#
H_{∞} Observer Based on Descriptor Systems Applied to Estimate the State of Charge

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- 1.
- The derivation of the piecewise function increased the order of the observer, which did not match the original system, and the observer error was not converged potentially;
- 2.
- The derivation of the current was ignored completely, so the dynamic performance of the observer became worse.

## 2. Battery Model

**Assumption**

**1.**

## 3. ${\mathit{H}}_{\infty}$ Observer

- 1.
- With $\omega =0$, the estimate error $e=x-\widehat{x}$ is asymptotically stable;
- 2.
- With $\omega \ne 0$, for a prescribed level of noise $\gamma >0$, ${\Vert e\Vert}_{{L}_{2}}\gamma {\Vert \omega \Vert}_{{L}_{2}}$ will be satisfied.

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

- 1.
- Model the battery system as a descriptor system (2);
- 2.
- Determine the matrix $\mathsf{\Phi}$ by $\mathsf{\Phi}E=\mathbf{0}$;
- 3.
- Determine the matrix ${N}^{{}^{\prime}}$ by the (9);
- 4.
- Choose the prescribed level of noise $\gamma $ by optimization problems (13);
- 5.
- Solve the feasible solution of (11) given by Theorem 1;
- 6.
- Calculate the matrices H, J, P, Q, R, ${\phi}_{1}$, and ${\phi}_{2}$;
- 7.
- Convert the virtual output into the actual measurable output by $\overline{y}=y-Du$.

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

REVs | Renewable energy vehicles |

SOC | State of charge |

OCVM | Open-circuit voltage method |

CCM | Coulomb counting method |

KF | Kalman filter |

SMO | Sliding-mode observer |

PI | Proportional-integral |

OCV | Open-circuit voltage |

RC | Resistance–capacitance |

LMI | Linear matrix inequality |

DST | Dynamic stress test |

RMSE | Root mean square error |

MAE | Maximum absolute error |

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**Figure 4.**SOC estimate error under the constant current discharge experiment: (

**a**) based on the ${H}_{\infty}$ observer; (

**b**) based on the PI observer; (

**c**) based on SMO.

**Figure 5.**SOC estimate error under the constant current discharge experiment with inaccurate initial SOC: (

**a**) based on the ${H}_{\infty}$ observer; (

**b**) based on the PI observer; (

**c**) based on SMO.

**Figure 10.**The real SOC and its estimate from the ${H}_{\infty}$ observer, PI observer, and SMO under parameter perturbations: (

**a**) full graph; (

**b**) zoomed graph.

${\mathit{C}}_{\mathit{N}}$ | ${\mathit{C}}_{\mathit{C}}$ | ${\mathit{R}}_{\mathit{e}}$ | ${\mathit{R}}_{\mathit{c}}$ | ${\mathit{R}}_{\mathit{t}}$ |
---|---|---|---|---|

18,000 F | 200 F | 0.003 $\mathsf{\Omega}$ | 0.003 $\mathsf{\Omega}$ | 0.001 $\mathsf{\Omega}$ |

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**MDPI and ACS Style**

Meng, S.; Li, S.; Chi, H.; Meng, F.; Pang, A.
*H*_{∞} Observer Based on Descriptor Systems Applied to Estimate the State of Charge. *Entropy* **2022**, *24*, 420.
https://doi.org/10.3390/e24030420

**AMA Style**

Meng S, Li S, Chi H, Meng F, Pang A.
*H*_{∞} Observer Based on Descriptor Systems Applied to Estimate the State of Charge. *Entropy*. 2022; 24(3):420.
https://doi.org/10.3390/e24030420

**Chicago/Turabian Style**

Meng, Shengya, Shihong Li, Heng Chi, Fanwei Meng, and Aiping Pang.
2022. "*H*_{∞} Observer Based on Descriptor Systems Applied to Estimate the State of Charge" *Entropy* 24, no. 3: 420.
https://doi.org/10.3390/e24030420