![Chapter 1
Introduction
1.1 Background
Since the 1940s, automatic control technology has been widely used in practical
engineering fields, such as robotics, chemical industry, aerospace, and nuclear power
plants, and continues to play a pivotal role in the new wave of industrialization in
the twenty-first century. As real systems grow in size and complexity, so does the
cost of control systems. Due to the complexity of the structure of the large-scale
control system and long-time working under high power and high load condition,
some equipments in the system, such as sensors and actuators, will inevitably age over
time, and may even evolve into failures. When failure occurs, the control performance
of the system will be dropped. In some cases even stability cannot be guaranteed,
which results in huge property damage and casualties. For example, in 2003, the U.S.
space shuttle Columbia exploded on its way back due to a defect in the installation of
the foam material on the surface of the external fuel tank, causing billions of dollars
in property damage and killing all seven astronauts. In 1986, the explosion of Unit
4 of the Chernobyl nuclear power plant in the former Soviet Union resulted in the
leakage of a large amount of radioactive material, which eventually killed more than
2,000 people and caused more than 3.2 million people to suffer serious consequences
of nuclear radiation hazards. Similar examples are common. Because a small fault
in the actual system may cause huge consequences, the research on the safety and
reliability of complex dynamic systems has become an extremely important topic.
In this context, fault diagnosis and fault-tolerant control technology [1–10] have
gradually developed and became one of the popular areas in the control field.
© Springer Nature Singapore Pte Ltd. 2022
J.-W. Zhu et al., Fault Estimation for Network Systems via Intermediate Estimator,
https://doi.org/10.1007/978-981-19-6321-6_1
1](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-17-320.jpg)
![2 1 Introduction
1.2 Basic Concepts in Fault Diagnosis
The International Federation of Automatic Controls (IFAC) defines a fault as a devi-
ation from the normal range of at least one observable or computable important
variable or characteristic in a control system. Faults can be roughly classified as
follows according to different angles, that is, time characteristics or occurrence loca-
tions: From the perspective of time characteristics, faults can be divided into the
following types [11–13]:
• Abrupt fault refers to fault in which some parameter values in the system suddenly
deviate greatly.
• Incipient fault refers to the fault in which some system parameters change slowly
with the change of time and environment, also known as soft fault.
• The AO-based FE have attracted much attention, but its existence conditions have
not been given explicitly yet.
• Intermittent fault refers to the fault caused by the aging of components, insufficient
error tolerance, or poor contact.
According to the analysis of the occurrence location, the faults can be divided
into the following categories:
• Component failure means that some components in the controlled object of the
system are abnormal, which makes the operation of the whole system to deviate,
resulting in the failure to accomplish the systems control goal precisely.
• Sensor fault refers to the fault that the sensor is stuck or has a certain degree of
deviation and cannot accurately measure the internal signal of the system.
• Actuator fault refers to the fault that the actuator cannot execute the control com-
mand correctly due to changes such as stuck, offset, or interruption.
Fault diagnosis generally refers to the following three directions:
• Fault detection: Determine whether a fault has occurred.
• Fault isolation: Determine the specific location or kind of failure.
• Fault identification: Determine the size and dynamics of the fault.
For the fault diagnosis system, according to different actual needs, we can define a
variety of performance indexes. For example, the rapidity of diagnosis, the sensitivity
and robustness of the fault diagnosis system, the rate of false positives and false
negatives of faults, etc.
In this book, we mainly study the problem of FE for network systems including
networked single system, MASs, and large-scale interconnected system, the goal of
FE is to obtain the exact information of the fault online, such as the size and shape
of the fault itself. Generally, it belongs to the direction of fault identification by
definition.](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-18-320.jpg)
![1.3 FE Method and Research Status 3
1.3 FE Method and Research Status
Unlike fault detection and isolation techniques, which aim to determine when and
where the faults occur, FE techniques seek to obtain more specific fault dynamics
online. Compared with fault detection and isolation, FE can obtain more fault infor-
mation. However, its requirements are usually stricter, and there are relatively more
constraints. In terms of technical form, it is also relatively monotonous. Generally,
FE is achieved by designing an observer. So far, a variety of different fault observer
design method have appeared, and each method is based on different ideas, so they
have different characteristics. The following is a brief overview of the current FE
methods.
• Sliding Mode Observer (SMO)
The SMO method starts from the idea of variable structure control [14–20]. A
typical literature on SMO design is Edwards and Spurgeon’s paper on the problem
of FE in Automatica [184], which is published in 2000. The basic idea of SMO
is the notion of equivalent output injection (equivalent output injection). The idea
of reconstructing the fault signal is based on the dynamic characteristics of the
sliding mode surface of the observer, and it constructs the estimation of the fault
signal by using the estimation error of the system output. Based on this theory,
SMOs have been widely used in the FE of various systems in the past decade [21–
24, 193, 195]. The technical feature of the SMO is to ensure that the estimation
error dynamics have better robustness against system uncertainties or external
disturbances by utilizing the design freedom of the nonlinear term of the observer
gain. However, SMO heavily depends on output estimation error information to
realize sliding mode dynamics. In its design, the strict observer matching condition
must be satisfied, and the upper bound of the fault itself and the first derivative
of the fault must be known. Moreover, the estimation performance of SMO often
suffers from chattering problems [25–30]. These constraints limit the application
scope of SMO to some extent.
• Adaptive Observer (AO)
AO achieves FE based on the idea of online compensation of FE error [31–33].
Technically, a strict equality constraint is introduced, then the observer is designed
by using the solution of the constraints of the formula to realize the compensation
of the FE error effect in the error system, thereby ensuring the eventual uniformly
bounded stability of the error system. FE methods based on AO mainly rely on
tuning parameters to ensure FE performance. So far, there are few related research
results. Authors of [186] designed a fault-tolerant control method based on AO for
Lipschitzian nonlinear systems with process faults. The result in [192] presents a
fault-tolerant control strategy based on AO for linear systems with actuator deflec-
tion faults. In general, the advantages of the AO method are that the observer
structure design is relatively simple, the applicability is strong, and the FE perfor-
mance is good. Through the stability analysis, the exponential convergence of the
error system state can be strictly proved and a clear upper bound of FE error can be
obtained. In addition, this kind of method can independently set the convergence](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-19-320.jpg)
![4 1 Introduction
rate of FE by adjusting parameters. However, it should be pointed out that in order
to ensure the solvability of the equality constraints, AO must satisfy the strictly
positive real (SPR) condition [34–36, 190], which is a sufficient and unnecessary
condition of the observer matching condition, so it is much more conservative
than the observer matching condition. Most practical systems cannot meet the
SPR condition, so the application potential of the AO is greatly restricted.
• Unknown Input Observer (UIO)
The idea of using UIO [37–41] for fault diagnosis of systems with uncertainty was
firstly proposed by Professor Wantanabe in 1982 [42], and the basic idea of the FE
method based on UIO is to use the redundant degrees of freedom of Luenberger
observer, so that the FE error can be decoupled from the disturbance or modeling
uncertainty, then it can effectively realize the robustness of the FE. Authors of [43]
proposed a class of UIO methods for linear systems with process faults, and gave
design conditions to ensure the stability of the error system. In general, UIOs are
also required to satisfy the observer matching condition [44–47].
• Robust Observer
The basic idea of the robust observer [48–53] is to combine Luenberger observer’s
design structure with the traditional robust performance optimization strategy, and
to suppress the effect of disturbances and fault derivative on estimation error by
introducing performance indexes, while its observer gain needs to be obtained by
solving a series of LMIs with performance optimization metrics. In [54], a class
of one-step robust observer design is proposed for continuous-time T-S fuzzy sys-
tems. On this basis, the approach in [171] improves the original one-step robust
observer for T-S fuzzy systems with time delay, and proposes a K-step robust
observer method. The step-iterative observer design further suppresses the influ-
ence of the fault derivative on the FE error, it also gives the design conditions to
ensure the stability of the error system. In addition, authors of [55] propose a robust
observer design for discrete-time systems. However, the above robust observers
have a common disadvantage, that is, the upper bound of the FE error often cannot
be obtained through the theoretical analysis of this method [56–62].
• High-gain Observer
The high-gain observer is widely used in various nonlinear systems. It makes the
observer robust to the uncertainty of system modeling by specially designing the
gain of the observer. In general, high-gain observer can only be used for estimation
of constant fault parameters. For the work in this area, readers can refer to [63–68].
The disadvantage of high-gain observers is that these methods require the system
matrix to satisfy an upper triangular structure, which cannot be met for most linear
systems. Apart from the above FE methods, there are also other techniques, such
as singular observer [187–189], proportional-integral observer [69–71], iterative
learning observer [72, 73], etc., which will not be described in detail here.](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-20-320.jpg)
![1.4 Research Status for FTC 5
1.4 Research Status for FTC
So far, FE technique is mainly further used for FTC. The idea of FTC started in the
1970s [74–77, 198], and the meaning of fault tolerance is to tolerate the effect of
faults in control systems. The necessary condition for a system to have fault tolerance
is that there is control redundancy in the system. For example, the fault tolerance
of the actuator requires redundancy of the actuator, the fault tolerance of the sensor
requires the redundancy of the sensor measurement, and the fault tolerance of the
components requires the presence of function redundancy of the components. The
design concept of fault-tolerant control is to use redundancy to tolerate faults. In
1985, Eterno divided FTC methods into two categories: passive fault tolerant and
active fault tolerant. Generally speaking, passive FTC [78–80] often regards faults
as a special form of disturbance or system uncertainty, and then uses robust control
technology to suppress the effect of faults on the system. The structure of its controller
is fixed, so passive FTC does not need to obtain any fault information online. Active
FTC system identifies fault parameters online by designing algorithms or attempts to
obtain fault information by fault diagnosis technique, and then changes the controller
structure to compensate the effect of faults. In contrast, active FTC has flexible design
ideas, and there are many methods so far. From the perspective of design structure,
activeFTCmethodscanbedividedintocontrolratereschedulingmethodsandcontrol
rate reconfiguration design methods. From the analysis of information sources, they
can be divided into state feedback FTC methods and output feedback FTC methods.
Although there have been a lot of research results in FTC, it should be pointed out
that most of the methods are based on the idea of state feedback control [81–84],
and in practical systems, the full state information cannot be obtained, so it is often
necessary to use the output feedback FTC strategy [85–89] to design the controller.
This book also studies the FTC problem based on output feedback. Below we briefly
summarize the current main methods in this area.
• Adaptive Output Feedback FTC
The adaptive output feedback control method mainly uses the output information
to design the adaptive updating law rate to achieve online compensation FTC [90–
92]. When the physical process failure occurs and causes the system to change
dynamically, an adaptive mechanism is triggered to mitigate the impact of the fail-
ure. [93] proposes an adaptive output feedback control method for linear systems
with actuator deflection, stuck and interrupted faults. On this basis, authors of
[94] further consider the design of a quantized output feedback adaptive controller
with actuator faults, whose advantage is that although it requires that there exists
the upper bound of the fault, it does not need to know the specific magnitude of
the upper bound of the fault, and at the same time, it can fully compensate the
influence of the fault and ensure that the system state is asymptotically stable. It
is necessary to meet SPR conditions or have certain requirements on the system
structure, such as strict feedback formal system [95–98], these constraints limit
the application of adaptive output feedback method in practical engineering.](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-21-320.jpg)
![6 1 Introduction
• Sliding mode output feedback FTC
The main feature of the sliding mode output feedback FTC method is that in
order to overcome the influence of the fault and let the system dynamically reach
the sliding mode surface, it needs to design the nonlinear feedback gain term
of the controller [99–101], and the design of this item needs to use the output
information and rely on a priori known information of the upper bound of the
fault. The result [102] considers the deflection of the actuator and combines with
the integral sliding mode surface design and the control distribution method, when
the fault occurs, the influence of the fault is overcome by redistributing the control
signal and the sliding mode control is realized. In [103], a sliding mode output
feedback FTC method is proposed for the output tracking control problem of
linear uncertain systems. Similar to sliding mode state feedback control, sliding
mode output feedback control has strong robustness, but in order to overcome the
influence of external disturbances or uncertainties to achieve better robustness, it is
often necessary to introduce extremely strict restrictions on system structure [104,
105], for instance, faults and disturbances must appear in the output channel.
