; Shenoy and Saini, 1986). The MFI
correlation equation is acquired from Rokudai and Okada (1980) study:

MFI = 1.06 x 1028( ̃ . )−6.00 (3.20)

The value of ̃ represents the branching index obtained from Dietrich et al. (2019)
study, as shown in Figure 3.4. The branching index information is essential since the
LDPE polymer has complex branches, and no general relationship can be obtained
unless the rheology of the polymer is characterized (Bremner et al., 1990).

Figure 3.4 Branching index for LDPE molecular weight (Dietrich et al., 2019)
53

In practice, the MFI values can be obtained from the online analyzer and
laboratory test (Skålén et al., 2016). Thus, if the MFI values are available, then a soft
sensor can be developed based on its relationship with available operating parameters.
Moreover, Eq 3.20 requires information regarding LDPE MWW. However, LDPE
MWW information is only available from model estimation or laboratory tests using
high-temperature Gel Permeation Chromatography (GPC). In this study, the LDPE
MWW information is obtained from the Aspen Dynamic simulation model based on the
method of moments (Katz and Saidel, 1967).

3.5 Model Analysis
Based on Hernjak et al. (2004), chemical processes can be classified into several

groups according to their behavior. To determine the process behavior, they have
suggested three types of tests, which are the degree of nonlinearity, dynamic
characteristic, and degree of interaction. Based on these tests, the chemical process's
necessary insight can be obtained to guide the control strategy selection decision. In this
work, several tests have been applied, which are parametric analysis, degree of
nonlinearity (Pearson, 2003), and nonlinearity index (Uddin et al., 2018). These tests
would be further explained in the following section.

3.5.1 Parametric Analysis

The parametric analysis aims to observe and study the dynamic behavior of
selected process outputs (control variables, CV) based on the perturbation in the process
inputs (manipulated variables, MV and disturbance variables, DV). All the selected
process variables are tabulated in Table 3.3. The initiators and CTA flow rates are
selected due to their role in controlling the reactor polymerization rate and polymer

54

quality. Most of the disturbances occur in a process that originates from the feed. Thus,
various feed stream conditions are studied to identify their effect on the reactor output.
For this case study, the polymerization reaction occurs in Zone 3 and Zone 5. Thus, it
is crucial to monitor the peak temperature inside those zones for safety purposes. The
LDPE composition (or polymer conversion) and MFI are essential reactor outputs in
maintaining polymer production and quality. The measurements for the reactor outputs
are taken at the exit of the reactor. In order to observe the process response, a step test
is conducted at time 0.5 minutes for a period of five minutes for the process inputs
mentioned in an open-loop simulation.

Table 3.3 Process input-output

Process Inputs Process Outputs
Initiator 1 flow rate
Initiator 2 flow rate Peak Temperature Zone 3

CTA flow rate Peak Temperature Zone 5
Feed temperature
LDPE composition
Feed pressure Weight average molecular
Feed composition
weight
Feed flow rate

3.5.2 Degree of nonlinearity

The degree of nonlinearity test is conducted to examine the dynamic behavior
of the process and its deviation from the linear condition. This assessment is conducted
based on several tests, which are asymmetric response, harmonic response, input
multiplicity, and output multiplicity (Pearson, 2003).

55

3.5.2(a) Asymmetric response

The asymmetric response test is similar to the parametric analysis in observing
the nonlinear behavior of the process. The asymmetric behavior can be considered as a
deviation of the normal symmetry of a linear system. By using symmetrical inputs, a
system that generates asymmetric responses can be considered as a nonlinear system.
However, it is essential to note that the violation of odd symmetry is only a sufficient
condition for nonlinearity and not a necessary condition (Pearson, 2003). In this case,
initiator 1, initiator 2, and CTA flow rate are changed similar to the parametric analysis
to observe their impact in peak temperature zone 3, LDPE conversion, and MFI.

3.5.2(b) Harmonic response

The response to periodic inputs (or sinusoidal inputs in particular) in general can
show the presence and nature of system nonlinearities. Based on Pearson (2003), one
of the responses of a nonlinear system to a sinusoidal input can be described as the
super-harmonic generation, where the output has the same period as the input, but its
shape is non-sinusoidal. Here, the sinusoidal input performed by initiator 1, initiator 2,
and CTA flow rate with the amplitude of its maximum and minimum value is tested on
the tubular reactor model.

