A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> We consider a class of stochastic linear systems that are subject to jumps of unknown magnitudes in the state variables occurring at unknown times. This model can be used when considering such problems as estimation for systems subject to possible component failures and the tracking of vehicles capable of abrupt maneuvers. Using Kalman-Bucy filtering and generalized likelihood ratio techniques, we devise an adaptive filtering system for the detection and estimation of the jumps. An example that illustrates the dynamical properties of our filtering scheme is discusssed in detail. <s> BIB001 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> The supervision of technical processes is the subject of increased development because of the increasing demands on reliability and safety. The use of process computers and microcomputers permits the application of methods which result in an earlier detection of process faults than is possible by conventional limit and trend checks. With the aid of process models, estimation and decision methods it is possible to also monitor nonmeasurable variables like process states, process parameters and characteristic quantities. This contribution presents a brief summary of some basic fault detection methods. This is followed by a description of suitable parameter estimation methods for continuous-time models. Then two examples are considered, the fault detection of an electrical driven centrifugal pump by parameter monitoring and the leak detection for pipelines by a special correlation method. <s> BIB002 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> We evaluate the effectiveness of the extended Kalman filter (EKF) for fault detection and diagnosis in closed-loop systems. A distillation column is used to illustrate the procedure. We describe the strategy for fault detection and diagnosis in the closed-loop system. We present the process models and dynamic analysis of accuracy of the model. We describe the implementation of EKF. We apply the strategy to a 30-stage binary distillation column with a partial condenser and a reboiler <s> BIB003 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> The fault detection problem for a navigation system is investigated. Mathematically this problem is formulated as that of the quickest detection of abrupt changes in a stochastic process. For solving this problem the modified cumulative sum algorithm (CUSUM) is proposed. The ability of the proposed algorithm to detect soft (drifting-type) faults and its application for the monitoring of the inertial and the inertial/radio navigation systems are demonstrated. Test results describing the application of CUSUM algorithms to experimental data from test flights of a new airplane are reported. <s> BIB004 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Abstract Detection of current faults in a D.C. motor with unknown load torques may not be feasible with linear methods and threshold logic. This paper suggests an FDI scheme with a bank of observers where a nonlinear observer is designed using the structure of the current fault. Multiple-hypothesis testing of the observer residuals is applied to discriminate between faults and unknown input. The strategy has been applied to a simulation of the diesel engine actuator benchmark problem with an encouraging progress on performance compared to other results. <s> BIB005 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> This paper presents novel approach on applying the standard observer based fault detection technique to the change detection of the output probability density functions for dynamic stochastic systems. For such systems, the control inputs of the system appear as a set of variables in the probability density functions of the system output, and these variables affect the shape of the probability density function of the system output. Using the B-splines approximation theory, the measured probability density functions of the system output are represented by a set of weights which are functions of the control inputs to the system. This leads to a unique expression of the dynamic characteristics of the output probability density functions for the system. Using this expression, it has been shown that standard observer based fault detection technique can be used to detect any unexpected changes caused by the additive type of fault in the dynamic part of the system. In specific, two observers are constructed for the fault detection purposes, where the first observer is based on the linear weighted integration of the output probability density functions whilst the second observer uses the non-linear residual signal generated from the integration of the output probability density functions. In both cases, the convergence of the observers have been proved under certain conditions when there is no fault in the system. An applicability study to the detection of unexpected changes of particle size (i.e. flocculation size) in paper-making is included to demonstrate the use of the proposed algorithm and desired results have been obtained. <s> BIB006 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> 1. Introduction.- 2. Model-based Fault Diagnosis Techniques.- 3. System Identification for Fault Diagnosis.- 4. Residual Generation, Fault Diagnosis and Identification.- 5. Fault Diagnosis Application Studies.- 6. Concluding Remarks.