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Control and Optimization of Batch Chemical Processes

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A batch process is characterized by the repetition of time-varying operations of finite duration. Due to the repetition, there are two independent "time" variables, namely, the run time during a batch and the batch index. Accordingly, the control and optimization objectives can be defined for a given batch or over several batches. This chapter describes the various control and optimization strategies available for the operation of batch processes. These include online and run-to-run control on the one hand, and repeated numerical optimization and optimizing control on the other. Several case studies are presented to illustrate the various approaches.
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Control and Optimization of Batch Chemical Processes
Dominique Bonvin1,and Gr´egory Fran¸cois2
1Laboratoire d’Automatique, Ecole Polytechnique F´ed´erale de Lausanne
EPFL - Station 9, CH-1015, Lausanne, Switzerland.
2Institute for Materials and Processes, School of Engineering
The University of Edinburgh, Edinburgh EH9 3FB, UK.
corresponding author : dominique.bonvin@epfl.ch
November 30, 2016
Abstract
A batch process is characterized by the repetition of time-varying operations of finite duration. Due to the
repetition, there are two independent “time” variables, namely, the run time during a batch and the batch
index. Accordingly, the control and optimization objectives can be defined for a given batch or over several
batches. This chapter describes the various control and optimization strategies available for the operation
of batch processes. These include online and run-to-run control on the one hand, and repeated numerical
optimization and optimizing control on the other. Several case studies are presented to illustrate the various
approaches
Keywords
Batch control, predictive control, iterative learning control, run-to-run control, batch process optimization,
dynamic optimization, optimizing control, run-to-run optimization.
1
1 Introduction
Batch processing is widely used in the manufacturing of goods and commodity products, in particular in
the chemical, pharmaceutical and food industries1. Batch operation differs significantly from continuous
operation. While in continuous operation the process is maintained at an economically desirable operating
point, in batch operation the process state evolves from an initial to a final time. In the chemical industry for
example, since the design of a continuous plant requires substantial engineering effort, continuous operation is
rarely used for low-volume production. Discontinuous operations can be of the batch or semi-batch type. In
batch operations, the products to be processed are loaded in a vessel and processed without material addition
or removal. This operation permits more flexibility than continuous operation by allowing adjustment of
the operating conditions and the final time. Additional flexibility is available in semi-batch operations,
where reactants are continuously added by adjusting the feedrate profiles, or products are removed via some
outflow. We use the term batch process to include semi-batch processes.
In the chemical industry, many batch processes deal with reaction and separation operations. Reac-
tions are central to chemical processing and can be performed in an homogeneous (single phase) or hetero-
geneous (multi-phase) environment. Separation processes can be of very different types, such as distillation,
absorption, extraction, adsorption, chromatography, crystallization, drying, filtration and centrifugation.
The operation of batch processes follows recipes developed in the laboratory. A sequence of operations is
performed in a pre-specified order in specialized process equipment, yielding a certain amount of product.
The sequence of tasks to be carried out on each piece of equipment, such as heating, cooling, reaction,
distillation, crystallization and drying, is predefined. The desired production volume is then achieved by
repeating the processing steps on a predetermined schedule.
This chapter describes the various control and optimization strategies available for the operation of
batch processes. Section 2 discusses the main features of batch processes, while Section 3 presents the two
types of models that are available for online and run-to run operation, respectively. The rest of the chapter
contains two parts, namely Part A comprising Sections 4-7 and concerned with control (online control, run-
2
to-run control, batch automation, and control applications), and Part B comprising Sections 8-10 and dealing
with optimization (numerical optimization, real-time optimization, and optimization applications). Finally,
a summary and outlook are provided in Section 11.
2 Features of Batch Processes
Process engineers have developed considerable expertise in designing and operating continuous processes.
On the other hand, chemists have been trained to develop new routes for synthesizing chemicals, often
using a batch or semi-batch mode of operation. The operation of batch processes requires considerable
attention, in particular regarding the coordination in time of various processing tasks such as charging,
heating, reacting, separation, cooling, and the determination of optimal temperature and feeding profiles.
Process control engineers have been eager to use their expertise in controlling and optimizing continuous
processes to achieve comparable success with batch processes. However, this is rarely possible, the reason
being significant differences between continuous and batch processes. The two main distinguishing features
are discussed first 2:
Distinguishing feature #1:No steady-state operating point. In batch processes, chemical and physical
transformations proceed from an initial state to a very different final state. In a batch reactor, for
instance, even if the reactor temperature is kept constant, the concentrations, and thus also the reaction
rates, change significantly over the duration of the batch. Consequently, there does not exist a steady-
state operating point around which the control system can be designed. The decision variables are
infinite-dimensional time profiles. Furthermore, important process characteristics—such as static gains,
time constants and time delays—are time varying.
Distinguishing feature #2: Repetitive nature. Batch processing is characterized by the frequent rep-
etition of batch runs. Hence, it is appealing to use the results from previous runs to optimize the
operation of subsequent ones. This has generated the industrially relevant topics of run-to-run control
and run-to-run optimization.
3
In addition to these two features, a number of issues tend to complicate the operation of batch processes:
Nonlinear behavior. Constitutive equations such as reaction rates and thermodynamic relationships
are typically nonlinear. Since a batch process operates over a wide range of conditions, it is not possible
to use, for the purpose of control design and optimization, models that have been linearized around a
steady-state operating point, as this is typically done for continuous processes.
Poor models. There is little time in batch processing for thorough investigations of the reaction, mixing,
heat- and mass-transfer issues. Consequently, the models are often poor. For example, it may well
happen that, in the production of specialty chemicals, the number of significant reactions is unknown,
not to mention their stoichiometry or kinetics.
Few specific measurements. The sensors that allow measuring concentrations online are rare. Chemical
composition is usually determined by drawing a sample and analyzing it offline, that is, by using invasive
and destructive methods. Furthermore, the available measurements—often physical quantities such as
temperature, pressure, torque, turbidity, reflective index and electric conductivity—might exhibit low
accuracy due to the wide range of operation that the measuring instrument has to cover.
Constrained operation. A process is typically designed to operate in a limited region of the state space.
In addition, the values that the inputs can take are upper and lower bounded. The presence of these
limitations (labeled constraints) complicates the design of operational strategies for two main reasons:
i) even if the process is linear or has been linearized along a reference trajectory, constraints make it
nonlinear, and ii) controllability might be lost when a manipulated variable hits a constraint. Due to
the wide operating range of batch processes, it is rarely possible to design the process so as to enforce
feasible operation close to constraints, as this is typically done for continuous processes. In fact, it has
been our experience that safety and operational constraints dominate the operation of batch processes.
Presence of disturbances. Operator errors (e.g. wrong stirrer or solvent choice, incorrect material
loading) and processing problems (e.g. fouling of sensors and reactor walls, insufficient mixing, incorrect
feeding profiles, sensor failures) represent major disturbances that, unfortunately, cannot be totally
4
ruled out. There are other unmeasured disturbances entering the process as the result of upstream
process variability such as impurities in the raw materials.
Irreversible behavior. In processes with history-dependent product properties, such as polymerization
and crystallization, it is often impossible to introduce remedial corrections once off-specification ma-
terial has been produced. This contrasts with continuous processes, where appropriate control action
can bring the process back to the desired steady state following an upset in operating conditions.
Limited corrective action. The ability to influence the process typically decreases with time. This,
together with the finite duration of a batch run, limits the impact of corrective actions. Often, if a
batch run shows a deviation in product quality, the charge has to be discarded.
3 Models of Batch Processes
Model-based control and optimization techniques rely on appropriate mathematical representations. The
meaning of the qualifier “appropriate” depends on the system at hand and the objectives of the study. Some
modeling aspects of batch processes are addressed next3.
3.1 What to Model?
Consider the example of a batch chemical reactor. The reactor system comprises the reactor vessel and
one or several reactions. For reasons mentioned above, the reactions are often poorly known in the batch
environment. Even when the desired reaction and the main side reactions are well documented, there
might exist additional poorly known or totally unknown reactions. For example, it is frequently observed
that the desired products and the known side products do not account for all the transformed reactants.
Hence, since the stoichiometry of the reaction system is not completely known, the kinetic models are often
of aggregate nature, that is, they encompass real and pseudo (lumped) reactions in order to describe the
observed concentrations. This way, the number of modeled reactions, and thereby the number of kinetic
5
parameters to be estimated, can be kept low. Furthermore, to determine the operational strategy for a
reactor, it is necessary to consider the reaction kinetics, reactor dynamics and operational constraints.
In industrial situations, the dynamics associated with the thermal exchange between the reactor and the
jacket are often dominant and must necessarily be included in the model. The reasoning developed here
for batch reactors4,5 extends similarly to other batch units such as batch distillation columns6,7 and batch
crystallizers 8,9 .
3.2 Model Types
The models used can be of several types, the characteristics of which are detailed next.
Data-driven black-box models. Despite their simplicity, empirical input-output models are often able
to satisfactorily represent the relationship between manipulated and observed variables. Linear and
nonlinear ARMAX-type models are readily used to represent dynamic systems10,11. Neural network
methods have also been proposed to model the dynamics of batch processes12. More recently, re-
searchers have investigated the use of multivariate statistical partial least-squares (PLS) models as a
means to predict product quality in batch reactors13. Experimental design techniques can help as-
sess and increase the validity of data-driven models14. Although simple and relatively easy to obtain,
data-driven input-output models have certain drawbacks:
1. They often exhibit good interpolative capabilities, yet they are inadequate for predicting the
process behavior outside the experimental domain in which the data were collected for model
building. Such a feature significantly limits the applicability of these models for optimization,
that is, the determination of better profiles that have not been seen before.
2. Input-output models represent a dynamic relationship only between variables that are manip-
ulated or measured. Unfortunately, some key variables, such as concentrations, often remain
unmeasured in batch processes and thus cannot be modeled via data-driven models.
Knowledge-driven white-box models. A mechanistic state-space representation based on energy and
6
material balances is the preferred approach for modeling batch processes. The rate expressions describe
the effect that the temperature and the concentrations have on the various rates. The model of a
given unit relates the independent variables (inputs and disturbances) to the states (concentrations,
temperature and volume) and the corresponding outputs.
The dynamic model of a batch process can be written as follows15:
˙xk(t) = Fxk(t), uk(t), xk(0) = x0,k ,(1)
yk(t) = Hxk(t), uk(t),(2)
zk=Zxk[0, tf], uk[0, tf],(3)
where tdenotes the run time, kthe batch or run index, xthe n-dimensional state vector, and uthe m-
dimensional input vector. There are two types of measured outputs, namely, the p-dimensional vector
of run-time outputs y(t) that are available online and the q-dimensional vector of run-end outputs z
that are available at the final time tf. Note that zis not limited to quantities measured at the final
time but can include quantities inferred from the profiles xk[0, tf] and uk[0, tf], such as the maximal
temperature reached during the batch.
In addition to the run-time dynamics specific to a given run, one has the possibility of updating the
initial conditions and the inputs on a run-to-run basis as follows:
x0,k+1 =Ixk[0, tf], uk[0, tf], x0,0=xinit ,(4)
uk+1[0, tf] = Kxk[0, tf], uk[0, tf], u0[0, tf] = uinit[0, tf].(5)
Note that, in this formulation, the final time tfis the same for all batches.
Mechanistic models are typically derived from physico-chemical laws. They are well suited for a wide
range of process operations. However, they are difficult and time-consuming to build for industrially
relevant processes. A sensitivity analysis can help evaluate the dominant terms in a model and retain
those that are most relevant to the processing objectives 16 . No realistic model is purely mechanistic,
as a few physical parameters typically need to be estimated from process data. As with data-driven
The initial conditions and inputs can be updated on the basis of several prior batches; the relationships proposed here consider
only the previous batch for simplicity of notation.