• Observer-based output feedback FTC
The observer-based output feedback FTC essentially integrates the design of FE
and fault-tolerant control strategies. The design idea is to first estimate the fault
by designing an appropriate observer, and then use the FE to design a controller to
compensate the fault online. At present, there are a large number of FTC methods
based on various major FE observers, such as the FTC method for singular SMOs
for stochastic systems [106–109], AO-based FTC method, [186, 192], the robust-
observer-based FTC scheme [54, 171], and so on. Different from the ideas of
adaptive control and sliding mode control, in general, the observer-based FTC
method has lower requirements on the system structure, but often cannot fully
Fig. 1.1 The flow chart of FE-based FTC](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-22-320.jpg)
![you cannot think that you are pleased when you are; and, being
bereft of your reasoning faculties, you cannot even have the power
to reckon that you will be pleased in future. You must live the life of
an oyster, or of some other of those living creatures, whose home is
the seas and whose souls are concealed in shelly bodies. Is all this
so, or can we think otherwise than this?
Protarchus. How can we?
Socrates. Well, then, can we think such a life desirable?
Protarchus. Socrates, your reasoning has left me utterly dumb.’
Socrates, we see, persuades Protarchus that Hedonism is absurd.
If we are really going to maintain that pleasure alone is good as an
end, we must maintain that it is good, whether we are conscious of
it or not. We must declare it reasonable to take as our ideal (an
unattainable ideal it may be) that we should be as happy as
possible, even on condition that we never know and never can know
that we are happy. We must be willing to sell in exchange for the
mere happiness every vestige of knowledge, both in ourselves and in
others, both of happiness itself and of every other thing. Can we
really still disagree? Can any one still declare it obvious that this is
reasonable? That pleasure alone is good as an end?
The case, it is plain, is just like that of the colours[18]
, only, as yet,
not nearly so strong. It is far more possible that we should some day
be able to produce the intensest pleasure, without any
consciousness that it is there, than that we should be able to
produce mere colour, without its being any particular colour. Pleasure
and consciousness can be far more easily distinguished from one
another, than colour from the particular colours. And yet even if this
were not so, we should be bound to distinguish them if we really
wished to declare pleasure alone to be our ultimate end. Even if
consciousness were an inseparable accompaniment of pleasure, a
sine quâ non of its existence, yet, if pleasure is the only end, we are
bound to call consciousness a mere means to it, in any intelligible
sense that can be given to the word means. And if, on the other](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-33-320.jpg)
![contemplation are mere parts of the good, and both of them equally
necessary parts. In short, Prof. Sidgwick’s argument here depends
upon the neglect of that principle, which I tried to explain in my first
chapter and which I said I should call the principle of ‘organic
relations[19]
.’ The argument is calculated to mislead, because it
supposes that, if we see a whole state to be valuable, and also see
that one element of that state has no value by itself, then the other
element, by itself, must have all the value which belongs to the
whole state. The fact is, on the contrary, that, since the whole may
be organic, the other element need have no value whatever, and
that even if it have some, the value of the whole may be very much
greater. For this reason, as well as to avoid confusion between
means and end, it is absolutely essential to consider each
distinguishable quality, in isolation, in order to decide what value it
possesses. Prof. Sidgwick, on the other hand, applies this method of
isolation only to one element in the wholes he is considering. He
does not ask the question: If consciousness of pleasure existed
absolutely by itself, would a sober judgment be able to attribute
much value to it? It is, in fact, always misleading to take a whole,
that is valuable (or the reverse), and then to ask simply: To which of
its constituents does this whole owe its value or its vileness? It may
well be that it owes it to none; and, if one of them does appear to
have some value in itself, we shall be led into the grave error of
supposing that all the value of the whole belongs to it alone. It
seems to me that this error has commonly been committed with
regard to pleasure. Pleasure does seem to be a necessary
constituent of most valuable wholes; and, since the other
constituents, into which we may analyse them, may easily seem not
to have any value, it is natural to suppose that all the value belongs
to pleasure. That this natural supposition does not follow from the
premises is certain; and that it is, on the contrary, ridiculously far
from the truth appears evident to my ‘reflective judgment.’ If we
apply either to pleasure or to consciousness of pleasure the only
safe method, that of isolation, and ask ourselves: Could we accept,
as a very good thing, that mere consciousness of pleasure, and
absolutely nothing else, should exist, even in the greatest quantities?](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-38-320.jpg)
![person’s happiness is singly the greatest amount of Universal Good
which can be realised at all.
61. It should be observed that, since this is so, ‘the relation of
Rational Egoism to Rational Benevolence,’ which Prof. Sidgwick
regards ‘as the profoundest problem of Ethics’ (III. XIII. § 5, n. 1),
appears in quite a different light to that in which he presents it.
‘Even if a man,’ he says, ‘admits the self-evidence of the principle of
Rational Benevolence, he may still hold that his own happiness is an
end which it is irrational for him to sacrifice to any other; and that
therefore a harmony between the maxim of Prudence and the
maxim of Rational Benevolence must be somehow demonstrated, if
morality is to be made completely rational. This latter view is that
which I myself hold’ (last Chap. § 1). Prof. Sidgwick then goes on to
shew ‘that the inseparable connection between Utilitarian Duty and
the greatest happiness of the individual who conforms to it cannot
be satisfactorily demonstrated on empirical grounds’ (Ib. § 3). And
the final paragraph of his book tells us that, since ‘the reconciliation
of duty and self-interest is to be regarded as a hypothesis logically
necessary to avoid a fundamental contradiction in one chief
department of our thought, it remains to ask how far this necessity
constitutes a sufficient reason for accepting this hypothesis[20]
’ (Ib.
§ 5). To ‘assume the existence of such a Being, as God, by the
consensus of theologians, is conceived to be’ would, he has already
argued, ensure the required reconciliation; since the Divine
Sanctions of such a God ‘would, of course, suffice to make it always
every one’s interest to promote universal happiness to the best of his
knowledge’ (Ib. § 5).
Now what is this ‘reconciliation of duty and self-interest,’ which
Divine Sanctions could ensure? It would consist in the mere fact that
the same conduct which produced the greatest possible happiness of
the greatest number would always also produce the greatest
possible happiness of the agent. If this were the case (and our
empirical knowledge shews that it is not the case in this world),
‘morality’ would, Prof. Sidgwick thinks, be ‘completely rational’: we
should avoid ‘an ultimate and fundamental contradiction in our](https://image.slidesharecdn.com/23443827-250511141532-01aa9207/85/Fault-Estimation-For-Network-Systems-Via-Intermediate-Estimator-Junwei-Zhu-47-320.jpg)
Fault Estimation For Network Systems Via Intermediate Estimator Junwei Zhu
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- 8. Jun-Wei Zhu Institute of Cyberspace Security Zhejiang University of Technology Hangzhou, China Guang-Hong Yang College of Information Science and Engineering Northeastern University Shenyang, China Xin Wang Heilongjiang University Harbin, Heilongjiang, China ISBN 978-981-19-6320-9 ISBN 978-981-19-6321-6 (eBook) https://doi.org/10.1007/978-981-19-6321-6 © Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore
- 9. Preface Because a small fault in the actual system may cause huge consequences, the research on the safety and reliability of complex dynamic systems has become an extremely important topic. In this context, fault estimation technology has gradually developed and become one of the popular areas in the control field. As we know, fault detection and isolation techniques aim to determine when and where the faults occur. Unlike them, fault estimation (FE) technique seeks to obtain more specific fault’s dynamics online. Compared with fault detection and isolation, FE can obtain more information of the fault. Generally, FE is achieved by designing an observer. So far, a variety of different fault observer design methods have appeared, and each method is based on different ideas, so they have different characteristics. However, the requirements of FE are normally stricter, and there are relatively more constraints. Therefore, there are still a number of challenging issues in FE problems, which greatly limit the application scope of the existing approaches. In this book, we mainly study the problem of FE for network systems including networked single system, multi-agent system (MAS), and large-scale interconnected system. A novel type of fault observer called Intermediate Estimator (IE) is proposed for Lipschitzian nonlinear system, and a series of modified IEs for network systems are also presented. Moreover, the fault tolerant control (FTC) problem for network systems is also studied. In Chap. 1, the motivation and history of the research and an overview on recent development of FE and FTC are presented. In Chap. 2, a new fault observer design method, i.e., IE is proposed for a class of Lipschitzian nonlinear systems. This observer has significantly improved the tradi- tional adaptive observer (AO), it overcomes the important constraint that the tradi- tional AO must satisfy SPR condition. Compared with other types of observers, this new type of fault observer has some important advantages, e.g., (a) it does not need to meet the observer matching conditions, (b) the convergence speed of the FE can be set and an explicit upper bound of the FE error can be obtained, (c) the norm upper bounds of the fault itself and the first derivative of the fault do not need to be known, (d) the system matrix is also not required to satisfy the upper triangular structure constraint. v
- 10. vi Preface In Chap. 3, the observer-based fault accommodation (FA) problem for linear systems with state perturbations and measurement errors is investigated. A novel IE isproposedtoestimatethestateandthefaultssimultaneously,wheretheeffectofstate perturbations and measurement errors is effectively eliminated by fully exploiting the design freedom of some specified parameters. Based on the estimation, an FTC scheme is synthesized. It is proved that the states of the resulting closed loop system are uniformly ultimately bounded with an explicit bound. In Chap. 4, the FA problem for linear systems with time-varying delay, system uncertainties and external disturbances is studied. A novel IE-based FTC scheme is proposed, where an IE is constructed to estimate the states and the faults simultane- ously, based on the estimation, an FTC compensation scheme is designed to mitigate the effects of faults, system uncertainties and time-varying delay. In Chap. 5, the robust distributed FE problem is considered for a network of dynamical systems with external disturbances. Based on local output measurements and state estimates from neighbors, a distributed IE is constructed for each node. By spectraldecompositionoftheLaplacianandproperscalingofthefaultsandthedistur- bances, a specially-structured global error system can be created. It is proved that the states of the global error system are uniformly ultimately bounded with an explicit error bound. The robustness of the global error system can be ensured by directly adjusting the design parameters of the IEs, without introducing any performance specification. In Chap. 6, we study the observer-based fault tolerant tracking control problem for linear MASs with multiple faults and mismatched disturbances. A novel distributed IE-based fault tolerant tracking protocol is presented. The leader’s input is nonzero and unavailable to the followers. By applying a projection technique, the mismatched disturbances are divided into matched and unmatched components. For each node, a tracking error system is established, for which an IE driven by the relative output measurements is constructed to estimate the sensor faults and a combined signal of the leader’s input, process faults, and matched disturbance component. Based on the estimation, a fault tolerant tracking protocol is designed to eliminate the effects of the combined signal. Besides, the effect of unmatched disturbance component can be attenuated by directly adjusting some specified parameters. In Chap. 7, the cooperative tracking control problem for MASs with unknown external disturbances under a directed graph is studied. The leader’s input is nonzero and unavailable to the followers. Based on the relative output measurements of neigh- boring agents, a novel observer-based cooperative tracking protocol is presented, where a group of IEs are constructed to estimate the tracking errors and the combined signal of the disturbances and the leader’s input simultaneously, then decentralized tracking protocols are designed based on the obtained estimates. In Chap. 8, we propose an improved observer-based fault-tolerant tracking control approach for industrial MASs. Firstly, a group of new distributed IEs are presented, where the design structure is modified to enhance the feasibility of the estimation scheme. It is shown that the both of the nominal distributed IE and the traditional extended state observer (ESO) are special cases of the proposed estimator. Secondly, theestimationperformancecanbeimprovedsignificantlyviaanonlinereinforcement
- 11. Preface vii learning estimation strategy, whose core is an adaptive switching mechanism inte- grated with a function block of source fault mode localization. Benefiting from satis- factory estimation results, good fault-tolerant tracking control performance can be guaranteeddespiteofmultiplefaultsanddisturbances.Theapplicationtoanetworked multi-axis motion control system demonstrates the effectiveness and superiority of the proposed method. In Chap. 9, the fault reconstruction problem of mobile robots is studied. To be specific, a novel IE is proposed to reconstruct the faults, where the estimation perfor- mance is guaranteed in the sense of driving the estimation errors to a small prescribed energy bound via a two-dimensional gain-regulation mechanism. It is shown that the proposed estimator is more accurate than the existing ESO and nominal IE. In Chap. 10, we consider the multiple attack reconstruction problem for CPSs. A new projected intermediate estimator is proposed, where the computation complexity problem is solved by reducing the number of loop iterations significantly via enhancing the designs of loop iteration, termination condition and parameter selec- tion strategy, without sacrificing the attack reconstruction performance. Compared with traditional IEs, the dependence of linear matrix inequality technique is completely removed, besides, the intrinsic heavy overshooting problem in transient period is effectively alleviated. Finally, conclusions and future research directions are presented in Chap. 11. Hangzhou, China Jun-Wei Zhu Xin Wang Guang-Hong Yang
