Identifying a Set of ARR Residuals Close to Zero

Alternatively, a DBG with non-ideal switches and mode invariant causalities can be developed that holds for all system modes. It is assumed that the system under consideration is healthy and that no faults occur during system mode identification. For simplicity it is assumed that mode dependent ARRs in closed symbolic form can be deduced from the DBG. Let s be the number of switches in the model and / V the number of physically feasible switch state combinations, i.e. denotes the number of system modes. Furthermore, let u = (u U2 - un) be the vector of N known system inputs and y = (yi Jm) the vector of M inputs into a DBG either obtained by measurements from the real system or by evaluating a behavioural model of the real system. Then each ARR residual n(r), / = 1, is the weighted sum of known system inputs, known measurements and derivatives of measurements. 

For each of the technically feasible system modes, there is a set of ARRs. Some of them may be system mode independent and can be discarded with regard to system mode identification. Furthermore, for each sampled time instant t, the weighting factors and are constants. 

ARRs are the ones that reflect the current system mode. Once some discrete switch states change, the system mode changes. Accordingly, output variables, obtained by measurement of a real system or by a numerical evaluation of a behavioural model that may encompass models of the sensors, change significantly. Accordingly, all entries of another column in the matrix become close to zero indicating the new system mode. 

In conclusion, there is a unique set of ARRs for each system mode. A system mode is characterised by a set of discrete switch states that are variables in the ARRs. When switches change their state, the system mode of operation changes. As a result, residuals of the current set of ARRs may take significantly different values invalidating the set of ARRs that characterises the current system mode. Sets of ARRs can be used to identify system modes and changes of system modes. 

This system mode identification may require considerable computational costs especially if ARRs cannot be deduced in closed symbolic form so that the entire DBG model is to be evaluated to obtain numerically the time history of ARR residuals. However, as it is one and same the problem with different sets of discrete switch state variables, this computation can be easily and efficiently performed in parallel on multicore processors or multiprocessor computers. 

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