# Real-Time Analysis of a Modified State Observer for Sensorless Induction Motor Drive Used in Electric Vehicle Applications

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## Abstract

**:**

## 1. Introduction

## 2. General Configuration of Model Reference Adaptive Systems and System Modeling

_{ref}and X

_{adap}are outputs of respective models. The adaptation mechanism makes use of the classical Proportional-Integral theory to process the speed tuning signal. A Lyapunov function candidate is used for the speed derivation mechanism.

#### 2.1. Structure of Sliding Mode Luenberger State Observer

_{eq}(t) and u

_{sw}(t) represent the control, equivalent control and the switching vector. For stability, the switching vector is obtained [17,26]:

#### 2.1.1. Reference Model (Motor)

- $\mathrm{x}={\left[{\mathrm{i}}_{\mathrm{ds}}^{\mathrm{s}},{\mathrm{i}}_{\mathrm{qs}}^{\mathrm{s}},{\mathsf{\psi}}_{\mathrm{dr}}^{\mathrm{s}},{\mathsf{\psi}}_{\mathrm{qr}}^{\mathrm{s}}\right]}^{\mathrm{T}}$, $\mathrm{A}=\left[\begin{array}{cc}{\mathrm{A}}_{11}& {\mathrm{A}}_{12}\\ {\mathrm{A}}_{21}& {\mathrm{A}}_{22}\end{array}\right]$,
- $\mathrm{B}={\left[\frac{1}{{\mathsf{\sigma}\mathrm{L}}_{\mathrm{s}}}\mathrm{I}0\right]}^{\mathrm{T}}$, $\mathrm{C}=\left[\mathrm{I},0\right]$, $\mathrm{u}={\left[{\mathrm{v}}_{\mathrm{ds}}^{\mathrm{s}}{\mathrm{v}}_{\mathrm{qs}}^{\mathrm{s}}\right]}^{\mathrm{T}}$,
- $\mathrm{I}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, $\mathrm{J}=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$,
- ${\mathrm{A}}_{11}=-\left[\frac{{\mathrm{R}}_{\mathrm{s}}}{{\mathsf{\sigma}\mathrm{L}}_{\mathrm{s}}}+\frac{1-\mathsf{\sigma}}{{\mathsf{\sigma}\mathrm{T}}_{\mathrm{r}}}\right]\mathrm{I}={\mathrm{a}}_{\mathrm{r}11}\mathrm{I},{\mathrm{A}}_{12}=\frac{{\mathrm{L}}_{\mathrm{m}}}{{\mathsf{\sigma}\mathrm{L}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{r}}}\left[\frac{1}{{\mathrm{T}}_{\mathrm{r}}}\mathrm{I}-{\mathsf{\omega}}_{\mathrm{r}}\mathrm{J}\right]={\mathrm{a}}_{\mathrm{r}12}\mathrm{I}+{\mathrm{a}}_{\mathrm{i}12}\mathrm{J}$,
- ${\mathrm{A}}_{21}=\frac{{\mathrm{L}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{r}}}\mathrm{I}={\mathrm{a}}_{\mathrm{r}21}\mathrm{I}$,
- ${\mathrm{A}}_{22}=\frac{-1}{{\mathrm{T}}_{\mathrm{r}}}\mathrm{I}+{\mathsf{\omega}}_{\mathrm{r}}\mathrm{J}={\mathrm{a}}_{\mathrm{r}22}\mathrm{I}+{\mathrm{a}}_{\mathrm{i}22}\mathrm{J}$.