- 12. Acknowledgements This work is dedicated to my wife Xiao-Ling Zhou and my Ph.D. student Cao-Yuan Gu. The authors would like to express especial thanks to Prof. Steven X. Ding, of the Institute for Automatic Control and Complex Systems (AKS), University of Duisburg-Essen, 47057 Duisburg, Germany for their valuable comments and encour- agement on the research work. Meanwhile, the authors would like to acknowledge the support of research grants, including National Natural Science Foundation of China (61803334, 61822311, 61703148, 61673351), Zhejiang Provincial Natural Science Foundation of China (LZ21F030004, LQ18F030012), the State Scholarship Fund of China Scholarship Council (201908330040), NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (U1709213). ix
- 13. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Basic Concepts in Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 FE Method and Research Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Research Status for FTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 IE for Lipschitzian Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Design of the IE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 The IE Design for Nonlinear Systems with Single Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.2 Extension of IE for Nonlinear Systems with Multiple Faults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.2 Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3 IE for Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Preliminaries and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 28 3.3 The Design of the IE-Based FA Strategy . . . . . . . . . . . . . . . . . . . . . 29 3.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 xi
- 14. xii Contents 4 IE for Time-Varying Delay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Problem Statement and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 IE-Based FTC Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Distributed IE for Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.2 Preliminaries and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.1 Basic Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 The Design of the Distributed IEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4.1 Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4.2 Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 6 Distributed IE for MASs with Undirected Graph . . . . . . . . . . . . . . . . . 77 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.2 Preliminaries and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.1 Basic Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.3 The Design of the Distributed IE Based Fault Tolerant Tracking Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7 Distributed IE for MASs with Directed Graph . . . . . . . . . . . . . . . . . . . 97 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2 Preliminaries and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 98 7.3 The Design of the IE-based Tracking Protocol . . . . . . . . . . . . . . . . 99 7.4 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 8 Intelligent IE for MASs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.2 Preliminaries and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 112 8.2.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 8.3.1 A Novel Observer-based CFTTC Protocol with Adaptive Switching Mechanism . . . . . . . . . . . . . . . . 114 8.3.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.3.3 CFTTC with Online Reinforcement Learning Estimation Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
- 15. Contents xiii 8.4 Application to Networked Multi-Axis Motion Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.4.1 Platform of Servo Motors . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.4.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9 IE for LPV Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 9.2.1 Kinematic Model of Mobile Robot . . . . . . . . . . . . . . . . . . 136 9.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9.3.1 Design of the Iterative IE with TDGRM . . . . . . . . . . . . . . 140 9.3.2 Stability Analysis of Estimation Error System . . . . . . . . . 143 9.4 Application to Mobile Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 10 Projected IE for CPSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 10.2 Preliminaries and Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 156 10.2.1 System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 10.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10.3.1 Observability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 10.3.2 The Design of Basic Steps for PIE . . . . . . . . . . . . . . . . . . 160 10.3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 10.3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.4 Application to Networked Motion Control System . . . . . . . . . . . . 167 10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 11 Conclusions and Future Research Directions . . . . . . . . . . . . . . . . . . . . . 173 11.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
- 16. Acronyms AO Adaptive observer CPS Cyber-physical system ESO Extended state observer FA Fault accommodation FE Fault estimation FTC Fault tolerant control IE Intermediate estimator LMI Linear matrix inequality MASs Multi-agent systems NIE Nominal intermediate estimator SMO Sliding mode observer SPR Strictly positive real TDGRM Two dimensional gain-regulation mechanism UIO Unknown input observer xv
- 17. Chapter 1 Introduction 1.1 Background Since the 1940s, automatic control technology has been widely used in practical engineering fields, such as robotics, chemical industry, aerospace, and nuclear power plants, and continues to play a pivotal role in the new wave of industrialization in the twenty-first century. As real systems grow in size and complexity, so does the cost of control systems. Due to the complexity of the structure of the large-scale control system and long-time working under high power and high load condition, some equipments in the system, such as sensors and actuators, will inevitably age over time, and may even evolve into failures. When failure occurs, the control performance of the system will be dropped. In some cases even stability cannot be guaranteed, which results in huge property damage and casualties. For example, in 2003, the U.S. space shuttle Columbia exploded on its way back due to a defect in the installation of the foam material on the surface of the external fuel tank, causing billions of dollars in property damage and killing all seven astronauts. In 1986, the explosion of Unit 4 of the Chernobyl nuclear power plant in the former Soviet Union resulted in the leakage of a large amount of radioactive material, which eventually killed more than 2,000 people and caused more than 3.2 million people to suffer serious consequences of nuclear radiation hazards. Similar examples are common. Because a small fault in the actual system may cause huge consequences, the research on the safety and reliability of complex dynamic systems has become an extremely important topic. In this context, fault diagnosis and fault-tolerant control technology [1–10] have gradually developed and became one of the popular areas in the control field. © Springer Nature Singapore Pte Ltd. 2022 J.-W. Zhu et al., Fault Estimation for Network Systems via Intermediate Estimator, https://doi.org/10.1007/978-981-19-6321-6_1 1
- 18. 2 1 Introduction 1.2 Basic Concepts in Fault Diagnosis The International Federation of Automatic Controls (IFAC) defines a fault as a devi- ation from the normal range of at least one observable or computable important variable or characteristic in a control system. Faults can be roughly classified as follows according to different angles, that is, time characteristics or occurrence loca- tions: From the perspective of time characteristics, faults can be divided into the following types [11–13]: • Abrupt fault refers to fault in which some parameter values in the system suddenly deviate greatly. • Incipient fault refers to the fault in which some system parameters change slowly with the change of time and environment, also known as soft fault. • The AO-based FE have attracted much attention, but its existence conditions have not been given explicitly yet. • Intermittent fault refers to the fault caused by the aging of components, insufficient error tolerance, or poor contact. According to the analysis of the occurrence location, the faults can be divided into the following categories: • Component failure means that some components in the controlled object of the system are abnormal, which makes the operation of the whole system to deviate, resulting in the failure to accomplish the systems control goal precisely. • Sensor fault refers to the fault that the sensor is stuck or has a certain degree of deviation and cannot accurately measure the internal signal of the system. • Actuator fault refers to the fault that the actuator cannot execute the control com- mand correctly due to changes such as stuck, offset, or interruption. Fault diagnosis generally refers to the following three directions: • Fault detection: Determine whether a fault has occurred. • Fault isolation: Determine the specific location or kind of failure. • Fault identification: Determine the size and dynamics of the fault. For the fault diagnosis system, according to different actual needs, we can define a variety of performance indexes. For example, the rapidity of diagnosis, the sensitivity and robustness of the fault diagnosis system, the rate of false positives and false negatives of faults, etc. In this book, we mainly study the problem of FE for network systems including networked single system, MASs, and large-scale interconnected system, the goal of FE is to obtain the exact information of the fault online, such as the size and shape of the fault itself. Generally, it belongs to the direction of fault identification by definition.
- 19. 1.3 FE Method and Research Status 3 1.3 FE Method and Research Status Unlike fault detection and isolation techniques, which aim to determine when and where the faults occur, FE techniques seek to obtain more specific fault dynamics online. Compared with fault detection and isolation, FE can obtain more fault infor- mation. However, its requirements are usually stricter, and there are relatively more constraints. In terms of technical form, it is also relatively monotonous. Generally, FE is achieved by designing an observer. So far, a variety of different fault observer design method have appeared, and each method is based on different ideas, so they have different characteristics. The following is a brief overview of the current FE methods. • Sliding Mode Observer (SMO) The SMO method starts from the idea of variable structure control [14–20]. A typical literature on SMO design is Edwards and Spurgeon’s paper on the problem of FE in Automatica [184], which is published in 2000. The basic idea of SMO is the notion of equivalent output injection (equivalent output injection). The idea of reconstructing the fault signal is based on the dynamic characteristics of the sliding mode surface of the observer, and it constructs the estimation of the fault signal by using the estimation error of the system output. Based on this theory, SMOs have been widely used in the FE of various systems in the past decade [21– 24, 193, 195]. The technical feature of the SMO is to ensure that the estimation error dynamics have better robustness against system uncertainties or external disturbances by utilizing the design freedom of the nonlinear term of the observer gain. However, SMO heavily depends on output estimation error information to realize sliding mode dynamics. In its design, the strict observer matching condition must be satisfied, and the upper bound of the fault itself and the first derivative of the fault must be known. Moreover, the estimation performance of SMO often suffers from chattering problems [25–30]. These constraints limit the application scope of SMO to some extent. • Adaptive Observer (AO) AO achieves FE based on the idea of online compensation of FE error [31–33]. Technically, a strict equality constraint is introduced, then the observer is designed by using the solution of the constraints of the formula to realize the compensation of the FE error effect in the error system, thereby ensuring the eventual uniformly bounded stability of the error system. FE methods based on AO mainly rely on tuning parameters to ensure FE performance. So far, there are few related research results. Authors of [186] designed a fault-tolerant control method based on AO for Lipschitzian nonlinear systems with process faults. The result in [192] presents a fault-tolerant control strategy based on AO for linear systems with actuator deflec- tion faults. In general, the advantages of the AO method are that the observer structure design is relatively simple, the applicability is strong, and the FE perfor- mance is good. Through the stability analysis, the exponential convergence of the error system state can be strictly proved and a clear upper bound of FE error can be obtained. In addition, this kind of method can independently set the convergence
- 20. 4 1 Introduction rate of FE by adjusting parameters. However, it should be pointed out that in order to ensure the solvability of the equality constraints, AO must satisfy the strictly positive real (SPR) condition [34–36, 190], which is a sufficient and unnecessary condition of the observer matching condition, so it is much more conservative than the observer matching condition. Most practical systems cannot meet the SPR condition, so the application potential of the AO is greatly restricted. • Unknown Input Observer (UIO) The idea of using UIO [37–41] for fault diagnosis of systems with uncertainty was firstly proposed by Professor Wantanabe in 1982 [42], and the basic idea of the FE method based on UIO is to use the redundant degrees of freedom of Luenberger observer, so that the FE error can be decoupled from the disturbance or modeling uncertainty, then it can effectively realize the robustness of the FE. Authors of [43] proposed a class of UIO methods for linear systems with process faults, and gave design conditions to ensure the stability of the error system. In general, UIOs are also required to satisfy the observer matching condition [44–47]. • Robust Observer The basic idea of the robust observer [48–53] is to combine Luenberger observer’s design structure with the traditional robust performance optimization strategy, and to suppress the effect of disturbances and fault derivative on estimation error by introducing performance indexes, while its observer gain needs to be obtained by solving a series of LMIs with performance optimization metrics. In [54], a class of one-step robust observer design is proposed for continuous-time T-S fuzzy sys- tems. On this basis, the approach in [171] improves the original one-step robust observer for T-S fuzzy systems with time delay, and proposes a K-step robust observer method. The step-iterative observer design further suppresses the influ- ence of the fault derivative on the FE error, it also gives the design conditions to ensure the stability of the error system. In addition, authors of [55] propose a robust observer design for discrete-time systems. However, the above robust observers have a common disadvantage, that is, the upper bound of the FE error often cannot be obtained through the theoretical analysis of this method [56–62]. • High-gain Observer The high-gain observer is widely used in various nonlinear systems. It makes the observer robust to the uncertainty of system modeling by specially designing the gain of the observer. In general, high-gain observer can only be used for estimation of constant fault parameters. For the work in this area, readers can refer to [63–68]. The disadvantage of high-gain observers is that these methods require the system matrix to satisfy an upper triangular structure, which cannot be met for most linear systems. Apart from the above FE methods, there are also other techniques, such as singular observer [187–189], proportional-integral observer [69–71], iterative learning observer [72, 73], etc., which will not be described in detail here.
