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Nonlinear Methods for Fault Diagnosis

Silvio SIMANI1 and Paolo CASTALDI2

1Department of Engineering, University of Ferrara, Emilia-Romagna, Italy

2Department of Electrical, Electronic and Information Engineering, University of Bologna, Italy

1.1. Introduction

The model-based approach to fault diagnosis in technical processes has been receiving more and more attention over the last four decades, in the contexts of both research and real plant application.

Stemming from this activity, a large number of methods can be found in current literature based on the use of mathematical models of the technical process under diagnosis and on exploiting advanced control theory.

Model-based fault diagnosis methods usually use residuals that indicate changes between the process and the model. One general assumption is that the residuals are changed significantly so that detection is possible. This means that the residual size after the appearance of a fault is large and long enough to be detectable.

This chapter provides an overview on different fault diagnosis strategies, with particular attention to the fault detection and isolation (FDI) methods related to the dynamic processes and application examples considered in this book.

For all of the methods considered, it is essential that the technical process can be described by a mathematical model. As there is almost never an exact agreement between the model used to represent the process and the plant, the model-reality discrepancy is of primary interest.

Hence, the most important issue in model-based fault detection concerns the accuracy of the model describing the behavior of the monitored system. This issue has become a central research theme over recent years, as modeling uncertainty has risen from the impossibility of obtaining complete knowledge and understanding of the monitored process.

The main focus of this chapter is the mathematical description aspects of the process whose faults are to be detected and isolated. The chapter also studies the general structure of the controlled system, its possible fault locations and modes. Residual generation is then identified as an essential problem in model-based FDI, because, if it is not performed correctly, some fault information could be lost. The general framework for the residual generation is also recalled.

Residual generators based on different methods, such as input-output, state and output observers, parity relations and parameter estimations, are just special cases in this general framework. In the following, some commonly used residual generation and evaluation techniques are discussed and their mathematical formulation is presented.

Finally, the chapter presents and summarizes special features and problems regarding the different methods.

1.2. Fault diagnosis tasks

According to the definitions available in the related literature, model-based FDI can be defined as the detection, isolation and identification of faults in a system by using methods that can extract features from measured signals and use a priori information on the process available in terms of mathematical models. Faults are, thus, detected by setting fixed or variable thresholds on residual signals generated from the difference between actual measurements and their estimates obtained by using the process model.

A number of residuals can be designed, with each having sensitivity to individual faults occurring in different locations of the system. The analysis of each residual, once the threshold is exceeded, then leads to fault isolation.

Figure 1.1 shows the general model-based FDI system. It comprises two main stages of residual generation and residual evaluation. This structure was first suggested in Chow and Willsky (1980) and now is widely accepted by the fault diagnosis community.

Figure 1.1. Fault diagnosis module

The blocks shown in Figure 1.1 perform the following tasks:

1. 1) The residual generation module generates residual signals using the available inputs and outputs from the process under diagnosis. This residual (or fault symptom) should indicate that a fault has occurred. It should normally be zero or close to zero under no fault condition, and significantly different from zero when a fault occurs. This means that the residual is characteristically independent of process inputs and outputs in ideal conditions. Referring to Figure 1.1, this block is called the residual generator.
2. 2) The residual evaluation module examines residuals for the likelihood of faults and a decision rule is then applied to determine if any faults have occurred. The residual evaluator block in Figure 1.1 may perform a simple threshold test (geometrical methods) on the instantaneous values or moving averages of the residuals. On the other hand, it may consist of statistical methods, for example, generalized likelihood ratio testing or sequential probability ratio testing (Isermann 1997; Willsky 1976; Basseville 1988; Patton et al. 2000).

Many works in the field of model-based FDI have focused on the residual generation problem, since the decision-making problem can be considered relatively straightforward if residuals are well designed. In the following, a number of different strategies oriented to solve the model-based residual generation problem have been addressed, with reference to multivariable dynamic processes and technical systems.

The first requirement of the model-based FDI approach consists of providing a mathematical description of the system under investigation that describes the possible fault cases as well.

The general scheme for model-based FDI considered in this chapter is depicted by Figure 1.2. The main components are the plant under diagnosis, the actuators and the sensors, which can be further sub-divided as input and output sensors, and finally, the controller. In the following, the process behavior will be monitored by analyzing its input u(t) and output y(t) measurements and the signals from the controller uR(t), which are supposedly completely available for FDI purposes. Moreover, as shown in Figure 1.2, the behavior of any controller that drives the system is inherently taken into consideration.

Figure 1.2. Model-based fault diagnosis strategy

It is worth noting that, when the signals uR(t) from the controller or measurements of plant inputs u(t) are not available, the controller plays an important role in the design of the FDI scheme, as a robust controller may reduce the effects of the faults, thus making the fault diagnosis more difficult.

Once the actual process inputs and outputs u*(t) and y*(t) (usually not available) are measured by the input and output sensors, FDI theory can be treated as an observation problem of u(t) and y(t). Concerning the occurrence of malfunctions, the location of faults and their modeling, the system under diagnosis can be separated into the different parts that can be affected by faults, as illustrated in Figure 1.2. With respect to previous works (see, e.g., Patton et al. 1989; Gertler 1998; Patton et al. 2000), it is necessary to distinguish between input and output sensors.

Figure 1.2 shows that the input and output signals u*(t) and y*(t) are acquired in order to obtain the measurements u(t) and y(t) from the sensors. Figure 1.2 also shows the situation where the controller can be affected by faults, since the monitored process consists of a closed-loop plant. However, because of technological reasons (e.g., the control action is performed by a digital computer), when the actuator is considered as a part or a component of the whole controller device, the former can be treated as subsystem where faults are likelier to occur, while the latter remains free from faults. Under these assumptions, as shown in Figure 1.2, when the monitored process is considered in view of fault location, since input and output measurements are supposed completely available for FDI purposes, the controller behavior in the design of a fault diagnosis scheme can be neglected, as well as the interconnection between the control system and the process.

In general, as shown in Figure 1.2, the actuation signals u*(t) are assumed to be measurable by neglecting input and output sensor noises. On the other hand, Figure 1.2 represents the situation where the uR signals are only measured by the input sensors.

Note that the general models for FDI represented in Figure 1.2 can be described in both the time and frequency domains, respectively, which have been widely accepted in the fault diagnosis literature (Patton et al. 1989, 2000; Chen and Patton 1999; Gertler 1998). Under these assumptions, the general model-based FDI problem treated here can be performed on the basis of the knowledge of only the measured sequences u(t) and y(t).

As mentioned in Volume 1, frequency domain descriptions are typically applied when the effects of faults, as well as the disturbances, have frequency characteristics which differ from each other and thus, the information in the...