# Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches

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## Abstract

**:**

## 1. Introduction

- Developing a pseudo-linearised representation of the PMSM system.
- Designing a controller for optimal tracking of the PMSM’s reference speed with high accuracy and a quick speed without a speed sensor.
- Estimating motor speed in a sensorless framework.
- Addressing the challenge of disturbances during the course of the control.
- Maintaining the function of estimation and speed control during all times of sampling and not just at the impulse sample times.
- Quantifying the effects of impulse intervals (the sample rate) and load torque.

## 2. State-Dependent Riccati Equation

## 3. Impulsive Observer

**Hypothesis**

**1.**

**Theorem**

**1.**

- -
- $V:{R}^{+}\times {S}_{\rho}\to {R}^{+}$, $V\in {V}_{0}$, $\rho >0$ in $\left(\left.{t}_{k-1},{t}_{k}\right]\right.$, and ${D}^{+}V\left(t,x\right)\le g\left(t,V\left(t,x\right)\right)$.
- -
- There is a ${\rho}_{0}>0$, so $x\in {S}_{{\rho}_{0}}$ and $x+\Delta x\in {S}_{{\rho}_{0}}$ for all $k$, and in $t={t}_{k}$, $V\left(t,x+\Delta x\right)\le {\psi}_{k}\left(V\left(t,x\right)\right)$.
- -
- $b\left(\Vert x\Vert \right)\le V\left(t,x\right)\le a\left(\Vert x\Vert \right)$ on ${R}^{+}\times {S}_{\rho}$ when $a,b\in \kappa $.

**Outcome**

**1.**

#### State-Dependent Impulsive Observer

## 4. Main Design: SDRE Controller Based on State-Dependent Impulsive Observer

**Hypothesis**

**2.**

**Theorem**

**2.**

**Proof**

**of Theorem 2.**

## 5. Case Study: Permanent Magnet Synchronous Motor

## 6. Simulation Results and Discussions

#### 6.1. The Main Simulation Results

#### 6.2. The Effect of Impulse Intervals

#### 6.3. The Effect of Load Torque

#### 6.4. Comparisons

_{1}, c c

_{2}, and c c

_{3}, which are the normalised mean squared errors and correlation coefficients across both the d and q axes, are lower for the LQR compared to the SDRE. Based on the performance metrics, this superiority is due to the SDRE being able to take care of the system’s intrinsic nonlinearities, especially when the working point changes during tracking.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Speed profiles of the PMSM, the state estimation, the tracking error, and the speed impulses, together with the zoomed-in areas.

**Figure 7.**Load torque impulsive observer gains, speed tracking and estimation with load torque, speed tracking error with load torque, and speed estimation error with load torque.

**Table 1.**The PMSM’s parameters [35].

Parameter | Value |
---|---|

Number of poles $\left({N}_{p}\right)$ | $4$ |

Stator resistance $\left({R}_{s}\right)$ | 0.0875 Ω |

Permanent magnetic flux $\left({\lambda}_{f}\right)$ | 1 Wb |

Inductance $\left({L}_{d}={L}_{q}\right)$ | $0.2155\hspace{0.17em}\mathrm{H}$ |

Moment of inertia $\left(J\right)$ | $0.0151\hspace{0.17em}{\mathrm{kgm}}^{2}$ |

Friction coefficient $\left(B\right)$ | $0.0378\hspace{0.17em}{\mathrm{kgm}}^{2}{\mathrm{s}}^{-1}$ |

Metrices | Impulse Intervals (Δ(s)) | ||||
---|---|---|---|---|---|

${10}^{-4}$ | $5\times {10}^{-4}$ | ${10}^{-3}$ | $5\times {10}^{-3}$ | $6\times {10}^{-3}$ | |

$NMSE$ | 45.6892 | 128.6873 | 177.1134 | 301.8423 | 2521.4897 |

$C{C}_{1}$ | 0.9992 | 0.9968 | 0.9925 | 0.9725 | 0.6850 |

$C{C}_{2}$ | 1 | 0.9999 | 0.9998 | 0.9994 | 0.7982 |

$C{C}_{3}$ | 1 | 1 | 0.9999 | 0.9812 | 0.8241 |

Δ(s) = 5 × 10^{−3} | The Proposed Method | LQR |
---|---|---|

$NMSE$ | 301.8423 | 3128.2159 |

$C{C}_{1}$ | 0.9725 | 0.6014 |

$C{C}_{2}$ | 0.9994 | 0.7102 |

$C{C}_{3}$ | 0.9812 | 0.7371 |

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**MDPI and ACS Style**

Kalamian, N.; Soltani, M.; Bouzari Liavoli, F.; Faraji Niri, M.
Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches. *Computers* **2024**, *13*, 142.
https://doi.org/10.3390/computers13060142

**AMA Style**

Kalamian N, Soltani M, Bouzari Liavoli F, Faraji Niri M.
Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches. *Computers*. 2024; 13(6):142.
https://doi.org/10.3390/computers13060142

**Chicago/Turabian Style**

Kalamian, Nasrin, Masoud Soltani, Fariba Bouzari Liavoli, and Mona Faraji Niri.
2024. "Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches" *Computers* 13, no. 6: 142.
https://doi.org/10.3390/computers13060142