- 21. 1.4 Research Status for FTC 5 1.4 Research Status for FTC So far, FE technique is mainly further used for FTC. The idea of FTC started in the 1970s [74–77, 198], and the meaning of fault tolerance is to tolerate the effect of faults in control systems. The necessary condition for a system to have fault tolerance is that there is control redundancy in the system. For example, the fault tolerance of the actuator requires redundancy of the actuator, the fault tolerance of the sensor requires the redundancy of the sensor measurement, and the fault tolerance of the components requires the presence of function redundancy of the components. The design concept of fault-tolerant control is to use redundancy to tolerate faults. In 1985, Eterno divided FTC methods into two categories: passive fault tolerant and active fault tolerant. Generally speaking, passive FTC [78–80] often regards faults as a special form of disturbance or system uncertainty, and then uses robust control technology to suppress the effect of faults on the system. The structure of its controller is fixed, so passive FTC does not need to obtain any fault information online. Active FTC system identifies fault parameters online by designing algorithms or attempts to obtain fault information by fault diagnosis technique, and then changes the controller structure to compensate the effect of faults. In contrast, active FTC has flexible design ideas, and there are many methods so far. From the perspective of design structure, activeFTCmethodscanbedividedintocontrolratereschedulingmethodsandcontrol rate reconfiguration design methods. From the analysis of information sources, they can be divided into state feedback FTC methods and output feedback FTC methods. Although there have been a lot of research results in FTC, it should be pointed out that most of the methods are based on the idea of state feedback control [81–84], and in practical systems, the full state information cannot be obtained, so it is often necessary to use the output feedback FTC strategy [85–89] to design the controller. This book also studies the FTC problem based on output feedback. Below we briefly summarize the current main methods in this area. • Adaptive Output Feedback FTC The adaptive output feedback control method mainly uses the output information to design the adaptive updating law rate to achieve online compensation FTC [90– 92]. When the physical process failure occurs and causes the system to change dynamically, an adaptive mechanism is triggered to mitigate the impact of the fail- ure. [93] proposes an adaptive output feedback control method for linear systems with actuator deflection, stuck and interrupted faults. On this basis, authors of [94] further consider the design of a quantized output feedback adaptive controller with actuator faults, whose advantage is that although it requires that there exists the upper bound of the fault, it does not need to know the specific magnitude of the upper bound of the fault, and at the same time, it can fully compensate the influence of the fault and ensure that the system state is asymptotically stable. It is necessary to meet SPR conditions or have certain requirements on the system structure, such as strict feedback formal system [95–98], these constraints limit the application of adaptive output feedback method in practical engineering.
- 22. 6 1 Introduction • Sliding mode output feedback FTC The main feature of the sliding mode output feedback FTC method is that in order to overcome the influence of the fault and let the system dynamically reach the sliding mode surface, it needs to design the nonlinear feedback gain term of the controller [99–101], and the design of this item needs to use the output information and rely on a priori known information of the upper bound of the fault. The result [102] considers the deflection of the actuator and combines with the integral sliding mode surface design and the control distribution method, when the fault occurs, the influence of the fault is overcome by redistributing the control signal and the sliding mode control is realized. In [103], a sliding mode output feedback FTC method is proposed for the output tracking control problem of linear uncertain systems. Similar to sliding mode state feedback control, sliding mode output feedback control has strong robustness, but in order to overcome the influence of external disturbances or uncertainties to achieve better robustness, it is often necessary to introduce extremely strict restrictions on system structure [104, 105], for instance, faults and disturbances must appear in the output channel. • Observer-based output feedback FTC The observer-based output feedback FTC essentially integrates the design of FE and fault-tolerant control strategies. The design idea is to first estimate the fault by designing an appropriate observer, and then use the FE to design a controller to compensate the fault online. At present, there are a large number of FTC methods based on various major FE observers, such as the FTC method for singular SMOs for stochastic systems [106–109], AO-based FTC method, [186, 192], the robust- observer-based FTC scheme [54, 171], and so on. Different from the ideas of adaptive control and sliding mode control, in general, the observer-based FTC method has lower requirements on the system structure, but often cannot fully Fig. 1.1 The flow chart of FE-based FTC
- 23. 1.5 Book Outline 7 compensate for time-varying faults, so it can only ensure that the system state to be uniformly ultimately bounded. At the same time, in order to ensure the FTC performance, the accuracy of the FE is required to be high. For reference, the flow chart of FE-based FTC is shown in Fig. 1.1. 1.5 Book Outline In the last two sections, we introduced the FE problem and the FTC method based on FE. It can be seen that researchers have obtained a lot of results in these two areas. However, there are still some shortcomings in the existing technology. First, from a technical point of view, most of the existing FE methods are based on the observer matching condition, which is not satisfied in most practical systems. In addition to the observer matching conditions, the general SMO also requires the upper bound of the fault itself and the first derivative of the fault to be known, but in practical systems, these information are often unknown. Although the AO does not require the upper bound information to be known and can freely set the convergence rate of the FE, it needs to satisfy the SPR condition, which is stricter than the observer matching condition. In addition, although the robust observer does not need to satisfy the observer matching condition, the upper bound of the FE error cannot be obtained through the theoretical analysis of this method. For nonlinear systems, the high- gain observer also has no observer matching condition constraints, but it requires the system matrix to satisfy the upper triangular structure constraint, which is not met by most nonlinear systems. On the other hand, from the perspective of FTC technology, the existing adaptive output feedback FTC methods all need to satisfy a strict equality constraint. However, most of the control system have no solution under this constraint. Although the sliding mode fault-tolerant control method does not require equality constraints, it still requires that the fault must appear in the output channel. Secondly, from the point of view of the problem, the existing technologies are generally developed for a single system. In comparison, there are few research results on the FE and fault-tolerant control problems of complex networks or MASs. This is mainly due to that the dynamic model of the network system is more complex, and the faults between the subsystems will affect each other through connections, so it is often difficult to accurately locate and estimate the fault. On the premise that only the measurement output information is available, the difficulty of FTC is also greatly increased. In this book, the above-mentioned problems are deeply studied. The details are listed as follows. In Chap. 2, a new fault observer design method is proposed for a class of Lipschitz nonlinear systems, which we call the IE. This observer is essentially an improvement of the traditional AO, which overcomes the important constraint that the traditional AO must satisfy SPR condition. Compared with other types of observers, this method has some important advantages, e.g., (a) it is not necessary to meet the observer matching conditions, (b) the convergence speed of the FE can be set and a clear
- 24. 8 1 Introduction upper bound of the FE error can be obtained, (c) the norm upper bound of the fault itself and the first derivative of the fault do not need to be known, and (d) the system matrix is also not required to satisfy the upper triangular structure constraints. In Chap. 3, the observer-based FA problem for linear systems with state perturba- tions and measurement errors is investigated. A novel IE is proposed to estimate the state and the faults simultaneously, where the effect of state perturbations and mea- surement errors can be effectively eliminated by fully exploiting the design freedom of some specified parameters. Based on the estimation, an FTC scheme is synthe- sized. It is proved that the states of the resulting closed-loop system are uniformly ultimately bounded with an explicit bound. Simulation examples verify the effec- tiveness of the proposed method. In Chap. 4, the FA problem for linear systems with time-varying delay, system uncertainties, and external disturbances is studied. A novel IE-based FTC scheme is proposed, where an IE is constructed to estimate the states and the faults simultane- ously, based on the estimation, an FTC compensation scheme is designed to eliminate the effects of faults, system uncertainties, and time-varying delay. It is proved that the states of the closed-loop system are uniformly ultimately bounded. A simulation example shows the effectiveness of the proposed method. In Chap. 5, the robust distributed FE problem is considered for a network of dynamical systems with external disturbances. Based on local output measurements and state estimates from neighbors, a distributed IE is constructed for each node. By spectral decomposition of the Laplacian and proper scalings of the faults and the disturbances, a specially structured global error system can be created. It is proved that the states of the global error system are uniformly ultimately bounded with an explicit error bound. Compared with the existing results, the proposed FE strategy has no requirements of the observer matching condition and the preliminary knowledge of the upper bounds of the faults; moreover, the robustness of the global error system can be ensured by directly adjusting the design parameters of the IEs, without introducing any performance specifications. Simulation examples verify the effectiveness of the proposed method. In Chap. 6, we study the observer-based fault-tolerant tracking control problem for linear multiagent systems with multiple faults and mismatched disturbances. A novel distributed IE-based fault-tolerant tracking protocol is presented. The leader’s input is nonzero and unavailable to the followers. By applying a projection technique, the mismatched disturbances are separated into matched and unmatched components. For each node, a tracking error system is established, for which an IE driven by the relative output measurements is constructed to estimate the sensor faults and a com- bined signal of the leaders input, process faults, and matched disturbance component. Based on the estimation, a fault-tolerant tracking protocol is designed to eliminate the effects of the combined signal. Besides, the effect of unmatched disturbance com- ponent can be attenuated by directly adjusting some specified parameters. Finally, a simulation example of aircraft demonstrates the effectiveness of the designed track- ing protocol. In Chap. 7, the cooperative tracking control problem for MASs with unknown external disturbances under a directed graph is studied. The leader’s input is nonzero
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- 26. as beauty out of relation to any mind whatsoever: but only that there is some standard of beauty valid for all minds. “It may, however, be said that beauty and other results commonly judged to be good, though we do not conceive them to exist out of relation to human beings (or at least minds of some kind), are yet so far separable as ends from the human beings on whom their existence depends, that their realization may conceivably come into competition with the perfection or happiness of these beings. Thus, though beautiful things cannot be thought worth producing except as possible objects of contemplation, still a man may devote himself to their production without any consideration of the persons who are to contemplate them. Similarly knowledge is a good which cannot exist except in minds; and yet one may be more interested in the development of knowledge than in its possession by any particular minds; and may take the former as an ultimate end without regarding the latter. “Still, as soon as the alternatives are clearly apprehended, it will, I think, be generally held that beauty, knowledge, and other ideal goods, as well as all external material things, are only reasonably to be sought by men in so far as they conduce (1) to Happiness or (2) to the Perfection or Excellence of human existence. I say ‘human,’ for though most utilitarians consider the pleasure (and freedom from pain) of the inferior animals to be included in the Happiness which they take as the right and proper end of conduct, no one seems to contend that we ought to aim at perfecting brutes except as a means to our ends, or at least as objects of scientific or æsthetic contemplation for us. Nor, again, can we include, as a practical end, the existence of beings above the human. We certainly apply the idea of Good to the Divine Existence, just as we do to His work, and indeed in a preeminent manner: and when it is said that, ‘we should do all things to the glory of God,’ it may seem to be implied that the existence of God is made better by our glorifying Him. Still this inference when explicitly drawn appears somewhat impious; and theologians generally recoil from it, and refrain from using the notion of a possible addition to the Goodness of the Divine Existence as a
- 27. ground of human duty. Nor can the influence of our actions on other extra-human intelligences besides the Divine be at present made matter of scientific discussion. “I shall therefore confidently lay down, that if there be any Good other than Happiness to be sought by man, as an ultimate practical end, it can only be the Goodness, Perfection, or Excellence of Human Existence. How far this notion includes more than Virtue, what its precise relation to Pleasure is, and to what method we shall be logically led if we accept it as fundamental, are questions which we shall more conveniently discuss after the detailed examination of these two other notions, Pleasure and Virtue, in which we shall be engaged in the two following Books.” It will be observed that in this passage Prof. Sidgwick tries to limit the range of objects among which the ultimate end may be found. He does not yet say what that end is, but he does exclude from it everything but certain characters of Human Existence. And the possible ends, which he thus excludes, do not again come up for consideration. They are put out of court once for all by this passage and by this passage only. Now is this exclusion justified? I cannot think it is. ‘No one,’ says Prof. Sidgwick, ‘would consider it rational to aim at the production of beauty in external nature, apart from any possible contemplation of it by human beings.’ Well, I may say at once, that I, for one, do consider this rational; and let us see if I cannot get any one to agree with me. Consider what this admission really means. It entitles us to put the following case. Let us imagine one world exceedingly beautiful. Imagine it as beautiful as you can; put into it whatever on this earth you most admire— mountains, rivers, the sea; trees, and sunsets, stars and moon. Imagine these all combined in the most exquisite proportions, so that no one thing jars against another, but each contributes to increase the beauty of the whole. And then imagine the ugliest world you can possibly conceive. Imagine it simply one heap of filth, containing everything that is most disgusting to us, for whatever reason, and the whole, as far as may be, without one redeeming
- 28. feature. Such a pair of worlds we are entitled to compare: they fall within Prof. Sidgwick’s meaning, and the comparison is highly relevant to it. The only thing we are not entitled to imagine is that any human being ever has or ever, by any possibility, can, live in either, can ever see and enjoy the beauty of the one or hate the foulness of the other. Well, even so, supposing them quite apart from any possible contemplation by human beings; still, is it irrational to hold that it is better that the beautiful world should exist, than the one which is ugly? Would it not be well, in any case, to do what we could to produce it rather than the other? Certainly I cannot help thinking that it would; and I hope that some may agree with me in this extreme instance. The instance is extreme. It is highly improbable, not to say, impossible, we should ever have such a choice before us. In any actual choice we should have to consider the possible effects of our action upon conscious beings, and among these possible effects there are always some, I think, which ought to be preferred to the existence of mere beauty. But this only means that in our present state, in which but a very small portion of the good is attainable, the pursuit of beauty for its own sake must always be postponed to the pursuit of some greater good, which is equally attainable. But it is enough for my purpose, if it be admitted that, supposing no greater good were at all attainable, then beauty must in itself be regarded as a greater good than ugliness; if it be admitted that, in that case, we should not be left without any reason for preferring one course of action to another, we should not be left without any duty whatever, but that it would then be our positive duty to make the world more beautiful, so far as we were able, since nothing better than beauty could then result from our efforts. If this be once admitted, if in any imaginable case you do admit that the existence of a more beautiful thing is better in itself than that of one more ugly, quite apart from its effects on any human feeling, then Prof. Sidgwick’s principle has broken down. Then we shall have to include in our ultimate end something beyond the limits of human existence. I admit, of course, that our beautiful world would be better still, if there were human beings in it to contemplate and enjoy its beauty. But that admission makes nothing against my point.
