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FDD	Number of contributions
Milling and grinding processes	91
Power plants and thermal processes	106
Fluid dynamic processes	67
Combustion engine and turbines	96
Automotive	68
Inverted pendulum	63
Miscellaneous	102
DC motors	121
Stirred tank reactor	77
Navigation system	75
Nuclear process	50

      Table I.6 shows that among mechanical and electrical processes, DC motor applications are mostly investigated. Parameter estimation and observer-based methods are used in the majority of applications in these kind of processes, followed by parity space and combined methods. Thermal and chemical processes are investigated less frequently.

      Table I.3 shows that parameter estimation and observer-based methods are used in nearly 70% of all applications considered. Neural networks, parity space and combined methods are applied notably less often.

      More than 50% of sensor faults are detected using observer-based methods, while parameter estimation, parity space and combined methods play a less important role. For the detection of actuator faults, observer-based methods are mostly used, followed by parameter estimation and neural network methods.

Parity space and combined methods are rarely applied. In general, there are fewer applications for actuator faults than for sensor or process faults. The detection of process faults is mostly carried out with parameter estimation methods. Nearly 50% of all the applications considered use parameter estimation-based methods for detection of process faults. Observer-based, parity space and neural network-based methods are used less often for this class of faults.

      Among all the described processes, linear models have been used much more than nonlinear models. In processes with nonlinear models, observer-based methods are mostly applied, but parity equations and neural networks also play an important role. In processes with linear or linearized models, parameter estimation and observer-based methods are mostly used. Parity space and combined methods are also used in several applications but not to the same extent as observer-based and parameter estimation methods.

      Taking into account the system considered, the number of nonlinear process applications using nonlinear models is decreasing. For linear processes, no significant change can be stated. The applications of fault-detection methods for nonlinear processes used mostly observer-based and parameter estimation, more than parity space methods. Also, the use of neural networks and combinations are important.

      Concerning the fault diagnosis methods, in recent years, the field of classification approaches, especially with neural networks and fuzzy logic, has steadily been growing. Also, rule-based reasoning methods are increasingly being based on fault diagnosis. A growing application of fuzzy rule-based reasoning can be stated. Applications using neural networks for classification are increasing and the trends are analogous to the increasing number of nonlinear process investigations. Nevertheless, the classification of generated residuals seems to remain the most important application area for neural networks.

      I.10. From FDI to FTC

      A conventional feedback control design for complex systems may result in unsatisfactory performance in the event of malfunction in input–output sensors, actuators and system components. A fault-tolerant closed-loop control system is very attractive because it can tolerate faults while also maintaining desirable performance.

      The conventional approach to the design of an FTC includes different steps and separate modules: modeling or identification of the controlled system, design of the controller, FDI scheme and a method for re-configuring the control system. Identification and design of the controller can be performed separately or using combined methods. Hence, the FDI and controller are linked through the reconfiguration module. The fundamental problem with such a system lies in the identification stage in the independent design of the control and FDI modules. Significant interactions occurring among these modules can be neglected. There is therefore a need for a research study into the interactions between system identification, control design, the FDI stage and the FTC design strategy.

Fault identification is the most important of all the fault diagnosis tasks. When a fault is estimated, detection and isolation can be easily achieved since the fault nature can improve the diagnosis process. However, the fault identification problem itself has not gained enough research attention.

      Most fault diagnosis techniques, such as parameter identification, parity space and observer-based methods, cannot be directly used to identify faults in sensors and actuators. Very little research has been done to overcome the fault identification problem. The Kalman filter for statistical testing and fault identification was proposed in Chen and Patton (1999). However, the statistical testing methods can impose a high computational demand. A fault identification scheme solving a system inversion problem was proposed in Chen and Patton (1999); Simani et al. (2003) and Simani and Farsoni (2018).

In the scheme, depicted in Figure I.3, fault identification is performed by estimating the nonlinear relationship between residuals and fault magnitudes. This is possible because robust residuals should only contain fault information.

Figure I.3. Fault estimation scheme FTC

      Such a nonlinear function approximation and estimation can be performed by using fuzzy systems, neural networks or an inversion of the transfer matrix between residuals and faults (Simani et al. 2003; Simani and Farsoni 2018). The central task in model-based fault detection is the residual generation. Most residual generation techniques are based on linear system models. For nonlinear systems, the traditional approach is to linearize the model around the system operating point. However, for systems with high nonlinearity and a wide dynamic operating range, the linearized approach fails to give satisfactory results.

One solution is to use a large number of linearized models corresponding to a range of operating points. This means that a large number of FDI schemes corresponding to each of the operating points is needed. Hence, it is important to study residual generation techniques that tackle nonlinear dynamic systems directly. There are some research studies on the residual generation of nonlinear dynamic systems, for example using nonlinear observers (Chen and Patton 1999; De Persis and Isidori 2001). There have been some attempts to use nonlinear observers to solve nonlinear

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