**1. Introduction**

Several schemes for speed-sensorless vector-controlled induction motor (IM) drives have been suggested for nearly a decade. Via flux and speed sensors positioned within the machine will deteriorate the machine's robustness and raise the associated maintenance expenses. Speed sensor cost is within the same range as the price of motor itself, at least for machines with ratings below 10 kW. In many applications, sensor mounting to the motor is an obstacle. The majority of estimators proposed in sensorless IM drives for combined flux and also speed estimation could be divided into three categories: (a) Modelreference-adaptive system (MRAS) [1,2]. Although MRAS-based estimators are favored due to simplicity, ease of implementation, and documented stability [3], they have certain drawbacks in the low-speed zone, where open-loop integration can result in instability due to stator resistance misestimating [4]. (b) Full-order observers (FOO) [5] providing both stator current and stator or rotor flux estimates. In FOO, alteration is made throughout the error between both measured stator current and its estimated value that is often used to adjust estimated speed in an adaptation law. (c) Reduced-order observers (ROO) [6], that are only rotor or stator flux estimators. It is the potential to make the ROO essentially sensorless [7], i.e., speed will not appear in the observer equations. The major drawback for sensorless velocity drives is their poor performance when IM operates at lowish velocity range or near zero-velocity point. However, low-velocity and zero-velocity operating case

**Citation:** Aziz, A.G.M.A.; Rez, H.; Diab, A.A.Z. Robust Sensorless Model-Predictive Torque Flux Controlfor High-Performance Induction Motor Drives. *Mathematics* **2021**, *9*, 403. https://doi.org/10.3390/ math9040403

Academic Editor: Vladimir Prakht

Received: 19 January 2021 Accepted: 15 February 2021 Published: 18 February 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

of IMs is the popular for industrial applications, like hoists (sometimes operating at 0.5 Hz to 1 Hz), marble cutting machines (operating at 0.3 Hz sometimes) and mine carrier cars (sometimes operating at 0 Hz). Therefore, sensorless drives performance at low-velocity ranges necessities to be improved.

To achieve an active decoupled control for flux and electromagnetic torque, the direct torque control (DTC) was offered to replace the field-oriented control (FOC) in the drive domain. DTC provides a simpler scheme, faster response, and lower machine parameter dependency than FOC. Besides that, it does not include coordinates transformation or current regulations [8,9]. The key problem of this technique is the excessive amount of flux and also electromagnetic torque ripples created through the variable switching frequency due to discrete nature of the hysteresis comparators and voltage vector selection look-up-table (LUT) [10].

In order to address some weaknesses in classical DTC techniques, an effective control approach called model predictive control (MPC), has recently been proposed. MPC's key advantages are its ability to recognize various control targets, the straightforward management of nonlinearities and control limits of the model, and easy implementation [11–14]. Recent works have implemented MPC method to avoid limitations of classical LUT-based DTC of IM drives [15–18]. The proposed MPDTC techniques implemented in these studies contributed to improving the drive's dynamic efficiency, but as an essential feature of the regulated variables, the ripple content was still present. There are different explanations for why the ripple phenomena in the MPDTC are presented. The foremost one would be originated in view of the technique principle of MPDTC itself. After the minimization of a cost function, it selects a voltage vector and inevitably applies this vector for the entire subsequent sampling duration, even though this is deleterious in certain situations, as it may occur that the regulated quantity (i.e., the torque) is therefore pushed out of reasonable limits before the end of the loop.

Different methods have been established to eliminate unwanted ripples, such as splitting the sampling interval into two parts and adding dual different voltage vectors in each interval [19–21], for example. This has essentially helped to limit the ripples. Otherwise, the complexities and computational burden of those methods are amplified and this would affect the dynamic digital control response, which is intended to be very rapid in many numbers of industrial applications.

