## 3 Proposed Fault Diagnosis Method  
This section presents a fast and reliable method for the detection and diagnosis of a faulty power device. The proposed fault diagnosis method is based on the definition of indexes related to the pattern of the winding currents. For the definition of these indexes, it will be used a monitoring method based on a statistical process. The statistical method it based on the entropy feature introduced initially by [PERSON] [37]. Although the concept of entropy was originally introduced in thermodynamics [38], its concepts and relevant principles have been used in many other areas such as information theory [39, 40, 41], biological diversity [42], statistical mechanism, financial markets [43], signal processing [44] and pattern recognition [45]. This statistic tool allows using a large number of data and measures the randomness or uncertainty of the system variables. To implement the entropy feature approach for the identification of the faulty patterns, it is necessary to acquire first the motor winding currents through current sensors. For the identification of the patterns it will be used a set of samples related to each winding current. On the other hand,  
Figure 4: Proposed Fault tolerant scheme for the drive used to supply the 8/6 SRM. Figure 5: Reconfiguration of the circuit in fault-tolerant mode after a failure (open or short-circuit) in power device \(S_{1A}\). to obtain the time behavior of the patterns, it is used a sliding window. Thus, a data sample matrix \(X\in\Re^{n\times m}\) of the currents will be created (1), with n samples and \(N\) variables. \[X\ =\ \left[ \begin{array}{cccc}i_{A}(t_{o})&i_{B}(t_{o})&\ldots&i_{N}(t_{o})\\ i_{A}(t_{o}+\Delta t)&i_{B}(t_{o}+\Delta t)&\ldots&i_{N}(t_{o}+\Delta t)\\ \vdots&\vdots&\vdots&\vdots\\ i_{A}(t_{o}+(n-1)\Delta t)&i_{B}(t_{o}+(n-1)\Delta t)&\ldots&i_{N}(t_{o}+(n-1) \Delta t)\end{array} \right] \tag{1}\]  
The identification of the patterns will be made through the entropy feature of each column of the matrix \(X\). Many different types of entropy methods have been developed for different application areas [46; 47]. Thus, from the data sample of each current signal of the matrix it will be computed the corresponding entropy at instant p determined by (2). \[H_{k}(i)\ =\ -\sum_{j=1}^{n}\frac{1}{n}\big{|}i_{k}(j)\big{|}\log_{2}\Big{(} \frac{1}{n}\big{|}i_{k}(j)\big{|}\Big{)},\ k=A,B,C,D \tag{2}\]  
where \(i=\{i_{A},i_{B},i_{C},i_{D}\}\) is a set of current phase variables with a data sample matrix with \(n\) samples for each current phase and \(k=A\), \(B\), \(C\), \(D\) is the index of each current phase. After the computation of the entropy feature value for each current at instant \(p\) it will be used normalized indexes (diagnostic variables) that are independent of the load and speed of the motor, in order to allow the diagnosis and identification of a faulty power device. These normalized indexes are defined as:  
\[IAV_{k}\ =\ \ 4\,H_{k}(i)\,k=A,B,C,D \tag{3}\]  
\[IAV_{T}\ =\ \sum_{k=A}^{D}H_{k}(i) \tag{4}\]  
Through Equations (3) and (4) it is possible to obtain the symmetry/asymmetry index (\(SI\)). However, several \(SI\) will be defined in order to associate the current patterns with each power device, connected to each motor winding. For the machine under study (8/6 \(SRM\)) four \(SI\) (\(A\), \(B\), \(C\), and \(D\)) will be considered and four \(SI\) indexes are defined in accordance with (5) (where \(k=A\), \(B\), \(C\), \(D\)):  
\[SI_{k}\ =\ \ \frac{IAV_{k}}{IAV_{T}} \tag{5}\]  
Since in (2) there is a logarithm of absolute of current value, if the currents are 0 then a computation problem could appear. Thus, in order to avoid this problem, it is considered a condition that if the current is 0 then it is computed the logarithm of absolute of _eps_ value. This parameter is a constant value and equal to 2-52. The different values of the diagnostic variables (\(SI\) index) will diagnose and identify the power device under failure.