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12.1 Introduction and Overview.............................. 12-1
12.2 Background: Polymerization Mechanisms
and Processes ....................................................12-2
Polymerization Reaction Mechanisms
• Polymerization Processes
12.3 Continuous Processes....................................... 12-5
Process Characteristics and Control Problems
• Base Regulatory Control • Advanced Control
Strategies I: Steady-State Operation • Advanced
Control Strategies II: Grade Transition
12.4 Discontinuous Processes ................................12-11
Characteristics of Discontinuous Processes
• Control of Batch Polymerization Processes I:
Feedback Control • Control of Batch
Polymerization Processes II: Optimal Control
12.5 Summary and Conclusions ............................12-21
University of Delaware
École Polytechnique Fédérale de Lausanne
École Polytechnique Fédérale de Lausanne
With an annual worldwide production well in excess of 100 million metric tons, synthetic polymers
constitute a significant part of the modern chemical process industry. Polymer reactors—operated in
continuous, batch, or semibatch mode—are therefore important processing units, but there are unique
problems associated with controlling them effectively. The most significant characteristics of polymer
reactors that make them one of the most challenging units to model, control, and optimize, are discussed
in this chapter; we also provide a survey of the strategies that have been proposed, and those that have
been successfully employed in industrial practice.
12.1 Introduction and Overview
The primary objective of polymerization processes is to produce polymers that will perform consistently
and acceptably in specific end-use applications, for example, in light switches, automobile bumpers,
fiber-optic cables, and so on. How well these polymer products perform is determined by such product
attributes as tensile strength, toughness, and UV resistance, which ordinarily cannot be measured during
the manufacturing process. But these product attributes themselves arise from the molecular and/or
macroscopic architecture of the polymer—characteristics that are determined during polymer synthesis.
Thus, meeting customer specifications on polymer product attributes (in addition to maintaining safe
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12-2 Control System Applications
process operation, and meeting production volume targets and environmental regulations) requires
effective control of the manufacturing process in a manner that is unique to this class of processes [1–3].
As discussed further below, polymerization processes (and there is a wide variety of them) are complex,
and exhibit significant nonlinear characteristics that are, in many cases, poorly understood; they also lack
online measurements of those product properties that are important to the product performance in
end-use. Furthermore, what constitutes an effective control strategy is highly dependent on the specific
characteristics of the polymer process in question. For example, large volume polymers are produced
more economically in continuous processes, where the primary objective is to start up the process as
quickly as possible and maintain it at an economically desirable steady-state operating condition. On the
other hand, low-volume specialty polymers are produced in batch and semibatch processes, where the
primary objective is to obtain acceptable product quality at the end of each batch cycle. These two distinct
operating modes present different sets of control problems unique to each mode of operation.
The principal difficulties in achieving good control of polymerization reactors are related to inadequate
online measurement, a lack of understanding of the dynamics of the process, the nonlinear behavior of
these reactors, and the lack of well-established techniques for controlling nonlinear processes. While
temperatures, pressures, flow rates, and reactant compositions are routinely measured online, important
product quality variables such as molecular weight distributions (MWDs) and copolymer composition
are usually measured offline, and typically with very long time delays. End-use polymer properties, which
are related to the molecular weight and composition distributions in the polymerization reactor according
to relations that are not entirely well understood, can only be measured after lengthy post-manufacturing
processing. Finally, each continuous industrial polymer reactor is typically used to manufacture a wide
variety of grades of the same basic product, thereby requiring frequent startups, transitions, and shut-
downs. Similarly, the same batch or semibatch reactor is often used for the production of many different
polymers from different sets of reactants; and while the equipment (reactor) remains the same, how the
process is operated and controlled often depends on the product currently being manufactured.
This chapter provides an overview of the key issues associated with controlling polymer reactors, along
with a discussion of techniques for addressing them. The rest of the chapter is organized as follows: in
Section 12.2, we provide a fundamental scientific context for the control of polymer reactors by presenting
a brief introduction to the mechanisms and processes by which polymers are produced; next, we discuss
the control of continuous processes in Section 12.3, and control of batch (and semibatch) processes in
Section 12.4. A summary and some conclusions are presented in Section 12.5.
12.2 Background: Polymerization Mechanisms and Processes
12.2.1 Polymerization Reaction Mechanisms
Polymers, very large molecules consisting of a huge number of monomer units linked in long chains,
are produced via many different reaction mechanisms. These mechanisms influence the fundamental
architecture of the final molecule, and hence the final product characteristics. Two of the most common
mechanisms, free radical and ionic polymerization, are summarized here.
188.8.131.52 Free-radical polymerization
As illustrated below, this mechanism consists of four steps: (1) Initiation, where an initiator molecule
decomposes to create two primary free radicals, and each radical reacts with a monomer unit to produce a
“live” polymer chain of length 1; (2) Propagation, where the live polymer molecule reacts rapidly with the
monomer to produce a growing polymer chain; (3) Termination, where a live polymer molecule reacts
with another live polymer molecule to form a dead polymer molecule, either by combination (into a single
polymer chain), or by disproportionation (into two dead polymer molecules); (4) Chain transfer, where
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Control of Polymerization Processes 12-3
the free radical at the end of a growing polymer chain is transferred to a chain transfer agent, a monomer
molecule, a solvent molecule, or even another polymer molecule.
Rk+M→Rk+1;k=1, 2, ...
•Termination by disproportionation
Rk+Rk→Pk+Pk;k,k=1, 2, ...
•Termination by combination
Rk+Rk→Pk+k;k,k=1, 2, ...
•Chain transfer to transfer agent
Rk+T→Pk+T•;k=1, 2, ...
•Chain transfer to monomer
Rk+M→Pk+R1;k=1, 2, ...
•Chain transfer to solvent
Rk+S→Pk+S•;k=1, 2, ...
•Chain transfer to polymer
Rk+Pm→Pk+Rm;k,m=1, 2, ...
Here Iand I•are the initiator molecule and the initiator radical, respectively; Mis a monomer molecule;
Rkand Pkrepresent growing (live) polymer and dead polymer molecules of length k, respectively; Tand S
are, respectively, transfer agent and solvent molecules with corresponding radicals represented as T•and
S•. A defining characteristic of polymer molecules is that they grow to varying lengths. Thus, the chain
length kindicated above is not fixed; it is a random quantity, determined by a wide variety of factors.
Polymers are, therefore, macromolecules with non-uniform molecular weights, which is why they are
primarily characterized by their MWDs. Note also from the mechanisms shown above that, with free-
radical polymerization, the resulting product molecular weight distribution (and hence average molecular
weights) can be controlled by manipulating (directly or indirectly) the rates of initiation, propagation,
chain transfer, and termination.
184.108.40.206 Ionic polymerization
With ionic polymerization, the intermediate species are positively or negatively charged ions—cations
or anions, respectively—rather than free-radicals. Furthermore, the reactions differ from the free-radical
polymerization reactions in that termination occurs only when the ions react with water. And since
termination cannot occur simply by the interaction of two ionized molecules, the average molecular weight
is more easily controlled in ionic polymerization than with free-radical polymerization. Furthermore,
because the initiation reaction has a low activation energy, ionic polymerization can be performed at low
temperatures. However, ionic reactions are more difficult to be carried out on an industrial scale, which
is why, whenever possible, free-radical polymerization is preferred.