7
models, experimental design techniques17 are useful tools for building sound models from a limited
amount of data.
Hybrid grey-box models. As a combination of the two extreme cases listed above, hybrid models can
be very helpful in certain situations. They typically possess a simple structure that is based on some
qualitative knowledge of the process 18 . The model parameters are particularly easy to identify if the
model structure can be reduced to an ARMAX-type form 19. For example, reaction lumping is often
exercized. Because of the aggregate structure of the model, it is necessary to adjust certain model
parameters online as, for example, the rate parameters might represent a different aggregation initially
than toward the end of the batch. This can be done continuously or in some ad-hoc fashion, but
clearly at a rate slower than that of the main dynamics. Furthermore, tendency models have been
proposed 20. These models retain the physical understanding of the system but are expressed in a
form that is suitable to be fitted to local behavior. Although tendency modeling appears to be of high
industrial relevance, it has received little attention in the academic research community.
3.3 Static View of a Batch Process
Due to the absence of a steady state, most signals in a batch process evolve with time. Consequently, the
decision variables (here the input profiles u[0, tf] for a given batch run) are of infinite dimension. It is
however possible to represent a batch process as a static map between a finite number of input parameters
(used to define the input profiles before batch start) and the run-end outputs (representing the outcome at
batch end). The key element is the parameterization of the input profiles as
u[0, tf] = Uπ, t,(6)
with the input parameters π. This parameterization is achieved by dividing the input profiles into time
intervals and using a polynomial approximation (of which the simplest form is a constant value) within each
interval. The input parameter vector πcan also include switching times between intervals21 .
With such a parameterization, the batch process can be seen as a static map between the finite set
8
of input parameters πand the run-end outputs z:
z=Mπ.(7)
This static model indicates that, once the input parameters πhave been specified, it is possible to compute
the run-end outputs z. For this, one needs to generate the input profiles u[0, tf] using Eq. (6), integrate
Eq. (1) and generate zvia Eq. (3). Alternatively, with an experimental set-up, one can generate the input
profiles u[0, tf], apply them to the batch process and collect information regarding the run-end outputs z.
Although the static model (7) looks simple, one has to keep in mind that the batch dynamics are hidden by
the fact that only the relationship between the input parameters (before the run) and the run-end outputs
(after the run) are considered. The dynamics are re presented implicitly in the static map between πand
z. These two different ways of looking at the operation of a given batch are illustrated in Figure 1. Very
naturally, the “Dynamic view” will be relevant for online operations, while the “Static view” will be more
suited for run-to-run operations.
x(t)
u(t) y(t)
z
x[0,tf]
u[0,tf]y[0,tf]z
π
Static view
Dynamic Model
(1) - (3)
Dynamic Model
(1) - (3)
Dynamic view
Static Model (7)
Input Parameterization
(6)
Figure 1: Two different views of the same batch run. Dynamic view: the dynamic model describes the
behavior of the states x(t), the run-time outputs y(t) and the run-end outputs zas functions of the inputs
u(t). Static view: the static model z=Mπexplicits the relationship between the input parameters πand
the run-end outputs z.
9
PART A. CONTROL
Batch process control has gained popularity over the past forty years. This is due to the development of novel
algorithms with proven ability to improve the performance of industrial processes despite uncertainty and
perturbations. But the development of new sensors and of automation tools also contributed to this success.
At this stage, it is important to distinguish between control and automation. In practice, industry often
tends to consider that control and automation are two identical topics and use the two words interchangeably.
Although arguable, we will consider hereafter that control corresponds to the theory, that is, the research
field, whereby new control methods, algorithms and controller structures are sought. On the other hand,
automation represents the technology (the set of devices and their organization) that is needed to program
and implement control solutions.
The control of batch processes differs from the control of continuous processes because of the two
main distinguishing features presented in Section 2. First, since batch processes have no steady-state operat-
ing point, the setpoints to track are time-varying trajectories. Second, batch processes are repeated over time
and are characterized by two independent variables, the run time tand the run index k. The independent
variable kprovides additional degrees of freedom for meeting the control objectives when these objectives do
not necessarily have to be completed in a single batch but can be distributed over several successive batches.
Batch process control aims to track two types of references:
a. The run-time references yr ef (t) or yref [0, tf] are determined offline either from accumulated experience
and knowledge of the process, or by solving a dynamic optimization problem (see Section 8.1). Exam-
ples of tra jectories to be tracked include concentration, temperature, pressure, conversion and viscosity
profiles. In general, trajectory tracking can be achieved in two ways: (i) via “online control” in a single
batch, whereby input adjustments are made as the batch progresses based on online measurements, or
(ii) via “run-to-run control” over several batches, where the input trajectories are computed between
batches using information gathered from previous batches. It is also possible to combine approaches
(i) and (ii) as will be detailed later.
10
b. The run-end references zref are associated with the run-end outputs that are only available at the end
of the batch. The most common run-end outputs are product quality, productivity and selectivity. As
with run-time references, the run-end references can be tracked online or on a run-to-run basis.
With these two types of control objectives and the two different ways of reaching them (online and over
several runs), there are four different control strategies as illustrated in Figure 2. It is interesting to see what
view (dynamic or static) each control strategy uses. It turns out that Strategies 1-3 use a dynamic view
of the batch process, with only Strategy 4 based on the static view. In the following, Section 4 addresses
online control (the first row in Figure 1, Strategies 1 and 2), while Section 5 deals with run-to-run control
(the second row in Figure 1, Strategies 3 and 4).
4 Online Control
The idea of “online control” is to take corrective action during the batch, that is, in run-time t. It is possible
to do so with two objectives in mind, namely, the run-time reference trajectories yref (t) and the run-end
references zref . This will be discussed in the next two subsections.
4.1 Feedback Control of Run-time Outputs (Strategy 1)
We will address successively the application of conventional feedback control to batch processes, the thermal
control of batch reactors, and the stability issue for batch processes.
4.1.1 Conventional feedback control
The application of online feedback control to batch processes has the peculiarity that the setpoints are often
time-varying trajectories. Even if some of the controlled variables, such as the temperature in isothermal
operation, remain constant, the key process parameters (static gains, time constants and time delays) can
11
Run-end references
zref
Run-time references
yref (t) or yref [0,tf]
Control objective
Feedback control
1
Iterative learning control
2
Run-to-run control
4
Implementation
aspect
Online
--> u(t), u[t, tf]
Run-to-run
--> u [0,tf]
Predictive control
3
u ( t ) y(t)
FbC
P
u[0, y[0,
tf]
tf]
ILC with run update
P
[0, z
tf]
u
RtR with run update
π
U (π)
P
E
x(t)
^
MPC
u[t,
tf]
P
y(t)
z
pred
(t)
Figure 2: Control strategies for batch processes. The strategies are classified according to the control
objective (horizontal division) and the implementation aspect (vertical division). Each objective can be met
either online or on a run-to-run basis depending on the type of measurements available. The signal u(t)
represents the input vector at time t,u[0, tf] the corresponding input trajectories, y(t) the run-time outputs
measured online at time t, and zthe run-end outputs available at the end of the batch. P stands for “plant”,
E for “state estimator”, FbC for “feedback control”, MPC for “model predictive control”, ILC for “iterative
learning control”, and RtR for “run-to-run control”. When online and run-to-run control are combined, the
run-to-run part u[0, tf] can be considered as the feedforward contribution, while the online part u(t) is the
feedback part.
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vary considerably during the duration of the batch.
Feedback control is implemented using PID techniques or more sophisticated alternatives such as
cascade control, predictive control, disturbance compensation (feedforward control) and time-delay compen-
sation22,23. The online feedback controller can be written formally as
u(t) = Ky(t), yref (t),(8)
where Kis the online control law for run-time outputs as illustrated in Figure 3.
Controller K
Batch Process y(t)
u(t)
yref(t)
Figure 3: Online feedback control of the run-time outputs yin a batch process.
One way to adapt for variations in process characteristics is via adaptive control24 . However,
adaptive control requires continuous process identification, which is often impractical in batch processes due
to the finite batch duration. Instead, practitioners use conventional feedback control and schedule the values
of some of the controller parameters (mostly the gains, thus leading to gain scheduling 25 ). Appropriate
values of the control parameters can be computed offline ahead of time for the various phases of the batch
run and stored for later use.
A few industrial applications of advanced feedback control are available in the literature. For exam-
ple, a successful implementation of temperature control in an industrial 35-m3semi-batch polymerization
reactor using a flatness-based two-degree-of-freedom controller has been reported26. Also, 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-liter jacketed batch suspension methyl methacrylate polymerization
reactor has been compared27. As expected, the performance of the standard PI controller was the poorest
since the controller parameters were fixed and not adapted to changing process characteristics. The self-
tuning PID control performed better because available measurements were used to adapt the controller to
13
the varying process characteristics. The two nonlinear controllers, which were based on differential-geometric
techniques requiring full-state measurement28, showed excellent performance despite significant uncertainty
in the heat-transfer coefficient.
4.1.2 Control of heat generation
The control of exothermic reactions in batch reactors is a challenging problem that is tackled using one of
various operation modes 29 :
1) Isothermal batch operation. Most reaction systems are investigated isothermally in the laboratory.
For safety and selectivity reasons, pilot-plant and industrial reactors are also run isothermally at the same
temperature specifications. However, the transfer function between the jacket inlet temperature and the
reactor temperature depends on the amount of heat evolved, and it can become unstable for certain combi-
nations of the reaction parameters29. Adaptation of the controller parameters is therefore necessary. Most
often, this is done by using pre-computed values, that is, via gain scheduling. An energy balance around the
reactor is sometimes used to estimate the heat generated by the chemical reactions, which represents the
major disturbance for the control system. This estimate can then be used very effectively in a feedforward
scheme. It is important to mention that isothermal batch operation often exhibits low productivity because
it is limited by the maximal heat-generation rate (corresponding to the maximal heat-removal capacity of
the reactor vessel), which typically occurs initially for only a short period of time.
2) Isothermal semi-batch operation. The productivity of isothermal batch reactors can be increased
through semi-batch operation. This way, the reactant concentrations can be increased once the initial
phase characterized by the heat-removal limitation is over. For strongly exothermic reactions, semi-batch
operation has the additional advantage of improving the effective heat-removal capacity through feeding of
cold reactants. This, in turn, increases the productivity of the reactor by allowing higher temperatures.
Hence, isothermal semi-batch operation is often the preferred way of running discontinuous reactors in
industry. The temperature is controlled by manipulating the feedrate of the limiting reactant. However, the
14
system exhibits non-minimum phase behavior, that is, a flowrate increase of the cold feed first reduces the
temperature before the temperature raises due to higher reaction rates, which significantly complicates the
control30. Furthermore, in most semi-batch operations, it is useful to finish off with a batch phase so as to
fully consume the limiting reactant. Without a kinetic model, it is not obvious when to switch from the
semi-batch to the batch mode. Finally, although isothermal, the operation of a semi-batch reactor should
be such that the heat-removal constraint is never violated. This has forced industrial practice to be rather
conservative, with a temperature setpoint often chosen from adiabatic considerations.
3) Constant-rate batch operation. Since a key objective in the operation of batch reactors is to master
the amount of heat produced by the chemical transformations, it is meaningful to maintain a constant reaction
rate. The temperature is kept low initially when the concentrations are high, to be increased with time as
the reactants deplete. This way, the reactor productivity is considerably higher than through isothermal
operation. However, large temperature excursions can have a detrimental effect on product selectivity. If
reaction kinetics are known, it is a simple matter to compute the temperature profile that results in the
desired heat generation. If the heat evolution can be measured or estimated online, feedback control can
maintain it at the (constant) setpoint. Laboratory-scale reaction calorimeters have been devised to measure
or estimate the heat-evolution profiles of (possibly unknown) reaction systems31. The temperature profiles
obtained in the laboratory can then be used in production environments, at least in qualitative terms,
provided that sufficient heat-removal capacity is available.
