# SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model

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## Abstract

**:**

## 1. Introduction

## 2. PMSM Model

## 3. SOS Optimization

## 4. SOS Based Nonlinear Observer Design

#### 4.1. SOS Optimization Stability Constraints

#### 4.2. SOS Optimization ISS Constraints

**Theorem**

**1.**

#### 4.3. SOS Based Observer Design

## 5. Observer Design for Speed Estimation

## 6. Observer Design for Load Estimation

## 7. Numerical Simulations

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Estimation performances: black line for true signal, red line for estimated signal. (

**a**) ${x}_{1}$ estimation, (

**b**) ${x}_{2}$ estimation, (

**c**) ${x}_{5}$ estimation, (

**d**) ${x}_{5}$ estimation.

**Figure 2.**Estimation performances: black line for true signal, red line for estimated signal. (

**a**) ${x}_{35}$ estimation, (

**b**) ${x}_{36}$ estimation, (

**c**) ${x}_{3}$ estimation, (

**d**) $w$ error signal.

**Figure 3.**Estimation performances: blue lines for perturbed parameters, red line for true parameters. (

**a**) ${x}_{3}$ estimation, (

**b**) $w$ error signal.

Time (s) | Voltage Inputs | Load Term |
---|---|---|

0–25 | ${u}_{1}=10\mathrm{sin}\left(2\pi \left[5\right]t\right)$ ${u}_{2}=10\mathrm{sin}\left(2\pi \left[5\right]t+\pi /2\right)$ | $w=-10$ |

25–50 | ${u}_{1}=10\mathrm{sin}\left(2\pi \left[5\right]t\right)$ ${u}_{2}=10\mathrm{sin}\left(2\pi \left[5\right]t+\pi /2\right)$ | $w=5$ |

50–60 | ${u}_{1}=20\mathrm{sin}\left(2\pi \left[5\right]t\right)$ ${u}_{2}=20\mathrm{sin}\left(2\pi \left[5\right]t+\pi /2\right)$ | $w=5$ |

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**MDPI and ACS Style**

Sel, A.; Sel, B.; Coskun, U.; Kasnakoglu, C.
SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model. *Sustainability* **2022**, *14*, 10650.
https://doi.org/10.3390/su141710650

**AMA Style**

Sel A, Sel B, Coskun U, Kasnakoglu C.
SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model. *Sustainability*. 2022; 14(17):10650.
https://doi.org/10.3390/su141710650

**Chicago/Turabian Style**

Sel, Artun, Bilgehan Sel, Umit Coskun, and Cosku Kasnakoglu.
2022. "SOS-Based Nonlinear Observer Design for Simultaneous State and Disturbance Estimation Designed for a PMSM Model" *Sustainability* 14, no. 17: 10650.
https://doi.org/10.3390/su141710650