- 29. If it be once admitted that the beautiful world in itself is better than the ugly, then it follows, that however many beings may enjoy it, and however much better their enjoyment may be than it is itself, yet its mere existence adds something to the goodness of the whole: it is not only a means to our end, but also itself a part thereof. 51. In the second passage to which I referred above, Prof. Sidgwick returns from the discussion of Virtue and Pleasure, with which he has meanwhile been engaged, to consider what among the parts of Human Existence to which, as we saw, he has limited the ultimate end, can really be considered as such end. What I have just said, of course, appears to me to destroy the force of this part of his argument too. If, as I think, other things than any part of Human Existence can be ends-in-themselves, then Prof. Sidgwick cannot claim to have discovered the Summum Bonum, when he has merely determined what parts of Human Existence are in themselves desirable. But this error may be admitted to be utterly insignificant in comparison with that which we are now about to discuss. “It may be said,” says Prof. Sidgwick (III. XIV. §§ 4-5), “that we may ... regard cognition of Truth, contemplation of Beauty, Free or Virtuous action, as in some measure preferable alternatives to Pleasure or Happiness—even though we admit that Happiness must be included as a part of Ultimate Good.... I think, however, that this view ought not to commend itself to the sober judgment of reflective persons. In order to shew this, I must ask the reader to use the same twofold procedure that I before requested him to employ in considering the absolute and independent validity of common moral precepts. I appeal firstly to his intuitive judgment after due consideration of the question when fairly placed before it: and secondly to a comprehensive comparison of the ordinary judgments of mankind. As regards the first argument, to me at least it seems clear after reflection that these objective relations of the conscious subject, when distinguished from the consciousness accompanying and resulting from them, are not ultimately and intrinsically desirable; any more than material or other objects are, when considered apart from any relation to conscious existence. Admitting
- 30. that we have actual experience of such preferences as have just been described, of which the ultimate object is something that is not merely consciousness: it still seems to me that when (to use Butler’s phrase) we ‘sit down in a cool hour,’ we can only justify to ourselves the importance that we attach to any of these objects by considering its conduciveness, in one way or another, to the happiness of sentient beings. “The second argument, that refers to the common sense of mankind, obviously cannot be made completely cogent; since, as above stated, several cultivated persons do habitually judge that knowledge, art, etc.,—not to speak of Virtue—are ends independently of the pleasure derived from them. But we may urge not only that all these elements of ‘ideal good’ are productive of pleasure in various ways; but also that they seem to obtain the commendation of Common Sense, roughly speaking, in proportion to the degree of this productiveness. This seems obviously true of Beauty; and will hardly be denied in respect of any kind of social ideal: it is paradoxical to maintain that any degree of Freedom, or any form of social order, would still be commonly regarded as desirable even if we were certain that it had no tendency to promote the general happiness. The case of Knowledge is rather more complex; but certainly Common Sense is most impressed with the value of knowledge, when its ‘fruitfulness’ has been demonstrated. It is, however, aware that experience has frequently shewn how knowledge, long fruitless, may become unexpectedly fruitful, and how light may be shed on one part of the field of knowledge from another apparently remote: and even if any particular branch of scientific pursuit could be shewn to be devoid of even this indirect utility, it would still deserve some respect on utilitarian grounds; both as furnishing to the inquirer the refined and innocent pleasures of curiosity, and because the intellectual disposition which it exhibits and sustains is likely on the whole to produce fruitful knowledge. Still in cases approximating to this last, Common Sense is somewhat disposed to complain of the mis-direction of valuable effort; so that the meed of honour commonly paid to Science seems to be
- 31. graduated, though perhaps unconsciously, by a tolerably exact utilitarian scale. Certainly the moment the legitimacy of any branch of scientific inquiry is seriously disputed, as in the recent case of vivisection, the controversy on both sides is generally conducted on an avowedly utilitarian basis. “The case of Virtue requires special consideration: since the encouragement in each other of virtuous impulses and dispositions is a main aim of men’s ordinary moral discourse; so that even to raise the question whether this encouragement can go too far has a paradoxical air. Still, our experience includes rare and exceptional cases in which the concentration of effort on the cultivation of virtue has seemed to have effects adverse to general happiness, through being intensified to the point of moral fanaticism, and so involving a neglect of other conditions of happiness. If, then, we admit as actual or possible such ‘infelicific’ effects of the cultivation of Virtue, I think we shall also generally admit that, in the case supposed, conduciveness to general happiness should be the criterion for deciding how far the cultivation of Virtue should be carried.” There we have Prof. Sidgwick’s argument completed. We ought not, he thinks, to aim at knowing the Truth, or at contemplating Beauty, except in so far as such knowledge or such contemplation contributes to increase the pleasure or to diminish the pain of sentient beings. Pleasure alone is good for its own sake: knowledge of the Truth is good only as a means to pleasure. 52. Let us consider what this means. What is pleasure? It is certainly something of which we may be conscious, and which, therefore, may be distinguished from our consciousness of it. What I wish first to ask is this: Can it really be said that we value pleasure, except in so far as we are conscious of it? Should we think that the attainment of pleasure, of which we never were and never could be conscious, was something to be aimed at for its own sake? It may be impossible that such pleasure should ever exist, that it should ever be thus divorced from consciousness; although there is certainly much reason to believe that it is not only possible but very
- 32. common. But, even supposing that it were impossible, that is quite irrelevant. Our question is: Is it the pleasure, as distinct from the consciousness of it, that we set value on? Do we think the pleasure valuable in itself, or must we insist that, if we are to think the pleasure good, we must have consciousness of it too? This consideration is very well put by Socrates in Plato’s dialogue Philebus (21 A). ‘Would you accept, Protarchus,’ says Socrates, ‘to live your whole life in the enjoyment of the greatest pleasures?’ ‘Of course I would,’ says Protarchus. Socrates. Then would you think you needed anything else besides, if you possessed this one blessing in completeness? Protarchus. Certainly not. Socrates. Consider what you are saying. You would not need to be wise and intelligent and reasonable, nor anything like this? Would you not even care to keep your sight? Protarchus. Why should I? I suppose I should have all I want, if I was pleased. Socrates. Well, then, supposing you lived so, you would enjoy always throughout your life the greatest pleasure? Protarchus. Of course. Socrates. But, on the other hand, inasmuch as you would not possess intelligence and memory and knowledge and true opinion, you would, in the first place, necessarily be without the knowledge whether you were pleased or not. For you would be devoid of any kind of wisdom. You admit this? Protarchus. I do. The consequence is absolutely necessary. Socrates. Well, then, besides this, not having memory, you must also be unable to remember even that you ever were pleased; of the pleasure which falls upon you at the moment not the least vestige must afterwards remain. And again, not having true opinion,
- 33. you cannot think that you are pleased when you are; and, being bereft of your reasoning faculties, you cannot even have the power to reckon that you will be pleased in future. You must live the life of an oyster, or of some other of those living creatures, whose home is the seas and whose souls are concealed in shelly bodies. Is all this so, or can we think otherwise than this? Protarchus. How can we? Socrates. Well, then, can we think such a life desirable? Protarchus. Socrates, your reasoning has left me utterly dumb.’ Socrates, we see, persuades Protarchus that Hedonism is absurd. If we are really going to maintain that pleasure alone is good as an end, we must maintain that it is good, whether we are conscious of it or not. We must declare it reasonable to take as our ideal (an unattainable ideal it may be) that we should be as happy as possible, even on condition that we never know and never can know that we are happy. We must be willing to sell in exchange for the mere happiness every vestige of knowledge, both in ourselves and in others, both of happiness itself and of every other thing. Can we really still disagree? Can any one still declare it obvious that this is reasonable? That pleasure alone is good as an end? The case, it is plain, is just like that of the colours[18] , only, as yet, not nearly so strong. It is far more possible that we should some day be able to produce the intensest pleasure, without any consciousness that it is there, than that we should be able to produce mere colour, without its being any particular colour. Pleasure and consciousness can be far more easily distinguished from one another, than colour from the particular colours. And yet even if this were not so, we should be bound to distinguish them if we really wished to declare pleasure alone to be our ultimate end. Even if consciousness were an inseparable accompaniment of pleasure, a sine quâ non of its existence, yet, if pleasure is the only end, we are bound to call consciousness a mere means to it, in any intelligible sense that can be given to the word means. And if, on the other
- 34. hand, as I hope is now plain, the pleasure would be comparatively valueless without the consciousness, then we are bound to say that pleasure is not the only end, that some consciousness at least must be included with it as a veritable part of the end. For our question now is solely what the end is: it is quite another question how far that end may be attainable by itself, or must involve the simultaneous attainment of other things. It may well be that the practical conclusions at which Utilitarians do arrive, and even those at which they ought logically to arrive, are not far from the truth. But in so far as their reason for holding these conclusions to be true is that ‘Pleasure alone is good as an end,’ they are absolutely wrong: and it is with reasons that we are chiefly concerned in any scientific Ethics.