#### 2.1.2. Estimation of Disturbance Torque from the Mechanical Model

#### 2.1.3. SMLO 1—Observer with Conventional Disturbance Rejection Mechanism (Adaptive Model)

- $\widehat{\mathrm{A}}=\left[\begin{array}{cc}{\mathrm{A}}_{11}& {\widehat{\mathrm{A}}}_{12}\\ {\mathrm{A}}_{21}& {\widehat{\mathrm{A}}}_{22}\end{array}\right]$,
- ${\widehat{\mathrm{A}}}_{12}=\frac{{\mathrm{L}}_{\mathrm{m}}}{{\mathsf{\sigma}\mathrm{L}}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{r}}}\left[\frac{1}{{\mathrm{T}}_{\mathrm{r}}}\mathrm{I}-{\widehat{\mathsf{\omega}}}_{\mathrm{r}}\mathrm{J}\right]={\mathrm{a}}_{\mathrm{r}12}\mathrm{I}+{\widehat{\mathrm{a}}}_{\mathrm{i}12}\mathrm{J}$,
- ${\widehat{\mathrm{A}}}_{22}=\frac{-1}{{\mathrm{T}}_{\mathrm{r}}}\mathrm{I}+{\widehat{\mathsf{\omega}}}_{\mathrm{r}}\mathrm{J}={\mathrm{a}}_{\mathrm{r}22}\mathrm{I}+{\widehat{\mathrm{a}}}_{\mathrm{i}22}\mathrm{J}$.

_{2}” are chosen in such a way that the eigenvalues of the observer are shifted more negative as compared to the eigenvalues of the motor. They also directly affect the dynamics and damping of the observer. “k

_{1}” is dependent on the motor parameters.

#### 2.1.4. SMLO 2—Observer with Modified Disturbance Rejection Mechanism (Adaptive Model)

#### 2.1.5. Adaptive Mechanism

#### 2.2. Stability Analysis of Both the Observers by Means of Pole Placement

#### 2.3. Structure of Current Regulated Vector Controller

_{c}is the speed error, kp and ki are the proportional and integral gains for tuning the speed error, and Ts is the sampling time. For operation in the motoring and flux weakening region, the rotor flux is constant for the former and as a function of the speed for the latter:

## 3. The Concept of Real-Time Simulation

## 4. Real-Time Simulation Results: Analysis and Discussion

#### 4.1. Performance at Flux Weakening

#### 4.2. Performance at Step Speed Command

#### 4.3. Performance at Low Speeds

#### 4.4. Effect of Parameter Detuning on the Dynamic Performance

#### 4.5. Switching Surface and Convergence of the Stator Current Error

#### 4.6. Pole Placement Plot of the Modified nd Conventional Disturbance Observers

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

i_{ds}^{s}, i_{qs}^{s}, i_{dr}^{r}, i_{qr}^{r} | Direct and quadrature axes stator and rotor current components in stationary and rotating frame |

v_{ds}^{s}, v_{qs}^{s} | Direct and quadrature stator voltages in stationary frame |

T_{r}, R_{s}, R_{r} | Rotor time constant, stator and rotor resistance |

σ, L_{r}, L_{m}, L_{s} | Leakage reactance, rotor, magnetizing and stator self inductance |

L_{ls}, L_{lr} | Stator and rotor leakage inductances |

${\mathsf{\omega}}_{\mathrm{r}},{\widehat{\mathsf{\omega}}}_{\mathrm{r}},{\mathsf{\omega}}^{*},{\mathsf{\omega}}_{\mathrm{bsync}}$ | Actual, estimated, reference and base synchronous speed |

ψ_{ds}^{s}, ψ_{qs}^{s}, ψ_{dr}^{s}, ψ_{qr}^{s} | Direct and quadrature axes stator and rotor flux linkages in stationary frame |

${\widehat{\mathsf{\phi}}}_{\mathrm{d}},{\widehat{\mathsf{\phi}}}_{\mathrm{q}}$ | Direct and quadrature axes estimated rotor flux linkages |

${\mathsf{\theta}}_{\mathrm{f}},{\mathsf{\theta}}_{\mathrm{sl}},{\mathsf{\theta}}_{\mathrm{r}}$, ${\mathrm{T}}_{\mathrm{e}}^{*}$ | Field, slip and rotor angles and Torque reference |

${{\mathrm{i}}_{\mathrm{ds}}}^{*},{\mathrm{i}}_{\mathrm{qs}}^{*}$ | Direct and quadrature axes stator currents in synchronously rotating frame |

${\mathrm{i}}_{\mathrm{as}}^{*},{\mathrm{i}}_{\mathrm{bs}}^{*},{\mathrm{i}}_{\mathrm{cs}}^{*}$ | Three-phase reference currents |

## Appendix A

_{s}= 0.087 Ω, R

_{r}= 0.228 Ω, L

_{ls}= L

_{lr}= 0.8 mH, L

_{m}= 34.7 mH, Inertia, J = 1.662 kgm

^{2}, friction factor = 0.1.

