1. Fault-Tolerant Control of Particulate Processes
Accounting for Implementation Issues
By
Trina G. Napasindayao
B.S. (De La Salle University, Philippines) 2008
Dissertation
Submitted in partial satisfaction of the requirements for the degree of
Doctor of Philosophy
in
Chemical Engineering
in the
Office of Graduate Studies
of the
University of California
Davis
Approved:
Nael H. El-Farra, Chair
Ahmet N. Palazoglu
William D. Ristenpart
Committee in Charge
2015
i
2. To God who makes all things possible.
Unless the LORD build the house, they labor in vain who build. (Psalm 127:1)
ii
6. List of Figures
2.1 Sampled-data control architecture. . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Timeline of measurement transmission and arrival times under measurement
sampling and delay. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Region of stability is larger with a propagation unit (δ = 0.3). Plots (a)-(b):
Contour plot of λmax(M) with (a) and without (b) a propagation unit. . . . 26
2.4 The closed-loop system can only be stabilized with a propagation unit (δ =
0.3, τ = 0.5h, ∆ = 1h). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Fault detection and accommodation maintains stability after a component
fault at Tf = 10h (δ = 0.3, τ = 0.5h, ∆ = 1h). Plots (a)-(d): Closed-loop
state profiles with (a)-(b) and without (c)-(d) fault detection and accommo-
dation. Plot (e): Closed-loop profiles of the manipulated input. Plot (f):
Fault detection based on the evolution of the residual. Note: Profiles in plots
(a)-(e) are in deviation variable form. Actual values are non-negative. . . . . 33
2.6 Fault accommodation using a contour plot of λmax(M) indicating the region
of stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 Fault detection and accommodation maintains the stability of the Particle
Size Distribution (PSD) in the presence of sensor measurement noise after a
component fault at Tf = 10h (δ = 0.3, τ = 0.5h, ∆ = 1h). Plot (a): Closed-
loop PSD profile with (a) and without (b) fault detection and accommodation.
Plot (c): Closed-loop profiles of the manipulated input in deviation variable
form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1 Overview of the integrated control architecture with fault identification and
accommodation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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7. 3.2 Region of stability based on actuator health was used to determine whether
equipment repair, fault accommodation, or system reconfiguration is required
(∆ = 6min). Contour plot of λmax(N) for pole values [−1 − 2 − 3 − 4 − 5 − 6]. 51
3.3 Actual and calculated values of the fault estimation parameters (∆ = 6min).
α1: inlet concentration (c0), α2: residence time (τr). Plots (a)-(b): Simulta-
neous faults. Plots (c)-(d): Consecutive faults. . . . . . . . . . . . . . . . . . 53
3.4 Fault accommodation logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Fault identification after a potentially destabilizing fault at 10h with ∆ =
6min. α1: inlet concentration (c0), α2: residence time (τr). . . . . . . . . . . 55
3.6 Fault identification and accommodation re-establishes stability after a poten-
tially destabilizing fault. Plot (a): Region of stability based on the health
of the actuator controlling the inlet concentration (c0), α1 and the first pole
value (λ) used to find the controller design parameter K (α2 = 1). Plots (b)-
(c): Dynamic profiles of (b) inlet concentration (c0), and (c) residence time
(τr) without fault accommodation. Plots (d)-(e): Dynamic profiles of (d) inlet
concentration (c0), and (e) residence time (τr) under fault accommodation. . 56
4.1 Overview of the integrated control architecture with fault identification and
accommodation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Fault accommodation logic with model uncertainty. . . . . . . . . . . . . . . 74
4.3 Fault accommodation logic without model uncertainty. . . . . . . . . . . . . 76
4.4 Plots (a)-(b): Region of stability is larger with a perfect model (a) com-
pared to one with model uncertainty (b). The feed concentration (c0) and
residence time (τr) are the manipulated variables (u1
(t) = [u1
1(t) u1
2(t)]T
=
[c0(t) τr(t)]T
). Contour plots of Γk
(∆) plotted against different values of the
fault parameter (α1
1) and fault model parameter (α1
1). . . . . . . . . . . . . . 79
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8. 4.5 Plots (a)-(d): Fault identification after a partial fault (α1 = 0.9) at t = 1h.
Plot (a): Dynamics of fault parameter (α1) and fault estimation parameter
(α∗
1). Plot (b): Region of stability with the estimation interval α1 = Ψ(α∗
1) =
[0.95, 1] for α1 = 1 (red line). Plots (c)-(d): Dynamics of the state (µ1)
(c) and the faulty actuator controlling the manipulated variable u1
1, the feed
concentration (c0) (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.6 Plots (a)-(d): Fault identification after a partial fault (α1 = 0.4) at t = 1h.
Plot (a): Dynamics of fault parameter (α1) and fault estimation parameter
(α∗
1). Plot (b): Region of stability with the estimation interval α1 = Ψ(α∗
1) =
[0.4, 0.475] for α1 = 1 (red line). Plots (c)-(d): Dynamics of the state (µ1)
(c) and the faulty actuator controlling the manipulated variable u1
1, the feed
concentration (c0) (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 Plots (a)-(b): Regions of stability used in selecting the best fault accommo-
dation strategy after a partial fault (α1 = 0.4) at t = 1h. Plot (a): Stability
region for different values of the fault parameter (α1
1) and the controller de-
sign parameter (p1) using the feed concentration (c0) and residence time (τr)
as the manipulated variables (α1
1 = 1). Plot (b): Stability region plotted
against the fault parameter (α2
1) and the fault model parameter (α2
1) using
the residence time (τr) as the only manipulated variable (u2
1). . . . . . . . . . 83
4.8 Plots (a)-(b): Dynamics of the state (µ1) (a) and the fall-back manipulated
variable u2
1 varying residence time (τr) (b) shows that fault accommodation
re-establishes stability after a potentially destabilizing fault. . . . . . . . . . 83
5.1 Sampling schedule of two sensors with different sampling rates. . . . . . . . . 92
5.2 Region of stability varies depending on the chosen manipulated input (δu =
0.2). Plots (a)-(b): Contour plots of λmax(N) when the manipulated variable
is (a) the inlet concentration, c0; and (b) the residence time, τ . . . . . . . . 99
viii
9. 5.3 Region of stability varies depending on the chosen manipulated input (δu =
0.2). Contour plot of λmax(N) when the coolant temperature, Tc, is the
manipulated variable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Closed-loop state profiles depend on the selected manipulated variable (δu =
0.2). Plots (a)-(b): Stability is reached using either (a) inlet concentration,
c0, or (b) residence time, τ, as manipulated variables (OP:∆1 = 0.002, ∆2 =
0.008). Plots (c)-(d): System stabilizes when (c) inlet concentration, c0, and
not (d) residence time, τ, is the manipulated variable (f1:∆1 = 0.002, ∆2 =
0.012). Plots (e)-(f): System becomes unstable by manipulating either (e)
inlet concentration, c0, or (f) residence time, τ (f2:∆1 = 0.011, ∆2 = 0.008). . 103
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10. List of Tables
2.1 Process parameters and steady-state values for the continuous crystallizer. . 17
3.1 Process parameters and steady-state values for the non-isothermal continuous
crystallizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Process parameters and steady-state values for the non-isothermal continuous
crystallizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1 Process parameters and steady-state values for the non-isothermal continuous
crystallizer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
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11. Abstract of the Dissertation
Fault-Tolerant Control of Particulate Processes
Accounting for Implementation Issues
Particulate processes comprise about 60% of commercial products. These processes are
defined by the co-presence of both a continuous and a dispersed phase. As a result, there is a
distributed characterization of the product properties. These differences across particles are
described using a particle size distribution which is an important product quality index since
controlling the shape of this distribution leads to quality control of the end product. A high-
dimensional population balance model is used to describe the particle size distribution which
makes it difficult to design control systems for these applications. There are limited studies
on fault accommodation and fault tolerant control for particulate processes. Moreover, var-
ious implementation issues arise in the design of any fault-tolerant control system. These
include model uncertainty, incomplete state measurements, measurement sampling and de-
lays. Measurement availability is constrained by inherent limitations on data collection and
the processing and transmission capabilities of the measurement sensors. In particulate
processes, sensor measurements are typically delayed and available only at discrete times.
These restrict controller implementation and process tracking which can, in turn, erode the
diagnostic capabilities of the fault-tolerant control system. Hence, it is crucial that these are
explicitly accounted for in designing the control system and in monitoring the process.
Motivated by the above considerations, this dissertation provides a unified framework
for fault-tolerant control of particulate processes with implementation issues. This frame-
work integrates fault detection/identification followed by fault accommodation wherein a
supervisor determines the best strategy for preserving closed-loop stability after a poten-
tially destabilizing fault has occurred. This strategy is based on a stability analysis on
the closed-loop system wherein the stability properties are given as functions of the control
xi
12. configuration, actuator gain, model uncertainty, fault parameters, and/or sampling period.
Fault accommodation is then carried out by controller reconfiguration, model update, or
actuator switching. These techniques are illustrated to be effective for a wide range of fault
scenarios using a simulated continuous crystallizer but may be generalized for particulate
processes.
xii
13. Acknowledgments
I am utterly grateful to my advisor, Prof. Nael H. El-Farra, for his invaluable patience and
guidance throughout my graduate studies. Thank you for believing in me and for giving me
that extra motivation when I needed it most.
I would also like to thank Prof. Ahmet N. Palazoglu and Prof. William D. Ristenpart
for taking the time to serve in my dissertation committee.
I want to acknowledge the professors that I have worked with for the many bits of wisdom
that they have imparted and for being a great source of inspiration.
I am extremely thankful for everyone in my research group who have been very obliging
and encouraging: Arthi, Sathyendra, Yulei, Ye, Zhiyuan, Xiaonan, Da, Shilpa. It is an honor
and a blessing to be counted as your cohort and friend.