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12-4 Control System Applications
12.2.2 Polymerization Processes
Whether operated in continuous or batch mode, a wide variety of processes are available for manufactur-
ing polymers, and each process has its own distinct characteristics. The processes most widely employed
in industrial practice are summarized below.
1. Bulk polymerization: This process, also known as mass polymerization, consists in the polymeriza-
tion of a (typically liquid) monomer, in the absence of any medium other than a catalyst, initiator,
or accelerator. An important feature of bulk polymerization is whether or not the polymer is
soluble in the monomer phase.
2. Solution polymerization: In this case, the polymerization reactions take place in the medium of a
solvent in which the monomer and catalyst are dissolved. The heat generated by the reactions is
absorbed by the solvent, making temperature control easier to achieve. In some cases, the solvent
must ultimately be removed from the polymer (e.g., via distillation), which may be rather costly.
3. Suspension polymerization: This process involves polymerization in the medium of a liquid (usually
water) in which the monomer is not soluble. Vigorous mechanical stirring and a stabilizing agent
are used to generate suspensions of monomer droplets within which the polymerization takes place.
As with solution polymerization, temperature control is easier than with bulk polymerization,
because part of the heat generated by the reaction is absorbed by the water.
4. Emulsion polymerization: This popular process refers to the free-radical polymerization of
monomer emulsions—water-insoluble monomer molecules dispersed into droplets that are sta-
bilized by a mono-layer of surfactant molecules at the water-monomer interface. Polymerization
is initiated via a water-soluble initiator, while propagation proceeds by monomer molecules dif-
fusing from droplets to growing polymer particles, where the presence of a surfactant prevents
aggregation of the particles.
The product of an emulsion polymerization is called a “latex,” and in many cases, virtually all
the monomer is consumed; furthermore, no solvents are involved. Finally, high molecular weight
latexes can be obtained at favorable reaction rates, something not possible with either bulk or
solution polymerization processes. These features confer significant economic and environmental
advantages and are mostly responsible for the popularity of emulsion polymerization in industry.
Bulk and solution polymerization are therefore classified as homogeneous processes while suspension
and emulsion polymerization processes are heterogeneous.
Another common classification in polymerization is based on the number of different monomers
involved in the reactions. In homopolymerization, the polymer is made from a single monomer, while in
copolymerization or in terpolymerization, two or three different monomers are involved in the formation
of the polymer product.
Observe therefore that the term “polymerization process” covers a broad spectrum of possible operat-
ing configurations, reaction mechanisms, and fundamental processes. For example, Figure 12.1 depicts a
semibatch, emulsion copolymerization process in which monomer B and surfactant are preloaded into
the reactor, with gradual addition of monomer A and initiator. The resulting polymer product is removed
at the end of the batch.
Despite such diversity, these manufacturing processes share some common characteristics around
which the challenges to effective control may be framed.
1. Polymerization processes exhibit complex steady state and nonlinear dynamic characteristics,
including multiple steady states, open-loop instability, and high parametric sensitivity.
2. Polymerization processes involve multiple strongly interacting variables.
3. A typical industrial continuous polymer reactor is used to manufacture a variety of grades of
the same basic product, thus necessitating frequent startups, online transitions, and shutdowns.
Similarly, the same batch or semibatch reactor is often used for the production of many different
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Control of Polymerization Processes 12-5
Monomer A Initiator
FIGURE 12.1 Typical semibatch emulsion copolymerization reactor.
polymers from different sets of reactants. While the equipment (reactor) remains the same, how
the process is operated and controlled often depends on the product currently being manufactured.
4. The important polymer product quality determinants (e.g., average molecular weights, MWD, melt
index, Mooney viscosity, etc.) can be measured only very infrequently. Product end-use properties
(e.g., tensile strength, UV resistance, etc.), which are dependent on product quality determinants,
can only be determined post manufacturing.
Thus, by themselves, classical linear, single-loop controllers with static structures based on frequently
available measurements, are often incapable of delivering effective control of product characteristics
without some sort of augmentation.
We are now in a position to discuss the strategies for controlling polymerization processes, beginning
with continuous processes.
12.3 Continuous Processes
12.3.1 Process Characteristics and Control Problems
Continuous processes, used mainly for high-volume production of commodity polymers, often exhibit
strong nonlinearities in the form of multiple steady states, parametric sensitivity, limit cycles, and so
on [4–6]. Especially with free-radical polymerization, a major source of the nonlinearity is the autocatalytic
nature of the polymerization reactions—the so-called “gel effect”—which frequently causes uncontrollable
reactions, resulting in excessive temperature rise, rapid conversion, and equipment plugging.
With continuous polymerization processes, there are four identifiable modes of operation:
2. Steady-state operation
3. Grade transition (i.e., transition from one steady-state to another)
During startups and shutdowns, the primary objective is safety; product quality control is mostly of
concern during steady-state operation and grade transitions where, in this latter case, the objective is to
transition from one steady-state operation to the next one as efficiently as possible. The control objectives,
and hence, appropriate control strategies, are different in each case.
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12-6 Control System Applications
The typical modern strategy involves a generic, two-level hierarchical structure, with “base regulatory
control” at the first level for controlling such process manipulated variables as monomer flow rates and
temperatures. The set-points for these process variables are determined at the higher, “advanced control
level,” in order to obtain desired product characteristics. The specific details of how the controllers in
each level are designed and implemented can vary depending on the specific problems at hand, and the
level of sophistication desired by the practitioner. Some general principles that are broadly applicable are
12.3.2 Base Regulatory Control
220.127.116.11 Temperature Control
As polymerization reactions are usually highly exothermic, temperature control is of universal impor-
tance, regardless of operating mode. In startup and shutdown mode, temperature control is primarily
for assuring safety; in the other two operating modes, temperature control is used indirectly to influence
polymer properties, since temperature has a strong effect on polymer properties. Temperature control
takes on added importance when the reactor must be operated at an unstable steady state. In highly
exothermic polymerization, in addition to controlling the actual value of the reactor temperature, the
rate of change of reactor temperature must also be carefully monitored, especially when a significant
amount of unreacted monomer is present in the reactor. This is because a sharp increase in temperature
in combination with a significant amount of unreacted monomer in the reactor poses considerable safety
With the most common equipment designs, heating and cooling is achieved by fluids flowing out-
side the reactor in a surrounding jacket, or else through a heating/cooling tube inside the reactor. In
industrial applications, effective reactor temperature control is typically achieved with a cascade control
scheme consisting of two proportional-integral-derivative (PID) controllers, where the outer temper-
ature controller sets the set-point for the inner cooling/heating fluid flow controller. Standard tech-
niques for cascade control systems design are found in process control textbooks (e.g., [7–9], etc.) are
therefore customarily employed for designing and implementing basic temperature control in polymer
18.104.22.168 Flow Control
For continuous polymerization reactors, the total sum of reactant (feed) flow rates (equivalent to the ratio
of the reactor volume to the reactor mean residence time), has a profound effect on the steady-state and
dynamic behavior of the reactor. As long as the reactor is operated at the desired steady state, this total
feed rate should be maintained as constant as possible. The monomer and solvent (inert) flow rates are
usually dominant; the flow rates of chain-transfer agents, cross-linking agents, and initiators, are typically
small relative to these dominant flow rates. As such, because changes in these “small” flow rates have
little or no effect on the reactor residence time, the “small” flow rates are used as manipulated inputs to
influence and fine-tune polymer properties.
