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Transcript of TGNDissertationMain
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
To God who makes all things possible.
Unless the LORD build the house, they labor in vain who build. (Psalm 127:1)
ii
Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background on monitoring and control of particulate processes . . . . . . . . 2
1.3 Objectives and organization of the dissertation . . . . . . . . . . . . . . . . . 6
2 Fault detection and accommodation in particulate processes with sampled
and delayed measurements 11
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Problem formulation and solution overview . . . . . . . . . . . . . . . 13
2.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Fault-free controller synthesis, analysis and implementation . . . . . . . . . . 18
2.3.1 Output feedback controller synthesis . . . . . . . . . . . . . . . . . . 19
2.3.2 Characterizing the minimum allowable sampling rate . . . . . . . . . 22
2.3.3 Application to the continuous crystallizer . . . . . . . . . . . . . . . . 25
2.4 Fault detection and accommodation . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Fault detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Fault accommodation and compensation . . . . . . . . . . . . . . . . 29
2.4.3 Application to the continuous crystallizer . . . . . . . . . . . . . . . . 31
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
iii
3 Data-based fault identification and fault accommodation in the control of
particulate processes with sampled measurements 38
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1.2 Problem formulation and solution overview . . . . . . . . . . . . . . . 40
3.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Fault identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.1 Fault model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Data-based fault identification . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Finite-dimensional sampled-data control system . . . . . . . . . . . . . . . . 47
3.4.1 State feedback controller synthesis . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Controller implementation under measurement sampling . . . . . . . 47
3.4.3 Closed-loop stability analysis . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Fault-tolerant control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.1 Fault identification under partial faults . . . . . . . . . . . . . . . . . 51
3.5.2 Fault identification and accommodation . . . . . . . . . . . . . . . . 53
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Model-based fault-tolerant control of uncertain particulate processes: in-
tegrating fault detection, estimation and accommodation 58
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 Problem formulation and solution overview . . . . . . . . . . . . . . . 60
4.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 Finite-dimensional sampled-data control system . . . . . . . . . . . . . . . . 65
4.3.1 State feedback controller synthesis . . . . . . . . . . . . . . . . . . . . 65
4.3.2 Implementation under measurement sampling . . . . . . . . . . . . . 65
iv
4.3.3 Fault model for continuous system . . . . . . . . . . . . . . . . . . . 66
4.4 Closed-loop stability analysis under measurement sampling . . . . . . . . . . 67
4.5 Data-driven actuator fault identification and accommodation . . . . . . . . . 69
4.5.1 Discrete fault model . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.2 Data-based fault identification . . . . . . . . . . . . . . . . . . . . . . 70
4.5.3 Fault accommodation . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Fault tolerant control application . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6.1 Fault identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5 Sensor fault accommodation strategies in the control of particulate pro-
cesses with multi-rate measurements and measurement sampling 85
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.1 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.2 Problem formulation and solution overview . . . . . . . . . . . . . . . 87
5.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3 Multi-rate sampling mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4 Finite-dimensional multi-rate sampled-data control system design . . . . . . 93
5.4.1 Output feedback controller synthesis . . . . . . . . . . . . . . . . . . 93
5.4.2 Controller implementation under multi-rate sampling . . . . . . . . . 94
5.4.3 Closed-loop stability analysis . . . . . . . . . . . . . . . . . . . . . . 94
5.5 Fault-tolerant control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
A Proofs of Chapter 2 104
v
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
vi
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) = [u11(t) u1
2(t)]T =
[c0(t) τr(t)]T ). Contour plots of Γk(∆) plotted against different values of the
fault parameter (α11) and fault model parameter (α1
1). . . . . . . . . . . . . . 79
vii
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 u11, 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 u11, 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 (α11) and the controller de-
sign parameter (p1) using the feed concentration (c0) and residence time (τr)
as the manipulated variables (α11 = 1). Plot (b): Stability region plotted
against the fault parameter (α21) and the fault model parameter (α2
1) using
the residence time (τr) as the only manipulated variable (u21). . . . . . . . . . 83
4.8 Plots (a)-(b): Dynamics of the state (µ1) (a) and the fall-back manipulated
variable u21 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
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
ix
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
x
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
0cκ(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
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
SensorActuator
)( τ−ty
)(ˆ ty
u
Plant
Model
Measurement
Reset
Local Control System
Cxy
fuBAxdt
dxc
=
++=
F
State Observer
)ˆ(ˆˆ ηηη CyLuBAdt
d−++=
η
Cwy
uBwAdt
dw
=
+=
ˆ
ˆˆ
wCy
uBwAdt
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
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)ǫk2e
(−k3
(c/cs−1)2
)
dc
dt=
(c0 − ρ)
ǫτr+
(ρ− c)
τr+
(ρ− c)
ǫ
dǫ
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−∫∞
0n(r, t)π 4
3r3dr 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
˙x0 = −x0 + (1− x3)Da e
(−F
y2c
)
˙xdm = −xdm + ycxdm−1, i = 1, 2, 3
˙yc =1− yc − (α− yc)ycx2
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 [xs0 xs
1 xs2 xs
3 ysc ] =
[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(−EgR·TI
)
cso = 1100 kg/m3 k2 = 7.958e(−EbR·TI
)
cs = 991.7125 kg/m3 k3 = 0.001217
τr = 1h σ = k1τ(cso − cs)
Eg = 2.2 kJ/mol Da = 8πσ3k2e(−EbR·TI
)τ
Eb = 0.00001 kJ/mol F =k3c
2s
(co − cs)2
R = 0.008314 kJ/mol ·K α =ρ− csco − cs
TI = 318K
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
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 − xs0 x1 −
xs1 x2 − xs
2 x3 − xs3 yc − ysc ]
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
(ysc)2 2FDa(1−xs
3)
(ysc)3 e
−F
(ysc)2
ysc −1 0 0 xs0
0 ysc −1 0 xs1
0 0 ysc −1 xs2
0 0 −ysc(α−ysc)1−xs
3
1−ysc−xs2y
sc(α−ysc)
(1−xs3)
2
−1−αxs2+2xs
2ysc
1−xs3
(2.6)
B =[0 0 0 0 1
1−xs3
]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
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 6= A and B 6= 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
B = [BT1 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) = A11w1(t) + A12w2(t) + B1u(t)
˙w2(t) = A21w1(t) + A22w2(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
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) = A11w1(t) + A12w2(t) + B1u(t)
˙w2(t) = A21w1(t) + A22w2(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
t0 t1- t1 t2t2-
)y(t)(ty 11
)(ty)(ty 11)(ty)(ty 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 measurementsampling 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 wT2 eT
]T, the augmented system can be
formulated as a combined discrete-continuous system of the general form:
22
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 wT2 (t0) 0
]T:= χ0 [47, 48]:
χ(t) =
eΛo(t−tj )M jχ0, t ∈ [tj , tj+1 − τ)
eΛo(t−tj+1+τ)IτeΛo(∆−τ)M jχ0, t ∈ [tj+1 − τ, tj+1)
(2.14)
with ∆ := tj+1 − tj and M := IoeΛoτIτe
Λo(∆−τ),
23
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
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
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 = k3c2s
(c0−cs)2 . There
is uncertainty in the actual value of k3 which is determined experimentally. Model uncer-
tainty is computed as δ =k3−km3
k3where 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
11
11
1
2
2
2
2
Sampling period, (hr)
Dela
y tim
e,
(h
r)
0 0.5 1 1.50
0.5
1 1
1
1
1
2
2
2
2 1
1
Sampling period, (hr)
Dela
y tim
e,
(h
r)
0 0.5 1 1.50
0.5
1
(a)
1
11
11
1
2
2
2
2
Sampling period, (hr)
Dela
y tim
e,
(h
r)
0 0.5 1 1.50
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
0 5 10 15-40
-20
0
20
40
60
Time (hr)
Cry
sta
l concentr
ation,
x0
With propagation unitWithout propagation unit
0 5 10 15-40
-20
0
20
40
60
Time (hr)
Cry
sta
l concentr
ation,
x0
With propagation unitWithout propagation unit
(a)0 10 20 30
-10
-5
0
5
10
15
Time (hr)
Tota
l part
icle
siz
e,
x1
With propagation unitWithout propagation unit
(b)
0 5 10 15 20 25
-0.1
-0.05
0
0.05
0.1
0.15
Time (hr)
Solu
te c
oncentr
ation,
yc
With propagation unitWithout propagation unit
(c)0 5 10 15 20
-0.4
-0.2
0
0.2
0.4
0.6
Time (hr)
Feed c
oncentr
ation,
u
With propagation unitWithout 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
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) 6= 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
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
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
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
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 = 8πσ3k2τr 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
0 10 20 30 40 50
-20
0
20
40
Time (hr)
Cry
sta
l concentr
ation,
x0
(a)0 5 10 15 20
-0.04
-0.02
0
0.02
Time (hr)
Solu
te c
oncentr
ation,
yc
(b)
0 10 20 30
-10
-5
0
5
10
Time (hr)
Cry
sta
l concentr
ation,
x0
(c)0 10 20 30
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
Time (hr)
Solu
te c
oncentr
ation,
yc
(d)
0 5 10 15 20 25
-0.3
-0.2
-0.1
0
0.1
0.2
Time (hr)
Feed c
oncentr
ation,
u
With SwitchingWithout Switching
(e)0 5 10 15
0
5
10
15
Time (hr)
Resid
ual, r
d(t
)
With SwitchingWithout SwitchingThreshold
Fault detectiont = 12 hr
(f)
Figure 2.5. Fault detection and accommodation maintains stability after a componentfault at Tf = 10h (δ = 0.3, τ = 0.5h, ∆ = 1h). Plots (a)-(d): Closed-loop state profileswith (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 ofthe residual. Note: Profiles in plots (a)-(e) are in deviation variable form. Actual valuesare non-negative.
33
1
1
1
1
11
2
2
2
Controller/Observer pole,
Resid
ence t
ime,
r
-3 -2 -1 00.9
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 regionof 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
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
01
23
0
20
400
0.2
0.4
Particle size, r (mm)Time (hr)
Pa
rtic
le S
ize
Dis
trib
utio
n, n
(r,t)
(a)0
12
3
0
20
400
0.2
0.4
Particle size, r (mm)Time (hr)
Pa
rtic
le S
ize
Dis
trib
utio
n,
n(r
,t)
(b)
0 5 10 15 20 25
-0.3
-0.2
-0.1
0
0.1
0.2
Time (hr)
Feed c
oncentr
ation,
u
With SwitchingWithout Switching
(c)
Figure 2.7. Fault detection and accommodation maintains the stability of the ParticleSize Distribution (PSD) in the presence of sensor measurement noise after a componentfault 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 ofthe manipulated input in deviation variable form.
36
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
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
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
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
0cκ(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
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
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 andaccommodation.
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
∂n
∂t= k1(cs − c)
∂n
∂r−
n
τr+ δ(r − 0)ǫk2e
(−k3
(c/cs−1)2
)
dc
dt=
(c0 − ρ)
ǫτr+
(ρ− c)
τr+
(ρ− c)
ǫ
dǫ
dt
dT
dt=
ρcHc
ρCp
dǫ
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 −∫
∞
0n(r, t)π 4
3r3dr is the volume of liquid per unit volume
of suspension; cs = −3T 2 + 38T + 964.9 is the concentration of the solute at saturation
computed using T = T−273333−273
; 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
dµ0
dt=
−µ0
τr+
(1−
4
3πµ3
)k2e
−k3
( ccs
−1)2
e−EbRT
dµv
dt=
−µv
τr+ vµv−1k1(c− cs)e
−EgRT , v = 1, 2, 3
dc
dt=
c0 − c− 4πk1e−EgRT τr(c− cs)µ2(ρ− c)
τr(1− 4
3πµ3
)
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 = [µs0 µs
1 µs2 µs
3 cs T s]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
Table 3.1. Process parameters and steady-state values for the non-isothermal continuouscrystallizer.
ρc = 1770 kg/m3 ρCp = 3000 J/m3 ·K
cso = 1000 kg/m3 Hc = −50 J/kg
τr = 1h U = 1800W/K ·m2
Eg = 1 kJ/mol Ac = 0.25m2
Eb = 0.00001 kJ/mol V = 0.01m3
T sc = 298K T s
o = 303K
R = 0.008314 kJ/mol ·K
k1 = 0.05064mm ·m3/kg · h k1 = k1e(−EgR·T
)
k2 = 7.957 (mm3 · h)−1 k2 = k2e(−EbR·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) +Bkαkuk(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{αk1, · · · , α
km} 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
45
Section 3.5, α1 represents the health of the actuator used to vary the inlet concentration (c0)
while αk2 is for the actuator used to adjust the residence time (τr). The elements of the fault
matrix αk take values between zero and one where zero denotes total actuator failure while
one denotes the fault-free state of the fault parameter in l-th actuator wherein αkl (t) ∈ [0, 1].
