基于MATLAB的非线性系统鲁棒故障诊断方法

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"Robust Observer-Based Fault Diagnosis for Nonlinear Systems Using MATLAB" 是一本专注于利用MATLAB进行非线性系统鲁棒观测器故障诊断的工业控制领域专业书籍。作者Jian Zhang, Akshya Kumar Swain 和 Sing Kiong Nguang都是电气与计算机工程领域的专家，来自新西兰奥克兰大学。在非线性系统中，故障诊断是一项挑战性的任务，因为系统的复杂行为使得故障模式难以预测和处理。这本书提供了一种基于鲁棒观测器的方法来解决这个问题。鲁棒观测器设计的目标是确保在系统存在不确定性或扰动的情况下，仍然能够准确地估计系统的状态，从而实现对故障的有效检测和隔离。 MATLAB作为一种强大的数值计算和建模工具，被广泛应用于控制系统的设计和分析。本书利用MATLAB的工具和功能，如Simulink，为读者提供了实现非线性系统故障诊断的实用步骤和示例。通过实例和代码，读者可以学习如何构建和应用鲁棒观测器，以识别并处理非线性系统中的潜在故障。书中的内容可能涵盖以下几个方面： 1. 非线性系统理论基础：介绍非线性动态系统的基本概念、特性以及与线性系统的主要区别。 2. 鲁棒观测器设计：详细阐述鲁棒观测器的数学模型，包括设计方法、稳定性分析以及如何考虑系统的不确定性。 3. MATLAB实现：解释如何利用MATLAB编程实现鲁棒观测器，包括编写脚本和使用Simulink建模。 4. 故障检测与隔离：讨论如何通过观测器的数据来识别系统故障，以及如何实现故障隔离，避免故障影响整个系统性能。 5. 实际应用案例：可能包含多个工业应用实例，如机械、航空航天或化工系统的故障诊断问题，展示理论在实际中的应用。 6. 验证与仿真：提供仿真实验，验证所提出的鲁棒观测器在不同条件下的性能。通过阅读本书，读者将能够掌握非线性系统故障诊断的关键技术和工具，从而在实际工程中有效地提升系统可靠性与安全性。此外，附带的在线材料可能包括额外的示例、代码和其他补充资源，以进一步增强学习体验。

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1.1 Fault Diagnosis Methodologies 3

physical redundancy dramatically increases system size, expense of extra equipment,

and maintenance cost [7]. And sometimes it may cause complex problems when

incorporated with other redundant devices.

Instead of using extra hardware to create redundancy, the analytical redundancy

approach uses analytical or mathematical model of the system to to track the changes

in the plant dynamics. The main idea behind this method is to generate directional

residuals by failure detection ﬁlters. Different fault effects can be mapped into differ-

ent directions or planes in the residual vector space so that FDI can be achieved [6].

The existing analytical redundancy fault diagnosis approaches can be broadly divided

into knowledge-based FDI methods, signal-based FDI methods, and model-based

FDI methods:

1. Knowledge-based FDI methods

The knowledge-based methods of fault diagnosis are essentially developed from

the heuristic symptoms. These are obtained either from expert human operators, or

from a qualitative model. In some of the knowledge-based methods, the model is

built up by expert reasoning [18], fuzzy reasoning [19 ], and neural networks [20,

21], for mapping the inputs and outputs of the unknown system. In many other

knowledge-based methods, measurement data is mapped to a known pattern which

includes different normal and abnormal operating conditions directly, so that the

system condition can be identiﬁed. One of the major advantages of knowledge-based

methods is that they do not require an explicit mathematical model of the monitored

system. However, they require, in advance, the knowledge of the monitored plants,

knowledge such as the training data which contains faults and the corresponding

symptoms, under different faulty conditions.