Multiple single-loop PID controllers are typically used to maintain the dominant flow rates constant,
and to set the desired values for the “small” flow rates when used as manipulated variables. Once again,
these controllers may be designed successfully using standard techniques discussed in process control
textbooks. Thus, base regulatory control for polymer reactors involves no more than the application of
standard single-loop and cascade control, specifically adapted to polymerization processes.
12.3.3 Advanced Control Strategies I: Steady-State Operation
While necessary (and adequate) for meeting safety and production volume requirements, base regula-
tory control of temperature and flows are insufficient (and inadequate) for achieving product quality
objectives. For the purpose of ensuring good product quality during steady-state operation and during
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Control of Polymerization Processes 12-7
transition from one steady state to another, control of continuous polymerization reactors requires the
judicious manipulation of the set-points to the regulatory controllers, a task that is usually carried out
with advanced control strategies. This task is particularly challenging primarily because measurements of
the variables to be controlled are not available online, or frequently enough via other analysis methods.
The predominant strategy, to be discussed shortly, is to use available online measurements to infer desired
polymer properties. But first, we note three major directions in which research efforts have been directed
toward making available reliable online measurements, from which polymer properties can be inferred.
• Development of new online sensors [2,10].
• Development of state estimation techniques for estimating nonmeasurable polymer properties from
available measurements (, and the references therein). A list of measurements from which certain
polymer properties can be observed and/or detected is given in .
• Understanding and exploiting the qualitative and/or quantitative relations between easily-available
online measurements such as density, viscosity and refractive index, and certain polymer properties
such as an average molecular weight [12–14]. See, for example,  for an approach to predicting
melt index and density in a fluidized bed ethylene copolymerization reactor from available online
temperature and gas composition measurements.
Control structures for implementing advanced control of polymerization reactors can be divided into
three major groups:
1. Multirate cascade control structure (Figure 12.2)
2. Multirate decentralized control structure (Figure 12.3)
3. Multirate control structure with multirate state estimation (Figure 12.4)
All these control structures reflect a key defining characteristic of advanced control of polymer reactors:
the availability of measurements at different sampling rates and with different time delays. While “fast”
measurements such as temperatures, pressures, and flow rates are available at high sampling rates and
with almost no time delays, “slow” measurements, which are usually directly related to product quality,
are measured at low sampling rates and with considerable time delays (time delays as long as 24 h between
the time a sample is taken and the time the sample analysis becomes available are not unusual). In the
subsequent discussion, the vector of fast output measurements is denoted by y, and the vector of slow
output measurements by Y.
22.214.171.124 Multirate Cascade Control Structure
As depicted in Figure 12.2, this control structure consists of two loops (or levels, or layers), for the case
where measurements are available at two different sampling frequencies—one loop for each sampling
frequency. The inner loop responsible for controlling the fast outputs is executed at the higher sampling
frequency of these fast measurements, while the outer loop (primary controller) is executed at the lower
sampling frequency of the slow measurements. The primary controller periodically (and infrequently)
adjusts the set-point values of the secondary (fast) controlled outputs. In general, the number of different
sampling frequencies at which the measurements are available determines the total number of distinct
FIGURE 12.2 Multirate cascade control structure.
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12-8 Control System Applications
FIGURE 12.3 Multirate decentralized control structure.
In the polymerization industry, the primary controller is often an operator or a process engineer who,
based on his/her experience and on data available from laboratory sample analysis, adjusts the set-point
values of the secondary controlled outputs in order to achieve the desired polymer product quality. In this
structure, the set-point values of the secondary controlled outputs are updated whenever measurements
of the primary controlled outputs are available. The primary and secondary controllers can be of any
type—from classical PID controllers to model predictive controllers.
As a simple illustrative example, consider a polymerization reactor in which reactor temperature,
measured online every second without delay, is y, and the rate of thermal energy added to or removed
from the reactor is the manipulated input, u. Let the polymer number-average molecular weight that is
measured offline in a laboratory once a day with a time delay of one day, be the secondary controlled
output, Y. Under the configuration in Figure 12.2, the reactor is under “continuous” temperature control,
and the temperature set-point is adjusted by the primary controller (possibly an operator or a process
engineer) once a day, depending on the difference between the measured number-average molecular
weight and its desired value.
This control structure is quite common in the polymerization industry. For such industrial systems,
at the “lower level,” the secondary loop is configured and implemented in the distributed control system
(DCS) for controlling pressure, temperature, level, and flow, whereas, at the “higher level,” the primary
control loop is configured in the supervisory computer for advanced control of polymer properties .
126.96.36.199 Multirate Completely Decentralized Control Structure
This control structure, depicted in Figure 12.3, consists of two “independent” feedback loops, for the
case in which measurements are available at two different sampling frequencies. Note that, when the
measurements are available at n>2 different sampling frequencies, there will be a commensurate number
of distinct feedback loops, n, one for each measurement frequency.
As with the structure in Figure 12.2, the secondary (fast) loop, which uses Controller 2 to control the
fast outputs y, operates at the higher sampling frequency of the fast output measurements. On the other
hand, the primary Controller 1 regulates the slow outputs (product quality variables) Yand operates at
the lower sampling frequency of the slow output measurements. However, unlike the structure in Figure
12.2, Controller 1 directly adjusts (infrequently) its own set of manipulated inputs, u1, that are paired with
the primary (slow) controlled outputs. As with the primary (master) controller in the cascade structure
of Figure 12.2, Controller 1 can be an operator or a process engineer. Both controllers (1 and 2) can
also be standard automatic controllers—from classical PID controllers to model predictive controllers.
The decentralized nature of this multirate control structure endows the control system with a measure of
robustness in the presence of extremely low-frequency measurements with very long delays. Compared to
the cascade control structure of Figure 12.2, implementing this control structure requires a larger number
of manipulated inputs.
As a simple illustrative example, consider a reactor for which the reactor temperature is measured
online every second without delay, while the polymer number-average molecular weight is measured
offline in a laboratory once a day with a time delay of one day. In this case, the latter is the primary (slow)
controlled output Y, while temperature is the secondary (fast) controlled output y. For this process, the
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Control of Polymerization Processes 12-9
manipulated inputs u1and u2are, respectively, the flow rate of a chain transfer agent (or a thermal
initiator), and the rate of thermal energy added to or removed from the reactor: The latter is used to
control temperature, while the former is used to control the number-average molecular weight. As in
the control structure of Figure 12.2, the reactor is under “continuous” temperature control; however,
unlike that earlier structure, the flow rate of the chain transfer agent or the thermal initiator stream is
adjusted by the primary controller (possibly an operator or a process engineer) only once a day, based on
the observed deviation between the measured number-average molecular weight and its desired value,
almost independent of the temperature control loop .
188.8.131.52 Multirate Control Structure with Multirate State Estimation
This control structure, depicted in Figure 12.4, is more general than the preceding structures, since it is
suitable for cases where measurements are available over a broad range of sampling frequencies. Because it
includes a multirate state estimator, the entire control scheme must be model-based, unlike the previously
discussed schemes where the controllers can take any form. The heart of the scheme is the multirate state
estimator which uses the infrequent measurements of the primary outputs Y, the frequent measurements
of the secondary outputs y, and information about the vector of manipulated inputs u, to calculate the
frequent, delay-free estimates ˆxof all the polymerization reactor state variables. Such estimates are then
used in a suitable control scheme to control the primary and secondary outputs simultaneously.