- References. <s> BIB007 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Abstract For the improvement of reliability, safety and efficiency advanced methods of supervision, fault-detection and fault diagnosis become increasingly important for many technical processes. This holds especially for safety related processes like aircraft, trains, automobiles, power plants and chemical plants. The classical approaches are limit or trend checking of some measurable output variables. Because they do not give a deeper insight and usually do not allow a fault diagnosis, model-based methods of fault-detection were developed by using input and output signals and applying dynamic process models. These methods are based, e.g., on parameter estimation, parity equations or state observers. Also signal model approaches were developed. The goal is to generate several symptoms indicating the difference between nominal and faulty status. Based on different symptoms fault diagnosis procedures follow, determining the fault by applying classification or inference methods. This contribution gives a short introduction into the field and shows some applications for an actuator, a passenger car and a combustion engine. <s> BIB008 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> In this work, a novel robust fault detection algorithm is investigated for stochastic distribution systems with multiple uncertainties, where the output is characterised by its measured output probability density function. By constructing an auxiliary augmented stochastic descriptor system, the original stochastic distribution system is transferred into a descriptor system subjected to model uncertainties, where a proportional and derivative descriptor estimator is developed to solve the fault detection problem. The system input and the output probability density function are used in the design of this estimator. Furthermore, the derivative gain of the estimator is chosen to attenuate the output uncertainties, and the free parameters embedded inside the proportional gain are selected to generate an optimally robust residual signal for fault detection so as to achieve a situation where this residual signal is sensitive to system faults while insensitive to model uncertainties, input disturbances and output noises. A numerical example is given, and the simulation result shows satisfactory detection performance. <s> BIB009 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> The paper deals with the robust fault detection problem for Takagi-Sugeno (T-S) fuzzy Ito stochastic systems. Our aim is to develop a robust fault detection approach to the T-S fuzzy systems with Brownian motion. By using a general observer-based fault detection filter as a residual generator, the robust fault detection is formulated as a filtering problem. Attention is focused on the design of both the fuzzy-rule-independent and the fuzzy-rule-dependent fault detection filters guaranteeing a prescribed noise attenuation level in an Hinfin sense. Sufficient conditions are proposed to guarantee the mean-square asymptotic stability with an Hinfin performance for the fault detection system. The corresponding solvability conditions for the desired fuzzy-rule-independent and fuzzy-rule-dependent fault detection filters are also established. Finally, a numerical example is provided to illustrate the effectiveness of the proposed theory. <s> BIB010 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Abstract The paper studies the problem of simultaneously estimating the state and the fault of linear stochastic discrete-time varying systems with unknown inputs. The fault and the unknown inputs affect both the state and the output. However, if the dynamical evolution models of the fault and the unknown inputs are available the filtering problem will be solved by the Optimal three-stage Kalman Filter (OThSKF). The OThSKF is obtained after decoupling the covariance matrices of the Augmented state Kalman Filter (ASKF) using a three-stage U–V transformation. Nevertheless, if the fault and the unknown inputs models are not perfectly known the Robust three-stage Kalman Filter (RThSKF) will be applied to give an unbiased minimum-variance estimation. Finally, a numerical example is given in order to illustrate the proposed filters. <s> BIB011 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> In this paper, a new estimation method for the inner combustion torque of internal combustion engines and a misfire detection algorithm are presented. The torque estimate is based on a parametric Kalman filter using the measurements available on commercial engine testbeds. The combustion torque is modeled as a superposition of specially designed basis functions. Excellent noise attenuation, phase-free estimation, and robust parametrization result from the chosen structure. Misfire detection is achieved by an interacting multiple model approach, where two dedicated Kalman filters for each cylinder represent regular operation and misfire, respectively. This approach enables reliable detection of the actual cylinder state early in the combustion phase. The functionality and performance of the new method are illustrated by means of simulation examples and test-bed results. <s> BIB012 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Structural damage to ball grid array interconnects incurred during vibration testing has been monitored in the prefailure space using resistance spectroscopy-based state space vectors, rate of change of the state variable, and acceleration of the state variable. The technique is intended for condition monitoring in high reliability applications where the knowledge of impending failure is critical and the risks in terms of loss of functionality are too high to bear. Future state of the system has been estimated based on a second-order Kalman Filter model and a Bayesian Framework. The measured state variable has been related to the underlying interconnect damage in the form of inelastic strain energy density. Performance of the prognostic health management algorithm during the vibration test has been quantified using performance evaluation metrics. The methodology has been demonstrated on leadfree area-array electronic assemblies subjected to vibration. Model predictions have been correlated with experimental data. The presented approach is applicable to functional systems where corner interconnects in area-array packages may be often redundant. Prognostic metrics including α - λ precision, β accuracy, and relative accuracy have been used to assess the performance of the damage proxies. The presented approach enables the estimation of residual life based on level of risk averseness. <s> BIB013 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> This paper addresses the fault detection problem for discrete-time Markovian jump systems with incomplete knowledge of transition probabilities, randomly varying nonlinearities and sensor saturations. For the Markovian mode jumping, the transition probability matrix is