2. Signal-based FDI methods

Many signal-based FDI methods have been developed and can be divided into two

categories: spectral analysis [22] (time-frequency, time-scale analysis, etc.) and sta-

tistical methods [23] (signal classiﬁcation, pattern recognition, etc.). These methods

extract proper signals or symptoms from the system such as spectral power densities,

correlation coefﬁcients, and covariances, for the analysis of faults. Although, unlike

knowledge-based methods, signal-based techniques do not require a complete ana-

lytical model, their efﬁciency is particularly limited for early fault detection and for

the detection of faults which occur during transient operation.

3. Model-based FDI methods

With the development of digital computers and system identiﬁcation techniques,

model-based FDI methods have received considerable attention in recent years [7,

24–26]. Unlike physical redundancy methods, where measurements from parallel

components are compared to each other, model-based FDI methods utilize analyt-

ically computed outputs and compare the values with the sensor measurements to

indicate the presence of faults in systems of interest. The resulting differences are

called residuals. These signals are supposed to be zero when the system is fault free

and nonzero when the system is faulty. Model-based FDI methods comprise two

principal steps: residual generation and residual evaluation. The most important and

4 1 Introduction

difﬁcult task in model-based fault diagnosis is the generation of residuals. Corre-

sponding to different residual generation techniques, model-based FDI methods that

have been developed in the literature can be divided into three groups [27]:

(1) Parity-equation approach [1, 28]: This approach is based on the test of consis-

tency of parity equations using measurements and inputs. Any inconsistency of

the parity equations can be used to detect the faults.

(2) Parameter-estimation approach [20, 29, 30]: This approach is based on the

assumption that the occurrence of any fault will change the values of the physical

system parameters such as mass, friction, resistance, etc. The parameters of the

actual process can be repeatedly estimated using on-line parameter-estimation

methods. A fault can be declared if there exist discrepancies between the true

values of the system parameters and their estimated values.

(3) Observer-based approach [8, 31–38]: Observer-based approach is the most exten-

sively used method in model-based fault diagnosis. The basic idea behind this

approach is to use observers or ﬁlters to estimate system outputs from measure-

ments. The output estimation errors are taken as residuals. Compared with other

approaches, the observer-based approach can be implemented using only on-line

measurements and offers more design freedom, and therefore attracted the most

interests during the past decade.

In this book, we will focus on model-based fault diagnosis methods. More specif-

ically, observer-based fault diagnosis methods will be the main concern.

1.2 Robust Observer-Based Fault Diagnosis: An Overview

Owing to the high dependence of the model-based fault diagnosis methods to the

corresponding mathematical models, a major downside of this class of approaches

is that they require an accurate mathematical model of the considered system, which

is often difﬁcult to obtain in many practical situations. Moreover, the system para-

meters often vary during the process and the characteristics of disturbances are also

unknown. The existence of system uncertainties and disturbances can cause a mis-

leading alarm and therefore make the model-based fault diagnosis system ineffec-

tive [39]. Consequently, it is vital to take the issue of robustness into account while

designing observer-based fault diagnosis systems. A robust fault diagnosis system

should have the ability to be sensitive to fault signals but insensitive to other signals.

During the past two decades, a lot of robust observer-based fault diagnosis methods

have been developed, and some of which are described as follows:

1. Beard–Jones fault detection ﬁlter (BJFDF)-based fault diagnosis. The devel-

opment of the observer-based FDI began in the early 1970 s when BJFDF was

ﬁrst introduced in [40, 41]. The main idea behind BJFDF is that each directional

residual is designed in correspondence to a particular fault or a particular group

of faults. This approach was reﬁned in a geometric framework in [42]. The design

1.2 Robust Observer-Based Fault Diagnosis: An Overview 5

problem of BJFDF was later investigated in [43] using a spectral approach and in

[44] using eigenstructure assignment. Some methods to improve the robustness

of BJFDF have been proposed in [45–47].

2. Unknown-input observer (UIO)-based fault diagnosis. The design of

observers for systems subject to unknown-inputs has attracted considerable atten-

tion in the past and different UIOs have been developed. For example, in [48]

full-order linear UIOs were designed while in [49] reduced-order linear UIOs

were designed; Nonlinear UIOs were designed for bilinear systems in [50] and

for Lipschitz nonlinear systems in [51, 52].