3.5.2(c) Input and Output multiplicity

Multiplicity is the existence of multiple steady states in the system. Input
multiplicity refers to systems for which one steady-state response can be produced by
more than one steady state input. Meanwhile, output multiplicity refers to systems for
which one steady state input corresponds to more than one steady-state output. In order
to detect input multiplicity, the input to the process is varied from minimum and

56

maximum operating value to observe the process output behavior (Ma et al., 2010).
Here, initiator 1 and initiator 2 feed flow rates are selected as inputs, and the outputs are
Zone 3 and Zone 5 reactor temperature. These temperature changes are observed along
the length of the reactor.

For output multiplicity, two operating steady state lines of a single input are
conducted to observe the process responses (Guttinger et al., 1997). The operating lines,
one ascending and one descending, are used to study the behavior of the output from
two different steady state lines. In this study, initiator 1, initiator 2, and CTA flow rates
are used to create the steady state operating line by using staircase input for the selected
period. The changes in peak temperature in Zone 3, LDPE composition, and MFI are
measured to observe its response.

3.5.3 Nonlinearity Index (NLI)

The nonlinearity index (NLI) was proposed by Uddin et al. (2018), serves as a
quantitative indicator to test the applicability of nonlinear model identification for a
system. The NLI technique is an improvement from the nonlinearity measure developed
by Du and Johansen (2017). Here, the NLI methodology is simplified and shown as the
following steps:

i. Generate sine wave signal as input to the process
ii. Collect the input and output response of the process
iii. Identify the number of data, N, and output means, ̅ . Calculate the process delay

or lags, τ using ‘finddelay’ command in Matlab
iv. Calculate the correlation function, :

57

2 ≈ 1 [( ( − ) − ̅ )( ( ) − ̅ )2] (3.21)

∑

=

v. Calculate the (0) and 2 2(0) based on the same equation with zero lag

vi. Calculate the NLI for each output:

= 0.5102√ − |√ ( 0 ) √ 2 ( ) 2 2(0)| (3.22)
vii. Calculate the NLI for the system with as the number of outputs: (3.23)

1

= ∏(1 + ) − 1

=1

3.6 Development of Neural Wiener MPC
A generic block diagram of the MPC workflow from Khaled and Pattel (2018)

is presented in Figure 3.5. From the figure, it is noted that the MPC depends on the
output measurement to update the observer. Then, the observer is used to predict the
plant's response using the plant model across the prediction horizon. The cost function
block refers to the objective function that needs to be solved using a real-time solver at
every sampling time. The solver solves the open-loop optimization problem by
computing the future sequences of the manipulated variables across the control horizon.
However, only the first value is sent to the plant for each time step. The new plant output
value is measured by the sensor and is sent back to the observer. Finally, the
optimization process is repeated to compute the optimal controller output to achieve the
desired target (or reference).

In this study, the development of the Neural Wiener MPC (NWMPC) follows a
similar structure to the presented MPC scheme with several modifications. Firstly, the
output measurement is used to update the process model directly without using an

58

observer since the Aspen model can provide the process state measurements. Secondly,
the current controller only uses output reference tracking and changes in the
manipulated variables in the optimizer’s cost function algorithm. In general, these two
cost functions are sufficient for the MPC to control the process (Khaled and Pattel,
2018). These cost functions are then solved using an online optimization technique to
produce the controller output’s trajectory. The main feature that differentiates a
Nonlinear MPC (NMPC) from a Linear MPC (LMPC) is the selection of the process
model type. In this work, the NMPC uses the Neural Wiener (NW) model as its process
model, which is now known as the Neural Wiener MPC (NWMPC)

59

60

Figure 3.5 Block diagram of MPC

C workflow (Khaled and Pattel, 2018)

3.6.1 Control structure

Figure 3.6 shows the implementation of the NW model in the MPC control
scheme. In the figure, the NW model functions as the ideal process condition compared
to the actual process by the Aspen Dynamic model. The difference between both outputs
is called a model mismatch, caused by process disturbance or uncertainty. This model
mismatch is sent to the controller with a set point (or reference) signal. The model
mismatch signal represents the process's current situation, whereas the setpoint signal
is the desired process condition. Both signals are computed inside the MPC algorithm,
and an optimization technique is used to generate the optimal process inputs that can
drive the process towards the desired condition.

