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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Udwadia, F.E., Bremen, H.F.: An efficient and stable approach for computation of Lyapunov characteristic exponents of continuous dynamical systems. Appl. Math. Comput. 121(2–3), 219–259 (2001)
Efimov, V., Prusov, A., Shokurov, M.: Seasonal instability of pacific sea surface temperature anomalies. Q. J. R. Meteorol. Soc. Suppl. B 123(538), 337–356 (1997)
Schmid, G., Dunkin, R.: Indications of nonlinearity intraindividual specificity and stability of human EEG—the unfolding dimension. Physica D 93(3–4), 165–1900 (1996)
Froeschle, C., Lega, E., Gonczi, R.: Fast Lyapunov indicators—application to asteroidal motion. Celest. Mech. Dyn. Astron. 1(67), 41–62 (1997)
Nicolis, G., Daems, D.: Nonequilibrium thermodynamics of dynamical systems. J. Phys. Chem. 100(49), 19187–19191 (1996)
Ramasubramanian, K., Sriram, M.S.: A comparative study of computation of Lyapunov spectra with different algorithms. Physica D 139(1–2), 72–86 (2000)
Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D Nonlinear Phenom. 16(3), 285–317 (1985)
Rosenstein, M.T., Collins, J.J., De Luca, J.: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D Nonlinear Phenom. 65(1–2), 117–134 (1993)
Sato, S., Sano, M., Sawada, Y.: Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic systems. Prog. Theor. Phys. 77(1), 1–5 (1987)
Eckmann, J.P., Ruelle, D.: Ergodic theory of chaos and strange attractors. In: Hunt, B.R., Kennedy, J.A., Li, T.Y., Nusse, H.E. (eds.) The Theory of Chaotic Attractors. Springer, New York (2004)
Farmer, J.D., Sidorowich, J.J.: Predicting chaotic time series. Phys. Rev. Lett. 59(8), 845 (1987)
Sano, M., Sawada, Y.: Measurement of the Lyapunov spectrum from a chaotic time series. Phys. Rev. Lett. 55(10), 1082 (1985)
Udwadia, F.E., von Bremen, H.F.: Computation of Lyapunov characteristic exponents for continuous dynamical systems. J. Appl. Math. Phys. 53(1), 123–146 (2002)
Huang, Z., Zhou, N., Diao, R., Wang, S., Elbert, S., Meng, D., Lu, S.: Capturing real-time power system dynamics: opportunities and challenges. In: Proceedings of IEEE Power and Energy Society General Meeting, Denver, CO. USA, 26–30 July (2015)
Farantatos, E., Stefopoulos, G.K., Cokkinides, G.J., Meliopoulos, A.P.: PMU-based dynamic state estimation electric power systems. In: Proceedings of IEEE Power and Energy Society General Meeting, Calgary, AB, Canada, 26–30 July (2009)
Ghahremani, E., Kamwa, I.: Online state estimation of a synchronous generator using unscented Kalman filter from phasor measurements units. IEEE Trans. Energy Convers. 26(4), 1099–1108 (2011)
Li, Y., Huang, Z., Zhou, N., Lee, B., Diao, R., Du, P.: Application of ensemble Kalman filter in power system state tracking and sensitivity analysis. In: Proceedings of IEEE PES Transmission and Distribution Conference and Exposition, Orlando, FL, USA, 7–10 May (2012)
Wang, S., Gao, W., Sakis Meliopoulos, A.P.: An alternative method for power system dynamic state estimation based on unscented transform. IEEE Trans. Power Syst. 27(2), 942–950 (2012)
Huang, Z., Du, P., Kosterev, D., Yang, S.: Generator dynamic model validation and parameter calibration using phasor measurements at the point of connection. IEEE Trans. Power Syst. 28(2), 1939–1949 (2013)
Fan, L., Wehbe, Y.: Extended Kalman filtering based real-time dynamic state and parameter estimation using PMU data. Electr. Power Syst. Res. 103, 168–177 (2013)
Zhou, N., Meng, D., Huang, Z., Welch, G.: Dynamic state estimation of a synchronous machine using PMU data: a comparative study. IEEE Trans. Smart Grid 6(1), 450–460 (2015)
Mandel, J.: A brief tutorial on the ensemble Kalman filter. Center for Computational Mathematics, University of Colorado at Denver (2007). http://www-math.ucdenver.edu/ccm/reports/rep242.pdf
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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
- Maximal Lyapunov exponents
- Nonlinear differential dynamic systems
- Inaccurate model
- Measurement with noise
- Nonlinear measurement functions