2. Faculty of Electrical Engineering, Computer Science and Telecommunications
University of Zielona Góra
Lecture Notes in Control and Computer Science
Volume 16
Editorial Board:
• Józef KORBICZ – Editor-in-Chief
• Marian ADAMSKI
• Alexander A. BARKALOV
• Krzysztof GAŁKOWSKI
• Roman GIELERAK
• Andrzej JANCZAK
• Eugeniusz KURIATA
• Sławomir NIKIEL
• Andrzej OBUCHOWICZ
• Krzysztof PATAN
• Andrzej PIECZYŃSKI
• Dariusz UCIŃSKI
• Marcin WITCZAK
4. Łukasz DZIEKAN
Institute of Control and Computation Engineering
University of Zielona Góra
ul. Podgórna 50
65-246 Zielona Góra, Poland
e-mail: L.Dziekan@issi.uz.zgora.pl
Supervisor:
• Marcin WITCZAK, University of Zielona Góra
Referees:
• Piotr TATJEWSKI, Warsaw University of Technology
• Andrzej PIECZYŃSKI, University of Zielona Góra
The text of this book is based on the author’s Ph.D. dissertation entitled
Neuro-Fuzzy-Based Takagi-Sugeno Modelling in Fault-Tolerant Control
Partially supported by the grant No. N N514 001436 of the Ministry of Science
and Higher Education in Poland
ISBN 978-83-7481-427-0
Camera-ready copy prepared in LATEX2ε by the author
Copyright c University of Zielona Góra Press, Poland, 2011
Copyright c Łukasz Dziekan, 2011
University of Zielona Góra Press
ul. Licealna 9, 65-417 Zielona Góra, Poland
tel./fax: +48 68 328 78 64, e-mail: oficynawydawnicza@adm.uz.zgora.pl
10. Acknowledgements
First of all. I would like to express my gratitude to my supervisor Prof. Marcin
Witczak. Without his constructive comments, his continuous support and useful
suggestions this dissertation would never have been completed.
I am also grateful to Profs. Vicenç Puig and Joseba Quevedo from Technical
University of Catalonia for their advice and support. Their knowledge greatly
contributed to the value of this work.
I also thank my friends and colleagues at the Institute of Control and Com-
putation Engineering for many enlightening discussions.
I would also like to thank my parents and my sisters for their unconditional
love and support.
11. Chapter 1
INTRODUCTION
Since the idea of the fuzzy set was first proposed by Zadeh (1965), a large progress
has been made in this area. Applications of this artificial intelligence technique
have been made in many diverse areas such as engineering, medicine, computer
vision, management, behavioural science, just to name a few. The application of
the fuzzy logic engages different technologies, such as fuzzy clustering on image
processing, classification of problems, augmenting decision making process in ex-
pert systems, identification and fault detection, extraction of expert knowledge to
make efficient fuzzy controllers in control systems, or fuzzy modelling of different
systems, solving fuzzy optimization design problems.
In the area of artificial intelligence the usage of the fuzzy systems is considered
as a way to represent knowledge. This representation belongs to the paradigm of
behavioural representation, in opposition to the structural representation given by
neural networks (Espinosa et al., 2004). The foundation of this paradigm states
that the intelligent behaviour can be obtained by the use of structures that do
not have to resemble the human brain. As a result, the fuzzy systems have a very
interesting characteristics, in particular they are capable to handle in the same
framework linguistic and numeric information. This property made these systems
very capable to handle expert control tasks, for example such as a task described
in detail below.
A continuous increase in the complexity, efficiency and reliability of mod-
ern industrial systems necessitates a continuous development in the control and
fault diagnosis theory and practice. A conventional feedback control design for
a complex system may cause an inadequate performance, or even instability, in
the event of improper functioning of actuators, sensors or other system compo-
nents. Hence, to subdue such disadvantages, new approaches to control system
synthesis have been developed that have the ability to tolerate component mal-
functions while providing desirable stability and performance properties. Such
properties are especially important for safety-critical systems, such as aircrafts,
spacecrafts, chemical plants processing hazardous substances and nuclear power
plants. In safety-critical systems, every fault in a system component, even a mi-
nor one, has a potential to cause chain effect that can be catastrophic. Hence,
the requirement of reliability, safety and fault tolerance is generally high. These
demands also extend to a new systems such as autonomous vehicles, rail systems,
automotive control, e.g., Anti-lock Braking Systems (ABS), etc. Therefore, it
is indispensable to develop control systems which are able to tolerate potential
12. 12 1. Introduction
faults in these systems, and by doing so improve the reliability and availability
while providing a desirable performance and increase safety margins, by stopping
a potential system wide failure. These classes of control systems are generally
known as Fault-Tolerant Control (FTC) systems. In other words, the FTC sys-
tems are control systems, which possess the ability to accommodate component
faults automatically (Zhang et al., 2003a). They capabilities allow maintaining the
overall system stability and acceptable performance (though possibly degraded)
in the event of such faults. Alternatively, a closed-loop control system capable
of tolerating component malfunctions, while maintaining stability properties and
desirable performance is said to be a fault-tolerant control system.
In general, the FTC systems are classified into two distinct classes (Zhang
et al., 2008): passive and active ones. In passive FTC (Chen et al., 1998; Liang
et al., 2000; Liao et al., 2002; Qu et al., 2003), controllers are designed to be robust
against a set of presumed faults, therefore there is no need for fault detection, but
such a design usually degrades the overall performance. In contrast to passive ones,
active FTC schemes, react to system components faults actively by reconfiguring
control actions, and by doing so the system stability and acceptable performance
is maintained (Blanke et al., 2006). In certain situations, degraded performance
must be accepted. To achieve fault tolerance, the control system relies heavily on
the Fault Detection and Diagnosis (FDD) systems, which allows an early detection
and maintenance of faults by control system.
Although individual research on the FTC systems has been carried out ex-
tensively, systematic concepts, design methods, and even terminology are still not
yet standardized (Blanke, 2000; Zhang et al., 2008). Lately, there have been some
attempts to unify some terminology (Blanke, 2001; Blanke et al., 2006; Isermann,
2006; Isermann et al., 1997). Additionally, because of historical reasons and the
complexity of the problem, most of the research on the FDD and reconfigurable
control was performed as a two separate subjects. To be more specific, the pri-
mary use of the FDD techniques were to provide a diagnostic or monitoring tool,
rather than being an integral part of the FTC systems. Hence, some existing FDD
techniques do not satisfy the need for controller reconfiguration. Similarly, recon-
figurable control is often synthesized assuming a perfect knowledge regarding the
FDD systems. So, a very little attention has been paid to the analysis and design
with the full integration of the FDD and control reconfiguration techniques, there
is a need for developing it as a holistic system. Additionally, Zhang et al., (2008)
points put other major issues in a current development of the FTC systems, in
particular a need for a research in the field of FTC for non-linear systems, as most
of the industrial systems are non-linear ones, and a somewhat related problem of
constraints of input and state/output variables in the FTC systems. As most of
the actuators have limited capabilities, so they are prone to enter into a saturation
zones, which introduces non-linear behaviour even in normally linear systems.
One way to effectively model non-linear systems is to use a soft computing
techniques such as a neuro-fuzzy Takagi-Sugeno (T-S) modelling, which are said
to be universal approximators (Espinosa et al., 2004). This property means that
the neuro-fuzzy systems are capable of approximating any continuous function
13. 13
into a compact domain with a certain level of accuracy. However, as previously
stated, the universal approximation property of the fuzzy models is not the only
exceptional attribute. Fuzzy modelling adds a new dimension to the information
that can be obtained from the model. This new dimension is the linguistic layer,
which grants intuitive (linguistic) descriptions over the behaviour of the modelled
system. There are many types of fuzzy models, the most exploited ones are the
rule-based fuzzy systems. In particular, one important class of this rule-based
systems are the Takagi-Sugeno fuzzy models (Takagi et al., 1985). According to this
model, non-linear dynamic systems can be linearised around a number of operating
points. Each of these linear models represents the local system behaviour around
the operating point. Thus, a fuzzy fusion of all linear model outputs describes the
global system behaviour.
Model Predictive Control (MPC) is one of the control techniques, that is
particularly viable for dealing with constrained systems. Additionally, predic-
tive control, is the only advanced control technique to have had a significant and
widespread impact on industrial process control and engineering (Maciejowski,
2002; Tatjewski, 2007). When there are no constraints it behaves similarly to a
linear quadratic regulator and can be computed off-line. Only when constraints
are considered the MPC controllers show their full potential, although at the cost
of the computational complexity. Some even argue that FTC is only possible by
the use of the MPC control schemes (Maciejowski, 2009). The MPC has a fault-
tolerant behaviour, but this behaviour is implicit and was not explicitly designed.
Yet, by explicitly combining a fault information and predictive control formulation,
it is possible to improve the results considerably, but how to change the formu-
lation problem when a fault occurs is not a trivial matter (Maciejowski, 2002).
Additionally, a non-linear MPC is an open research subject. As it was previously
mentioned MPC is one of the few widely accepted control strategies applied in
industrial applications and several commercial applications exist.
