# Sensorless Direct Torque Control of Surface-Mounted Permanent Magnet Synchronous Motors with Nonlinear Kalman Filtering

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## Abstract

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## 1. Introduction

## 2. Dynamics of Surface-Mounted Permanent Magnet Synchronous Motors

## 3. Direct Torque Control

## 4. Nonlinear Estimation

#### 4.1. Extended Kalman Filter

#### 4.2. Resilient Extended Kalman Filter

- ${x}_{k}\in {\mathcal{R}}^{n}$ state vector
- ${v}_{k}\in {\mathcal{R}}^{n}$ system noise
- ${y}_{k}\in {\mathcal{R}}^{n}$ measurement vector
- ${w}_{k}^{i}\in \mathcal{R}$ measurement noise in each phasormeasurement unit and ${w}_{k}={[{w}_{k}^{1},{w}_{k}^{2},\dots ,{w}_{k}^{p}]}^{T}$
- $f,h$ differentiable non-linear vector functions

- Initialization$$\begin{array}{c}\hfill {\widehat{x}}_{o}=E[{x}_{o}]\\ \hfill {P}_{o}=E[({x}_{o}-{\widehat{x}}_{o}){({x}_{o}-{\widehat{x}}_{o})}^{T}]\end{array}$$
- Computation of Jacobian matrices$$\begin{array}{c}\hfill {A}_{k}=\frac{\partial f}{\partial x}{{|}_{x={\widehat{x}}_{k}},{C}_{k}=\frac{\partial h}{\partial x}|}_{x={\widehat{x}}_{k}}\end{array}$$
- For time steps $k=1,2,3,\dots $, the estimator propagates by calculating the feedback gain$$\begin{array}{cc}\hfill {K}_{k}^{o}=& ({A}_{k}{P}_{k}{C}_{k}^{T}{\overline{\Gamma}}_{k}^{T})[{\overline{\Gamma}}_{k}{C}_{k}{P}_{k}{C}_{k}^{T}{\overline{\Gamma}}_{k}^{T}+\hfill \\ & \mathrm{{\rm Y}}\otimes (h({\widehat{x}}_{k}){h}^{T}({\widehat{x}}_{k})+{C}_{k}{P}_{k}{C}_{k}^{T})+{W}_{k}{]}^{-1}\hfill \end{array}$$$$\begin{array}{c}{P}_{k+1}={A}_{k}{P}_{k}{A}_{k}^{T}+{V}_{k}+{\lambda}_{max}\{{\overline{\Gamma}}_{k}{C}_{k}{P}_{k}{C}_{k}^{T}{\overline{\Gamma}}_{k}^{T}+{W}_{k}+\hfill \\ \mathrm{{\rm Y}}\otimes (h({\widehat{x}}_{k}){h}^{T}({\widehat{x}}_{k})+{C}_{k}{P}_{k}{C}_{k}^{T})\}\delta I\hfill \\ -({A}_{k}{P}_{k}{C}_{k}^{T}{\overline{\Gamma}}_{k}^{T})[{\overline{\Gamma}}_{k}{C}_{k}{P}_{k}{C}_{k}^{T}{\overline{\Gamma}}_{k}^{T}+\hfill \\ \mathrm{{\rm Y}}\otimes (h({\widehat{x}}_{k}){h}^{T}({\widehat{x}}_{k})+{C}_{k}{P}_{k}{C}_{k}^{T})+{W}_{k}{]}^{-1}({\overline{\Gamma}}_{k}{C}_{k}{P}_{k}{A}_{k}^{T})\hfill \end{array}$$$$\begin{array}{c}\hfill {\widehat{x}}_{k+1}=f\left({\widehat{x}}_{k}\right)+({K}_{k}^{o}+{\Delta}_{k})({y}_{k}-{\overline{\Gamma}}_{k}h\left({\widehat{x}}_{k}\right))\end{array}$$$$\begin{array}{c}\hfill \mathrm{{\rm Y}}=diag[{\pi}_{1}(1-{\pi}_{1}),{\pi}_{2}(1-{\pi}_{2}),\dots ,{\pi}_{p}(1-{\pi}_{p})]\\ \hfill =\left(\begin{array}{cccc}{\pi}_{1}(1-{\pi}_{1})& 0& \cdots & 0\\ 0& \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0\\ 0& \cdots & 0& {\pi}_{p}(1-{\pi}_{p})\end{array}\right)\end{array}$$