The estimator can be Kalman filter-based , or Luenberger observer-based . However, because
of the nonlinear dynamics associated with polymer reactors, these filters and observers must be nonlinear;
in addition, sufficiently accurate dynamic models of the polymer reactors are required in order to obtain
sufficiently accurate estimates of the unmeasured states. In recognition of the fact that accurate process
models are not easy to develop (especially for industrial processes), it is important that the estimator be
robust to plant-model mismatch and unmeasured disturbances. One of the more straightforward means
of ensuring such robustness is to estimate a set of model parameters along with the state variables. This
makes the estimator adaptive, at the cost of solving a larger estimation problem. Because of the coupling
among variables and the heavy reliance on the estimation of the states of a complex process, this control
scheme is more likely to be less robust to the late arrival of measurements than the other two schemes.
To illustrate, let us consider the same simple polymerization reactor example used earlier, in which the
reactor temperature is measured online at a high sampling frequency and without delay; furthermore, the
polymer number-average molecular weight is measured offline in a laboratory once a day, with a time
delay of one day. The manipulated inputs u1and u2are, respectively, the flow rate of a chain transfer
agent (or a thermal initiator), and the rate of thermal energy added to or removed from the reactor.
In this case, a multirate state estimator uses all the available information to obtain a high-frequency,
delay-free estimate of the number-average molecular weight, which the controller (a multivariable or a
fully decentralized controller) subsequently uses to determine appropriate control action. Note that the
control action is determined and implemented at the high sampling rate of the secondary output y. In this
FIGURE 12.4 Multirate control structure with multirate state estimation.
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12-10 Control System Applications
control structure, “continuous” control of number-average molecular weight is achieved using the high-
frequency, delay-free estimates of number-average molecular weight produced by the state estimator.
The control system performance therefore depends heavily on the performance of this multirate state
Note that one can also use a multirate estimator with the control structures shown in Figures 12.2 and
12.3, calculate delay-free frequent estimates of the primary outputs, Y, and use the delay-free frequent
estimates in the feedback loops instead of the delayed infrequent measurements of the primary outputs. In
such control schemes, both controllers (the primary and secondary controllers in Figure 12.2; Controllers
1 and 2 in Figure 12.3) operate at the higher sampling frequency of the fast output measurements.
Nevertheless, “continuous” control for both feedback loops is achieved at the cost of requiring a robust
12.3.4 Advanced Control Strategies II: Grade Transition
A single continuous polymerization process is often used in industry to produce several different grades of
the same polymer product. For such processes, each “campaign” involves operating at a particular steady-
state condition until the desired amount of the corresponding product grade has been manufactured;
thereafter, the next “campaign” starts by “transitioning” to the new operating conditions required to
make the next product type in the production cycle. Operating such processes effectively, clearly requires
that transitions between grades be carried out smoothly and as quickly as possible. Slow transitions lead
to the production of considerable amounts of off-specification polymer, and the resultant waste of energy
The intrinsic nature of the control problem associated with grade transitions (driving process outputs
from an initial state to a different final desired state in minimum time) makes transition control of
continuous polymer reactors an ideal dynamic optimization problem. Such problems may be solved in
several different ways.
One approach is to compute, offline, the optimal input (feed flow rates and temperature) profiles
via numerical optimization using a nominal process model, and then implement such input profiles
online, either in an open-loop fashion (as computed with no feedback or mid-course correction), or in
a closed-loop fashion, where the optimal input profiles are enforced using temperature and feed flow
rate feedback controllers. The primary disadvantage with this dynamic optimization approach is that the
grade transition ceases to be optimal in the presence of plant-model mismatch and unmodeled process
disturbances. Nevertheless, it is still possible to obtain good performance even under such nonideal
conditions. An example application is contained in . Optimal open-loop policies/trajectories were
determined for reactor temperature, bleed stream flow, catalyst feed rate, and bed level, via offline dynamic
optimization studies. A differential geometric controller was used to regulate instantaneous melt index
and density, and to provide servo control during grade changeovers. Hydrogen and butene feed rates
were manipulated to force the “measured” product properties onto the desired trajectories.
An alternative approach involves formulating the problem as a model predictive control (MPC) prob-
lem designed to minimize a desired cost function (typically the amount of off-spec material, the transi-
tion time, or both). The resulting optimization problem is typically nonconvex, because of the nonlinear
dynamics of the reactors. An example application to the problem of grade transition in a continuous
methyl methacrylate polymerization process, and also in a gas-phase polyethylene process, using non-
linear MPC and a Luenberger observer, can be found in .
A third approach is a hybrid of the first two. It involves the offline numerical computation of the
optimal transition using a nominal model of the process. The computed profiles are by no means optimal
for the plant due to uncertainty in the form of plant-model mismatch and disturbances. However, the
resulting optimal transition can be specified in terms of the succession and the types of arcs—arcs that are
either in the interior or on the boundaries of the feasible region, reflecting respectively, the fact that they
either force a sensitivity to zero or keep a constraint active. The result is a “solution model” that expresses
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Control of Polymerization Processes 12-11
in very practical terms the necessary conditions of optimality (NCO) that need to be satisfied by the
plant. These NCOs are enforced using feedback and appropriate online measurements. In other words,
these arcs and the corresponding switching times between them are adapted online via feedback control
using typically a multiloop PID control structure. The approach has been referred to as NCO-tracking
and falls within the class of self-optimizing controllers. Applications of this third approach to industrial
polymerization processes can be found in [21–23].
12.4 Discontinuous Processes
12.4.1 Characteristics of Discontinuous Processes
A significant number of polymers are low-volume specialty materials manufactured for tailored appli-
cations. Such products are manufactured most efficiently using discontinuous (batch and semibatch)
processes because of the intrinsic flexibility that these processes afford: batch and semibatch processes
can be operated over relatively short periods of time, repeatedly, making them convenient for manufactur-
ing a wide variety of low-volume products. The repetitive nature also permits batch-to-batch adjustments,
facilitating quick adaptation to changes in quality specifications.
Although batch and semibatch polymerization processes share the common characteristics that they
are not operated continuously, there are some important differences between them:
• In a batch process, the reactants are loaded into the reactor vessel at the beginning; the reactions
then proceed without the further addition or removal of material until a prespecified reaction time
has elapsed, after which the product is removed.
• In a semibatch process, one or several reactants can be added progressively to the reaction mixture
or, in rare cases, products can be progressively removed from the reactor during the course of the
Nevertheless, within the context of this discussion, the overriding distinguishing characteristic of this
class of processes is the repetitive, discontinuous operating policy, that is, the fact that they are not
operated continuously. Hence, from this point on, the term “batch” will be used in a general sense that
also encompasses semibatch processes, except when it is necessary to distinguish one from the other.