In the absence of faults, αk = I where I is the identity matrix.
The continuous equation in (3.6) is converted into discrete form in order to more read-
ily compare these discrete estimates to the historical input and state measurements. The
modified system takes the form:
x[j + 1] = Ax[j] + Bαu[j], j ∈ {0, 1, · · · } (3.7)
where A = eA∆, B = A−1(eA∆ − I)B, ∆ = τj+1 − τj is the update period which represents
the time interval between discrete consecutive measurements, and j is the update instance.
3.3.2 Data-based fault identification
Data-based fault identification involves estimating the value of the fault parameter matrix
α and comparing this against the values generated by the fault model in (3.7). This is
done using historical data of the state measurements and the manipulated input. The cost
function is:
J(ζj , α) =
j−NI+1∑
p=j
(∥∥∥x[p + 1]− Ax[p]− Bαu[p]∥∥∥2)
(3.8)
where ζj = {(x[j − p], u[j − p])|p = 1, 2, · · · , NI} denotes the past NI historical data of
the state measurements and the manipulated inputs for each jth sampling instance. Using
a large value for NI results in higher accuracy for calculated values obtained for the fault
estimation matrix α; however, this may also result in a high computational load as well as
discontinuities in the values of α particularly right after a fault has occurred since the pool
of I/O data used in the calculations will involve data both before and after the fault. This
46
parameter should therefore be selected appropriately.
Using the cost function in (3.8), finite-horizon least squares optimization problem was
then developed as follows:
minα
J(ζj, α)
s.t. 0 ≤ α1,2 ≤ 1
Note that the calculated values of the fault matrix α may slightly differ from the actual
values α particularly at the onset of the fault.
3.4 Finite-dimensional sampled-data control system
3.4.1 State feedback controller synthesis
The control system design involved first synthesizing a state feedback controller that sta-
bilizes the finite-dimensional system when the sensors continuously transmit data to the
controller. We considered the following discrete controller for simplicity:
u(t) = Kx(t), t ∈ [τj , τj+1) (3.9)
The controller gain K is chosen to ensure that the eigenvalues of A + BK lie in the open
left half of the complex plane.
3.4.2 Controller implementation under measurement sampling
The implementation of the controller of (3.9) requires continuous availability of the sensor
measurements. Due to measurement sampling, the controller cannot be directly implemented
since the output measurements are only available at discrete time instances. To compen-
sate for the unavailability of continuous measurements, sample-and-hold scheme is used to
provide the controller with estimates of the states when measurements are not available.
At each sampling time, the corresponding values of the measured states are instantaneously
transmitted to the controller and are used to update the corresponding model states. The
model-based state feedback controller is implemented as follows:
47
u(t) = Kx(t), t ∈ [τj , τj+1)
˙x(t) = 0, t ∈ [τj , τj+1)
x(τj) = x(τj), j ∈ {0, 1, · · · }
(3.10)
where x is a model used in generating the discrete control by utilizing previously held state
values x(τj) until the next state measurement x(τj+1) is available, j denotes each sampling
instance, and τj are the update times when values of the state are collected.
3.4.3 Closed-loop stability analysis
To investigate the stability properties of the finite-dimensional sampled-data closed-loop
system, we first define the model estimation error as e = x(t) − x(t), where e represents
the difference between the model output given in (3.10) and the actual measured state.
Then, defining the augmented state vector ξ(t) = [x(t) x(t) e(t)]T , the finite-dimensional
sampled-data closed-loop system is formulated as a switched system and written in the form:
ξ(t) = Fξ(t), t ∈ [τj , τj+1)
e(τj) = 0, j ∈ {0, 1, · · · }(3.11)
where F is a matrix defined as:
F =
A+BαK −BαK
A+BαK −BαK
, (3.12)
The following proposition characterizes the sampled-data closed-loop system behavior in
terms of the sampling rate, the controller design parameters, and the actuator health.
Proposition 3.1. The augmented closed-loop system described by (3.11)-(3.12), subject to the
initial condition ξ(0) = [x(τ0) x(τ0) e(τ0)]T := ξ0, has a response of the form:
ξ(t) = eF (t−τj)N jξ0 (3.13)
for t ∈ [τj , τj+1), ∀ j ∈ {0, · · · }, where N is given by:
N = IseF∆ (3.14)
48
where Is = diag{I, O} is a diagonal matrix that accounts for the model update at each trans-
mission time which also resets the estimation error e to zero. The null matrix O accounts
for this update.
Based on (3.13)-(3.14), the following proposition provides a necessary and sufficient con-
dition for stability of the finite-dimensional sampled-data closed-loop system.
Proposition 3.2. Consider the sampled-data closed-loop system, (3.10), and the augmented
system of (3.11)-(3.12) whose solution is given by (3.13)-(3.14). Then the zeros solution,
ξ = [x x e]T = [0 0 0]T , is exponentially stable if and only if r(N(∆)) < 1.
Remark 3.1. An examination of the structure of N in (3.14) indicates that its spectral radius
is dependent on the sampling period, ∆, and F (which, in turn, depends on the actuator
health and the controller gain). All these factors are tied together through the stability con-
dition of Proposition 3.2 which can, therefore, be used to examine and quantify the various
interdependencies between these factors. For instance, if the sampling rate of a particular
sensor is fixed by some performance requirement, one can determine the level of actuator
malfunction that the system can still handle without leading to instability.
Remark 3.2. The requirement that the spectral radius of N be strictly less than one ensures
stability by limiting the growth of the closed-loop state within each update period ∆ as the
measurement sampling is repeatedly executed over time.
3.5 Fault-tolerant control
A non-isothermal continuous crystallizer example is selected to illustrate the proposed fault-
tolerant scheme. Discrete measurements of the moments of the particle size distribution (µ0,
µ1, µ2, µ3), concentration (c), and temperature (T ), are used to control the system. The
inter-sample state estimator is used to estimate values of the states using held values of past
49
state measurements when actual sensor measurements are unavailable. A sampling period
of ∆ = 6min is chosen.
The system is controlled by simultaneously manipulating the inlet solute concentra-
tion (c0), and residence time (τr). The stability regions are obtained using the condition
λmax(N) < 1 derived from the closed-loop stability analysis of the test matrix N in (3.14)
(Fig. 3.2). These stability conditions are obtained as an explicit function of the controller
gain (K), sampling period (∆), and the fault parameter matrix (α).
These regions are plotted as a function of the health status of the actuators–with α1
corresponding to the manipulated inlet concentration (c0) and α2 to that of the residence
time (τr). These are obtained for a sampling period of ∆ = 6min. The blue area enclosed by
the unit contour line shows the region where the process is stable since λmax(N) < 1, while the
pink region denotes process conditions that lead to instability where λmax(N) > 1. Such plots
are useful in predicting the behavior of the process and in determining the appropriate fault-
tolerant response once a fault is identified. A partial malfunction in any of the actuators could
possibly occur such that the operating point is shifted somewhere within the stability region.
This will result in behavior that is not detrimental to process performance and product
quality which does not warrant immediate fault accommodation or control reconfiguration.
Based on this knowledge, the plant supervisor is then able to strategically prioritize which
specific control loop or plant equipment requires maintenance or replacement through this
stability-based closed-loop analysis. For the case where there are more variables to consider
(e.g., a larger number of manipulated variables), instead of a two-dimensional contour plot,
a look-up table with values of the spectral radius of N for varying magnitudes of the process
parameters may be generated offline and then used in the event of a process malfunction in
determining whether a fault warrants an immediate response.
Among the highlights of the data-based fault identification scheme proposed in this chap-
ter is the added capability of identifying partial malfunctions that do not result in system
50
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
1
α2
α1
Figure 3.2. Region of stability based on actuator health was used to determine whetherequipment repair, fault accommodation, or system reconfiguration is required (∆ =6min). Contour plot of λmax(N) for pole values [−1− 2− 3− 4− 5− 6].
instability. Past studies made use of stability-based alarm threshold and are only able to
detect faults resulting in instability. This is approach is proactive in dealing with malfunc-
tions as opposed to reactive schemes where action is triggered only when there is threat of
instability. Machine repair carried out at an early stage may prove to be less costly and
time-consuming as opposed to the urgent repairs or fault accommodation following a more
severe and destabilizing malfunction.
The next subsection illustrates the simulation results of the fault identification for a
system with partial faults that do not lead to instability.
3.5.1 Fault identification under partial faults
For both the simultaneous and continuous faults shown below, the controller gain (K) is
calculated by specifying the location of the poles of A + BK at [−1 − 2 − 3 − 4 − 5 −
6]. Fault identification is carried out by comparing the past 10 data points of the state
measurements and manipulated input, NI = 10, to the discrete model generated in (3.7) by
51
solving the optimization problem in (3.3.2). Actuator faults in both manipulated variables
are investigated and the simulation is carried out under a sampling period of 6min.
The first case involves a simultaneous fault that occurred after 10h where α1 = 0.8 and
α2 = 0.5 where the actuator handling the inlet concentration becomes 20% effective while
the other actuator used in varying the residence time (τr) suffers 50% loss in performance
(Fig. 3.2a-b). The second case involves a consecutive fault occurring after 5h followed by
another one at 10h. The first fault causes an 90% step change in the performance of the
actuator responsible for adjusting the residence time (τr) while the second fault is a gradual
one that causes a linear decline in the control action of the actuator manipulating the inlet
concentration (c0) (Fig. 3.2c-d).
In both cases, the fault identification scheme is shown to be effective and capable of
almost instantaneously locating and quantifying the fault as soon as it happens. However,
jumps in the calculated values of the fault estimation parameters are occasionally observed
right after a fault has taken place. These jumps are visible even in the plot of a fault
estimation parameter that was not directly affected by the fault. This behavior is due to
sudden disruptions in the data points used in the data-based identification method which
included values of the state and the manipulated variable before and after the fault. This
is why the optimization horizon (NI) has to be properly selected—small values result in
inaccuracies in the fault identification while large values lead to sharp jumps or prolonged
settling time. Due to this behavior, plant response should be suspended until the fault
identification scheme settles to a final value. These sharp discontinuities; however, may
provide insight with regards to the health status of a neighboring actuator. In the case of
the consecutive faults, two separate faults were introduced at 10h and 15h. A plot of the
fault estimation parameter devoted to the actuator responsible for manipulating the inlet
concentration (c0) revealed a step change at 10h owing to a malfunction in that actuator;
however, it also revealed a spike at exactly 15h which is when another malfunction affected a
52
neighboring actuator (Fig. 3.3a). Based on that plot alone, one can infer that a malfunction
has occurred within the system.
0 20 40 60 800
0.2
0.4
0.6
0.8
1
Time (h)
α1
Actual valueComputed value
(a)0 20 40 60 80
0
0.2
0.4
0.6
0.8
1
Time (h)
α2
Actual valueComputed value
(b)
0 10 20 30 400
0.2
0.4
0.6
0.8
1
Time (h)
α1
Actual valueComputed value
(c)0 10 20 30 40
0.5
0.6
0.7
0.8
0.9
1
Time (h)
α2
Actual valueComputed value
(d)
Figure 3.3. Actual and calculated values of the fault estimation parameters (∆ = 6min).α1: inlet concentration (c0), α2: residence time (τr). Plots (a)-(b): Simultaneous faults.Plots (c)-(d): Consecutive faults.
3.5.2 Fault identification and accommodation
Fig. 3.4 is a schematic diagram of the fault accommodation logic embedded within the
control architecture. It shows the various responses to a fault depending on whether it
results in instability within the system. If it is not a destabilizing fault, the fault location
and magnitude is noted for upcoming repair or replacement but no immediate action is made
since the system is still stable. If the fault was found to lead to instability, the controller is
reconfigured or the system switches to a different or back-up actuator. The idea is to restore
stability in spite of the fault while avoiding the introduction of disruptions in normal process
53
:
1
1
0
k
l
Set
l
i
*
:Solve
)(
1)(
*
?
l
k
l
k h
)(
1)(
*
?
K
hk
:
2
1k
l
Set
l
1ii*
:
KK
Update
Y
N
Y
N
Fault
Identification
Fault
Accommodation
Figure 3.4. Fault accommodation logic.
operation.