4.1.3 Run-time stability of batch processes
Stability is of uppermost importance to prevent runaway-type behavior. However, run-time asymptotic
stability is of little value since tfis finite. Hence, one can say that “instability” can be tolerated for a batch
process because of its finite duration.
Yet, an important issue in run time is reproducibility, which addresses the question of whether the
trajectories of various runs with sufficiently close initial conditions and identical input profiles will remain
15
close during the run. The problem of stability is in fact related to the sensitivity with respect to perturbations.
The basic framework for assessing this sensitivity is as follows:
the system is perturbed by either variations of the initial conditions or external disturbances that affect
the system states,
sensitivity is assessed as a norm indicating the relative effect of the perturbations.
For a system to be stable, this norm must remain bounded for a bounded perturbation, and it must go to
zero with time for a vanishing perturbation. However, when dealing with finite-time systems, it is difficult
to infer stability from this norm since, except for some special cases such as finite escape time, boundedness
is guaranteed. Also, the vanishing behavior with time cannot be analyzed since tfis finite. The element of
interest is in fact the numerical value of the norm, and the issue becomes quantitative rather than binary
(yes-no). Details can be found elsewhere 15.
4.2 Predictive Control of Run-end Outputs (Strategy 2)
With a sufficiently accurate process model, and in the absence of disturbances, tracking the profiles de-
termined offline is often sufficient to meet the batch-end product quality requirements32. However, in the
presence of disturbances, following pre-specified profiles is unlikely to lead to the desired product quality.
Hence, the following 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 to compute the required corrective
control action. Model predictive control (MPC) is well suited for the task of controlling future quantities that
need to be predicted 33 . This approach to batch control can, therefore, be formulated as an MPC problem
with a shrinking prediction horizon (equal to the remaining duration of the batch) and an objective function
that penalizes deviations from the desired product quality at batch end34.
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4.2.1 Model predictive control
With the MPC strategy, two steps need to be implemented online at each sampling time, namely, the
estimation of the current states and the computation of piecewise-constant profiles of future control moves.
First step: Online estimation of x(t). This is done via state estimation using Model (1)-(2) and measure-
ments of the run-time outputs y(t)35,36. The resulting state estimate is denoted ˆx(t). However, this task
is difficult to implement with sufficient accuracy because of some of the limiting characteristics of batch
processes, in particular their nonlinear behavior, the presence of large model inaccuracies and the lack of
specific measurements.
Second step: Online prediction of zand computation of u[t, tf]. In the context of batch operation, predic-
tive control consists in determining the inputs u[t, tf] to meet zr ef by solving at time ta finite-horizon
open-loop optimal control problem. Concretely, we consider the dynamic model given by Eqns (1) and (3)
and solve the following optimization problem at the discrete time instant ti:
min
u[ti,tf]J:= 1
2zT(ti)Pz(ti) + 1
2Ztf
ti
uT(t)R u(t)dt (9)
s.t. ˙x(t) = Fx(t), u(t), x(ti) = ˆx(ti),(10)
zpred(ti) = Zx[0, tf], u[0, tf],(11)
x(tf) X ,(12)
where zpred(ti) are the run-end outputs predicted at time ti,z(ti) := zpred(ti)zref the predicted run-end
errors, Pand Rare positive-definite weighting matrices of appropriate dimensions, Xis the bounded region
of state space where the final state should lie, tiis the current discrete time instant at which the optimization
is performed, and tfthe final time. Note that the profiles u[0, tf] and x[0, tf] include (i) the parts u[0, ti]
and x[0, ti] that were determined prior to the time ti, and (ii) the the parts u[ti, tf] and x[ti, tf] that are
computed at ti. The integral term in the objective function (9) is optional and serves to penalize large
control efforts. Numerical optimization yields the control sequence, u[ti, tf], of which only the first part,
u[ti, ti+h], is applied to the plant in an open-loop fashion, where his the sampling period. Numerical
optimization is then repeated at every sampling instant. Note that it is important to include a bounded
17
region for the final states for the sake of stability33.
The control law can be written formally as
u[t, tf] = Pzpred(t), zr ef ,(13)
where Pis the predictive control law for run-end outputs and zpred (t) is the prediction of zavailable at time
tas illustrated in Figure 4.
Model Predictive
Controller P
Batch Process y(t)
z
ref
State Estimation
x(t)
z
u[t, tf]
Figure 4: Online predictive control of the run-end outputs zin a batch process. The model in the “Model
Predictive Controller” block is used to compute zpred(t) from ˆx(t).
Linear MPC is widely used in industry, more so than nonlinear MPC. However, one may ask whether
linear MPC is sufficient to deal with highly nonlinear batch processes? Note that nonlinear MPC is an active
area of research. For example, a nonlinear MPC scheme has been proposed to regulate run-end molecular
properties within bounds in the context of batch polymerization reactors 37.
4.2.2 Alternative ways of meeting zr ef online
In MPC, zis predicted online as zpred(t) from the state estimates ˆx(t), and then the control profile u[t, tf]
is computed via numerical optimization. This is rather demanding, in particular regarding the availability
of an accurate process model for both state estimation and input computation. This requirement can be
reduced significantly with the approaches proposed next.
Use of data-driven PLS models. When the detailed dynamic model (1) is not available, it is possible to use
partial least-squares (PLS) regression models identified from historical data and a few additional experi-
18
ments38 . Then, in real time during the progress of the batch, it is possible to use the PLS model to estimate
the run-end outputs and track zref by adjusting u[t, tf] via run-to-run control (see Section 5.2).
Another less ambitious but very efficient control approach has also been proposed, whereby, if the prediction
of zfalls outside a predefined no-control region, a mid-course correction is implemented39. Note that the
approach can easily be modified to accommodate more than one mid-course correction.
Still another variant of the use of PLS models involves controlling the process in the reduced space (scores)
of a latent variable model rather than in the space of the discretized inputs40 .
Use of online tracking. An alternative for meeting run-end references using online measurements consists
in tracking some feasible trajectory, zref (t), whose main purpose is to enforce zref at final time, that is,
zref (tf) = zref 41 . This necessitates to be able to measure or estimate z(t) during the batch.
5 Run-to-run Control
Run-to-run control takes advantage of the repetition of batches that is characteristic of batch processing.
One uses information from previous batches to improve the performance of the next batch. This can be
achieved with respect to both run-time and run-end objectives, as discussed in the next two subsections.
5.1 Iterative Learning Control of Run-time Profiles (Strategy 3)
The time profiles of the manipulated variables can be generated using Iterative Learning Control (ILC),
which exploits information from previous runs to improve the performance of the current run 42 . This
strategy exhibits the limitations of open-loop control with respect to the current run, in particular the
fact that there is no feedback correction for run-time disturbances. Nevertheless, this scheme is useful for
generating time-varying feedforward input terms. The ILC controller has the formal structure
uk+1[0, tf] = Iyk[0, tf], yr ef [0, tf],(14)
19
where Iis the ILC law for run-time outputs. ILC uses the entire profiles of the previous run to generate the
input profiles for the next run as illustrated in Figure 5. ILC has been successfully applied in robotics43,44
and in batch chemical processing 45,46.
ILC
Controller I
Batch Process
uk [0, tf]
k+1 --> k
uk+1[0, tf] y
k
[0, t
f
]
y
ref
[0, t
f
]
Figure 5: ILC of the run-time profiles yk[0, tf] in a batch process. The run update is indicated by k+ 1 k.
5.1.1 ILC problem formulation
A repetitive batch process is considered in operator notation:
yk[0, tf] = Guk[0, tf], xk(0),(15)
where uk[0, tf] and xk(0) represent the input profiles and the initial conditions in run k,Gis the operator
representing the system. This relationship can be obtained from Model (1)-(2) as follows:
With the inputs uk[0, tf] and the initial conditions xk(0) specified, integrate Eq. (1) to obtain xk[0, tf].
Eq. (2) allows computing yk[0, tf] from xk[0, tf] and uk[0, tf].
For the case where the system operator is linear and the signals are in discrete time, Eq. (15) can
be expressed in matrix form as follows:
yk=G uk+y0,k,(16)
with yk∈ ℜNp,uk∈ ℜN m and G∈ ℜN p×N m, where Nis the finite number of time samples, pthe number
of outputs, and mthe number of inputs. y0,k ∈ ℜN p represents the response to the initial conditions xk(0).
ILC tries to improve trajectory following by utilizing the previous-cycle tracking errors. The ILC
update law for the inputs uk+1 is given as
uk+1 =Auk+Bek, ek=yr ef yk,(17)
20
where yref ∈ ℜN p are the references to be tracked, and A∈ ℜN m×N m and B∈ ℜN m×Np are operators
applied to the previous-cycle inputs and tracking errors, respectively. In the remainder of this section
dealing with ILC, the signals without an explicit time dependency are expressed as vectors containing the
Nfinite time samples, that is, uk∈ ℜN m ,yk∈ ℜNp and ek∈ ℜNp .
5.1.2 ILC convergence and residual errors
An ILC law is convergent if the following limits exist:
lim
k→∞ yk=yand lim
k→∞ uk=u.(18)
One way to ensure convergence is that the relation between ukand uk+1 be a contraction mapping in some
appropriate norm:
kuk+1k ≤ ρkukk,0ρ < 1.(19)
Using (16) in Eq. (17) gives:
uk+1 = (ABG)uk+B(yref y0,k ).(20)
Note that (yref y0,k ) does not represent errors as in Eq. (17), but the differences between the reference
trajectories and the responses to the initial conditions. Also, in contrast to Eq. (17), where ekdepends on
ukthrough yk, (yref y0,k ) is independent of uk. Hence, the convergence of the algorithm depends on the
homogenous part of Eq. (20) and the condition for convergence is:
kABGk<1.(21)
If the iterative scheme converges, with uk=uk+1 =uand y0,k =y0,k+1 =y0, Eq. (20) gives:
u= (IA+BG)1B(yref y0).(22)
Since the converged outputs yare not necessarily equal to the desired outputs yref , the final tracking
errors can be different from zero. The final tracking errors are given by
e=yref (Gu+y0) = IG(IA+BG)1B(yref y0).(23)
21
If BG is invertible, this equation can be rewritten as
e=G(BG)1(IA) (IA+BG)1B(yref y0).(24)
Note that the residual errors are zero when A=Iand non zero otherwise. Hence, the choice A=Iprovides
integral action along the run index k. Zero errors imply that perfect system inversion has been achieved.
5.1.3 ILC with current-cycle feedback
In conventional ILC schemes, only the tracking errors of the previous cycle are used to adjust the inputs in
the current cycle. To be able to reject within-run perturbations, the errors occurring during the current cycle
can be used as well, which leads to modified update laws. However, from the points of view of convergence
and error analysis, these modified schemes simply correspond to different choices of the operators Aand B
in Eq. (17) as shown next. Let
uk+1 =uff
k+1 +ufb
k+1,(25)
with uff
k+1 =¯
Auf f
k+¯
Bek, uf b
k+1 =Cek+1 ,(26)
where the superscripts (·)ff and (·)f b are used to represent the feedforward and feedback parts of the inputs,
and ¯
A,¯
Band Care operators.