In view of computer-aided analysis and synthesis, the efficiency of a method
involves two main aspects: minimizing storage requirements and keeping the com-
putational complexity as low as possible. The first requirement is equivalent to
keeping the storage proportional to the amount of the data defining the prob-
lem instance, however as a larger amount of memory is today easily obtainable
in modern computer workstations it can often be neglected. Regarding the other
requirement, some key concepts from the theory of the complexity must first be
introduced (Arora et al., 2009). The class of problems so-called P is algorithmically
solvable with polynomial time complexity, i.e., the running time of an algorithm on
any problem instance of size n increases no faster than some polynomial function
in n. The problems belonging to such a class are considered as efficiently solvable
ones. On the other hand, the problem is allocated to the NP (non-deterministic
polynomial time) class if it is verifiable (but not necessary solvable) in the poly-
nomial time, i.e., the proposed solution can be verified in polynomial time if it
is correct. Other problems are shown to be NP-hard (Arora et al., 2009), that is
though they may be algorithmically solvable, no polynomial time algorithm exists1
.
1Under the assumption of the validity of a long-standing conjecture in computer science
14. 14 1. Introduction
For example, the algorithms for synthesis of the problems considered in this book
can be described in terms of Bilinear Matrix Inequality (BMI) or Linear Matrix
Inequalities (LMIs) (Paszke, 2005; Toker et al., 1995). These two problem formula-
tions differ from each other. The BMIs formulation is considered to be NP-hard as
no polynomial-time algorithm has been found so far for solving them (the convex
optimization methods cannot be utilized) (Goh et al., 1994). Whereas the LMIs
can be solved with great theoretical and practical efficiency using interior-point
algorithms (Boyd et al., 2004), which are polynomial-time algorithms.
However, many NP-hard problems can be regularly solved in practice, either
exactly or approximately using the various methods, though generally they are
only limited to low scale problems. Similarly, P problems can be solvable with a
great difficulty if the order of the polynomial bounding the computational com-
plexity is high enough, i.e., the time required to obtain the solution is surpassing
the allocated time frame. Thus, an important branch of computer science is con-
cerned with further optimizations of these polynomial time algorithms, to make
them applicable for problems at hand. For example, sorting algorithm are consid-
ered P problems, but much work has been done to optimize them and currently
quick-sort and heap-sort are considered as the fastest ones (average complexity
O(n log n)) (Cormen et al., 2009). However, in the worst case scenario heap-sort
complexity is still O(n log n), whereas for the quick-sort it is O(n2
), yet heap-sort
is assumed to be on average somewhat slower than quick-sort. Hence, it is required
to further optimize algorithms presented in this book, to make them applicable to
the problems where allocated time frame is small enough.
Thus, for the reasons stated above this book is concerned with the develop-
ment of an effective soft computing algorithms, utilizing Takagi-Sugeno fuzzy mod-
elling that are suitable to provide fault tolerance for constrained non-linear systems
with a possibly low computational effort requirements (i.e., the time required for
the technique is acceptable for applications at hand). Hence, the objective of this
work can be stated as follows:
The objective of this work is twofold. The first subject concerns the
development of a soft computing neuro-fuzzy based Takagi-Sugeno tech-
nique, that provides an integrated fault-tolerant control capabilities for
non-linear dynamic systems and is able to overcome faults induced
by actuators and sensors. The second subject focuses on designing a
fuzzy predictive fault tolerant control algorithms, which allows control-
ling constrained systems effectively. In particular, the main problem
is to develop algorithms, which allow computing a fast solution to the
quadratic programming problem involved in the fuzzy predictive fault
tolerant problem statement.
The following thesis can be formulated:
Neuro-fuzzy based Takagi-Sugeno modelling makes it possible to effi-
ciently solve the fault-tolerant control problems for nonlinear dynamic
system in a time required in practical applications.
(P=NP), which is currently believed to be true.
15. 15
To confirm this thesis, the following problems have been addressed:
Theoretical aspects:
• development of a computer implementable formulations of the neuro-
fuzzy based Takagi-Sugeno modelling that allows integrated fault-tole-
rant control, based on the use of the so-called fuzzy virtual actuators
and fuzzy virtual sensors for:
– linear dynamic systems,
– non-linear dynamic systems,
• development of a fuzzy predictive virtual actuator which allows effective
computing of the control law for a faulty and constrained non-linear dy-
namic systems described by the neuro-fuzzy Takagi-Sugeno modelling,
in particular:
– stating the problem in an explicit way in the form of a quadratic
programming problem,
– providing an approximate, but heavily optimised solution to an
original quadratic programming problem, that allows implement-
ing a predictive virtual actuator for a wide range of problems, not
previously implementable due to the high computational cost,
Application aspects:
• neuro-fuzzy fault tolerant control of simulated and constrained real sys-
tems, in particular a non-linear laboratory model of a tunnel furnace.
The following outlines the structure of the book and shortly highlights its contri-
bution.
• Chapter 2: This chapter presents principal concepts of the fault tolerant
control. At first introducing the definition of faults and its effects on the
system, followed by the classification of the fault-tolerant control systems,
and concludes by providing the current research topics in the fault-tolerant
control systems.
• Chapter 3: In this chapter, the Takagi-Sugeno fuzzy models, which have
become a standard tool used to model non-linear systems behaviour, will be
described in details. It will be done by providing a theoretical background
about fuzzy systems and the theory of constructing the Takagi-Sugeno fuzzy
system and basic guidelines about the experiment design to identify such
models. Moreover, a Takagi-Sugeno fuzzy model identification example will
be provided, based on the experimental data.
• Chapter 4: This chapter focuses on providing an effective scheme for design-
ing a fuzzy virtual actuator and fuzzy virtual sensor for the Takagi-Sugeno
fuzzy systems. Such a design of the computer implementable FTC system
allows including it in any control loop without making any modifications or
16. 16 1. Introduction
reconfigurations of the loop controllers and is able to control the system in a
faulty state and (if it is possible) to drive it back to the region of the nominal
performance. Additionally, the techniques are tested in a simulation of var-
ious non-linear systems, including a test of virtual sensors with the real-life
experimental data obtained from the laboratory model of the tunnel furnace.
• Chapter 5: The objective of this chapter is to demonstrate that combining
the model predictive control with the Takagi-Sugeno fault tolerant system is
a viable option. The first part presents some background information about
the MPC and its extension to the Takagi-Sugeno fuzzy systems. Afterwards,
a concept of a fuzzy predictive virtual actuator is introduced. However,
such a technique can be too complex for using effectively in the faster sys-
tems. Thus, its possible optimization is considered, which allows an effective
computer implementation. As a result approximate, but fast solution for
the MPC problem is provided. Finally the results of the experiments are
described.
• Chapter 6: This chapter discusses the major results and the contribution
of this work. Several directions for future research are identified.
• Appendix A gives the details of the hardware specification of the laboratory
model of the tunnel furnace used in the experiments presented in this book.
17. Chapter 2
FAULT-TOLERANT CONTROL
“And oftentimes excusing of a fault doth make the fault the worse by
the excuse.”
William Shakespeare
This chapter collects briefly the basic fundamentals of the FTC. At first in-
troducing definition of faults and their effects on the system, followed by the
classification of the FTC systems. It concludes by providing the current research
topics in fault tolerant control systems.
2.1. Faults and fault tolerance
Generally speaking, a fault is something that modifies the behaviour of a system
in such a way that the system does not longer satisfy its purpose. The nature of
such a fault can be of an internal event in the system in question, for example a
breakage of the pump, leak in the pipe, or simply breaking an information link,
etc. It may also be a change in the environmental conditions that in the end
causes a stop of some chemical reaction or even destroys the reactor itself. Finally,
it can be a wrong decision or control action given by the human operator that
eventually stops the system or, alternatively, it can be an error in the overall
design of the control system that remains undetected until the system is working
under the specific conditions and/or the operating point. In all of these cases, the
fault is the primary reason of changes in the system parameters or structures that
eventually leads to a degraded system performance, or in the worst case to the
complete loss of the system function or even its destruction.
In large systems, even a fault in a single component usually changes the
performance of the overall system. It is so, because every component is designed to
accomplish a certain goal and the overall system performance is dependent on the
union of these goals. Hence, faults have to be detected as quickly as possible and a
further propagations of their effect must be stopped or the overall production will
quickly deteriorate or even causes some damage of machines and human workers.
These measures should be preferably carried out by the control equipment. The
aim should be to achieve a fault tolerance of the system. By successfully achieving
that goal, the control algorithm adapts to the faulty plant and the overall system
18. 18 2. Fault-tolerant control
function is maintained even in the face of a fault, though with a possible short
time of the degraded performance.
If a dynamical system is considered the following definition of a fault can be
provided (Blanke et al., 2006):
Definition 2.1. A fault in a dynamical system is a deviation of the system struc-
ture or the system parameters from the nominal situation.
For example, structural changes include: the blockage of an actuator, the
disconnection of a system component, or a failure of a sensor. In all of these cases,
the fault is changing the set of interacting components of the plant or the interface
between the controller and the plant. On the other hand, parametric changes are
brought about, e.g., by a wear or a damage. All these faults cause divergences
of the dynamical input/output properties of the plant from the nominal ones.
