Dynamic threshold generators for robust fault detection in linear systems with parameter uncertainty

Abstract

The problem of developing robust thresholds for fault detection is addressed. An inequality for the solution of a linear system with uncertain parameters is provided and is shown to be a valuable tool for developing dynamic threshold generators for fault detection. Such threshold generators are desirable for achieving robustness against model uncertainty in combination with sensitivity to small faults.

The usefulness of the inequality is illustrated by developing an algorithm for detection of sensor faults in a turbofan engine. The proposed algorithm consists of a state observer with integral action. A dynamic threshold generator is derived under the assumption of parametric uncertainty in the process model. Successful simulations with measurement data show that the algorithm is capable of detecting faults without generating false alarms.

Introduction

Technical systems are inherently exposed to faults such as leaking valves, broken bearings, faulty sensors, etc. In most applications, it is vital that these faults are detected and isolated in an early stage and accommodated for. In e.g. a single engine aircraft, a fault that leads to a shut down of the engine will inevitably have hazardous consequences. This is of course not allowed to happen, instead continuous operation despite any occurring faults must be ensured. As faults in mechanical systems mainly are due to wear, stress or poor quality, it is not likely that there will be less faults in the future so a good strategy for detection and handling is of great concern.

The idea behind model-based fault detection is to use the redundancy in information obtained from measurement in combination with a process model. If the measured output does not match the expected output produced by a process model, then the presence of a fault can be deduced. The two main properties that are desired in a fault detection algorithm are robustness and sensitivity. The former, in this context, means that the fault detection system does not produce false alarms due to disturbances and modelling errors while the latter should be understood as sensitivity to faults. Obviously, these two properties are conflicting.

When an analytical process model is available, fault detection methods based on analytical redundancy may be employed. During the past three decades, extensive research has been carried out in this area and many methods have been developed (Frank & Ding, 1997). All of these consist essentially of two steps, residual generation and residual evaluation. The purpose of the first step is to generate a signal, the residual, which is supposed to be nonzero in the presence of fault and zero otherwise. This problem has been treated extensively in the literature and solutions based on e.g. state observers, parity equations (Viswanadham, Taylor, & Luce, 1987), or on-line identification algorithms (Isermann, 1997) have been suggested. The issues of sensitivity and robustness have been addressed by optimizing the residual generator to be sensitive to faults and insensitive to disturbances.

However, the residual is almost always nonzero due to disturbances and model uncertainty, even if there is no fault. The purpose of the second step of the fault detection algorithm is thus to evaluate the residual and draw conclusions on the presence of a fault. This is done by comparing some function of the residual, the evaluation signal, to a threshold and then to declare the presence of a fault if the former exceeds the latter. The method selected for this second step depends mostly on the assumptions on the disturbances, e.g. whether they are stochastic or deterministic. The residual evaluation problem considering stochastic disturbances has been treated in e.g. Basseville and Nikiforov (1993) but the main source for nonzero residuals is often model uncertainty which is more convenient to describe in a deterministic setting. Instrumental for achieving robustness in this context is a robust detection threshold, i.e. an upper bound for the evaluation signal under admissible model uncertainty while sensitivity implies that this threshold should be as tight as possible.

Detection thresholds that are robust against frequency domain uncertainty are developed in e.g. Emami-Naeini, Akhter, and Rock (1988) and Frank and Ding (1994). However, the thresholds that result from this kind of uncertainty description are generally functions only of some signal norm of the known inputs and are thus essentially constant. In contrast, experience shows that the residual in a real fault detection application is often correlated with the inputs, as a result of model uncertainty. It is therefore desirable that also the detection threshold varies with the inputs in order to be as tight as possible. The threshold generator should thus be a dynamic system with the measurement signals as inputs. Furthermore, for fault detection, accuracy of the process model is always of crucial importance. This makes it desirable to be able to use all available sources of knowledge of the process behavior, i.e. identified models but also models based on process physics. The latter kind, however, are often expressed in the form of nonlinear differential equations of high order. In conclusion, this motivates the search for methods to be able to use uncertainty descriptions in the time-domain.

