An alternative approach for MLE calculation in nonlinear continuous dynamic systems

Abstract

As an important metric to tell whether a nonlinear dynamic system has a singular attractor or divergent trajectory, the maximal Lyapunov exponent (MLE) can be calculated from either system models or time series of state variable measurement. However, in the real world, due to inaccurate models, measurement noise, and the fact that sometimes state variables cannot be measured directly, it is very difficult to get an accurate MLE, which limits its application in, for example, in prediction of a nonlinear physical system (e. g. power systems) behavior. To overcome these factors, this paper proposed a trajectory estimation-based MLE calculation approach. The proposed approach addressed how to calculate the MLE when state variables cannot be accessed directly, and uncertainties in system models, as well as noise in measurements. The simulation results show that the proposed approach is able to handle well the nonlinear measurement functions between state variables and measurements, and get better results than pure model-based approaches or measurement-based approaches in front of measurement noise and model uncertainties.

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Correspondence to Shaobu Wang.

Appendix

Appendix

Codes for step 1

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Codes for step 2–4

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Wang, S., Huang, Z. An alternative approach for MLE calculation in nonlinear continuous dynamic systems. Nonlinear Dyn 95, 2591–2603 (2019). https://doi.org/10.1007/s11071-018-4712-1

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Keywords

  • Maximal Lyapunov exponents
  • Nonlinear differential dynamic systems
  • Inaccurate model
  • Measurement with noise
  • Nonlinear measurement functions