To illustrate the fault accommodation capabilities of the control architecture, a destabi-
lizing fault was introduced after 10h of operation, causing the actuator controlling the inlet
concentration (c0) to drop its effectiveness from 100% to 45%. Initially, the fault-free system
has a controller gain K such that the poles of A + BK are at [-9.5 -2 -3 -4 -5 -6] under a
sampling period of ∆ = 6min. A fault occurs and the system identifies both the location
(α1) and magnitude of the fault (Fig. 3.5). Using the original parameter values and the
new faulty condition, the spectral radius of N is calculated and the fault is determined to be
destabilizing. The new operating point is now inside the pink unstable region (Fig. 3.6a). To
avoid instability, fault accommodation is carried out using the plot of the region of stability
(Fig. 3.6a). The x-axis in the plot is based on the first pole location which is the only pole
that was modified in order to vary the controller gain. The controller gain (K) is adjusted
by shifting the location of the poles of A + BK to [−6.4 − 2 − 3 − 4 − 5 − 6]. This shifted
54
the operating point into the stable region once again. The simulation plots of the dynamics
of the manipulated variables demonstrate the destabilizing effect of the fault in the absence
of fault accommodation. These undesired effects are avoided through fault identification
and fault accommodation which was able to re-establish stability even after a potentially
disruptive fault.
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
Time (h)
α1
Actual valueComputed value
(a)0 10 20 30 40 50
0.5
0.6
0.7
0.8
0.9
1
Time (h)
α2
Actual valueComputed value
(b)
Figure 3.5. Fault identification after a potentially destabilizing fault at 10h with ∆ =6min. α1: inlet concentration (c0), α2: residence time (τr).
3.6 Conclusions
This chapter dealt with the design and analysis of a system with sampled state measurements
to demonstrate a fault-tolerant control strategy for particulate processes which included a
data-based fault identification scheme and a fault accommodation set-up that handles desta-
bilizing faults. Fault-tolerance is important since faults or even non-destabilizing malfunc-
tions can lead to process instability and/or inferior product quality. This fault identification
scheme provides a timely technique for a fault-tolerant system that is able to readily deal
with faults and malfunctions right after they are detected. This may be carried out by
scheduling system maintenance, fault accommodation, or system reconfiguration. To en-
sure robustness against faults, a stability region was constructed to analyze the stability
properties of the system as a function of the controller design parameter (K), sampling
time (∆), and actuator health (α). After a fault was detected, this stability region was
55
−10 −9.5 −9 −8.5 −8 −7.5 −7 −6.5 −6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
11
1
1
Pole value, λ
α1
Faultoccurrence
Fault identificationand accommodation
λ = −9.5α
1=1
λ = −6.4α
1=0.45
λ = −9.5α
1=0.45
(a)
0 20 40
994
996
998
1000
1002
Time (h)
Fe
ed
co
nce
ntr
atio
n,
c o (
kg/m
m3)
(b)0 20 40
298.28
298.3
298.32
298.34
298.36
298.38
Time (h)
Te
mp
era
ture
, T
(K
)
(c)
0 20 40
998.5
999
999.5
1000
Time (h)
Fe
ed
co
nce
ntr
atio
n,
c o (
kg/m
m3)
(d)0 20 40
298.31
298.32
298.33
298.34
Time (h)
Te
mp
era
ture
, T
(K
)
(e)
Figure 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 actuatorcontrolling the inlet concentration (c0), α1 and the first pole value (λ) used to find thecontroller design parameter K (α2 = 1). Plots (b)-(c): Dynamic profiles of (b) inlet con-centration (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 faultaccommodation. 56
used to determine the appropriate control action which may involve: fault accommodation
by switching to a different controller gain (K), varying the sampling time (∆), or simply
scheduling future equipment maintenance or repair when the minor malfunction is not found
to be destabilizing. A future expansion in this area will include accounting for the case when
only output measurements are available instead of full state measurements. This is a more
realistic case since the measurements of the concentration (c) and temperature (T ) are more
readily available when compared to the moments of the particle size distribution (µ0, µ1, µ2,
µ3).
57
Chapter 4
Model-based fault-tolerant control of
uncertain particulate processes:
integrating fault detection, estimation
and accommodation
This chapter is on fault identification and accommodation in particulate processes with
discretely-sampled measurements and plant-model mismatch. This is done by designing a
state feedback controller based on a reduced-order model describing the dominant dynamics
of the infinite-dimensional system. While measurements are unavailable, a model generates
state estimates which are updated each sampling time. A moving-horizon least-squares pa-
rameter optimization is utilized for online actuator fault identification using historical data.
The closed-loop stability properties of the discrete-continuous system is used in fault accom-
modation by updating parameters in the model and/or the controller. This is illustrated
using a non-isothermal continuous crystallizer.
This chapter is structured accordingly: Section 4.1.2 contains the problem formulation
and solution overview which is followed by the design of finite-dimensional sampled-data
control system in Section 4.3. This system is then subjected to a closed-loop stability analysis
under measurement sampling in Section 4.4. Results from this analysis are then used in
58
Section 4.5 to derive strategies for data-driven fault identification and accommodation. These
techniques are applied to a simulated non-isothermal continuous crystallizer in Section 4.6
and some concluding remarks are discussed in Section 4.7. Findings in this chapter were
first presented in [51].
4.1 Preliminaries
4.1.1 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)) (4.1)
z = f(z) + g(z)u+ Az
∫ rmax
0
q(n, z, r)dr (4.2)
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, (4.1)
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 (4.2) is
59
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
0cκ(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.
4.1.2 Problem formulation and solution overview
The control objective is to stabilize the process at some desired equilibrium state in the pres-
ence of actuator faults using discretely-sampled state measurements. The problems under
consideration include: fault-free process regulation using discretely-sampled measurements,
timely identification and isolation of the actuator faults in the presence of model uncertainty,
fault compensation to maintain the desired stability and performance characteristics through
fault accommodation. To address these problems, we consider the following methodology:
• Model reduction: Derive a finite-dimensional model through model reduction tech-
niques. This reduced-order model captures the dominant dynamics of the infinite-
dimensional system describing the continuous crystallizer.
• Controller synthesis: Use the reduced-order model to synthesize a model-based state
feedback controller that regulates the process states at the desired steady-state in the
absence of faults. To account for measurement sampling, the controller calculates the
control action based on state estimates generated by a model.
• Analysis: Obtain an explicit characterization of the maximum allowable sampling pe-
60
riod that guarantees fault-free stability and residual convergence. This stability condi-
tion is given in terms of the model accuracy, the sampling period, the fault estimation
parameter/s in the model, and the controller design parameters.
• Fault identification: Derive a strategy for data-based fault identification using moving
horizon estimation wherein an optimization problem is solved at each sampling time
to calculate estimates of the fault parameter/s in each actuator. Use bounds on the
model uncertainty to obtain a confidence interval for each fault parameter.
• Fault accommodation: Formulate a fault accommodation logic that determines the
appropriate fault accommodation strategy in response to a fault based on the confi-
dence intervals obtained for each fault parameter following fault identification. Possible
responses to a fault include: no action, model update, controller reconfiguration, or
actuator switching. The goal is to preserve stability and plant performance while
introducing minimal process interference.
Figure 4.1 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 structure.
4.2 Motivating example
A well-mixed non-isothermal continuous crystallizer is used throughout the paper. The
model-based fault detection and accommodation framework is designed and implemented on
this system but is also applicable to other processes modeled by infinite-dimensional partial
integro-differential equations.
61
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 4.1. Overview of the integrated control architecture with fault identification andaccommodation.
Particulate processes are characterized by the co-presence of a continuous and dispersed
phase. The continuous phase is the fluid phase which surrounds the distributed, dispersed
phase. The dispersed phase is described by a particle size distribution whose shape reflects
some property of the end-product which we would like to control. Due to the combined
presence of the two phases in this system, apart from the mass balance for the continuous
phase, it is necessary to include a population balance on the dispersed phase. Under the as-
sumptions of spatial homogeneity, constant volume, mixed suspension, nucleation of crystals
of infinitesimal size, mixed product removal, and a single internal particle coordinate—the
particle size (r); a dynamic crystallizer model can be derived:
62
∂n
∂t= k1(cs − c)
∂n
∂r−
n
τr+ δ(r − 0)ǫk2e
(−k3
(c/cs−1)2
)
dc
dt=
(c0 − ρ)
ǫτr+
(ρ− c)
τr+
(ρ− c)
ǫ
dǫ
dt
dT
dt=
ρcHc
ρCp
dǫ
dt−
UAc
ρCpV(T − Tc) +
(T0 − T )
τr
(4.3)
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 −∫
∞
0n(r, t)π 4
3r3dr is the volume of liquid per unit volume
of suspension; cs = −3T 2 + 38T + 964.9 is the concentration of the solute at saturation
computed using T = T−273333−273
; 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 process dynamics
is characterized by an unstable steady-state surrounded by a stable periodic orbit. This
results from the highly oscillatory behavior exhibited by the crystallizer due to the relative
nonlinearity of the nucleation rate as compared to the growth rate. The control objective
is to suppress this oscillatory behavior in the presence of actuator faults by stabilization
at an unstable steady-state that corresponds to the desired particle size distribution by
manipulating the solute feed concentration (c0) and/or residence time (τr).
Through method of moments, a sixth-order ordinary differential equation system is de-
rived to describe the temporal evolution of the first four moments of the particle size dis-
tribution, solute concentration, and temperature (see [7] for a detailed derivation). The
reduced-order model can be cast in the following form:
63
dµ0
dt=
−µ0
τr+
(1−
4
3πµ3
)k2e
−k3
( ccs
−1)2
e−EbRT
dµv
dt=
−µv
τr+ vµv−1k1(c− cs)e
−EgRT , v = 1, 2, 3
dc
dt=
c0 − c− 4πk1e−EgRT τr(c− cs)µ2(ρ− c)
τr(1− 4
3πµ3
)
dT
dt= −
ρHc
ρCp
dµ3
dt−
UAc
ρCpV(T − Tc) +
(T0 − T )
τr
(4.4)
The global phase portrait of the system of (4.4) has a unique unstable equilibrium point
surrounded by a stable limit cycle at:
xs = [µs0 µs
1 µs2 µs
3 cs T s]T
= [0.0047 0.0020 0.0017 0.0022 992.95 298.31]T(4.5)
Sampled measurements of the moments (µ0, µ1, µ2, µ3), solute concentration (c), and tem-
perature (T ) are the manipulated variables used to control the process. These state measure-
ments are collected discretely and sent to the controller where the control action is calculated
and finally sent to the actuator.
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) +Bkuk(t) (4.6)
where t ∈ [0,∞) is the time; x(t) is the vector of state variables; uk(t) = [uk1(t) · · · uk
m(t)]T
is the vector of manipulated inputs in deviation variable form, m is the number of ma-
nipulated inputs, k is a discrete variable denoting the active control actuator configu-
ration. 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 ; and A and Bk are constant matrices given by:
A=∂f
∂x
∣∣∣∣(xs,uk
s )
, Bk =∂f
∂uk
∣∣∣∣(xs,uk
s )
where uks denotes the steady state values for the available
manipulated inputs in the k-th control configuration. Throughout the paper, we use the
norm notations | · | and ‖·‖ to represent the standard Euclidean norm and the L2 norm,
64
respectively. Furthermore, the notation x(τ−k ) will be used to denote the limit limt→τ−kx(t).
Table 4.1 gives the process parameters and steady state values used in the simulated crys-
tallizer example.
Table 4.1. Process parameters and steady-state values for the non-isothermal continuouscrystallizer.
ρc = 1770 kg/m3 ρCp = 3000 J/m3 ·K
cso = 1000 kg/m3 Hc = −50 J/kg
τr = 1h U = 1800W/K ·m2
Eg = 1 kJ/mol Ac = 0.25m2
Eb = 0.00001 kJ/mol V = 0.01m3
T sc = 298K T s
o = 303K
R = 0.008314 kJ/mol ·K
k1 = 0.05064mm ·m3/kg · h k1 = k1e(−EgR·T
)
k2 = 7.957 (mm3 · h)−1 k2 = k2e(−EbR·T
)
k3 = 0.001217 k3 = k3
4.3 Finite-dimensional sampled-data control system
4.3.1 State feedback controller synthesis
The control system design involves first synthesizing a state feedback controller that stabilizes
the finite-dimensional system when the sensors continuously transmit data to the controller.
