Diagnosis and Fault-tolerant Control 1. Группа авторов
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The estimation is generated by a mathematical model of the plant considered. The comparison is done using the residual quantities that are computed as differences between the measured signals and the corresponding signals generated by the mathematical model (Patton et al. 1989; Chen and Patton 1999).
Figure I.1. Comparison between hardware and analytical redundancy schemes
In practice, the most frequently used diagnosis method is to monitor the level (or trend) of the residual and take action when the signal reaches a given threshold. This method of geometrical analysis, while simple to implement, has a few drawbacks. The most important is that, in the presence of noise, input variations and change of operating point of the monitored process, false alarms are possible.
The major advantage of the model-based approach is that no additional hardware components are required to implement an FDI algorithm as well as FTC. A model-based FDI algorithm can be implemented via software on a process control computer. In many cases, the measurements necessary to control the process are also sufficient for the FDI algorithm, so no additional sensors have to be installed (Patton et al. 1989; Basseville and Nikiforov 1993; Chen and Patton 1999).
Analytical redundancy uses a mathematical model of the system under investigation and therefore it is often referred to as the model-based approach to fault diagnosis.
I.4. Model-based fault diagnosis
This diagnosis task detects faults in the technical process, including actuators, components and sensors by measuring the available input and output variables u(t) and y(t). The principle of model-based fault diagnosis is depicted in Figure I.2.
Figure I.2. Scheme for the model-based fault diagnosis
Basic process model-based FDI methods have been described by Patton et al. (1989, 2000), Basseville and Nikiforov (1993), Gertler (1998) and Chen and Patton (1999), which include the following steps:
1 1) output observers (OO, estimators, filters);
2 2) parity equations;
3 3) identification and parameter estimation.
These methods generate residuals for output variables with fixed parametric models using step 1, fixed parametric or non-parametric models using step 2 and adaptive non-parametric or parametric models using step 3.
An important aspect of these methods is the kind of fault to be detected. As noted above, one can distinguish between additive faults, which influence the variables of the process by summation, and multiplicative faults, which are products of the process variables. The basic methods show different results, depending on the type of fault.
If only output signals y(t) can be measured, signal model-based methods can be applied, for example, vibrations can be detected, which are related to rotating machinery or electrical circuits. Typical signal model-based methods of fault detection are as follows:
1 1) bandpass filters;
2 2) spectral analysis (FFT);
3 3) maximum-entropy estimation.
The characteristic quantities or features from fault detection methods show stochastic behavior with mean values and variances. Deviations from the normal behavior must then be detected by methods of change detection (residual analysis, Figure I.2), such as:
1 1) mean and variance estimation;
2 2) likelihood-ratio test, Bayes decision;
3 3) run-sum test.
I.5. Model uncertainty and fault detection
Model-based FDI makes use of mathematical models of the system. However, a perfectly accurate mathematical model of a physical system is not possible. Usually, the parameters of the system may vary with time, and the characteristics of the disturbances and noises are unknown, so they cannot be modeled accurately. Hence, there is always a mismatch between the actual process and its mathematical model, even under no fault conditions. Such discrepancies cause difficulties in FDI applications, in particular, since they act as sources of false alarms and missed alarms. Therefore, the effect of modeling uncertainties, disturbances and noise is the most crucial point in the model-based FDI concept, and the solution to this problem is the key for its practical applicability (Chen and Patton 1999).
To overcome these problems, a model-based FDI scheme has to be insensitive to modeling uncertainty. Sometimes, a reduction of the sensitivity to modeling uncertainty does not solve the problem, because the sensitivity reduction may be associated with a reduction of the sensitivity to faults (Gertler 1998; Chen and Patton 1999). A more meaningful formulation of the FDI problem is to increase the insensitivity to modeling uncertainty in order to provide increasing fault sensitivity.
The difficulties introduced by model uncertainties, disturbances and noises in model-based FDI have been widely considered during the last 10 years by both academia and industry (Gertler 1998). A number of methods have been proposed to tackle this problem, for example, the Unknown Input Observer (UIO), eigenstructure assignment and parity relation methods.
An important task of the model-based FDI scheme is to be able to diagnose incipient faults in a system. With respect to abrupt faults, incipient faults may have a small effect on residuals and can be hidden by disturbances. On the other hand, hard faults can be detected more easily because their effects are usually larger than modeling uncertainties and a simple fixed threshold is usually enough to diagnose their occurrence by residual analysis.
The presence of incipient faults may not necessarily degrade the performance of the plant, however, they may indicate that the component should be replaced before the probability of more serious malfunctions increases. The successful detection and diagnosis of incipient faults can therefore be considered a challenge for the design and evaluation of FDI algorithms.
I.6. Robust fault diagnosis
In the context of automatic control, the term robustness is used to describe the insensitivity or invariance of the performance of control systems with respect to disturbances, model–plant mismatches or parameter variations. Fault diagnosis schemes, on the other hand, must, of course, also be robust to the mentioned disturbances, but, in contrast to automatic control systems, they must not be robust to actual faults. On the contrary, while generating robustness to disturbances, the designer must maintain or even enhance the sensitivity of fault diagnosis schemes to faults. Furthermore, the robustness, as well as the sensitivity properties, must be independent of the particular fault and disturbance mode. Generally, the problem of robust FDI can be divided into the tasks of robust residual generation followed by robust residual evaluation.
In many cases, the disturbances and model–plant mismatches to which robustness must be generated are due to the use of linear models for describing dynamic behavior of nonlinear processes. Modeling errors can be avoided from the very beginning by focusing on robust residual generation methods using linear and nonlinear process models. This, in turn, simplifies the problem of residual evaluation without reducing the sensitivity to actual faults.
Effective tools for robust residual generation and even complete decoupling from external