Using Eq. (26) in Eq. (25) and combining it with Eqns (16)-(17) gives:
uk+1 = (I+CG)1¯
A(I+CG)¯
BGuk+ (I+CG)1¯
B+C¯
AC(yr ef y0)
=ABGuk+B(yref y0),(27)
where A= (I+CG)1(¯
A+CG) and B= (I+C G)1¯
B+ (I¯
A)C. The convergence condition and
residual errors can be analyzed using the operators Aand B, similarly to what was done in Section 5.1.2.
5.1.4 ILC with improved performance
The standard technique for approximate inversion is to introduce a forgetting factor in the input update 47 .
This causes the residual tracking errors to be non zero over the entire interval. Note, however, that the
22
main difficulty with the feasibility of inversion arises during the first part of the trajectory due to unmatched
initial conditions48. After a certain catch-up time, trajectory following is relatively easy. Hence, the idea is
to allow non-zero tracking errors early in the run and have the errors decrease progressively with run time t,
which can be achieved with an input shift. This is documented next as part of an ILC implementation that
includes (i) input shift, (ii) shift of the previous-cycle errors, and (iii) current-cycle feedback49.
The iterative update law is written as follows:
uk+1(t) = uf f
k+1(t) + Kf b ek+1(t),(28)
where Kf b ∈ ℜNm×N p are the proportional gains of the online feedback controller. The feedforward part of
the current inputs, uf f
k+1(t), consists of shifted versions of the feedforward part of the previous inputs and
the previous-cycle tracking errors:
uff
k+1(t) = uf f
k(t+δu) + Kf f ek(t+δe),(29)
where Kf f ∈ ℜNm×N p are the proportional gains of the feedforward controller, δuthe time shift of the
feedforward input trajectories and δethe time shift of the previous-run error trajectory. The remaining parts
of the inputs and errors are kept constant, that is, uf f
k[tfδu, tf] = uff
k(tfδu) and ek[tfδe, tf] = ek(tfδe),
respectively.
5.2 Run-to-run Control of Run-end Outputs (Strategy 4)
In this case, the interest is to steer the run-end outputs ztowards zref using the input parameters π. For
this, one uses the static model (7) that results from the input parameterization (6). The run-to-run feedback
control law can be written as
πk+1 =Rzk, zref ,(30)
uk[0, tf] = Uπk, t,(31)
23
where Urepresents the input parameterization and Ris the run-to-run feedback control law for run-end
outputs as illustrated in Figure 6. For example, one can use the discrete integral control law
πk+1 =πk+K(zref zk),(32)
where Kis an appropriate gain matrix. To be able to meet exactly all the references, it is necessary to have
at least as many inputs parameters as there are run-end outputs.
Run-to-run
Controller R
Batch
Process
zk
uk[0, tf]
Input
Generator
Output
Selector
yk[0, tf]
πk
k+1 --> k
Static Map
zref
πk+1
Figure 6: Run-to-run control of the run-end outputs zin a batch process. The input generator constructs
uk[0, tf] from πkaccording to Eq. (31), while the output selector selects the run-end outputs.
5.2.1 Control algorithm
Run-to-run control proceeds as follows:
1. Initialization. Parameterize the input profiles u(t) as Uπ , tin such a way that the input parameter
vector πhas the same dimension as the vector of run-end outputs zin order to generate a square
control problem. Set k= 1 and select an initial guess for πk.
2. Iteration. Construct the inputs uk[0, tf] using Eq. (31) and apply them to the process. Complete the
run and determine zkfrom the measurements.
3. Update the input parameters using the control law (30). Set k:= k+ 1 and repeat Steps 2-3 until
convergence.
24
5.2.2 Convergence analysis
The convergence of the run-to-run algorithm can be determined by analyzing the closed-loop error dynamics,
where the errors are ek:= zref zk.
We discuss next a convergence analysis that considers a linearized version of System (7) and the
linear integral control law (32). With the linear model,
zk=S πk,(33)
where S=∂M
∂π
¯πis the sensitivity matrix computed at the nominal operating point corresponding to ¯π, the
linearized error dynamics become:
ek+1 =ekS Kek= (IS K)ek.(34)
The eigenvalues of the matrix (IS K) should be within the unit circle for the algorithm to converge.
Convergence is determined by both the sensitivity matrix Sand the controller gains K. Convergence analysis
of run-to-run control algorithms is also possible when the relationship between πkand zkis nonlinear, but
it typically requires additional assumptions regarding the nature of the nonlinearities50 .
5.2.3 Run-to-run stability
The interest in studying stability in the run index karises from the necessity to guarantee convergence of
run-to-run control schemes. Here, the standard notion of stability applies as the independent variable kgoes
to infinity. The main conceptual difference with the stability of continuous processes is that “equilibrium”
refers to entire trajectories. Hence, the norms have to be defined in the space of functions Lsuch as the
integral squared norm l2of the signal x(t):
kx[0, tf]kl2=Ztf
0x(t)2dt1/2.(35)
For studying stability with respect to the run index k, System (1) is considered under closed-loop
operation, that is, with all possible online and run-to run feedback loops. Hence, we no longer restrict our
25
attention to the input-output run-to-run π-zbehavior, but we consider the entire state vector. For simplicity,
let us assume that tfis constant for all runs. When dealing with the kth run, the tra jectories of the (k1)st
run are known, which fixes uk[0, tf] according to the ILC control law (14). These input profiles, along with
the online feedback law (8) or (13), are applied to System (1) to obtain xk(t) for all tand thus xk[0, tf]. All
these operations can be represented formally as:
xk[0, tf] = Fxk1[0, tf], x0[0, tf] = xinit[0, tf],(36)
where xinit[0, tf] are the initial state tra jectories obtained by integration of Eq. (1) for k= 0, x0,0=xinit
and u0[0, tf] = uinit[0, tf]. Eq. (36) describes the run-to-run dynamics associated with all control activities.
Run-to-run stability is considered around the equilibrium trajectory computed from (36), that is ¯x[0, tf] =
F¯x[0, tf], and is investigated using a Lyapunov-function method51.
6 Batch Automation
Although batch processing was dominant until the 30’s, batch process control received significant interest
only 50 years later 52 . This is mainly due to the fact that, when control theory started to emerge in the 40’s,
there was a shift from the dominance of batch processing to continuous operations. Hence, the dynamics and
control of batch processes became a subject of investigation in the 80’s. Engineers then tried to carry over
to batch processes the experience gained over the years with the control of continuous processes. However,
the specificities of batch processes make their control quite challenging.
This is also true at the implementation level. If continuous processes typically exhibit four main
regimes, namely, start-up, continuous operation, possibly grade transition and shut down, batch processes
are much more versatile since, in the absence of a steady state, the system evolves freely from a set of initial
conditions to a final state. During this transient operation, the recipe can be rather varied, ranging from
basic isothermal operation to a succession of complex operations that can involve discrete decisions. Also,
batch processes are often repeated over time, with or without changes between batches, which makes the
task of designing a control strategy much more involved than for continuous operations.
26
The control of batch operations relies on various types of controllers that include stand-alone con-
trollers, programmable logic controllers (PLCs), distributed control systems (DCS), and personal computers
(PCs). These four platforms are discussed next.
6.1 Stand-alone Controllers
The basic stand-alone controllers are single-loop controllers (SLCs)53. These devices generally embed a
microprocessor with fixed functionalities such as PID control. The popularity of SLCs arose from their
simplicity, low cost and small size. Originally, stand-alone controllers were capable of integrating dual-
loop control, that is, they can handle two control loops with basic on/off and PID control. More recently,
these devices have incorporated self-tuning algorithms and, more importantly for batch processes, time
scheduling and sequencing functionalities. Still, the main advantage of SLCs is their low cost per control
loop. Furthermore, they are often used in parallel or in combination with more advanced control structures
such as DCS, mainly because they can achieve very acceptable control performance for minimal investment
regarding cost, maintenance and required knowledge.
6.2 Programmable Logic Controllers
PLCs were introduced in the process industries in the early 70’s, following the development in the automotive
industry, as computing systems “that had the flexibility of a computer, yet could be programmed and main-
tained by plant engineers and technicians”53. Before PLCs, relays, counters and timers were mainly used,
but they offered much less flexibility. PLCs can perform complex computations as they have a computational
power that is comparable to a small PC 54 .
Considerable progress was made with the introduction of micro-PLCs, which can be installed close to
the process at a much lower cost. The development of Supervisory Control and Data Acquisition (SCADA)
constitutes another breakthrough that has increased the scope of applications. It is possible to handle
numerous operations distributed over a large distance, thereby opening up the applicability of PLCs to cases
27
where batch process operations are fully integrated in the plant-wide operation of a production site.
6.3 Distributed Control Systems
DCS are control systems with control elements spread around the plant. They generally propose a hierarchical
approach to control, with several interconnected layers operating at different time scales. At the top of the
hierarchy is the production/scheduling layer, where the major decisions are taken on the basis of market
considerations and measurements collected from the plant. These decisions are sent to an intermediate layer,
where there are used, together with plant measurements and estimated fluctuations on price and raw-material
quality, to optimize the plant performance. These decisions can take the form of setpoint trajectories for
low-level controllers. This intermediate layer, which is often referred to as the real-time optimization layer, is
implemented via DCS55. Because of their distributed nature, DCS are helpful to integrate batch operations
into the continuous operation of a plant, which has considerable impact on the planning, scheduling and
real-time optimization tasks 56 . The control and optimization hardware has improved continuously over the
past decades, thereby keeping pace with the development of advanced control and optimization methods.
Although the differences between DCS and SCADA can appear to be subtle, one could state the
following56: DCS (i) are mainly process driven, (ii) are focused on small geographic areas, (iii) are suited
to large integrated chemical plants, (iv) rely on good data quality, and (v) incorporate powerful closed-loop
control hardware. On the other hand, SCADA is generally more “event driven”, which makes it more suited
to the supervision of multiple independent systems.
6.4 Personal Computers
PCs can also be used for the control and optimization of batch processes. It has been reported that 25% of
batch process control involve personal computers. Of course, this figure does not mean that 25% of batch
process control involves only PCs, as PCs are often used in combination with other controllers such as SLCs
or PLCs. In industry, PCs are not only used to program and run control algorithms, but they include
28
many other elements such as interfaces, communication protocols and networking53. The development of
user-friendly software interfaces, of simulation tools, and of fast and reliable optimization software has been
key to the development of batch processing. With the relatively low cost of PCs, it has become economically
viable to use them routinely for control. While a PC gives a lot of flexibility, especially for taking educated
decisions in the context of a multi-layered control structure, it can also be used as a stand-alone advanced
controller or even as an online optimization tool. In practice, however, PC-based control is often limited to
supervisory control, where the PC sends recommendations to low-level controllers. This can take the form of
input profiles to implement or of output tra jectories to track during batch operation. These recommendations
can then be implemented directly by means of low-level controllers, or manually by an operator who can
take the final decision of implementing or not the recommendations53.
7 Control Applications
7.1 Control of Temperature and Final Concentrations in a Semi-batch Reactor
7.1.1 Reaction system
Consider the reaction of pyrrole Awith diketene Bto produce 2-acetoacetyl pyrrole C. Diketene is a very
aggressive compound that reacts with itself and other species to produce the undesired products D,E, and
F57. The desired and side reactions are
A+BC,
2BD,
C+BE,
BF.
The reactions are highly exothermic, that is, they produce heat. The reactor is operated in semi-batch mode
with Apresent initially in the reactor and Badded with the feedrate profile u(t). Moreover, the reaction
29
system is kept isothermal by removing the heat produced by the chemical reactions through a cooling jacket
surrounding the reactor. The flowrate of cooling fluid is adjusted to keep the reactor temperature at its
desired setpoint.