We consider the following discrete controller for simplicity:
uk(t) = Kx(t), t ∈ [τj , τj+1) (4.7)
4.3.2 Implementation under measurement sampling
The implementation of the controller of (4.7) requires continuous availability of the sensor
measurements. Due to measurement sampling, the controller cannot be directly implemented
65
since the output measurements are only available at discrete time instances. To compensate
for the unavailability of continuous measurements, an inter-sample state estimator is used in
the control design. At each sampling time, the corresponding values of the measured states
are instantaneously transmitted to the controller and are used to update the corresponding
model states. The model-based state feedback controller is implemented as follows:
uk(t) = Kx(t), t ∈ [τj , τj+1)
˙x(t) = Ax(t) + Bkuk(t), t ∈ [τj , τj+1)
x(τj) = x(τj), j ∈ {0, 1, · · · }
(4.8)
where x is the vector of estimated state variables which is used in generating the inter-sample
control by utilizing model estimates of the state x(τj) until the next state measurement
x(τj+1) is available, j denotes each sampling instance, and τj are the update times when
values of the state are collected. The A and Bk matrices in the linearized plant model
are approximated by the constant matrices A and Bk. Note that, in general, A 6= A and
Bk 6= Bk to account for plant-model mismatch. The model uncertainty can be explicitly
represented by ∆A and ∆Bk such that A = A+∆A and Bk = Bk +∆Bk.
It is assumed that the model-based state feedback controller of the above form enforces
closed-loop stability when sensor readings are unavailable such that the origin of the closed-
loop model satisfies:
‖x(t)‖ ≤ γ‖x(τ0)‖e−ϕ(t−τ0) (4.9)
where γ ≥ 1 and ϕ > 0.
4.3.3 Fault model for continuous system
To model the fault, the reduced, linearized system dynamics is written in the following form:
x(t) = Ax(t) +Bkαkuk(t)
˙x(t) = Ax(t) + Bkαkuk(t)(4.10)
where αk = diag{αk1, · · · , α
km} is a diagonal fault parameter matrix that accounts for the
presence of actuator faults or malfunctions in the system. Each of the diagonal elements
66
in the fault parameter matrix (αk) characterizes the local health status of the individual
actuators, where m is the total number of actuators in the k-th configuration. The entries
of the fault matrix (αk) take values between zero and one where zero denotes total actuator
failure while one denotes the fault-free state. In the absence of faults, αk = I where I is the
identity matrix. The diagonal fault model parameter matrix αk = diag{αk1, · · · , α
km} is used
by the model to account for the faults that occur in the plant. Entries of this matrix are
decision variables generated from the fault identification and accommodation scheme which
will be discussed in later sections. The controller gain (K) is chosen to ensure that the
eigenvalues of A+ BkαkK lie in the open left half of the complex plane.
4.4 Closed-loop stability analysis under measurement
sampling
To simplify the analysis, we consider the case when the update period is constant and equal
for all the sensors, i.e., all the state measurements are available to the controller every ∆
hours. Defining the model estimation error as e(t) = x(t) − x(t), the overall closed-loop
system can be formulated as a discrete-continuous system:
x(t) = (A−∆A)(t) + (Bk −∆Bk)αkuk(t)
˙x(t) = Ax(t) + Bkαkuk(t)
e(τj) = 0, j = 0, 1, 2, · · ·
(4.11)
By analyzing the obtained discrete-continuous system, the following theorem is obtained
which provides a sufficient condition for stability of the networked closed-loop finite-dimensional
system in terms of the update period, model uncertainty, and controller design parameters.
Theorem 4.1. Consider the closed-loop system of (4.6) subject to the control and update law
of (4.8), if ∆ is chosen such that:
Γk(∆) := γ
(e−ϕ∆ +
LA
ϕ+ LBk
(eLA∆ − e−ϕ∆)
)< 1 (4.12)
67
then the networked closed-loop states satisfy:
‖x(τ−j+1)‖ < ‖x(τj)‖, ∀j = 0, 1, 2, · · · (4.13)
where LA = ‖A − A‖ and LBk = [‖Bk‖‖αk − αk‖ + ‖∆Bk‖‖αk‖]‖K‖. Furthermore,
limj→∞‖x(t)‖ = 0.
Proof. From the definition of the error e(t), we have e(t) = ˙x(t)− x(t). Substituting for ˙x(t)
and x(t) from (4.10) yields:
e(t) = (A−A)e(t) + [Bk(αk − αk) + ∆Bkαk]uk(t) (4.14)
Solving the above equation, we have, for t ∈ [τj , τj+1):
e(t) = e(τj) +
∫ t
τj
(A− A)e(τ)dτ
+
∫ t
τj
[Bk(αk − αk) + ∆Bkαk]Kx(τ )dτ
(4.15)
Taking the norm on both sides and using e(τj) = 0:
‖e(t)‖ ≤ ‖A− A‖
∫ t
τj
‖e(τ)‖dτ + [‖Bk‖‖αk − αk‖
+‖∆Bk‖‖αk‖]‖K‖
∫ t
τj
‖x(τ)‖dτ
:= LA
∫ t
τj
‖e(τ )‖dτ + LBk
∫ t
τj
‖x(τ)‖dτ
(4.16)
where LA = ‖A− A‖ and LBk = [‖Bk‖‖αk − αk‖+ ‖∆Bk‖‖αk‖]‖K‖. Substituting for ‖x‖
in (4.9):
‖e(t)‖ ≤LBkγ
ϕ‖x(τj)‖(1− e−ϕ(t−τj ))
+LA
∫ t
τj
‖e(τ)‖dτ(4.17)
Applying Gronwall-Bellman inequality for t ∈ [τj , τj+1):
‖e(t)‖ ≤ ‖x(τj)‖LBkγ
ϕ+ LA
(eLA(t−τj ) − e−ϕ(t−τj ))
:= ‖x(τj)‖ Ξk(t− τj)
(4.18)
68
From the above inequality, we note that the estimation error will be zero if ∆ is zero.
With this bound on model estimation error and the bound on model state given in (4.9),
a bound on the state of the system of (4.6) using the triangular inequality is obtained
‖x(t)‖ ≤ ‖x(t)‖+ ‖e(t)‖ and we can show that, for t ∈ [τj , τj+1):
‖x(t)‖ ≤ ‖x(τj)‖Γk(t− τj) (4.19)
where Γk(∆) = γe−ϕ∆ + Ξk(∆). This implies that the closed-loop state remains bounded
between update times, i.e.,
‖x(t)‖ ≤ µ‖x(τj)‖ (4.20)
where µ is the maxima of the continuous function Γk(·) over the interval [0,∆]. Using (4.19)
to calculate ‖x(τ−j+1)‖, we finally obtain:
‖x(τ−j+1)‖ − ‖x(τj)‖ ≤ (Γk(∆)− 1)‖x(τj)‖ (4.21)
Clearly, if Γk(∆) < 1, then ‖x(τ−j+1)‖ − ‖x(τj)‖ < 0 and (4.13) holds which implies that
limj→∞‖x(τj)‖ = 0. Substituting this estimation into (4.20), we finally have limj→∞‖x(t)‖ =
0.
Remark 4.1. It can be seen from (4.12) and the definition of LA and LBk that the given
bound on the minimum stabilizing communication rate is dependent on the degree of plant-
model mismatch and the controller design parameters. The stability condition can therefore
be used to explicitly characterize the relationship between these various factors.
4.5 Data-driven actuator fault identification and ac-
commodation
4.5.1 Discrete fault model
The continuous equation in (4.10) is converted into a discrete form to compare discrete
estimates to the historical input and state measurements. The modified system takes the
69
form:
x[j + 1] = Adx[j] +Bkdα
kuk[j]
x[j + 1] = Adx[j] + Bkd α
kuk[j](4.22)
subject to the update law:
x[j] = x[j], j ∈ {0, 1, · · · } (4.23)
where x[j] = x(τj) is the vector of discrete process states, x[j] = x(τj) is the vector of
discrete state estimates, and uk[j] = uk(τj) the vector of discrete input data. The update
period ∆ = τj+1 − τj is the time interval between discrete consecutive measurements, j is
the update instance, and Ad, Bkd , Ad and Bk
d are discrete versions of the constant matrices
A, Bk, A, and Bk; respectively.
4.5.2 Data-based fault identification
Fault identification involves estimating the value of the fault parameter matrix αk using past
state measurements and manipulated inputs. These values are fitted to the fault model in
(4.22) using the cost function:
J(ζj, α∗) =
j−NI+1∑
p=j
(∥∥∥x[p + 1]− Adx[p]− Bkdα
∗uk[p]∥∥∥2)
(4.24)
where ζj = {(x[j − p], uk[j − p])|p = 1, 2, · · · , NI} denotes the past NI historical data of
the state measurements and the manipulated inputs for each jth sampling instance. Using
a large value for NI results in higher accuracy for calculated values obtained for the fault
estimation matrix α∗. However, this may also result in a high computational load as well as
discontinuities in the values of α∗ particularly right after a fault has occurred since the pool
of I/O data used in the calculations will involve conflicting data taken both before and after
the fault. This parameter should therefore be selected appropriately. Note that the fault
identification scheme makes use of the constant matrices, Ad and Bkd , from the inter-sample
model since plant dynamics are not fully known by the system in actual applications.
70
Using the cost function in (4.24), a finite-horizon least squares optimization problem is
developed:
minα∗
J(ζj, α∗) (4.25)
s.t. 0 ≤ α∗
1,2,··· ,m ≤ 1
Note that calculated values of the fault estimation matrix α∗ may slightly differ from the
actual values αk even without model uncertainty at the onset of the fault.
Due to the discrepancy between the model and process (i.e., A 6= A and Bk 6= Bk leads to
Ad 6= Ad and Bkdα
k 6= Bkd α
k), the optimal solution α∗ of the optimization problem of (4.25)
is not exactly the same as the actual fault parameter αk. However, Ψ(α∗), an estimation
interval of αk dependent on α∗ may be obtained. This result provides an estimate of the size
of the fault and can be used for fault detection. For a given update period ∆ that satisfies
the stability condition of (4.12), a fault on the l-th actuator can be detected at Td when the
upper bound of Ψ(α∗
l ) is less than 1; since, in that situation, we can easily obtain a fault
parameter such that αkl < 1.
Remark 4.2. Compared with the fault detection approach in [52], the fault identification
method proposed in this section is not only capable of determining the presence of a fault,
but is also useful in locating faulty actuators. This fault isolation mechanism is very useful
especially when several actuators are used in control.
Remark 4.3. When the continuous model of (4.6) is exactly the same as that of the process
in (4.8) (e.g., A = A and Bk = Bk), it is easy to show that the discrete model and the
process take the same form (e.g., Ad = Ad and Bkd (·) = Bk
d(·)) and the optimization problem
of (4.25) used for fault identification can be reduced as:
minα∗
NI−1∑
p=0
‖(Bkdα
k −Bkdα
∗)uk‖2 (4.26)
71
As we can see from the above formulation, α∗ = αk is always an optimal solution. Therefore,
if a perfect model is used for both control and fault identification, the estimation interval of
αkl collapses to a single point which is the same as the value of the actual fault parameter of
the process, αkl . In this case, we declare that a fault takes place at the l-th actuator when
α∗
l < 1, and the extent of the fault is determined as αk = α∗.
4.5.3 Fault accommodation
Following the detection of a fault in the operating control configuration, we need to deter-
mine whether corrective action (e.g., updating αk using α∗ or using a new feedback gain K∗)
is required to preserve closed-loop stability. When the partial fault is not significant enough
to impair the stability properties of the process, switching to a new control configuration is
not necessary. Considering this situation, we develop a stability-based fault accommodation
logic which is formulated in Algorithm 3.1. The key idea is to maintain the current control
configuration if all the elements of Ψ(α∗
l ); l = 1, · · · , m satisfy the stability condition of
(4.12), otherwise, the system should switch to a new control configuration which guarantees
the stability of the closed-loop system. Fig. 4.2-4.3 are schematic diagrams of the fault ac-
commodation logic embedded within the control architecture with and without plant-model
mismatch.
