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Learning Model Discrepancy of an Electric Motor with Bayesian Inference^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Electric Motor Model

## 3. Bayesian Inference Solution Process

#### 3.1. Bayesian Model 1 (BM1): Measurement Noise

#### 3.2. Bayesian Model 2 (BM2): Measurement Noise and Model Discrepancy

## 4. Numerical Results

**Definition 1.**

**Definition 2.**

#### 4.1. Results Bayesian Model 1

#### 4.2. Results Bayesian Model 2

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Noisy synthetic measurements of electric current $I\left(t\right)$ and angular velocity $\omega \left(t\right)$.

**Figure 2.**Results obtained with BM1 (noted by index BM1) and BM2 for $K=0,\cdots ,20$. Moments of the marginal posterior distributions of R are displayed on the left and moments of ${\sigma}_{I},{\sigma}_{\omega}$ on the right.

**Figure 3.**Remaining discrepancy ${\mathit{d}}^{\mathit{\epsilon}}\left(\widehat{R}\right)=[{\mathit{d}}_{I}^{\mathit{\epsilon}},{\mathit{d}}_{\omega}^{\mathit{\epsilon}}]\left(\widehat{R}\right)=\mathit{y}-{\mathcal{G}}_{\varphi}\left(\widehat{R}\right)$, for $\widehat{R}$ obtained with BM1.

**Figure 4.**Posterior moments of model discrepancy ${\mathit{\delta}}^{K}\left(\mathit{a}\right)=[{\mathit{\delta}}_{I}^{K},{\mathit{\delta}}_{\omega}^{K}]\left(\mathit{a}\right)$ for $K=9$ in comparison to the reference discrepancy ${\mathit{d}}_{0}=[{\mathit{d}}_{I,0},{\mathit{d}}_{\omega ,0}]$.

**Figure 5.**Bias-variance tradeoff: Bias $Bia{s}_{K}\left(R\right)$, variance ${V}_{K}\left[R\right|\mathit{y}]$ and mean square error $MS{E}_{K}\left(R\right)$ of R with respect to ${R}_{0}$, depending on K, for $K=0,\cdots ,20$ from BM2. The index BM1 denotes the previous results without $\mathit{\delta}$, obtained via Bayesian model 1.

**Table 1.**Marginal posterior mean and relative error of parameters $R,{\sigma}_{I}$ and ${\sigma}_{\omega}$ for BM1 and BM2 with $K=9$. The relative errors are with respect to the reference values ${R}_{0}=0.1,{\sigma}_{I,0}=2$, and ${\sigma}_{\omega ,0}=10$.

Marginal Posterior Mean | Relative Error | |||||
---|---|---|---|---|---|---|

$\widehat{\mathit{R}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{I}}$ | ${\widehat{\mathit{\sigma}}}_{\mathit{\omega}}$ | ${\mathit{\u03f5}}_{\mathit{rel}}\left(\widehat{\mathit{R}}\right)$ | ${\mathit{\u03f5}}_{\mathit{rel}}\left({\widehat{\mathit{\sigma}}}_{\mathit{I}}\right)$ | ${\mathit{\u03f5}}_{\mathit{rel}}\left({\widehat{\mathit{\sigma}}}_{\mathit{\omega}}\right)$ | |

BM1 | 9.03 × ${10}^{-2}$ | 6.29 × ${10}^{0}$ | 1.03 × ${10}^{1}$ | 9.66 × ${10}^{-2}$ | 2.14 × ${10}^{0}$ | 2.79 × ${10}^{-2}$ |

BM2 ($K=9$) | 9.92 × ${10}^{-2}$ | 2.02 × ${10}^{0}$ | 9.95 × ${10}^{0}$ | 8.48 × ${10}^{-3}$ | 1.03 × ${10}^{-2}$ | 4.89 × ${10}^{-3}$ |

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**MDPI and ACS Style**

John, D.N.; Schick, M.; Heuveline, V.
Learning Model Discrepancy of an Electric Motor with Bayesian Inference. *Proceedings* **2019**, *33*, 11.
https://doi.org/10.3390/proceedings2019033011

**AMA Style**

John DN, Schick M, Heuveline V.
Learning Model Discrepancy of an Electric Motor with Bayesian Inference. *Proceedings*. 2019; 33(1):11.
https://doi.org/10.3390/proceedings2019033011

**Chicago/Turabian Style**

John, David N., Michael Schick, and Vincent Heuveline.
2019. "Learning Model Discrepancy of an Electric Motor with Bayesian Inference" *Proceedings* 33, no. 1: 11.
https://doi.org/10.3390/proceedings2019033011