Model-predictive current control (MPCC) was presented in [22,23] as a solution to limit the use of the cost function weighting value. Two prediction horizons were suggested in [23] to predict stator currents, which resulted in the reduction of the accompanying noise. However, measurement time was increased, otherwise amplifying the commutations of the inverter.

Due to its intuitive definition, high versatility, and simple incorporation of constraints, model-predictive torque control (MPTC) has recently gained significant attention within the academic and industrial communities [24,25]. Despite the MPTC's intuitive definition and rapid response, as stator flux and also torque control variables have different amplitudes and units, a proper stator flux weighting factor should be built to achieve adequate performance [13,26,27].

Model-predictive flux control (MPFC) is suggested in [28] and compared with the conventional MPTC in [29] to prevent nontrivial weighting factor adjustment effort for MPTC. While in [13], the prediction errors are translated into ranking values for different control variables. As the finest one, the voltage vector leading to the minimum average ranking value is chosen.

Model-Predictive torque and flux control (MPTFC) [30] for variable speed drives is a significant member of the finite control set model-predictive control (FCS-MPC) family. A significant level of robustness to model parameter deviations could be achieved by directly controlling machine magnetization and electromagnetic torque [31]. Throughout the cost function, the expected errors of torque and stator flux magnitude are enhanced. A corresponding objective function, subject to inverter and machine model, is minimized

online to this end. This results in an optimum switch position (i.e., control input) applied to the power converter [32].

Another feature of IM drives is that they exhibit greater efficiency when working at rated load [33,34]. However, owing to the persistence of iron losses and a part of copper losses, efficiency deteriorates under light load operation. This factor would harm the machine service life; accordingly, a criterion should be adopted to mitigate losses, particularly at light loads. Techniques for efficiency optimization are typically categorized into two types: search methods and model-based methods centered on how to decide the steadystate-optimal control variable [35]. Search-based methods are techniques of perturbation and observation that push the control command in the direction of minimal power losses. If closed-form solutions are not available, machine losses could either be calculated via loss model or measured using a power analyzer. Search-based strategies still struggle from a slow convergence rate and undesirable torque dynamics and fluctuations [36], despite attempts to raise the convergence rate using mathematical algorithms.

As a consequence, approaches focused on search are limited to steady-state processes. Multiple flux modes exist for industrial applications. The control minimizing loss is only switched on at steady-state, and during transients, the rated flux is usually linked to fast torque dynamics. There is no clear identification of transitions between flux modes during torque transients [37]. When the stator flux linkage shifts rapidly during torque transients, major losses are caused. Consequently, for highly dynamic load profiles, the steady-state optimal solution causes even more losses than constant rated flux operation.

An adaptive observer for combined flux and speed guesstimate in rotor speed reference frame is suggested in this paper. Rather than the traditional rotor flux plus stator current IM model, the rotor and stator flux models are utilized in the rotor speed reference frame. This makes it probable to apply the suggested observer in both stator and rotor-flux vector-controlled IM drives and in DTC-IM drives. It should also be mentioned that both stator and rotor resistance values and estimated velocity are the uncertainties considered in this article.