#### 4.3. Unscented Kalman Filter

- $f({x}_{k},{u}_{k})$ process model
- ${x}_{k}$ state vectors
- ${u}_{k}$ input state vectors
- $h\left({x}_{k}\right)$ output model
- ${y}_{k}$ output state vectors
- ${v}_{k}$ process WGN
- ${w}_{k}$ measurement WGN

- Initialization$$\begin{array}{c}\hfill {\widehat{x}}_{o}=E[{x}_{o}]\\ \hfill {P}_{o}=E[({x}_{o}-{\overline{x}}_{o}){({x}_{o}-{\overline{x}}_{o})}^{T}]\end{array}$$
- Define sigma points ${\chi}_{k}^{\left(s\right)}$ and weights ${\varpi}^{s}$ for $s=1,\cdots ,2n$ as follows:$$\begin{array}{c}\hfill {\chi}_{k}^{\left(0\right)}={\widehat{x}}_{k}\end{array}$$$$\begin{array}{cc}\hfill {\chi}_{k}^{\left(s\right)}={\widehat{x}}_{k}+{\tilde{x}}_{k}^{\left(s\right)}& s=1,\cdots ,2n\end{array}$$$$\begin{array}{cc}\hfill {\tilde{x}}_{k}^{\left(s\right)}={{\left(\sqrt{n{P}_{k}}\right)}_{s}}^{T}& s=1,\cdots ,n\end{array}$$$$\begin{array}{cc}\hfill {\tilde{x}}_{k}^{(n+s)}=-{{\left(\sqrt{n{P}_{k}}\right)}_{s}}^{T}& s=1,\cdots ,n\end{array}$$The weighing coefficients are determined by$$\begin{array}{cc}\hfill {\varpi}^{s}={\displaystyle \frac{1-{\varpi}^{o}}{2n}}& s=1,\cdots ,2n\end{array}$$$$\sum _{s=0}^{2n}{\varpi}^{s}=1.$$
- Process UpdateThe priori mean and covariance of the estimated value ${\widehat{x}}_{k+1}^{-}$ can be obtained using the transformed sigma points as follows:$$\begin{array}{c}\hfill {\widehat{x}}_{k+1}=\sum _{s=0}^{2n}{\varpi}_{s}\xb7f({\chi}_{k}^{(s)},{u}_{k})\end{array}$$$$\begin{array}{c}\hfill {P}_{k+1}^{-}=\sum _{s=0}^{2n}{\varpi}_{s}\xb7(f({\chi}_{k}^{(s)},{u}_{k})-{\widehat{x}}_{k+1}^{-}){(f({\chi}_{k}^{(s)},{u}_{k})-{\widehat{x}}_{k+1}^{-})}^{T}+{V}_{k}\end{array}$$
- Output Covariance UpdateThe predicted measurement is$$\begin{array}{c}\hfill {\widehat{y}}_{k+1}=\sum _{s=0}^{2n}{\varpi}_{s}\xb7h({\chi}_{k}^{(s)})\end{array}$$$$\begin{array}{c}\hfill {P}_{y}=\sum _{s=0}^{2n}{\varpi}_{s}\xb7(h({\chi}_{k}^{(s)})-{\widehat{y}}_{k+1}){(h({\chi}_{k}^{(s)})-{\widehat{y}}_{k+1})}^{T}+{W}_{k+1}\end{array}$$
- Cross-correlation UpdateThe cross-correlation ${P}_{xy}$ is determined by$$\begin{array}{c}\hfill {P}_{xy}=\sum _{s=0}^{2n}{\varpi}_{s}\xb7(f({\chi}_{k}^{(s)})-{\widehat{x}}_{k+1}){(h({\chi}_{k}^{(s)})-{\widehat{y}}_{k+1})}^{T}\end{array}$$
- Measurement UpdateThe final measurement update can be performed using normal Kalman filter equations as: The Kalman gain ${K}_{k}$ can be written as follows:$$\begin{array}{c}\hfill {K}_{k+1}={P}_{xy}P{{}_{y}}^{-1}\end{array}$$The posteriori covariance matrix ${P}_{k+1}$ and the estimated state variable ${\widehat{x}}_{k+1}$ can be expressed as follows:$$\begin{array}{c}\hfill {\widehat{x}}_{k+1}={\widehat{x}}_{k+1}^{-}+{K}_{k+1}({y}_{k+1}-{\widehat{y}}_{k+1})\end{array}$$$$\begin{array}{c}\hfill {P}_{k+1}={P}_{k+1}^{-}-{K}_{k+1}{P}_{y}{K}_{k+1}^{T}\end{array}$$

