This chapter addresses the problem of Fault Detection and Isolation (FDI) of switching systems where faulty behaviors are represented as faulty modes. The objectives of the online FDI are to identify accurately the current mode and to estimate the switching time. A data-based method is considered here to generate residuals for FDI. Conditions of two important properties, namely discernability between modes and switching detectability, are established. These conditions are different for the two properties. More specifically, it will be shown that a switching occurrence may be detected even if the two considered modes are not discernible. A vehicle rollover prevention example is provided for illustration purpose.
Top1 Introduction
In this chapter we deal with a class of hybrid systems, namely switching systems. Switching systems consist of dynamical modes which are modeled in state space form, and a switching logic that governs the transitions between modes. Switching may be due to internal variables (input, output, state variables, system faults), or external actions (human operators, environment conditions).
The dynamical modes may depict faulty behaviors. The objectives of online fault detection and isolation (FDI) are thus to recognize accurately the current mode and to estimate the switching time.
Indeed, if the system switches to a healthy mode, the system can continue to operate, but if the new mode is a failure mode, it has to be recognized as soon as possible and then a corresponding action must be done to prevent the failure consequences on the system itself or its environment. FDI is an important task to ensure the system dependability in providing online pertinent informations on the system state not only to the human operators through a supervision interface, but also directly to the control system that may be re-configured adequately. Several methods related to FDI design methods have been proposed and many results are scattered in the literature. Model-based and model-free approaches are generally distinguished:
- •
In model-based methods, the principle is to generate mode-dedicated fault indicator called residual. This residual is calculated as a difference between the model and the process behaviors. A residual is close to zero in fault free case and differs from zero if a fault occurs. Several model-based FDI techniques (Patton, 1994; Simani et al., 2003) have been proposed in the literature: state observers have been used in (Palma, 2002), the parity-space approach has been proposed in (Chow and Willsky, 1984) and extended for switching system in (Cocquempot et al., 2004) and (Domlan et al., 2007), Kalman filters has been developed in (Frank and Ding, 1997).
- •
In model-free methods, the online available input and output data are analyzed to determine some behavior characteristics. Several techniques have been proposed in the literature as learning procedures in (Fussel and Balle, 1997), pattern recognition techniques in (Casimir et al., 2003) and Principal Component Analysis techniques in (Ding et al., 2010; Harkat et al., 2003). In some cases, the model structure (for instance, as in this chapter, linear structure) is known, but parameters are not known or are not easily identifiable.
This chapter presents a novel residual generation data-based method for FDI in linear switching systems where the parameters values of the linear models are not known. Previous publications have introduced the proposed method for different model structures (bilinear (Hakem et al., 2011) and linear (Pekpe, 2004). The evaluation form of the data-based residual is obtained by projecting an output matrix onto the kernel of an input Hankel matrix. The first contribution of that chapter is to extend the technique in the multiple mode case. Each mode is supposed to be linear and stable. It is shown how this residual can be used for switching detection, mode recognition and sensor FDI.
It is classically adopted that a necessary condition for mode recognition and switching detection, is that the modes must be discernable (or distinguishable). The discernability conditions between modes are actually the subject of intensive studies (Cocquempot et al., 2003; Domlan et al., 2007). The detectability of the switching is often likened to the discernability between modes. The second contribution of the chapter is to establish discernability and detectability conditions. Moreover, it will be proved that discernability and detectability have two different sets of conditions. As a consequence, even if two modes are not discernable, it will be possible to detect the mode switching.