Thus, they change the performance of the closed-loop system, which can result in
a further degradation or even a loss of functionality of the system.
Disturbances and model uncertainties also change the plant behaviour. Hence,
to distinguish between them, let us consider a system described by an analytical
model. For this class of systems, faults are usually depicted by the use of addi-
tional external signals or parameter deviations. So, in the former case, the faults
are called additive faults and represented by an unknown input that enters the
model equation as an addend. Whereas in the latter case, the system parameters
depending on the fault size are multiplied with the system state or input, and thus
these faults are called multiplicative faults.
Indeed, model uncertainties and disturbances have comparable effects on the
system as faults. Disturbances are usually represented by the unknown input sig-
nals, that are added to the system output. Whereas model uncertainties change
the system parameters in a related way as multiplicative faults do. The differ-
entiation is given by the aim of the fault-tolerant control. The faults should be
detected and their impact should be removed by corrective actions, whereas model
uncertainties and disturbances are annoyances, which are known to exist but at
the same time their effects are countered by suitable measures like filtering or
robust design. The control theory has proved that controllers can be synthesized
so as to attenuate disturbances and tolerate model uncertainties up to a certain
degree. Faults, on the other hand, cause more severe changes. Their effects cannot
be suppressed by a fixed controller, without compromising nominal performance.
Hence, the FTC goal is to change the control law so as to minimize the effects of
the faults, or at least to attenuate them to a tolerable level.
Generally one can classify faults, according to their origin, as follows (also
shown in Fig. 2.1):
• Actuator faults: The plant properties are not influenced, but due to the
degraded actuator performance, the influence of the controller on the plant
is interrupted or altered, e.g., a decreased power of a motor, a malfunction
of a pump, stuck rudder, etc.
• Process faults: These type of faults change the dynamical I/O properties
19. 2.1. Faults and fault tolerance 19
Figure 2.1: Classification of faults.
of the system, e.g., a leak in a tank, a wrongly diluted chemical compound,
etc.
• Sensor faults: The plant properties are not influenced, but the sensor
readings have considerable errors, e.g., a disconnected sensor, a sensor being
stuck, etc.
Due to the position of the sensor and actuator faults at the beginning or the
end of the cause-effect chain of the plant, there are specific methods for detecting
them. For example this book is only concerned with the sensor and actuator
faults (see Sec. 4 on page 47 for details), although some process faults can be
modelled to some degree by the actuator faults. However, the sensor and actuator
faults left unchecked, could in the end cause process faults. For example, let us
consider a chemical plant with a sensor reading a concentration of some chemical
compound. If this sensor at some point would start showing lower concentration
of this compound, then the controller being not aware of the fault, could try to
correct the readings by increasing the concentration of the compound. By doing so,
the resulting concentration would be higher than required, changing the dynamics
of the chemical process (i.e., a process fault), and in the end could make the entire
chemical batch useless.
Finally, faults can be discerned concerning their size and temporal behaviour.
The fault can occur abruptly, for example, a disconnected sensor, while on the
other hand, steadily increasing faults are caused by wear, and intermittent faults
(these faults are particularly difficult to detect) are due an intermittent electrical
connection, etc.
A short reminder is needed here regarding the differentiation of the concept of
fault and failure, especially in respect to their current use in the engineering ter-
minology. As described above, a fault brings about a change in the characteristics
of a component, in such a way that the performance of the component or its mode
of operation is changed in an unwanted way. Thus, the system performance does
not meet the required specifications. Despite, that an FTC can “work around” a
fault and keep the faulty system operational.
While on the other hand, the concept of a failure describes the inability of a
system or a component to accomplish its function. As the failure is an irrecoverable
20. 20 2. Fault-tolerant control
event, the system or a component has to be shut off. Taking this into account, the
following definition of the FTC can be stated as follows (Blanke et al., 2006):
Definition 2.2. Fault-tolerant control has to prevent a fault from causing a failure
at the system level.
2.1.1. Requirements and properties of systems subject to faults
Engineers have investigated the occurrence and impact of faults for a long time, due
to their potential to cause substantial damage on machinery and risk for human
health or life. Different notions have been established and investigated and their
short summaries are given below:
• Safety depicts the absence of danger. A safety system is a part of the
control equipment with a sole purpose to protect a technological system
from permanent damage (or to prevent human casualties). It enables a
controlled shut-down, which halts the technological process into a safe state.
It is capable of doing so, by evaluating the information about critical signals
and enables dedicated actuators to stop the process under special conditions.
Hence, the overall system is called a fail-safe system.
• Reliability is the probability that a system executes its intended function for
a specified time period under normal conditions. Reliability studies appraise
the frequency with which the system is faulty, but they are not able to provide
information about the current fault status. The FTC cannot change the
reliability of the plant components, but it can improve the overall reliability,
because the FTC allows the overall system to remain functional even after
the occurrence of faults.
• Availability is the probability that a system stays operational when needed.
As opposed to reliability it is also dependent on the maintenance policies,
which are applied to the system components.
• Dependability accumulates together the three properties of safety, relia-
bility and availability. A dependable system is a fail-safe system with high
availability and reliability.
As was defined before, a fault tolerant system has a property that faults do not
develop into a failure. In the strict form, the system is said to be fail-operational,
because the performance remains the same. Whereas in a reduced form, the system
is operational after a fault occurrence, but with a degraded performance. Such a
system is then called to be fail-graceful.
The relation of the safety and fault tolerance will be elaborated in more details,
because of its importance. Let us assume that the system performance can be
described by different regions, as shown in Fig. 2.2.
The system is satisfying its function in the region of required performance. In
this region, the system should remain during the time of being operational. The
controller job is to hold the nominal system in this region, in spite of uncertainties
21. 2.2. Classification of the fault-tolerant control 21
Figure 2.2: Regions of the system performance.
and disturbances. It may even hold it during small faults, however it is not its
primary aim. Indeed, by doing so the effect of faults is being masked, therefore
the fault diagnostic system may not detect them.
On the other hand, the region of degraded performance is the region where the
faulty system is allowed to remain, however the performance of such a system is
(substantially) degraded and does not satisfy the nominal performance levels. The
system goes from the region of required performance to the degraded one due to
faults. The FTC controller should have the capability to initiate recovery actions
that prevents a further degradation of the performance towards the unacceptable
or even dangerous levels and it should work towards bringing the system back
to the region of required performance. At the borderline of these two regions,
the supervision system involved, which diagnoses the faults and reconfigures the
controller into the new circumstances.
The region of an unacceptable performance should be avoided at all costs,
by means of FTC. This region lies between the region of degraded performance
and the region of danger, which could lead to disaster if it was ever reached. The
system goes to this region by either a sudden failure, or by the FTC system not
doing its function, i.e., preventing faults to develop into failures.
To avoid danger for the system and its environment, the safety system stops
the operation of the overall system. If the outer border of the region of an unac-
ceptable performance is crossed, the safety system should be immediately involved.
This clearly shows that the FTC controller and the safety system work in separate
regions of the system performance and fulfil a complementary role. For example,
in industrial standards safety systems and supervision systems are executed as
separate units. Due to this separation, the design of the FTC does not need to
meet safety standards (Blanke et al., 2006).
23. 2.2. Classification of the fault-tolerant control 23
2.2. Classification of the fault-tolerant control
In general, the FTC systems are classified into two distinct classes (Zhang et al.,
2008): passive and active ones. In the passive FTC (Chen et al., 1998; Liang et al.,
2000; Liao et al., 2002; Qu et al., 2003; Zhang et al., 2003a), controllers are designed
to be robust against a set of presumed faults, therefore there is no need for fault
detection, but such a design usually degrades the overall performance. Hence, the
passive FTC sets the control aim in a context, where the ability of the system to
achieve its given objective is preserved, using the identical control law, whichever
the system situation (faulty or healthy). Indeed, the control law is not changed
when faults occur, so the system is able to achieve its control goal, in general,
only for objectives associated with a very low level of performances (sometimes
called conservative approach). Further, such controller works sub-optimally for
the nominal plant because its parameters are prearranged so as to get a trade-off
between the performance and the fault tolerance. It should be noted that a passive
fault-tolerant controller is similar to the robust approach when uncertain systems
are considered. Although the difference lies not only in the size and interpretation
of faults versus uncertainties, but also in the structure of the constraints resulting
from the faults (Blanke et al., 2006). An overall structure of the passive FTC can
be seen in Fig. 2.3. In the literature, the passive FTC system is also known
as reliable control systems or control systems with integrity. However, a further
discussion of the scope of the passive FTC is beyond the scope of this book and
interested readers are referred to the previously mentioned papers and references
therein.