I wish to thank my peers in the Chemical Engineering Department especially those with
whom I have taken some of the graduate-level coursework: Ben, Alvin, Josh, Claudia, Salem,
Pinghong, Jorgen. Our efforts have finally borne fruit. I am glad to have shared this journey
with you.
I want to express my deepest gratitude to my family who was there from the very start.
I could not have done it without all of you. Your love has kept me going. Finally, words
cannot express how thankful I am to Daniel–you came just in time.
xiii
14. Chapter 1
Introduction
1.1 Motivation
Chemical engineering deals with processes that convert raw materials into more valuable
products while satisfying requirements based on safety, environmental regulations, eco-
nomics, and production specifications. This is carried out by making efficient use of time,
energy, and raw materials to maximize profit by improving quality and increasing yield while
minimizing costs in the form of expenditures, environmental impact, and safety hazards. Pro-
cess Control is a discipline that focuses on the architectures, mechanisms, and algorithms
that are necessary to ensure that these severalsometimes conflictingrequirements are met.
Through Process Control, the process is steered towards desired behavior by ensuring stable
and optimum performance while suppressing the influence of external disturbances.
Particulate processes comprise about 60% of commercial products and encompass a wide
range of fields including the agricultural, chemical, food, mineral, and pharmaceutical indus-
tries. A high-dimensional model which, coupled with the complex dynamics and nonlineari-
ties in the system, makes it difficult to design fault tolerant control systems for particulate
process. Moreover, various implementation issues arise in the design of any fault-tolerant
control system. These include model uncertainty, incomplete state measurements, measure-
ment sampling and delays. These restrict controller implementation and process tracking
which can, in turn, erode the diagnostic capabilities of the fault-tolerant control system.
1
15. Hence, it is crucial that these are explicitly accounted for in designing the control system
and in monitoring the process. The remainder of the chapter will be on the origin and
implications of the above-mentioned topics and will provide an overview of relevant work in
this area.
1.2 Background on monitoring and control of particu-
late processes
Particulate processes are defined by the co-presence of both a continuous and a dispersed
phase. The dispersed phase is composed of particulates dispersed throughout the continuous
phase which is usually a fluid medium. As a result, there is a distributed characterization
of the product properties, such as size, morphology, porosity,etc. The physico-chemical and
mechanical properties of such materials are strongly dependent on the differences across
particles which is described using a Particle Size Distribution (PSD). For example, a nearly
mono-disperse PSD is required for titania pigments to obtain the maximum hiding power
per unit mass. In coatings, the product composition, molecular weight and PSD often
need to be within in a specific range to ensure that the material has the desired level of
film formation, film strength, and gloss. In all of these examples, the PSD provides the
critical link between the product quality indices and the process operating variables; and,
therefore, the ability to effectively manipulate the PSD is essential for our ability to control
the quality of the end products made in these processes. A high-dimensional population
balance model is used to describe the particle size distribution which is coupled with the
complex dynamics and nonlinearities in the system. Hence, such models cannot be used
directly for the synthesis of practically implementable controllers. An effort to address
these problems was initiated where a methodology for the detection and handling of control
actuator faults in particulate processes was developed based on low-order models that capture
the dominant process dynamics [1]. These results were generalized to address the problems
of fault isolation and robustness against model uncertainty [2].
2
16. Significant research work has been carried out on the synthesis and implementation of
feedback control systems on particulate processes. These include: the use of conventional PI
and PID controllers, nonlinear analytic model-based control, optimization-based control [3–
19]. For a more rigorous review of results in this area, refer to [20, 21]. Despite the significant
number of studies that have been carried out, there is limited research on designing and
implementing fault diagnosis and fault-tolerant control systems for particulate processes.
This problem is significant since faults are inevitable and a control system that ignores
faults, carries out an incorrect fault diagnosis, and/or improperly handles malfunctions can
negatively affect the particle size distribution and ultimately harm the end product. In
the production of specialty chemicals, for instance, the end-product utility is dependent on
stringent product specifications. Hence, control system faults may result in off-spec products
and lead to substantial production losses.
The successful design and implementation of active fault-tolerant control systems require
the integration of two basic steps. The first is fault diagnosis, and involves the detection and
identification of faults with sufficient accuracy on the basis of which remedial action can be
taken. There are several ways in which this can be done. In the subsequent chapters, fault
diagnosis is carried out by either fault detection or fault identification. Fault detection is
carried out by using residuals that are based on the dynamics of the fault-free plant. When
this threshold is breached, a fault is declared. This technique which makes use of residuals is
primarily useful for determining if a destabilizing fault has occurred but generally does not
locate the origin and magnitude of the fault. Faults that do not have negative impacts on
the stability properties of the system are left undetected but this does not have undesired
implications on the system performance since such malfunctions do not require immediate
fault accommodation. This is where fault identification comes in. Fault identification, in
contrast to fault detection, allows one to identify and isolate the source of the fault–including
those that do not lead to instability. As such, fault identification may be used in determin-
3
17. ing the best response or approach with regards to the fault be it equipment maintenance or
replacement, model update, or control system reconfiguration. Once the faults have been
identified, the second step in fault-tolerant control is that of fault handling which is typically
accomplished through reconfiguration of the control system structure (through switching
between redundant actuator/sensor configurations) to cancel the effects of the faults or to
attenuate them to an acceptable level. The problems of fault diagnosis and fault-tolerant
control have been studied extensively in process control literature [22–31]. However, most of
the existing methods have been developed for lumped parameter processes described by sys-
tems of ordinary differential equations (ODEs). The dynamic models of particulate processes
are typically obtained through the application of population, material and energy balances
and consist of systems of nonlinear partial integro-differential equations that describe the
evolution of the PSD, coupled with systems of nonlinear ordinary differential equations that
describe the evolution of the state variables of the continuous phase [32, 33]. Thus, the
conventional approach used for fault-tolerant control for lumped parameter systems can-
not be applied to particulate processes which are modeled by complex, infinite-dimensional
equations.
Moreover, various implementation issues arise in the design of any fault-tolerant control
system. These include model uncertainty, incomplete state measurements, measurement
sampling and delays. Typical sources of model uncertainty include unknown or partially
known time-varying process parameters, exogenous disturbances, and un-modeled dynamics
(such as fast actuator and sensor dynamics not included in the process model). It is impor-
tant to account for the plant-model mismatch since ignoring them in the controller design
may lead to severe deterioration of the nominal closed-loop performance or even to closed-
loop instability. Research on robust control of nonlinear distributed chemical processes with
uncertainty has mainly focused on transport-reaction processes described by nonlinear partial
differential equations (PDEs). In this area, important contributions include the development
4
18. of Lyapunov-based robust control methods for hyperbolic and parabolic PDEs [34–37]. An
alternative approach for the design of controllers for PDE systems with time-invariant un-
certain variables involves the use of adaptive control methods [38–41]. Despite this, there
is no general framework for the synthesis of practically implementable nonlinear feedback
controllers for particulate processes that allow attaining desired particle-size distributions in
the presence of significant model uncertainty.
Measurement availability is constrained by inherent limitations on data collection and
the processing and transmission capabilities of the measurement sensors. In particulate
processes, sensor measurements of the dispersed (e.g., obtained using light scattering tech-
niques) and the continuous phase variables (e.g., solute concentration) are typically delayed
and available only at discrete times. These restrict controller implementation and process
tracking which can, in turn, erode the diagnostic capabilities of the fault-tolerant control
system. Hence, it is crucial that these are explicitly accounted for in designing the control
system and in monitoring the process. Furthermore, fault-tolerant control systems have to
consider the type of fault that occurs to ensure proper handling. Faults are classified as
sensor, actuator, or component faults depending on where they appear in a process plant.
Existing methods for distributed parameter systems only considered actuator failure diagno-
sis and compensation [42–45]. Despite this, component and sensor faults are also commonly
encountered and requires the use of alternative fault accommodation techniques.
Motivated by the above considerations, this dissertation provides a unified framework
for fault-tolerant control of particulate processes with implementation issues. This frame-
work integrates fault detection/identification followed by fault accommodation wherein a
supervisor determines the best strategy for preserving closed-loop stability even after a po-
tentially destabilizing fault has occurred. This strategy is based on a stability analysis on
the closed-loop system wherein the stability properties are given as functions of the control
configuration, actuator gain, model uncertainty, fault parameters, and/or sampling period.
5
19. Fault accommodation is then carried out by controller reconfiguration, model update, or
actuator switching. These techniques were illustrated to be effective for a wide range of fault
scenarios–component, actuator, and sensor faults—using a simulated continuous crystallizer
example.
1.3 Objectives and organization of the dissertation
Motivated by the considerations highlighted in previous sections, the broad objectives of this
dissertation are:
• To develop an architecture that integrates monitoring and control of particulate pro-
cesses.
• To address practical implementation issues within the integrated monitoring and con-
trol architecture, such as uncertain and nonlinear process dynamics, unavailability of
complete and/or continuous state measurements, and delayed sensor measurements.
• To demonstrate the effectiveness of the developed methods using a simulated continu-
ous crystallizer.
The rest of the dissertation is organized as follows:
Chapter 2 synthesizes a model-based framework for component fault detection and accom-
modation in particulate processes described by population balance equations with discretely-
sampled and delayed measurements. Model reduction techniques are used to derive a finite-
dimensional system that captures the dominant dynamics of the particulate process. An
observer-based output feedback controller is then designed using this system to stabilize the
fault-free process. To compensate for the discrete measurements, an inter-sample model
predictor is included within the control system to provide the observer with process output
estimates when sensor measurements are unavailable. The model state is then updated when
measurements are received at discrete times. To compensate for the measurement delay, the
6
20. control system includes a propagation unit that estimates the current output from the out-
dated measurements using the low-order model together with the past values of the control
input. Estimates from the propagation unit are used to update the inter-sample model pre-
dictor which, together with the controller, generates the control signal for the process. For
fault detection, the current plant behavior is compared with the ideal fault-free behavior.
Significant discrepancies between the two indicate that there is a fault in the system. To
characterize the ideal behavior, the minimum allowable sampling rate for fault-free stability
is obtained by formulating the closed-loop system as a combined discrete-continuous system.