Remark 4.4. Besides considering the stability requirement to compensate for the destabi-
lizing effect of the actuator fault, it is possible to incorporate performance criteria in the
accommodation logic to determine the optimal solution between all stabilizing backup con-
trol configurations. Also note that Algorithm 3.1 not only considers the case of a single
fault, but also can be applied in the case of multiple and simultaneous faults. In this case,
the solution of the optimization problem of (4.25) also determines which of the actuators is
faulty. The fault may be accommodated following the similar approach in Algorithm 3.1, by
72
Algorithm 3.1
1. Choose ∆ that satisfies (4.12) and set αk = αk = 1, j = 0
2. Solve (4.25) for α∗ and estimate Ψ(α∗
l ) for each α∗
l , l = 1, · · · , m
3. If for any ϑl ∈ Ψ(α∗
l ), (4.12) is violated with αkl = ϑl
4. If for all ωl ∈ Ψ(α∗
l ), α∗
l satisfies (4.12) with αkl = ωl and αk
l = α∗
l
5. Update αkl using α∗
l at next transmission time and GOTO step 12
6. Else if any K∗ satisfies (4.12)
7. Update K using K∗ at next transmission time and GOTO step 12
8. Else
9. Replace l-th actuator with a new actuator that satisfies (4.12), set αkl = αk
l = 1
at next transmission time and GOTO step 12
10. End if
11. Else
12. Implement the next sequence j = j + 1 and GOTO step 2
13. End if
73
updating all α∗
l associated with the faulty actuators or by using a new feedback gain.
Remark 4.5. As mentioned in Remark 4.3, when a perfect model is implemented for control
and fault identification (e.g., ∆A = 0 and ∆Bk = 0), the estimation interval of αkl , Ψ(α∗
l )
will shrink to a point, α∗
l . In this case, the fault accommodation logic will be modified as in
Algorithm 3.2.
Remark 4.6. The algorithms differ in steps 2-4. In a perfect model, the fault identification
and accommodation policy is simplified, since the stability condition (4.12) only needs to be
satisfied for a specific point α∗
l , instead of all values in the estimation interval Ψ(α∗
l ).
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:
KK
Update
K
Y
N
Y Y
N N
Fault
Identification Fault Accommodation
Figure 4.2. Fault accommodation logic with model uncertainty.
4.6 Fault tolerant control application
A non-isothermal continuous crystallizer is selected to illustrate the proposed fault-tolerant
scheme. Discrete measurements of the moments of the particle size distribution (µ0, µ1, µ2,
74
Algorithm 3.2
1. Choose ∆ that satisfies (4.12) and set αk = αk = 1, j = 0
2. Solve (4.25) for α∗
3. If for any α∗
l , (4.12) is violated with αkl = α∗
l
4. If for any α∗
l , α∗
l satisfies (4.12) with αkl = αk
l = α∗
l
5. Update αkl using α∗
l at next transmission time and GOTO step 12
6. Else if any K∗ satisfies (4.12)
7. Update K using K∗ at next transmission time and GOTO step 12
8. Else
9. Replace l-th actuator with a new actuator that satisfies (4.12), set αkl = αk
l = 1
at next transmission time and GOTO step 12
10. End if
11. Else
12. Implement the next sequence j = j + 1 and GOTO step 2
13. End if
75
:
1
1
0
k
l
Set
l
i
*
:Solve
)(
1)(
*
?
l
k
l
k h
)(
1)(
*
?
K
hk
:
2
1k
l
Set
l
1ii*
:
KK
Update
Y
N
Y
N
Fault
Identification
Fault
Accommodation
Figure 4.3. Fault accommodation logic without model uncertainty.
µ3), concentration (c), and temperature (T ), are used to control the system. The system is
controlled by simultaneously manipulating the inlet solute concentration (c0), and residence
time (τr). The inter-sample state estimator is used to estimate values of the states when
sensor measurements are unavailable. Simulations are performed under a sampling period
(∆) of 0.1h.
The stability regions are obtained using the condition Γk(∆) < 1 which is derived from
the closed-loop stability analysis of the discrete-continuous system in (4.12) (Fig. 4.4). These
stability conditions are obtained as an explicit function of the controller gain (K), sampling
period (∆), and the fault parameter matrix (αk), plant model mismatch (∆A, ∆B), and
control configuration (Bk, Bk).
Each of the diagonal elements in the fault matrix (α) characterizes the local health status
of the individual actuators. In the initial control configuration (k = 1), two actuators are
utilized for control: α1 represents the health of the actuator related to the first manipulated
76
variable (u11) used to vary the inlet concentration (c0) and α2 is for the other actuator used to
adjust the second manipulated variable (u12), the residence time (τr). The regions of stability
are plotted as a function of the health status of the first actuator (α1) against the fault
model parameter (α1). The blue area enclosed by the unit contour line shows the region
where the process is guaranteed to be stable since Γk(∆) < 1. These two contour plots are
useful when there is a single fault in the actuator controlling the inlet concentration since
these are generated by setting the fault parameters of other actuators equal to one thereby
signifying their fault-free status. Similar plots may be generated for other conditions. Such
plots are useful in predicting the behavior of the process and in determining the appropriate
fault-tolerant response once a fault is identified. A partial malfunction in any of the actuators
could possibly occur such that the operating point is shifted somewhere within the stability
region. Such faults are not detrimental to process performance and product quality and;
therefore, do not warrant immediate fault accommodation or control reconfiguration. Based
on this knowledge, the plant supervisor is then able to strategically prioritize which specific
control loop or plant equipment requires maintenance or replacement through this stability-
based closed-loop analysis. In cases where there are more variables to consider (e.g., a larger
number of manipulated variables), instead of a two-dimensional contour plot, a look-up table
with values of Γk(∆) for varying magnitudes of the process parameters may be generated
offline and then used to judge if an identified fault requires urgent attention.
Among the highlights of the data-based fault identification scheme proposed in this chap-
ter is the added capability of identifying partial malfunctions that do not result in system
instability. This is an improvement over past studies which utilize a stability-based alarm
threshold and is; therefore, a more proactive approach to malfunctions as opposed to the
reactive method wherein action is triggered only when there is a threat of instability. This
allows for timely machine repair which may prove to be less costly and time-consuming
compared to the urgent repairs or equipment replacement following a severe destabilizing
77
malfunction.
This chapter proposes a general approach for data-based fault identification which is
applicable to systems where model uncertainties are present. Previous studies were carried
out with the assumption that a perfect model of the system was in place. This simplifies
the fault identification scheme but does not represent actual plant conditions where model
inaccuracies are commonplace and may have drastic effects on the stability of the closed-loop
system. The contour plots of the region of stability for different systems with and without
plant-model mismatch show that uncertainties can significantly limit the range of parameters
under which a system is still stabilizable (Fig. 4.4). When there is a perfect model, the system
is more tolerant to differences in values of α1 and α1 and is still closed-loop stable under
severe malfunctions.
In the hypothetical fault scenario, model uncertainty results in inaccurate values for the
k3 parameter, an experimentally determined constant that influences the growth rate of
particles in the continuous crystallizer in (4.3). Both regions of stability take the form of a
diagonal figure that is symmetric on the α1 = α1 axis. This is reasonable since, in the best
case scenario, the fault model parameter α1 should be equal to the actual fault parameter
α1.
4.6.1 Fault identification
For both the simultaneous and continuous faults introduced, the controller gain (K) is cal-
culated by specifying the location of the poles of A+ BkαkK at [−1 − 2 − 3 − 4 − 5 − 6].
Fault identification is carried out by solving for the fault estimation matrix (α∗) in the op-
timization problem in (4.25). This is done by comparing the past 20 data points (NI = 20)
of the state measurements and manipulated input to values generated by the discrete model
in (4.22). Actuator faults in the manipulated variable responsible for controlling the inlet
solute concentration (c0) are investigated and the simulation is carried out under a sampling
period (∆) of 0.1h. All faults are introduced after 1h of operation.
78
0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
1
1
1 1
1
1F
au
lt p
ara
me
ter
Fault model parameter (a)0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1 1
1
1
1
1
Fa
ult
pa
ram
ete
r
Fault model parameter (b)
Figure 4.4. Plots (a)-(b): Region of stability is larger with a perfect model (a) comparedto one with model uncertainty (b). The feed concentration (c0) and residence time (τr)are the manipulated variables (u1(t) = [u11(t) u12(t)]
T = [c0(t) τr(t)]T ). Contour plots
of Γk(∆) plotted against different values of the fault parameter (α11) and fault model
parameter (α11).
In both cases, the fault identification scheme is shown to be effective in quantifying and
almost instantaneously locating faults—limited primarily by the measurement sampling.
However, jumps in the calculated values of the fault estimation parameters are occasionally
observed right after a fault. This is attributed to sudden disruptions in the data points used in
the data-based identification method which includes values of the state and the manipulated
variable before and after the fault occurred. This is why the optimization horizon (NI) has
to be properly selected—small values result in inaccuracies in the fault identification while
large values lead to sharp jumps or prolonged settling time. Hence, the selection of the
appropriate fault accommodation strategy should be suspended until the fault identification
scheme settles to a final value for the fault estimation parameter.
The first case involves a malfunction wherein the actuator controlling the feed concen-
tration (c0), the first manipulated variable (u11), becomes 90% effective (α1 = 0.9) while the
other actuator used in varying the residence time (τr) remains fault-free (α2 = 1) (Fig.4.5a).
Prior to the fault, the fault parameters (α1, α2) and fault model parameters (α1, α2) for all
the actuators are equal to one. The fault is almost immediately identified and is reflected
by changes in the calculated values of the fault estimation parameter (α∗
1) which eventually
79
settles to a final value of 0.8246. The offset in the fault magnitude (α1) and the fault es-
timation (α∗
1) is due to the model uncertainty. Since there is a plant-model mismatch, the
estimated values of the fault are unreliable but yield some information about the range of
possible values of the fault parameter: α1 = Ψ(α∗
1) = [0.95, 1] (Fig. 4.5b). The red vertical
line in Fig. 4.5b indicates the estimation interval for the fault parameter. In the absence of
model uncertainty, this line will shrink to a point in the stability region. An examination
of the region of stability shows that this range of values for α1 is still within the region of
stability at the original value of the fault model parameter α1 = 1. Hence, the fault will not
result in system instability and plant operations may resume without having to modify the
fault model parameter α1. This is verified by the dynamics of the total particle size (µ1)
and inlet solute concentration (c0) which reveal that the fault does not significantly disrupt
plant performance and the states eventually settle to their steady state values (Fig. 4.5c-d).
In the second scenario, a fault causes a 40% drop in the performance of u11, the actuator
modifying the feed concentration (c0); this time leading to plant instability. The fault param-
eter matrix then shifts from α = diag{α1, α2} = diag{1, 1} to α = diag{0.4, 1} (Fig.4.6a).
The fault identification scheme is able to estimate the fault at α∗
1 = 0.3675 which is used to de-
termine the range of possible fault parameter values in the system: α1 = Ψ(α∗
1) = [0.4, 0.475]
(Fig. 4.6a-b). From the contour plot, one can observe that for α∗
1 = 0.3675, there are values
of the fault parameter (α1) that fall outside the stability region for all values of the fault
model parameter (α1) (Fig. 4.6b). There is, therefore, no guarantee of closed-loop stability
for this fault. Plots of the evolution of the state and the faulty manipulated input reveal that
the plant eventually becomes unstable (Fig.4.6c-d). To avoid plant instability, the supervisor
must then determine the best approach for fault accommodation.
Since there is no stabilizing α1 value for α1 = 0.4, the next alternative is to search
for a stabilizing K∗ value when α1 = 0.4 and α1 = 1 (Fig. 4.7a). The stability region is
then plotted as a function of the fault parameter (α1) and the control design parameter.
80
0 5 100.75
0.8
0.85
0.9
0.95
1
1.05
Time (h)
Fa
ult
Ide
ntif
ica
tion
α1
α1*
(a)0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
1
11
1
Fa
ult
pa
ram
ete
r
Fault model parameter (b)
0 5 10
1.85
1.9
1.95
2
x 10−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(c)0 5 10
988
990
992
994
996
998
1000
Time (h)
Fe
ed
co
nce
ntr
atio
n,
c o (
kg/m
m3)
(d)
Figure 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 (redline). Plots (c)-(d): Dynamics of the state (µ1) (c) and the faulty actuator controllingthe manipulated variable u11, the feed concentration (c0) (d).
Recall that the feedback gain K is selected using pole placement such that the poles of the
system A + BkαK are at p = [λ p2 p3 p4 p5] = [−1 −2 −3 −4 −5 −6]. The
first pole value (λ) is selected as the control design parameter used to adjust K for fault
accommodation. The stability region generated reveals that there is no stabilizing feedback
gain (K∗) when α1 = 0.4 and α1 = 1. The next option is reverting to a different control
configuration that does not use the faulty actuator responsible for controlling u11, the feed
concentration (c0). The residence time (τr), the second manipulated variable in the original
control configuration (u12), becomes the sole manipulated variable (u2
1) in the second control
configuration. This causes a change in the stability properties of the system (Fig. 4.7b).