In Zhang, Polycarpou, and Parisini (2002) and Zhang, Parisini, and Polycarpou (2003), unstructured uncertainty in a nonlinear state-space process representation is treated for some special cases e.g. full state measurement. Another way of representing the model uncertainty is to assume uncertain parameters. This kind of uncertainty description has been considered in e.g. Ding and Frank (1993) for a special class of nonlinear systems and more recently in Johansson and Medvedev (2000) and also Ding, Frank, and Ding (2002), Ding, Zhang, Frank, and Ding (2003). Uncertain parameters in discrete-time models have been handled with interval analysis techniques in e.g. Puig, Saludes, and Quevedo (2003) and Adrot, Maquin, and Ragot (2000). In this approach, upper and lower bounds for the residual are found by numerical optimization. In fact, it is not necessary to generate a residual, since bounds may be calculated for the measurement signals directly. The drawback of the interval analysis approach is that the numerical optimization is generally very time consuming, except for special cases, e.g. full state measurement. The contribution in this paper may be considered as an analytic solution to a linear, continuous time interval analysis problem.

An uncertain parameter in a general, nonlinear system, can clearly affect the system response in many different ways. However, considering a Taylor approximation of a state-space representation of the system motivates distinguishing between additive and multiplicative parameters. In this context, it is clear that parameters entering mutiplicatively with the state constitutes the main difficulty. Therefore, we will here consider systems of the type

$$\dot{x}=\mathit{Ax}+P(\pi \otimes x)+E\pi +F+\mu \text{,}$$

$$y=\mathit{Cx}+\nu \text{,}$$

where $\otimes$ denotes the Kronecker product and $A\in {\mathbb{R}}^{n\times n}$, $P\in {\mathbb{R}}^{n\times \mathit{nm}}$, and $C\in {\mathbb{R}}^{k\times n}$ are constant matrices. The vector $x(t)\in {\mathbb{R}}^n$ represents the state and $y(t)\in {\mathbb{R}}^k$ is the measured output while $\mu (t)\in {\mathbb{R}}^n$ and $\nu (t)\in {\mathbb{R}}^k$ are some additive disturbances. Measured inputs of (1) are allowed to enter via the time-varying $E(t)\in {\mathbb{R}}^{n\times m}$ and $F(t)\in {\mathbb{R}}^n$. The parameter uncertainties $\pi (t)\in {\mathbb{R}}^m$ are assumed to be bounded by $|\pi (t)|⩽\Pi \in {\mathbb{R}}^m$ and enter bilinearly with the state x. To guarantee existence of a unique solution to (1a) it is assumed that E, F, and $\mu$ are integrable in some sense.

When applying a Luenberger observer to a system of the form (1), then the error system, i.e. the dynamics of the estimation error, also take the form (1) with $F(t)\equiv 0$ as will be shown in the sequel. As the output of this error system is a residual, it is motivated to search for an upper bound for the modulus of y in (1). In this paper, such an upper bound is provided as a dynamic system with E, $\mu$, and $\nu$ as input. Using this upper bound, a fault detection threshold can be developed if some bounds on $\mu$ and $\nu$ are known.

Two very different fault detection problems have, so far, been successfully addressed with the described results. In Johansson and Bask (2005), an algorithm for detecting incipient clogging in the valves of a flotation process is reported. In the sequel, an algorithm for detecting increased sensor noise in a turbofan jet engine is presented.

The continuation of this paper begins with some mathematical notations and preliminaries in Section 2 followed by motivation for the problem under study in Section 3. In Section 4, the main result is presented. It consists of an inequality for the solution of a linear system with uncertain parameters. In Section 5, some comments on the general residual evaluation problem are given. The results are, in Section 6, applied to the problem of sensor fault detection in a jet engine. The paper ends with some conclusions and suggestions for future work in Section 7 and acknowledgements in Section 8.