Passive robust fault detection using fuzzy parity equations

In this paper, a new approach for robust fault detection based on fuzzy parity equations is presented. The problem of robustness is handled using a passive approach based on generating a fuzzy adaptive threshold. The fuzzy adaptive threshold will be generated using fuzzy parity equations. Fuzzy parity equations are a generalisation of crisp parity equations but in this case the parameters in the used model are modelled using fuzzy parameters. Then, the residuals produced by fuzzy parity equations will also be fuzzy numbers. The extension from crisp parity equations to fuzzy parity equations is based on the extension principle of Zadeh [Inform. Contr. 8 (1965) 338; Inform. Sci. 89 (1975) 43]. Two possible crisp parity equations can be considered, according to Gertler [Fault Detection and Diagnosis in Engineering Systems, Marcel Dekker, New York, 1998] the moving average (MA) parity equation and autoregressive-moving average (ARMA) parity equations. When generalising these crisp parity equations to the fuzzy case, different problems appear when MA or ARMA parity equation approach is used. The underlying approach of fuzzy parity equations is the approach based on interval parity equations. Interval parity equations is an extension of crisp parity equations in the case of substituting crisp parameters in model equations by intervals. These intervals reflect the uncertainty on model parameters. When applying the extension principle of Zadeh [Inform. Contr. 8 (1965) 338; Inform. Sci. 89 (1975) 43] to extend crisp parity equations to fuzzy parity equations, a set of interval parity equations must be evaluated at different α-cuts. Interval parity equations have been extensively studied and applied in the literature, using two approaches: the simulation approach [Robust model-based fault diagnosis: the state of the art, in: Proceedings of the IFAC Symposium (SAFEPROCESS’94), 1994] and the prediction approach [IEEE Expert Intell. Syst. Appl. (1997)]. These two approaches can be deduced directly from crisp parity equations, in particular, from ARMA and MA approaches. Properties of interval parity equations (MA and ARMA) will be presented and compared. Finally, the proposed fuzzy parity equations will be tested on an example.