SP Model based MV Aspen Dynamic Yp
controller with simulation model

optimizer

Neural Wiener Model
Ym +

L NL
-

model mismatch

Figure 3.6 Neural Wiener model in MPC control scheme
3.6.2 Neural Wiener Model

The Neural Wiener (NW) model development consists of selecting the model
input-output variables, generating a set of process data, selecting the model structure,
and identifying the model using Matlab system identification.

61

3.6.2(a) Model I/O selection

Based on the review in section 2.4, most of the control variables in the LDPE
reactor control strategy focus on product quality variables (i.e., Melt Flow Index or
Gloss Index) and reactor temperature. In practice, temperature control is utilized by
manipulating the initiator or/and jacket feed flow rates to maintain the reactor’s
temperature and prevent temperature runaway (Lopez et al., 1996). The initiator is
injected at various points along the tubular reactor, producing an exothermic reaction
that peaks in the temperature and then decays again until the polymer reaches the next
initiator injection point. Increasing the initiator amount would increase the polymer
reaction resulting in a higher peak temperature and increased conversion (Naidoo et al.,
2007). The product quality is commonly controlled by using the CTA flowrate by
inhibiting the production of large polymer chains and encouraging the formation of
short polymer molecules (Asteasuain et al., 2001b). A considerable number of short-
chain molecules would make the polymer easy to flow and increase the MFI value.

3.6.2(b) Data generation

The identification technique requires the input-output data of the process to
identify its parameters based on the desired model type. In this study, a set of training
and validation data is generated from the Aspen Dynamic model using the excitation
signals from the Matlab Simulink. Here, initiator 1, initiator 2, and CTA flowrates are
excited with minimum and maximum changes from their steady state value using a
uniform random number generator (Zhu, 2009). These minimum and maximum values
are presented in Table 3.4. The values are obtained based on the conducted parametric
analysis results.

62

Table 3.4 Maximum and minimum values for MV

Parameter Steady state Max Min
Initiator 1 (kg/h) 3.6 4.6 2.0
Initiator 2 (kg/h) 0.576 2.5 0.3
480 0.5
CTA (kg/h) 27.432

The dataset is generated for 2000 minutes, with 6 minutes of change time and 0.1
minutes of sampling time. The small sampling time is selected to capture the process's
dynamic more accurately (Tenny, 2002). Furthermore, in order to avoid the ill-
conditioned modeling of the system, all the parameters are scaled as follows (Uddin et
al., 2018):

= − (3.24)
−

Where is the original value, is the steady state value of the parameter, is the

maximum value of the parameter and is the minimum value of the parameter. The

maximum and minimum values are determined from this work parametric analysis.

3.6.2(c) Model structure

The Wiener model structure is presented in Figure 3.7, which consists of a linear
dynamic block cascaded with a static nonlinear block. Here, ( ) is an intermediate
signal that does not necessarily have a physical meaning.

( ) ( ) Nonlinear static ( )

Linear dynamic

block block

Figure 3.7 Wiener model (L-N) structure

63

3.6.2(d) Model identification
In general, the Wiener model identification technique can be grouped into

several approaches, which are L-N, N-L, and simultaneous approaches (Iqbal and Aziz,
2011). In this work, the L-N approach is followed as it is a straightforward method, and
an accurate description of static nonlinearity can be established (Cervantes et al., 2003).
The steps of the L-N identification approach are presented in Figure 3.8. For this
approach, a set of dynamic data is generated in the beginning. Then, the dynamic linear
block is identified using a linear identification technique, and the intermediate signal, v
is produced. The intermediate signal is used to connect the output of the linear model
and the input of the nonlinear model identification. Thus, the output signals from the
dynamic linear model serve as input signals for the static nonlinear model and are
estimated using the nonlinear identification technique.