- 35. 53. It seems, then, clear that Hedonism is in error, so far as it maintains that pleasure alone, and not the consciousness of pleasure, is the sole good. And this error seems largely due to the fallacy which I pointed out above in Mill—the fallacy of confusing means and end. It is falsely supposed that, since pleasure must always be accompanied by consciousness (which is, itself, extremely doubtful), therefore it is indifferent whether we say that pleasure or the consciousness of pleasure is the sole good. Practically, of course, it would be indifferent at which we aimed, if it were certain that we could not get the one without the other; but where the question is of what is good in itself—where we ask: For the sake of what is it desirable to get that which we aim at?—the distinction is by no means unimportant. Here we are placed before an exclusive alternative. Either pleasure by itself (even though we can’t get it) would be all that is desirable, or a consciousness of it would be more desirable still. Both these propositions cannot be true; and I think it is plain that the latter is true; whence it follows that pleasure is not the sole good. Still it may be said that, even if consciousness of pleasure, and not pleasure alone, is the sole good, this conclusion is not very damaging to Hedonism. It may be said that Hedonists have always meant by pleasure the consciousness of pleasure, though they have not been at pains to say so; and this, I think is, in the main, true. To correct their formula in this respect could, therefore, only be a matter of practical importance, if it is possible to produce pleasure without producing consciousness of it. But even this importance, which I think our conclusion so far really has, is, I admit, comparatively slight. What I wish to maintain is that even consciousness of pleasure is not the sole good: that, indeed, it is absurd so to regard it. And the chief importance of what has been said so far lies in the fact that the same method, which shews that consciousness of pleasure is more valuable than pleasure, seems also to shew that consciousness of pleasure is itself far less valuable than other things. The supposition that consciousness of pleasure is
- 36. the sole good is due to a neglect of the same distinctions which have encouraged the careless assertion that pleasure is the sole good. The method which I employed in order to shew that pleasure itself was not the sole good, was that of considering what value we should attach to it, if it existed in absolute isolation, stripped of all its usual accompaniments. And this is, in fact, the only method that can be safely used, when we wish to discover what degree of value a thing has in itself. The necessity of employing this method will be best exhibited by a discussion of the arguments used by Prof. Sidgwick in the passage last quoted, and by an exposure of the manner in which they are calculated to mislead. 54. With regard to the second of them, it only maintains that other things, which might be supposed to share with pleasure the attribute of goodness, ‘seem to obtain the commendation of Common Sense, roughly speaking, in proportion to the degree’ of their productiveness of pleasure. Whether even this rough proportion holds between the commendation of Common Sense and the felicific effects of that which it commends is a question extremely difficult to determine; and we need not enter into it here. For, even assuming it to be true, and assuming the judgments of Common Sense to be on the whole correct, what would it shew? It would shew, certainly, that pleasure was a good criterion of right action— that the same conduct which produced most pleasure would also produce most good on the whole. But this would by no means entitle us to the conclusion that the greatest pleasure constituted what was best on the whole: it would still leave open the alternative that the greatest quantity of pleasure was as a matter of fact, under actual conditions, generally accompanied by the greatest quantity of other goods, and that it therefore was not the sole good. It might indeed seem to be a strange coincidence that these two things should always, even in this world, be in proportion to one another. But the strangeness of this coincidence will certainly not entitle us to argue directly that it does not exist—that it is an illusion, due to the fact that pleasure is really the sole good. The coincidence may be susceptible of other explanations; and it would even be our duty to
- 37. accept it unexplained, if direct intuition seemed to declare that pleasure was not the sole good. Moreover it must be remembered that the need for assuming such a coincidence rests in any case upon the extremely doubtful proposition that felicific effects are roughly in proportion to the approval of Common Sense. And it should be observed that, though Prof. Sidgwick maintains this to be the case, his detailed illustrations only tend to shew the very different proposition that a thing is not held to be good, unless it gives a balance of pleasure; not that the degree of commendation is in proportion to the quantity of pleasure. 55. The decision, then, must rest upon Prof. Sidgwick’s first argument—‘the appeal’ to our ‘intuitive judgment after due consideration of the question when fairly placed before it.’ And here it seems to me plain that Prof. Sidgwick has failed, in two essential respects, to place the question fairly before either himself or his reader. (1) What he has to shew is, as he says himself, not merely that ‘Happiness must be included as a part of Ultimate Good.’ This view, he says, ‘ought not to commend itself to the sober judgment of reflective persons.’ And why? Because ‘these objective relations, when distinguished from the consciousness accompanying and resulting from them, are not ultimately and intrinsically desirable.’ Now, this reason, which is offered as shewing that to consider Happiness as a mere part of Ultimate Good does not meet the facts of intuition, is, on the contrary, only sufficient to shew that it is a part of Ultimate Good. For from the fact that no value resides in one part of a whole, considered by itself, we cannot infer that all the value belonging to the whole does reside in the other part, considered by itself. Even if we admit that there is much value in the enjoyment of Beauty, and none in the mere contemplation of it, which is one of the constituents of that complex fact, it does not follow that all the value belongs to the other constituent, namely, the pleasure which we take in contemplating it. It is quite possible that this constituent also has no value in itself; that the value belongs to the whole state, and to that only: so that both the pleasure and the
- 38. contemplation are mere parts of the good, and both of them equally necessary parts. In short, Prof. Sidgwick’s argument here depends upon the neglect of that principle, which I tried to explain in my first chapter and which I said I should call the principle of ‘organic relations[19] .’ The argument is calculated to mislead, because it supposes that, if we see a whole state to be valuable, and also see that one element of that state has no value by itself, then the other element, by itself, must have all the value which belongs to the whole state. The fact is, on the contrary, that, since the whole may be organic, the other element need have no value whatever, and that even if it have some, the value of the whole may be very much greater. For this reason, as well as to avoid confusion between means and end, it is absolutely essential to consider each distinguishable quality, in isolation, in order to decide what value it possesses. Prof. Sidgwick, on the other hand, applies this method of isolation only to one element in the wholes he is considering. He does not ask the question: If consciousness of pleasure existed absolutely by itself, would a sober judgment be able to attribute much value to it? It is, in fact, always misleading to take a whole, that is valuable (or the reverse), and then to ask simply: To which of its constituents does this whole owe its value or its vileness? It may well be that it owes it to none; and, if one of them does appear to have some value in itself, we shall be led into the grave error of supposing that all the value of the whole belongs to it alone. It seems to me that this error has commonly been committed with regard to pleasure. Pleasure does seem to be a necessary constituent of most valuable wholes; and, since the other constituents, into which we may analyse them, may easily seem not to have any value, it is natural to suppose that all the value belongs to pleasure. That this natural supposition does not follow from the premises is certain; and that it is, on the contrary, ridiculously far from the truth appears evident to my ‘reflective judgment.’ If we apply either to pleasure or to consciousness of pleasure the only safe method, that of isolation, and ask ourselves: Could we accept, as a very good thing, that mere consciousness of pleasure, and absolutely nothing else, should exist, even in the greatest quantities?
- 39. I think we can have no doubt about answering: No. Far less can we accept this as the sole good. Even if we accept Prof. Sidgwick’s implication (which yet appears to me extremely doubtful) that consciousness of pleasure has a greater value by itself than Contemplation of Beauty, it seems to me that a pleasurable Contemplation of Beauty has certainly an immeasurably greater value than mere Consciousness of Pleasure. In favour of this conclusion I can appeal with confidence to the ‘sober judgment of reflective persons.’ 56. (2) That the value of a pleasurable whole does not belong solely to the pleasure which it contains, may, I think, be made still plainer by consideration of another point in which Prof. Sidgwick’s argument is defective. Prof. Sidgwick maintains, as we saw, the doubtful proposition, that the conduciveness to pleasure of a thing is in rough proportion to its commendation by Common Sense. But he does not maintain, what would be undoubtedly false, that the pleasantness of every state is in proportion to the commendation of that state. In other words, it is only when you take into account the whole consequences of any state, that he is able to maintain the coincidence of quantity of pleasure with the objects approved by Common Sense. If we consider each state by itself, and ask what is the judgment of Common Sense as to its goodness as an end, quite apart from its goodness as a means, there can be no doubt that Common Sense holds many much less pleasant states to be better than many far more pleasant: that it holds, with Mill, that there are higher pleasures, which are more valuable, though less pleasant, than those which are lower. Prof. Sidgwick might, of course, maintain that in this Common Sense is merely confusing means and ends: that what it holds to be better as an end, is in reality only better as a means. But I think his argument is defective in that he does not seem to see sufficiently plainly that, as far as intuitions of goodness as an end are concerned, he is running grossly counter to Common Sense; that he does not emphasise sufficiently the distinction between immediate pleasantness and conduciveness to pleasure. In order to place fairly before us the question what is good as an end
- 40. we must take states that are immediately pleasant and ask if the more pleasant are always also the better; and whether, if some that are less pleasant appear to be so, it is only because we think they are likely to increase the number of the more pleasant. That Common Sense would deny both these suppositions, and rightly so, appears to me indubitable. It is commonly held that certain of what would be called the lowest forms of sexual enjoyment, for instance, are positively bad, although it is by no means clear that they are not the most pleasant states we ever experience. Common Sense would certainly not think it a sufficient justification for the pursuit of what Prof. Sidgwick calls the ‘refined pleasures’ here and now, that they are the best means to the future attainment of a heaven, in which there would be no more refined pleasures—no contemplation of beauty, no personal affections—but in which the greatest possible pleasure would be obtained by a perpetual indulgence in bestiality. Yet Prof. Sidgwick would be bound to hold that, if the greatest possible pleasure could be obtained in this way, and if it were attainable, such a state of things would be a heaven indeed, and that all human endeavours should be devoted to its realisation. I venture to think that this view is as false as it is paradoxical. 57. It seems to me, then, that if we place fairly before us the question: Is consciousness of pleasure the sole good? the answer must be: No. And with this the last defence of Hedonism has been broken down. In order to put the question fairly we must isolate consciousness of pleasure. We must ask: Suppose we were conscious of pleasure only, and of nothing else, not even that we were conscious, would that state of things, however great the quantity, be very desirable? No one, I think, can suppose it so. On the other hand, it seems quite plain, that we do regard as very desirable, many complicated states of mind in which the consciousness of pleasure is combined with consciousness of other things—states which we call ‘enjoyment of’ so and so. If this is correct, then it follows that consciousness of pleasure is not the sole good, and that many other states, in which it is included as a part, are much better than it. Once we recognise the principle of organic
- 41. unities, any objection to this conclusion, founded on the supposed fact that the other elements of such states have no value in themselves, must disappear. And I do not know that I need say any more in refutation of Hedonism. 58. It only remains to say something of the two forms in which a hedonistic doctrine is commonly held—Egoism and Utilitarianism. Egoism, as a form of Hedonism, is the doctrine which holds that we ought each of us to pursue our own greatest happiness as our ultimate end. The doctrine will, of course, admit that sometimes the best means to this end will be to give pleasure to others; we shall, for instance, by so doing, procure for ourselves the pleasures of sympathy, of freedom from interference, and of self-esteem; and these pleasures, which we may procure by sometimes aiming directly at the happiness of other persons, may be greater than any we could otherwise get. Egoism in this sense must therefore be carefully distinguished from Egoism in another sense, the sense in which Altruism is its proper opposite. Egoism, as commonly opposed to Altruism, is apt to denote merely selfishness. In this sense, a man is an egoist, if all his actions are actually directed towards gaining pleasure for himself; whether he holds that he ought to act so, because he will thereby obtain for himself the greatest possible happiness on the whole, or not. Egoism may accordingly be used to denote the theory that we should always aim at getting pleasure for ourselves, because that is the best means to the ultimate end, whether the ultimate end be our own greatest pleasure or not. Altruism, on the other hand, may denote the theory that we ought always to aim at other people’s happiness, on the ground that this is the best means of securing our own as well as theirs. Accordingly an Egoist, in the sense in which I am now going to talk of Egoism, an Egoist, who holds that his own greatest happiness is the ultimate end, may at the same time be an Altruist: he may hold that he ought to ‘love his neighbour,’ as the best means to being happy himself. And conversely an Egoist, in the other sense, may at the same time be a Utilitarian. He may hold that he ought always to direct his
