Nonlinear Dynamics

, Volume 95, Issue 2, pp 1165–1188 | Cite as

On choosing state variables for piecewise-smooth dynamical system simulations

  • Jin-Song PeiEmail author
  • Joseph P. Wright
  • François Gay-Balmaz
  • James L. Beck
  • Michael D. Todd
Original Paper


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.


Restoring force model Bilinear hysteresis model Bouc–Wen model BWBN model Hybrid dynamical system State event location algorithm Bond graph theory Flow-controlled system Effort-controlled system 



Dr. Pei would like to acknowledge the hospitality of California Institute of Technology during her sabbatical leave for the completion of this study. Dr. Gay-Balmaz is partially supported by the ANR project GEOMFLUID, ANR-14-CE23-0002-01.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Civil Engineering and Environmental ScienceUniversity of OklahomaNormanUSA
  2. 2.Division of Applied ScienceWeidlinger Associates Inc.New YorkUSA
  3. 3.CNRS LMD IPSLEcole Normale Supérieure de ParisParisFrance
  4. 4.Department of Computing and Mathematical SciencesCalifornia Institute of TechnologyPasadenaUSA
  5. 5.Department of Structural EngineeringUniversity of California, San DiegoLa JollaUSA

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