7.1.2 Model of the reactor
The nonlinear dynamic model of the reaction system is obtained from a material balance for each species
and a total mass balance 58 :
dcA
dt =u
VcAk1cAcB,(37)
dcB
dt =u
V(cin
BcB)k1cAcB2k2c2
Bk3cCcBk4cB,(38)
dcC
dt =u
VcC+k1cAcBk3cCcB,(39)
dcD
dt =u
VcD+k2c2
B,(40)
dcE
dt =u
VcE+k3cCcB,(41)
dcF
dt =u
VcF+k4cB,(42)
dV
dt =u, (43)
where ciis the concentration of the ith species, Vis the reactor volume, kjis the rate constant of the jth
reaction, cin
Bis the inlet concentration of B, and uis the volumetric feedrate of B.
7.1.3 Control objective
One would like to operate the reactor safely and efficiently. If the heat produced cannot be removed by
the cooling jacket, the reactions accelerate and produce even more heat. This positive feedback effect,
known as thermal runaway, can result in high temperature and pressure and thus lead to an explosion;
this is precisely what caused the Bhopal disaster59. The performance of the reactor is evaluated by its
productivity, namely, the amount of desired product available at final time, as well as its selectivity, namely,
the portion of converted reactant Bthat forms the desired product C. Note that the performance criteria
30
(productivity and selectivity) are evaluated at the final time, whereas the manipulated variable (feedrate
of B) is a run-time profile. The control objective is to operate isothermally at 50 C and match the final
concentrations cB,ref and cD,ref obtained in the laboratory
7.1.4 Online control, ILC and run-to run control
The control strategies, illustrated in Figure 7, include (i) a cascade scheme for online feedback control of
Tr(t) and ILC update of Tf f
j,ref [0, tf], and (ii) run-to-run control of the final concentrations through update
of the input parameters u1and u2.
cooling fluid
Slave
B
Master
TT
TT
R2R
c
B (
t
f
) c
D
( t
f
)
u1 u2
ILC Tr
T
ff [0,tf]
j,ref
[0,tf]
T
fb
j,ref
T
r,ref
T
j,ref
T
r
T
j
Figure 7: Online and run-to-run strategies for controlling the reactor temperature Tr(t) and the final con-
centrations cB(tf) and cD(tf).
Control of the reactor temperature Tris implemented by means of a PI/P-cascade scheme that
adjusts the flowrate of cooling fluid. The master loop contributes a feedback term to the reference value for
the jacket temperature Tj. This reference value, which also includes a feedforward term, reads
Tj,ref,k (t) = Tf f
j,ref,k (t) + Tf b
j,ref,k (t),(44)
Tfb
j,ref,k (t) = KRek(t) + 1
τIZt
0
ek(τ),(45)
Tff
j,ref,k+1 [0, tf] = Tf f
j,ref,k [0, tf] + KILC ek[0, tf],(46)
31
where ek(t) := Tr,ref (t)Tr,k(t), KRis the proportional gain, and τIis the integral time constant of the PI
master controller. KI LC is the gain of the ILC controller. Eq. (45) computes the feedback term Tf b
j,ref,k (t)
using a PI feedback law, while Eq. (46) implements run-to-run adaptation of the feedforward term Tf f
j,ref,k (t)
based on ILC.
The second manipulated variable is u, the feedrate of reactant B, through which the two reactions
can be steered and brought to the desired final concentrations. Since these final concentrations are measured
only at the end of the batch, the feedrate of Bis adjusted on a run-to-run basis. Two input parameters are
needed to control these two concentrations. Hence, the feedrate profile u[0, tf] is parameterized using the
two feedrate levels u1and u2, each valid over half the batch time. The sensitivities of the final concentrations
z= (cB(tf), cD(tf))Twith respect to π= (u1, u2)Tis evaluated experimentally, which gives S=∂z
∂π
k. With
this notation, the discrete integral control law has the form
πk+1 =πk+KRtR S1[zref zk],(47)
where KRtR is the 2 ×2 diagonal gain matrix of the run-to-run controller.
The contributions of ILC and run-to-run control to the control of reactor temperature and final
concentrations are illustrated in Figures 8 and 9, respectively. One sees that adjustment of Tf f
j,ref using ILC
reduces the maximal temperature excursion from 57.1C to 52.7C.
7.2 Scale-up via Feedback Control
Short times to market are required in the specialty chemicals industry. One way to reduce this time to
market is by skipping the pilot-plant investigations. However, due to scale-related differences in operating
conditions, direct extrapolation of conditions obtained in the laboratory is often impossible, especially when
terminal objectives must be met and path constraints respected. In fact, ensuring feasibility at the industrial
scale is of paramount importance. This section presents an example for which the combination of online and
run-to-run control allows meeting production requirements over a few batches.
32
01234
-10
10
30
50
70
T [°C]
t [h]
tf
01234
-10
10
30
50
70
T [°C]
t [h]
tf
Tj,ref
ff
Tr
Tj,ref
Tj,ref
Tr
Tj,ref
ff
Tj
Tj
Figure 8: Temperature control around the constant setpoint Tr,ref = 50 C, without ILC (left), and after
three ILC iterations (right). The red and grey solid lines represents the reactor temperature Trand the
jacket temperature Tj, respectively. The blue and green dashed lines represent the feedforward term of the
jacket temperature setpoint Tf f
j,ref and the resulting jacket temperature setpoint Tj,r ef, respectively.
0 1 2 3 4
10
20
30
40
50
u [l/h]
t [h]
tf
01234
0
0.1
0..2
0..3
t [h]
tf
cB,ref
u1
u2
0 1 2 3 4
0
0.1
0..2
0..3
t [h]
tf
cD,ref
cD
cB
Figure 9: Run-to-run control for meeting the run-end concentrations cB,ref and cD,ref . The inlet feedrate is
parameterized using the two levels u1and u2(left). The concentration profiles for cB(t) and cD(t) are shown
in the center and at right, respectively. The dashed lines show the initial profiles, while the solid lines show
the corresponding profiles after two iterations. The references cB,ref and cD,ref are reached via run-to-run
adjustment of u1and u2.
33
7.2.1 Problem formulation
Consider the following parallel reaction scheme60:
A+BC, 2BD .
The desired product is C, while Dis undesired. The reactions are exothermic. A 1-L reactor is used in the
laboratory, while a jacketed reactor of 5 m3is used in production. The manipulated inputs are the feedrate
F(t) and the coolant flowrate through the jacket Fj(t). The operational requirements can be formulated as:
Tj(t)10C,(48)
yD(tf) = 2nD(tf)
nC(tf) + 2 nD(tf)0.18,(49)
where Tj(t) is the jacket temperature, and nC(tf) and nD(tf) denote the numbers of moles of Cand Dat
final time.
7.2.2 Laboratory recipe
The recipe obtained in the laboratory proposes to initially fill the reactor with A, and then feed B at some
constant feedrate ¯
F, while maintaining the reactor isothermal at Tr= 40C. As cooling is not an issue for
the laboratory reactor equipped with an efficient jacket, experiments were carried out using a scale-down
approach, that is, the cooling rate was artificially limited so as to anticipate the limited cooling capacity of
the industrial reactor. Scaling down is performed by the introduction of an operating constraint that limits
the cooling capacity; in this case, the maximal cooling capacity of the industrial reactor is simply divided
by the scale-up factor:
[qc,max]lab =[(TrTj,min )UA]prod
r,(50)
where r= 5000 is the scale-up factor and U A = 3.7×104J/mol C is the estimated heat-transfer capacity
of the production reactor. With TrTj,min = 30C, the maximal cooling rate is 222 J/min. Table 1
summarizes the key parameters of the laboratory recipe and the (simulated) laboratory results.
34
Table 1: Laboratory recipe and results for the scale-up problem.
Recipe parameters Laboratory results
Tr= 40CcB,in = 5 mol/L nC(tf) = 0.346 mol
cA,o = 0.5 mol/L cB,o = 0 mol/L yD(tf) = 0.170
V0= 1 L tf= 240 min max
tqc(t) = 182.6 J/min
¯
F= 4 ×104L/min
7.2.3 Scale-up via online control, ILC and run-to run control
The goal of scale-up is to reproduce in production the productivity and selectivity that are obtained in the
laboratory, while enforcing the desired reactor temperature. With selectivity and productivity as run-end
outputs, the feedrate profile F[0, tf] is parameterized using two adjustable parameters, namely, the constant
feedrate levels F1and F2, each one valid over half the batch time. Temperature control is done via a combined
feedforward/feedback scheme as illustrated in Section 7.1.
A control problem can be formulated with the following manipulated variables (MVs), controlled variables
(CVs) and setpoints (SPs):
MVs: u(t) = Tj,ref (t), π=F1, F2T
CVs: y(t) = Tr(t), z=nC(tf), yD(tf)T
SPs: yref = 40C, zref =1630 mol,0.17 T
.
The setpoints are chosen as follows:
Tr,ref = 40C is the value used in the laboratory.
nC,ref is the value obtained in the laboratory multiplied by the scale-up factor rminus a 100-mol backoff
to account for scale-related uncertainties and run-time disturbances. The introduction of backoff makes
the control problem more flexible.
yD,ref is the value obtained in the lab, which respects the upper bound of 0.18.
35
The control scheme is shown in Figure 10. The input profiles are updated using (i) the cascade feedback
controller Kto control the reactor temperature Tr(t) online, (ii) the ILC controller Ito improve the reactor
temperature by adjusting Tf f
j,ref (t) , and (iii) the run-to-run controller Rto control z= [ nC(tf), yD(tf)]T
by adjusting π= [ F1, F2]T. Details regarding the implementation of the different control elements can
be found elsewhere 60 .
Online
Measurements
Online
R
Run-to-run
K
I
Run-end
Measurements
Batch
Process
Trajectory
Generation
Run
Update
Tj,ref,k+1[0,tf]
ff
Tj,ref,k(t)
ff
Tj,ref,k(t)
fb
Tj,ref,k(t)
Tj,ref,k(t)
F1,k+1, F2,k+1
F1,k, F2,k
Fk(t)
Tr,ref
Tr,k(t)
nC,ref yD,ref
Tr,ref
xk(t)
xk[0,tf]
ek(t)
ek[0,tf]
+
+
nC,k(tf)
yD,k(tf)
Figure 10: Control scheme for scale-up implementation. Notice the distinction between online and run-to-
run activities. The symbol represents the concentration/expansion of information between a profile (e.g.
xk[0, tf]) and an instantaneous value (e.g. xk(t)).
7.2.4 Simulation results
The recipe presented in Table 1 is applied to the 5-m3industrial reactor, equipped with a 2.5-m3jacket.
In this simulated case study, uncertainty is introduced artificially by modifying the two kinetic parame-
ters, which are reduced by 25% and 20%, respectively. Also, Gaussian noise with standard deviations of
0.001 mol/L and 0.1 C is considered for the measurement of the final concentrations and for the reactor
temperature, respectively. It turns out that, for the first run, application of the laboratory recipe with
36
π=r¯
F , r ¯
FTviolates of the final selectivity of D in the first batch. Upon adapting the MVs with the
proposed scale-up algorithm, the free parts of the recipe are modified iteratively to achieve the production
targets for the industrial reactor, as illustrated in Figure 11.
yD(tf)
5 10 15
0.165
0.17
0.175
0.18
0.185
Batch index
nC(tf)
1520
1560
1600
1640
1480
Figure 11: Evolution of the production of C,nC(tf), and the yield of D,yD(tf), for the large-scale industrial
reactor. Most of the improvement is achieved in the first 7 runs (the dashed lines represent the target values).
7.3 Control of a Batch Distillation Column
A binary batch distillation column is used to illustrate in simulation the application of iterative learning
control. This example is described in detail elsewhere61.