While continuous polymerization processes are predominantly operated at economically desirable
steady states, batch polymerization processes operate permanently in “transient” mode, with process
conditions and product characteristics constantly evolving from start to finish, so that there is no “steady-
state” to speak of. Therefore, in addition to ensuring safe and economic operation, the main objective of
batch polymerization reactor design and operation is to obtain, at the end of each batch cycle, material
with acceptable product quality characteristics. As such, while under continuous operation the control
system is designed to drive the polymerization process to the desired steady state operating point as
quickly as possible—and maintain it there, the objective for batch polymerization is to follow time-varying
policies/trajectories designed to produce, at the end of the batch, polymers with desired properties. Any
effective batch control system must contend with this characteristic nonsteady-state process operation.
The implications are as follows: with continuous processes, linear controllers have a reasonable chance
of being effective precisely because of the operating objective of maintaining the process within a small
neighborhood of the steady-state operating condition, where linear approximations are reasonably valid.
With no such steady state for batch operation, linear approximations will be largely invalid and linear
controllers largely ineffective.
In addition to the absence of a steady state, some other key features of batch polymerization processes
that influence control system effectiveness include:
1. Broad operating ranges: batch operation spans conditions that extend from the beginning of the
batch, with only reactants in the reactor, to the end of the batch, with mostly finished products.
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2. Single equipment, multiple products: the same batch or semibatch reactor is usually used for the
production of many different polymer products.
3. Repetitiveness: batch processes are used for low-volume production, necessitating frequent repeti-
tions of batch runs.
The first two features pose challenges to effective control by demanding the use of sensors capable of
covering a broad range of values, and also by placing stringent robustness and performance demands
on controllers that must function over such broad ranges of operation. Because polymerization reactor
dynamics depend strongly on the chemical composition and type of the feed (reactants), the second
feature requires the use of control schemes and/or controller tunable parameters that can be modified
appropriately for each set of reactants, even when the same equipment (reactor) is used. The third feature,
on the other hand, can be advantageous because the results from prior runs can be used to improve the
operation of subsequent ones (through “run-to-run” control and optimization schemes to be discussed
The issue of lack of online measurement and its negative effect on control system design and imple-
mentation is also commonly found in batch processing.
12.4.2 Control of Batch Polymerization Processes I: Feedback Control
As noted previously, the diversity of reaction mechanisms and polymerization processes creates a wide
variety of problem-specific issues. Nevertheless, there is a fundamental, defining characteristic of the
control of batch polymerization processes: the requirement to track desired trajectories from the beginning
to the end of the run, with the objective that the final product has desired properties. Consequently, as
discussed in greater detail, effective control strategies involve the following two-steps:
1. Determine, offline, the input and output profiles (trajectories) required to manufacture a product
with the desired quality.
2. Design a control scheme to track selected desired profiles as closely as possible.
What distinguishes one strategy from another is how each step is realized in practice. In general, the
reference trajectories are determined either from accumulated experience or knowledge of the process
and the product, or by solving a dynamic optimization problem. The computation of trajectories via
optimization will be addressed in Section 12.4.3. Here, we focus on the control strategies that can be used
to follow prescribed trajectories.
In general, trajectory tracking can be achieved in one of two ways: (1) via “online control” in an
individual batch, whereby input adjustments are made online as the batch progresses, or (2) via “run-to-
run” control over several batches, where the adjustments are not made online during a batch, but rather
input trajectories are computed between batches, using information gathered from the previous batches
to determine how to operate the next one.
In the context of these two control approaches, one needs to distinguish between run-time outputs,
which are output variables that evolve with “run-time” during a batch, and run-end outputs, which consist
of process values available at the end of the batch, such as the product quality (measured offline), batch
time, and the maximum reactor temperature.
184.108.40.206 Online Control
For an individual batch, online control consists in tracking profiles that have been predetermined offline.
Although the entire activity involved in implementing this strategy is repeated for each batch, what
is done for one batch does not (formally) carry over to what is done for the next one. However, the
objective is to track profiles that were pre-computed using nominal models, since the resulting control
system performance will not be optimal in the presence of plant-model mismatch and unmodeled process
disturbances. How the reference profiles can be adapted in the presence of such uncertainties is the focus
of Section 12.4.3.
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Controller Batch reactor
ysp + u y
FIGURE 12.5 Online control of the run-time outputs yin a batch reactor.
With online control of run-time outputs, the most popular strategy consists primarily in tracking the
pre-computed temperature profile by adjusting the reactor’s heating/cooling rate. Sometimes, a second
reference profile for the product average molecular weight (as an example) can also be tracked, typically
by adjusting the feed rate of chain transfer agent. However, such a control strategy requires the use
of an observer to reconstruct the average molecular weight of the product in the reactor from other
(possibly infrequent) measurements. Figure 12.5 shows a generic block diagram representation of online
control of the run-time outputs yin a batch reactor. Depending on the amount of process knowledge
available, the controller can take several forms, from simple linear controllers to more elaborate nonlinear
A few industrial applications of advanced control within the context of online control of run-time
outputs are available in the literature. For example, a successful implementation of temperature control
in an industrial 35 m3semibatch polymerization reactor using a flatness-based two-degree-of-freedom
controller is reported in . The performance of four different controllers (standard PI control, self-
tuning PID control, and two nonlinear controllers) for regulating the reactor temperature in a 5 L jacketed
batch suspension methyl methacrylate polymerization reactor is compared in . As expected, the
performance of the standard PI controller was the poorest since the controller parameters were fixed
and not adapted to match the changing process characteristics. The self-tuning PID control, based on
adaptive pole cancelation , performed better because available measurements were used to adapt the
controller to the varying process characteristics. The two nonlinear controllers were based on differential-
geometric techniques, which requires full-state measurement ; this necessitates the implementation
of an extended Kalman filter to estimate the unavailable states from the available measurements. The
two nonlinear controllers, which differ in the models on which each one is based, showed excellent
performance despite significant uncertainty in the heat-transfer coefficient.
With a sufficiently accurate process model, and in the absence of disturbances, tracking the profiles
determined offline is often sufficient to meet the batch-end product quality requirements. However, in the
presence of disturbances, following prespecified profiles is unlikely to lead to the desired product quality.
Hence the question arises: Is it possible to design an online control scheme for effective control of run-end
outputs using run-time measurements? Since such an approach amounts to controlling a quantity that
has not yet been measured, it is necessary to predict run-end outputs in order to compute the requisite
corrective control action, using, for example, MPC. This approach to batch control may, therefore, be
formulated as an MPC problem with a shrinking prediction horizon (equal to the time remaining to the
end-of-batch), and an objective function that penalizes deviations from the desired product quality at
batch end. With this strategy, at each sampling time during the batch, a piecewise-constant profile of
future control moves is computed as the solution of the MPC optimization; and, in classic, MPC fashion,
the first control move is implemented, and the states are reinitialized at the next sampling instant using
process measurements or state estimates. The procedure is repeated until the end of the batch.
Within the context of batch process control, linear MPC is widely used in industry; not so with
nonlinear MPC, but it remains an active area of research. For example, modifications of nonlinear MPC
have been proposed for maintaining run-end molecular properties within bounds (in line with industry
specifications) as opposed to controlling them at fixed values . The recommended methodology has
been applied in simulation to the control of batch styrene emulsion polymerization.
