The Weighted Nonlinear Least Square Problem

Measurements y t,0) obtained by n sensors from the real system carry some uncertainty. Therefore, the residuals e t, 0) = y t, 0) — y(t, 0) between measured and computed vectors, the cost function and the estimated parameters 0 are uncertain. Each residual ej 0) = e tk-q- +j, 0) has an error Sj. It is assumed that all these errors ej, j = 1. q + 1, are independent and that each of its n components is normally distributed with mean zero and a known variance i = 1. n. If the known variances are quite different, a weighted least squares problem may be considered. 

The weighting matrix Wj may be chosen as a diagonal matrix with wj = l/(cr/), i = That is, less uncertain residuals will have a stronger influence on the 

4 Multiple Fault Isolation by Least Squares ARR Residuals Minimisation 

The parameter estimation presented in the previous section is based on a least squares minimisation of the errors between measured system outputs and outputs of a system model evaluated by using estimated parameter values. If the real system is replaced by a model in preferred integral causality, measured outputs can be obtained by solving the model equations for given initial conditions and can be used for offline parameter estimation in order to isolate multiple faults deliberately introduced into the system model. In real-time FDI, initial conditions are either not known or difficult to obtain. Therefore, in online parameter estimation, they have to be considered as additional unknowns that are to be estimated. 

Alternatively, multiple faults may be isolated by least squares minimisation of ARR residuals. The latter are indicators for the errors between measurements from a faulty system and outputs of a model computed by using estimated parameters. ARRs obtained from a DBG do not depend on initial conditions but use derivatives of measurements with respect to time which entails the drawback that differentiation carried out in discrete time amplifies noise if not properly filtered. 

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