It is explicitly characterized in terms of the plant-model mismatch, the controller and ob-
server design parameters, and the measurement delay. The fault-free closed-loop behavior
from this analysis was used to derive rules for fault detection and accommodation. The state
observer serves as a fault detection filter by comparing its output with the current plant out-
put estimates generated by the propagation unit at each sampling time. The discrepancy
is used as a residual and compared with a time-varying alarm threshold from the stability
analysis to detect faults. Faults are accommodated by adjusting the controller and observer
design parameters to preserve closed-loop stability and minimize performance deterioration.
In Chapter 2, fault detection is carried out by designing a fault-free time-varying alarm
threshold offline and later comparing this with values of the residual for the entire duration
of the process. This scheme for fault detection is stability-based, leaving small malfunctions
undetected when these do not lead to instability. In designing this threshold, there are
competing design requirements that need to be considered–there is the need to tighten the
threshold for timely fault detection; however, an extremely tight bound may result in false
alarms. It is also assumed that a fault identification scheme was already in place which was
able to determine the nature and location of the fault. This is needed in devising an appro-
priate response for fault accommodation. After each fault, a new alarm threshold has to be
used since the system will have different stability properties after the fault accommodation.
7
21. Chapter 3 develops a fault identification methodology that allows for immediate detec-
tion of actuator faults and/or malfunctions while determining its location and magnitude.
Another advantage of the proposed scheme is that it may still be used for fault identification
even after the fault accommodation has taken place. This allows for timely fault detection
right after a fault has been accommodated. This is an advantage over the previous detection
schemes where a new alarm threshold has to be calculated after every fault accommodation.
This recalculation of a new alarm threshold may result in a delay in fault detection preceding
a fault. Timely or even instantaneous fault identification is crucial even for faults that do
not immediately result in unstable behavior since these malfunctions may later on result in
poor plant performance or even instability. In addition, this timely detection will also allow
for systematic scheduling of plant maintenance and equipment repair or replacement.
In Chapter 3, we develop a model-based framework for fault-tolerant control of sampled-
data particulate processes under sensor faults under state feedback and a data-based fault
identification mechanism. These particulate processes are described by complex population
balance equations. Model reduction techniques are, therefore, applied to derive a finite-
dimensional model used in designing a stabilizing sample-and-hold state feedback controller.
This controller uses past values of the state measurements in between sampling times. The
controller is then updated once measurements are received at discrete times. Stability analy-
sis is then carried out to obtain an explicit characterization of the behavior of the system as a
function of the controller design parameters, update time, and actuator health. This scheme
shall be used in determining the appropriate post-fault response once a fault is detected.
Fault identification is achieved out by solving a data-based moving horizon optimization
problem. Data from the fault identification is used in the fault accommodation which in-
volves modifying the controller design parameter based on the stability plots generated from
the stability analysis.
8
22. The timely fault identification from Chapter 3 allows for systematic scheduling of plant
maintenance and equipment repair or replacement; however, this identification strategy was
constructed based on a perfect plant model. This assumption is unrealistic since model
uncertainties are always present and could lead to inaccurate diagnosis of actuator status. In
addition, the system is controlled using a sample-and-hold model because of the measurement
sampling. This approach is simplistic and may lead to limited control capabilities especially
for large sampling periods. Thus, Chapter 4 aims to generalize techniques in Chapter 3 by
introducing an inter-sampling state estimator while accounting for model uncertainties.
In Chapter 4, we propose a model-based framework for fault-tolerant control of sampled-
data particulate processes with model uncertainty and actuator faults using state feedback
and a data-based fault identification mechanism. Model reduction techniques were applied
to derive a finite-dimensional model used in designing a state feedback controller. This
controller used inter-sample state estimates in between sampling times. The inter-sample
state estimator is updated when sensor readings are received. Through stability analysis,
an explicit characterization of the behavior of the system is obtained as a function of the
controller design parameters, update time, model uncertainty, and actuator health. These
findings are used for fault accommodation. Fault identification is carried out by solving
a data-based moving horizon optimization problem. The fault is then accommodated by
either modifying the fault model parameter matrix in the inter-sample state estimator or
the controller design parameter based on the stability analysis for all values within the
estimation interval.
Chapter 5 presents a model-based framework for fault-tolerant control of multi-rate
sampled-data particulate processes under sensor faults. These particulate processes are de-
scribed by complex population balance equations. Model reduction techniques are, therefore,
applied to derive a finite-dimensional model used in designing a stabilizing observer-based
output feedback controller. To compensate for the discrete measurements, an inter-sample
9
23. model predictor provided the observer with process output estimates. The model states
were updated when measurements were received at discrete times. For fault tolerance, the
stabilizing output sampling rates are calculated and explicitly characterized in terms of the
plant-model mismatch, controller and observer design parameters, and the manipulated in-
put. Conditions from the closed-loop stability analysis were used to obtain a region of
stability for a given manipulated input. These regions are plotted as a function of the sam-
pling period of the outputs and are used in predicting the behavior of the system under a
certain set of operating conditions. The plots are then used in determining the appropriate
scheme for fault tolerance. Passive fault-tolerance is achieved by selecting a manipulated
input based on its robustness to a particular type of fault using knowledge of the nature of
future sensor faults. Active fault tolerance is attained by: returning to the original operating
point by reverting to a back-up sensor with the same sampling period as the faulty one, by
switching to a different sensor with a sampling period that shifted the operating point back
into the region of stability, or choosing a different manipulated variable such that the new
operating point was within the new stability region.
Finally, the proposed fault-tolerant control frameworks in all chapters are illustrated
using a simulated model of a continuous crystallizer but may be generalized for particulate
processes modeled by partial-integro differential equations.
10
24. Chapter 2
Fault detection and accommodation
in particulate processes with sampled
and delayed measurements
In this chapter, a model-based framework is developed for component fault detection and
accommodation in particulate processes with discretely-sampled and delayed measurements.
An observer-based output feedback controller is initially designed based on a suitable reduced-
order model that captures the dominant process dynamics. The controller includes an inter-
sample model predictor that compensates for measurement intermittency, and a propagation
unit that compensates for the delays. The inter-sample model predictor provides the observer
with process output estimates between sensor measurements, and the model states are up-
dated using current output estimates obtained from the propagation unit. The fault-free
stability properties are characterized in terms of model accuracy, sampling rate and delay
size, and is used to derive appropriate rules for fault detection and accommodation. The
difference between the output estimates from the state observer and the propagation unit is
compared against a time-varying alarm threshold for fault detection. Once the threshold is
breached, controller design parameters are adjusted to preserve closed-loop stability.
The rest of the chapter is organized as follows: The class of systems is described in
Section 2.1, followed by the problem formulation and solution overview. In Section 2.2, the
11
25. continuous crystallizer is first introduced as a representative example of a particulate process
which will be used to illustrate the proposed control scheme. This is then reduced to a low-
order Moments Model. In Section 2.3, a controller is designed for the system with sampled
and delayed measurements in the absence of faults. This fault-free closed-loop behavior is
used to derive appropriate rules that are used for fault detection and accommodation in
Section 2.4. Some concluding remarks are then given in Section 2.5. The results of this
chapter were first published in [46].
2.1 Preliminaries
2.1.1 System description
We focus on spatially homogeneous particulate processes with simultaneous particle growth,
nucleation, agglomeration and breakage, and consider the case of a single internal particle
coordinate–the particle size. Applying a population balance to the particle phase, as well
as material and energy balances to the continuous phase, we obtain the following general
nonlinear system of partial integro-differential equations:
∂n
∂t
= −
∂(G(z, r) · n)
∂r
+ wn(n, z, r), n(0, t) = b(z(t)) (2.1)
˙z = f(z) + g(z)u + Az
rmax
0
q(n, z, r)dr (2.2)
where n(r, t) ∈ L2[0, rmax) is the particle size distribution function which is assumed to be a
continuous and sufficiently smooth function of its arguments (L2[0, rmax) denotes a Hilbert
space of continuous functions defined on the interval [0, rmax)), r ∈ [0, rmax) is the particle
size (rmax is the maximum particle size, which may be infinity), t is the time, z ∈ Rn
is
the vector of state variables that describe properties of the continuous phase (e.g., solute
concentration, temperature and pH in a crystallizer), u ∈ R is the manipulated input, (2.1)
is the population balance where G(z, r) is the particle growth rate from condensation, and
12
26. wn(n, z, r) accounts for the net rate of introduction of new particles into the system, i.e., it
includes all the means by which particles appear or disappear within the system including
particle agglomeration, breakage, nucleation, feed, and removal. The z-subsystem of (2.2) is
derived from material and energy balances in the continuous phase. In this subsystem, f(z),
g(z), q(n, z, r) are smooth nonlinear vector functions and Az is a constant matrix. The term
containing the integral represents mass and heat transfer from the continuous phase to all
the particles in the population.
To express the desired control objectives, such as regulation of the total number of
particles, mean particle size, temperature, pH, etc., we define the controlled outputs as:
yι(t) = hι
rmax
0
cκ(r)n(r, t)dr, z , ι = 1, · · · , ˜m where hι(·) is a smooth nonlinear function
of its arguments and cκ(r) is a known smooth function of r which depends on the desired
performance specifications. For simplicity, we will consider that the controlled outputs are
available as online measurements.
2.1.2 Problem formulation and solution overview
The control objective is to stabilize the process at some desired equilibrium state in the
presence of component faults using discretely-sampled and delayed measurements of the
output. The problems under consideration therefore include: fault-free process regulation us-
ing discretely-sampled and delayed measurements, timely detection of the component faults,
fault compensation to maintain the desired stability and performance characteristics through
fault accommodation. To address these problems, we consider the following methodology:
• Model reduction: Initially use model reduction techniques to derive a finite-dimensional
model that captures the dominant dynamics of the infinite-dimensional system describ-
ing the continuous crystallizer.