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220.127.116.11 Run-to-run Control
The objective of run-to-run control is to exploit one of the peculiarities of batch processes—their repetitive
nature—in such a way that the procedure for controlling the current batch explicitly incorporates relevant
information from previous batches. The input profiles are not adjusted online; instead, at the end of the
kth batch, the input profiles uk[0, tf]and the resulting product quality are used to determine the next
input profiles uk+1[0, tf], where tfrepresents the final time.
The run-to-run adjustments serve to meet run-end targets and can be implemented in several different
ways. For instance, the difference between predicted and actual run-end outputs can be used to refine
the process model, and the updated model is subsequently used to adapt the controller parameters.
Alternatively, the input profiles can be parameterized and the input parameters can be adapted from one
run to the next in order to enforce run-end objectives. This latter run-to-run control approach, depicted
schematically in Figure 12.6, can be expressed algorithmically as follows:
1. Parameterize the input profiles, uk[0, tf]=U(πk), where πkrepresents the vector of input param-
2. Start with the first run: set k=1 and initialize πk.
3. Implement the entire kth input profile, open loop; at the end of the batch, measure the run-end
polymer properties zk.
4. Determine the difference between the measured and desired run-end outputs and compute the
(k+1)th values of the input parameters. For example, with integral run-to-run control, πk+1=
I(zk,zsp), so that uk+1[0, tf]=U(πk+1).
5. Set k:= k+1 and return to (3).
Thus, while each batch is operated in open-loop fashion, feedback is introduced by updating the input
profiles between the kth and (k+1)th batches. However, it is important to note that although a batch
reactor is a highly dynamic process, the input parameters, πk, have been related to the run-end outputs,
zk, in this control formulation via a static map. In actual fact, because the input parameters are fixed at
the beginning of the batch, and the run-end outputs are evaluated at the end of the same batch, the plant
dynamics, which are responsible for transforming the reactants available at the beginning of the batch to
the products obtained at the end of the batch, have been incorporated into this static map. The process
dynamics are therefore implicit in this apparently static relation. Note also that, because of its recursive
nature, an important issue for this strategy is the convergence of the run-to-run control law.
Run-to-run control of run-end outputs has the drawback that it does not use any online information
that is otherwise available through the run-time outputs. One way by which run-time information from
previous batches can be used to control the run-time outputs of the current batch is through active
and progressive “learning” of the run-time characteristics of the process. Such an approach, known as
iterative learning control (ILC), is depicted in Figure 12.7. With ILC, the input profiles are adapted from
one run to the next on the basis of the error between the reference output trajectories ysp[0, tf]and
each observed trajectory yk[0, tf]with the objective of reducing this error at each iteration. Although the
processing objective is still to meet run-end product requirements, ILC focuses on the run-time outputs
as an indirect means of achieving run-end objectives.
uk(0, tf)yk(0, tf)zk
k+1 --> k
FIGURE 12.6 Run-to-run control of the run-end outputs zin a batch reactor.
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Control of Polymerization Processes 12-15
ILC – controller Batch reactor
k+1 --> k
uk(0, tf)yk(0, tf)
FIGURE 12.7 Iterative learning control of the run-time output profiles y[0,tf]in a batch reactor.
An example of successful industrial application of ILC can be found in , where an adaptive mech-
anism is used to learn about and compensate for variations in the reactor heat-transfer coefficient,
resulting in a shorter heat-up phase for subsequent batches. This approach has now become standard in
many industrial applications .
The main drawback of all run-to-run approaches is that they are unable to reject true run-time
disturbances because the run-to-run strategy makes no provision for online feedback correction dur-
ing the batch run. The obvious implication is that a combination of online and run-to run control
approaches should provide improved performance, especially in the presence of significant run-time
18.104.22.168 Combined Online and Run-to-Run Control
The fact that the control strategies discussed in the previous two sections complement each other well
seems to suggest that one should be able to combine them judiciously in order to meet both run-time and
run-end output objectives. A word of caution is necessary here, however: since both schemes attempt to
adjust the same input variables, one must be careful to ensure that the online and between-run corrections
do not oppose each other. Such a hybrid strategy is discussed in , where online MPC and run-to-run
control are combined to control particle-size distribution of an emulsion polymerization product, both
in simulation and experimentally.
Another example of the successful combination of online and run-to-run control is presented in .
The temperature profile of a batch polymerization reactor is separated into two sequential arcs tracked
using online control, while the switching time between the two arcs, and the final time of the batch, are
each adjusted run-to-run.
12.4.3 Control of Batch Polymerization Processes II: Optimal Control
As an alternative to “feedback” approaches discussed in Section 12.4.2, both within-batch and batch-end
objectives can be met simultaneously and directly by formulating—and solving—the batch polymerization
control problem as an optimization problem. This approach takes full advantage of the rich literature
on general optimal control, appropriately adapted for batch polymerization processes. A comprehensive
survey is given in , which contains nearly 140 references dealing with various aspects of polymerization
modeling, control, and optimization. The main conceptual difference between the objectives discussed
in Section 12.4.2 and those of the current section lies in the introduction of an economic performance
criterion to be optimized. Run-time and run-end objectives are now seen as constraints. While optimal
control strategies for batch polymerization processes have been widely studied in the literature, only
a handful of successful applications to industrial reactors have been reported. One such application is
presented later in this section as an illustrative example.
22.214.171.124 Problem Formulation
The optimization of a batch polymerization process can be formulated mathematically as follows:
L(xk(t), uk(t), t)dt (12.1)
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s.t.: ˙xk(t)=F(xk(t), uk(t), ρ); xk(0) =xk,0(ρ) (12.2)
S(xk(t), uk(t), ρ)≤0 (12.3)
P(xk(tf), ρ)≤0 (12.4)
where Jkis the scalar cost to be minimized for the kth batch, S, the run-time constraints, P, the run-end
constraints, ρ, the vector of time-invariant decision variables, and tf, the “fixed” or “free” final time. If tf
is free, in the sense that it is to be determined as part of the optimization (rather than fixed, in the sense
of having been determined a priori), then it is included in ρ.φis the scalar cost associated with the final
states, and Lis the Lagrangian function. Note that the initial conditions can also be considered as decision
For a specific process, Pwill represent a set of bounds on, for example, the batch-end weight-average
molecular weight (the polymer product property of interest), while Swill represent bounds on the manip-
ulated variables and operational constraints such as physical limits on the heat-removal capacity of the
reactor. For the purpose of illustration, a specific optimization problem could be formulated as follows:
s.t. ˙x(t)=F(x(t), Tj,in(t)), x(0) =x0(12.6)
where Tris the reactor temperature and Tj,in, the jacket inlet temperature. Frepresents the process model
equations, with the n-dimensional state vector xand the associated initial conditions x0. It is usual to
divide the system constraints into two categories:
•Terminal constraints (Equations 12.7 and 12.8): These are constraints on the final values taken by
certain variables at the end of the batch. Xmin is the lower bound on the final conversion, X(tf),
Mw,min is the lower bound on the final weight-average molecular weight, ¯
Mw(tf). The lower
bound on conversion is to ensure the production of an adequate amount of polymer in addition
to preventing the accumulation of a toxic monomer; the lower bound on the average molecular
weight guarantees the quality of the polymer.
