*5.2. Experimental Results*

The performance of the proposed sensorless DTC strategy has been experimentally surveyed using DSP platform, programmed through Code Composer Studio (CCS v.3.3) and MATLAB. The IQmath and digital motor control (DMC) libraries have been used to provide optimized code. A 10 kHz sampling frequency with a 2 μs dead-band has been adopted. The experimental results have been captured using an Advantech PCI-1716 data acquisition card (DAQ) and serial port with LABVIEW and MATLAB, respectively. The serial communications interface (SCI) module has been employed to provide a serial connection between host PC and DSP. An incremental shaft encoder has been used to verify the performance of the speed estimation algorithm. All of the experiments have been carried out in sensorless mode as well as closed-loop adaptation of the stator resistance under various test scenarios, emphasizing on the low-speed region.

The experimental results of the proposed parallel estimation system of stator resistance and rotor speed under 50% initial stator resistance mismatch are shown in Figure 6. The speed command is 7% rated speed under rated load torque. In this test, the electric drive is allowed to start with a wrong stator resistance. This causes an error in estimated electromagnetic torque and actual speed. However, the estimated speed and the stator flux follow their reference values because of the controller action. It can be seen that the estimation error of the speed and the electromagnetic torque due to detuned stator resistance are removed within short seconds after activation of the stator resistance estimator at *t* = 5 s.

**Figure 6.** Experimental results of the proposed parallel estimation system under initial mismatch of stator resistance.

As already mentioned, the proposed parallel estimation system has the merit of avoiding overlap between stator resistance and rotor speed estimators, whereby the stator resistance is independently estimated from rotor speed using additional freedom degrees of 6PIM. The experimental results of estimated stator resistance under speed changes and load change are shown in Figure 7a,b, respectively. In Figure 7a, the speed command is changed as a step function from a very low speed to 17% rated speed, and, in Figure 7b, a load torque is suddenly applied to the motor at *t* = 2 s. It can be clearly adjudged that the adaptation process of stator resistance is independent of speed and load torque changes.

Disturbance-free operation of the ADRC-based speed controller is evaluated through a comparative study of its performance and the conventional PI regulator. The experimental results for the estimated speed under sudden load torque changes at 7% rated speed when the conventional PI and introduced ADRC are utilized as speed controllers are shown in Figure 8. As can be seen, applying the external load torque to the 6PIM leads to a larger overshoot (undershoot), when the conventional PI regulator is employed. The ADRC properly improves the disturbance rejecting capability, which in turn provides a robust performance against load torque changes.

**Figure 7.** Experimental results of the estimated stator resistance under (**a**) speed changes (**b**) load torque change.

**Figure 8.** Experimental results of the estimated speed with PI and ADRC-based speed controllers under load changes.

#### **6. Conclusions**

Multiphase electrical machines and drives have different advantages over their traditional three phase counterparts. In recent years, multiple research works have been published to explore the specific advantages of multiphase machines and drives. In this regard, a parallel estimation system of the stator resistance and the rotor speed for direct torque-controlled 6PIM was proposed in this paper. The speed estimator is based on an adaptive full-order observer, which estimates the speed signal using the 6PIM model in the *α* − *β* subspace, while the stator resistance estimator employs the 6PIM model in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace. Hence, the stator resistance is identified independently of the rotor speed. The rotor speed- and the stator resistance-adaptation laws were derived using the Lyapunov stability theorem. The performance of the proposed sensorless DTC was experimentally investigated, where the obtained results confirmed its capabilities in terms of accuracy as well as no overlap between the stator resistance and the rotor speed estimators. In order to provide a robust performance for the DTC technique against external load torques, the PI regulator was replaced by an ADRC, as a well-known disturbance-free controller. The better performance of the DTC scheme based on ADRC was verified through a comparative study with the conventional PI regulator.

**Author Contributions:** Conceptualization, methodology, validation, formal analysis, writing–original draft preparation, H.H., M.H.H.; writing—review and editing, resources, A.R.; project administration, T.V.; investigation, A.K.; funding acquisition, D.V.L.; supervision, A.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research has been supported by the Estonian Research Council under grant PSG453 "Digital twin for propulsion drive of autonomous electric vehicle" and was financially supported by Government of Russian Federation (Grant 08-08).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. The Design of Adaption Law for Stator Resistance Estimation**

The quadratic Lyapunov function for asymptotic stability of the proposed stator resistance estimation system is defined as

$$\mathbf{V}\_r = \mathbf{e}\_r{}^T \mathbf{e}\_r + \frac{\Delta R\_s^2}{\lambda\_r} \tag{A1}$$

where *λ<sup>r</sup>* is a positive constant, Δ*Rs* = *R*ˆ *<sup>s</sup>* <sup>−</sup> *Rs*, *<sup>R</sup>*<sup>ˆ</sup> *s* is the estimated stator resistance, *Rs* is the real stator resistance, and *e<sup>r</sup>* is the error matrix of the state variables in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace as

$$\mathbf{e}\_{I} = \mathbf{x}\_{2} - \mathbf{\hat{x}}\_{2} = \begin{bmatrix} \dot{i}\_{sz1} - \hat{i}\_{sz1} & \dot{i}\_{sz2} - \hat{i}\_{sz2} \end{bmatrix}^{T} \tag{A2}$$

