**Appendix A**

In this section, the performance of the suggested AFO observer with feedback gains is analyzed to prove the robustness of the estimated speed algorithm. In addition to calculating the observer feedback gains to achieve high performance over a wide range of speed.

The suggested estimate speed in (20) could be rewritten in rotary reference (d − q) frame as:

$$\mathbf{d}\boldsymbol{\omega}\_{\rm f} = \left(\mathbf{K}\_{\rm P\omega} + \frac{\mathbf{K}\_{\rm l\omega}}{\rm s}\right) \left(\stackrel{\rightarrow}{\mathbf{i}}\_{\rm qs} - \mathbf{i}\_{\rm qs}\right) \left.\Psi\right|\_{\rm qs} \tag{A1}$$

A closed-loop schematic diagram of estimate speed is seen as Figure A1:

**Figure A1.** Proposed speed estimation algorithm.

It is potential to derive the open-loop transfer function (OLTF) between both estimated and actual rotor speed from Figure A1.

$$\mathbf{G}\_{\rm op} = -\boldsymbol{\Psi}\_{\rm ds} \left( \mathbf{K}\_{\rm P\omega} + \frac{\mathbf{K}\_{\rm l\omega}}{\rm s} \right) \mathbf{G}\_{\rm q}(\mathbf{s}) \tag{A2}$$

Feedback gains can indeed be obtained on the basis of calculated speed stability as:

$$\begin{cases} \begin{array}{c} \mathbf{h}\_{1} = \sigma \, \mathrm{L}\_{\mathsf{A}} \left[ \frac{-\mathrm{R}\_{\mathsf{s}}}{\sigma \, \mathrm{L}\_{\mathsf{s}}} - \frac{(1-\sigma)\, \mathrm{R}\_{\mathsf{r}}}{\sigma \, \mathrm{L}\_{\mathsf{r}}} + \mathrm{K} \frac{\mathrm{R}\_{\mathsf{r}}}{\mathrm{L}\_{\mathsf{s}}} + \frac{\mathrm{R}\_{\mathsf{s}} \mathrm{L}\_{\mathsf{r}}^{2}}{\mathrm{L}\_{\mathsf{r}}^{2}} \right] \\ \mathrm{h}\_{2} = -\mathrm{K} \, \sigma \, \mathrm{L}\_{\mathsf{A}} \, \mathrm{\dot{\underline{\alpha}}}\_{\mathsf{I}} \\ \mathrm{h}\_{3} = \frac{\mathrm{R}\_{\mathsf{s}} \mathrm{L}\_{\mathsf{r}\mathsf{a}}}{\mathrm{L}\_{\mathsf{s}}} \\ \mathrm{h}\_{4} = 0 \end{array} \tag{A3}$$

where, Δ<sup>ω</sup>r = <sup>ω</sup>r<sup>∗</sup> − ω<sup>ˆ</sup> r and Gq(s) is designated as:

$$\mathbf{G}\_{\mathbf{q}}(\mathbf{s}) = \frac{\mathbf{i}\_{\mathbf{q}\mathbf{s}} - \hat{\mathbf{i}}\_{\mathbf{q}\mathbf{s}}}{\Delta w\_{\mathbf{i}}} = \frac{\mathbf{s}^3 + \mathbf{q}\_2\mathbf{s}^2 + \mathbf{q}\_1\mathbf{s} + \mathbf{q}\_0}{\mathbf{B}} \tag{A4}$$

$$\mathbf{B} = \mathbf{s}^4 + \mathbf{b}\_3 \mathbf{s}^3 + \mathbf{b}\_2 \mathbf{s}^2 + \mathbf{b}\_1 \mathbf{s} + \mathbf{b}\_0 \tag{A5}$$

$$\mathbf{q}\_2 = \frac{\mathbf{L}\_\mathbf{r} (\mathbf{R}\_\mathbf{s} + \mathbf{h}\_1) + \mathbf{R}\_\mathbf{r} \mathbf{L}\_\mathbf{s} - \mathbf{h}\_3 \mathbf{L}\_\mathbf{m}}{\sigma \, \mathbf{L}\_\mathbf{s} \, \mathbf{L}\_\mathbf{r}} = \mathbf{x} \tag{A6}$$

$$\mathbf{q}\_1 = (\omega\_\mathbf{e})^2 + \frac{\mathbf{h}\_1 \mathbf{R}\_\mathbf{l} + \mathbf{R}\_\theta \mathbf{R}\_\mathbf{l}}{\sigma \, \mathbf{L}\_\theta \, \mathbf{L}\_\mathbf{l}} - \frac{\mathbf{h}\_2 \omega\_\mathbf{e}}{\sigma \, \mathbf{L}\_\theta} = (\omega\_\mathbf{e})^2 + \mathbf{y} \tag{A7}$$