In contrast to passive ones, active FTC schemes react to the system compo-
nents faults actively by reconfiguring control actions, and by doing so the system
stability and acceptable performance is maintained. In certain situations, degraded
performance must be accepted. An active FTC (referred from here on simply as
the FTC, unless some reference to the passive FTC must be made) in the liter-
ature is sometimes also called as self-repairing, reconfigurable, restructurable, or
self-designing control systems. To achieve a fault tolerance, the control system
relies heavily on the Fault Detection and Diagnosis (called FDD, to differentiate
between FDI — Fault Detection and Isolation) to provide the most up-to-date
information about real status of the system (Korbicz et al., 2004; Li et al., 2007;
Witczak, 2007). Hence, the main goal in an FTC system is to design controller with
an appropriate architecture, that allows stability and satisfactory performance, not
only when all control components are healthy, but also in cases when there are
faults in sensors, actuators, or other system components.
The design objectives of the active FTC must include not only the transient
and the steady-state performance for the nominal (healthy) system, but addi-
tionally for a faulty system. However, the emphasis with respect to the system
behaviour in both cases is quite different. During healthy conditions, the empha-
sis lies with the performance and overall quality of the system, whereas in the
presence of a fault the prevailing objective is to keep the system from further
degradation, even when the nominal performance cannot be achieved (though it
24. 24 2. Fault-tolerant control
should be regained as far as it is possible).
Usually, as depicted in Fig. 2.4, the FTC system can be divided into four
sub-systems (Zhang et al., 2008):
• a reconfigurable controller,
• an FDD scheme,
• a controller reconfiguration scheme,
• a command/reference governor.
It should be noted that the inclusion of both FDD and reconfigurable con-
troller within system structure is the main difference between the active and the
passive FTC system. Hence, key issues of successful FTC scheme is to design a
controller which can be easily reconfigured and an FDD scheme that is able to
detect faults quickly, yet at the same time being robust to model uncertainties,
external disturbances and changing operating conditions. Lastly, a reconfiguration
mechanism must be able to recover as much as possible of the pre-fault system
performance, while working under uncertainties and time-delays intrinsic in the
FDD, yet at the same time do not cross the control input and the system state con-
straints. The key issue in every FTC system is the limited time frame allotted for
the FDD and for the reconfiguration of the system controller. Moreover, efficient
employment and supervision of the available redundancy (in software, hardware
and communication networks), while at the same time stabilizing the faulty plant
with some performance goals, are some of the main issues to take into account in
the FTC.
As shown in Fig. 2.4, the FDD must provide information on-line in the real
time about an every detected fault, which then have to be isolated, and its size,
parameters, etc. estimated. Based on this information, the reconfiguration block
must take into consideration the current system state and the outputs, as well as to
construct an appropriate post-fault system model, afterwards the reconfiguration
data for the controller should be designed, in such a way that a currently faulty
system is stabilized and a fault propagation stopped. The second objective is to
recover as much of the nominal performance as possible. Also there is often a need
for synthesizing a feedforward controller in order to guarantee that the closed-
loop system tracks a future trajectory during its faulty state. Yet, at the same
time the actuator saturation and other system constraints should be taken into
consideration, and system trajectories adjusted if needed. Such an FTC system is
often classified as a reconfigurable one, though some authors call it accommodation
scheme (Blanke et al., 2006).
However, in some cases reconfiguration of the controller is not enough to
stabilize the faulty system. In such cases, the structure of the new controller must
be changed. This restructure also uses an alternative input and output signals in
the new controller configuration. Afterwards, a new control law has to be designed
on-line. Such an FTC controller is called restructurable fault-tolerant controller,
and can be seen in Fig. 2.5. This type of the FTC is also sometimes called
25. 2.2. Classification of the fault-tolerant control 25
reconfiguration (Blanke et al., 2006), but to avoid confusion the former terms will
be used, i.e., reconfigurable versus restructurable. Restructure of the controller is
necessary after an occurrence of severe faults, that lead to serious changes of plant
dynamics:
• Actuator failures interrupt the normal means of controlling the plant and
could make the plant partially uncontrollable. Alternative (or redundant)
actuators have to be used.
• Sensor failures disrupt the information flow between the controller and the
plant. They may make the plant partially unobservable. Alternative mea-
surements have to be chosen and used in such a way that the control task is
still possible.
• Plant faults alter the dynamical behaviour of the overall system. If these
alterations cannot be tolerated by any existing control law, the overall control
loop has to be redesigned and a new control law computed.
The necessity of the control restructuring is apparent if actuator or sensor
failures are contemplated. The total failure of these components leads to a break-
down of control loop. Hence, a simple adaptation of the controller parameters to a
new situation is no longer possible and hence, alternative sensors or actuators have
to be taken into consideration, preferably the ones that have a similar interactions
with the plant and are not under fault influence. By doing so, it is possible
to design a controller that satisfies the performance specification of the nominal
system (Blanke et al., 2006).
2.2.1. Classification of the existing active fault-tolerant control techniques
Currently, the existing reconfigurable fault tolerant control design methods can be
classified as a one of the following approaches (Zhang et al., 2003a; Zhang et al.,
2008):
• Linear quadratic (Yang et al., 2000);
• Pseudo-inverse/control mixer (Bajpai et al., 2001);
• Intelligent control using expert systems (Liu, 1996), neural networks (Ho et
al., 2002), fuzzy logic (Ichtev, 2003) and learning methodologies (Diao et al.,
2002);
• Gain scheduling/linear parameter varying (Shin et al., 2004);
• Adaptive control (model reference) (Kim et al., 2003);
• Model following (Zhang et al., 2002);
• Multiple-model (Yen et al., 2003);
• Integrated diagnostics and control (Zhang et al., 2001);
26. 26 2. Fault-tolerant control
Classification
of
FTC methods
Mathematical
design tools
Linear Quadratic (LQ)
Pseudo-Inverse (PI)
Intelligent Control (IC)
Gain scheduling (GS)/Linear Parameter Varying (LPV)
Model Following (MF)
Adaptive Control (AC)
Multiple Model (MM)
Eigenstructure Assignment (EA)
Feedback Linearisation (FL)/Dynamic Inversion (DI)
H∞ and other robust control techniques
Model Predictive Control (MPC)
Quantitative Feedback Theory (QFT)
Linear Matrix Inequality (LMI)
Variable Structure Control (VSC)/Sliding Model Control
Generalized Internal Model Control (GIMC)
Design
approaches
Pre-computed control laws
GS/LPV
MM
QFT
LMI
GIMC
On-line automatic redesign
LQ
PI
MF/AC
EA
FL/DI
VSC/SMC
MPC
Reconfiguration
mechanisms
Optimizations
LQ
H∞/µ synthesis
LMI
MPC
Switching
MM
GS/LPV
VSC/MSC
Matching
PI–System matrix
EA–Eigenstructure
Following
MF–State/Output
MPC–Set-point/Output
Compensation
Additive compensation
Adaptive compensation
Type of
systems
dealt with
Linear systems LQ; PI; MF; EA; MM; MPC; QFT; GIMC
Non-linear systems GS/LPV; MM; FL/DI; LMI; VSC; IC
Figure 2.6: Classification of active FTC systems (Zhang et al., 2008).
Control
Structure
Adaptive
Indirect
Direct
Switching
GS/LPV
MM
VSC/SMC
Following
Explicit MF
Implicit MF
Interaction
←−−−−−−−−
−−−−−−−−−→
Combination
Control
Algorithms
Optimization
LQ
H∞
LMI
MPC
Matching EA
Inversion
PI
DI
Figure 2.7: Combination of reconfigurable control algorithms in the active FTC
(Zhang et al., 2008).
27. 2.2. Classification of the fault-tolerant control 27
• Eigenstructure assignment (Konstantopoulos et al., 1999);
• Feedback linearisation or dynamic inversion (Doman et al., 2002);
• H∞ and other robust controls (Yang et al., 2001);
• Model predictive control (Kale et al., 2005);
• Linear Matrix Inequality (Ganguli et al., 2002);
• Variable structure and sliding mode control (Hess et al., 2003);
• Generalized internal model control (Campos-Delgado et al., 2003);
The FTC methods, as shown in Fig. 2.6, can be also classified in accordance
with the following criteria: mathematical design tools, design approaches, recon-
figuration mechanisms and type of systems to be dealt with. The methods, shown
in Fig. 2.6, were listed approximately in a chronological order to emphasize the
historical evolution of the FTC design techniques.
Yet, in most cases and practical applications, the FTC systems rarely use
only one of these methods and to obtain the best possible results a combination of
several methods is usually more appropriate. Hence, Fig. 2.7 shows a combinations
of different control structures and control design algorithms frequently used in the
successful FTC control schemes.
Additionally, many of currently used FTC design methods rely on ideas orig-
inally developed for other control objectives. However, using those well-known
control techniques, does not mean that new problems and challenges will not ap-
pear, besides the standard problems found in the conventional controller synthesis.
Finally, in order to judge the adequateness of a control method for the FTC,
its ability to be implemented in an on-line real-time setting and yet at the same
time being able to maintain an acceptable (nominal or degraded) performance,
is one of the most important criteria. Hence, the following requirements for any
technique used in the FTC systems can be proposed (Zhang et al., 2008):
• control reconfiguration must be computed reliably under the real-time con-
straints;
• the reconfigurable controller should be synthesized automatically with as
little as possible of trail-and-error and human interactions;
• the selected methods must always provide a solution even if the obtained
solution is suboptimal.