allowed to have partially unknown entries, while the cases with completely known or completely unknown transition probabilities are also investigated as two special cases. The randomly varying nonlinearities and the sensor saturations are introduced to reflect the limited capacity of the communication networks resulting from the noisy environment, probabilistic communication failures, measurements of limited amplitudes, etc. Two energy norm indices are used for the fault detection problem in order to account for, respectively, the restraint of disturbance and the sensitivity of faults. The purpose of the problem addressed is to design an optimized fault detection filter such that 1) the fault detection dynamics is stochastically stable; 2) the effect from the exogenous disturbance on the residual is attenuated with respect to a minimized H∞ -norm; and 3) the sensitivity of the residual to the fault is enhanced by means of a maximized H∞-norm. The characterization of the gains of the desired fault detection filters is derived in terms of the solution to a convex optimization problem that can be easily solved by using the semi-definite programme method. Finally, a simulation example is employed to show the effectiveness of the fault detection filtering scheme proposed in this paper. <s> BIB014 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> In this paper a Robust Adaptive Kalman Filter (RAKF) is introduced. The RAKF incorporates measurement and process noise covariance adaptation procedures (R and Q adaptation respectively) and utilizes adaptive factors in order to adapt itself against sensor/actuator faults. Thus the filter stands robust against the faults and even in case of sensor/actuator failure keeps providing accurate estimation results. In a single algorithm, the RAKF detects the fault, isolates it and applies the required adaptation process such that the estimation characteristic is not deteriorated. The performance of the proposed RAKF is investigated by simulations for the state estimation procedure of an Unmanned Aerial Vehicle. <s> BIB015 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Multiple-model adaptive estimation techniques have been previously successfully applied to fault diagnosis of microsystems. Their diagnosis performance highly depends on the accuracy of modeling techniques used in representing faults. This paper presents the application of a self-tuning forgetting factor technique in the modeling of faults in MEMS and its effects on diagnosis performance compared with the application of Kalman filters and fixed gain estimation techniques. The self-tuning-based modeling used in the diagnosis algorithm was experimentally implemented. It demonstrated superior results compared to Kalman filter and fixed gain estimation techniques by accelerating the diagnosis process. <s> BIB016 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Interior permanent-magnet synchronous motors (IPMSMs) become attractive candidates in modern hybrid electric vehicles and industrial applications. Usually, to obtain good control performance, the electric drives of this kind of motor require one position, one dc link, and at least two current sensors. Failure of any of these sensors might lead to degraded system performance or even instability. As such, sensor fault resilient control becomes a very important issue in modern drive systems. This paper proposes a novel sensor fault detection and isolation algorithm based on an extended Kalman filter. It is robust to system random noise and efficient in real-time implementation. Moreover, the proposed algorithm is compact and can detect and isolate all the sensor faults for IPMSM drives. Thorough theoretical analysis is provided, and the effectiveness of the proposed approach is proven by extensive experimental results. <s> BIB017 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> The detection of changes in the eigenstructure of a linear time invariant system by means of a subspace-based residual function has been proposed previously. While enjoying some success in its applicability in particular in the context of vibration monitoring, the robustness of this framework against changes in the noise properties has not been properly addressed yet. In this paper, a new robust residual is proposed and the robustness of its statistics against changes in the noise covariances is shown. The complete theory for hypothesis testing for fault detection is derived and a numerical illustration is provided. <s> BIB018 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Abstract The paper presents an online strategy for sensor and/or actuator fault detection and isolation applied to a dam-gallery. A recursive subspace identification algorithm is used to estimate the dam-gallery model parameters. The main contribution consists in developing a specific identification scheme, insensitive to a certain type of faults. That is, the identified parameters are invariant to the faults. A fault estimation procedure is proposed to detect potential faults. The proposed approach appears to be suitable for open channel systems for which the characteristics are not easily measurable. <s> BIB019 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> This paper deals with the problem of fault estimation and fault-tolerant control for Takagi-Sugeno (T-S) fuzzy stochastic systems with sensor failures, where the system under consideration contains Ito-type stochastic disturbances. A new robust observer technique is presented to obtain the estimates of the system states and the sensor faults simultaneously, and a fuzzy fault-tolerant control scheme is developed to guarantee the closed-loop system to be exponentially stable in mean square. Sufficient conditions are obtained for the existence of admissible controllers, and it is shown that the reachability of the sliding-mode dynamics can be guaranteed under the proposed control techniques. Finally, a numerical example is provided to illustrate the effectiveness and applicability of the theoretic results obtained. <s> BIB020 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Fault occurrence can be embodied by the physical parameter variations of the hydraulic servo system. Faults