If uncertainties are treated as unknown-inputs, UIOs can be readily used for fault

diagnosis. In [53–55], fault diagnosis schemes based on UIO were proposed for

linear systems with uncertainties. For special classes of nonlinear systems such

as bilinear systems and Lipschitz nonlinear systems, UIO-based fault diagnosis

were designed in [56, 57]. Linear UIO was extended to a more general class of

nonlinear systems using a nonlinear state transformation, and was applied to fault

diagnosis for uncertain nonlinear systems in [58]. It is worth noting that most

existing UIO-based schemes assume that the fault distribution matrix i s known,

which is often not the case, and many of them are only devoted to FD or single FI.

3. Adaptive observer (AO)-based fault diagnosis. The AO has the ability that

they can estimate not only system states, but also the slowly varying unknown

parameters of the observed systems. In [59–61], systems are assumed to be known

and faults can be properly parameterized. In [62, 63], AO was designed for systems

with unknown parameters. The main disadvantage of AO-based fault diagnosis

methods is that the they are normally only suitable for the constant fault case.

However, faults are often time-varying and even fast time-varying sometimes.

4. Sliding-mode observer (SMO)-based fault diagnosis. Due to the inherent

robustness of sliding-mode algorithms to unknown modeling uncertainties and

disturbances, the fault diagnosis methods based on SMOs have been widely stud-

ied in recent years. Considerable success has been achieved in many areas; for

example, see [8, 39, 64–70].

Early work on applying the SMO for fault diagnosis was shown in [71] where an

SMO is designed with the assumption that the states of the system are available. In

[66, 67], the authors attempted t o design an SMO for systems with uncertainties.

When a fault occurs, the sliding motion will be destroyed and the residual will

deviate from zero. On the other hand, the SMO proposed in [8], which is similar

to that of [72], can maintain the sliding mode even after the presence of faults by

selecting an appropriate gain. Therefore, the constant actuator faults and sensor

faults can be reconstructed by the so-called equivalent output injection concept

under certain conditions. This result was extended to a more general case in [73]

where the derivative of the sensor fault is nonzero. However, the requirement

of a complicated coordinate transformation and that the system is accurately

known and limits its application. The assumption of open-loop stability in [73]

was later relaxed in [68] to achieve robust sensor-fault estimation using a linear

matrix inequality (LMI) formulation. In [74

], a nonlinear diffeomorphism was

introduced to explore the system structure and the sensor fault was transformed

6 1 Introduction

into a pseudo-actuator-fault scenario using a ﬁlter. An SMO was then designed

to estimate the sensor fault based on the ﬁltered system. In [75], the authors

proposed a scheme to estimate incipient sensor faults for both open-loop stable

and unstable systems. The problem of early detection of incipient faults using

SMOs is also studied in [76]. A high-order SMO was designed in [69] to estimate

sensor faults. It has been shown in [77, 78] that if certain conditions on fault and

uncertainty distribution matrices are met, then the system uncertainties can be

perfectly decoupled from faults and the reconstruction of actuator faults can be

achieved. If this condition is not satisﬁed, the approach must settle for minimizing

the effect of the uncertainties on the fault estimation. In [68], an SMO-based FDI

scheme which can minimize the L

gain between the uncertainty and the fault

reconstruction signal was proposed for a class of linear systems with uncertainties.

1.3 Outline of the Book

In this book, we restrict our attention to design novel observers to detect, isolate

and estimate both actuator and sensor faults. As most plants are inherently nonlinear

and the faults may often amplify the nonlinearities by driving the plants from a

relatively linear operating point into a more nonlinear operating region, the study

of fault diagnosis for nonlinear systems has become a very active research topic

for both theoretical and practical reasons. The presence of nonlinearities in control

systems constitutes a major challenge to observer-based fault diagnosis. Due to the

importance and practicality of Lipschitz nonlinear systems, we consider this class

of nonlinear system throughout this book. Detailed discussions on the backgroud of

each problem and literature surveys may be found in the introduction of each chapter.