2.2.2. Classification of the existing FDD techniques
The FDD is an important part in every active FTC systems, also one should not
forget that its primary use is in diagnostic systems. Hence, a lot of research has
been done in the area of the FDD in the last three decades and many new FDD
schemes were developed. Nevertheless, the majority of research in the FDD area
28. 28 2. Fault-tolerant control
is still focusing on monitoring or diagnostics purposes, rather than control appli-
cations. So in the context of the FTC, there is a comparatively low number of
research about the role of the FDD in the overall FTC scheme and methodology
of designing FDD to be appropriate for the FTC purposes (Patton, 1997; Zhang
et al., 2008). Early researches in (Jiang, 1994; Jiang et al., 1997; Patton, 1997)
have shown that most suitable for fault detection are the state estimation based
schemes, due to their intrinsic speed and very short time delay in the real-time
decision-making process, especially when compared with the parameter estimation
approach. Nevertheless, the information obtained from the state estimation tech-
niques may not be elaborated enough for the following control system reconfigura-
tion, because there is a need for determining the fault-induced parameter changes
or even a new system model. For this purpose, parameter based schemes are more
suitable. Hence, a combination of both the state- and the parameter-estimation-
based schemes is more suitable (Patton, 1997; Zhang et al., 2008). Although, the
parameter estimation techniques are often preferred for reconfigurable flight con-
trol (Zhang et al., 2008). A comprehensive review on the subject of the FDD is
beyond the scope of this book and interested readers are referred to the following
survey papers (Isermann, 1997; Isermann et al., 1997; Patton, 1997; Venkatasub-
ramanian et al., 2003; Zhang et al., 2008) and books (Basseville et al., 1993; Chen
et al., 1999; Isermann, 2006; Patton et al., 2000; Pieczyński, 2003; Simani et al.,
2002; Witczak, 2007).
An FDD system has three primary tasks (Isermann, 2006):
• Fault detection: provides decision whether or not a fault occurred, also
the time of the fault occurrence;
• Fault isolation: provides information about the location and the type of
the fault (which component is faulty);
• Fault identification: identify the fault and provides estimate of its mag-
nitude. It determines the category of the fault and its severity.
The existing FDD techniques can be in a general manner classified into two
categories: data-based (model-free) and model-based techniques; each of these
methods can be classified in addition as qualitative and quantitative approaches
(Zhang et al., 2008).
In essence, a quantitative model-based FDD approach employs mathematical
model (sometimes called the analytical redundancy, in contrast to the hardware
redundancy) to perform the FDD tasks in real-time. The most commonly used
techniques are based on: state estimation, parameter estimation, parity space and
some combination of these methods. Due to the fact that most of control schemes
are model-based, so the majority of fault tolerant controllers are designed based
on mathematical model of the system being analysed, in particular its post-fault
counterpart.
An FDD suitable for the FTC can be selected based on the following crite-
ria: its capacity to deal with different type of faults (actuator, process and sensor
faults), capacity to supply quick detection, its isolability and identifiability, ease of
29. 2.3. Current research in the active fault-tolerant control 29
Table 2.1: Comparison of the FDD techniques (Zhang et al., 2008).
Criteria/method State estimation
Single Multiple
Observer Kalman fil-
ter
Observers Kalman fil-
ter
Sensor fault
Actuator fault + + +
Type structure + + +
Speed of detection
Isolability × ×
Identifiability × × + ♦
Suitability for FTC × ×
Multiple faults identifia-
bility
− −
Non-linear systems × × +
Robustness − − ♦ ♦
Computational complex-
ity
♦ ♦
Criteria/method Parameter estima-
tion RLS and vari-
ants
Simultaneous parameter
and state estimation
Parity space
Extended
Kalman
filter
Two stage
Kalman
filters
Sensor fault ♦
Actuator fault +
Type structure +
Speed of detection ♦
Isolability
Identifiability ♦
Suitability for FTC ×
Multiple faults identifia-
bility
♦
Non-linear systems +
Robustness + + +
Computational complex-
ity
♦
Notation: ( ) favourable; (♦) less favourable; (×) not favourable; (+) applicable;
(−) not applicable.
its integration with an FTC scheme, its ability to identify multiple faults, robust-
ness to uncertainties and noise, and lastly computational complexity. The com-
parison of the existing quantitative model-based approaches can be seen in Table
2.1. It should be noted that no single method is capable to satisfy all these goals.
Though, it can be concluded that multiple-model based, parameter estimation,
simultaneous state and parameter estimation techniques are more appropriate to
the framework of the active FTC.
2.3. Current research in the active fault-tolerant control
The FDD techniques are required to design the post-fault system model for the
FTC synthesis, even though the nature and seriousness of faults are usually un-
known a priori and the post-fault system dynamics are not always known. The
performance of the FTC system relies on many factors, such as preciseness of the
FDD scheme, the remaining functional actuators, its utilization of hardware and
analytical redundancy, reconfiguration strategy and overall integration of all these
30. 30 2. Fault-tolerant control
components, not mentioning the computational constraints due to the on-line na-
ture of the FTC control. Hence the following sections will briefly deal with a
current research problems in the field of the FTC, though for a wider discussion
on the subject refer to (Zhang et al., 2003a; Zhang et al., 2008).
2.3.1. Integrated design of the FDD and reconfigurable control
In order to construct a functional FTC system, careful analysis of all its subsystems
to guarantee that they can work in a harmony, is of a great significance. To be
more exact, to compute a reasonable control law the reconfigurable controller must
depend on some kind of information from the FDD, whereas the FDD must be
able to provide this information. If there are some discrepancies between what was
expected and generated, the overall system may not function as expected. If the
fault data from the FDD is incorrect or significantly delayed not only the overall
performance will be impaired, but it may result in an instability of the overall
system. At best, the control law obtained from an incorrect fault information will
lead to the undesirable behaviour.
Although, combining different subsystems in the FTC system appears to be
straightforward task in essence, yet this is never the case in reality. The princi-
pal difficulty exists in the fact that each individual subsystem, despite the fact
that it is able to operate perfectly by itself, is almost incapable to provide reliable
and instantaneous decisions or actions for other subsystems. Hence, a seamless
integration of an FDD scheme and a suitable reconfigurable control scheme is of
paramount importance, yet it still poses major challenges in practice, and deserve
further investigation (Zhang et al., 2008). The mitigation of the adverse interplay-
ing between each subsystem is an important research topic (Eberhardt et al., 1999;
Zhang et al., 2006), along with balancing the robustness of the performance during
the nominal operation versus the fault sensitivity at the faulty state of a system
(Wu, 1997). Additionally, as indicated in (Morari et al., 1999) integration of diag-
nosis and performance monitoring with model predictive controllers for industrial
applications remains one of the future research topics. Further discussion about
the issues on the integration of the FDD and reconfigurable control in the FTC
can be found in a (Zhang et al., 2006).
2.3.2. The FTC design for non-linear systems
Most of the systems are non-linear, therefore there is a need for developing the
FTC for non-linear systems. A standard practice to solve a non-linear reconfig-
urable control problem is to synthesize nominal and reconfigurable controller based
on the linearised models around certain operating points (equilibrium points), e.g.,
gain scheduling (Shin et al., 2004), multiple-model (Yen et al., 2003), sliding mode
control (Hess et al., 2003). Nevertheless, most of the work in this matter mostly
considered either fault scenarios, or operating point changes, rarely both. Ex-
cluding particular cases, like aeronautics, where it is straightforward for the gain
scheduling type approaches to take into consideration alterations brought by both
operating conditions variations and the fault induced, it is generally non trivial
31. 2.3. Current research in the active fault-tolerant control 31
to design an active FTC, which can operate efficiently in the entire range of the
general non-linear systems. Additionally, the general method to distinguish alter-
ations induced by fault or by changing operating conditions still remains an open
research subject. Several methods for dealing with non-linear systems have been
developed recently, e.g., backstepping (Zhang et al., 2001), feedback linearisation
(Ochi, 1993), non-linear dynamic inversion (Doman et al., 2002), neural networks
(Ho et al., 2002), Lyapunov methods (Qu et al., 2003) and non-linear regulator
(Bajpai et al., 2002). Yet, effective design methods for tackling the issues intro-
duced by the non-linear FTC systems are not currently available (Zhang et al.,
2008).
2.3.3. Dealing with input, state, and output constraints
The extent of the system control redundancy and the available actuator capabil-
ities are reduced, when an actuator fails. If the nominal performance is still to
be sustained, the remaining actuators will be forced to work beyond their nor-
mal obligations to compensate for the handicaps caused by the failed actuators.
