Fault diagnosis for nonlinear systems using a bank of neural estimators

Abstract

A model-based method to detect faults in nonlinear systems is proposed. Fault diagnosis is accomplished by means of a bank of estimators, which provide estimates of parameters that describe actuator, plant, and sensor faults. These estimators perform according to a receding-horizon strategy and are designed using models of the failures. The problem of designing such estimators for general nonlinear systems is solved by searching for optimal estimation functions. These functions are approximated by feedforward neural networks and the problem is reduced to find the optimal neural weights. The learning can be split into two phases. In the first one, any possible “a priori” knowledge on the statistics of the random variables is used to initialize the neural estimation functions off line. In the second one, the optimization (or training) continues on line. Both off and on line learning rely on stochastic approximation. The performances obtained in the estimation of the fault parameters by the proposed neural estimators and by the extended Kalman filters are compared by means of simulations with an application to underwater robotics.

Introduction

Model-based fault diagnosis relies on mathematical models of the plant (with sensors and actuators) to identify a discrepancy between the nominal plant and the plant when a fault occurs [1], [2]. The detection of the fault is usually accomplished by evaluating the residuals, i.e. variables over-sensitive to the occurrence of faults in the system. In this context, fault-detection based on a bank of observers enables one to improve the isolation capabilities by imposing selected inputs and outputs in the residual generation (see, among others, [3]). Most model-based fault-detection techniques use linear models of the plant and only few methods can address problems for nonlinear systems. Actually, many approaches rely on linearized models of the plant and try to design a fault-detection scheme based on the well-settled theory for linear systems. Some solutions are proposed for particular nonlinearities, thus reducing the number of potential applications [4]. In realistic conditions, when complex nonlinearities and unmodeled dynamics are present, the linearization-based methods may suffer from poor performances.

In order to estimate all the variables related to the occurrence of faults, a bank of neural receding-horizon estimators is proposed. As far as plant faults are concerned, we shall consider changes in the dynamics described by means of a parameter vector; also actuator and sensor faults can be viewed as disturbances modeled by parameter variations. All these quantities are unknown variables to be estimated according to the approaches reported in the literature (see, for example, [5], [6]) and we shall call them fault parameters. The problem of estimating a fault variable for linear continuous-time systems has been more formally faced in [7].

With respect to the existing approaches (some cited above), the method proposed in this paper enables to estimate the fault parameters for a quite general class of systems, including those with nonlinear dynamics and multiplicative faults, without particular assumptions on the system and measurement equations. The applicability of the approach stems from the generality of the design technique for receding-horizon estimation reported in [8], [9], which will be briefly recalled here for the sake of the reader’s comprehension. Receding-horizon estimation is based on the idea of imposing that the information older than a given period of time stages do not influence the current estimate. Such a problem was originally stated in [10] for nonlinear systems, and has been successfully applied to problems for which the extended Kalman filter performs poorly or diverges due to modeling errors (see also [11]). In [12] the focus is on sliding-window techniques to accomplish residual generation for linear systems, both in continuous- and discrete-time setting. In [13] receding-horizon filters to detect faults in continuous-time linear systems have been investigated in details.

Any estimation problem is easily resolvable by means of the Kalman filter under the hypotheses of Gaussian random noises and on the linearity of the dynamic and measurement equations, otherwise the solution remains difficult to find. The same issues apply to fault diagnosis. An approach to the solution of nonlinear estimation problems is well-suited to dealing with fault-detection in general cases. Specifically, well-established results (see [9]) enable one to tackle the difficulties by casting the problem into the estimation of fault parameters. Estimation is stated as the minimization of a cost functional, according to a least-squares approach, and by solving it in an approximate way. For each type of fault (i.e. for each set of fault parameters), we first restrict the class of admissible estimation functions that provides the optimal estimates. Then we constrain these functions to take on given structures by means of parameterized approximators, where such parameters can be chosen optimally [8], [9], [14]. This methodology has been successfully applied to numerous functional optimization problems (see, for an introduction, [15]).

Among various approximators, we choose multilayer feedforward neural networks. This choice is motivated by both the excellent approximating properties of such nonlinear approximators [16], [17], [18] and the availability of effective optimization algorithms (see, among others, [19]). The algorithms for optimizing the neural weights rely on stochastic approximation and are performed off line (using “a priori” information) and on line (using sequentially collected data).

The paper is organized as follows. Section 2 is devoted to the description of fault models and introduces the problem of constructing a bank of observers for the purpose of fault diagnosis. The problem of designing such estimators using a receding-horizon technique is considered in Section 3. A solution based on the approximation of the related optimal estimation functions by means of feedforward neural networks is presented in Section 4, where, in addition, algorithms for the neural training are described. In Section 5, an application of the proposed approach to fault diagnosis for underwater vehicles is presented. The conclusions are drawn in Section 6.