81
0 5 10
0.4
0.6
0.8
1
Time (h)
Fa
ult
Ide
ntif
ica
tion
α1
α1*
(a)0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
1
1
11
1
Fa
ult
pa
ram
ete
r
Fault model parameter (b)
0 5 10 15 20 250.5
1
1.5
2
2.5x 10−3
Time (h)
To
tal p
art
icle
siz
e, µ
1 (
mm
−2)
(c)0 10 20 30
600
700
800
900
1000
1100
Time (h)
Fe
ed
co
nce
ntr
atio
n,
c o (
kg/m
m3)
(d)
Figure 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 (redline). Plots (c)-(d): Dynamics of the state (µ1) (c) and the faulty actuator controllingthe manipulated variable u11, the feed concentration (c0) (d).
Thus, through fault accommodation, the system is able to maintain system stability after a
potentially destabilizing fault (Fig. 4.8).
Note that the regions of stability are not only useful in determining the appropriate
control action once a fault has occurred but may also provide insight in selecting the best
control design parameters for fault-tolerance. The stability region plotted against values of
the fault parameter (α1) and the controller design parameter (λ) reveal that the stability
region is less robust to faults for small values of (λ) when the fault model parameter (α1) is
equal to one. This is the basis for initializing the controller gain K using λ = −1 (Fig. 4.7a).
82
−10 −8 −6 −4 −20.8
0.85
0.9
0.95
1
1
11
1
1
α1
Pole value, λ (a)0.2 0.4 0.6 0.8 1
0.2
0.4
0.6
0.8
11
1
1
1
1
1
Fa
ult
pa
ram
ete
r
Fault model parameter (b)
Figure 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 fordifferent values of the fault parameter (α1
1) and the controller design parameter (p1) usingthe 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 modelparameter (α2
1) using the residence time (τr) as the only manipulated variable (u21).
0 5 10
1.9
1.95
2
x 10−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(a)0 5 10
0.4
0.6
0.8
1
Time (h)
Re
sid
en
ce t
ime
, τ r (
h−
1)
(b)
Figure 4.8. Plots (a)-(b): Dynamics of the state (µ1) (a) and the fall-back manipulatedvariable u21 varying residence time (τr) (b) shows that fault accommodation re-establishesstability after a potentially destabilizing fault.
83
4.7 Conclusions
A unified framework was presented for fault identification and accommodation for particu-
late processes modeled by high-dimensional equations with nonlinearities and complex dy-
namics. These techniques were applied taking into account practical implementation issues
such as model uncertainty and measurement sampling. The model-based feedback control
architecture included a model-based feedback controller that made use of state estimates
when measurements are not available, a fault identifier that generated estimates of the fault
parameter while accounting for plant-model mismatch, and a supervisor that determines
the appropriate fault accommodation strategy in response to a fault that could potentially
destabilize the system. Possible fault strategies included: model update, control parameter
adjustment, actuator switching. The idea is to determine the appropriate fault accommo-
dation strategy based on the worst case behavior predicted by the stability properties of
the confidence interval generated for the fault parameter estimates in the fault identification
procedure. These were derived from the stability analysis of the closed-loop sampled-data
system wherein stability was explicitly characterized as a function of the sampling period,
fault parameter/s, and controller design parameter. These techniques were applied to a simu-
lated continuous crystallizer model. Two fault scenarios were introduced in the system. The
first case was a partial malfunction that did not lead to instability and did not necessitate
immediate action. The second was a potentially destabilizing fault. It was found that fault
accommodation structure in place was able to identify both faults and was able maintain
stability and avert undesired process behavior for the case of the potentially destabilizing
fault.
84
Chapter 5
Sensor fault accommodation strategies
in the control of particulate processes
with multi-rate measurements and
measurement sampling
This chapter focuses on the problem of handling sensor faults in particulate processes with
multiple outputs sampled at different sampling rates. This is carried out by reducing the
infinite-dimensional equation describing the particulate process to a finite-dimensional model
that captured the dominant dynamics of the system. This reduced-order model is used to
design an observer-based output feedback controller. An inter-sample model predictor is
utilized to compensate for the measurement intermittency by providing state estimates. The
model predictor is updated once the output measurements were available. The combined
discrete-continuous system is explicitly characterized in terms of a feasible combination of
output sampling rates that will lead to stability. This condition is based on the model
accuracy as well as the controller and observer design parameters. Appropriate rules for
passive and active fault tolerant control are derived based on this stability analysis. These
techniques are illustrated using a simulated model of a non-isothermal continuous crystallizer
to show how faults and failures in the sensors may be handled through the careful selection
85
of a stabilizing controller or sensor.
This chapter is structured accordingly: The problem formulation and solution overview
is given in Section 5.1.2. Some preliminary analysis and definition of terms are introduced in
Section 5.3 to facilitate the analysis of the multi-rate sampling mechanism. In Section 5.4,
the finite-dimensional multi-rate sampled-data control system is designed and subjected to
a closed-loop stability analysis. These techniques are applied to a simulated non-isothermal
continuous crystallizer in Section 5.5 wherein different scenarios were introduced to the sys-
tem to demonstrate the passive and active fault accommodation capabilities of the proposed
framework. Some final remarks are given in Section 5.6. Findings in this chapter were first
presented in [53].
5.1 Preliminaries
5.1.1 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)) (5.1)
z = f(z) + g(z)u+ Az
∫ rmax
0
q(n, z, r)dr (5.2)
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
86
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, (5.1)
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 (5.2) 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
0cκ(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.
5.1.2 Problem formulation and solution overview
The control objective is process stabilization at the desired equilibrium state in the pres-
ence of sensor faults using discretely-sampled state measurements. The problems under
consideration include: fault-free process regulation using discretely-sampled measurements,
passive and/or active sensor fault accommodation strategies to maintain the desired stability
and performance characteristics. We consider the following methodology to address these
problems:
• Model reduction: Derive a finite-dimensional model through model reduction. This
reduced-order model captures the dominant dynamics of the infinite-dimensional sys-
tem describing the continuous crystallizer which is used as an illustrative example to
87
represent particulate processes.
• Sampling mechanism: Characterize the multi-rate sampling mechanism to simplify the
presentation and analysis of results. This is done by defining the Shortest Time Unit
(STU) and Basic Time Unit (BTU) based on the sampling period of the different
sensors.
• Controller synthesis: Design an observer-based output feedback controller based on
the reduced-order model. The observer is used to generate state estimates from the
output measurements. A low order model of the system is used in the controller to
account for the measurement sampling. This model is updated at each sampling time.
• Analysis: Obtain an explicit characterization of the maximum allowable sampling pe-
riod that guarantees fault-free stability and residual convergence. This stability con-
dition is given in terms of the sampling period, the control configuration based on the
active manipulated input, and the controller and observer design parameters.
• Fault accommodation: Formulate a fault accommodation logic that determines the
appropriate fault accommodation strategy in response to a fault that changes the
sensor sampling period. Results from the closed-loop stability analysis are useful in
predicting the behavior of the process and in selecting an appropriate manipulated
variable when the model uncertainty and sampling periods for the output are known.
Thus, passive fault accommodation ensures that the system is robust to the most
probable fault within the system. Active fault accommodation is utilized when selecting
an appropriate sensor sampling period that will maintain stability after a potentially
destabilizing fault has already taken place.
5.2 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.
88
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:
∂n
∂t= k1(cs − c)
∂n
∂r−
n
τr+ δ(r − 0)ǫk2e
−k3
( ccs
−1)2
dc
dt=
(c0 − ρ)
ǫτr+
(ρ− c)
τr+
(ρ− c)
ǫ
dǫ
dt
dT
dt=
ρcHc
ρCp
dǫ
dt−
UAc
ρCpV(T − Tc) +
(T0 − T )
τr
(5.3)
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 −∫
∞
0n(r, t)π 4
3r3dr is the volume of liquid per unit volume
of suspension; cs = −3T 2 + 38T + 964.9 is the concentration of the solute at saturation
computed using T = T−273333−273
; c0 is the concentration of solute entering the crystallizer; k1 and
k2 are temperature-dependent variables; k3 is a constant; 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 that is
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 sensor
faults. This is carried out by stabilizing it at an unstable steady-state that corresponds to a
89
desired crystal size distribution by manipulating the solute feed concentration (c0), residence
time (τr), or coolant temperature (Tc). Measurements of the solute concentration (c) and the
temperature (T ) in the continuous crystallizer are collected discretely at different sampling
times 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.
Through method of moments, a sixth-order ordinary differential equation system is de-
rived to describe the temporal evolution of the first four moments of the crystal size dis-
tribution, the solute concentration, and the temperature (see [7] for a detailed derivation).
The reduced-order model can be cast in the following form:
dµ0
dt=
−µ0
τr+
(1−
4
3πµ3
)k2e
−k3
( ccs
−1)2
e−EbRT
dµv
dt=
−µv
τr+ vµv−1k1(c− cs)e
−EgRT , v = 1, 2, 3
dc
dt=
c0 − c− 4πk1e−EgRT τr(c− cs)µ2(ρ− c)
τr(1− 4
3πµ3
)
dT
dt= −
ρcHc
ρCp
dµ3
dt−
UAc
ρCpV(T − Tc) +
(T0 − T )
τr
(5.4)
The global phase portrait of the system of (5.4) has a unique unstable equilibrium point
surrounded by a stable limit cycle at xs = [µs0 µs
1 µs2 µs
3 cs T s]T =
[0.0047 0.0020 0.0017 0.0022 992.95 298.31]T . Multi-rate sampled measurements
of the solute concentration (c) and temperature (T ) are used to control the process.
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) +Blul(t), y(t) = Cx(t) (5.5)
where x(t) = [x1(t) x2(t)]T is the vector of state variables; x2(t) := y(t) is the measured out-
put vector; ul, l ∈ {1, 2, 3}, is the active manipulated input. The state vector is in deviation
variable form, x(t) = χ(t) − xs, where χ(t) = [µ0(t) µ1(t) µ2(t) µ3(t) c(t) T (t)]T ;
90
and A, Bl, and C are constant matrices given by:
A =∂f
∂x
∣∣∣∣(xs,us)
=
A11 A12
A21 A22
, Bl =
∂f
∂ul
∣∣∣∣(xs,us)
= [BTl,1 BT
l,2]T , and C =
0 0 0 0 1 0
0 0 0 0 0 1
,
where us denotes the steady state values for the available manipulated inputs. Table 5.1 gives
the process parameters and steady state values used in the simulated crystallizer example.
Over the next two sections, we describe how the control strategy is tailored to explicitly
account for the effects of multi-rate measurement sampling.
Table 5.1. Process parameters and steady-state values for the non-isothermal continuouscrystallizer.
ρc = 1770 kg/m3 ρCp = 3000 J/m3 ·K
cso = 1000 kg/m3 Hc = −50 J/kg
τr = 1h U = 1800W/K ·m2
Eg = 1 kJ/mol Ac = 0.25m2
Eb = 0.00001 kJ/mol V = 0.01m3
T sc = 298K T s
o = 303K
R = 0.008314 kJ/mol ·K
k1 = 0.05065mm ·m3/kg · h k1 = k1e(−EgR·T
)
k2 = 7.958 (mm3 · h)−1 k2 = k2e(−EbR·T
)
k3 = 0.001217 k3 = k3
5.3 Multi-rate sampling mechanism
Before designing and analyzing the control system with multiple measurement sampling
rates, the time units (or intervals) in the multi-rate sampling mechanism are defined to
simplify the presentation and analysis of the results. We first define the sampling periods
for the different sensors as ∆1 = δ1τ , · · · , ∆m = δmτ , where δ1, · · · , δm are some positive
91
integers. The following time units are obtained:
• Shortest time unit (STU): τs = gcd(δ1, δ2, · · · , δm)τ , where gcd(·) represents the great-
est common divisor.
• Basic time unit (BTU): τB = lcm(δ1, δ2, · · · , δm)τ , where lcm(·) represents the least
common multiple.
Using these two time units to analyze the multi-rate measurement sampling logic, output
measurements may be collected and transmitted at a certain τkj = (kN + j)τs, where τj is
a possible sampling time (PST), k ∈ {0, 1, · · · }, j ∈ {0, 1, · · · ,M − 1}, and M = τB/τs.