## 5. Computer Simulation Studies and Hardware Implementations

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

${i}_{a},{i}_{b},{i}_{c}$ and ${v}_{a},{v}_{b},{v}_{c}$ | 3-phase currents and voltages |

${R}_{s},{L}_{s},{i}_{s},{v}_{s}$ | stator resistance, inductance, current and voltage |

${v}_{d},{v}_{q}$ | direct and quadrature axis voltages |

${i}_{d},{i}_{q}$ | direct and quadrature axis currents |

${\lambda}_{s},{\lambda}_{m}$ | stator and rotor magnetic flux linkages |

${\lambda}_{d},{\lambda}_{q}$ | direct and quadrature axis flux linkages |

${\lambda}_{\alpha},{\lambda}_{\beta}$ | $\alpha $ and $\beta $ axis flux linkages |

${L}_{d},{L}_{q}$ | direct and quadrature axis inductances |

${\omega}_{e},{\omega}_{m}$ | electrical and mechanical angular speed |

P, J, D | number of pole, moment of inertia, and viscous friction coefficient |

${\theta}_{e}$ | electrical angular position |

${\tau}_{e},{\tau}_{l},{\tau}_{est},{\tau}_{ref}$ | electrical, load, estimated, and reference torques |

${\gamma}_{s}$ | computed angular position |

$d{\lambda}_{est},d{\tau}_{est}$ | digitized variables for flux and torque controller |

${H}_{\lambda},{H}_{\tau}$ | flux and torque tolerance bands |

${P}_{k}^{-}$, ${x}_{k}^{-}$ | priori covariance and priori state estimate |

${P}_{k+1}$ | posteriori covariance matrix |

${\widehat{x}}_{k+1}$ | estimated state variable |

${K}_{k}$ | Kalman gain |

${v}_{k}$, ${w}_{k}$ | process and measurement noise |

${V}_{k},{W}_{k}$ | covariance matrix of process and measurement noise $v,w$ at the ${k}^{th}$ time step |

${\gamma}_{k}^{i}$ | scalar binary random variables following the Bernoulli-distribution |

${\Delta}_{k}$ | additive uncertainty in Kalman gain |

${P}_{xy}$ | cross-correlation matrix |

${\chi}_{k}^{\left(s\right)}$ | sigma points |

${\varpi}^{s}$ | weighing coefficients |

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$\mathit{d}\mathit{\lambda}$ | $\mathit{d}{\mathit{\tau}}_{\mathit{est}}$ | Number of Sectors (N) | |||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

$d\lambda =1$ | $d{\tau}_{est}=1$ | ${V}_{2}$ | ${V}_{3}$ | ${V}_{4}$ | ${V}_{5}$ | ${V}_{6}$ | ${V}_{1}$ |