Generate a set of dynamic data:
u(k), t(k)

Set v(k) = t(k)

Identify linear model
v(k) = f(u(k))

Identify nonlinear model
y(k) = ϕ(v(k))

Figure 3.8 Flow chart for Wiener model identification using L-N approach

64

The dynamic linear model is identified using the state space model identification
technique based on the subspace method (Favoreel et al., 2000). The state space model
equation is shown in equation 3.25:

( + 1) = ( ) + ( − ) (3.25)
( ) = ( )

where ( ), ( ), and ( ) are the state, input, and output of the model at time instant
, respectively. , , and are the system matrices with defined dimensions based on
the input-output of the process, and is the dead time. The prediction error estimate
(PEM) technique is also employed to improve the model accuracy during the
identification process (Gumussoy et al., 2018). The PEM uses the numerical
optimization technique to minimize the weighted norm of the prediction error ( ) (as
a cost function), which is the difference between the measured output ( ) and the
predicted output of the model ( ). For simplicity, the technique can be defined as
follows:

(3.26)

( , ) = ∑ 2( )

=1

The static nonlinear model is developed using a neural network (NN) model.

Dynamic features such as time delay and feedback or recurrent connection are not used

to develop a static neural network model. Here, a fitting neural network (fitnet) is

utilized to map the intermediate input and process output of the Wiener model. The

fitting neural network is a type of feedforward neural network model that is used to fit

an input-output relationship. This neural Wiener modeling approach is developed based
on Lawryńczuk (2013). The structure of the neural network model is presented in Figure

3.9.

65

Figure 3.9 The structure of the neural network model used as the nonlinear part of
the Wiener model (Lawryńczuk, 2013)

The output of the neural network model can be written as:

(3.27)

( ) = 0 + ∑ 2 ( ( ))

=1

where,

( ) = 1 ,0 + 1 ,1 ( ) (3.28)

where 0 is the network bias, and the weight of the network is denoted by 1 , and 2 ,
for the first and second layers, respectively. represents the nonlinear transfer function
(i.e., hyperbolic tangent sigmoid), and is the number of hidden neurons. The output

for the neural wiener model can be described as: (3.29)



( ) = 0 + ∑ 2 ( 1 ,0 + 1 ,1[ ( )])

=1

In order to find the best number of hidden neurons, an iterative validation
method is implemented (Sheela and Deepa, 2013). In this method, the neural network
model is trained using a set of hidden neurons to obtain the one producing the least
validation error. This process is repeated (or iterated) several times to find the best set
of weights and biases for a particular hidden neuron. Typically, this process is stopped

66

when the validation error becomes constant or decreases with the increased number of
hidden neurons.

3.6.3 Optimizer

Mayne (2016) had emphasized that the study on the MPC controller, especially
for robust and stochastic MPC, is too complicated for current implementation in the
process industries. Stochastic optimization techniques such as Simulated Annealing
(SA) and Particle Swarm Optimization (PSO) have the advantage of producing
improved optimal solutions than deterministic optimization. However, the cost of
computational effort and time makes it unappealing to be used in the industry. Thus, in
this work, the application of successive quadratic programming (SQP) is adopted as the
current MPC optimizer. SQP is preferred over QP since the latter method can only solve
the linear optimization problem, which is not suitable for handling nonlinear processes.

SQP works by searching for the optimum solution of an optimization problem
by sequentially solving a QP problem. At each SQP iteration, the gradient and the
hessian of the objective function at the current point are calculated to form the QP
problem. Then, the resulting QP problem is solved to obtain the direction to the next
point. The optimum step size to the calculated search direction is then obtained using
the line search algorithm or trust-region algorithm. The next point can now be calculated
from the optimum step size and the search direction. The SQP solution (controller
output) is implemented in a moving horizon framework, i.e., only the first control
interval is implemented at each sampling instant. The optimization process is repeated
at each sampling time based on the updated information from the plant. Here, the
fmincon function inside Matlab is used to run the SQP optimization program.

67

The quadratic objective function used in the MPC control scheme is shown in

equation 3.30.