In practice, such a situation is highly undesirable as actuators have a limited
physical capabilities. Such a design may lead to an actuator saturation, or even
cause a further damage. Hence, trade-offs between possible performance and an
attainable actuator capability must be made. This situation is often referred to
as graceful performance degradation. So, the objective of the control reconfigura-
tion or re-allocation is to select a configuration of the control actuators to meet a
specified objective, subject to the saturation constraints. Therefore, dealing with
non-linearity introduced by the constraints of the input and state/output variables
is another challenging issue and an active research topic, specifically in the domain
of the controller design dealing with the actuator amplitude and rate saturations
(Kapila et al., 2002). In practice, there are two classes of approaches to deal with
constraints, one connected with controller design (Mhaskar et al., 2006) and the
other using command (reference) management techniques, e.g., command (refer-
ence) governor (Zhang et al., 2003b), or command shaping and limiting (Eberhardt
et al., 1999). Nevertheless, there are still many open problems in the domain of
constraints of the input and state/output variables, especially in the case of the
multi-input and multi-output systems (Zhang et al., 2008).
2.3.4. Real-time issues in the fault-tolerant control systems
In consequence of the dynamic character of a control system and real-time en-
vironment of the control reconfiguration and the FDD, an active FTC system is
required to detect, identify and accommodate faults as soon as possible. Conse-
quently, every subsystem in active FTC should be able to operate in an on-line
and real-time manner. Hence, at least considering this aspect, active FTCs are
real-time systems. A hard deadline is required for the controller reconfiguration
and its resulting control law in order to avoid bringing the overall system into a
potentially dangerous situation. Also, the FDD scheme is required to supply accu-
rate and the most recent information (including post-fault system models) about
32. 32 2. Fault-tolerant control
the system in real-time, otherwise the successful control system reconfiguration
may not be possible(Zhang et al., 2003a). Additionally, in order to preserve the
desired stability margins with acceptable performance within the allocated time
the reconfiguration mechanism must be able to synthesize the reconfigured con-
troller as quickly as possible, also the design must take into the account constraints
of the control inputs and states/outputs. The compromises between various design
objectives need for executing on-line in real-time as well. These kind of real-time
issues have not been taken care of to an adequate level, in spite of being a critical
issue for any real-time system (Kopetz, 1997; Zhang et al., 2008).
2.3.5. Practical consideration in applications of the FTC systems
Although a considerable work has been done recently in the field of the FTC,
many new promising algorithms and methods have been constructed in different
application areas, that could possibly provide improvements in a current FTC
architecture. From a theoretical standpoint, there is a need for developing uni-
fied, systematic theory and design techniques. Whereas in practical applications
the important topics for research include: real-time fault propagation and recon-
figurability analysis, efficient redundancy management, reconfigurable controller
synthesis with the deliberation of some practical problems, integration of the FDD
and the reconfigurable control, and likewise practical implementation connected
to the software structure, redundant hardware and fault-tolerant communication
networks. With a rapid development in microelectronics and mechatronics tech-
nologies, intelligent actuators and sensors possessing self-diagnostic attributes are
accessible. Thus, the use of such components will have a considerable impact
on the overall design and implementation of the FTC and this extra layer of di-
agnostic should be fully exploited in future FTC systems. At the same time, a
rapid progress of the control systems development, from a simple loop controllers
to the distributed control systems discloses the limitations and inadequacy of the
existing FTC systems. Hence, new techniques for integrated designs of the entire
FTC systems along with the related implementation platforms (i.e., hardware,
software, computing platforms, and communication protocols) are pressingly de-
manded (Zhang et al., 2008).
To summarise, the FTC is a complex interdisciplinary research subject, cov-
ering not only control engineering disciplines, such as modelling and identification,
but also engine applied statistics and mathematics, stochastic system theory, re-
liability and risk analysis, computing, communication, control, signal processing,
in addition to hardware and software implementation issues. Indeed, a functional
active FTC system must be considered as whole, including not only the recon-
figurable controller and the connected FDD schemes, but also methods linked to
the real-time computing, communication, reconfigurable hardware/software im-
plementation and redundancy management (Zhang et al., 2008).
33. 2.4. Concluding remarks 33
2.4. Concluding remarks
In this chapter, the elementary fundamentals for fault-tolerant control were ex-
plained. At first, the definition of fault and its effects on the system were provided,
followed by the classification of faults. Later, basic properties and requirements of
a fault tolerance were explained. Finally, the FTC systems and their classification
into passive and active ones were introduced. This book is concerned only with
the active ones, capable of adapting themselves to a current fault-state thanks to
their connection to an FDD scheme. For each of the schemes, i.e., reconfigurable
control design and the FDD design, the currently used techniques were provided
and classified. Finally, this chapter is finalised by giving information about most
pressing and open research issues of the currently existing active fault-tolerant
control systems.
34. Chapter 3
TAKAGI-SUGENO FUZZY SYSTEMS
“So far as the laws of mathematics refer to reality, they are not certain.
And so far as they are certain, they do not refer to reality.”
Albert Einstein
3.1. Fuzzy logic
Fuzzy logic is a superset of conventional (Boolean) logic that was extended to
handle the concept of partial truth – truth values between “completely true” and
“completely false”. It was introduced by Dr. Lotfi Zadeh (Zadeh, 1965) as a means
to model the vagueness of a natural language. At first, it encountered scepticism,
and it took a long time until it was finally accepted. Nowadays, fuzzy logic systems
are widespread, and has found numerous applications, especially in the domain of
system control, identification as well as modern computer science.
Just as there is a strong relationship between Boolean logic and the concept
of a subset, there is a similar strong relationship between fuzzy logic and fuzzy
subset theory.
Definition 3.1. Classical set F is a set of ordered pairs
F = {(w, IF (w)) | ∀w ∈ W}, (3.1)
defined by indicator function IF (w) ∈ {0, 1}.
The value zero of indicator function is used to represent non-membership, and
the value one is used to represent membership.
But, for example, if the set of young people F is described, as a crisp interval
of people younger than, say 20 years, i.e., F = [0, 20]. Then the question arises:
why is somebody on his 20th birthday young and right on the next day not young?
Obviously, this is a structural problem, if the upper bound of the range is moved
from 20 to an arbitrary point the same question can be posed.
A more natural way to construct the set F would be to relax the strict sep-
aration between young and not young. This can be accomplished by allowing not
only the crisp decision YES he/she is in the set of young people or NO he/she is
not in the set of young people but more flexible phrases like: Well, he/she belongs
a little bit more to the set of young people or NO, he/she belongs nearly not to
35. 3.1. Fuzzy logic 35
(a) Crisp set (b) Fuzzy set A (c) Fuzzy set B
(d) Intersection of A and B (e) Union of A and B (f) Complement of A
Figure 3.1: Fuzzy sets and operations on them, here µ(·) is a membership function
(Tizhoosh, 2004).
the set of young people. Thus, the concept of fuzzy set is introduced, and young
is described as a linguistic variable1
, which represents humans cognitive category
of “age”.
Definition 3.2. A fuzzy set F is a set of ordered pairs
F = {(w, µF (w)) | ∀w ∈ W}, (3.2)
defined by membership function 0 µF (x) 1.
A membership function provides a measure of the degree of similarity of an
element in W to the fuzzy subset. In practice, the terms “membership function”
and fuzzy subset get used interchangeably.
Now, an idea of what fuzzy sets are and basic operations on fuzzy sets can
be introduced. In fuzzy logic, union, intersection and complement are defined
in terms of membership functions and are motivated by their crisp counterparts.
Let fuzzy sets F1 and F2 be described by their membership functions µF1 (w) and
µF2 (w). One definition of fuzzy intersection leads to the membership function
µF1∩F2
(w) = min[µF1
(w), µF2
(w)] ∀w ∈ W (3.3)
and one definition of fuzzy union leads to the membership function
µF1∪F2
(w) = max[µF1
(w), µF2
(w)] ∀w ∈ W. (3.4)
1However, Mendel (2003) demonstrated that to use a (type 1) fuzzy set to model a word is
scientifically incorrect, because word is uncertain whereas a fuzzy set is certain. To do so a type
2 fuzzy set is required, for an example the reader is referred to (Dziekan et al., 2007).
36. 36 3. Takagi-Sugeno fuzzy systems
Table 3.1: Nomenclature for computational complexity of general fuzzy logic.
Symbol Description
r The number of inputs
p The number of input fuzzy sets
NID The number of of discretization of input universe of discourse
M The number of rules
NOF The number of output fuzzy sets
NOD The number of discretization of output universe of discourse
m The number of outputs
Table 3.2: Number of operations in fuzzy logic controller (Kim et al., 2000).
Method Operations
Fuzzification Triangular (59 + 31p)r
Non-specific (70p + 29pNID + 8)r
Inference Minimum inference (63NOD + 37r + 19)M + 6
Product inference (88NOD + 37r + 20)M + 6
Defuzzification Mean of maximum (25NOD + 5)M + 15
Center of Gravity (39NOD + 5)M + 15
Additionally, the membership function for fuzzy complement is
µ ¯F1
(w) = 1 − µF1
(w) ∀w ∈ W. (3.5)
The “max” and “min” operators are not the only ones that could have been
chosen to model fuzzy union and fuzzy intersection. Other operators, which have
an axiomatic basis, can be used—t-conorm operator for fuzzy union (also known as
an s-norm, and denoted S), e.g., bounded sum, drastic sum, and t-norm operators
for fuzzy intersection (denoted T ), e.g., product, drastic product (Mendel, 1995).
