normal operation, in a neighborhood of a certain known operating point (Chen and Patton 1999; Patton et al. 2000)
It is clear that, as almost every process system is nonlinear, the modeling errors almost always reduce the accuracy of the linear model and therefore the performance of the FDI algorithm is compromised. Various methods for generating robustness to linearization have been proposed in the literature and the reader is referred to (Patton et al. 2000, Chapter 7) for a comprehensive treatment of this subject.
This book also surveys the state of the art of robustness methods and presents some important ideas concerning the development of the use of nonlinear models and predictors for FDI. For example, observer-based approaches to robust FDI and FTC for dynamic systems are considered in more detail. The available model-based approaches are generalized, and thus extended to a wider class of dynamic systems.
In order to accommodate the application of robust FDI concepts, disturbances and parameter uncertainties of the monitored plants, as well as faults, are modeled in the form of unknown input signals. It is shown that, provided certain conditions can be met, complete decoupling of the residual from disturbances as well as from the parameter uncertainties of the process model can be achieved, while the sensitivity of the residual to faults is maintained. As the faults are also modeled in the form of external signals, this method additionally provides tools for the purpose of fault isolation. Fault isolation requires the decoupling of the effects of different faults from the residual (Chen and Patton 1999) and this, in turn, allows for decisions on which fault or faults out of a given set of possible faults has actually occurred.
These residual properties must be completely independent of the magnitude or frequency of the unknown inputs and the faults. This is crucial in cases where no a priori knowledge about these properties is available. For systems where the complete decoupling of the remaining unknown inputs or faults from the residual proves impossible, a threshold selection method, employing functional analytic methods and appropriate vector and operator norms, can be exploited. This technique provides a tool for the robust evaluation of the residuals, which have been generated by UIOs. Using the same functional analysis methods as employed for threshold selection, a performance index can be defined that allows for performance evaluation and, to a certain degree, also allows for optimal residual generator design (Patton et al. 2000).
I.7. Data–driven approaches to robust FDI
In previous sections, we have seen that model-based FDI methods formally require a high accuracy mathematical model of the monitored system. The better the model is as a representation of the dynamic behavior of the system, the better the FDI performance will be. It is difficult to develop a highly accurate model of a complex system and hence the interesting question is: “what is a reasonable model to enable good performance in FDI to be guaranteed?”.
It would be attractive to develop a robust FDI technique which is insensitive to modeling uncertainty, that is, so that a highly accurate mathematical model is no longer required. However, in order to design a robust FDI scheme, we should have a description (i.e. some information) about the uncertainty, for example, its distribution matrix and spectral bandwidth, and so on. Furthermore, this description should provide assistance for a robust FDI design, that is, it can be handled in a systematic manner. This book will show how a typical uncertainty description makes use of the concept of “unknown inputs” acting upon a nominal linear model of the system. These unknown disturbances describe the uncertainties acting upon the system but disturbance distribution matrices are assumed to be known since they can be estimated by identification schemes.
It is clear that disturbances and faults act on the system in the same way, and thus we cannot easily discriminate between these excitation signals unless we know the structure of the disturbance distribution matrix. Once the disturbance distribution matrix is known, we can generate the residual with the disturbance decoupling (robust) property, that is, the residual is decoupled from the disturbance (uncertainty). The robust residual can then be used to achieve reliable FDI and FTC.
The theories underlying robust FDI approaches have been very well developed, but for real applications the following problems remain unsolved:
– estimation of the reliable model for the monitored process;
– modeling accuracy of the real uncertainty by means of identified disturbance terms when no knowledge of the uncertainty is available;
– estimation of the disturbance terms and the structure of distribution matrices.
This book addresses these unsolved problems. Some simulation and real examples are given to test some of the theoretical results. These problems have to be addressed, otherwise the application domain of the disturbance decoupling approach for robust FDI is very limited. In fact, few researchers and contributions have presented the application results of robust fault diagnosis to real processes.
As mentioned above, a primary requirement for model-based and disturbance decoupling approaches to robust FDI is that both the system model and disturbance distribution matrices must be known. It is interesting that, within the framework of international research on this subject, there have been few attempts to address the problem by means of the identification approach. This lack of information has obstructed the application of robust FDI in real engineering systems. Therefore, we present the research developments surrounding the joint estimation of system and disturbance matrices in order to solve the robust fault diagnosis problem.
Concerning the data-driven schemes developed and exploited throughout the book, when all observed variables of a dynamic process are affected by uncertainties, the parameter estimation task can be performed by the so-called errors-in-variables methods. On the other hand, equation error methods can be developed in the case of exactly known plant variables (Simani et al. 2000). It is worthwhile noting that less attention has been paid to errors-in-variables schemes.
Under these considerations, this book presents the robust FDI results concerning the description of monitored plants by means of equation error and error-in-variables identified models in the presence of variable uncertainties. Moreover, for the examples presented, estimates obtained by the proposed data-driven approaches and parameter estimates will be computed and compared.
I.8. Data-driven methods for fault diagnosis
If several symptoms change differently for certain faults, an initial way of determining them is to use classification methods which indicate changes of symptom vectors.
Some classification methods are as follows (Patton et al. 1989; Basseville and Nikiforov 1993; Babuška 1998; Gertler 1998; Chen and Patton 1999):
1 1) geometrical distance and probabilistic methods;
2 2) artificial neural networks;
3 3) fuzzy clustering.
When more information about the relations between symptoms and faults is available in the form of diagnostic models, methods of reasoning can be applied. Diagnostic models then exist in the form of symptom–fault causalities, for example, in the form of symptom–fault trees. The causalities can be expressed as IF–THEN rules. Then analytical as well as heuristic symptoms (from operators) can be processed. By considering these symptoms as vague facts, probabilistic or fuzzy set descriptions lead to a unified symptom representation. By using forward and backward reasoning, probabilities or possibilities of faults are obtained as a result of diagnosis. Typical approximate reasoning methods are as follows (Basseville and Nikiforov 1993; Chen and Patton 1999):
1 1) probabilistic reasoning;