• Controller synthesis: Use the reduced-order model to design an observer-based output
feedback controller that regulates the process states at the desired steady-state in the
13
27. absence of faults. To compensate for the lack of continuous measurements, an inter-
sample model predictor is included within the control system to provide the observer
with an estimate of the output when measurements are not available from the sensors.
To compensate for the measurement delay, we incorporate within the control system a
propagation unit that uses the process model and the past values of the control input
to estimate the current process output from the delayed measurements.
• Analysis: Obtain an explicit characterization of the minimum allowable sampling rate
that guarantees stability and residual convergence in the absence of faults in terms of
the model accuracy, the delay size, and the controller and observer design parameters.
• Monitoring: Use the state observer as a fault detection filter by comparing its output
with that of the process at the times that the measurements are available and using
the discrepancy as a residual. Derive a time-varying alarm threshold for the residual
based on its fault-free behavior.
• Fault accommodation: Derive a fault accommodation logic that determines the set of
feasible values for the controller and observer design parameters that can be used to
preserve closed-loop stability and minimize performance deterioration under a given
measurement sampling rate and delay time.
Figure 2.1 is a schematic diagram showing the different components of the control system
design that compensates for measurement sampling and delays. In the structure, a model is
embedded which estimates plant outputs when measurements are unavailable. To compen-
sate for delays, a propagation unit is also included which estimates the current output at
sampling times. The values from the propagation unit are used to reset the model output
once the delayed sensor measurements are received. Model estimates, in turn, are utilized
by the state observer which estimates the state measurements which are used by the model-
based output feedback controller to generate the appropriate control action to be applied to
the plant.
14
28. SensorActuator
)( τ−ty
)(ˆ ty
u
Plant
Model
Measurement
Reset
Local Control System
Cxy
fuBAx
dt
dx
c
=
++=
F
State Observer
)ˆ(ˆˆ ηηη CyLuBA
dt
d −++=
η
Cwy
uBwA
dt
dw
=
+=
ˆ
ˆˆ
wCy
uBwA
dt
wd
=
+= ˆˆ
Propagation
)(ty
Figure 2.1. Sampled-data control architecture.
2.2 Motivating example
A well-mixed isothermal continuous crystallizer, a spatially homogeneous particulate process,
is used throughout the paper to illustrate the design and implementation of model-based fault
detection and accommodation. Crystallization is widely used in producing fertilizer, proteins,
and pesticides. Particulate processes are characterized by the co-presence of a continuous
and dispersed phase. The dispersed phase is described by a particle size distribution whose
shape influences the product properties and ease of product separation. Hence, a population
balance on the dispersed phase coupled with a mass balance for the continuous phase is
necessary to accurately describe, analyze, and control particulate processes. Under the
assumptions of constant volume, mixed suspension, nucleation of crystals of infinitesimal
15
29. size, mixed product removal, and a single internal particle coordinate–the particle size; a
dynamic crystallizer model can be derived:
∂n
∂t
= ¯k1(cs − c)
∂n
∂r
−
n
τr
+ δ(r − 0)¯ǫ¯k2e
−¯k3
(c/cs−1)2
dc
dt
=
(c0 − ρ)
¯ǫτr
+
(ρ − c)
τr
+
(ρ − c)
¯ǫ
d¯ǫ
dt
(2.3)
where n(r, t) is the number of crystals of radius r ∈ [0, ∞) at time t per unit volume of
suspension; τr is the residence time; c is the solute concentration in the crystallizer; ρ is the
particle density; ¯ǫ = 1 −
∞
0
n(r, t)π4
3
r3
dr is the volume of liquid per unit volume of suspen-
sion; cs is the concentration of solute at saturation; c0 is the concentration of solute entering
the crystallizer; ¯k1, ¯k2 and ¯k3 are constants; and δ(r −0) is the standard Dirac function. The
term containing the Dirac function accounts for the nucleation of crystals of infinitesimal size
while the first term in the population balance represents the particle growth rate. The crys-
tallizer exhibits highly oscillatory behavior due to the relative nonlinearity of the nucleation
rate as compared to the growth rate. This results in process dynamics that are characterized
by an unstable steady-state surrounded by a stable periodic orbit. The control objective is
to suppress the oscillatory behavior of the crystallizer in the presence of component faults.
This is carried out by stabilizing it at an unstable steady-state that corresponds to a desired
crystal size distribution by manipulating the solute feed concentration. Measurements of the
crystal concentration in the continuous crystallizer are collected at discrete sampling times
with a delay time of τ and sent to the controller where the control action is calculated and
then sent to the actuator to affect the desired change in the process state.
Through method of moments, a fifth-order ordinary differential equation system is derived
to describe the temporal evolution of the first four moments of the crystal size distribution
and the solute concentration. Using dimensionless variables, the reduced-order model can
be cast in the following form:
16
30. ˙˜x0 = −˜x0 + (1 − ˜x3)Da e
− ˜F
˜y2
c
˙˜xdm = −˜xdm + ˜yc ˜xdm−1, i = 1, 2, 3
˙˜yc =
1 − ˜yc − (˜α − ˜yc)˜yc ˜x2
1 − ˜x3
+
u
1 − ˜x3
(2.4)
where ˜xdm, dm = 0, 1, 2, 3, are the dimensionless moments of the crystal size distribution;
˜yc is the dimensionless concentration of the solute in the crystallizer; u is the dimensionless
concentration of the solute in the feed stream; ˜F = 0.1021, ˜α = 7.187, and Da = 2719
are the dimensionless constants computed from the process parameters [7]. At these values
and at the nominal steady-state operating condition of ˜us = 0, the global phase portrait of
the system of (2.4) has a unique unstable equilibrium point at [˜xs
0 ˜xs
1 ˜xs
2 ˜xs
3 ˜ys
c] =
[46.73 6.62 0.94 0.13 0.14], which is surrounded by a stable limit cycle. Only
measurements of the crystal concentration, ˜x0, are considered to be available online. These
can be obtained, for example, via light scattering techniques.
Table 2.1. Process parameters and steady-state values for the continuous crystallizer.
ρ = 1770 kg/m3 ¯k1 = 0.05065e
(
−Eg
R·TI
)
cs
o = 1100 kg/m3 ¯k2 = 7.958e
(
−Eb
R·TI
)
cs = 991.7125 kg/m3 ¯k3 = 0.001217
τr = 1 h σ = ¯k1τ(cs
o − cs)
Eg = 2.2 kJ/mol ˜Da = 8πσ3¯k2e
(
−Eb
R·TI
)
τ
Eb = 0.00001 kJ/mol F =
k3c2
s
(co − cs)2
R = 0.008314 kJ/mol · K ˜α =
ρ − cs
co − cs
TI = 318 K
For simplicity, we consider the problem on the basis of the linearization of the process
around the desired steady state. The linearized process model takes the form:
17
31. ˙x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
(2.5)
where x is the vector of state variables, u is the manipulated input, and y is the measured
output vector. The state vector is defined by x = [x0 x1 x2 x3 yc] = [˜x0 − ˜xs
0 ˜x1 −
˜xs
1 ˜x2 − ˜xs
2 ˜x3 − ˜xs
3 ˜yc − ˜ys
c]T
, where the superscript denotes the steady-state value;
and A, B, and C are constant matrices given by:
A =
−1 0 0 −Da e
− ˜F
(ys
c)2 2 ˜F Da(1−xs
3)
(ys
c )3 e
− ˜F
(ys
c)2
ys
c −1 0 0 xs
0
0 ys
c −1 0 xs
1
0 0 ys
c −1 xs
2
0 0 −ys
c (˜α−ys
c )
1−xs
3
1−ys
c −xs
2ys
c (˜α−ys
c )
(1−xs
3)2
−1−˜αxs
2+2xs
2ys
c
1−xs
3
(2.6)
B = 0 0 0 0 1
1−xs
3
T
, C = [1 0 0 0 0] (2.7)
Over the next two sections we describe how the control strategy is tailored to explicitly
account for the effects of measurement sampling, measurement delays, and component faults.
We begin with the design and analysis of the fault-free control system. The results will serve
as the basis for tackling the fault detection and fault accommodation problems in the later
sections.
2.3 Fault-free controller synthesis, analysis and imple-
mentation
The objective of this section is to design an output feedback controller that enforces closed-
loop stability in the absence of faults using sampled and delayed output measurements. The
second objective is to characterize the minimum allowable sampling rate necessary for closed-
loop stability. The design and analysis of the fault-free control system will serve as the basis
for tackling the fault detection and fault accommodation problems in the next sections.
18
32. 2.3.1 Output feedback controller synthesis
We consider an observer-based output feedback controller of the form:
u(t) = Fη(t)
˙η(t) = Aη(t) + Bu(t) + L(y(t) − Cη(t))
(2.8)
where F is the feedback gain; η is the state of an observer that generates estimates of x using
y; A and B are constant matrices that represent models of A and B, respectively; L is the
observer gain. In general, A = A and B = B to allow for plant-model mismatch. When the
output measurements are transmitted continuously without delay, and in the special case
that A = A, B = B, a necessary and sufficient condition for the stability of the closed-loop
system of (2.5)-(2.8) (with fc ≡ 0) is to have the eigenvalues of both A + BF and A − LC
in the left half of the complex plane.