7.3.1 Model of the column
A binary batch distillation column with pequilibrium stages is considered (see Figure 12). Using standard
assumptions62 and writing molar balance equations for the holdup in the reboiler and for the liquid on the
37
Stage p
Reboiler
Stage 2
Condenser
Distillate
Measurement
Input r = L/V
M2
Mp
Mp+1
V, y1
V, yp-1
V, y2
V, yp
M1, x
1
L, xp
L, x3
L, x2
L, x
p+1
D, xp+1
fD(t)
xD(t)
Figure 12: Binary batch distillation column.
various stages and in the condenser, the following model of order (p+ 2) is obtained:
˙
M1= (r1) V, (51)
˙x1=V
M1
(x1y1+rx2),(52)
˙xi=V
Miyi1yi+r(xi+1 xi), i = 2,...,p, (53)
˙xp+1 =V
Mp+1
(ypxp+1),(54)
where xiis the molar liquid fraction, yithe molar vapor fraction, Mithe molar holdup on Stage i,Vthe
vapor flowrate, and Dthe distillate flowrate. Stage 1 refers to the reboiler, Stage pto the top of the column,
and Stage p+ 1 to the condenser. The internal reflux ratio r=L
V=VD
Vis considered as the manipulated
variable. The vapor-liquid equilibrium relationship is:
yi=α xi
1 + (α1)xi
, i = 1,··· , p , (55)
38
where αis the relative volatility. The composition of the accumulated distillate, xD, which is measured with
the sampling time h= 0.1h, is given by:
xD(t) = Pp
i=1 xi(t)Mi(t)xi(0)Mi(0)
M1(t)M1(0) .(56)
The model parameters, the initial conditions and the control objective xD,ref are given in Table 2.
Table 2: Model parameters and initial conditions, i= 2,··· , p.
p10 M1(0) 100 kmol
α1.6x1(0) 0.5
Mi0.2 kmol xi(0) 0.5
Mp+1 2 kmol xp+1(0) 0.5
V15 kmol/h xD,ref 0.9
tf10 h h0.1 h
7.3.2 Operational objective
A batch is divided into 2 operational phases:
1. Start-up phase with full reflux up to time ts:r= 1, t= [0, ts], ts= 1.415 h.
2. Distillation phase: r[0,1], t= (ts, tf], tf= 10 h.
The objective is to adjust the reflux ratio to get the distillate purity xD(tf) = 0.9 at the end of the
batch. This will be obtained by tracking an appropriate reference trajectory for the distillate purity xD(t)
in the second phase. This trajectory, which has the property of ending at the desired distillate purity at
final time, is chosen to be linear, with xD,r ef (ts) = 0.925 and xD,ref (tf) = 0.9. The accumulated distillate
purity xD(t) is measured in Phase 2.
In order to obtain a realistic test scenario, the following uncertainty is considered:
39
- Perturbation: The vapor rate fluctuates every 0.5hfollowing an uniform distribution in the range
V= [13,17] kmol/h.
- Measurement noise: 5% multiplicative Gaussian noise is added to the product composition xD(t).
The values of the squared tracking error Ptf
tse2(t) and the final tracking error e(tf) upon convergence
are averaged over 20 realizations of the perturbation and measurement noise. Also, the variance ve(tf) of
the final tracking error is calculated from 20 realizations.
7.3.3 Trajectory tracking via ILC
Trajectory tracking involving a single output, xD(t), and a single input, r(t), is implemented on a run-to-run
basis via ILC. The initial input trajectory used for ILC is linear with r(ts) = 0.898 and r(tf) = 0.877. The
residual tracking error cannot be reduced to zero for all times because of the non-zero tracking error at
time tsarising from uncertainties in the start-up phase, that is, xD(ts)6=xD,r ef (ts). As a consequence, the
ILC schemes that try to enforce zero residual tracking error do not converge in this case. Instead, non-zero
tracking error has to be tolerated to enforce convergence. This can be accomplished by applying a forgetting
factor or a time shift to the feedforward trajectory. Three ILC schemes without current-cycle feedback are
considered, with N= 100 time samples:
β= 0.999. A forgetting factor βis applied to the feedforward input trajectory in Eq. (26) with ¯
A=β I
and ¯
B=K.
δu= 0.25 h. A small shift is applied to the feedforward input trajectory in Eq. (29).
δu= 1 h, δe= 1 h. The same large shift is applied to the feedforward input and the error trajectory in
Eq. (29).
The proportional gain K= 0.1Iis determined as a compromise between robustness and perfor-
mance. ILC with forgetting factor converges after 30 runs, while the schemes with time shift of the trajec-
tories converge after 25 runs as illustrated in Figure 13. The final tracking error is slightly smaller with the
40
latter methods, especially when the time shift of the feedforward trajectory is reduced to δu= 0.25 has
shown in Table 3. Figure 14 shows the tracking performance of ILC with input shift after 20 runs. Note
that, since it is necessary to have distillate in the product tank to be able to take measurements, tracking
starts one sampling time after the start of Phase 2, that is, at at ts+h.
1 5 10 15 20 25 30
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
Batch index k
β = 0.999
δu = 0.25 h
δu = δ
e = 1 h
Squared tracking error
Figure 13: Evolution of the squared tracking error for three ILC schemes.
0 2 4 6 8 10
0.8
0.85
0.9
0.95
1
Time t [h]
xD
Reference
δu = 0.25 h
tsts+h tf
Figure 14: Tracking performance using ILC with input shift.
41
Table 3: Comparison of the three tracking schemes after 30 runs in terms of the squared tracking error
Ptf
tse2(t), the tracking error at final time e(tf), and its variance ve(tf).
Strategy Ptf
tse2(t)|e(tf)| × 103ve(tf)×105
β= 0.999 0.8738 6.5 4.1
δu=0.25h0.8745 0.5 3.0
δu=1h, δe= 1h0.8833 2.2 3.5
42
PART B. OPTIMIZATION
Process optimization is the method of choice for improving the performance of chemical processes while
enforcing operational constraints63. Long considered as an appealing tool but only applicable to academic
problems, optimization has now become a viable technology64,65. Still, one of the strengths of optimization,
namely, its inherent mathematical rigor, can also be perceived as a weakness, since engineers might some-
times find it difficult to obtain an appropriate mathematical formulation to solve their practical problems.
Furthermore, even when process models are available, the presence of plant-model mismatch and process
disturbances makes the direct use of model-based optimal inputs hazardous.
In the last 30 years, the field of real-time optimization (RTO) has emerged to help overcome the
aforementioned modeling difficulties. RTO integrates process measurements into the optimization framework.
This way, process optimization does not rely exclusively on a (possibly inaccurate) process model but also on
process information stemming from measurements. The first widely available RTO approach was the two-
step approach that adapts the model parameters on the basis of differences between predicted and measured
outputs and uses the updated process model to re-compute the optimal inputs66,67. However, in the presence
of structural plant-model mismatch, this method is very unlikely to drive the plant to optimality68,69 . Hence,
alternatives to the two-step approach have recently been developed. For example, the modifier-adaptation
scheme also proposes to solve a model-based optimization problem, but using a plant model with fixed
model parameters 69 . Correction for uncertainty is made via modifier terms that are added to the cost and
constraint functions of the optimization problem. As the modifiers include information on the differences
between the predicted and the plant necessary conditions of optimality (NCO), this approach is prone to
reach the plant optimum upon convergence. Yet another field has emerged, for which numerical optimization
is not used online. With the so-called self-optimizing approaches70–72 , the optimization problem is recast
as a control problem that uses measurements to enforce certain optimality features for the plant. However,
note that self-optimizing control70 and extremum-seeking control71, which have been developed to optimize
continuous plants, are not suited to the optimization of batch processes.
43
The optimization of a batch process falls naturally in the category of dynamic optimization, that is,
with infinite-dimensional time profiles as decision variables. However, the introduction of input parameteri-
zation can transform the dynamic optimization problem into a static optimization problem with a countable
number of decision variables, a so-called nonlinear program (NLP).
This optimization part is organized as follows. Various aspects of numerical optimization for both
static and dynamic optimizations are presented first, followed by a discussion of two RTO classes, namely, re-
peated numerical optimization and optimizing control. The theoretical developments will then be illustrated
by three case studies.
8 Numerical Optimization
Apart from very specific cases, the standard way of solving an optimization problem is via numerical opti-
mization, for which a model of the process is required. As already mentioned, the optimization of a batch
process can be formulated as a dynamic optimization problem. In this chapter, we will consider the following
constrained dynamic optimization problem:
min
u[0,tf]J:= φx(tf)(57)
s.t. ˙x(t) = Fx(t), u(t), x(0) = x0,(58)
Sx(t), u(t)0,(59)
Tx(tf)0,(60)
where φis the terminal-time cost functional to be minimized, x(t) the n-dimensional vector of state profiles
with the known initial conditions x0,u(t) the m-dimensional vector of input profiles, Sthe nS-dimensional
vector of path constraints that hold during the interval [0, tf], Tthe nT-dimensional vector of terminal
constraints that hold at tf, and tfthe final time, which can be either free or fixed. The optimization
If tfis free, it becomes a decision variable. Note that it is also possible to have additional constant decision variables. In this
case, the decision variables would include the vector of input profiles u[0, tf] and the vector of constant quantities, ρ. This
general case will not be considered here for simplicity of exposition, see72 for details.
44
problem (57)-(60) is said to be in the Mayer form, that is, Jis a terminal-time cost. When an integral cost
is added to φ, the corresponding problem is said to be in the Bolza form, while when it only incorporates
the integral cost, it is referred to as being in the Lagrange form. Note that these three formulations can be
made equivalent by the introduction of additional states 73.
The nonlinear dynamic optimization problem (57)-(60) is difficult to solve due to the time depen-
dency of the various variables. In particular, the inputs u(t) represent an infinite-dimensional decision
vector. This time dependency can be removed via time discretization using, for example, the method of
orthogonal collocation74, which however results in a large vector of decision variables. Another approach
consists in viewing the batch process as a static map between a small number of input parameters πand a
small number of run-end outputs zas discussed in Section 3.3. These approximations reduce the dynamic
optimization problem into a static problem. Next, the following problems will be addressed successively:
dynamic optimization, reformulation of a dynamic optimization problem as a static optimization problem,
static optimization, and effect of uncertainty.
8.1 Dynamic Optimization
Consider the optimization problem (57)-(60), for which we will present the necessary conditions of optimality.
8.1.1 Pontryagin’s minimum principle
The NCO for a dynamic optimization problem are given by Pontryagin’s minimum principle (PMP). The
application of PMP provides insight in the optimal solution and generates explicit conditions for meeting
active path and terminal constraints and forcing certain sensitivities to zero.
Let us define the Hamiltonian function H(t),
H(t) = λT(t)Fx(t), u(t)+µT(t)Sx(t), u(t),(61)
45
and the augmented terminal cost
Φ(tf) = φx(tf)+νTTx(tf),(62)
where λT(t) are the adjoint variables such that
˙
λT(t) = ∂H
∂x (t), λT(tf) = Φ
∂x(tf),(63)
µ(t)0 are the Lagrange multipliers associated with the path constraints, and ν0 are the Lagrange
multipliers associated with the terminal constraints. In dynamic optimization problems, there are both
path and terminal objectives to satisfy, with these objectives being either constraints or sensitivities. PMP
and the introduction of the Hamiltonian function H(t) allow (i) assigning the NCO separately to path and
terminal objectives, and (ii) expressing them in terms of meeting constraints and sensitivities, as illustrated
in Table 4. The solution is generally discontinuous and consists of several intervals or arcs73,75. Each arc is
characterized by a different set of active path constraints, that is, this set changes between successive arcs.
Table 4: NCO for a dynamic optimization problem 73 .