can, therefore, be diagnosed according to the model coefficient variations of the hydraulic servo system. This paper proposes an approach for fault diagnosis based on the unscented Kalman filter (UKF) with a mathematical model of the hydraulic servo system. The mathematical model is established using the dynamic equations of the hydraulic servo system. Based on the fault mechanism analysis results, several important system model parameters that can separately represent different faults in different components of the hydraulic servo system are chosen. Discrete state space equations are derived from the dynamic equations. The UKF algorithm is used to estimate the important system model parameters of the hydraulic servo system by utilizing the discretized state space model. According to the variations of these model parameters, the fault modes and locations of the hydraulic servo system can be diagnosed and isolated. Two types of faults, namely, abrupt fault in servovalve gain and slow wear fault in hydraulic cylinder piston, which cannot be directly detected from the system output, are introduced individually to the hydraulic servo system in this work. By comparing with the extended Kalman Filter, three different experimental cases are used to validate the effectiveness of the UKF for hydraulic servo system fault diagnosis. <s> BIB021 </s> A Survey of Fault Diagnosis and Fault-Tolerant Techniques—Part I: Fault Diagnosis With Model-Based and Signal-Based Approaches <s> B. Stochastic Fault Diagnosis Methods <s> Abstract In this paper, an anti-disturbance fault diagnosis scheme is proposed for non-Gaussian stochastic distribution systems (SDSs) with multiple disturbances. The available driven information for fault diagnosis is probability density functions (PDFs) of output rather than output value. Using B-spline expansion technique, the output PDFs can be approximated in terms of dynamic weights of B-spline neural network by which a nonlinear model can be established between input and weights. Therefore, the concerned problem is transformed into fault diagnosis problem of the weighting system presented by an uncertain nonlinear system with multiple disturbances and time-varying fault. Different from most of the existing results, the multiple disturbances are supposed to include unknown disturbance modeled by an exo-system and norm bounded uncertain disturbances. In the proposed approach, a disturbance observer is designed to estimate and compensate the modeled disturbance, and H ∞ optimization technology is applied to attenuate the norm bounded disturbance. Finally, simulation results are given to show the efficiency of the proposed approach. <s> BIB022
In parallel with the development of the fault diagnosis for deterministic systems, stochastic approaches were also developed for fault diagnosis in the early 1970s. A general fault detection and diagnosis procedure was first proposed in  by using residuals (or innovations) generated by Kalman filters with similar structure to observers, where the faults were diagnosed by statistic testing on whiteness, mean, and covariance of the residuals. A variety of statistical tools, such as generalized likelihoods BIB001 , χ 2 testing  , cumulative sum algorithms BIB004 , and multiple hypothesis test BIB005 , were further developed for testing Kalman-filter-based residuals in order to check the likelihood that a particular fault occurs. Further research studies have led to a couple of modified Kalman filter techniques for fault diagnosis, such as extended Kalman filters (EKFs), unscented Kalman filters (UKFs), adaptive Kalman filters, and augmented state Kalman filters. Unlike the conventional Kalman filters, the EKF can be used to diagnose the faults in a nonlinear industrial process BIB003 . The UKF, depending on a more accurate stochastic approximation, i.e., unscented transform, can better capture the true mean and covariance, leading to better diagnosis performance BIB021 ,  . Adaptive Kalman filters can be employed to tune process noise covariance matrix, or measurement noise covariance matrix in order to obtain satisfactory fault diagnosis BIB015 , BIB016 . The augmented state Kalman filters can be utilized to simultaneously estimate system states and fault signals BIB011 . Recent application examples of Kalman-filter-based fault diagnosis can be found in BIB012 BIB013 BIB017 for combustion engines, electronic systems under mechanical shock, and permanentmagnet synchronous motors, respectively. Another important stochastic fault diagnosis method is parameter estimation on the basis of system identification techniques (e.g., least square error and its derived methods), which was initialized by  . In this approach, the faults are assumed to be reflected in system parameters, and only the model structure is needed to be known. The basic idea of the detection method is to identify the parameters of the actual process online, which are compared with the reference parameters obtained initially under healthy conditions. The parameterestimation-based fault diagnosis methods are very straightforward if the model parameters have an explicit mapping with the physical coefficients. This method was well reviewed in the early survey papers BIB002 , BIB008 and book BIB007 . Recent development of this approach can be found in  BIB018 BIB019 . Motivated by combustion processes, paper making systems, and chemical processes, the monitored system outputs can be described by probability density functions. For this class of stochastic systems, fault diagnosis was first addressed by BIB006 , where the probability density function outputs are approximated by using B-spline expansion techniques, and random noises/errors can be non-Gaussian. In order to improve robustness against measurement noises, modeling errors, and process disturbances, an integration of descriptor estimator and parametric eigenstructure assignment was utilized to detect faults in BIB009 . Recent development of fault diagnosis for nonlinear systems can be found in BIB022 . In addition, fault diagnosis methods were also developed for other classes of stochastic systems such as stochastic processes with Brownian motions BIB010 , BIB020 and Markovian jumps BIB014 .