When the output measurements are delayed and sampled discretely, the observer in (2.8)
cannot be implemented directly. To compensate for the lack of continuous measurements,
a dynamic model of the process of (2.5) is embedded to provide the observer with an esti-
mate of the measured output when sensor measurements are unavailable. The state of the
model is then updated when the actual output measurements are received. The computa-
tional capabilities of modern control systems justifies and supports the computational load
associated with this approach (e.g., solving the model equations and performing the control
calculations). Specifically, we consider an inter-sample model predictor of the form:
˙w1(t) = A11w1(t) + A12w2(t) + B1u(t)
˙w2(t) = A21w1(t) + A22w2(t) + B2u(t)
(2.9)
where w = [w1 w2]T
, w1 := y is an estimate of the process output (e.g., the crystal concen-
tration) and w2 is a vector of the remaining unmeasured process states, A =
A11 A12
A21 A22
,
19
33. B = [BT
1 BT
2 ]T
. The model output is updated when the output measurements are transmit-
ted and received by the controller at discrete times. In addition to measurement sampling,
we also consider the case when the measurements are delayed. For a constant delay time τ
and a sampling period ∆, the sensor output that the controller receives at times tj = j∆ is
the output value sampled τ hours earlier, i.e., y(j∆ − τ). To compensate for the measure-
ment delay, a propagation unit is embedded in the control system which uses the process
model and the past values of the control input to calculate current output estimates from
the delayed measurements. This is then used to update the inter-sample model predictor
which, together with the controller, generates the process control signal. The propagation
unit can be described by:
˙¯w1(t) = A11 ¯w1(t) + A12 ¯w2(t) + B1u(t)
˙¯w2(t) = A21 ¯w1(t) + A22 ¯w2(t) + B2u(t)
¯y(t) = ¯w1(t)
¯y(tj+1 − τ) = y(tj+1 − τ)
(2.10)
where ¯w = [ ¯w1 ¯w2]T
, ¯w1 := ¯y is an estimate of the current process output calculated from
the delayed output measurement y, ¯w2 is the estimate of the current value of unmeasured
process states, tj is the j-th sampling instance, and ∆ := tj+1 − tj is the sampling period.
With the aid of the inter-sample model predictor and the propagation unit, the output
feedback controller can be implemented as follows:
20
34. u(t) = Fη(t)
˙η(t) = Aη(t) + Bu(t) + L(y(t) − Cη(t))
˙w1(t) = A11w1(t) + A12w2(t) + B1u(t)
˙w2(t) = A21w1(t) + A22w2(t) + B2u(t), t ∈ (tj, tj+1]
y(t) = w1(t)
˙¯w1(t) = A11 ¯w1(t) + A12 ¯w2(t) + B1u(t)
˙¯w2(t) = A21 ¯w1(t) + A22 ¯w2(t) + B2u(t), t ∈ (tj+1 − τ, tj+1]
¯y(t) = ¯w1(t)
¯y(tj+1 − τ) = y(tj+1 − τ)
y(tj) = ¯y(tj), j = 0, 1, 2, · · ·
(2.11)
The mechanism of how the propagation unit and inter-sample model predictor are reset
at the respective transmission and arrival times is shown in Fig.2.2. Note that only the
output of the model is re-set using current output estimates generated by the propagation
unit. This is calculated from the delayed measurements received at each sampling time.
Furthermore, the choice of A11 = O, A12 = O, B1 = O; corresponds to the special case of
sample-and-hold where the last available measurement is kept between consecutive sampling
times until the next one is available.
Note that unlike state feedback, the control action in (2.8) depends on the state of the
observer and not that of the inter-sample model predictor. Under this formulation, the
control action is a function of the observer state which is continuous. This scheme was
selected to prevent jumps in the value of the manipulated variable whenever the state of the
inter-sample model predictor is reset. Such behavior is undesired since it requires an almost
instantaneous action from the actuator which is typically subject to input rate constraints.
21
35. t0 t1- t1 t2t2-
)y(t)(ty 11
)(ty)(tyˆ 11
)(ty)(tyˆ 22
)y(t)(ty 22
Transmission time:
Propagation unit reset
Arrival time:
Inter-sample model reset
Figure 2.2. Timeline of measurement transmission and arrival times under measurement
sampling and delay.
2.3.2 Characterizing the minimum allowable sampling rate
To simplify the analysis, we focus on the typical case when the sampling period and the delay
time are constant (or at least bounded; extensions to the case of time-varying sampling
periods and delay times are possible and the subject of other research work). We also
consider that the sampling period is greater than the delay time (∆ > τ). To characterize
the maximum allowable sampling period or the minimum sampling rate between the sensors
and the controller; the model estimation error is defined as e(t) = y(t)− ¯y(t) = w1(t)− ¯w1(t),
where e is the difference between the output of the model and the estimate of the current
process output generated by the propagation unit. Similarly, the propagation estimation
error is ¯e(t) = ¯y(t) − y(t) = ¯w1(t) − Cx(t), where ¯e represents the difference between the
estimate of the current output and the actual current output of the process. After defining
the augmented state vector χ = xT
ηT
¯wT
2 ¯eT
wT
2 eT T
, the augmented system can be
formulated as a combined discrete-continuous system of the general form:
22
36. ˙χ(t) = Λoχ(t), t ∈ (tj, tj+1)
e(tj) = ¯e(tj+1 − τ) = 0, j = 0, 1, 2, · · · ,
(2.12)
where
Λo =
A BF O O O O
LC A + BF − LC O L O L
A21C B2F A22 A21 O O
A11C − CA B1F − CBF A12 A11 O O
A21C B2F O A21 A22 A21
O O −A12 O A12 A11
(2.13)
is a constant matrix. Note that while the process state x, the observer state η, the prop-
agation estimate of the unmeasured states ¯w2, and the model predictor state of the un-
measured states w2 all evolve continuously over time, the error e is re-set to zero at each
transmission instance, tj, since the output of the model is updated every ∆ seconds using
the estimate of the current output measurement, and the error ¯e is re-set to zero at tj+1 − τ
since the estimate of the current output is updated using the actual output measurement
at that time. It can be shown from Proposition A.1 in Appendix A that the system de-
scribed by (2.12) has the following solution for j = 0, 1, 2, · · · with the initial condition
χ(t0) = xT
(t0) ηT
(t0) ¯wT
2 (t0) 0 wT
2 (t0) 0
T
:= χ0 [47, 48]:
χ(t) =
eΛo(t−tj )
Mj
χ0, t ∈ [tj, tj+1 − τ)
eΛo(t−tj+1+τ)
Iτ eΛo(∆−τ)
Mj
χ0, t ∈ [tj+1 − τ, tj+1)
(2.14)
with ∆ := tj+1 − tj and M := IoeΛoτ
Iτ eΛo(∆−τ)
,
23
37. Io =
Ip×p O O O O O
O Ip×p O O O O
O O I(p−q)×(p−q) O O O
O O O Iq×q O O
O O O O I(p−q)×(p−q) O
O O O O O O
(2.15)
Iτ =
Ip×p O O O O O
O Ip×p O O O O
O O I(p−q)×(p−q) O O O
O O O O O O
O O O O I(p−q)×(p−q) O
O O O Iq×q O Iq×q
(2.16)
where I is the identity matrix. Based on (2.14), it can be shown that for the stability of
the fault-free sampled-data closed-loop plant, it is necessary and sufficient to have all the
eigenvalues of the matrix M strictly inside the unit circle (see Theorem A.1 in Appendix A).
The augmented system satisfies a bound of the following form:
χ(t) ≤ α χ0 e−β(t−t0)
(2.17)
for some constants α > 1 and β > 0, if and only if λmax(M) < 1, where λmax(M) is the
maximum eigenvalue magnitude of the matrix M.
It can be seen from the structure of Λo in (2.13) that the minimum stabilizing sampling
rate is dependent on the accuracy of the inter-sample model predictor, the delay time, and the
controller and observer design parameters. This dependence can be used to systematically
investigate the tradeoffs that exist between these various factors in influencing closed-loop
stability. It can also be shown that the requirement on the spectral radius of the test matrix
24
38. M to be strictly less than unity is not only sufficient but also necessary to guarantee closed-
loop stability. Note that while the above analysis was carried out for the case when the delay
time is smaller than the sampling period, a similar analysis can be applied to the case when
the sampling period is less than the delay time (∆ < τ). In the latter case, however, multiple
propagation units are required to account for every instance of measurement transmission
that occurs within each subinterval. In addition, because the update pattern within each
subinterval is different for the two cases, the structure of the stability test matrix M will
differ which, in turn, affects the stability criterion. This results in a larger augmented system
depending on the relative size of the delay time and the sampling period. Despite these
differences, a general analysis may be carried out for both cases wherein the repeating pattern
is determined and the stability analysis is carried out for each subinterval. In addition, these
results may also be extended for more general cases involving multiple measurement outputs
that are sampled at different rates and will be the subject of future research work.
It should also be noted that the ideas of using a process model and a propagation unit
to compensate for the lack of continuous measurements and the delay, respectively, are
inspired by the results obtained in the context of networked control systems [47, 48]. In
these works, however, the sensor-controller communication is limited due to the presence
of a bandwidth-limited network, while here it is limited by the constraints on the sensor
sampling rate. Furthermore, the control architecture presented here differs in that: (a) the
controller, observer, propagation unit, and model are all co-located on the controller side,
(b) the control action is calculated using the observer state (and not the model state), and
(c) the model is used only by the observer, and its output is reset by the estimate of the
current process output at the sampling times.
2.3.3 Application to the continuous crystallizer
An output feedback controller of the form (2.8) is designed to stabilize the continuous crys-
tallizer in the absence of faults where the controller and observer gains are chosen such that
25
39. the poles of A − BF and A − LC are at (−1.001, −2.001, −3.001, −4.001, −2.5). We
consider the case of parametric uncertainty in the dimensionless constant ˜F in (2.4) to in-
vestigate the effect of model uncertainty on the stability of the sampled-data system. This
results from the dependence of ˜F on ¯k3 based on the following relation: ˜F =
¯k3c2
s
(c0−cs)2 . There
is uncertainty in the actual value of ¯k3 which is determined experimentally. Model uncer-
tainty is computed as δ =
¯k3−¯km
3
¯k3
where ¯k3 is the actual value and ¯km
3 is the value used in the
model. Any other source of model uncertainty can be considered and analyzed in a similar
fashion.
1
1
1
1
1
1
2
2
2
2
Sampling period, (hr)
Delaytime,(hr)
0 0.5 1 1.5
0
0.5
1 1
1
1
1
2
2
2
2 1
1
Sampling period, (hr)
Delaytime,(hr)
0 0.5 1 1.5
0
0.5
1
(a)
1
1
1
1
1
1
2
2
2
2
Sampling period, (hr)
Delaytime,(hr)
0 0.5 1 1.5
0
0.5
1
(b)
Figure 2.3. Region of stability is larger with a propagation unit (δ = 0.3). Plots (a)-(b):
Contour plot of λmax(M) with (a) and without (b) a propagation unit.