The current study introduces an efficient MPTFC technique for the IM drive in which some of the MPTC shortcomings can be mitigated, in specific by reducing the ripples of torque, stator flux and current. Furthermore, a loss minimization criterion (LMC) is suggested to improve the drive efficiency, particularly under light loads and low-speed operations, extending the IM's lifetime. Drive efficiency is validated by comprehensive simulation testing, in which the feasibility of the MPTFC strategy is demonstrated. The combination of prediction and estimation, in this work, is the additional challenge of sensorless MPTFC approach versus traditional sensorless approaches. **The paper's contributions could be summarized as follows**; the work intends to eliminate and replace the mechanical sensor with novel soft sensorless algorithms to minimize the drive cost and boost its reliability. Moreover, the paper offers a novel predictive torque-flux control approach for IM drive. Representing the IM model with consideration of core-loss and illustration of the used 2 −level voltage source inverter has been implemented. Developing a novel AFOO, which contributes to reducing the fluctuations and thereby enhancing the method torque-flux and prediction, with the compensation of core-loss, for; rotor speed estimation, stator flux estimation, rotor flux estimation, stator and rotor resistance estimation, stator current estimation, electromagnetic torque estimation has been introduced. The analysis and design of the proposed MPTFC strategy are described in a straightforward manner that clarifies the suggested drive basic operation. To minimalize the copper and iron losses, particularly at low-velocity and light loading, which enhances the IM efficiency, an LMC is provided in steady-state operation. In order to assess IM dynamics under the proposed MPTFC and MPTC approaches, comprehensive simulation tests are conducted. Test results approve the suggested MPTFC reliability to be used as a stronger alternative to the MPTC. All gain selections of controllers and controllers that have been designed to ensure overall drive stability shall be found in Appendix A.

The paper starts with the implementation of the IM mathematical model and 2 −level inverter, with consideration of core-loss. Then, the proposed AFOO is designed and explained. The suggested MPTFC method is subsequently presented and explained indepth, and the proposed LCM is then presented and systematically evaluated. Eventually, the complete system design and performance test results are introduced and discussed.

#### **2. Mathematical Model of IM and Inverter**

In the stationary ( α − β) reference frame, indicated by the apex s for the stator and rotor amounts, Figure 1 demonstrates the IM model. No transformation of stator amounts (indicated by index s) is necessary for this frame, while the rotor amount (index rs) simply refers to the stator. Even though all of them remain sinusoidal during steady-state activity, this model makes it easier to formulate the predictive torque and flux control model proposed. Stator and rotor voltage balance equations in Figure 1 are provided as in (1) and (2) in the continuous time domain [38,39]:

$$\stackrel{\rightarrow}{\mathbf{p}}\stackrel{\rightarrow}{\Psi\_{\mathbf{s}}} = \stackrel{\rightarrow}{\mathbf{v}\_{\mathbf{s}}} - \stackrel{\rightarrow}{\mathbf{R}\_{\mathbf{s}}}\stackrel{\rightarrow}{\mathbf{i}}\_{\mathbf{s}}\tag{1}$$

$$\mathbf{p}\,\stackrel{\rightarrow}{\Psi}\_{\text{rs}} = \begin{array}{c} -\text{R}^{\prime}\_{\text{r}} \stackrel{\rightarrow}{\text{i}}\_{\text{rs}} + \text{j} \,\omega\_{\text{me}} \stackrel{\rightarrow}{\Psi}\_{\text{rs}} \end{array} \tag{2}$$

where p is a differential operator, ωme = P ω m , with P is pole pairs and ω m is the shaft speed. Additionally, Lσs and Rs are stator leakage inductance and resistance, respectively; L - σr and R - r are rotor leakage inductance and resistance, respectively, both referred to the stator; and Rfe, L m are the equivalent resistance representing the iron losses, and the magnetizing inductance, respectively. The relationships among currents and fluxes and are assumed linear (no saturation) and are listed as follows:

$$\stackrel{\rightarrow}{\Psi}\_{\text{s}} = \text{L}\_{\text{cos}} \stackrel{\rightarrow}{\text{i}}\_{\text{s}} + \text{L}\_{\text{m}} \stackrel{\rightarrow}{\text{i}}\_{\text{m}} = \text{L}\_{\text{cos}} \stackrel{\rightarrow}{\text{i}}\_{\text{s}} + \stackrel{\rightarrow}{\Psi}\_{\text{m}} \tag{3}$$

