Sliding mode observers for fault detection and isolation

Abstract

This paper considers the application of a particular sliding mode observer to the problem of fault detection and isolation. The novelty lies in the application of the equivalent output injection concept to explicitly reconstruct fault signals. Previous work in the area of fault detection using sliding mode observers has used disruption of the sliding motion to detect faults. A design procedure is described and nonlinear simulation results are presented to demonstrate the approach.

Introduction

It is well known that the core element of model-based fault detection in control systems is the generation of residual signals which act as indicators of faults. The residual signals are generated using estimates of and a comparison with real measured quantities. For the design of residual generators, various approaches have been discussed in the literature. Fault detection using the parity space approach is considered in (Gertler, 1991; Patton, 1988; Patton & Chen, 1991; Patton, Frank & Clark, 1989) whilst (Ge & Fang, 1988; Massoumnia, 1986) employ fault detection filters. Besides these approaches, the class of observer-based approaches have been the most widely considered (Chen, Patton & Zhang, 1996; Ding & Frank, 1990; Ding, Guo & Frank, 1994; Duan, Patton, Chen & Chen, 1997; Frank 1990, Frank 1994; Ge & Fang, 1988; Gertler 1988, Gertler 1991; Hou and Müller 1992, Hou and Müller 1994; Patton and Chen 1991, Patton and Chen 1997). Patton and Chen (1994) have shown that the observer-based and parity space residual generators are related mathematically and under certain conditions are identical. An additional concept in the fault diagnosis literature is the failure detection filter in which the structure of the observer (filter) is specially designed to provide powerful fault isolation properties (Wilsky, 1976).

The basic idea behind the use of the observer for fault detection is to estimate the outputs of the system from the measurements by using some type of observer, and then construct the residual by a properly weighted output estimate error. The residual is then examined for the likelihood of faults by using a fixed or adaptive threshold. Certain decision rules can then be applied to determine if a fault has occurred. A decision process may be based on a simple threshold test on the instantaneous values or moving averages of the residuals. When the system under consideration is subject to unknown disturbance or unknown inputs, to achieve effective fault detection, the effect of the disturbance has to be de-coupled from the residual signal to avoid ‘false alarms’ in detection. This problem is known in the literature as robust fault detection or fault detection using unknown input observers (Patton, 1997) and has been considered by many authors, for example, Chen et al. (1996), Dalton, Patton and Chen (1996), Hou and Müller 1992, Hou and Müller 1994, Patton and Chen 1991, Patton and Chen 1992, Patton et al. (1989), Patton and Willcox (1987). Chen et al. (1996) discuss extensions to the failure detection filter theory and provide an important link between this theory and the use of unknown input observers as a robustness problem. When, in the observer-based approaches, a full order observer is used in residual generator design, the main design procedure in fault detection in fact becomes an equivalent state feedback control problem because of the dual relation between the state feedback control and the full order observer design. Based on this idea, some well-established approaches for state feedback control can be readily applied to robust fault detection using full order unknown input observers.

This paper considers the use of a particular class of nonlinear observer — a so-called sliding mode observer — and attempts to reconstruct the fault rather than detect the presence of a fault through a residual signal. This problem of fault estimation is a powerful alternative to the detection of a fault via the use of a residual signal as long as the location of the fault effect on the system is known. The residual approach is more suited to the combined problem of fault detection and fault isolation, when the structure of the fault influence on the system is not perfectly known. A bank of dissimilar (but redundant) residual signals can then be used to infer the location of the fault in the system. On the other hand, the fault estimation approach is a direct way of providing fault information which, when compared with other fault estimation signals (from the same system), can be used to isolate all faults. The fault estimation method also provides a direct estimate of the size and severity of the fault, which can be important in many applications.

There has been a substantial body of new work in the field of fault estimation using a number of deterministic approaches, based upon observers with input signal reconstruction or de-convolution (Patton, Chen & Zhang, 1992) and $\text{H}{}_{\mathrm{\infty }}$ estimation (Stourstrup & Grimble, 1997). More recently, Patton and Hou (1998) have provided necessary and sufficient conditions for fault observability and reconstruction, using a matrix pencil approach and based on a study of input signal reconstruction for the unknown input observer problem. They use a numerically stable orthogonal transformation method to generate the required estimator. There is no loss of generality in their approach in the sense that it can be extended for application to certain nonlinear system problems. A disadvantage, however, is the requirement for derivatives of measurement signals (in continuous time).

An alternative strategy is to make direct use of nonlinear observer structures (Patton, Frank & Clark, 1998). However, nonlinear observer approaches are limited in that the structure and parameters of the model must be accurately known. This paper describes an alternative philosophy of using variable structure and sliding mode theory to obviate some of the restrictions that must apply to most of the methods of fault estimation found in the literature.

This paper provides some new developments in the use of sliding mode observer theory for de-coupling the effects of fault signals from the response of the system estimated outputs. The work is based upon the sliding mode observer theory proposed by Edwards and Spurgeon (1994). Sliding modes have been previously used for fault detection: Sreedhar, Fernández and Masada (1993) consider a model-based sliding mode observer approach although in their design procedure it is assumed that the states of the system are available; a different approach is adopted by Hermans and Zarrop (1996), who attempt to design an observer in such a way that in the presence of a fault the sliding motion is destroyed. This paper considers the practical situation when the system states are not available. The observer is designed to maintain a sliding motion even in the presence of faults which are detected by analysing the so-called equivalent output injection. The novelty lies in the manipulation of the equivalent output injection signal to explicitly reconstruct fault signals. (This may be allied to the equivalent control signal which appears in the analysis of sliding mode based feedback control systems.)

The paper provides an example of the application of the theory to a nonlinear laboratory inverted pendulum problem. In addition to the nonlinear dynamics, this example has coulomb friction in the cart bearings with the result that the cart and pendulum have a limit cycle oscillation, providing further uncertainty and a challenge to the robust fault estimation problem (the fault signal must be estimated in the presence of modelling uncertainties). The nonlinearities and friction are not modelled in the design of the sliding mode observer.

Throughout this paper the notation ||·|| will be used to represent the Euclidean norm for vectors and the (induced) spectral norm for matrices.