- 42. efforts towards getting pleasure for himself on the ground that he is thereby most likely to increase the general sum of happiness. 59. I shall say more later about this second kind of Egoism, this anti-altruistic Egoism, this Egoism as a doctrine of means. What I am now concerned with is that utterly distinct kind of Egoism, which holds that each man ought rationally to hold: My own greatest happiness is the only good thing there is; my actions can only be good as means, in so far as they help to win me this. This is a doctrine which is not much held by writers now-a-days. It is a doctrine that was largely held by English Hedonists in the 17th and 18th centuries: it is, for example, at the bottom of Hobbes’ Ethics. But even the English school appear to have made one step forward in the present century: they are most of them now-a-days Utilitarians. They do recognise that if my own happiness is good, it would be strange that other people’s happiness should not be good too. In order fully to expose the absurdity of this kind of Egoism, it is necessary to examine certain confusions upon which its plausibility depends. The chief of these is the confusion involved in the conception of ‘my own good’ as distinguished from ‘the good of others.’ This is a conception which we all use every day; it is one of the first to which the plain man is apt to appeal in discussing any question of Ethics: and Egoism is commonly advocated chiefly because its meaning is not clearly perceived. It is plain, indeed, that the name ‘Egoism’ more properly applies to the theory that ‘my own good’ is the sole good, than that my own pleasure is so. A man may quite well be an Egoist, even if he be not a Hedonist. The conception which is, perhaps, most closely associated with Egoism is that denoted by the words ‘my own interest.’ The Egoist is the man who holds that a tendency to promote his own interest is the sole possible, and sufficient, justification of all his actions. But this conception of ‘my own interest’ plainly includes, in general, very much more than my own pleasure. It is, indeed, only because and in so far as ‘my own
- 43. interest’ has been thought to consist solely in my own pleasure, that Egoists have been led to hold that my own pleasure is the sole good. Their course of reasoning is as follows: The only thing I ought to secure is my own interest; but my own interest consists in my greatest possible pleasure; and therefore the only thing I ought to pursue is my own pleasure. That it is very natural, on reflection, thus to identify my own pleasure with my own interest; and that it has been generally done by modern moralists, may be admitted. But, when Prof. Sidgwick points this out (III. XIV. § 5, Div. III.), he should have also pointed out that this identification has by no means been made in ordinary thought. When the plain man says ‘my own interest,’ he does not mean ‘my own pleasure’—he does not commonly even include this—he means my own advancement, my own reputation, the getting of a better income etc., etc. That Prof. Sidgwick should not have noticed this, and that he should give the reason he gives for the fact that the ancient moralists did not identify ‘my own interest’ with my own pleasure, seems to be due to his having failed to notice that very confusion in the conception of ‘my own good’ which I am now to point out. That confusion has, perhaps, been more clearly perceived by Plato than by any other moralist, and to point it out suffices to refute Prof. Sidgwick’s own view that Egoism is rational. What, then, is meant by ‘my own good’? In what sense can a thing be good for me? It is obvious, if we reflect, that the only thing which can belong to me, which can be mine, is something which is good, and not the fact that it is good. When, therefore, I talk of anything I get as ‘my own good,’ I must mean either that the thing I get is good, or that my possessing it is good. In both cases it is only the thing or the possession of it which is mine, and not the goodness of that thing or that possession. There is no longer any meaning in attaching the ‘my’ to our predicate, and saying: The possession of this by me is my good. Even if we interpret this by ‘My possession of this is what I think good,’ the same still holds: for what I think is that my possession of it is good simply; and, if I think rightly, then the truth is that my possession of it is good simply—not, in any sense,
- 44. my good; and, if I think wrongly, it is not good at all. In short, when I talk of a thing as ‘my own good’ all that I can mean is that something which will be exclusively mine, as my own pleasure is mine (whatever be the various senses of this relation denoted by ‘possession’), is also good absolutely; or rather that my possession of it is good absolutely. The good of it can in no possible sense be ‘private’ or belong to me; any more than a thing can exist privately or for one person only. The only reason I can have for aiming at ‘my own good,’ is that it is good absolutely that what I so call should belong to me—good absolutely that I should have something, which, if I have it, others cannot have. But if it is good absolutely that I should have it, then everyone else has as much reason for aiming at my having it, as I have myself. If, therefore, it is true of any single man’s ‘interest’ or ‘happiness’ that it ought to be his sole ultimate end, this can only mean that that man’s ‘interest’ or ‘happiness’ is the sole good, the Universal Good, and the only thing that anybody ought to aim at. What Egoism holds, therefore, is that each man’s happiness is the sole good—that a number of different things are each of them the only good thing there is—an absolute contradiction! No more complete and thorough refutation of any theory could be desired. 60. Yet Prof. Sidgwick holds that Egoism is rational; and it will be useful briefly to consider the reasons which he gives for this absurd conclusion. ‘The Egoist,’ he says (last Chap. § 1), ‘may avoid the proof of Utilitarianism by declining to affirm,’ either ‘implicitly or explicitly, that his own greatest happiness is not merely the ultimate rational end for himself, but a part of Universal Good.’ And in the passage to which he here refers us, as having there ‘seen’ this, he says: ‘It cannot be proved that the difference between his own happiness and another’s happiness is not for him all-important’ (IV. ii. § 1). What does Prof. Sidgwick mean by these phrases ‘the ultimate rational end for himself,’ and ‘for him all-important’? He does not attempt to define them; and it is largely the use of such undefined phrases which causes absurdities to be committed in philosophy.
- 45. Is there any sense in which a thing can be an ultimate rational end for one person and not for another? By ‘ultimate’ must be meant at least that the end is good-in-itself—good in our undefinable sense; and by ‘rational,’ at least, that it is truly good. That a thing should be an ultimate rational end means, then, that it is truly good in itself; and that it is truly good in itself means that it is a part of Universal Good. Can we assign any meaning to that qualification ‘for himself,’ which will make it cease to be a part of Universal Good? The thing is impossible: for the Egoist’s happiness must either be good in itself, and so a part of Universal Good, or else it cannot be good in itself at all: there is no escaping this dilemma. And if it is not good at all, what reason can he have for aiming at it? how can it be a rational end for him? That qualification ‘for himself’ has no meaning unless it implies ‘not for others’; and if it implies ‘not for others,’ then it cannot be a rational end for him, since it cannot be truly good in itself: the phrase ‘an ultimate rational end for himself’ is a contradiction in terms. By saying that a thing is an end for one particular person, or good for him, can only be meant one of four things. Either (1) it may be meant that the end in question is something which will belong exclusively to him; but in that case, if it is to be rational for him to aim at it, that he should exclusively possess it must be a part of Universal Good. Or (2) it may be meant that it is the only thing at which he ought to aim; but this can only be, because, by so doing, he will do the most he can towards realising Universal Good: and this, in our case, will only give Egoism as a doctrine of means. Or (3) it may be meant that the thing is what he desires or thinks good; and then, if he thinks wrongly, it is not a rational end at all, and, if he thinks rightly, it is a part of Universal Good. Or (4) it may be meant that it is peculiarly appropriate that a thing which will belong exclusively to him should also by him be approved or aimed at; but, in this case, both that it should belong to him and that he should aim at it must be parts of Universal Good: by saying that a certain relation between two things is fitting or appropriate, we can only mean that the existence of that relation is absolutely good in itself (unless it be so as a means, which gives case (2)). By no possible meaning, then, that can be
- 46. given to the phrase that his own happiness is the ultimate rational end for himself can the Egoist escape the implication that his own happiness is absolutely good; and by saying that it is the ultimate rational end, he must mean that it is the only good thing—the whole of Universal Good: and, if he further maintains, that each man’s happiness is the ultimate rational end for him, we have the fundamental contradiction of Egoism—that an immense number of different things are, each of them, the sole good.—And it is easy to see that the same considerations apply to the phrase that ‘the difference between his own happiness and another’s is for him all- important.’ This can only mean either (1) that his own happiness is the only end which will affect him, or (2) that the only important thing for him (as a means) is to look to his own happiness, or (3) that it is only his own happiness which he cares about, or (4) that it is good that each man’s happiness should be the only concern of that man. And none of these propositions, true as they may be, have the smallest tendency to shew that if his own happiness is desirable at all, it is not a part of Universal Good. Either his own happiness is a good thing or it is not; and, in whatever sense it may be all- important for him, it must be true that, if it is not good, he is not justified in pursuing it, and that, if it is good, everyone else has an equal reason to pursue it, so far as they are able and so far as it does not exclude their attainment of other more valuable parts of Universal Good. In short it is plain that the addition of ‘for him’ ‘for me’ to such words as ‘ultimate rational end,’ ‘good,’ ‘important’ can introduce nothing but confusion. The only possible reason that can justify any action is that by it the greatest possible amount of what is good absolutely should be realised. And if anyone says that the attainment of his own happiness justifies his actions, he must mean that this is the greatest possible amount of Universal Good which he can realise. And this again can only be true either because he has no power to realise more, in which case he only holds Egoism as a doctrine of means; or else because his own happiness is the greatest amount of Universal Good which can be realised at all, in which case we have Egoism proper, and the flagrant contradiction that every
- 47. person’s happiness is singly the greatest amount of Universal Good which can be realised at all. 61. It should be observed that, since this is so, ‘the relation of Rational Egoism to Rational Benevolence,’ which Prof. Sidgwick regards ‘as the profoundest problem of Ethics’ (III. XIII. § 5, n. 1), appears in quite a different light to that in which he presents it. ‘Even if a man,’ he says, ‘admits the self-evidence of the principle of Rational Benevolence, he may still hold that his own happiness is an end which it is irrational for him to sacrifice to any other; and that therefore a harmony between the maxim of Prudence and the maxim of Rational Benevolence must be somehow demonstrated, if morality is to be made completely rational. This latter view is that which I myself hold’ (last Chap. § 1). Prof. Sidgwick then goes on to shew ‘that the inseparable connection between Utilitarian Duty and the greatest happiness of the individual who conforms to it cannot be satisfactorily demonstrated on empirical grounds’ (Ib. § 3). And the final paragraph of his book tells us that, since ‘the reconciliation of duty and self-interest is to be regarded as a hypothesis logically necessary to avoid a fundamental contradiction in one chief department of our thought, it remains to ask how far this necessity constitutes a sufficient reason for accepting this hypothesis[20] ’ (Ib. § 5). To ‘assume the existence of such a Being, as God, by the consensus of theologians, is conceived to be’ would, he has already argued, ensure the required reconciliation; since the Divine Sanctions of such a God ‘would, of course, suffice to make it always every one’s interest to promote universal happiness to the best of his knowledge’ (Ib. § 5). Now what is this ‘reconciliation of duty and self-interest,’ which Divine Sanctions could ensure? It would consist in the mere fact that the same conduct which produced the greatest possible happiness of the greatest number would always also produce the greatest possible happiness of the agent. If this were the case (and our empirical knowledge shews that it is not the case in this world), ‘morality’ would, Prof. Sidgwick thinks, be ‘completely rational’: we should avoid ‘an ultimate and fundamental contradiction in our