It was previously shown that λmax(M) < 1 is the necessary and sufficient condition for
fault-free closed-loop plant stability. A contour plot is used to show how λmax(M) varies
depending on the delay time τ and sampling period ∆ (Fig. 2.3). Since the contour lines
signify different values of λmax(M), then the area enclosed by the unit contour lines denotes
the stability region of the linearized plant. Given the delay time, the minimum allowable
sampling rate or maximum sampling period corresponds to values along the unit contour
lines that bound the stability region. As expected, the range of values of the sampling period
that lead to stable behavior shrinks as the delay time is increased. For comparison, a contour
plot is also generated for a similar system without the aid of a propagation unit (Fig. 2.3(b)).
26
40. 0 5 10 15
-40
-20
0
20
40
60
Time (hr)
Crystalconcentration,x0
With propagation unit
Without propagation unit
0 5 10 15
-40
-20
0
20
40
60
Time (hr)
Crystalconcentration,x0
With propagation unit
Without propagation unit
(a)
0 10 20 30
-10
-5
0
5
10
15
Time (hr)
Totalparticlesize,x1
With propagation unit
Without propagation unit
(b)
0 5 10 15 20 25
-0.1
-0.05
0
0.05
0.1
0.15
Time (hr)
Soluteconcentration,y
c
With propagation unit
Without propagation unit
(c)
0 5 10 15 20
-0.4
-0.2
0
0.2
0.4
0.6
Time (hr)
Feedconcentration,u
With propagation unit
Without propagation unit
(d)
Figure 2.4. The closed-loop system can only be stabilized with a propagation unit (δ =
0.3, τ = 0.5h, ∆ = 1h).
In this second case, the inter-sample model predictor is updated at each sampling instance
using the delayed output measurements, instead of the current output estimates generated
by the propagation unit. The stability region is larger when a propagation unit is used.
This indicates that accounting for the measurement delays increases the range of values
for the delay time and sampling period that will still lead to stability in the system. In
addition to the previously mentioned assumptions on the sampling period and delay time,
we consider the case when both values are known. This is not generally the case in actual
practice; however, knowing that the operating point is within a given range that lies inside
the stability region will suffice for practical applications (Fig. 2.3).
The operating point selected is inside the stability region predicted by Fig. 2.3(a) but
27
41. outside the stability region in Fig. 2.3(b). These findings are confirmed by the closed-loop
evolution of the states and manipulated input at a delay time of τ = 0.5h and sampling
period of ∆ = 1h (Fig. 2.4). It is evident from this example that the process can only be
stabilized at the desired steady-state when the control system is operated with the aid of a
propagation unit.
2.4 Fault detection and accommodation
In this section, the fault-free closed-loop behavior characterized in the previous section is
used to derive appropriate rules for fault detection and accommodation. The idea is to use
the state observer in (2.11) as a fault detection filter and to compare its output with the
actual output of the system when measurements are available to ascertain the health status
of the process.
2.4.1 Fault detection
Consider the closed-loop system of (2.5) and (2.11) with no component fault (fc ≡ 0), and
consider the augmented system of (2.12)-(2.13) where the sampling period is chosen such
that λmax(M) < 1. The residual defined by rd = y − Cη can then be shown to satisfy a
time-varying bound of the following form for all t ≥ t0:
rd(t) ≤ ¯α χ0 e− ¯β(t−t0)
(2.18)
where ¯α = 2 C α and ¯β = β. This bound can be obtained directly from the fact that
x(t) ≤ χ(t) , η(t) ≤ χ(t) , and the fact that χ(t) satisfies (2.17) in the absence of
faults. Thus, for a sampling rate that is stabilizing in the absence of faults, the bound in
(2.18) can be used as a time-varying alarm threshold. A fault is declared when the residual
breaches this threshold, i.e.,
rd(Td) > ¯α χ0 e− ¯β(Td−t0)
=⇒ fc(Td) = 0 (2.19)
for some Td > 0. Note, however, that even though η is available continuously, the residual
can only be evaluated discretely regardless of when the fault actually occurs (i.e., faults
28
42. can be detected only at tj + τ, j = 0, 1, 2, · · ·). This is because the output measurements
are sent discretely at each sampling instance and are received τ hours after transmission.
Detection delays can be minimized by proper tuning of the controller and observer design
parameters and appropriate selection of the constants ¯α and ¯β such that the alarm threshold
is sufficiently tight. In principle, one could calculate appropriate values of ¯α and ¯β from the
proof but this would result in conservative figures that are not restrictive enough. To prevent
detection delays, the fault-free closed-loop behavior may be simulated and values of ¯α and ¯β
are obtained based on the profile generated. However, detection delays can only be minimized
to some extent since their values are ultimately constrained by the feasible sampling rate
and the delay time of the measurement sensors. While it is desirable to minimize detection
delays, there should be an appropriate balance in the selection of the alarm threshold such
that it is tight enough to detect faults without resulting in false alarms.
It should be noted that the above fault detection scheme can be used for fault detection
for incipient and abrupt faults and other faults that influence the evolution of the process
state.
2.4.2 Fault accommodation and compensation
Once a fault is detected, corrective action needs to be taken to compensate thereby main-
taining closed-loop stability and ensuring fault-tolerance. Using the known values for the
model parameters, sampling period, and delay time; stabilizing feedback and observer gains
are selected. This is based on the necessary and sufficient condition for stability where
λmax(M) < 1 has to be satisfied. The matrix M in (2.14) depends on Λo which, in turn,
is a function of the feedback and observer gains as shown in (2.13). This is the basis for
the fault accommodation logic which involves adjusting the controller and observer design
parameters. Hence, this ensures that the control system remains stabilizing in the presence
of faults for the given sampling period and delay time.
The implementation of the fault accommodation logic requires a characterization of the
29
43. regions of stability which does not necessitate a graphical depiction. Note that this region
of stability is based on the stability condition for fault-free sampled-data closed loop plant
that all the eigenvalues of the matrix M be strictly inside the unit circle. Contour plots
of the region of stability may be generated for illustrative purposes to enhance clarity with
regards to the fault accommodation technique; however, the construction of such plots is not
required for the implementation of the fault accommodation logic. All that is needed is the
calculation of matrix M. Such contour plots are possible in the case of a single component
fault but become more involved in the case of multiple component faults. Nonetheless, the
same principle applies wherein the stability is determined based on the eigenvalues of the
matrix M.
The same logic is also applicable when multiple consecutive faults take place. This
control architecture makes use of a stability-based fault-detection scheme which does not
handle faults that are not severe or destabilizing, as in the case when multiple faults offset
each other. Such faults do not necessitate fault accommodation since they do not affect
stability. Prior to fault detection, the fault time and nature of the fault is unknown. The
dynamics of the fault should propagate through the filter until it violates the alarm threshold.
When multiple destabilizing faults occur at different times, fault accommodation is handled
the same way. This is best understood when the resolution time exceeds the time required
for fault detection. On the other hand, multiple simultaneous faults or faults that are not
sufficiently temporally resolved become indistinguishable from each other and are treated
as a single fault. In both cases, fault accommodation is carried out as soon as the fault
registers in the filter. Note that a new alarm threshold needs to be obtained following each
fault accommodation event to detect possible faults in the new design. This is carried out
since the residual depends on the nominal fault-free behavior of the system as shown in
(2.18). This behavior is, in turn, affected by the controller and observer design parameters
which were modified following fault accommodation.
30
44. The fault accommodation strategy is event-based and triggered only when an abnormality
is detected through the alarm threshold. Since the architecture makes use of a single residual,
it is not possible to detect different faults at different times. A breach in the alarm threshold
could be caused by a single fault or the combined effect of several faults. For the second
case, one can only distinguish among the faults once the fault isolation is carried out. Upon
fault detection, the fault isolation scheme assumed to be in place determines the nature
and location of the fault. Fault accommodation is then carried out after determining the
appropriate parametric values that satisfy the stability condition given the new operating
point. Once the threshold is exceeded, the fault detection filter is unable to detect succeeding
faults and a new residual has to be put in place.
2.4.3 Application to the continuous crystallizer
To illustrate the fault detection and handling capabilities of the fault-tolerant control system,
the continuous crystallizer is initialized at a residence time of τr = 1h. Since the controller
and observer gain values are calculated by first specifying the desired location for the poles
of A+BF and A−LC, the gain values may be controlled indirectly by changing the location
of one of these poles. The variable closed-loop poles for both gains are chosen to be initially
at λ = −2.5. The sampling period is set to ∆ = 1h with a time delay of τ = 0.5h. An
inter-sample model predictor is used to estimate the evolution of the states between sampling
instances. The fault-free residual behavior along with results from (2.18) are used to derive
the following time-varying bound on the residual rd(t) ≤ 13e−0.08(t−t0)
. This serves as an
alarm threshold for fault detection. Alarm thresholds need not be time-varying; however,
this feature ensures timely recovery from faults by minimizing detection delays. The shape
of the alarm threshold in (Fig. 2.5(f)) is based on the desired fault-free dynamic behavior of
the augmented system in (2.3) which should decay exponentially thereby leading to stability.
A fault is modeled by introducing a malfunction in the mechanism responsible for regu-
lating the flow rate τr at Tf = 10h. This leads to a change in the residence time. This event
31
45. is modeled as a component or parametric fault since it leads to a change in the values of
the process parameter, Da as follows: Da = 8πσ3¯k2τr where σ = ¯k1τr(c0 − cs). Note that
this is different from an actuator fault since it does not affect the feed concentration—the
manipulated input of the control loop of interest. As such, this fault cannot be handled
through controller reconfiguration since switching to a different actuator or choosing a dif-
ferent manipulated input will not address the source of the fault. In fact, in this specific
example, controller reconfiguration is not possible since the feed concentration is the only
variable that is manipulated.
