# A Novel Sensorless Approach for Speed and Displacement Control of Bearingless Switched Reluctance Motor

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*Applied Sciences*: Invited Papers in Electrical, Electronics and Communications Engineering Section)

## Abstract

**:**

## 1. Introduction

## 2. Operation and Modelling of the Bearingless Switched Reluctance Motor

#### 2.1. Operating Principle

#### 2.2. Rotor Modelling (Suspension Control)

_{1}, X

_{2}, Y

_{1}and Y

_{2}are chosen as the state variables from Equations (1) and (2). Hence, the desired tracking rotor displacement states are modelled as:

#### 2.3. The Speed Control of the BSRM

#### 2.4. Switching Control Strategy

## 3. Modelling of the Proposed Controller and Observer

#### 3.1. Design of the DSMC

- (a)
- The system will achieve stability even with incomplete information from the state observer or absence of the state observer.
- (b)
- The system will get stability even under the accommodation of unmatched disturbances.
- (c)
- The chattering can be reduced to a great extent.

_{x}> 0 and λ

_{y}> 0). When ${\sigma}_{x}$= 0 and ${\sigma}_{y}$= 0, ${\stackrel{\u2022}{S}}_{x}+{\lambda}_{x}{S}_{x}$= 0 and ${\stackrel{\u2022}{S}}_{y}+{\lambda}_{y}{S}_{y}$= 0 are asymptotically stable; therefore, the error and its first-order differential functions tend to zero. (The design of λ

_{x}and λ

_{y}are given in Appendix A.2.2).

_{2}and Y

_{2}to their desired values as t tends to infinity (the designs of ${\sigma}_{x},$ and ${\sigma}_{y},$ are given in Appendix A.2.2).

_{w}and ${\lambda}_{w}$ are the properly chosen positive design parameters. The stepwise procedure and block diagram for the implementation of the designed DSMC for the BSRM is shown in Figure 3 and Figure 4. The design parameters of the DSMC speed control are presented in Table A1 (see in Appendix A.2).

#### 3.2. Design of the SMO

_{Ψ}= 200, K

_{W}= 30 and K

_{θ}= 25 are the design parameters for the SMO, which are called the switching gain constants used as gain multipliers to the current error signal [21,22,23]. The equivalent structure of the sliding mode observer is shown in Figure 5. The sufficient condition of the sliding mode surface (${S}^{T}\stackrel{\u2022}{S}\le 0$) is considered to get the perfect system sliding mode property, even though under the presence of any faults. The switching dynamics are defined as shown in Equations (21) and (22):

_{x}* and F

_{y}*. These command force signals are given to the current hysteresis control logic block, which excites the four-phase asymmetric converter. According to the generated current hysteresis control signals, the four-phase converter directs the controlled currents to the suspension windings. Finally, the controlled suspension magnetic force is produced, and it suspends the rotor to the centre position. The speed is calculated from the rotor position using the speed estimator block, which is further compared with the speed command W *(reference speed). The proposed controller in the speed loop produces the current command signal Im* according to the resultant speed error input signal. The current hysteresis control block generates the sign-enabled signals, which are applied to the two-phase asymmetric converter. This two-phase converter excites the stator main windings according to the control signals.

## 4. Results and Discussions

#### 4.1. Measurements of the BSRM Parameters in Normal Conditions

#### 4.2. Suspension Control When There Is a Change in Suspension Loads

#### 4.3. Sensorless Speed Control When the BSRM Is in Healthy Conditions

#### 4.4. Sensorless Speed Control When Reference Speeds Are Changed

#### 4.5. Sensorless Speed Control When There Is a Change of the Load Torque

#### 4.6. Sensorless Speed Control Change in the Moment of Inertia

#### 4.7. Varying of the Supply Voltage and Phase Resistances

#### 4.8. Sensorless Speed Control When There Are Changes in the Switching Angle

## 5. Comparisons of Speeds with the Conventional SMC-Based SMO

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$m$ | Rotor weight in kg |

$g$ | Gravitational force |

$Fx$, $Fy$ | Suspension forces, N |

${F}_{dx}$,${F}_{dy}$ | Uncertain disturbances, N |

${K}_{X},{K}_{Y}$ | Equivalent constant matrices of suspension forces |

K_{xxp}, K_{yyp} K_{xyp}, K_{yxp}, K_{xxn}, K_{yyn}, K_{xyn}, K_{yxn} | Equivalent suspension force constants in both positive and negative directions, respectively |

${I}_{x}$,${I}_{Y}$ | Suspension current vectors |

ixp, iyp, ixn, iyn | X and Y-directional individual suspension currents |

$\psi $ | Per phase flux linkages |

$r$ | Phase resistance, ohms |

$N$ | Phase-inductance reversal vector |

${N}_{0}$, ${\mathrm{M}}_{0}$ | Initial reversal flux and position vector constants, respectively |

