The Conjugate Gradient (CG) method is often used to solve a positive definite linear system Ax = b. This paper analyzes two hard cases for CG or any Krylov subspace type methods by either analytically finding the residual formulas or tightly bound the residuals from above and below, in contrast to existing results which only bound residuals from above. The analysis is based on a general framewo...

This paper deals with fault detection methods taking model uncertainties into account. It focuses on dynamic and structured uncertain models, where parameter uncertainties are described by bounded variables. In order to de-couple residuals from unknown physical variables, a parity space approach is proposed, where the parity matrix depends on uncertain parameters. In this way, residuals are not...

Artificial intelligence techniques such as neural networks and fuzzy logic have been widely used in fault detection and diagnosis. Combining these two techniques, referred to as neurofuzzy networks, provides a powerful tool for modelling. B-spline neurofuzzy networks are used to model the residuals. The weights of the networks are trained online using recursive least squares method. Fuzzy rules...

Concerned is residual generation in systems with mild nonlinearities, that is, nonlinearities which involve the inputs, outputs and faults but not the state. It is shown how structured residuals can be designed by the parity space method, in discrete time, if mild nonlinearities are present. Next a number of fundamental links between the parity space and the input-output representations are exp...

Residuals are frequently used to evaluate the validity of the assumptions of statistical models and may also be employed as tools for model selection. For standard (normal) linear models, for example, residuals are used to verify homoscedasticity, linearity of effects, presence of outliers, normality and independence of the errors. Similar uses may be envisaged for three types of residuals that...

In this paper, the approximation of nonlinear systems using unscented Kalman filter (UKF) is discussed, and the conditions for the convergence of the UKF are derived. The detection of faults from residuals generated by the UKF is presented. As fault detection often reduced to detecting irregularities in the residuals, such as the mean, the local approach, a powerful statistical technique to det...

Time series linear regression model with the stationary residuals has been studied in many fields, and well established. However, the stationary assumption on the residuals seems to be restrictive. Therefore, we extend the model to the case when the residuals are locally stationary. The best linear unbiased estimator (BLUE) of coefficient vector contains the residual covariance matrix which is ...