In order to improve the readability of the book and facilitate the partial readings, we

write the chapters in a self-sufﬁcient format. The rest of the book is organized as

follows:

The book starts with an introduction of actuator FDI in Chap. 2.AnSMOis

proposed based on a constrained Lyapunov equation and the sufﬁcient conditions

for the design of such an observer have been derived. The fault signal is estimated

by employing the concept of equivalent output error injection. The estimated fault

signal provides information about the occurrence of a fault as well as about the size

and severity of the fault. Under certain conditions, the actuator fault can be precisely

reconstructed. The effectiveness of the proposed scheme is illustrated considering

the example of a modiﬁed seventh-order aircraft model.

Compared with actuator FDI, very little research has been carried out on the

sensor fault diagnosis, especially for the incipient sensor fault diagnosis. Therefore,

a robust sensor FDI scheme is proposed in Chap. 3. The essential idea behind the

proposed scheme is to employ a coordinate transformation s uch that the sensor faults

can be separated from system uncertainties. More speciﬁcally, after the coordinate

transformation of the original system, one of the subsystems is only subject to sensor

faults , but without modeling uncertainties, which only appears in another subsystem.

1.3 Outline of the Book 7

Based on the transformed system and the integral observer-based approach, multiple

sensor faults are detected using the classical Luenberger observer and then isolated

using a bank of SMOs.

FDI can only indicates when and where a fault occurs in the system, but can not

provide the magnitude, shape, and duration of the fault. In the following Chaps. 4–8,

a more comprehensive topic, fault estimation, is studied. We start to discuss this topic

with a comparatively simpler situation where there occur only actuators or sensor

faults occurring at a particular time. In Chap. 4, actuator FE is ﬁrst discussed. A new

SMO-based FE scheme which is inspired by the work presented in [78] is developed.

Then the results are extended to sensor FE. In Chap. 5, two methods are proposed

for sensor FE. In the ﬁrst method, the sensor fault is estimated using SMOs, while

in the second method, it is estimated based on adaptive technique.

Chapters 4 and 5 deal separately with the actuator faults and sensor faults. In

Chap. 6, the situation when both actuator faults and sensor faults occur in a process

simultaneously is studied. By integrating H

∞

criteria, the proposed observer is not

only capable of estimating the states and faults, but also minimizes the H

∞

gain

between the estimation error and system uncertainties. Since the integrated H

∞

ﬁltering ensures more accurate state estimations, the fault estimation is much more

robust against the system uncertainties, compared with the fault estimation obtained

from observers without H

∞

ﬁltering.

The essence behind the methods of Chap. 6 is to employ two cascade coordinate

transformations to transform the system matrices to a special structure. However, the

use of such transformations may bring complexities to the design of observers. In

Chap. 7, we develop a different FE method which consists of an SMO and an UIO

for the same faulty system.

A descriptor system approach is introduced to investigate the simultaneous estima-

tion of actuator fault and sensor fault in Chap. 8. The descriptor system is constructed

by taking sensor faults as auxiliary states. An observer together with adaptive tech-

nique is designed for the augmented system to estimate faults. Asymptotic estimates

of the original system states and the sensor faults can be directly obtained from

the descriptor observer. Meanwhile, the actuator faults are obtained using adap-

tive observer-based approach. The difference between the method developed in this

chapter and those proposed in Chaps. 4–7 is that, this approach does not require any

coordinate transformation and there is no need to determine matrices, which satisfy

the matching condition.

Concluding remarks and suggestions for future research are discussed in Chap. 9.

References

1. Gertler JJ (1998) Fault detection and diagnosis in engineering systems. Marcel Dekker, New

York

2. Venkatasubramanian V, Rengaswamy R, Yin K, Kavuri SN (2003) A review of process fault

detection and diagnosis part 1: quantitative model-based methods. Comput Chem Eng 27:293–

311

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