$\theta $ | Position of the rotor per phase in degree |

$V$ | Phase voltage vector, volts |

$w$ | Speed of the rotor, rpm |

${T}_{e},{T}_{l}$ | Net actual torque and applied load torque |

$B,J$ | Damping coefficient, actual moment of inertia of rotor |

$i$ | Main phase current vector, amps |

${X}_{ref}$, ${Y}_{ref},{X}_{d}$,${Y}_{d}$ | Rotor reference and desired displacements |

${X}_{2}$,${Y}_{2},{\widehat{X}}_{2}$,${\widehat{Y}}_{2}$ | Actual and estimated rotor displacements |

${K}_{\psi}$,${K}_{w}$,${K}_{\theta}$ | Flux linkage, speed and position observer coefficients |

${e}_{x},{e}_{y},{e}_{w},{e}_{\psi},{e}_{\theta}$ | Displacement errors, error speed, flux linkage and positions, respectively |

${C}_{x},{C}_{y},{C}_{w}$ | Rotor X and Y displacements and speed-switching function positive constants, respectively |

${S}_{x},{S}_{y},{S}_{w}$ | Rotor X and Y displacements and speed-sliding mode switching equations, respectively |

${U}_{x},{U}_{y},{U}_{w}$ | Rotor X and Y displacements and speed control equations, respectively |

${\sigma}_{x},{\sigma}_{y},{\sigma}_{w}$ | New dynamic sliding mode switching constants of rotor displacements and speed, respectively |

${\lambda}_{x},{\lambda}_{y},{\lambda}_{w}$ | Rotor displacements and speed control design constants |

${\mathsf{\u0220}}_{x},{\mathsf{\u0220}}_{y},{\mathsf{\u0220}}_{w}$ | Signum function constants |

## Appendix A

#### Appendix A.1. Asymmetric Converter

**Figure A1.**(

**a**) Asymmetric converter for torque windings. (

**b**) Asymmetric converter for suspending force windings.

#### Appendix A.2. DSMC Controller Design Parameters

Parameters | Values |
---|---|

Cx, λx, Ƞx | 5, 20, 78 |

Cy, λy, Ƞy | 15, 20, 108 |

Cw, λw, Ƞw | 10, 15, 130 |

#### Appendix A.2.1. Design of Switching the Function Constant

#### Appendix A.2.2. Design of the new switching function constant

- (a)
- $\stackrel{\u2022}{\sigma}\sigma =\sigma \left(\stackrel{\u2022}{D}\left(t\right)+(c+\lambda D\left(t\right)-\mathsf{\u0220}\mathrm{sgn}\left(\mathsf{\sigma}\right)\right)$, where $\sigma $, $D$ and $\mathsf{\u0220}$ are the dynamic switching mode, uncertain function and signum function constants, respectively.
- (b)
- $\mathsf{\u0220}>D+\left(c+\lambda \right)\left({D}_{0}\right)$, where ${D}_{0}$ is initial value of the uncertain function constant.

#### Appendix A.3. SMO Controller Design Parameters

SMO Design Parameters | Values |
---|---|

Kx2, Ky2 | 250, 400 |

K_{Ψ} | 200 |

K_{w} | 30 |

K_{θ} | 25 |

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**Figure 3.**Stepwise flow chart for the implementation of the dynamic sliding mode control of the bearingless switched reluctance motor (BSRM).

**Figure 16.**(

**a**) Rotor X and Y displacements, (

**b**) actual and estimated X-directional suspension forces when suspension loads are applied, and (

**c**) actual and estimated Y-directional suspension forces when suspension loads are applied.

**Figure 19.**(

**a**) Estimated and actual speeds, (

**b**) speed tracking error, and (

**c**) estimated and actual net torque values when load torques varied.

**Figure 20.**(

**a**) Estimated and actual speeds when there are increments in the moment of inertia, (

**b**) Estimated and actual speeds when there are decrements of the moment of inertia, and (

**c**) Speed tracking error for change of the moment of inertia.

**Figure 21.**(

**a**) Regulated main phase voltages, (

**b**) estimated and actual speeds when changes in the supply voltage, (

**c**) Estimated and actual speeds when changes in phase resistances, and (

**d**) speed tracking error when the phase resistances are varied.

**Figure 22.**(

**a**) Estimated and actual speeds when switching angle is changed to 11 degrees and (

**b**) Estimated and actual speeds when switching angle is changed to 13 degrees.

**Figure 23.**(

**a**) Speed-tracking error when switching angle is changed to 11 degrees (

**b**) Speed-tracking error when switching angle is changed to 13 degrees.

**Figure 24.**Changes in the supply voltage. [Note: ACT_SMC: actual speed under sliding mode controller; EST_SMC: estimated speed under sliding mode controller; ACT_DSMC: actual speed under dynamic sliding mode controller; EST_DSMC: estimated speed under dynamic sliding mode controller; SPEED_REF: reference speed].

**Figure 25.**Changes in the load torque. [Note: ACT_SMC: actual speed under sliding mode controller; EST_SMC: estimated speed under sliding mode controller; ACT_DSMC: actual speed under dynamic sliding mode controller; EST_DSMC: estimated speed under dynamic sliding mode controller; SPEED_REF: reference speed].