- 48. apparent intuitions of what is Reasonable in conduct.’ That is to say, we should avoid the necessity of thinking that it is as manifest an obligation to secure our own greatest Happiness (maxim of Prudence), as to secure the greatest Happiness on the whole (maxim of Benevolence). But it is perfectly obvious we should not. Prof. Sidgwick here commits the characteristic fallacy of Empiricism— the fallacy of thinking that an alteration in facts could make a contradiction cease to be a contradiction. That a single man’s happiness should be the sole good, and that also everybody’s happiness should be the sole good, is a contradiction which cannot be solved by the assumption that the same conduct will secure both: it would be equally contradictory, however certain we were that that assumption was justified. Prof. Sidgwick strains at a gnat and swallows a camel. He thinks the Divine Omnipotence must be called into play to secure that what gives other people pleasure should also give it to him—that only so can Ethics be made rational; while he overlooks the fact that even this exercise of Divine Omnipotence would leave in Ethics a contradiction, in comparison with which his difficulty is a trifle—a contradiction, which would reduce all Ethics to mere nonsense, and before which the Divine Omnipotence must be powerless to all eternity. That each man’s happiness should be the sole good, which we have seen to be the principle of Egoism, is in itself a contradiction: and that it should also be true that the Happiness of all is the sole good, which is the principle of Universalistic Hedonism, would introduce another contradiction. And that these propositions should all be true might well be called ‘the profoundest problem in Ethics’: it would be a problem necessarily insoluble. But they cannot all be true, and there is no reason, but confusion, for the supposition that they are. Prof. Sidgwick confuses this contradiction with the mere fact (in which there is no contradiction) that our own greatest happiness and that of all do not seem always attainable by the same means. This fact, if Happiness were the sole good, would indeed be of some importance; and, on any view, similar facts are of importance. But they are nothing but instances of the one important fact that in this world the quantity of good which is attainable is ridiculously small compared to that which
- 49. is imaginable. That I cannot get the most possible pleasure for myself, if I produce the most possible pleasure on the whole, is no more the profoundest problem of Ethics, than that in any case I cannot get as much pleasure altogether as would be desirable. It only states that, if we get as much good as possible in one place, we may get less on the whole, because the quantity of attainable good is limited. To say that I have to choose between my own good and that of all is a false antithesis: the only rational question is how to choose between my own and that of others, and the principle on which this must be answered is exactly the same as that on which I must choose whether to give pleasure to this other person or to that. 62. It is plain, then, that the doctrine of Egoism is self- contradictory; and that one reason why this is not perceived, is a confusion with regard to the meaning of the phrase ‘my own good.’ And it may be observed that this confusion and the neglect of this contradiction are necessarily involved in the transition from Naturalistic Hedonism, as ordinarily held, to Utilitarianism. Mill, for instance, as we saw, declares: ‘Each person, so far as he believes it to be attainable, desires his own happiness’ (p. 53). And he offers this as a reason why the general happiness is desirable. We have seen that to regard it as such, involves, in the first place, the naturalistic fallacy. But moreover, even if that fallacy were not a fallacy, it could only be a reason for Egoism and not for Utilitarianism. Mill’s argument is as follows: A man desires his own happiness; therefore his own happiness is desirable. Further: A man desires nothing but his own happiness; therefore his own happiness is alone desirable. We have next to remember, that everybody, according to Mill, so desires his own happiness: and then it will follow that everybody’s happiness is alone desirable. And this is simply a contradiction in terms. Just consider what it means. Each man’s happiness is the only thing desirable: several different things are each of them the only thing desirable. This is the fundamental contradiction of Egoism. In order to think that what his arguments tend to prove is not Egoism but Utilitarianism, Mill must think that he
- 50. can infer from the proposition ‘Each man’s happiness is his own good,’ the proposition ‘The happiness of all is the good of all’; whereas in fact, if we understand what ‘his own good’ means, it is plain that the latter can only be inferred from ‘The happiness of all is the good of each.’ Naturalistic Hedonism, then, logically leads only to Egoism. Of course, a Naturalist might hold that what we aimed at was simply ‘pleasure’ not our own pleasure; and that, always assuming the naturalistic fallacy, would give an unobjectionable ground for Utilitarianism. But more commonly he will hold that it is his own pleasure he desires, or at least will confuse this with the other; and then he must logically be led to adopt Egoism and not Utilitarianism. 63. The second cause I have to give why Egoism should be thought reasonable, is simply its confusion with that other kind of Egoism—Egoism as a doctrine of means. This second Egoism has a right to say: You ought to pursue your own happiness, sometimes at all events; it may even say: Always. And when we find it saying this we are apt to forget its proviso: But only as a means to something else. The fact is we are in an imperfect state; we cannot get the ideal all at once. And hence it is often our bounden duty, we often absolutely ‘ought,’ to do things which are good only or chiefly as means: we have to do the best we can, what is absolutely ‘right,’ but not what is absolutely good. Of this I shall say more hereafter. I only mention it here because I think it is much more plausible to say that we ought to pursue our own pleasure as a means than as an end, and that this doctrine, through confusion, lends some of its plausibility to the utterly different doctrine of Egoism proper: My own greatest pleasure is the only good thing. 64. So much for Egoism. Of Utilitarianism not much need be said; but two points may seem deserving of notice. The first is that this name, like that of Egoism, does not naturally suggest that all our actions are to be judged according to the degree in which they are a means to pleasure. Its natural meaning is that the standard of right and wrong in conduct is its tendency to
- 51. promote the interest of everybody. And by interest is commonly meant a variety of different goods, classed together only because they are what a man commonly desires for himself, so far as his desires have not that psychological quality which is meant by ‘moral.’ The ‘useful’ thus means, and was in ancient Ethics systematically used to mean, what is a means to the attainment of goods other than moral goods. It is quite an unjustifiable assumption that these goods are only good as means to pleasure or that they are commonly so regarded. The chief reason for adopting the name ‘Utilitarianism’ was, indeed, merely to emphasize the fact that right and wrong conduct must be judged by its results—as a means, in opposition to the strictly Intuitionistic view that certain ways of acting were right and others wrong, whatever their results might be. In thus insisting that what is right must mean what produces the best possible results Utilitarianism is fully justified. But with this correct contention there has been historically, and very naturally, associated a double error. (1) The best possible results were assumed to consist only in a limited class of goods, roughly coinciding with those which were popularly distinguished as the results of merely ‘useful’ or ‘interested’ actions; and these again were hastily assumed to be good only as means to pleasure. (2) The Utilitarians tend to regard everything as a mere means, neglecting the fact that some things which are good as means are also good as ends. Thus, for instance, assuming pleasure to be a good, there is a tendency to value present pleasure only as a means to future pleasure, and not, as is strictly necessary if pleasure is good as an end, also to weigh it against possible future pleasures. Much utilitarian argument involves the logical absurdity that what is here and now, never has any value in itself, but is only to be judged by its consequences; which again, of course, when they are realised, would have no value in themselves, but would be mere means to a still further future, and so on ad infinitum. The second point deserving notice with regard to Utilitarianism is that, when the name is used for a form of Hedonism, it does not commonly, even in its description of its end, accurately distinguish
- 52. between means and end. Its best-known formula is that the result by which actions are to be judged is ‘the greatest happiness of the greatest number.’ But it is plain that, if pleasure is the sole good, provided the quantity be equally great, an equally desirable result will have been obtained whether it be enjoyed by many or by few, or even if it be enjoyed by nobody. It is plain that, if we ought to aim at the greatest happiness of the greatest number, this can only, on the hedonistic principle, be because the existence of pleasure in a great number of persons seems to be the best means available for attaining the existence of the greatest quantity of pleasure. This may actually be the case; but it is fair to suspect that Utilitarians have been influenced, in their adoption of the hedonistic principle, by this failure to distinguish clearly between pleasure or consciousness of pleasure and its possession by a person. It is far easier to regard the possession of pleasure by a number of persons as the sole good, than so to regard the mere existence of an equally great quantity of pleasure. If, indeed, we were to take the Utilitarian principle strictly, and to assume them to mean that the possession of pleasure by many persons was good in itself, the principle is not hedonistic: it includes as a necessary part of the ultimate end, the existence of a number of persons, and this will include very much more than mere pleasure. Utilitarianism, however, as commonly held, must be understood to maintain that either mere consciousness of pleasure, or consciousness of pleasure together with the minimum adjunct which may be meant by the existence of such consciousness in at least one person, is the sole good. This is its significance as an ethical doctrine; and as such it has already been refuted in my refutation of Hedonism. The most that can be said for it is that it does not seriously mislead in its practical conclusions, on the ground that, as an empirical fact, the method of acting which brings most good on the whole does also bring most pleasure. Utilitarians do indeed generally devote most of their arguments to shewing that the course of action which will bring most pleasure is in general such as common sense would approve. We have seen that Prof. Sidgwick
- 53. appeals to this fact as tending to shew that pleasure is the sole good; and we have also seen that it does not tend to shew this. We have seen how very flimsy the other arguments advanced for this proposition are; and that, if it be fairly considered by itself, it appears to be quite ridiculous. And, moreover, that the actions which produce most good on the whole do also produce most pleasure is extremely doubtful. The arguments tending to shew it are all more or less vitiated by the assumption that what appear to be necessary conditions for the attainment of most pleasure in the near future, will always continue so to be. And, even with this vicious assumption, they only succeed in making out a highly problematical case. How, therefore, this fact is to be explained, if it be a fact, need not concern us. It is sufficient to have shewn that many complex states of mind are much more valuable than the pleasure they contain. If this be so, no form of Hedonism can be true. And, since the practical guidance afforded by pleasure as a criterion is small in proportion as the calculation attempts to be accurate, we can well afford to await further investigation, before adopting a guide, whose utility is very doubtful and whose trustworthiness we have grave reason to suspect. 65. The most important points which I have endeavoured to establish in this chapter are as follows. (1) Hedonism must be strictly defined as the doctrine that ‘Pleasure is the only thing which is good in itself’: this view seems to owe its prevalence mainly to the naturalistic fallacy, and Mill’s arguments may be taken as a type of those which are fallacious in this respect; Sidgwick alone has defended it without committing this fallacy, and its final refutation must therefore point out the errors in his arguments (36-38). (2) Mill’s ‘Utilitarianism’ is criticised: it being shewn (a) that he commits the naturalistic fallacy in identifying ‘desirable’ with ‘desired’; (b) that pleasure is not the only object of desire. The common arguments for Hedonism seem to rest on these two errors (39-44). (3) Hedonism is considered as an ‘Intuition,’ and it is pointed out (a) that Mill’s allowance that some pleasures are inferior in quality to others implies both that it is an Intuition and that it is a false one (46-48);
- 54. (b) that Sidgwick fails to distinguish ‘pleasure’ from ‘consciousness of pleasure,’ and that it is absurd to regard the former, at all events, as the sole good (49-52); (c) that it seems equally absurd to regard ‘consciousness of pleasure’ as the sole good, since, if it were so, a world in which nothing else existed might be absolutely perfect: Sidgwick fails to put to himself this question, which is the only clear and decisive one (53-57). (4) What are commonly considered to be the two main types of Hedonism, namely, Egoism and Utilitarianism, are not only different from, but strictly contradictory of, one another; since the former asserts ‘My own greatest pleasure is the sole good,’ the latter ‘The greatest pleasure of all is the sole good.’ Egoism seems to owe its plausibility partly to the failure to observe this contradiction—a failure which is exemplified by Sidgwick; partly to a confusion of Egoism as doctrine of end, with the same as doctrine of means. If Hedonism is true, Egoism cannot be so; still less can it be so, if Hedonism is false. The end of Utilitarianism, on the other hand, would, if Hedonism were true, be, not indeed the best conceivable, but the best possible for us to promote; but it is refuted by the refutation of Hedonism (58-64).
- 55. CHAPTER IV. METAPHYSICAL ETHICS. 66. In this chapter I propose to deal with a type of ethical theory which is exemplified in the ethical views of the Stoics, of Spinoza, of Kant, and especially of a number of modern writers, whose views in this respect are mainly due to the influence of Hegel. These ethical theories have this in common, that they use some metaphysical proposition as a ground for inferring some fundamental proposition of Ethics. They all imply, and many of them expressly hold, that ethical truths follow logically from metaphysical truths— that Ethics should be based on Metaphysics. And the result is that they all describe the Supreme Good in metaphysical terms. What, then, is to be understood by ‘metaphysical’? I use the term, as I explained in Chapter II., in opposition to ‘natural.’ I call those philosophers preeminently ‘metaphysical’ who have recognised most clearly that not everything which is is a ‘natural object.’ ‘Metaphysicians’ have, therefore, the great merit of insisting that our knowledge is not confined to the things which we can touch and see and feel. They have always been much occupied, not only with that other class of natural objects which consists in mental facts, but also with the class of objects or properties of objects, which certainly do not exist in time, are not therefore parts of Nature, and which, in fact, do not exist at all. To this class, as I have said, belongs what we mean by the adjective ‘good.’ It is not goodness, but only the things or qualities which are good, which can exist in time—can have
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