A mathematical framework for new fault detection schemes in nonlinear stochastic continuous-time dynamical systems

Abstract

In this work, a mathematical unifying framework for designing new fault detection schemes in nonlinear stochastic continuous-time dynamical systems is developed. These schemes are based on a stochastic process, called the residual, which reflects the system behavior and whose changes are to be detected. A quickest detection scheme for the residual is proposed, which is based on the computed likelihood ratios for time-varying statistical changes in the Ornstein-Uhlenbeck process. Several expressions are provided, depending on a priori knowledge of the fault, which can be employed in a proposed CUSUM-type approximated scheme. This general setting gathers different existing fault detection schemes within a unifying framework, and allows for the definition of new ones. A comparative simulation example illustrates the behavior of the proposed schemes.

Fault isolation schemes for a class of continuous-time stochastic dynamical systems

“…Under some additional assumptions such as the availability of an estimate of T 0 , or a partial knowledge of fault functions (e.g. their profile) it is possible to derive the approximations leading to the efficient isolation schemes proposed in this paper (see Zufiria (2012) for more details).…”

“…Such tests are applied on the residual mean vector, in order to check which fault the actual residual mean vector corresponds to. The residual mean estimators given in Castillo and Zufiria (2009) have been shown in Zufiria (2012) to be valid overall likelihood ratio approximators, and consequently a valid residual feature; so they have been also successfully employed in Castillo and Zufiria (2009 can be applied to check if the actual residual mean corresponds to the expected mean when each one of the faults in the given list was affecting the system. Substituting <f>(t) by 4> k {t), k = 1 /, in (3), the distribution of the residual given the occurrence of the kth fault can be determined, so that the mentioned hypotheses tests for the means {£[e(t); <P^] £[e(í);<P¡]} can be appropriately constructed (Fukunaga, 1990;Gertler, 1998).…”

“…Unfortunately, these overall likelihood functions are very difficult to obtain in general. Their computation has been addressed under some specific assumptions in Zufiria (2012), where the problems of classical detection and quickest detection (Kailath and Poor, 1998;Poor and Hadjiliadis, 2009) have been studied using also the residual e(t) of Eq. (3).…”

“…In fact, the likelihood ratio between the null (no fault) and the fault hypotheses is calculated there (generalizing some results in Arató (1982)) for the case of deterministic faults and fixed interval of time. Also in Zufiria (2012), this likelihood ratio is employed as the basis of some online suboptimal hypotheses tests for building fault detection schemes. Nevertheless, these results in Zufiria (2012) may not be easily extended to the general isolation problem, specially for the case of stochastic faults.…”