•Path constraints (Equations 12.9 and 12.10): These are constraints on the values of the process
variables during the course of the batch. In this specific example, Tj,in,min is the minimal value
that the jacket inlet temperature can take, and Tr,max is the maximal allowed value of the reactor
Solving this specific problem will produce a jacket inlet temperature profile which, in the absence of
disturbances or process-model mismatch, minimizes batch time, while guaranteeing that the conversion
and average molecular weight specifications are met.
126.96.36.199 Solving Dynamic Optimization Problems
There exist several approaches for solving dynamic optimization problems, each one with its own peculiar
features. For example, for low-order dynamical systems, variational approaches such as Pontryagin’s
maximum principle, or differential-geometric techniques, lead to analytical expressions for the various
arcs that constitute the optimal control profiles. In practice, however, optimization problems are generally
solved numerically using a variety of techniques that include sequential quadratic programming (SQP),
genetic algorithms and stochastic optimization.
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Regardless of the specific solution technique employed, it is important to keep in mind that the resulting
solution, because it is based on a nominal model, will be valid and effective only to the extent that the
model idealization matches reality. In practice, there is plenty of uncertainty in the form of plant-model
mismatch and disturbances. Under these conditions, for the optimal control approach to be effective,
appropriate corrective measures must be incorporated into the implementation strategies. These issues,
which are central to the practice of optimal control of batch polymerization processes, are discussed
188.8.131.52 Implementation Strategies
Once the model-based optimization problem (Equations 12.1 through 12.4) has been solved numerically
offline, open-loop implementation of the computed input trajectories will only be “optimal” if there
are no disturbances and the process model is perfect. In actual practice, in the face of disturbances, it
will be necessary to augment this basic strategy with active feedback and track the computed optimal
output trajectories. An application of this approach to an emulsion copolymerization process is given
On the other hand, the repercussions of plant-model mismatch are that the input and output trajec-
tories, computed via optimization based on a nominal model, will no longer be optimal for the plant.
Even worse, these trajectories may correspond to infeasible paths that violate safety or operational con-
straints. Thus, one either needs to consider the effect of uncertainty explicitly in the optimization problem
(robust optimization) or else adapt the trajectories online using process measurements (online optimiza-
184.108.40.206.1 Robust Optimization
To prevent constraint violation, it may be necessary to account for uncertainty explicitly in the com-
putation of the optimal profiles. For example, by formulating the optimization problem with a set of
possible values for the uncertain parameters, a robust optimal solution may be determined that guaran-
tees that the constraints are satisfied over the entire set of values specified for the uncertain parameters.
This approach, known as robust optimization, obviously endows the solution with robustness and can
lead to satisfactory results when the uncertainty region is small. With significant model uncertainty, the
robust optimal solution is often unnecessarily conservative, thus leading to poor performance. Because
polymerization processes are difficult to model, it is not unusual that the uncertainties associated with
the model structure and the parameter values will be significant and, therefore, robust optimization will
enjoy only limited effectiveness.
220.127.116.11.2 Online Optimization
An alternative strategy for avoiding the inevitable conservatism of robust optimization is to deal with the
uncertainty actively by incorporating measurements into the optimization framework, with the premise
that, when available, measurements provide the best up-to-date information about the real process behav-
ior. This strategy can be implemented in different ways, as described next.
1. Repeated, updated optimization: In this two-step approach, the idea is to use measurements to
update the optimization problem and repeat the optimization. There are different ways of updating
the optimization problem:
• Update the initial conditions of the subsequent optimization problem—as with MPC, when
the output measurements or state estimates serve as new initial conditions for the subsequent
• Identify the uncertain model parameters and update the process model.
• Identify specific deviations between the plant and the model prediction (e.g., the constraint
values), and correct the optimization problem formulation accordingly.
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None of these methods is a panacea; they all have strengths and weaknesses . Nevertheless,
with careful consideration of the problem at hand, it is often possible to apply these repeated
optimization approaches successfully to actual batch polymerization reactors, as reported in .
As was the case with feedback control, the repetitive nature of batch processes can also be exploited
in the context of repeated optimization. The optimization follows essentially the same steps as
discussed above, the main difference being that the update step is performed between consecutive
batches instead of at the sampling times during a batch. This way, much more data are available
(e.g., from the previous completed batches).
2. Self-optimizing control: In self-optimizing-control, optimality is achieved via feedback control, not
by repeatedly solving an optimization problem. The key feature is the design of a feedback control
structure that enforces optimal plant performance; and a primary means of enforcing optimality
through feedback is by choosing the set-points as the NCO of the optimization problem.
The optimal solution of a dynamic optimization problem consists of one or several arcs, with each
one either enforcing an active constraint or else forcing a sensitivity to zero. The control structure
is constructed as follows: (i) use a plant model to compute, offline, the nominal optimal solution
(which will not be optimal for the plant because of uncertainty); (ii) design a multiloop control
system such that each loop regulates a specific element of the NCO to zero; and (iii) implement
this feedback control using measurements or estimates of the NCO elements. This approach has
been labeled NCO tracking because it is designed to satisfy the NCO of the plant .
The main difficulty with the implementation of NCO tracking is the real-time computation
of the NCO elements. However, this difficulty is offset by the following facts: (i) constrained
quantities are typically measured and thus directly available for feedback control, and (ii) accurate
estimation of sensitivities is often not needed as there is generally much more to gain in terms
of cost improvement by enforcing the set of active constraints than by forcing the sensitivities to
An alternative to online self-optimizing control is run-to-run self-optimizing control, which
involves reformulating the optimization problem as a control problem. With this run-to-run
alternative, there is the additional advantage of being able to wait until the end of the batch and
thus accumulate more data. By nature, the run-to-run self-optimizing control strategy mimics the
way the performance of batch processes is improved in industry. For instance, in the case of an
isothermal batch polymerization process, one typically tries to increase the reactor temperature
gradually from batch to batch in order to reduce the final time. This procedure stops when the
operator ascertains that the process is sufficiently close to its constraints, and that increasing the
reactor temperature any further will lead to off-spec products.
It must be clear from the above that, although very attractive conceptually, self-optimizing
control can be limited by the lack of appropriate measurements. Hence, state estimation is essential
for successful implementation of this strategy. One such successful industrial application of self-
optimizing control is discussed next.
18.104.22.168 Illustrative Example: Optimization of an Industrial Batch Polymerization Reactor
The problem involves a batch inverse-emulsion copolymerization process, where the objective is to min-
imize the reaction time . (An inverse-emulsion polymerization process is so-called because, contrary
to standard emulsion polymerization processes, which involves oil-soluble monomers and water-soluble
initiators, here the monomers are water-soluble and emulsified in the oil-phase, while the polymeriza-
tion is initiated with an oil-soluble initiator.) The reaction is highly exothermic; as such, higher reactor
temperatures will speed up the reaction and hence reduce reaction time. However, the resulting heat
generation will significantly increase the risk of thermal runaway, in addition to being potentially detri-
mental to the final product quality (especially average molecular weight). As a result, the prevalent
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Control of Polymerization Processes 12-19
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
FIGURE 12.8 Normalized measured profiles in an industrial batch reactor. Tris the reactor temperature, Tj, the
jacket temperature, and X, the molar conversion. The solid conversion line corresponds to the prediction of a simple
model of the polymerization reactor (Adapted from G. Francois et al., Ind. Eng. Chem. Res., vol. 43, no. 23, pp.