This order of sensor transmissions is repeated in a periodic fashion for each τB wherein all
sensors are activated in the same patten in each t ∈ [τkj , τk+1j ). Specifically, only at any τk0 ,
k ∈ {0, 1, 2, · · · }, will all the sensors be activated concurrently. It should also be noted that
the sensors can measure the outputs only at a PST; however, for some τkj , not all the sensors
are necessarily active. To indicate the sensor sampling status, we define a binary function
ς(i, j) to show whether the i-th sensor is active or dormant at each PST τkj :
ς(i, j) =
1, if j is divisible by δi
0, otherwise(5.6)
where ς(i, j) = 1 if the i-th sensor transmits a measurement, while ς(i, j) = 0 if the i-th
sensor is dormant.
0
0
1
0
2
0
3
0
4
0
5
0
0
1
s
1
2
B
× × × ×
×××
s1
s2
……t
Figure 5.1. Sampling schedule of two sensors with different sampling rates.
92
To illustrate the utility of this time unit system, consider a simple example involving two
sensors with different sampling periods, ∆1 = 0.2 and ∆2 = 0.3. Based on the time unit
notions introduced above, δ1 = 2, δ2 = 3, τ = 0.1, and thus the STU is τs = 0.1 while the
BTU is τB = 0.6. Based on this, the PSTs are τkj = 0.1(6k+ j) = 0.1i, for j ∈ {0, 1, · · · , 5},
k ∈ {0, 1, · · · }, i = 6k + j. Comparing this result with the actual sampling times, it can
be seen that the set of actual sampling times is a subset of the set of PSTs. This is the
conclusion after investigating the sampling status at the first six PSTs in Fig.5.1. At t = 0,
both sensors transmit their measurements to the controller; at t = 0.1 and t = 0.5, none
were active. Then at t = 0.2 and t = 0.4 only the first sensor is active, and at t = 0.3, sensor
2 is the only one that takes an output measurement. This sampling pattern will be repeated
over each τB = 0.6 for all future times. This is only a specific example. Since the magnitude
of the time units depend on the sampling periods of the sensors, each process will have a
unique periodic transmission schedule.
5.4 Finite-dimensional multi-rate sampled-data control
system design
5.4.1 Output feedback controller synthesis
The control system design involves first synthesizing an output feedback controller that
stabilizes the finite-dimensional system when the sensors continuously transmit data to the
controller. We consider an observer-based feedback controller:
ul(t) = Kη(t)
η(t) = Aη(t) + Blul(t) + L(y(t)− Cη(t))(5.7)
where η denotes the estimate of x, and A =
A11 A12
A21 A22
and Bl = [BT
l,1 BTl,2]
T are
approximate models of A and Bl. Note that, in general, A 6= A and Bl 6= Bl to allow for
possible model uncertainty. The controller (K) and observer gains (L) are chosen to ensure
that the eigenvalues of A+ BlK and A− LC lie in the open left half of the complex plane.
93
5.4.2 Controller implementation under multi-rate sampling
The implementation of the controller of (5.7) requires continuous availability of all the mea-
sured outputs (y) from the sensors. The observer cannot be directly implemented since the
output measurements are only partly available at discrete time instances due to multi-rate
sampling. To compensate for the unavailability of continuous measurements, a low-order
model of the system is included in the controller to provide the observer with estimates of
the measured outputs when they are not available. In this case, however, not all the sensors
send their measurements at a given time; instead, different sensors may transmit their data
at different rates. When one or more sensors are active at a possible sampling time, the cor-
responding values of the measured outputs are assumed to be instantaneously transmitted
to the controller and are used to update the corresponding model outputs and the model
states. The model-based output feedback controller is then implemented as follows:
ul(t) = Kη(t), t ∈ [τkj , τkj+1)
ω(t) = Aω(t) + Blul(t), y(t) = Cω(t)
η(t) = Aη(t) + Blul(t) + L(y(t)− Cη(t))
yi(τkj ) = yi(τkj ), ∀ ς(i, j) = 1
i ∈ {1, 2, · · · , m}, j ∈ {0, 1, · · · ,M − 1}
(5.8)
where ω = [ω1(t) ω2(t)]T is the vector of model states which provides an estimate of x,
ω2(t) := y is the model output which provides an estimate of y, yi denotes the i-th element
of y and yi represents the actual measured output of the i-th sensor.
5.4.3 Closed-loop stability analysis
To investigate the stability properties of the finite-dimensional sampled-data closed-loop
system, we first define the model estimation error as ei(t) = yi(t)−yi(t), for i ∈ {1, 2, · · · , m},
where ei represents the difference between the i-th model output given in (5.8) and the actual
measured i-th output. Then, introducing the error vector e(t) = [e1(t) e2(t) · · · em(t)]T and
defining the augmented state vector ξ(t) = [x(t) η(t) ω1(t) e(t)]T , the finite-dimensional
94
sampled-data closed-loop system is formulated as a switched system and written in the form:
ξ(t) = Fξ(t), t ∈ [τkj , τkj+1)
ei(τkj ) = 0, ∀ ς(i, j) = 1
i ∈ {1, 2, · · · , m}, j ∈ {0, 1, · · · ,M − 1}
(5.9)
where F is a matrix defined as:
F =
A BlK O O
LC D O L
A12C Bl,1K A11 A12
A22C −A21I O A21 A22
, (5.10)
D = A+ BK − LC, A22 = A22 −A22, and I = [I O] such that x1 = Ix where I is the iden-
tity matrix. The following proposition characterizes the multi-rate sampled-data closed-loop
system behavior in terms of the different sampling rates, the controller and observer design
parameters, and the model uncertainty.
Proposition 5.1. The augmented closed-loop system described by (5.9)-(5.10), subject to the
initial condition ξ(0) = [x(τ 00 ) η(τ00 ) ω1(τ
00 ) e(τ
00 )]
T := ξ0, has a response of the form:
ξ(t) = eF (t−τkj )RjNkξ0 (5.11)
for t ∈ [τkj , τkj+1), ∀ j ∈ {0, · · · ,M − 1}, k ∈ {0, 1, · · · }, where
Rj = Πjµ=1I
j−µ+1s eFτs, for j ≥ 1
R0 = diag{I, I, I}
Ijs = diag{I, I, I − Sj}
Sj = diag{ς(1, j), ς(2, j), · · · , ς(m, j)},
(5.12)
and N is given by:
N = I0s eFτsRM−1 (5.13)
95
Based on (5.11)-(5.13), the following proposition provides a necessary and sufficient condi-
tion for stability of the finite-dimensional sampled-data closed-loop system.
Proposition 5.2. Consider the sampled-data closed-loop system, (5.8), and the augmented
system of (5.9)-(5.10) whose solution is given by (5.11))-(5.13). Then the zeros solution,
ξ = [x η e]T = [0 0 0]T , is exponentially stable if and only if r(N(∆1, · · · ,∆m)) < 1.
Remark 5.1. An examination of the structure of N in (5.13) indicates that its spectral radius
is dependent on the sampling periods, ∆j , j ∈ {1, · · · , m}, and F (which, in turn, depends
on the choice of the model and the controller and observer gains). All these factors are tied
together through the stability condition of Proposition 5.2 which can, therefore, be used to
examine and quantify the various interdependencies between these factors. For instance, if
the sampling rate of a particular sensor is fixed by some performance requirement, one can
determine the minimum allowable sampling rates of other sensors.
Remark 5.2. The requirement that the spectral radius of N be strictly less than one ensures
stability by limiting the growth of the closed-loop state within each basic time unit of size
τB as the measurement sampling is repeatedly executed over time.
5.5 Fault-tolerant control
A non-isothermal continuous crystallizer example is selected to illustrate the proposed fault-
tolerant scheme. Discrete measurements of the concentration (c) and temperature (T ), which
are available at different sampling rates, were used to control the system. The inter-sample
model predictor is used to estimate values of the states as well as the output when sensor
measurements are unavailable. To account for plant-model mismatch, the model is designed
with an uncertainty of δu = 0.2 for the parameters kw, w = 1, 2, 3 where kw = kw(1 + δu) is
96
the approximate value used in the plant model.
The controller gain (K) is calculated by specifying the location of the poles of A+ BlK
at [−1 − 2 − 3 − 4 − 5 − 6] while the observer gain (L) is chosen such that the poles
of A + LC are at [−10 − 11 − 12 − 13 − 14 − 15]. The system is controlled using one of
three possible manipulated inputs: inlet solute concentration (c0), coolant temperature (Tc),
residence time (τ). The stability regions are obtained for all possible manipulated variables
using the condition λmax(N) < 1 which is derived from the closed-loop stability analysis
of the test matrix N in (5.13) (Fig. 5.2-5.3). These regions, plotted as a function of the
sampling period for concentration (c) and temperature (T ), differ significantly depending on
the selected manipulated input. The yellow area enclosed by the unit contour line shows the
region where the process is unstable since λmax(N) > 1. Such plots are useful in predicting
the behavior of the process and in selecting an appropriate manipulated variable when the
model uncertainty and sampling periods for the output are known. Several conclusions may
be derived from a close inspection of the contour plots generated. In this particular example,
comparing the stability regions for the two cases when the coolant temperature (Tc) and inlet
solute concentration (c0) is manipulated, it is evident that the stability region for the first
case falls within the second (Fig. 5.2a, Fig. 5.3). For this reason, the discussion on fault-
tolerant control will focus on the use of the inlet solute concentration (c0) and residence time
(τ) as manipulated variables. In addition, it is observed that the contour plots generated
when the concentration (c0) and residence time (τ) are manipulated indicate an almost
anti-correlation. This behavior may be utilized in actuator reconfiguration wherein actuator
switching is carried out to maintain stability when a back-up actuator is unavailable.
The initial operating point (OP), set at a sampling period of ∆1 = 0.002h for the con-
centration sensor and ∆2 = 0.008h for the temperature sensor, is stable when either the
inlet solute concentration (c0) or residence time (τ) is chosen as the manipulated variable.
Both configurations are within the regions of stability for the two manipulated inputs and
97
thereby satisfy the condition obtained from the closed-loop stability analysis (Fig. 5.2). This
is further verified by simulations showing the dynamic behavior of the total particle size
(Fig. 5.4a-b).
Although either may be used to control the crystallizer, robustness to faults in the output
sensor is another criterion for selecting the best manipulated input. For instance, close
inspection of the two different regions of stability indicates that the process will be more
robust to faults in the concentration sensor when the residence time (τ) is chosen as the
manipulated variable (Fig. 5.2). This is because this configuration has a wider range of
possible sampling periods for the concentration sensor (∆2 < 1h) that will lead to process
stability at fast sampling rates for the temperature sensor (∆2 < 0.010h) (Fig. 5.2b). In
contrast, manipulating the inlet solute concentration (c0) results in greater tolerance for
faults in the temperature sensor and, therefore, larger sampling periods (Fig. 5.2a).
Two scenarios are used in the discussion to show that fault tolerance is achieved when the
operating point lies within the region of stability which, in turn, is a function of the active
manipulated variable, the initial operating conditions, and the magnitude and direction of
the fault (i.e., a change in the sensor sampling period). In these examples, faults are modeled
by introducing a malfunction in one of the sensors resulting in a larger sampling period in
either the concentration (c) or temperature (T ) sensor. Different schemes are then proposed
on how to best deal with each malfunction so as to maintain stability.