$$\stackrel{\rightarrow}{\Psi}\_{\text{rs}} = \stackrel{\prime}{\mathcal{L}}\_{\text{err}}^{\prime} \stackrel{\rightarrow}{\mathcal{i}}\_{\text{rs}} + \stackrel{\rightarrow}{\mathcal{L}}\_{\text{m}} \stackrel{\rightarrow}{\mathcal{i}}\_{\text{m}} = \stackrel{\prime}{\mathcal{L}}\_{\text{err}}^{\prime} \stackrel{\rightarrow}{\mathcal{i}}\_{\text{rs}} + \stackrel{\rightarrow}{\Psi}\_{\text{m}} \tag{4}$$

where → Ψ m and → i m are magnetizing air-gap flux and magnetizing current, respectively. Furthermore,

$$\mathbf{R\_{fe}}\stackrel{\rightarrow}{\mathbf{i}}\_{\text{fe}} = \mathbf{L\_m}\stackrel{\rightarrow}{\mathbf{p}}\stackrel{\rightarrow}{\mathbf{i}}\_{\text{m}}\tag{5}$$

$$
\stackrel\frown{\mathbf{i}}\_{\mathbf{m}} + \stackrel\frown{\mathbf{i}}\_{\mathbf{fe}} = \stackrel\frown{\mathbf{i}}\_{\mathbf{s}} + \stackrel\frown{\mathbf{i}}\_{\mathbf{rs}} \tag{6}
$$

**Figure 1.** IM model in stationary (α − β) reference frame taking iron losses into account.

With some managemen<sup>t</sup> of the equations, Equation (2) can be replaced by

$$\mathbf{p}\stackrel{\rightarrow}{\mathbf{l}}\_{\text{S}} = \frac{\mathbf{L}\_{\text{th}}}{\mathbf{L}\_{\text{th}}\mathbf{L}\_{\text{cyl}}^{\prime} + \mathbf{L}\_{\text{cyl}}\left(\mathbf{L}\_{\text{cyl}}^{\prime} + \mathbf{L}\_{\text{th}}\right)} \left\{ \left(\frac{\mathbf{L}\_{\text{cyl}}^{\prime} + \mathbf{L}\_{\text{th}}}{\mathbf{L}\_{\text{m}}}\right)^{\rightarrow} \stackrel{\rightarrow}{\mathbf{v}}\_{\text{s}} - \left[\mathbf{R}\_{\text{b}} \left(\frac{\mathbf{L}\_{\text{cyl}}^{\prime\prime} + \mathbf{L}\_{\text{th}}}{\mathbf{L}\_{\text{m}}}\right) + \mathbf{R}\_{\text{r}}^{\prime}\right] \stackrel{\rightarrow}{\mathbf{v}}\_{\text{s}} + \mathbf{R}\_{\text{t}}^{\prime} \left(\stackrel{\rightarrow}{\mathbf{l}}\_{\text{m}} + \stackrel{\rightarrow}{\mathbf{l}}\_{\text{ft}}\right) - \left[\mathbf{w}\_{\text{ma}}\stackrel{\rightarrow}{\mathbf{l}}\_{\text{r}}\right] \left(\mathcal{T}\_{\text{s}}^{\prime} \right) \stackrel{\rightarrow}{\mathbf{v}}\_{\text{s}} \right\}$$

Although dynamic IM performance is well represented in Equations (1) and (7), it is very multifaceted to appreciate. We suppose to rewrite the following Equations (8)–(11) to characterize the IM state-space model in terms of equivalent core-loss resistance, Rm, which is included in Rfe where Rm is presumed to be proportional to ω1.6 [40].