Examples of fuzzy sets, and operations on them are shown in Fig. 3.1.
3.1.1. Computational complexity of the general fuzzy logic control
In (Kim et al., 2000) the computational complexity of general fuzzy logic were con-
sidered. This paper analysed the number of operations and parameters of general
fuzzy logic control algorithms. Also limitations of loop controllers to implement
the fuzzy logic control were investigated in terms of the computation time and
the required memory. Generally speaking, control algorithms for a loop controller
should have a small number of tuning parameters and short computation time due
to the limited memory and slow processors. There are many tuning parameters
in the membership functions and control rules of general fuzzy logic control. This
37. 3.2. Discrete-time Takagi-Sugeno fuzzy systems 37
results in a long computation time since it performs fuzzification, inference, and
defuzzification processes in determining control inputs. Thus, it is difficult for
control inputs of general fuzzy logic control to be computed within the sampling
time of a loop controller. Hence, the simplifications method of fuzzy logic for loop
controllers were also presented, for the details the reader is refereed to there (Kim
et al., 2000).
The results of analysis of general fuzzy logic can be seen in Table 3.2, with
nomenclature presented in Table 3.1. The analysis were performed for a general
fuzzy logic control program that is implemented in the assembly language of an
MC68000 microprocessor. Addition, subtraction, multiplication, and division are
considered as basic operations when the number of operations is calculated. As a
result of the program analysis, about 50% of the operations are move operations,
and the remaining 50% are computing operations addition, subtraction, multipli-
cation, and division, and program flow control operations such as branch and jump
(Kim et al., 2000).
3.2. Discrete-time Takagi-Sugeno fuzzy systems
A non-linear dynamic system can be described in a simple way by a Takagi-Sugeno
fuzzy model, being a branch of general fuzzy framework, which uses series of locally
linearised models from the non-linear system, parameter identification of an a
priori given structure or transformation of a non-linear model using the non-linear
sector approach (see, e.g., (Korbicz et al., 2004; Takagi et al., 1985; Tanaka et al.,
2001; Tatjewski, 2007)). According to this model, a non-linear dynamic systems
can be linearised around a number of operating points. Each of these linear models
represents the local system behaviour around the operating point. Thus, a fuzzy
fusion of all linear model outputs describes the global system behaviour.
Let us consider a non-linear model affine in control given by the expression:
xk+1 = g1(wk)xk + g2(wk)uk,
yk = g3(wk)xk,
(3.6)
with gi(·), i = 1, 2, 3 being non-linear functions, xk the state vector, uk the input
vector, yk the output vector and wk a vector supposed measurable.
A methodical way to deal with (3.6) is the Takagi-Sugeno modelling (Takagi
et al., 1985). Depending on the ‘point of view’, two approaches are available leading
to a unique framework (Guerra et al., 2009).
Historically speaking, the first approach stems from the fuzzy rule-based con-
trol area and their property of being the universal approximator (Castro et al.,
1996). In this class of fuzzy modelling, T-S fuzzy model act as an approximation
of (3.6), thus allowing to describe non-linear dynamical system by a set of Linear
Time Invariant (LTI) models interconnected with non-linear functions. Each of
LTI models, are then associated by rule at the consequent part of a weighting
function established from the premises. It has a base of M rules, each having p
38. 38 3. Takagi-Sugeno fuzzy systems
antecedents, where ith rule is expressed as (viewed in a state-space representation):
Ri
: IF w1
k is Fi
1 and . . . and wp
k is Fi
p,
THEN
xi
k+1 = Ai
xi
k + Bi
uk,
yi
k = Ci
xi
k,
(3.7)
in which xi
k ∈ Rn
stands for the state, yi
k ∈ Rm
is the output (note that each model
had an individual state and output), and uk ∈ Rr
denotes the nominal control in-
put, also i = 1, . . . , M, Fi
j (j = 1, . . . , p) are fuzzy sets and wk =[w1
k, w2
k, . . . , wp
k] is
a known vector of premise variables (Takagi et al., 1985). In general manner, these
models are obtained via an identification procedure, according to the universal
approximation property (Gasso et al., 2001; Margaliot et al., 2003).
Whereas, the second point of view uses directly the non-linear expression
of the model and can be expressed in a rule-based form, although not strictly
equivalent to (3.7), where the ith rule is described as
Ri
: IF w1
k is Fi
1 and . . . and wp
k is Fi
p,
THEN
xk+1 = Ai
xk + Bi
uk,
yk = Ci
xk.
(3.8)
Given a pair of (wk, uk) and a product inference engine, the final output of
the normalized T-S fuzzy model can be inferred as:
xk+1 =
M
i=1 hi(wk)[Ai
xk + Bi
uk],
yk =
M
i=1 hi(wk)Ci
xk,
(3.9)
where hi(wk) are normalized rule firing strengths (non-linear functions of wk)
defined as
hi(wk) =
T p
j=1µF i
j
(wj
k)
M
i=1(T p
j=1µF i
j
(wj
k))
, (3.10)
and T denotes a t-norm (e.g., product). The term µF i
j
(wj
k) is the grade of mem-
bership of the premise variable wj
k. Moreover, the rule firing strengths hi(wk)
(i = 1, . . . , M) satisfy the following constraints (the convex sum property)
M
i=1 hi(wk) = 1,
0 hi(wk) 1, ∀i = 1, . . . , M.
(3.11)
Hence, (3.9) also corresponds to a quasi-Linear Parameter Varying (LPV) form
(Lu et al., 2000).
Nevertheless, the choice ultimately depends on the way to obtain the T-S
model.
39. 3.3. Construction of the T-S fuzzy model 39
Nonlinear system
Identification using
input-output data
Physical model
Takagi-Sugeno
fuzzy model
Fuzzy controller
Parallel distributed
compensation (PDC)
Figure 3.2: Model-based fuzzy control design.
3.3. Construction of the T-S fuzzy model
Figure 3.2 illustrates the model-based fuzzy control design approach discussed in
this thesis. In order to design a fuzzy controller, a Takagi-Sugeno fuzzy model
for a non-linear system is needed. Hence the construction of a fuzzy model is of
paramount importance and basic procedure in this approach. In general, there are
two approaches for constructing fuzzy models:
• Identification (fuzzy modelling) using input-output data
• Derivation from given non-linear system equations
There has been an extensive literature on fuzzy modelling using input-output
data following Takagi’s, Sugeno’s, and Kang’s excellent work (Sugeno et al., 1988).
The procedure essentially consists of two parts: structure identification and pa-
rameter identification. The identification approach to fuzzy modelling is suitable
for plants that are unable or too difficult to be represented by analytical and/or
physical models (Tanaka et al., 2001). A very interesting paper on the experiment
design for identification can be found here (Johansen et al., 2000).
In reality, input-output identification methods allow finding a model in the
form (3.7) or an equivalent one using linear/non-linear estimation (Gasso et al.,
2001; Margaliot et al., 2003) or clustering methods (Espinosa et al., 2004; Li et
al., 2009). The identification problem also demands the selection of the premise
variables. The premise variables are the variables that govern the changes of
dynamical regime. Practically speaking, to guarantee a smooth behaviour of the
40. 40 3. Takagi-Sugeno fuzzy systems
(a) (b)
Figure 3.3: (a) Global sector non-linearity. (b) Local sector non-linearity.
model the premise variables are selected to be slowly varying. The combination
of the local models demands consistency among them. Unfortunately, very few of
them are useful for control (Sala et al., 2005) mainly because the conclusion parts
of the rules do not share the same state vector.
However, one way to guarantee the consistency of rules using state-space
models is to identify all the local models with the same order and convert all of
them to the so-called observer canonical form. In this way all the states of the
local model will be consistent and in the form of (3.7), whereas their evolution
will be perfectly synchronized (Espinosa et al., 2004). For the example of such
modelling see Sec. 3.4.
On the other hand, non-linear dynamic models for physical systems can be
readily obtained by, for example, the the Newton-Euler method and Lagrange
method (Tanaka et al., 2001). In such cases, the second approach, which derives a
fuzzy model, in the form (3.9), from given non-linear dynamical models, is more
appropriate. This approach utilizes the concept of “sector non-linearity”, “local
approximation”, or a combination of them to construct fuzzy models. The latter
is based on the linearisation around several set points of the non-linear plant.
In this case, the resulting model is only an approximation and the membership
functions µF i
j
(wj
k) are chosen as triangular or sigmoid functions (Tanaka et al.,
2001). Whereas, the former technique represents exactly the considered non-linear
model in a compact set of state variables. They are composed of linear models
blended together with non-linear functions. The following section focuses on this
approach.
41. 3.3. Construction of the T-S fuzzy model 41
3.3.1. Sector non-linearity
Sector non-linearity is based on the following idea (Tanaka et al., 2001). Consider a
simple non-linear system ˙x(t) = f(x(t)), where f(0) = 0. The aim is to find global
sector such that ˙x(t) = f(x(t)) ∈ [a1, a2]x(t), i.e., a1x(t) f(x(t)) a2x(t). Fig.