The fault causes a 10% increase in the residence time, τr, shifting it from 1h to 1.1h.
Since the fault is modeled by a change in the residence time and fault accommodation is
carried out by modifying the pole values, the stability region needs to be characterized based
on these two variables. This is carried out using the condition for fault-free closed-loop plant
stability, λmax(M) < 1, and the fact that M is a function of the residence time and pole
values. The matrix M is related to Λ0 based on (2.13) which, in turn, is affected by the
pole values as shown in (2.14). Using this relationship, a contour plot is created describing
how the maximum eigenvalue magnitude of the matrix M, λmax(M), changes depending on
the residence time and pole values (Fig. 2.6). This plot, which shows the stability region
bounded by the unit contour lines, is instrumental in the fault accommodation process once
a fault is declared. The operating point corresponding to a residence time, τr, of 1h and a
closed-loop pole value, λ, of −2.5 was initially within the stability region (Fig. 2.6). The new
process condition resulting from the parametric fault pushes the operating point outside the
region bounded by the unit contour line (i.e., τr = 1.1h, λ = −2.5). A pole value, λ, of −2.5
at a residence time, τr, of 1h satisfies the condition for fault-free closed-loop plant stability,
λmax(M) < 1, and is, therefore, expected to be stabilizing; while the same pole value at the
new residence time results in instability since the maximum eigenvalue magnitude of the
matrix M exceeds one, λmax(M) > 1 (Fig. 2.6).
32
46. 0 10 20 30 40 50
-20
0
20
40
Time (hr)
Crystalconcentration,x
0
(a)
0 5 10 15 20
-0.04
-0.02
0
0.02
Time (hr)
Soluteconcentration,yc
(b)
0 10 20 30
-10
-5
0
5
10
Time (hr)
Crystalconcentration,x0
(c)
0 10 20 30
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
Time (hr)
Soluteconcentration,y
c
(d)
0 5 10 15 20 25
-0.3
-0.2
-0.1
0
0.1
0.2
Time (hr)
Feedconcentration,u
With Switching
Without Switching
(e)
0 5 10 15
0
5
10
15
Time (hr)
Residual,r
d
(t)
With Switching
Without Switching
Threshold
Fault detection
t = 12 hr
(f)
Figure 2.5. Fault detection and accommodation maintains stability after a component
fault at Tf = 10h (δ = 0.3, τ = 0.5h, ∆ = 1h). Plots (a)-(d): Closed-loop state profiles
with (a)-(b) and without (c)-(d) fault detection and accommodation. Plot (e): Closed-
loop profiles of the manipulated input. Plot (f): Fault detection based on the evolution of
the residual. Note: Profiles in plots (a)-(e) are in deviation variable form. Actual values
are non-negative.
33
47. 1
1
1
1
11
2
2
2
Controller/Observer pole,
Residencetime,
r
-3 -2 -1 0
0.9
1
1.1
1.2
1.3
Fault occurrence
Fault accommodation
r
= 1
= -2.5
r
= 1.1
= -0.5
Figure 2.6. Fault accommodation using a contour plot of λmax(M) indicating the region
of stability.
The fault is detected at Td = 12h when it causes the residual to breach the time-varying
alarm threshold (Fig. 2.5(f)). In this particular example, ¯α = 13 and ¯β = 0.08 using the
techniques in Section 2.4. There are several existing studies on fault detection and isolation
mechanisms which are used to determine the source and magnitude of a fault in a given
system [27, 49]. In this example, a fault estimation scheme is assumed to be available and
is used to approximate the nature and size of the fault. This information is then utilized
to estimate the change in the values of the process parameter Da and, in turn, the new
residence time. The control system then modifies the faulty controller design settings based
on the calculated value so that it does not disturb future process operation. This is achieved
by selecting a point with a stabilizing closed-loop pole value of λ = −0.5 corresponding to
the new residence time. This new operating point (λ = −0.5, τr = 1.1h) now lies within the
stability region (Fig. 2.6). Any arbitrary pole value may be selected as long as it is within
the unit contour lines for a residence time of τr = 1.1h. Changing the pole values alters the
controller and observer design parameters and moves the new operating point into the stable
34
48. region. The closed-loop profiles confirm the predicted behavior and show how fault detection
and accommodation prevents the instability that would have resulted from the component
fault (Fig. 2.5).
The efficacy of the sampled-data controller coupled with the fault detection and accom-
modation scheme was also evaluated by applying the results to the nonlinear crystallizer of
(2.3), where the behavior of the particle size distribution was simulated using finite differ-
ences with 4000 temporal discretization points and 100 spatial discretization points (Fig. 2.7).
Grid-independence was ensured after obtaining identical results from higher order discretiza-
tion. The simulations were carried out in the presence of ±1% sensor measurement noise to
account for non-ideal behavior in actual conditions. It is shown that the system stabilizes
after a component fault occurs at Tf = 10h. In this case, the measurement noise results in
an additional delay in the fault detection which occurs at Td = 13h instead of at Td = 12h
(Fig. 2.7). Note that, following the fault accommodation, a new alarm threshold has to be
used to detect possible future faults.
In cases where fault accommodation cannot satisfy the stability requirement, the problem
can be addressed either by control reconfiguration (i.e., switching to a different manipulated
input) or by switching to an alternative set of sensors or actuators that have the required
sampling period and delay time. In extreme situations when all measures fail and the control
system cannot recover from the fault, the fault diagnosis information must be reported to
a higher-level supervisor which acts to ensure a graceful shutdown of the faulty process.
Another option would involve utilizing a safe-parking approach to steer the process to a
different operating point while the actuator is being repaired [49].
2.5 Conclusions
A model-based framework for fault detection and accommodation for particulate processes
subject to discretely-sampled and delayed measurements was presented. The control sys-
tem included an inter-sample model predictor and a propagation unit to account for the
35
49. 0
1
2
3
0
20
40
0
0.2
0.4
Particle size, r (mm)Time (hr)
ParticleSizeDistribution,n(r,t)
(a)
0
1
2
3
0
20
40
0
0.2
0.4
Particle size, r (mm)Time (hr)
ParticleSizeDistribution,n(r,t)
(b)
0 5 10 15 20 25
-0.3
-0.2
-0.1
0
0.1
0.2
Time (hr)
Feedconcentration,u
With Switching
Without Switching
(c)
Figure 2.7. Fault detection and accommodation maintains the stability of the Particle
Size Distribution (PSD) in the presence of sensor measurement noise after a component
fault at Tf = 10h (δ = 0.3, τ = 0.5h, ∆ = 1h). Plot (a): Closed-loop PSD profile with
(a) and without (b) fault detection and accommodation. Plot (c): Closed-loop profiles of
the manipulated input in deviation variable form.
36
50. effects of the measurement sampling and delays. By formulating the closed-loop system as a
combined discrete-continuous system, an explicit characterization of the fault-free behavior
was obtained and used to derive rules for fault detection and accommodation. The results
were illustrated using a continuous crystallizer example. Due to the closure of the moments
of the crystal size distribution in this example, the reduced-order moments model used for
controller design captured the exact crystallizer dynamics. This implies that the closed-loop
stability analysis and the fault detection and accommodation logic, derived on the basis
of the reduced-order model, can be applied directly to the infinite-dimensional system. In
general, when the reduced-order model only captures the approximate process dynamics,
modifications in the fault detection alarm thresholds and fault accommodation logic are
necessary to account for approximation errors when the finite-dimensional control system is
implemented. These modifications can be obtained using regular perturbation techniques
[1].
37
51. Chapter 3
Data-based fault identification and
fault accommodation in the control of
particulate processes with sampled
measurements
This chapter deals with the problem of fault identification and accommodation in particu-
late processes with discretely sampled measurements. The methodology involves reducing
the infinite-dimensional equation describing the particulate process to a finite-dimensional
model that captures the dominant dynamics of the system. A state feedback controller is
designed based on the reduced-order model. A zero-order hold, inter-sample model pre-
dictor is used to compensate for the measurement intermittency. This model is updated
at each sampling time once actual measurements are available. The location and magni-
tude of actuator faults are calculated at each sampling time by solving a moving-horizon
least-squares parameter estimation scheme online. The closed-loop stability properties of
the discrete-continuous system is explicitly characterized in terms of the sampling period,
controller design parameters, and actuator effectiveness (absence or extent of malfunction).
These are used in the fault accommodation approach which is critical in maintaining stability
after a fault occurs in the system. The ability of the proposed methodology to identify and
38
52. handle simultaneous and consecutive, as well as full and partial, faults are illustrated using
a non-isothermal continuous crystallizer.
The rest of the chapter is organized as follows: In Section 3.3, the model for fault identi-
fication is introduced wherein fault parameters are defined for each actuator in the system.
The next step in Section 3.4 is defining a state feedback controller under measurement sam-
pling whose stability properties are analyzed and characterized as a function of the controller
design parameters, sampling time, and the fault parameters of individual actuators. These
stability properties are utilized in Section 3.5 using a simulated model of a non-isothermal
continuous crystallizer. The data-based fault identification and fault accommodation struc-
ture was found to be effective in maintaining stability even when subject to various types of
fault scenarios. The results of this chapter were first presented in [50].