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**Figure 3.**Basic configuration of parameter adaptive model reference adaptive system (MRAS) scheme with parallel disturbance torque estimation.

**Figure 6.**(

**a**) Estimated speed of conventional sliding mode luenberger observer (SMLO) 1; and (

**b**) zoomed version of (

**a**).

**Figure 10.**(

**a**) Estimated speed of Modified Sliding Mode Luenberger Observer (SMLO 2); and (

**b**) zoomed version of (

**a**).

**Figure 20.**Speed tuning signal for 50% incorrect setting of stator resistance (R

_{s}) and rotor time constant (T

_{r}) for (

**a**) SMLO 1, and (

**b**) SMLO 2.

**Figure 21.**Speed tuning signal for nominal setting of stator resistance (R

_{s}) and rotor time constant (T

_{r}) for (

**a**) SMLO 1; and (

**b**) SMLO 2.

**Figure 22.**Speed tuning signal for 150% incorrect setting of stator resistance (R

_{s}) and rotor time constant (T

_{r}) for (

**a**) SMLO 1; and (

**b**) SMLO 2.

**Figure 23.**SMLO 1 (

**a**) sliding surface; (

**b**) flux component of stator current; (

**c**) torque component of stator current; and (

**d**) stator current error.

**Figure 24.**SMLO 2 (

**a**) sliding surface; (

**b**) flux component of stator current; (

**c**) torque component of stator current; and (

**d**) stator current error.

Test Cases | SMLO 1 | SMLO 2 |
---|---|---|

Low flux weakening region | Maximum speed oscillation of around 70 rad/s (around 42% of the reference value). Speed oscillations do not die out. | Initial maximum speed oscillation of around 20 rad/s (around 12% of the reference value). Speed Oscillations gradually reduce with time. |

Step speed command | Very high overshoot and undershoot observed at the instance of fast deceleration. | Smoother tracking during fast acceleration and deceleration. |

Low speed operation | Does not track, becomes unstable and speed convergence goes out of bounds. | Tracks well, initial undershoot and overshoot, which results for a very small interval of time. |

Disturbance torque | Higher torque pulsations as a result of high stator current pulsation. | Comparatively lower torque pulsation resulting in better torque holding capability. |

Speed and Stator error convergence | Slower convergence, higher speed and stator current error | Faster convergence, resulting in smoother tracking |

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**MDPI and ACS Style**

Krishna S., M.; Daya J.L., F.; Padmanaban, S.; Mihet-Popa, L.
Real-Time Analysis of a Modified State Observer for Sensorless Induction Motor Drive Used in Electric Vehicle Applications. *Energies* **2017**, *10*, 1077.
https://doi.org/10.3390/en10081077

**AMA Style**

Krishna S. M, Daya J.L. F, Padmanaban S, Mihet-Popa L.
Real-Time Analysis of a Modified State Observer for Sensorless Induction Motor Drive Used in Electric Vehicle Applications. *Energies*. 2017; 10(8):1077.
https://doi.org/10.3390/en10081077

**Chicago/Turabian Style**

Krishna S., Mohan, Febin Daya J.L., Sanjeevikumar Padmanaban, and Lucian Mihet-Popa.
2017. "Real-Time Analysis of a Modified State Observer for Sensorless Induction Motor Drive Used in Electric Vehicle Applications" *Energies* 10, no. 8: 1077.
https://doi.org/10.3390/en10081077