# Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods and Problem Definition

#### 2.1. Nonlinear Model Selection for Dynamic Equations of Engineering Structures

#### 2.2. Initialising, Scaling and Bounding for the Parameters

#### 2.3. Assessed Optimisation Methods

#### 2.4. Parameters of Optimisation and Model Selection Algorithms

#### 2.5. Comparison of Optimisation Methods

- CPU time
- Number of function evaluations (Equation (4))
- Number of iterations

- Number of the function added or eliminated
- Whether the functions selected are the true nonlinear function
- Complexity by number of terms overall
- How accurate are the prediction of the nonlinear model: error (MSE), dispersion of error (standard deviation of errors)

## 3. Benchmark Problems

## 4. Results and Discussion

#### Performance of Hybrid Optimisation Methods

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Cantilever beam with a nonlinear attachment benchmark (

**a**) configuration, (

**b**) the first ten normalized mode shape functions ${\varphi}_{n}(x/L)$, (

**c**) Amplitude spectrum of acceleration response at the tip of the beam.

**Figure 4.**Model selection convergence curves for FB algorithm for the cases: (

**a**) SDOF, (

**b**) 3DOF, (

**c**) Cbeam systems.

**Figure 5.**Efficiency criteria of optimisation methods for nonlinear model selection (

**a**) CPU time, (

**b**) function calls, (

**c**) number of iterations; (*: the number of vertical axes should be multiplied by ten).

**Figure 6.**Evaluation of the number of terms processed by model selection algorithm using different optimisation methods (

**a**) SDOF terms added and removed, (

**b**) SDOF number of terms delivered, (

**c**) 3DOF terms added and removed, (

**d**) 3DOF number of terms delivered, (

**e**) Cbeam terms added and removed, (

**f**) Cbeam number of terms delivered; (The red dashed lines present the true number of the terms).

**Figure 7.**Delivering the true nonlinear model for each benchmark. Color tiles: successful; white tiles: unsuccessful at delivering the true model.

**Figure 8.**Accuracy of the delivered model using different optimisation methods in predicting the response (

**a**) MSE of nonlinear force; (

**b**) standard deviation (SD) of error.

**Figure 9.**Nonlinear modal force-displacement responses fitted based on LM optimisation algorithm (

**a**) mode 1, (

**b**) mode 2, (

**c**) mode 3, (

**d**) mode 4.

Global Methods | Local Methods |
---|---|

Nelder–Mead Simplex (NMS) | Quasi-Newton (QN) |

Particle Swarm (PS) | Sequential Quadratic Programming (SQP) |

Particle Swarm (PS) + local method | Active-Set (AS) |

Multi-start (MS) + local method | Interior Point (IP) |

Trust-Region-Reflective (TRR) | |

Levenberg-marquardt (LM) |

No. | Nonlinear Term ${\mathit{f}}_{\mathit{nl}}\left(\mathit{q}\right)$ | No. | Nonlinear Term ${\mathit{f}}_{\mathit{nl}}\left(\mathit{q}\right)$ |
---|---|---|---|

1 | $\left|q\right|q$ | 7 | $\mathrm{sign}\left(q\right)\sqrt{\left|q\right|}$ |

2 | ${q}^{3}$ | 8 | $q\sqrt{\left|q\right|}$ |

3 | $\left|q\right|{q}^{3}$ | 9 | $\left|q\right|q\sqrt{\left|q\right|}$ |

4 | ${q}^{5}$ | 10 | ${q}^{3}\sqrt{\left|q\right|}$ |

5 | $\left|q\right|{q}^{5}$ | 11 | ${F}_{f}\left(\frac{2}{\left(1+{\mathrm{e}}^{(-{\sigma}_{g}q)}\right)}-1\right)+{K}_{f}q$ |

6 | ${q}^{7}$ | 12 | ${K}_{d}\left(q-\mathrm{sign}\left(q\right)\frac{d}{2}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}\left|q\right|\ge d/2$ |

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Criteria | SDOF-FB | SDOF-ES | ||
---|---|---|---|---|

MS-LM | PS-LM | MS-LM | PS-LM | |

CPU time (sec) | 6652 | 5168 | 2027 | 5248 |

MSE | $5\times {10}^{-13}$ | $4\times {10}^{-7}$ | $6\times {10}^{-13}$ | $4\times {10}^{-8}$ |

Success | Yes | No | Yes | Yes |

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**MDPI and ACS Style**

Safari, S.; Monsalve, J.L.
Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems. *Vibration* **2021**, *4*, 648-665.
https://doi.org/10.3390/vibration4030036

**AMA Style**

Safari S, Monsalve JL.
Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems. *Vibration*. 2021; 4(3):648-665.
https://doi.org/10.3390/vibration4030036

**Chicago/Turabian Style**

Safari, Sina, and Julián Londoño Monsalve.
2021. "Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems" *Vibration* 4, no. 3: 648-665.
https://doi.org/10.3390/vibration4030036