POD-SOD methods for signal processing

POD, also referred to as Principal Component Analysis (PCA), is a statistical approach that can be used to represent multidimensional data as orthogonal basis vectors. It is a technique extensively used for reduced-order modeling. Smooth Orthogonal Decomposition (SOD) can be seen as an extension of POD, performing a projection of the data matrix to basis vectors such that the projection of the data onto these vectors has both maximal variance and minimal roughness.

We leverage POD and SOD methods to identify the dynamically relevant subspaces of the system. The directly measured vibration signals are projected onto the identified subspaces, and appropriate features are extracted from the mapped signals in the subspaces, instead of the relatively complicated raw measurement space. These obtained features pave the way for developing an Artificial Neural Network (ANN) to effectively classify different gear conditions and detect defects.

The top corner picture depicts the geometric interpretation of POD. In order to obtain POMs (Proper Orthogonal Models) ϕ𝑖, the norm of the projections of data onto this vector (𝑦𝑛⃗ϕ̂𝑖). We obtain Ο•1 as the solution of the optimization problem and choose the orthogonal vector Ο•2 as the second POM.The geometric interpretation of POD where Target to obtain SOMs (Smooth Orthogonal Models) πœ‘π‘– to maximize the norm of the projections of data onto this vector (π‘¦π‘‘βƒ—πœ‘Μ‚π‘–) meanwhile minimize the norm of the projections of the velocity of each data point (π‘£π‘‘βƒ—πœ‘Μ‚π‘–). This optimization problem has two solutions πœ‘1 and πœ‘2.

In this work, we consider features directly related to POD/SOD-based subspaces for the detection of gear system defects, as shown in the figures above. For the POD technique, a feature set is extracted from each data set, including: (1) POMs, (2) POVs, and (3) the angles lying between each POM and the positive x-axis. For the SOD technique, the feature set from each data set consists of (1) SOMs, (2) SOVs, and (3) the angles lying between each SOM and the positive x-axis. POMs, POVs, SOMs, and SOVs are exclusive features of the POD and SOD methods and can describe POD-based and SOD-based subspaces. Note that the angles (features of category 3) characterize important aspects of the transformation from measurement space to subspaces, providing informative insights.

The above pictures show POD-based features: POM ϕ𝑖, POV πœ†π‘– , and angles between POMs and x-axis πœΆπ’Š, 𝑖=1,…,𝑛, and SOD-based features: SOM Ο†i, SOV Ξ»i, and angles between SOMs and x-axis 𝛂𝐒,i=1,…,n.

These features are used to train a neural network to identify single crack gears. The results demonstrate that the POD/SOD-based features can detect single crack gears with 100% accuracy [1].

Sample Publications
  1. Zihan Liu, T. Haj Mohamad, Shahab Ilbeigi, C. Nataraj, "Early Detection of Cracks in a Gear-Train System Using Proper and Smooth Orthogonal Decompositions," Advances in Nonlinear Dynamics: Proceedings of the Second International Nonlinear Dynamics Conference (NODYCON 2021), vol. 2, pp. 451-461, 2021.