$$\text{s.p.}\ i\_{\text{as}} = -\left(\frac{R\_{\text{s}}}{L\_{\text{s}}} + \frac{R\_{\text{r}}}{L\_{\text{s}}} \frac{L\_{\text{m}}^{2}}{L\_{\text{r}}^{2}\sigma}\right)i\_{\text{as}} + \left[\frac{R\_{\text{r}}L\_{\text{m}} + R\_{\text{W}}(sL\_{\text{m}} - L\_{\text{r}})}{L\_{\text{s}}}\right] \Psi\_{\text{ar}} + \frac{L\_{\text{m}}}{L\_{\text{s}}}\omega\_{\text{r}}\,\Psi\_{\text{\hat{\mu}}\text{r}} + \frac{1}{L\_{\text{s}}\sigma}v\_{\text{as}} \tag{8}$$

$$\text{p } i\_{\text{fds}} = -\left(\frac{R\_{\text{s}}}{L\_{\text{s}}}\frac{R\_{\text{r}}}{\sigma} + \frac{R\_{\text{r}}}{L\_{\text{s}}}\frac{L\_{\text{m}}^{2}}{L\_{\text{r}}^{2}\sigma}\right)i\_{\text{fds}} + \left[\frac{R\_{\text{r}}L\_{\text{m}} + R\_{\text{m}}(sL\_{\text{m}} - L\_{\text{r}})}{L\_{\text{s}}}\right]\Psi\_{\text{fds}} - \frac{L\_{\text{m}}}{L\_{\text{s}}}\omega\_{\text{r}}\Psi\_{\text{a}} + \frac{1}{L\_{\text{s}}\sigma}v\_{\text{fds}} \tag{9}$$

$$\text{ip }\boldsymbol{\Psi}\_{\alpha\mathbf{r}} = \frac{\mathbf{R}\_{\mathbf{r}} \mathbf{L}\_{\mathbf{m}}}{\mathbf{L}\_{\mathbf{r}}} \mathbf{i}\_{\alpha\mathbf{s}} - \left(\frac{\mathbf{R}\_{\mathbf{r}} - \mathbf{s}}{\mathbf{L}\_{\mathbf{r}}}\right) \boldsymbol{\Psi}\_{\alpha\mathbf{r}} - \boldsymbol{\omega}\_{\mathbf{r}} \,\, \boldsymbol{\Psi}\_{\beta\mathbf{r}} \tag{10}$$

$$\text{lp } \Psi\_{\beta \text{r}} = \frac{\text{R}\_{\text{r}} \text{ L}\_{\text{m}}}{\text{L}\_{\text{r}}} \text{ i}\_{\beta \text{s}} - \left(\frac{\text{R}\_{\text{r}} - \text{s}}{\text{L}\_{\text{r}}}\right) \Psi\_{\beta \text{r}} + \omega\_{\text{r}} \Psi\_{\alpha \text{r}} \tag{11}$$

where ωr is the rotor speed, s is the slip and ω is the excitation frequency. Cross product of rotor and stator flux linkages vectors states the developed torque [28] as:

$$\mathbf{T\_{o}} = \frac{3}{2} \text{ P} \, \frac{\text{L}\_{\text{m}}}{\text{L}\_{\text{r}} \text{ L}\_{\text{s}} - \text{ L}\_{\text{m}}^{2}} \left( \stackrel{\rightarrow}{\Psi}\_{\text{s}} \times \stackrel{\rightarrow}{\Psi}\_{\text{r}} \right) \tag{12}$$

For both MPTC and MPTFC methods, a 2−level voltage source inverter (VSI) is used in this work. The inverter topology and its feasible voltage vectors are shown in Figure 2. With every phase A, B and C the responding switching state S can be described as:

$$\mathbf{S} = \frac{2}{3} \left( \mathbf{S\_a} + \mathbf{a} \, \mathbf{S\_b} + \mathbf{a}^2 \, \mathbf{S\_c} \right) \tag{13}$$

where A, B and C are motor terminals, a = ej2π/3, Si = 1 means Si on, Si means off and DC-bus voltage is Vdc.