−1
1)]}2
min = ∑{ [ ̂ ( + + ∑ { [∆ ( + 1)]}2 (3.30)

∆ ( ) =1 =1

where; ̂ is the predicted error, is the prediction horizon, is the control horizon,

is the objective function, is the tuning weight for setpoint tracking, is the tuning

weight for MV rate and ∆ is the effect of current input, ∆ ( ) = ∆ ( ) − ∆ ( −

1). The predicted error term, ̂ can be defined as:

̂ ( + 1) ≜ ( + 1) − ̃ ( + 1) (3.31)

where is the desired set point and ̃ is the corrected prediction over the prediction

horizon, P. ̃ can be acquired by:

̃ ( + ) ≜ ̂ ( + ) + ( + ) (3.32)

where ̂ ( + ) is referred to as an uncorrected prediction and ( + ) = ( ) − ̂ ( )

is the model mismatch term. The full Matlab script for the NWMPC using the s-function

block is presented in Appendix B.

3.6.4 MPC Tuning

A typical MPC has four parameters that need to be tuned, i.e., prediction horizon
(P), control horizon (M), output weighting (or error penalty) matrix, and input rate
weighting (or move suppression) matrix. The procedure for tuning the MPC controller
is presented in Figure 3.10. Based on the figure, the value of the prediction horizon and
control horizon is selected initially. Here, guidelines from Seborg et al. (2004) are
applied.

68

The guidelines are 5 ≤ ≤ 20 and /3 < < /2, where is the model
horizon. The model horizon can be calculated from ∆ = , where ∆ is the sampling
time and is the settling time for the open-loop response. The increase of control
horizon M can make the MPC become more aggressive and requires more
computational effort. The prediction horizon is usually selected to be = + so
that the full effect of the last input move is considered. A decrease in the control horizon
tends to make the controller more aggressive.

Figure 3.10 Flow chart for MPC tuning procedure
69

After finalizing P and M values, the MPC controller is tuned based on a linear
model of the process using Matlab MPC Tuning Advisor. In the software, the output
weighting and input-rate weighting matrix are adjusted to achieve the best performance
using the Integral Squared Error (ISE) performance function:



= ∑ (∑( )2 + ∑( )2 + ( ∆ ∆ )2) (3.33)

=1 =1 =1

To simulate the scenario of MFI grade changes in the LDPE polymerization process,

three different grades, which are low (MFI 1.5), medium (MFI 5), and high (MFI 22),

are used during the tuning process. Thus, the selected tuning values must be able to

achieve the entire grades with the smallest error. After achieving a satisfying

performance with the tuning software, the preliminary tuning parameters are evaluated

with the SSMPC and the NWMPC during the online simulation. At this stage, the

controller is tuned using a heuristic approach, where only the output tuning weight is

adjusted (Jacob and Dhib, 2011).

3.6.5 Integrate with Aspen Dynamic

The online control scheme is accomplished by integrating the Matlab Simulink
environment with Aspen Dynamic via a program called AMSimulation. The program
script is developed into the s-function block to directly access the Aspen Dynamic
model in the Simulink environment, as shown in Figure 3.11. Thus, the process input
from the Aspen Dynamic model can be transmitted to the MPC control scheme and vice
versa in a connected online platform.

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Set point AMSimulation

MPC
Controller

NW Model LDPE tubular reactor
-
+

Matlab Simulink Aspen Dynamic

Figure 3.11 NWMPC control scheme with Aspen Dynamic

3.7 Development of soft sensor
The soft sensor is developed in two parts, which are the modeling and the online

scheme. In the modeling part, an empirical model that can describe the desired delayed
response is constructed. An input selection process is made to determine the most
suitable input for the soft sensor model. In the online scheme part, the completed soft
sensor model is placed within the NWMPC control scheme. In order to update the
current output with the delayed measurement signal, a bias update scheme is
implemented.

3.7.1 Soft sensor modeling

The soft sensor model is developed based on a neural network (NN)
identification technique available in Matlab. A feedforward NN (FFNN) model with
input delays is adopted as the model scheme, as presented in Figure 3.12. This type of
NN model is also known as time delayed NN (TDNN). The application of input delay
allows the network to have a finite dynamic response to the time series input data. The
model has a single hidden layer and output layer, with hyperbolic tangent sigmoid and

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