The asymptotic stability of the stator resistance estimator is assured when the Lyapunov candidate function *Vr* is positive definite as well as its time derivative *pVr* is negative definite. The time derivative of the Lyapunov candidate function is calculated as

$$p\mathbf{V}\_{\mathbf{r}} = \mathbf{e}\_{r}^{T} p\mathbf{e}\_{r} + p\mathbf{e}\_{r}^{T} \mathbf{e}\_{r} + \frac{2}{\lambda\_{r}} \Delta R\_{s} p\hat{\mathcal{R}}\_{s} \tag{A3}$$

With some mathematical manipulation, Equation (A3) can be written as

$$\begin{split}p\mathbf{V}\_{\mathbf{r}} &= \mathbf{e}\_{\mathbf{r}}^{\mathrm{T}}(A\_{2} + A\_{2}^{\mathrm{T}})\mathbf{e}\_{\mathbf{r}} - \left[\mathbf{e}\_{\mathbf{r}}^{\mathrm{T}}\Delta A\_{2}\hat{\mathbf{x}} + \hat{\mathbf{x}}\_{2}^{\mathrm{T}}\Delta A\_{2}^{\mathrm{T}}\mathbf{e}\_{\mathbf{r}}\right] \\ &+ \frac{2}{\lambda\_{\mathrm{r}}}\Delta R\_{\mathrm{s}}p\hat{\mathbf{R}}\_{\mathrm{s}}\end{split} \tag{A4}$$

The first term of Equation (A4) is inherently negative definite. The stability of the system is eventually assured, when the sum of the last two terms of Equation (A4) is zero as

$$\frac{2}{\lambda\_r} \Delta R\_s p \vec{R}\_s - \left[ \mathbf{e}\_r^T \Delta A\_2 \mathbf{\hat{x}}\_2 + \mathbf{\hat{x}}\_2^T \Delta A\_2^T \mathbf{e}\_r \right] = 0 \tag{A5}$$

which leads to

$$
\hat{\mathcal{R}}\_{\mathbb{S}} = -\frac{\lambda\_r}{2} \int \epsilon\_{\mathbb{R}\mathbb{S}} dt \tag{A6}
$$

where the tuning signal *RS* is

$$
\epsilon\_{R\_S} = \hat{t}\_{sz1}(i\_{sz1} - \hat{t}\_{sz1}) + \hat{t}\_{sz2}(i\_{sz2} - \hat{t}\_{sz2}) \tag{A7}
$$

A PI regulator is employed to enhance the dynamic behaviour of the proposed estimator, instead of Equation (A6) as

$$\mathcal{R}\_s = K\_{pr} \varepsilon\_{R\_S} + K\_{ir} \int \varepsilon\_{R\_S} dt \tag{A8}$$

where *Kir* and *Kpr* are the integral and proportional constants.

#### **Appendix B. The Design of Adaption Law for Speed Estimation**

The Lyapunov candidate function for asymptotic stability of the speed estimation system is

$$\mathbf{V}\_{\omega} = \mathbf{e}\_{\omega}^{T} \mathbf{e}\_{\omega} + \frac{\Delta \omega\_{r}^{2}}{\lambda\_{\omega}} \tag{A9}$$

where *λω* is a positive constant, Δ*ω<sup>r</sup>* = *ω*ˆ *<sup>r</sup>* − *ωr*, and *e<sup>ω</sup>* is the error matrix of the estimated and real values in *α* − *β* subspace as

$$\begin{aligned} \mathbf{e}\_{\omega} &= \mathbf{x}\_{1} - \hat{\mathbf{x}}\_{1} \\ &= \begin{bmatrix} \hat{\mathbf{i}}\_{sa} - \hat{\mathbf{i}}\_{sa} & \hat{\mathbf{i}}\_{s\beta} - \hat{\mathbf{i}}\_{s\beta} & \Psi\_{ra} - \hat{\Psi}\_{ra} & \Psi\_{r\beta} - \hat{\Psi}\_{r\beta} \end{bmatrix}^{T} \end{aligned} \tag{A10}$$

In this case, the first-order time derivative of Lyapunov function can be deduced as

$$pV\_{\omega} = \mathbf{e}\_{\omega}^{T}[(A\_1 - G\_1\mathbf{C}\_1) + (A\_1 - G\_1\mathbf{C}\_1)^T]\mathbf{e}\_{\omega} \tag{A11}$$

$$+ \left(\mathbf{e}\_{\omega}\Delta A\_2\mathbf{\hat{x}}\_2 + \mathbf{\hat{x}}\_2\Delta A^T\mathbf{e}\_{\omega}\right) + \frac{2}{\lambda\_{\omega}}\Delta\omega\_r p\mathbf{\hat{x}}\_r$$

The first term of Equation (A11) is guaranteed to be negative definite by suitable adopting of observer gain matrix *G*1. The Lyapunov stability criterion is satisfied, if the sum of second and third terms of Equation (A11) is zero. With some calculations, the adaptation law for speed estimator is acquired as

$$
\hat{\omega}\_r = K\_{p\omega} \epsilon\_\omega + K\_{i\omega} \int \epsilon\_\omega dt \tag{A12}
$$

where the tuning signal *ω* is

$$
\epsilon\_{\omega} = (\dot{\mathfrak{i}}\_{sa} - \hat{\mathfrak{i}}\_{sa})\hat{\mathfrak{p}}\_{r\mathfrak{f}} - (\dot{\mathfrak{i}}\_{s\mathfrak{f}} - \hat{\mathfrak{i}}\_{s\mathfrak{f}})\hat{\mathfrak{p}}\_{ra} \tag{A13}
$$

#### **References**


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