In the first case, a malfunction occurs in the temperature sensor that shifts its sampling
period from ∆2 = 0.008h to ∆2 = 0.012h (f1). This pushes the operating point (OP:
∆1 = 0.002h, ∆2 = 0.008h) to a different location (f1: ∆1 = 0.002h, ∆2 = 0.012h) in
the stability regions (Fig. 5.2). The new point is still within the region of stability when
the inlet solute concentration (c0) is manipulated (Fig. 5.2a). This is not the case when the
residence time (τ) is chosen as the manipulated variable (Fig. 5.2b). Closed-loop simulations
of the dynamics of the system under the different manipulated inputs are in agreement with
98
1
1
1Con
cent
ratio
n sa
mpl
ing
perio
d,
1 (h)
Temperature sampling period, 2 (h)
5 10 15
x 10-3
2
4
6
8
10
12
14
16
x 10-3
OP:1=0.002,
2=0.008
f1 :
1=0.002,
2=0.012
f2 :
1=0.011,
2=0.008
f2
f1
(a)
1
1
1
Con
cent
ratio
n sa
mpl
ing
perio
d,
1 (h)
Temperature sampling period, 2 (h)
5 10 15
x 10-3
2
4
6
8
10
12
14
16
x 10-3
f2
f1
OP:1=0.002,
2=0.008
f1 :
1=0.002,
2=0.012
f2 :
1=0.011,
2=0.008
(b)
Figure 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) theinlet concentration, c0; and (b) the residence time, τ
99
1
1
1
1
1
1Con
cent
ratio
n sa
mpl
ing
perio
d,
1 (h)
Temperature sampling period, 2 (h)
5 10 15
x 10-3
2
4
6
8
10
12
14
16
x 10-3
(c)
Figure 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 manipulatedvariable.
these predictions (Fig. 5.4a-d). When the process is initially operating using the inlet solute
concentration (c0) as the manipulated variable, the system is still stabilizable even after the
fault took place. In fact, a comparison of closed-loop state profiles show that the system
stabilizes much faster at the new operating point (Fig. 5.4a,c). This shows how carefully
selecting the manipulated input results in passive fault tolerance. Furthermore, the fault
occurrence pushes the process into an operating point that now lies within the stability
region when the coolant temperature (Tc) is chosen as the manipulated input (Fig. 5.3). The
coolant temperature (Tc) may then be used as a back-up actuator in case of a malfunction in
the actuator manipulating the inlet solute concentration (c0). However, to maintain stability
after the fault when manipulating the residence time (τ); the process has to either revert to
a redundant temperature sensor with the original sampling period of ∆2 = 0.008h or switch
to an actuator that manipulates the inlet solute concentration (c0) or coolant temperature
(Tc).
The second scenario involves a malfunction in the concentration sensor wherein its sam-
pling period is driven from ∆1 = 0.002h to ∆1 = 0.011h (f2). This moves the operating point
100
(OP: ∆1 = 0.002h, ∆2 = 0.008h) to a different location (f2: ∆1 = 0.011h, ∆2 = 0.008h)
(Fig. 5.2). This new point is still within the region of stability when the crystallizer is ma-
nipulated using the residence time (τ) and unstable otherwise (Fig. 5.4e-f). In the event of
a malfunction in the mechanism used to manipulate the residence time (τ), there is no other
option but active reconfiguration which may be carried out by either reverting to a back-up
sensor to return to the initial operating point or switching to a different set of outputs that
will move the operating point into a region of stability. Actuator reconfiguration may only
be carried out by switching to a redundant actuator that manipulates the residence time
(τ) since the new operating point is within the region of instability when the inlet solute
concentration (c0) or coolant temperature (Tc) is manipulated.
For the case where the process is initially controlled using the inlet solute concentra-
tion (c0), the fault may be dealt with a number of ways: by switching to an actuator that
manipulates the residence time (τ), returning to the original operating point using a redun-
dant sensor, or using a different sensor with a different sampling period such that operating
conditions are within a stable region. The last strategy is carried out by either selecting a
temperature sensor with a larger sampling period or a concentration sensor with a faster
sampling rate (Fig. 5.2b).
The contrast between the first and second scenarios indicates that a priori knowledge
of the nature of future faults provides insight as to which manipulated input will be more
robust to faults. An examination of the regions of stability indicates that the process is more
robust to faults in the concentration sensor when the residence time (τ) is selected as the
manipulated variable. If faults in the concentration sensor are projected to occur, then it
would be wise to control the process by varying the residence time (τ) (Fig. 5.2b). If large
faults in the temperature sensor are more likely to occur, then it is better to manipulate the
inlet solute concentration (c0) to make the process more robust to such faults (Fig. 5.2a).
The same logic may be used with regard to knowledge of the sensors that are available
101
for process monitoring. For instance, if temperature sensors with fast sampling rates are
available, then it is better to manipulate the residence time (τ) while slow sampling rates
for the temperature sensor work best when the inlet solute concentration (c0) or coolant
temperature (Tc) is manipulated.
5.6 Conclusions
This chapter dealt with the design and analysis of a system with multi-rate output mea-
surements to demonstrate passive and active fault-tolerant control strategies for particulate
processes under measurement sampling. Fault-tolerance is important since faults can lead
to process instability and/or inferior product quality. Fault tolerant control was carried out
using passive and active strategies. Passive fault-tolerance may be used as an additional
criterion to ensure that the system is robust to the faults that are most likely to occur in
the system. This is achieved, for instance, by selecting the manipulated variable when all
other variables are known. Active fault-tolerant control was carried out by sensor or actuator
reconfiguration or by switching to a back-up sensor or actuator. To achieve these objectives,
a unified framework for model-based fault-tolerant control was derived. This framework
involves an observer-based output feedback controller that compensates for measurement
sampling using a dynamic model that estimates the output measurements when they are
unavailable. The stability properties of this controller are used to explicitly characterize
the maximum stabilizing sampling periods for the sensors in the system. These properties
are finally utilized to select the appropriate fault accommodation strategy under multi-rate
sampling to ensure that the process is robust to faults.
102
0 5 10 15 20
1.6
1.8
2
2.2
x 10−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(a)0 10 20 30 40
2
2.01
2.02
2.03
x 10−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(b)
0 5 10
1.6
1.7
1.8
1.9
2
2.1x 10
−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(c)0 10 20 30 40 50
1.98
2
2.02
2.04x 10
−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(d)
0 1 2 3
1
2
3
4
x 10−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(e)0 10 20 30
2
2.01
2.02
2.03
x 10−3
Time (h)
Tot
al p
artic
le s
ize,
µ1 (
mm
−2 )
(f)
Figure 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 be-comes unstable by manipulating either (e) inlet concentration, c0, or (f) residence time,τ (f2:∆1 = 0.011,∆2 = 0.008).
103
Appendix A
Proofs of Chapter 2
In order to derive precise conditions for closed-loop stability in terms of the sampling pe-
riod and the delay time, the closed-loop response must be expressed as a function of these
parameters in the absence of faults. The following proposition provides the needed charac-
terization.
Proposition A.1. Consider the system described by (2.12)-(2.13) with the initial condition
χ(t0) =[xT (t0) ηT (t0) wT
2 (t0) 0 wT2 (t0) 0
]T:= χ0. Then:
(a) For t ∈ [tj , tj+1 − τ), j = 0, 1, 2, · · · ; the closed-loop system response is given by:
χ(t) = eΛo(t−tj ) (M)j χ0 (A.1)
(b) For t ∈ [tj+1 − τ, tj+1), j = 0, 1, 2, · · · ; the closed-loop system response is given by:
χ(t) = eΛo(t−tj+1+τ)IτeΛo(∆−τ) (M)j χ0 (A.2)
where ∆ := tj+1 − tj, M := IoeΛoτIτe
Λo(∆−τ),
104
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
, (A.3)
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
, (A.4)
and I is the identity matrix.
Proof. First, we have from (2.12) that at times t = tj only the error e(t) is reset to zero.
This can be represented by writing χ(tj) = Ioχ(t−
j ), where Io is given in (A.3). While at
times t = tj+1 − τ only the error e(t) is reset to zero, and this can be represented by writing
χ(tj+1−τ) = Iτχ((tj+1−τ)−), where Iτ is given in (A.4), since at that time e = y−y = e+ e.
Then on the interval t ∈ [tj , tj+1 − τ), j = 0, 1, 2, · · · ; the system response is given by:
χ(t) = eΛo(t−tj )χ(tj) = eΛo(t−tj )Ioχ(t−
j ) (A.5)
while on the interval t ∈ [tj+1 − τ, tj+1), the system response is given by:
χ(t) = eΛo(t−tj+1+τ)χ(tj+1 − τ)
= eΛo(t−tj+1+τ)Iτχ((tj+1 − τ)−)(A.6)
Using (A.5) to calculate χ((tj+1 − τ)−) we get:
105
χ(t) = eΛo(t−tj+1+τ)IτeΛo(∆−τ)Ioχ(t
−
j ) (A.7)
which can be used to write:
χ(t) = eΛo(t−tj+1+τ)IτeΛo(∆−τ)
[Ioe
ΛoτIτeΛo(∆−τ)
]Ioχ(t
−
j−1)
...
= eΛo(t−tj+1+τ)IτeΛo(∆−τ)
[Ioe
ΛoτIτeΛo(∆−τ)
]jχ(t0)
= eΛo(t−tj+1+τ)IτeΛo(∆−τ)(M)jχ0, t ∈ [tj+1 − τ, tj+1), j = 0, 1, 2, · · ·
In the same way, we obtain from (A.5) that:
χ(t) = eΛo(t−tj)[Ioe
ΛoτIτeΛo(∆−τ)
]Ioχ(t
−
j−1)
...
= eΛo(t−tj)[Ioe
ΛoτIτeΛo(∆−τ)
]jχ(t0)
= eΛo(t−tj)(M)jχ0, t ∈ [tj , tj+1 − τ)
This completes the proof of the proposition.
Having characterized the fault-free closed-loop response in terms of the sampling period
and the size of the delay, the main result of this section will now be presented. The following
theorem provides a condition for the stability of the sampled-data closed-loop system in the
absence of faults.
Theorem A.1. Referring to the system of (2.12)-(2.13) with initial condition χ(t0) = χ0; the
zero solution, χ =[xT ηT wT
2 eT wT2 eT
]T= [0 0 0 0 0 0]T , is globally exponentially stable
if the eigenvalues of the matrix M := IoeΛoτIτe
Λo(∆−τ) are strictly inside the unit circle.
Proof. Evaluating the norm of the response of the closed-loop system described in Proposi-
tion 1, we have from (A.2) that for t ∈ [tj+1 − τ, tj+1), j = 0, 1, 2, · · · :
‖χ(t)‖ ≤ ‖eΛo(t−tj+1+τ)‖‖Iτ‖‖eΛo(∆−τ)‖‖(M)j‖‖χ0‖
Since ‖Iτ‖‖eΛo(∆−τ)‖ is a positive constant, we can write:
106
‖χ(t)‖ ≤ k1‖eΛo(t−tj+1+τ)‖ · ‖(M)j‖ · ‖χ0‖ (A.8)
where k1 = ‖Iτ‖‖eΛo(∆−τ)‖. Analyzing the first term on the right hand side of (A.8):
‖eΛo(t−tj+1+τ)‖ ≤
∞∑
i=0
‖1
i!(Λo)
i(t− tj+1 + τ)i‖
=∞∑
i=0
1
i!(t− tj+1 + τ)i(σ)i
= eσ(t−tj+1+τ) ≤ eστ := k2
(A.9)
where σ is the largest singular value of Λo. In general, this term can always be bounded
since the time difference t− tj+1+ τ is always smaller than τ . In other words, even if Λo has
eigenvalues with positive real parts, ‖eΛo(t−tj+1+τ)‖ can only grow a certain amount, and this
growth is independent of j. The second term on the right hand side of (A.8) is bounded if
and only if all the eigenvalues of M lie inside the unit circle, i.e.,:
‖(M)j‖ ≤ αe−βj = αeβe−βtj+1−t0
∆
for some α,β > 0, where we have used the fact that j =tj−t0∆
to establish the equality. Since
t ∈ [tj+1 − τ, tj+1) and tj+1 > t:
‖(M)j‖ < αeβe−βt−t0∆ := αeβe−β(t−t0) (A.10)
where β = β/∆ > 0. Combining (A.8) with (A.9) and (A.10), the following bound is
obtained:
‖χ(t)‖ ≤ k1k2αeβ‖χ0‖e
−β(t−t0), t ∈ [tj+1 − τ, tj+1) (A.11)
In a similar fashion, one can show that on the interval t ∈ [tj , tj+1 − τ), j = 0, 1, 2, · · · ;
the closed-loop response satisfies a bound of the form ‖χ(t)‖ ≤ k3αβe ‖χ0‖e
−β(t−t0), where
k3 := eσ(∆−τ) > 0. This, together with (A.11), implies that the origin of the networked
closed-loop system is globally exponentially stable, and the augmented system satisfies a
bound of the following form:
‖χ(t)‖ ≤ α‖χ0‖e−β(t−t0) (A.12)
where α = max{k1k2αeβ, k3αe
β}. This completes the proof of the theorem.
107
Remark A.1. 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, as well as the controller and observer design parameters. This dependence can be used
to systematically explore the tradeoffs that exist between these various factors. It can also
be shown that the requirement on the spectral radius of the test matrix M to be strictly less
than one is not only sufficient but also necessary to guarantee closed-loop stability.
Remark A.2. 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 was limited due to the presence of the network, while
here it is limited by the sensor sampling constraints. Furthermore, the control architecture
presented here differs in that: (a) the controller, observer, propagation unit, and model are
all co-located, (b) the control action is calculated using the observer 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.
108
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