$$\begin{split} \mathbf{q}\_{0} &= \frac{\mathbf{I}\_{\mathbf{e}} (\mathbf{R}\_{\mathbf{e}} + \mathbf{h}\_{\mathbf{l}}) + \mathbf{R}\_{\mathbf{e}} \mathbf{I}\_{\mathbf{e}} - \mathbf{h}\_{\mathbf{l}} \mathbf{I}\_{\mathbf{m}}}{\sigma \, \mathbf{I}\_{\mathbf{e}} \, \mathbf{I}\_{\mathbf{e}}} \left( \boldsymbol{\omega}\_{\mathbf{e}} \right)^{2} - \left[ \frac{\mathbf{h}\_{2} \mathbf{R}\_{\mathbf{e}}}{\sigma \, \mathbf{I}\_{\mathbf{e}} \, \mathbf{I}\_{\mathbf{e}}} + \frac{\mathbf{h}\_{1} + \mathbf{R}\_{\mathbf{e}}}{\sigma \, \mathbf{I}\_{\mathbf{e}}} \boldsymbol{\omega}\_{\mathbf{r}} \right] \boldsymbol{\omega}\_{\mathbf{e}} \\ &= \boldsymbol{\chi} (\boldsymbol{\omega}\_{\mathbf{e}})^{2} + \mathbf{z} \boldsymbol{\omega}\_{\mathbf{e}} \end{split} \tag{A8}$$

All zeros of the OLTF must be placed within the left plane to ensure the stability of the speed estimation. With (A2) and (A4) and using the Routh-Hurwitz criterion, where Routh-Table is shown in Table A1, it is potential to obtain the required and adequate conditions for estimate speed stability as [41]:

$$\begin{cases} \quad \text{x } > 0\\ \quad \text{xy} - \text{z}\omega\_{\theta} > 0\\ \quad \text{x}(\omega\_{\theta})^2 + \text{z}\omega\_{\theta} > 0 \end{cases} \tag{A9}$$

Feedback gains H values are difficult to acquire because values of x, y and z in (A8) are complicated. Deserting the second state in (A9) for easy analysis, the gains of H in (A3) can be provided. To satisfy the conditions in (A9), in this study, the conditions of (A10) are achieved:

$$\begin{array}{l} \text{x } > 0\\ \text{y } > 0\\ \text{z } = 0 \end{array} \tag{A10}$$

(a) Based on (z = <sup>0</sup>), the relation among h1 and h2 can be optimized:

$$\mathbf{h}\_2 = \frac{-\mathbf{L}\_\mathbf{f} (\mathbf{h}\_1 + \mathbf{R}\_\mathbf{s})}{\mathbf{R}\_\mathbf{f}} \boldsymbol{\omega}\_\mathbf{r} \tag{A11}$$

(b) Based on (y > <sup>0</sup>), a stability range of h1 for speed estimate can be achieved:

⎧⎨⎩

$$
\Lambda\_1 > -\mathbb{R}\_8 \tag{A12}
$$

(c) Based on (x > <sup>0</sup>), relation among h1 and h2 can be optimized:

$$\mathbf{h}\_1 > \frac{\mathbf{L}\_m}{\mathbf{L}\_F} \mathbf{h}\_3\tag{A13}$$
 
$$\mathbf{h}\_1 > \frac{\mathbf{L}\_m}{\mathbf{L}\_F} \mathbf{h}\_3\tag{A13}$$

Finally, it is potential to ge<sup>t</sup> the required and adequate conditions for stable estimation of speed:

$$\begin{cases} \text{ h}\_{2} = \frac{-\mathcal{L}\_{\text{r}}(\text{h}\_{1} + \mathcal{R}\_{\text{s}})}{\mathcal{R}\_{\text{g}}} \text{ } \boldsymbol{\omega}\_{\text{F}}\\ \text{ h}\_{1} > \frac{\mathcal{L}\_{\text{r}}}{\mathcal{L}\_{\text{g}}} \text{h}\_{3} > -\mathcal{R}\_{\text{s}} \end{cases} \tag{A14}$$

If gains of feedback H satisfy (A14), speed estimation stability can be assured at all IM speed ranges. It could be seen through (A14), that h4 does not affect the speed estimate stability. For simplicity, feedback gains being given in our study as:

$$\begin{cases} \text{ } \mathbf{h}\_1 = \mathbf{h}\_3 = 0.048\\ \text{ } \mathbf{h}\_2 = \frac{-\mathbf{L}\_c(\mathbf{h}\_1 + \mathbf{R}\_s)}{\mathbf{R}\_r} \boldsymbol{\omega}\_r\\ \text{ } \mathbf{h}\_4 = 0 \end{cases} \tag{A15}$$

As an 4th−order system, the observer has 2−pairs of conjugate poles. To guarantee the observer's stability, each pole must be in the left-hand s-plane side. Observer pole placement with (A15) could be seen in Figure A2a, where the observer's dynamic performance appears to be strong. Via (A2) and (A15), the OLTF zeros with a full load is shown in Figure A2b (−<sup>40</sup> π ≤ ωe ≤ 40 <sup>π</sup>).

It is observed that there are no unstable zeros in the low-speed area, even in regenerating mode.Moreover, velocity estimation can indeed be stable within full load throughout all velocity range.

**Table A1.** Routh-Table.


**Figure A2.** S-plane (**a**) observer poles placement (**b**) transfer function zeros of estimated speed.