**Figure 2.** Inverter topology and its feasible voltage vectors (**a**) 2−level VSI circuit; (**b**) Voltage vectors.

The relation between inverter output voltage vector →vsαβ,<sup>k</sup> and the switching state S is stated as:

$$
\stackrel{\rightarrow}{\mathbf{v}}\_{\ast \alpha \circledast} = \mathbf{V}\_{\text{d} \circledast} \mathbf{S} \tag{14}
$$

Voltage space vectors will regulate the output torque for the DTC technique, where two voltage vectors are selected in each sector to decrease or increase stator flux amplitude. There are eight switching states and seven separate voltage vectors V0, V1, V2, ... ., V7 for a 2−level inverter-fed IM drive, as seen in Figure 2b. Cost function G is evaluated for each voltage vector value, and the vector generating minimum G is chosen as the best one.

#### **3. Adaptive Full Order Observer Modification**

In this work, accurate state estimation is a major step towards achieving good performance of MPTFC in dynamic implementation. Owing to its precision and insensitivity to parameter variance over a broad speed range [41], AFOO is approved for rotor speed-flux estimation. By compensating for the impact of core-loss, AFOO mathematical model can be revealed as [42]:

$$\mathbf{p} \cdot \mathbf{\hat{x}} = \mathbf{\hat{A}} \cdot \mathbf{\hat{z}} + \mathbf{B} \,\mathbf{v}\_{\\$} + \mathbf{H} \left(\mathbf{\hat{i}}\_{\\$} - \mathbf{i}\_{\\$}\right) + \mathbf{D} \,\Psi\_{\text{I}} \tag{15}$$

where xˆ = [ˆisΨ<sup>ˆ</sup> r]T is the estimated state for stator current and rotor flux, vs is stator voltage vector, I = 1 0 01, J = 0 −1 10, B = [ 1Lsσ I 0 ]T, [ D ] is the core-loss compensating term,

$$
\begin{bmatrix}
\hat{\mathbf{A}} & \mathbf{I} \\
\hat{\mathbf{A}} & \mathbf{I}
\end{bmatrix} = \begin{bmatrix}
\hat{\mathbf{A}}\_{11} & \hat{\mathbf{A}}\_{12} \\
\hat{\mathbf{A}}\_{21} & \hat{\mathbf{A}}\_{22}
\end{bmatrix} = \begin{bmatrix}
& \frac{\mathbf{R}\_{\mathbf{e}}\mathbf{I}\_{\mathbf{e}}}{\mathbf{I}\_{\mathbf{e}}}\mathbf{I} & -\frac{\mathbf{R}\_{\mathbf{e}}}{\mathbf{I}\_{\mathbf{e}}}\mathbf{I} + \hat{\mathbf{A}}\_{\mathbf{e}}\mathbf{I}
\end{bmatrix}, \mathbf{D} = \begin{bmatrix}
\mathbf{D}\_{1}\mathbf{I} \\
\mathbf{D}\_{2}\mathbf{I}\end{bmatrix} = \mathbf{0}
$$

Rm#− ( Lr−<sup>s</sup> Lm) (σLs Lr<sup>2</sup>) I − sLr I \$Tand the observer gain matrix, see appendix, is given as:

$$\mathbf{H} = \begin{bmatrix} \mathbf{h}\mathbf{1} & \mathbf{h}\mathbf{2} & \mathbf{h}\mathbf{3} & \mathbf{h}\mathbf{4} \\ -\mathbf{h}\mathbf{2} & \mathbf{h}\mathbf{1} & -\mathbf{h}\mathbf{4} & \mathbf{h}\mathbf{3} \end{bmatrix}^{\mathrm{T}} \tag{16}$$