Path Terminal
Constraints µT(t)Sx(t), u(t)= 0, µ(t)0νTTx(tf)= 0, ν 0
Sensitivities ∂H
∂u (t) = 0
8.1.2 Solution methods
Solving the dynamic optimization problem (57)-(60) corresponds to finding the best input profiles u[0, tf]
such that the cost functional is minimized, while meeting both the path and terminal constraints. Since the
decision variables u[0, tf] are infinite dimensional, the inputs need to be parameterized using a finite set of
parameters to be able to use numerical techniques. These techniques can be classified in two main categories
If the decision variables include the constant vector ρ, there will be terminal sensitivity elements for the determination of ρ,
see72 for details.
46
according to the underlying formulation73, namely, direct optimization methods that solve the optimization
problem (57)-(60) directly, and PMP-based methods that attempt to satisfy the NCO given in Table 4.
Direct optimization methods are distinguished further depending on whether the system equations
are integrated explicitly or not 73 . In the sequential approach, the system equations are integrated explicitly,
and the optimization is carried out in the space of the decision variables only. This corresponds to a
“feasible-path” approach since the differential equations are satisfied at each step of the optimization. A
piecewise-constant or piecewise-polynomial approximation of the inputs is often used. The computationally
expensive part of the sequential approach is the accurate integration of the system equations, which needs
to be performed even when the decision variables are far from the optimal solution!
In the simultaneous approach, an approximation of the system equations is introduced to avoid explicit
integration for each candidate set of input profiles, thereby reducing the computational burden. Since
the optimization is carried out in the full space of discretized inputs and states, the differential equations
are satisfied only at the solution 74,76 . This is therefore called an “infeasible-path” approach. The direct
approaches are by far the most commonly used. However, note that the input parameterization is often
chosen arbitrarily by the user, which can affect both the efficiency and the accuracy of the approach77 .
On the other hand, PMP-based methods try to satisfy the first-order NCO given in Table 4. The
NCO involve the state and adjoint variables, which need to be computed via integration. The differential
equation system is a two-point boundary-value problem since initial conditions are available for the states
and terminal conditions for the adjoints. The optimal inputs can be expressed analytically from the NCO
in terms of the states and the adjoints, that is, u[0, tf] = Ux[0, tf], λ[0, tf]. The resulting differential-
algebraic system of equations can be solved using a shooting approach78, that is, the decision variables
include the initial conditions λ(0) that are chosen in order to satisfy λ(tf).
47
8.2 Reformulation of a dynamic optimization problem as a static optimization
problem
Consider the optimization problem (57)-(60). Numerically, this problem can be solved by control vector
parameterization (CVP) 76 , whereby the infinite-dimensional inputs u(t) are parameterized using a finite
number of parameters π. Typically, the inputs are divided into small intervals, and a polynomial approxima-
tion of the inputs is used within each interval. The input parameter vector πcan also include the switching
times between the intervals. It follows that the inputs can be expressed as u(t) = U(π, t) and the optimization
performed with respect to π, where πis the nπ-dimensional vector of constant decision variables.
The difficulty with such a parameterization is that the path constraints S(x, u) have to be discretized
as well, resulting in as many constraints as there are discretization points (typically a large number). However,
if structural information regarding the shape of the optimal solution is available, the path constraints can
be handled differently using an alternative parameterization as explained next.
The solution to a constrained terminal-time dynamic optimization problem is typically discontinuous
and consists of various arcs. For a given input, the ith arc ηi(t) can be of two types:
1. ηi,path : Arc ηi(t) is determined by an active path constraint (constraint-seeking arc),
2. ηi,sens : Arc ηi(t) is not governed by an active path constraint, it is sensitivity seeking.
Let us assume that the path constraints are met using either (i) assignment of input values in
the case of active input bounds, or (ii) online feedback control via the measurement or estimation of the
constrained quantities and their regulation to track the constraint values. The other parts of the inputs are
parameterized as in CVP as u(t) = U(π, t). Since the path constraints are handled directly, they do not
involve the decision variables π. Note that the knowledge of the active path constraints in various intervals
is crucial for this parameterization. The final states, which affect the objective function and the terminal
constraints, can be expressed in terms of the decision variables πas follows:
x(tf) = x0+Ztf
0
Fx(t),U(π, t)dt := F(π),(64)
48
as tfis an element of π. Hence, the dynamic optimization problem (57)-(60) can be reformulated as:
min
πφF(π)(65)
s.t. TF(π)0,(66)
since the system dynamics are included in Fand the path constraints are taken care of as mentioned above.
It follows that the dynamic optimization problem can be reformulated as the following static NLP:
min
πJ:= Φ(π) (67)
s.t. G(π)0.(68)
The scalar cost function to be minimized is written Φ(π) := φF(π), and the ng-dimensional vector of
constraints is written generically G(π) := TF(π).
Some remarks are necessary at this point:
This reformulation is a mathematical way of viewing the dynamic process as a static input-output
π-(Φ, G)” map. However, the process remains dynamic, with the terminal cost and the terminal
constraints being affected by the process dynamics.
This reformulation is clearly an approximation since it relies on: (i) a parameterization of the inputs,
and (ii) approximate tracking of the state constraints.
NLP (67)-(68) is a complex problem if one wants to predict Φ and Gfrom πsince the system dynamics
need to be integrated according to Eq. (64). However, the situation is much simpler in the experimental
context, as one simply apply the inputs πto the plant and measure both Φ and G. In this case, the
model used to predict Φ and Gis replaced by the plant itself. This is a perfect example of the power
of measurement-based optimization compared to numerical optimization, as will be detailed later.
8.3 Static Optimization
The optimization problem is of algebraic nature with a finite number of constant decision variables.
49
8.3.1 KKT necessary conditions of optimality
With the formulation (67)-(68) and the assumption that the cost function Φ and the constraint functions G
are differentiable, the Karush-Kuhn-Tucker (KKT) conditions read 79:
G(π)0,(69)
Φ(π) + (ν)TG(π) = 0,(70)
ν0,(71)
(ν)TG(π) = 0,(72)
where πdenotes the solution, νthe ng-dimensional vector of Lagrange multipliers associated with the
constraints, Φ(π) the nπ-dimensional row vector denoting the cost gradient evaluated at π, and G(π)
the (ng×nπ)-dimensional Jacobian matrix computed at π. For these equations to be necessary conditions,
πneeds to be a regular point for the constraints, which calls for linear independence of the active constraints,
that is, rank{∇Ga(π)}=ng,a , where Garepresents the set of active constraints, whose cardinality is ng ,a.
Condition (69) is the primal feasibility condition, Condition (71) the dual feasibility condition, and
Condition (72) the complementarity slackness condition. The stationarity condition (70) indicates that, at
the solution, collinearity between the cost gradient and a linear combination of the gradients of the active
constraints prevents from finding a search direction that would result in cost reduction while still keeping
the constraints satisfied.
8.3.2 Solution methods
Static optimization can be solved by state-of-the-art nonlinear programming techniques. In the presence
of constraints, the three most popular approaches are 80: (i) penalty-function methods, (ii) interior-point
methods, and (iii) sequential quadratic programming. The main idea in penalty-function methods is to
replace the solution to a constrained optimization problem by the solution to a sequence of unconstrained
optimization problems. This is made possible by incorporating the constraints in the ob jective function via
50
a penalty term, which penalizes any violation of the constraints, while guaranteeing that the two problems
share the same solution (by selecting sufficiently large weights). Interior-point methods also incorporate the
constraints in the objective function 81 . However, the constraints are approached from the feasible region,
and the additive terms increase to become infinitely large at the value of the constraints, thereby acting more
like a barrier than a penalty term. A clear advantage of interior-point methods is that feasible iterates are
generated, while for penalty function methods, feasibility is only guaranteed upon convergence. Note that a
barrier-penalty function that combines the advantages of both approaches has also been proposed82. Another
way of computing the solution to a static optimization problem is to solve the set of NCO, for example
iteratively using sequential quadratic programming (SQP). SQP methods solve a sequence of optimization
sub-problems, each one minimizing a quadratic approximation to the Lagrangian function L:= Φ + νTG
subject to a linear approximation of the constraints. SQP typically uses Newton’s or quasi-Newton methods
to solve the KKT conditions.
8.4 Effect of Uncertainty
8.4.1 Plant-model mismatch
The model used for optimization consists of a set of equations that represent an abstract view, yet always
a simplification, of the real process. Such a model is built based on conservations laws (for mass, numbers
of moles, energy) and constitutive relationships that express kinetics, equilibria and transport phenomena.
The simplifications that are introduced at the modeling stage to obtain a tractable model affect the quality
of the process model in two ways:
(i) some physical or chemical phenomena are assumed to be negligible and are discarded, and
(ii) some dynamic equations are assumed to be at quasi-steady state.
Hence, the structure of the model that is used differs from the “true” model structure. This di-
chotomy gives rise to the so-called structural plant-model mismatch. Furthermore, the model involves a
51
number of physical parameters, whose values are not known accurately. These parameters are identified
using process measurements and, consequently, are only known to belong to some interval with a certain
probability. For the sake of simplicity, we will consider thereafter that all modeling uncertainties, although
unknown, can be incorporated in the vector of uncertain parameters θ.
8.4.2 Model adequacy
Uncertainty is detrimental to the quality of both model predictions and optimal solutions. If the model
is not able to predict the process outputs accurately, it will most likely not be able to predict the NCO
correctly. On the other hand, even if the model is able to predict the process outputs accurately, it is often
unable to predict the NCO correctly since it has been trained to predict the outputs and not, for instance,
the gradients that are key constituents of the NCO. Hence, numerical optimization is capable of computing
optimal inputs for the model, but it often fails to reach plant optimality.
The property that ensures that a model-based optimization problem will be able to determine the
optimal inputs for the plant is referred to in the literature as “model adequacy”. For a given model-based
RTO scheme, a model is adequate if the RTO scheme is able to predict the correct set of active plant
constraints and the correct alignment of plant gradients. Model adequacy represents a major challenge
in process optimization since, as discussed earlier, models are trained to predict the plant outputs rather
than the plant NCO. In practice, application of model-based optimal inputs leads to suboptimal, and often
infeasible, operation.
9 Real-time Optimization
In the presence of modeling errors and process disturbances, the control trajectories computed offline lose
their optimal character. One way to reject the effect of uncertainty on the overall performance (with respect
to both optimality and feasibility) is by adequately incorporating process measurements in the optimization
52
framework. This is the field of real-time optimization. Measurements can be incorporated in two different
ways as illustrated in Figure 15:
a. Adapt the process model and repeat the optimization. At each iteration, the model parameters are
updated and the optimization problem solved numerically to generate u[0, tf]. This adaptation and
optimization can also be repeated online, in a single run, to estimate at time tthe current states ˆx(t)
and compute the inputs u[t, tf] for the remaining part of the batch.
b. Adapt the inputs through optimizing feedback control. Here, the optimization is implicit since opti-
mality and feasibility are enforced via feedback control to satisfy the necessary conditions of optimality.
The control scheme often involves (i) online elements that generate parts of the inputs, u
a(t), and (ii)
run-to-run elements that generate the other part of the inputs, u
b[0, tf], via the input parameters π.
These RTO schemes will be discussed in the following subsections.
9.1 Repeated Numerical Optimization
As additional information about the process becomes available, either during or at the end of the run,
this information can be used to improve future operations. This will be documented next for both online
implementation using economic MPC and run-to-run implementation via the two-step approach.
9.1.1 Economic MPC (Strategy 1)
The approach is similar to MPC discussed in Section 4.2.1, but for the fact that the cost function is no longer
tailored to tracking run-end references, but rather to minimize an economic cost function. The problem can
53
Economic MPC
1
2
3aOnline NCO tracking
3b
Two-step approach
Optimizing feedback control
(track optimality conditions)
Repeated num. optim.
