# An alternative approach for MLE calculation in nonlinear continuous dynamic systems

## Abstract

As an important metric to tell if 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, etc.) 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 shows 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.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1504439

- Report Number(s):
- PNNL-SA-126104

Journal ID: ISSN 0924-090X

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Journal Name:
- Nonlinear Dynamics

- Additional Journal Information:
- Journal Volume: 95; Journal Issue: 3; Journal ID: ISSN 0924-090X

- Publisher:
- Springer

- Country of Publication:
- United States

- Language:
- English

- Subject:
- maximal Lyapunov exponents, nonlinear differential dynamic systems, inaccurate model, measurement with noise

### Citation Formats

```
Wang, Shaobu, and Huang, Zhenyu.
```*An alternative approach for MLE calculation in nonlinear continuous dynamic systems*. United States: N. p., 2019.
Web. doi:10.1007/s11071-018-4712-1.

```
Wang, Shaobu, & Huang, Zhenyu.
```*An alternative approach for MLE calculation in nonlinear continuous dynamic systems*. United States. doi:10.1007/s11071-018-4712-1.

```
Wang, Shaobu, and Huang, Zhenyu. Tue .
"An alternative approach for MLE calculation in nonlinear continuous dynamic systems". United States. doi:10.1007/s11071-018-4712-1.
```

```
@article{osti_1504439,
```

title = {An alternative approach for MLE calculation in nonlinear continuous dynamic systems},

author = {Wang, Shaobu and Huang, Zhenyu},

abstractNote = {As an important metric to tell if 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, etc.) 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 shows 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.},

doi = {10.1007/s11071-018-4712-1},

journal = {Nonlinear Dynamics},

issn = {0924-090X},

number = 3,

volume = 95,

place = {United States},

year = {2019},

month = {1}

}