On choosing state variables for piecewise-smooth dynamical system simulations

Abstract

Choosing state variables in a state-space representation of a nonlinear dynamical system is a nonunique procedure for a given input–output relationship and therefore a potentially challenging task. It can be even more challenging when there are piecewise-defined restoring forces, as in bilinear hysteresis or Bouc–Wen models, which are just two of many such engineering mechanics models. Using various piecewise-smooth models, we make use of flow- and effort-controlled system concepts, common to bond graph theory, to initiate our state variable selection task, and we view numerical simulation as being within the framework of hybrid dynamical systems. In order to develop accurate and efficient time integration, we incorporate MATLAB’s state event location algorithm, which is a mathematically sound numerical solver that deserves to be better known in the engineering mechanics community. We show that different choices of state variables can affect state event implementation, which in turn can significantly affect accuracy and efficiency, as judged by tolerance proportionality and work–accuracy diagrams. Programming details of state event location are included to facilitate application to other models involving piecewise-defined restoring forces. In particular, one version of the Bouc–Wen–Baber–Noori (BWBN) class of models is implemented as a demonstration.