(of an updated model)
Run-to-run
Online
Implementation
aspect
Optimization via
Run-to-run NCO tracking
u
[0,t
f
]
u
(t), u
[t, t
f
]
u[0, y[0,
tf]
tf]
Id. & Opt. for next run
P
ua
( t ) y(t)
track path objectives
P
u
b
[0, z
tf]
P
track terminal objectives
on a run-to run basis
eMPC
u[t,
tf]
P
E
x(t)
^
y(t)
Figure 15: RTO schemes for batch processes. The schemes are classified according to whether the opti-
mization is implemented via repeated numerical optimization or feedback control (horizontal division) and
whether it is implemented online or on a run-to-run basis (vertical division). eMPC stands for “economic
model predictive control”. Note that NCO tracking can be implemented using the four control approaches
given in Figure 2 to track the path and terminal objectives either online or on a run-to-run basis. Most often,
the path constraints are implemented online, while the terminal objectives are implemented on a run-to-run
manner (see the similarities with Table 4).
54
be formulated as follows:
min
u[ti,tf]J:= φx(tf)(73)
s.t. ˙x(t) = Fx(t), u(t), x(ti) = ˆx(ti),(74)
Sx(t), u(t)0,(75)
Tx(tf)0,(76)
x(tf) X ,(77)
where tiis the discrete time instant at which the optimization is performed, ˆx(ti) is the state estimate at that
time, and Xis the bounded region of state space where the final state should lie. Numerical optimization
yields the control sequence, u[ti, tf], of which only the first part, u[ti, ti+h], is applied in an open-loop
fashion to the plant. Numerical optimization is then repeated at every sampling instant. The scheme is
illustrated in Figure 16.
Economic
MPC
Batch Process y(t)
State Estimation
x(t)
z
u [t, tf]
*
Figure 16: Economic MPC to optimize a batch process repeatedly online.
Note that re-optimization performed during a run requires a valid process model, including estimates
of the current states. Similarly, correction through state feedback requires knowledge of the current states or
their estimates. Hence, the main engineering challenge in the context of batch processes lies in estimating the
states and the parameters from a few noisy measurements. Although the use of extended Kalman filters83
or moving-horizon estimation 84 has become increasingly common and successful for continuous processes,
these methods are difficult to use for batch processes due to the large variations that the states typically go
through and the limited time of operation. Good accounts of these difficulties are available85,86.
55
The weakness of this method is clearly its reliance on the model; if the model parameters are not
updated, model accuracy plays a crucial role. However, if the model is updated, there is a conflict between
parameter estimation and optimization, the so-called dual control problem87, since parameter estimation
requires persistency of excitation, that is, the inputs must be sufficiently varied to uncover the unknown
parameters, a condition that is usually not satisfied when near-optimal inputs are applied. Finally, note that
the scheme is called “economic MPC” because MPC typically addresses the tracking of given trajectories.
Here, there are no optimal trajectories to track, but rather some economic cost function to minimize. This
topic has gained a lot of interest in recent years88 .
9.1.2 Two-step approach (Strategy 2)
In the two-step approach, measurements are used to refine the model, which is then used to optimize the
process 67. The two-step approach has gained popularity over the past thirty years mainly because of its
conceptual simplicity. Yet, the two-step approach is characterized by certain intrinsic difficulties that are
often overlooked. In its iterative version, the two-step approach involves two optimization problems, namely,
one each for parameter identification and process optimization as shown in Figure 17 and described next:
Identification ˆ
θk:= arg min
θkyp,k[0, tf]yk[0, tf]kl2(P1)
s.t. ˙xk(t) = Fxk(t), u
k(t), θ, xk(0) = x0,k ,
yk(t) = Hxk(t), u
k(t), θ,
θΘ
Optimization u
k+1[0, tf] := arg min
u[0,tf]φxk(tf)(P2)
s.t. ˙xk(t) = Fxk(t), u(t),ˆ
θk, x(0) = x0,k,
S(xk(t), u(t),ˆ
θk)0,
Txk(tf),ˆ
θk0,
where Θ indicates the set in which the uncertain parameters θare assumed to lie. The subscript (·)pis
used to indicate that the outputs are the plant measurements yp,k [0, tf], by opposition to the values pre-
dicted by the model, yk[0, tf].
56
Identification
Optimization
Plant
Updated Model Parameters
Updated Inputs
Process Performance
Uncertainty
Run Delay
and
OK?
yes
no
Stop
uk [0,tf]
*
θk
ˆ
yp,k [0,tf]
yp,k [0,tf]
Figure 17: Basic idea of the two-step approach with its two optimization problems.
The first step identifies best values for the uncertain parameters by minimizing the l2-norm of the
output prediction errors. The updated model is then used for computing the optimal inputs for the next
iteration. Algorithmically, the optimization of the performance of a batch process proceeds as follows:
1. Initialization. Set k= 1 and select an initial guess for the input profiles u
k[0, tf].
2. Iteration. Apply the inputs u
k[0, tf] to the plant. Complete the batch while measuring the run-time
outputs yp,k[0, tf].
3. Compute the distance between the predicted and measured outputs and continue if this distance exceeds
the tolerance, otherwise stop.
4. Solve the identification problem (P1) and compute ˆ
θk.
5. Solve the optimization problem (P2) and compute u
k+1[0, tf]. Set k:= k+ 1 and repeat Steps 2-5.
The two-step approach suffers from two main limitations. First, the identification problem requires
sufficient excitation, which is however rarely the case since the inputs are computed for optimality rather
57
than for the sake of identification. Hence, one has to make sure that there is sufficient excitation, for example
via a dual optimization approach that adds a constraint to the optimization problem regarding the accuracy
of the estimated parameters 89 . The second limitation is inherent to the philosophy of the method. Since
the adjustable handles are the model parameters, the method assumes that (i) all the uncertainty (including
process disturbances) can be represented by the set of uncertain parameters, which is rarely the case.
9.2 Optimizing Feedback Control
The second class of RTO methods proposes to adapt the process inputs via feedback control, that is, without
repeating the numerical optimization. Since feedback control is used to enforce the necessary conditions of
optimality of the dynamic optimization problem, the resulting scheme is called “NCO tracking”.
9.2.1 NCO tracking (Strategy 3)
NCO tracking is a feedback control scheme, where the controlled variables (CVs) correspond to measurements
or estimates of the plant NCO, and the manipulated variables (MVs) are appropriate elements of the input
profiles. Enforcing the plant NCO is indeed an indirect way of solving the optimization problem for the
plant, along the lines of the PMP-based methods discussed in Section 8.1.2.
Necessary conditions of optimality. The NCO of the dynamic optimization problem (57)-(60) encompass four
parts as shown in Table 4: (i) the path constraints, (ii) the path sensitivities, (iii) the terminal constraints,
and (iv) the terminal sensitivities. There are as many NCO as there are degrees of freedom in the optimization
problem, thus making the system of equations perfectly determined. NCO tracking proposes to enforce the
four NCO parts via tailored feedback control. In particular, some objectives are met using online control,
while others are met via run-to-run control, thereby fully exploiting the versatility of control approaches
for batch processes, as discussed in Sections 4 and 5. The NCO-tracking scheme is therefore a two-level
(online and run-to-run) multivariable feedback control problem, as illustrated in Figure 18. The design of
the NCO-tracking controller is supported by the concept of “solution model” 72.
58
Control
Design
Numerical
Optimization
PROCESS zua(t)
model
solution
model
process
NCO-tracking
controller
Real-time (online & run-to-run)
Offline
Characterization solution
optimal
ub[0,tf]y(t) y[0,tf]
NCO-tracking
Controller
*
*
Figure 18: NCO-tracking scheme. The offline activities include the generation of a solution model and the
design of a NCO-tracking controller. The real-time activities involve online and run-to-run control to enforce
the plant NCO; u
a(t) and u
b[0, tf] are the input parts that are generated online and on a run-to-run basis,
respectively.
Solution model. The solution model is a qualitative model of the optimal solution, which includes (i) the
types and sequence of arcs, thus assuming no change in active constraints, (ii) the degrees of freedom (MVs)
that one would like to use for control, and (iii) the corresponding NCO (CVs). It is important to understand
that the solution model is simply a tool that helps solve the problem at hand in an efficient way. Although
there may be an exact solution model for the process model at hand, this is certainly not the case for the
unknown plant. Hence, the designer will be able to propose alternative (from simple to more complex)
solution models to tackle the design of the multivariable control problem. This aspect, which represents one
of the strengths of NCO tracking, will be detailed in the first case study.
The development of a solution model involves three main steps:
1. Characterize the optimal solution in terms of the types and sequence of arcs by performing numerical
optimization using the best available plant model. This plant model need not be very accurate, but it
ought to provide the correct types and sequence of arcs. One typically performs a robustness analysis
59
to ensure that the qualitative solution remains structurally valid in presence of uncertainty.
2. Select a finite set of input arcs and parameters to represent (or approximate) the input profiles, and
formulate the NCO for this choice of degrees of freedom. Note that the NCO will change with the
choice of the degrees of freedom.
3. Pair the MVs and the NCO to form a multivariable control problem.
Steps 2 and 3 might require iterations since they greatly affect the implementation that is discussed next.
Implementation aspects. In its general form, NCO tracking involves both online and run-to-run control,
as determined by the solution model and the resulting MV-CV pairing. Some NCO elements are typically
implemented online, while others are easier on a run-to-run basis. For example, a path constraint is easily
enforced online via constraint control, while both terminal constraints and terminal sensitivities are easier
to meet iteratively over several runs. The decision regarding which NCO elements to implement online and
which on a run-to-run basis depends on the nature of the various arcs. For a given input, the various arcs
can be of two types:
(i) A constraint-seeking input arc is associated with a path constraint being active in a given time interval.
If the path constraint is an input bound, the input is simply set at the bound, while in the case of a
state constraint, feedback control can be used to track the value of the constrained quantity.
(ii) A sensitivity-seeking input arc requires ∂H
∂u (t) = 0, which is difficult to implement as such since H(t)
is a function of the adjoint variables λ(t). Hence, one tries to approximate the sensitivity-seeking
arc using a parsimonious input parameterization with only a few constant parameters. This way, the
number of degrees of freedom are no longer the infinite-dimensional u(t), but rather the few constant
input parameters. Furthermore, with the input parameterization, the batch process is viewed as the
static map z=Mπgiven by Eq. (7). Consequently, the path condition ∂H
∂u (t) = 0 is replaced by
terminal conditions of the type φ(x(tf))
∂π = 0, which can be enforced on a run-to-run basis. Note that
the effect of the approximations introduced at the solution level can be assessed in terms of optimality
loss.
60
In summary, the ease of implementation and therefore the success of NCO tracking depends on the
quality of the approximations that are introduced to generate the solution model (choice of MVs, correspond-
ing NCO, and pairing MV-NCO). These are true engineering decisions that are made with plant optimality
in mind!
10 Optimization Applications
10.1 Semi-batch Reactor with Safety and Selectivity Constraints
A simple semi-batch reactor with jacket cooling is considered to illustrate the NCO-tracking approach and,
in particular, the generation of alternative solution models72.
10.1.1 Problem formulation
Reaction system: A+BC, 2BD, isothermal, exothermic reactions.
Objective: Maximize the amount of Cat a given final time.
Manipulated input: Feedrate of B.
Path constraints: Input bounds; heat-removal constraint expressed as a lower bound on the cooling
jacket temperature.
Terminal constraint: Upper bound on the amount of Dat final time.
61
Model equations. Assuming perfect control of the reactor temperature through adjustment of the cooling