7238–7242, 2004.). Note the loss of controllability in the interval [0.6, 0.8]when the manipulated variable Tjis at its
lower bound. This loss of controllability was compensated for manually by reducing the initiator feed.
practice in industry has been to operate isothermally at a safe (relatively low) temperature. Reactor per-
formance can be improved by determining the reactor temperature profile that provides the best possible
tradeoff between productivity on the one hand, and safety and quality on the other. Figure 12.8 shows
a set of temperature and molar conversion profiles representative of industrial practice in this 1 m3
While a detailed discussion of the design and implementation of a run-to-run measurement-based
optimization strategy for this problem can be found in , the main points are summarized here.
1. Numerical optimization based on a simple process model indicates qualitatively that the first
phase of the reaction must be nearly isothermal so as not to violate the heat removal constraint,
while the temperature is allowed to increase in the latter part of the reaction in order to reduce
reaction time. Indeed, once conversion has reached a certain value, the reaction rate decreases
naturally and the reactor temperature can be increased (to reduce the batch time) while still
meeting the terminal specification on the average molecular weight. The resulting temperature
policy is therefore approximately semiadiabatic, whereby an initial isothermal phase is followed
by an adiabatic phase.
2. The temperature at the final time, tf, must respect a prescribed limit, above which the polymer
starts to coagulate. Since the maximum temperature that will be reached depends on the amount of
unreacted monomers remaining in the reactor when the control policy switches from isothermal to
adiabatic operation, the switching time between the two phases is adjusted to meet this temperature
Consequently, the run-to-run optimization task amounts to specifying two scalar parameters:
i. ts, the switching time between isothermal and adiabatic operations
ii. tf, the final time
in order to meet two terminal constraints: (a) the desired conversion of monomer Xdes, and (b) the
maximum reactor temperature Tr,max. Since, for this process, Xdes is nearly 100%, the desired conversion
and the maximum temperature occur simultaneously. Therefore, the optimization problem reduces to a
run-to-run adaptation of only the switching time, ts, to satisfy Tr(tf)=Tr,max. Such an adaptation scheme
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tsqj (0, tf)
FIGURE 12.9 Run-to-run optimization scheme showing how the final reactor temperature target can be met by
adjusting the switching time ts. The reactor is initially operated isothermally at the normalized temperature Tr,ref =1.
Adiabatic operation starts at ts, a value that is adjusted on a run-to-run basis to achieve Tr(tf)=Tr,max. The true
manipulated input is the flow rate of cooling medium in the jacket qj[0, tf], which is determined in the isothermal
phase by regulation of Tr(t) around Tr,ref =1 for t<ts; in the adiabatic phase, qj[ts,tf]is simply set to 0.
is conveniently implemented with the discrete integral control law:
where Kis the gain of the run-to-run integral controller. Figure 12.9 shows a block diagrammatic repre-
sentation of the overall run-to-run scheme.
0 0.2 0.4 0.6 0.8 1
FIGURE 12.10 Normalizedtemperature profiles in the industrial reactor obtained with the run-to-run optimization
scheme that adjusts the switching time between isothermal and adiabatic phases. Compared to the normalized reaction
time of 1 for the current-practice isothermal operation, the reaction time is successively reduced to 0.78, 0.72, and 0.65
in three consecutive batches. Note that a slight “back-off” from Tr,max is implemented for safety purposes, mainly to
account for run-time disturbances that cannot be handled by this approach.
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Control of Polymerization Processes 12-21
The improvement obtained with this scheme in three consecutive batches in the 1 m3industrial reactor
is shown in Figure 12.10. The batch time is reduced by about 35%, while the final average molecular weight
specifications are satisfied for all batches.
12.5 Summary and Conclusions
Process control continues to play an increasingly central role in the manufacture of high quality polymer
products, safely and in an environmentally friendly fashion. In particular, because end-use properties of
a polymer product strongly depend on the molecular and/or macroscopic architecture of the polymer—
characteristics that are determined during polymer synthesis in a polymerization reactor and which
cannot be altered in downstream processes or by blending—effective control of polymerization reactors
is of great importance to the modern economic production of polymers. However, effective control of
polymerization reactors is difficult because these reactors are typically highly exothermic, exhibit complex
and highly nonlinear—and poorly understood—dynamics that are strongly dependent on the chemical
type and composition of the reactants. Furthermore, the molecular properties that determine the polymer
product performance in end-use are usually measured offline with very long time delays, so that these
measurements are rarely available for online control.
Many of the modern methodologies employed for controlling polymer processes have been reviewed
in this chapter. While the various techniques are different in specific details, their development, use, and
implementation share the following three common steps, irrespective of the polymerization reactor type,
whether continuous, batch, or semibatch.
1. Given a set of end-use properties desired of a polymer product, the first step is to determine the
molecular and/or macroscopic architecture of the polymer (i.e., average molecular weights, func-
tional group distribution, and copolymer composition—when two or more different monomers
are involved) that will combine to yield the specified end-use properties.
2. As these molecular and macroscopic characteristics are rarely measured online, the second step is
to determine the reactor temperature and feed flow rate profiles which, once implemented, will lead
to the production of a polymer product with the desired molecular and macroscopic characteristics.
These profiles can be piece-wise constant as in the case of continuous reactors, or entirely time-
varying as in the case of semibatch or batch reactors. Of course, these profiles must be implemented
in such a way that the reactor is not steered into unsafe operating regimes. If an accurate reactor
model is available, the temperature and feed profiles can be calculated systematically offline and/or
online using optimal control techniques and the model. The optimization methods reviewed in
this chapter can be used to perform such calculations.
3. In the final step, feedback control is used to implement these profiles, which are presented as
set-points for the temperature and feed flow rate feedback control system. If any infrequent
measurements of polymer properties are available, they should be used in the feedback control
system to improve performance.
While several of the techniques presented here have been implemented successfully in industrial prac-
tice, many challenging problems still remain. Here are a few of them. First, advances in optimization
techniques in general, and in MPC in particular, have undoubtedly influenced industrial implementa-
tion; but these techniques depend on the availability of polymer reactor models of reasonably high fidelity
and modest complexity. Achieving such an intrinsically contradictory balance in models of processes of
practical importance remains difficult: industrial processes are, by nature, complex; capturing the essential
components of such complex processes with models of sufficient high fidelity almost inevitably demands
a minimum level of complexity. Methods of nonlinear model reduction capable of representing complex
process dynamics effectively with reduced-order, control-relevant models will provide a significant boost
to the industrial application of advanced control and optimization techniques. Next, while advances in
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state estimation have enabled the estimation of infrequently measured product characteristics—and hence
made online control of product characteristics possible—such estimates can never completely replace the
actual measurements themselves. And as the manufacturing chain becomes inexorably more tightly
integrated, each successive downstream customer will place increasingly stringent end-use performance
demands on the polymer products they receive from each of their suppliers. Meeting these demands will
ultimately require polymer process control system performance levels that are unattainable via unavoid-
ably imperfect inferred product attributes. Advances in sensors, analyzers and ancillary measurement
technology will be required in order to make available actual measurements of product attributes more
frequently than is currently possible. Finally, as the polymer process control system structures acquire
more complexity, theoretical analyses of model dynamics, overall control system stability, and achievable
performance (both nominal and robust), will be essential, especially for providing guidance in selecting
the best alternative for each problem.
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