$d{\tau}_{est}=0$ | ${V}_{7}$ | ${V}_{0}$ | ${V}_{7}$ | ${V}_{0}$ | ${V}_{7}$ | ${V}_{0}$ | |

$d{\tau}_{est}=-1$ | ${V}_{6}$ | ${V}_{1}$ | ${V}_{2}$ | ${V}_{3}$ | ${V}_{4}$ | ${V}_{5}$ | |

$d\lambda =0$ | $d{\tau}_{est}=1$ | ${V}_{3}$ | ${V}_{4}$ | ${V}_{5}$ | ${V}_{6}$ | ${V}_{1}$ | ${V}_{2}$ |

$d{\tau}_{est}=0$ | ${V}_{0}$ | ${V}_{7}$ | ${V}_{0}$ | ${V}_{7}$ | ${V}_{0}$ | ${V}_{7}$ | |

$d{\tau}_{est}=-1$ | ${V}_{5}$ | ${V}_{6}$ | ${V}_{1}$ | ${V}_{2}$ | ${V}_{3}$ | ${V}_{4}$ |

Rated Power | 400 W |

Rated Torque | 180 oz.in |

Rated Voltage | 220 V |

Rated Current | 2.7 A |

Stator resistance, ${R}_{s}$ | $4.7\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}$ |

Stator inductance, ${L}_{s}$ | $13.3$ mH |

Rotor magnetic flux, ${\lambda}_{m}$ | $0.0785$ Wb |

Number of rotor poles, P | 8 |

Moment of inertia, J | $0.00439\text{}oz.in.{s}^{2}$ |

Time Period (s) | EKF Estimation Error (N/m) | REKF Estimation Error (N/m) | UKF Estimation Error (N/m) |
---|---|---|---|

0 s–0.5 s | 0.6089 | 0.6692 | 0.6675 |

0.5 s–1.0 s | 0.2188 | 0.1987 | 0.1348 |

1.0 s–1.5 s | 0.2180 | 0.1951 | 0.1653 |

1.5 s–2.0 s | 0.2197 | 0.1938 | 0.1067 |

Time Period (s) | EKF Estimation Error (A) | REKF Estimation Error (A) | UKF Estimation Error (A) |
---|---|---|---|

0 s–0.5 s | 22.3889 | 22.4353 | 13.775 |

0.5 s–1.0 s | 0.3985 | 0.3629 | 0.1491 |

1.0 s–1.5 s | 0.3917 | 0.3718 | 0.2004 |

1.5 s–2.0 s | 0.3971 | 0.3619 | 0.1842 |

Time Period (s) | EKF Estimation Error (rad/s) | REKF Estimation Error (rad/s) | UKF Estimation Error (rad/s) |
---|---|---|---|

0 s–0.5 s | 10.7434 | 11.2876 | 1.7615 |

0.5 s–1.0 s | 4.3493 | 3.0354 | 0.5825 |

1.0 s–1.5 s | 4.3622 | 2.9833 | 0.7320 |

1.5 s–2.0 s | 4.3790 | 3.0036 | 0.7345 |

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**MDPI and ACS Style**

Park, J.B.; Wang, X.
Sensorless Direct Torque Control of Surface-Mounted Permanent Magnet Synchronous Motors with Nonlinear Kalman Filtering. *Energies* **2018**, *11*, 969.
https://doi.org/10.3390/en11040969

**AMA Style**

Park JB, Wang X.
Sensorless Direct Torque Control of Surface-Mounted Permanent Magnet Synchronous Motors with Nonlinear Kalman Filtering. *Energies*. 2018; 11(4):969.
https://doi.org/10.3390/en11040969

**Chicago/Turabian Style**

Park, Joon B., and Xin Wang.
2018. "Sensorless Direct Torque Control of Surface-Mounted Permanent Magnet Synchronous Motors with Nonlinear Kalman Filtering" *Energies* 11, no. 4: 969.
https://doi.org/10.3390/en11040969