3.1 Preliminaries
3.1.1 System description
We focus on spatially homogeneous particulate processes with simultaneous particle growth,
nucleation, agglomeration and breakage, and consider the case of a single internal particle
coordinate–the particle size. Applying a population balance to the particle phase, as well
as material and energy balances to the continuous phase, we obtain the following general
nonlinear system of partial integro-differential equations:
∂n
∂t
= −
∂(G(z, r) · n)
∂r
+ wn(n, z, r), n(0, t) = b(z(t)) (3.1)
˙z = f(z) + g(z)u + Az
rmax
0
q(n, z, r)dr (3.2)
where n(r, t) ∈ L2[0, rmax) is the particle size distribution function which is assumed to be a
continuous and sufficiently smooth function of its arguments (L2[0, rmax) denotes a Hilbert
space of continuous functions defined on the interval [0, rmax)), r ∈ [0, rmax) is the particle
39
53. size (rmax is the maximum particle size, which may be infinity), t is the time, z ∈ Rn
is
the vector of state variables that describe properties of the continuous phase (e.g., solute
concentration, temperature and pH in a crystallizer), u ∈ R is the manipulated input, (3.1)
is the population balance where G(z, r) is the particle growth rate from condensation, and
wn(n, z, r) accounts for the net rate of introduction of new particles into the system, i.e., it
includes all the means by which particles appear or disappear within the system including
particle agglomeration, breakage, nucleation, feed, and removal. The z-subsystem of (3.2) is
derived from material and energy balances in the continuous phase. In this subsystem, f(z),
g(z), q(n, z, r) are smooth nonlinear vector functions and Az is a constant matrix. The term
containing the integral represents mass and heat transfer from the continuous phase to all
the particles in the population.
To express the desired control objectives, such as regulation of the total number of
particles, mean particle size, temperature, pH, etc., we define the controlled outputs as:
yι(t) = hι
rmax
0
cκ(r)n(r, t)dr, z , ι = 1, · · · , ˜m where hι(·) is a smooth nonlinear function
of its arguments and cκ(r) is a known smooth function of r which depends on the desired
performance specifications. For simplicity, we will consider that the controlled outputs are
available as online measurements.
3.1.2 Problem formulation and solution overview
The control objective is to formulate a unified framework for data-based fault identifica-
tion and accommodation that will enforce closed-loop stability under actuator faults using
sampled state measurements. The problems under consideration include: process regula-
tion using discretely-sampled measurements in the absence of faults, timely isolation and
identification of actuator faults, and fault accommodation to maintain the desired stabil-
ity and performance characteristics. To address these problems, we consider the following
methodology:
• Model reduction: Initially use model reduction techniques to derive a finite-dimensional
40
54. model that captures the dominant dynamics of the infinite-dimensional system describ-
ing the continuous crystallizer.
• Controller synthesis: Design a model-based feedback controller that stabilizes the pro-
cess states at the desired steady-state in the absence of faults. To compensate for the
lack of continuous measurements, a zero-order hold model is used wherein past state
measurements are held until the next sampling period when new state measurements
are available.
• Analysis: Obtain an explicit characterization of the minimum allowable sampling rate
that guarantees stability and residual convergence in the absence of faults in terms
of the sampling period, fault parameter/s for each actuator, and the controller design
parameter.
• Fault identification: Obtain estimates of the fault parameter via moving horizon es-
timation by comparing state estimates generated by a discrete model to the set of
previous state data.
• Fault accommodation: Derive a fault accommodation logic to preserve closed-loop
stability and minimize performance deterioration for the given sampling period and
fault parameters. The supervisor then determines the appropriate accommodation
strategy: no action, controller reconfiguration, or actuator switching.
Figure 3.1 is a schematic depiction of the different layers in the hierarchical structure for
fault identification and accommodation. This architecture shows the main components in
the design: controller, process, fault identifier, supervisor. At each sampling time, the fault
identifier updates its set of data with the current sensor measurement and uses it to calculate
estimates of the fault parameter. This information is sent to the supervisor which determines
the appropriate control action. The next sections provide a detailed description of the design
and implementation of the proposed hybrid monitoring structure.
41
55. Particulate Process
Sensors
Fault Identifier
Continous-time
model
Controller
Actuators
Fault
accommodation
strategy
Supervisor
Discrete-time
model
Optimization
problem
Data storage
u
xi
x(tj)
Figure 3.1. Overview of the integrated control architecture with fault identification and
accommodation.
3.2 Motivating example
A well-mixed non-isothermal continuous crystallizer is used throughout the paper to illus-
trate the design and implementation of model-based fault detection and accommodation.
Particulate processes are characterized by the co-presence of a continuous and dispersed
phase. The dispersed phase is described by a particle size distribution whose shape influ-
ences the product properties and ease of product separation. Hence, a population balance
on the dispersed phase coupled with a mass balance for the continuous phase is necessary
to accurately describe, analyze, and control particulate processes. Under the assumptions of
spatial homogeneity, constant volume, mixed suspension, nucleation of crystals of infinitesi-
mal size, mixed product removal, and a single internal particle coordinate—the particle size
(r); a dynamic crystallizer model can be derived:
42
56. ∂n
∂t
= ¯k1(cs − c)
∂n
∂r
−
n
τr
+ δ(r − 0)¯ǫ¯k2e
−¯k3
(c/cs−1)2
dc
dt
=
(c0 − ρ)
¯ǫτr
+
(ρ − c)
τr
+
(ρ − c)
¯ǫ
d¯ǫ
dt
dT
dt
=
ρcHc
ρCp
d¯ǫ
dt
−
UAc
ρCpV
(T − Tc) +
(T0 − T)
τr
(3.3)
where n(r, t) is the number of crystals of radius r ∈ [0, ∞) at time t per unit volume
of suspension; τr is the residence time; c is the solute concentration in the crystallizer; ρ
is the particle density; ¯ǫ = 1 −
∞
0
n(r, t)π4
3
r3
dr is the volume of liquid per unit volume
of suspension; cs = −3 ¯T2
+ 38 ¯T + 964.9 is the concentration of the solute at saturation
computed using ¯T = T−273
333−273
; c0 is the concentration of solute entering the crystallizer; ¯k1,
¯k2 and ¯k3 are constants; and δ(r − 0) is the standard Dirac function. The term containing
the Dirac function accounts for the nucleation of crystals of infinitesimal size while the
first term in the population balance represents the particle growth rate. The crystallizer
exhibits highly oscillatory behavior due to the relative nonlinearity of the nucleation rate as
compared to the growth rate. This results in process dynamics characterized by an unstable
steady-state surrounded by a stable periodic orbit. The control objective is to suppress the
oscillatory behavior of the crystallizer in the presence of actuator faults. This is carried
out by stabilizing it at an unstable steady-state that corresponds to a desired particle size
distribution by manipulating the solute feed concentration (c0) and residence time (τr).
Through method of moments, a sixth-order ordinary differential equation system was
derived to describe the temporal evolution of the first four moments of the particle size
distribution, the solute concentration, and the temperature (see [7] for a detailed derivation).
The reduced-order model can be cast in the following form:
43
57. dµ0
dt
=
−µ0
τr
+ 1 −
4
3
πµ3
¯k2e
−¯k3
( c
cs
−1)
2
e
−Eb
RT
dµv
dt
=
−µv
τr
+ vµv−1
¯k1(c − cs)e
−Eg
RT , v = 1, 2, 3
dc
dt
=
c0 − c − 4π¯k1e
−Eg
RT τr(c − cs)µ2(ρ − c)
τr 1 − 4
3
πµ3
dT
dt
= −
ρHc
ρCp
dµ3
dt
−
UAc
ρCpV
(T − Tc) +
(T0 − T)
τr
(3.4)
The global phase portrait of the system of (3.4) has a unique unstable equilibrium point
surrounded by a stable limit cycle at xs
= [µs
0 µs
1 µs
2 µs
3 cs
Ts
]T
=
[0.0047 0.0020 0.0017 0.0022 992.95 298.31]T
. Sampled measurements of
the moments (µ0, µ1, µ2, µ3), the solute concentration (c), and temperature (T) are used
to control the process. These state measurements are collected discretely and sent to the
controller where the control action is calculated and then sent to the actuator to effect the
desired change in the process state.
For simplicity, we consider the problem on the basis of the linearization of the process
around the desired steady state. The linearized process model takes the form:
˙x(t) = Ax(t) + Bu(t) (3.5)
where x(t) is the vector of state variables; u is the manipulated input; A and B are constant
matrices given by: A =
∂f
∂x (xs,us)
, B =
∂f
∂u (xs,us)
where us
denotes the steady state values
for the available manipulated inputs. The state vector is expressed as a deviation variable,
x(t) = χ(t) − xs
, where χ(t) = [µ0(t) µ1(t) µ2(t) µ3(t) c(t) T(t)]T
.
Table 3.1 gives the process parameters and steady state values used in the simulated
crystallizer example. Over the next sections, we describe the control architecture and fault
identification scheme.
44
58. Table 3.1. Process parameters and steady-state values for the non-isothermal continuous
crystallizer.
ρc = 1770 kg/m3
ρCp = 3000 J/m3
· K
cs
o = 1000 kg/m3
Hc = −50 J/kg
τr = 1 h U = 1800 W/K · m2
Eg = 1 kJ/mol Ac = 0.25 m2
Eb = 0.00001 kJ/mol V = 0.01 m3
Ts
c = 298 K Ts
o = 303 K
R = 0.008314 kJ/mol · K
k1 = 0.05064 mm · m3
/kg · h ¯k1 = k1e(
−Eg
R·T
)
k2 = 7.957 (mm3
· h)−1 ¯k2 = k2e(
−Eb
R·T
)
k3 = 0.001217 ¯k3 = k3
3.3 Fault identification
3.3.1 Fault model
To model the fault, the reduced, linearized system dynamics is written in the following form:
˙x(t) = Ax(t) + Bk
αk
uk
(t) (3.6)
where x(t) is the vector of state variables; u is the manipulated input. The state vector is a de-
viation variable, x(t) = χ(t) − xs
, where χ(t) = [µ0(t) µ1(t) µ2(t) µ3(t) c(t) T(t)]T
;
and A and Bk
are constant matrices given by: A =
∂f
∂x (xs,us)
, Bk
=
∂f
∂uk
(xs,uk,s)
where
uk,s
denotes the steady state values for the available manipulated inputs, k is the active
control configuration and m is the total number of actuators. For fault identification,
αk
= diag{αk
1, · · · , αk
m} is a diagonal matrix that is used to account for the presence of
actuator faults or malfunctions in the system. Each of the diagonal elements in αk
char-
acterizes the local health status of the individual actuators. In the illustrative example in
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