Signal Based Fault Detection and Diagnosis for Rotating Electrical Machines: Issues and Solutions - Complex System Modelling and Control Through Intelligent Soft Computations - page 275

2.1 Recalled Results

In this section authors present the algorithms used to develop the FDD procedure.
They extract patterns by current signals using PCA and KDE. Then K - L divergence

compares these patterns to extract the motor health index.

2.1.1 Principal Component Analysis

PCA is a dimensionality reduction technique that produces a lower dimensional
representation in a way that preserves the correlation structure between the process
variables capturing the variability in the data (Jolliffe 2002 ). PCA rotates the
original coordinate system along the direction of maximum variance. Considering a
data matrix X

N m of N sample rows and m variable columns that are nor-
malized to zero mean with mean values vector

2 R

l . The matrix X can be decomposed

as follows:

¼ X þ X X ; ð 1 Þ

where X is the projection on the Principal Component Subspace (PCS) S d , and X,
the residual matrix, is the projection on the Residual Subspace (RS) S r (see Misra
et al. 2002 ). De

ning the loading matrix P, whose columns are the right singular
vectors of X, and selecting the columns of the loading matrix P

m d , which

2 R

correspond to the loading vectors associated with the first d singular values, it

follows that:

X

XPP T ¼ 2 S d : ð 2 Þ

X, is the difference between the data matrix X and its

The residuals matrix

projection into the first d principal components retained in the PCA model:

X

PP T

¼ X ð I Þ2 S r ; ð 3 Þ

therefore the residual matrix captures the variations in the observations space
spanned by the loading vectors associated with the r

d smallest singular
values. The projections of the observations into the lower-dimensional space are
contained in the score matrix:

¼ m

N d

T ¼ XP 2 R : ð 4 Þ

Here, PCA is applied to the three-phase currents of induction motors in order to
reduce the inputs space from the three original dimensions to two because the
currents are highly correlated. Indeed for healthy motor, with three-phase without
neutral connection,

ideal conditions and a balanced voltage supply, the stator