**Figure 26.**Increments of the moment of inertia by 10%. [Note: ACT_SMC: actual speed under sliding mode controller; EST_SMC: estimated speed under sliding mode controller; ACT_DSMC: actual speed under dynamic sliding mode controller; EST_DSMC: estimated speed under dynamic sliding mode controller; SPEED_REF: reference speed].

**Figure 27.**Decrements of the moment of inertia by 10%. [Note: ACT_SMC: actual speed under sliding mode controller; EST_SMC: estimated speed under sliding mode controller; ACT_DSMC: actual speed under dynamic sliding mode controller; EST_DSMC: estimated speed under dynamic sliding mode controller; SPEED_REF: reference speed].

**Figure 28.**Changes in the switching angle to 11 degrees. [Note: ACT_SMC: actual speed under sliding mode controller; EST_SMC: estimated speed under sliding mode controller; ACT_DSMC: actual speed under dynamic sliding mode controller; EST_DSMC: estimated speed under dynamic sliding mode controller; SPEED_REF: reference speed].

**Figure 29.**Change in the switching angle to 13 degrees. [Note: ACT_SMC: actual speed under sliding mode controller; EST_SMC: estimated speed under sliding mode controller; ACT_DSMC: actual speed under dynamic sliding mode controller; EST_DSMC: estimated speed under dynamic sliding mode controller; SPEED_REF: reference speed].

Parameters | Value |
---|---|

Rated power(motor) | 1 kW |

Maximum motor Current/phase | 4 amp |

Voltage/phase | 250 volts |

Net torque | 1 Nm |

Speed | 9000 rpm |

Toque winding per phase resistance | 0.86 ohms |

Suspension winding per phase resistance | 0.32 ohms |

Suspension voltage | 250 volts |

Maximum suspension current | 4 amp |

12/14 BSRM | Number of Power Switches | Total |
---|---|---|

Torque winding (two-phase) | 2 per phase | 4 |

Suspending force winding (four-phase/four poles) | 2 per pole | 8 |

**Table 3.**Switching state rule of the hysteresis control method for the bearingless switched reluctance motor (BSRM).

Desired Force | Suspending Force Poles Selection | Enable Is1 | Enable Is2 | Enable Is3 | Enable Is4 |
---|---|---|---|---|---|

$\mathrm{If}\text{}{F}_{x}\ge 0,{F}_{y}\ge o$ | Is1 and Is2 | 1 | 1 | 0 | 0 |

$\mathrm{If}\text{}{F}_{x}\ge 0,{F}_{y}\le o$ | Is2 and Is3 | 0 | 1 | 1 | 0 |

$\mathrm{If}\text{}{F}_{x}\le 0,{F}_{y}\le o$ | Is3 and Is4 | 0 | 0 | 1 | 1 |

$\mathrm{If}\text{}{F}_{x}\le 0,{F}_{y}\ge o$ | Is4 and Is1 | 1 | 0 | 0 | 1 |

Parameters | SMC Based-SMO | DSMC-Based SMO | ||
---|---|---|---|---|

Change of system parameters | Difference in speed | % of Chattering | Difference in speed | % of Chattering |

Change of Switching angle to 13 degree | 3000 | $\pm 8$ | 1000 | $\pm 3$ |

Varying of supply voltage | 3500 | $\pm 8$ | 1500 | $\pm 5$ |

Varying in load Torque | 3500 | $\pm 8$ | 1500 | $\pm 5$ |

Decrease of moment of inertia by 10% | 3000 | $\pm 5$ | 800 | $\pm 4$ |

Increase of moment of inertia by 10% | 2000 | $\pm 5$ | 800 | $\pm 4$ |

Change of Switching angle to 11 degree | 2000 | $\pm 8$ | 1000 | $\pm 3$ |

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**MDPI and ACS Style**

Nageswara Rao, P.; Manoj Kumar, N.; Padmanaban, S.; Subathra, M.S.P.; Chand, A.A.
A Novel Sensorless Approach for Speed and Displacement Control of Bearingless Switched Reluctance Motor. *Appl. Sci.* **2020**, *10*, 4070.
https://doi.org/10.3390/app10124070

**AMA Style**

Nageswara Rao P, Manoj Kumar N, Padmanaban S, Subathra MSP, Chand AA.
A Novel Sensorless Approach for Speed and Displacement Control of Bearingless Switched Reluctance Motor. *Applied Sciences*. 2020; 10(12):4070.
https://doi.org/10.3390/app10124070

**Chicago/Turabian Style**

Nageswara Rao, Pulivarthi, Nallapaneni Manoj Kumar, Sanjeevikumar Padmanaban, M. S. P. Subathra, and Aneesh A. Chand.
2020. "A Novel Sensorless Approach for Speed and Displacement Control of Bearingless Switched Reluctance Motor" *Applied Sciences* 10, no. 12: 4070.
https://doi.org/10.3390/app10124070