3.3(a) illustrates the (global) sector non-linearity approach. This approach guaran-
tees an exact fuzzy model construction. Nevertheless, it is sometimes troublesome
to find global sectors for general non-linear systems. In this case, local sector non-
linearity can be considered. This is reasonable as variables of physical systems
are always bounded. Fig. 3.3(b) shows the local sector non-linearity, where two
lines become the local sectors under −d < x(t) < d. The fuzzy model exactly
represents the non-linear system in the “local” region, that is, −d < x(t) < d.
In a formal way, the sector non-linearity approach can be described as follows
(Guerra et al., 2009). Consider the non-linear model (3.6) and a compact set of
wk ∈ C ⊂ Rw
on which they are bounded. If nli(·) ∈ [nli, nli], i ∈ {1, . . . , q} are
the non-linear functions of gi(·), then in the set of C ⊂ Rw
: nli(·) = nli · µi
0(·) +
nli · µi
1(·) with:
µi
0(·) = (nli − nli(·))/(nli − nli),
µi
1(·) = (nli(·) − nli)/(nli − nli).
(3.12)
In this case the weighted functions share the following properties µi
0(·) 0, µi
1(·)
0, µi
0(·)+µi
1(·) = 1. Finally, the hi(wk), i = 1, . . . , M of the T-S model are defined
as:
h1+i0+i1×2+···+iq−1×2q−1 (·) =
q
j=1
µj
ij
(·). (3.13)
Therefore, the membership functions are a combination of the non-linear functions
coming from the model. The convex sum property straightforwardly holds.
Remark 3.1. The number of linear models (2q
) exponentially increases with the
number q of non-linear functions of (3.6) (Tanaka et al., 2001).
Thus, prior to applying the sector non-linearity approach, it is often a good
practice to simplify the original non-linear model as much as possible. This step is
important for practical applications because it always leads to the reduction of the
number of model rules, which reduces the effort for analysis and design of control
systems.
The following example illustrates the concrete steps to construct the fuzzy
model (Guerra et al., 2009).
Example 3.1. Let us consider the following model with αk ∈ [α, α]:
xk+1 =
1 cos(x1,k)
αk −1
xk +
cos(x1,k)
2
uk. (3.14)
42. 42 3. Takagi-Sugeno fuzzy systems
By using w = [cos(x1,k) αk]T
as premise vector, a T-S representation of (3.14)
can be obtained. Hence, directly applying the sector non-linearity approach, the
following membership functions are defined:
µ1
0(cos(x1,k)) =
1 − cos(x1,k)
2
, µ1
1(cos(x1,k)) =
cos(x1,k) + 1
2
,
µ2
0(αk) =
α − αk
α − α
, µ2
1(αk) =
αk − α
α − α
.
Afterwards using (3.13):
h1(wk) = µ1
0(cos(x1,k)) · µ2
0(αk), h2(wk) = µ1
1(cos(x1,k)) · µ2
0(αk),
h3(wk) = µ1
0(cos(x1,k)) · µ2
1(αk), h4(wk) = µ1
1(cos(x1,k)) · µ2
1(αk).
Hence, a possible representation of (3.14) with a T-S model is:
xk+1 =
4
i=1
hi(wk)[Ai
xk + Bi
uk],
where
A1
=
1 1
α −1
, A2
=
1 −1
α −1
,
A3
=
1 1
α −1
, A4
=
1 −1
α −1
,
B1
=B3
=
1
2
, B2
=B4
=
−1
2
.
It is an exact representation of (3.14) in the compact set αk ∈ [α, α].
3.4. Example of the fuzzy model identification — a T-S fuzzy
model of a tunnel furnace
The selected non-linear system is a laboratory model of a tunnel furnace. The
considered tunnel furnace is a laboratory counterpart of the real industrial tunnel
furnaces, which can be applied in the food industry or production of ceramic among
others. The furnace is equipped with three electric heaters and four temperature
sensors. Details about hardware setup can be found in Appendix A on page 135.
The considered system is a distributed parameter system (i.e., a system whose
state space is infinite-dimensional), thus any resulting model from input-output
data will be at best an approximation. Hence, to achieve a good approximation,
43. 3.4. Example of the fuzzy model identification — a tunnel furnace 43
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
50
100
150
200
250
300
Time [s]
Temperature
T1 model
T1 kiln
(a)
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
50
100
150
200
250
300
350
Time [s]
Temperature
T2 model
T2 kiln
(b)
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
50
100
150
200
250
300
Time [s]
Temperature
T3 model
T3 kiln
(c)
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
40
60
80
100
120
140
160
180
200
220
240
260
Time [s]
Temperature
T4 model
T4 kiln
(d)
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time [s]
Controlsignal
u1
(e)
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time [s]
Controlsignal
u2
(f)
Figure 3.4: Tunnel furnace modelling results: (a)-(d) trajectories of modelled
outputs and real outputs, (e) first control trajectory, (f) second control trajectory.
44. 44 3. Takagi-Sugeno fuzzy systems
0 2000 4000 6000 8000 10000 12000 14000 16000 18000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time [s]
Controlsignal
u3
(g)
100 150 200 250 300
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Premise variable
Membershipvalue
F1
F2
F3
F4
F5
(h)
Figure 3.4: Tunnel furnace modelling results cont.: (g) third control trajectory,
(h) fuzzy sets used in Takagi-Sugeno model.
the optimal experiment design is of paramount importance and is often an iter-
ative process. An interesting paper dealing specifically with identification of T-S
fuzzy models can be found in (Johansen et al., 2000) and its recommendations are
partially repeated in a book dealing with fuzzy systems (Espinosa et al., 2004).
In non-linear system identification, both the amplitude and frequency contents
of the input signals are of major importance. Thus, for identifying T-S fuzzy
models containing both equilibrium and off-equilibrium local affine models, it is
recommended that input signals should be designed according to the following
guidelines, to be useful if the model is intended for control design (Johansen et al.,
2000):
• The system in question, should be brought through a sequence of equilibria
that includes the equilibria of the local models. At each equilibrium the
system should be excited by super-positioned Pseudorandom, Binary Signal
(PRBS), i.e., a pseudorandom, usually binary signal added to the original
input signal. The PRBS signals should have a frequency content that covers
an interval from the inverse rise time to above the bandwidth of the closed-
loop system.
• For each off-equilibrium local model, several transient trajectories should be
generated. The corresponding input signals should contain both large ampli-
tude steps and perturbations, so both the trend and perturbation dynamics
of the off-equilibrium local models could be determined. Also the frequency
contents should typically be higher compared to the frequency content of the
equilibrium data to prevent the system from settling at some equilibrium.
Of course, these are general guidelines, so in practical applications there will
be some constraints that will often limit the number of transitions, frequency con-
tent, amplitudes, and length of the experiment. Also depending strongly on the
45. 3.4. Example of the fuzzy model identification — a tunnel furnace 45
application, the requirements in terms of accuracy of the off-equilibrium local mod-
els should be considered. Sometimes, equilibrium local models can be extrapolated
into transient operating regions without significant loss of accuracy.
Other excitation signals are the multisine signals with variable frequency and
the swept sinus with random frequencies. These signals are frequently used in the
identification of mechanical systems (Espinosa et al., 2004).
Thus in order to identify a model for the tunnel furnace, the input signals
were defined as follows:
• Five operating points were considered, at 20%, 40%, 60%, 80% and 100% of
maximum power of heaters, respectively.
• Heating phase: at each operating point for the first 1800th seconds, the
constant input signal values were used (to heat the furnace to a desired
temperature).
• Perturbation phase: After the heating phase for the next 1620th seconds, for
each of the input signal individually, and independently a perturbation signal
were applied as follows. The perturbation was an uniformly distributed
pseudo-random signal in the range of [−10%, 10%] of maximum power, super-
positioned on the signal generated in the heating phase (for 100% operating
point the range was [−10%, 0%]). Whereas, each signal duration was chosen
at random in the range [5,15] seconds. Afterwards a new value and duration
were generated.
• Short cooling phase: After the perturbation phase, for a short time period
of 120th second the heaters were disabled.
• Short heating phase: After the cooling phase, for sixty seconds a maximum
power for all the heaters was applied.
• Thus, a cycle for each operating point lasted 3600th seconds, and after five
full cycles, for the remaining time of simulation, an uniformly distributed
pseudo-random signal in the range of [0%, 100%] of maximum power and
duration in the range [5,15] seconds were applied for each heater, individually.
Thus giving a total duration of experiment equal 19000th seconds.
Afterwards, the resulting experimental data were cut into appropriate seg-
ments for each of the operating points. From each input-output data of the
operating point a local system-state model were built, using subspace methods.
Subspace methods originate in a mix between system theory, geometry and nu-
merical linear algebra. These subspace methods are successfully used for model
identification for industrial processes (Favoreel et al., 2000). The N4SID (Ljung,
1999) algorithm were selected for the task of model identification, with the order
of the models equal four, as greater orders provided little information based on
the singular values of the Hankel matrices of the impulse response for different
orders (Ljung, 1999). Subsequently, to guarantee the consistency of fuzzy rules,
all of the resulting local models were converted into the observer canonical form
