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**Chapter 3 ****Detection and isolation of incipient sensor faults for uncertain nonlinear systems**

In this chapter, a sensor FDI scheme based on SMOs is proposed for the same class of nonlinear systems considered in Chapter 2. The research is carried out to solve not only sensor fault detection, but also sensor fault isolation problems by employing SMOs design.

**Introduction**

With the development of modern technology, autonomous systems are more and more dependent on sensors to acquire system information and signals from sensors often carry the most important information in automated/feedback control systems. A sensor fault may lead to poor regulation or tracking performance, or even affect the stability of the control system. Therefore the study of sensor FDI is becoming increasingly important. However, compared with the study of actuator FDI using SMOs, the research on sensor FDI is less studied in this realm.

Almost all the SMO-based approaches develpoed in the past mainly focus on rela-tively large-sized faults [8, 12, 72, 77, 78, 91]. The research on the detection and isolation of incipient faults has been less studied and still remains a challenge to model-based FDI techniques because they are almost unnoticeable during their ini-tial stage and their effects to residuals are most likely to be concealed by system uncertainties. Incipient faults can cause very serious problems although they may develop slowly and be tolerable when they first appear. Hence it is necessary to de-tect and isolate incipient faults as early as possible to maintain the reliability of the system and this motivates the research reported in this chapter.

The proposed idea is inspired by the work presented in [58]. In [58], two coordinate transformations are introduced such that the original system can be decomposed into two subsystems. Based on the transformed systems, an SMO and a Luenberger observer are designed to eliminate the effects of disturbances and to detect incip-ient actuator faults, respectively. However, the proposed method is only applicable for linear systems and the problem of fault isolation still remains unsolved. In this chapter, the result in [58] for actuator fault detection for LTI systems is extended to sensor fault detection and isolation for Lipschitz nonlinear systems. The proposed method essentially transforms the original system into two subsystems (subsystem-1 and 2) where subsystem-1 includes the effects of system uncertainties but is free from sensor faults and subsystem-2 has sensor faults but without any uncertainties. Sensor faults in subsystem-2 are treated as actuator faults by using integral observer based approach [92]. For the purpose of fault detection, a traditional Luenberger observer is designed for this subsystem. The sensor fault is detected by consider-ing the output estimation error of subsystem-2 as the residual. When this residual goes over a predefined threshold, a fault is detected. The most distinct feature of the proposed FDI scheme is that, by imposing a coordinate transformation to the original system, the effects of system uncertainties to the residual of subsystem-2 are com-pletely de-coupled, which makes the scheme sensitive to incipient faults while robust to modelling uncertainty. Thus, early detection can be achieved and a false alarm caused by modelling uncertainties can be totally avoided.

After a fault is being detected, the next step is to determine the location of the fault, namely fault isolation. In principle the use of one single observer may permit the isolation of faults if their effect has independent projections onto the residual space. However if the system has significant nonlinearities, it is difficult to assure this inde-pendence. Therefore, a bank of observers is needed to isolate faults if they occur on different sensors. There are two schemes for fault isolation. The first one is called dedicated observer scheme [93]. In this scheme, *N* observers are designed to generate *N* residuals and the *ith* residual is expected to be only sensitive to the *ith *fault but insensitive to others. The other scheme is called generalized observer scheme [5], where *N* observers are also designed to produce *N* residuals. However, the difference is that the *ith* residual is sensitive to all possible faults except the *ith* one. In this chapter, the sensor fault isolation is carried out using the modified ded-icated observer scheme to subsystem-2. Multiple observers, one for each possible sensor fault, are used to generate the estimated output vector. The estimated output vector is then compared with the actual output vector in order to determine which sensor is affected by the fault.

