23. Approximate Bayesian Inference for Piecewise-Linear Stiffness Systems

This paper considers the problem of simultaneous model selection and parameter estimation for dynamical systems with piecewise-linear (PWL) stiffnesses. PWL models are a series of locally linear models that specify or approximate nonlinear systems over some defined operating range. They can be used to model hybrid phenomena common in practical situations, such as, systems with different modes of operation, or systems whose dynamics change because of physical limits or thresholds. Identifying PWL models can be a challenging problem when the number of operating regions and the boundaries of the regions are unknown. This study focusses on the joint problem of identifying the regions (their number and boundaries) as well as their associated parameters. PWL-stiffness models with up to four regimes are considered, and the identification problem is treated as a combined model selection and parameter estimation problem, addressed in a Bayesian framework. Because of the varying number of parameters across the PWL models, traditional Bayesian model selection would typically require reversible-jump Markov chain Monte Carlo (RJ-MCMC) for switching between model spaces. Here instead, a likelihood-free Approximate Bayesian Computation (ABC) scheme with nested sampling is followed, which simplifies the jump between model spaces. To illustrate its performance, the algorithm has been used to select models and identify parameters from four PWL-stiffness systems—linear, bilinear, trilinear, and quadlinear stiffnesses. The results demonstrate the flexibility of using ABC for identifying the correct model and parameters of PWL-stiffness systems, in addition to furnishing uncertainty estimates of the identified parameters.