Modeling and analysis of the coupled dynamics of machine degradation and repair processes using piecewise affine stochastic differential equations

Abstract

This paper introduces an approach to model the coupled dynamics of recurring degradation and restoration processes that take place in real-world systems, such as manufacturing machines, in the form of nonlinear (piecewise affine) differential equations. Unlike previous methods, interactions between degradation and repair dynamics that influence downtime distributions in such manufacturing systems can be explicitly considered and dependencies beyond correlations between the time between failures (TBF) and the time to repair (TTR) can be captured. The periodic solutions of the model capture the progressive evolution of long timescale failure and repair patterns. The distribution of short timescale failure-repair cycles can be captured by providing a class of random perturbations to certain model parameters. We provide sufficient conditions for the existence and stability of the resulting nonlinear stochastic differential equation (n-SDE) model solutions that mimic the breakdown and repair patterns observed in many real-world manufacturing systems, namely, fairly regular (periodic) large breakdown and repair cycles, interspersed with highly right skewed distributions of short cycles. We also define the basin of attraction for the periodic orbit. The n-SDE model was parametrized using real-world data sets acquired from an automotive manufacturing assembly line segment, and the model solutions were compared with actual observations of TBF and TTR patterns, as well as the performance of the process. Our approach reduces the computation time by about 25% when compared to a discrete-event simulation model, which uses conventional TBF and TTR distributions, implemented on a commercial platform. Experimental investigations also suggest that the model can capture the correlations and nonlinear coupled dynamics that exist in real-world operations among TBF and TTR, which are typically ignored in traditional approaches.