The rest of the chapter is organized as follows: Following the introduction, section-3.2 briefly describes the mathematical preliminaries required for designing observers. Section-3.3 proposes a sensor fault detection scheme and derives the stability con-dition of the proposed observers based on Lyapunov approach. The scheme of iso-lating multiple sensor faults is presented in section-3.4. The results of simulation are shown in section-3.5 with conclusions in section-3.6.

residual change is observed. If we design sliding mode obervers directly for the orig-inal system, then the effect of incipient sensor faults on state estimation errors could be attenuated or even eliminated by the variable structure term [58] (the magnitude of the residual obtained will be within the chattering amplitude or smaller than the predefined threshold for a certain length of time if the gain *k*(*:*) in (3.20) is chosen too large). The early detection, even detection of incipient sensor faults therefore be-comes difficult. Observing the structure of subsystem-2 in (3.11), it is found that the state *z*_{0} is neither subject to system uncertainties nor faults before the occurrence of any sensor fault. If we can design an observer for this particular subsystem and take the output estimation error *w*_{3} *¡* *w*^_{3} (*w*^_{3} is the estimation of *w*_{3}) as the residual, then the problem caused by designing conventional sliding mode observers for the origi-nal system can be solved. This intuition inspires the proposed fault detection scheme which is described in this section.

*p ¡ r *components of* e*_{0}, namely* e*_{z}_{3} directly. More specifically, if there occurs a fault,* e*_{z}_{3} will definitely change. The situation that the sensor fault only affects the first n-r components of *e*_{0} does not exist. From Lemma 3.4, *e*_{0} will approach to zero if there is no sensor fault. After the occurrence of any sensor fault, the last *p* *¡r* components of *e*_{0} will deviate from zero. Therefore* ke*_{w}_{3}*k *=* kC*_{0}*e*_{0}*k *=* ke*_{z}_{3}* k *provides a good choice as the residual to detect the occurrence of sensor faults. Accordingly, the sensor fault detection scheme can be devised as follows:

**Sensor fault detection scheme**: Sensor faults can be detected if the residual *ke*_{w}_{3} *k* exceeds a predefined threshold *&*. Otherwise the system is healthy within the consid-ered time. The detection time *t** _{d}* (

*t*

_{d}*‚*

*t*

*) is defined as the first time instant such that*

_{f}*ke*

_{w}_{3}

*k*is observed greater than

*&*.

**Remark**From (3.46), it is easy to see that before the occurrence of any sensor fault, the norm bound of the state estimation error

*e*

_{0}(

*t*) depends on the bound of the unknown initial condition

*e*

_{0}(0). Since

*ke*

_{0}(0)

*k*is multiplied by

*e*

^{(c}

^{0}

^{L}*f*2

^{kT}

^{¡}^{1}

^{k¡a}^{0}

^{)t}, the effect of this bound will decrease exponentially and

*e*

_{0}will approach to zero. It implies that a small threshold

*&*can be selected and the value does not significantly affect the performance of the fault detection scheme.

1 Introduction

1.1 Basic concept of fault diagnosis

1.2 Fault diagnosis methodologies

1.3 Complexities in Model-based fault diagnosis

1.4 Sliding mode observer based fault diagnosis: an overview

1.5 Objectives of the thesisz

2 Reconstruction of actuator faults for uncertain nonlinear systems

2.1 Introduction

2.2 Problem Formulation

2.3 Sliding mode observer design

2.4 Estimation of actuator fault

2.5 Simulation Results

2.6 Conclusions

3 Detection and isolation of incipient sensor faults for uncertain nonlinear systems 37

3.1 Introduction

3.2 Problem Formulation

3.3 Sensor fault detection scheme

3.4 Sensor fault isolation scheme

3.5 Simulation results

3.6 Conclusions

4 Estimation of sensor faults for uncertain nonlinear systems

4.1 Introduction

4.2 Problem Formulation

4.3 Sensor fault reconstruction using sliding mode observers with adaption laws

4.4 Sensor fault estimation using adaptive observer

4.5 Simulation results

4.6 Conclusions

5 Robust H1 filtering for uncertain nonlinear systems with fault estimation synthesis

5.1 Introduction

5.2 Problem Formulation

5.3 Fault estimation using SMO

5.4 Fault estimation using SMO and AO

5.5 Simulation results

5.6 Conclusions

6 Estimation of actuator and sensor faults for uncertain nonlinear systems using a descriptor system approach

6.1 Introduction

6.2 Problem Formulation

6.3 Design of the fault estimation observer

6.4 Simulation results

6.5 Conclusion

7 Conclusions and future work

7.1 Conclusions

7.2 Future work

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ROBUST OBSERVER BASED FAULT DIAGNOSIS FOR NONLINEAR SYSTEMS