#### *3.1. Stator Flux Estimation*

DTC stator flux estimate is among the primary challenges that determine the accuracy and stability of the drive's operation, since it's not measured. The simplest solution is the pure integrator-based stator flux estimator voltage model, but it is susceptible to numerous problems [43]: (i) sensitivity in respecting to the dc-drift present in the pure integrator input that causes saturation of the integrator, and (ii) pure integrator initial conditions that cause an unwanted dc deviation in the estimation stator flux signal. We can rewrite AFOO in Equation (15) for IM model assumed in Equations (8)–(11) with the core-loss compensation concept as follows, in order to prevent the dc offset issue:

$$\begin{cases} \begin{array}{rcl} p \stackrel{\rightarrow}{\mathsf{I}}\_{s} = \mathsf{\dot{A}}\_{11} \stackrel{\rightarrow}{\mathsf{I}}\_{s} + (\mathsf{\dot{A}}\_{12} + \mathsf{D}\_{1}) \stackrel{\rightarrow}{\mathsf{V}}\_{\mathsf{r}} + B \,\mathrm{v}\_{\mathsf{s}} + \mathsf{H}\_{\mathsf{s}} \,(\mathsf{\dot{I}}\_{\mathsf{s}} - \mathsf{i}\_{\mathsf{s}})\\ \begin{array}{rcl} p \stackrel{\rightarrow}{\mathsf{V}}\_{\mathsf{r}} = \mathsf{\dot{A}}\_{21} \stackrel{\rightarrow}{\mathsf{I}}\_{s} + (\mathsf{\dot{A}}\_{22} + \mathsf{D}\_{2}) \stackrel{\rightarrow}{\mathsf{V}}\_{\mathsf{r}} + B \,\mathrm{v}\_{\mathsf{s}} + \mathsf{H}\_{\mathsf{r}} \,(\mathsf{\dot{A}}\_{\mathsf{s}} - \mathsf{i}\_{\mathsf{s}})\\ \end{array} \end{cases} \tag{17}$$

where, L1 = Ls Lr−Lm<sup>2</sup> Lr 0 0 Ls Lr−Lm<sup>2</sup> Lr and L2 = LmLr 0 0 LmLr .

 can

The first benefit of the estimated stator flux in Equation (17) is that there is no need for motor speed statistics to estimate the flux. This reduces any additional errors, particularly at lower frequencies, associated with calculating or even measuring such signals. Another benefit is that estimated stator flux independent on machine resistances that improve drive reliability. Estimated stator flux → Ψ ˆ s components in stationary (α − β) reference frame depending on estimated rotor flux → Ψ ˆ r and → ˆiinEquation(17)bestated

 as:

$$\begin{cases} \begin{array}{c} \Psi\_{\alpha\circledast} = \frac{\mathbf{l}\_{\alpha\circledast}}{\mathbf{L}\_{\gamma}} \Psi\_{\alpha\pi} + \frac{\mathbf{l}\_{\alpha}\cdot\mathbf{L}\_{\gamma} - \mathbf{L}\_{\alpha\pi}}{\mathbf{L}\_{\gamma}} \mathbf{f}\_{\alpha\circledast} \\\ \Psi\_{\beta\circledast} = \frac{\mathbf{l}\_{\alpha\epsilon}}{\mathbf{L}\_{\gamma}} \Psi\_{\beta\pi} + \frac{\mathbf{l}\_{\alpha}\cdot\mathbf{L}\_{\gamma} - \mathbf{L}\_{\alpha\pi}}{\mathbf{L}\_{\gamma}} \mathbf{f}\_{\beta\circledast} \end{array} \tag{18}$$

## *3.2. Rotor Speed Estimation*

 s

stator current

The cost and complexity of the system will be effectively minimized by a sensorless system where the velocity is determined instead of measured. One of the key reasons for the success of inverter fed IM drives is that it is possible to use any standard IM without modifications. It is suggested with current and voltage measuring devices that IM speed can be guessed without the need for speed or flux sensors to be mounted. Through traditional AFOO, rotor speed estimate is obtained via conventional adaptation law as in (15) [5,42]:

$$\mathfrak{A}\_{\mathsf{F}} = \mathsf{K}\_{\mathsf{I}\mathsf{a}\mathsf{b}} \cdot \int\_{0}^{\mathsf{t}} \left[ \hat{\Psi}\_{\mathsf{f}\mathsf{F}} \left( \mathsf{i}\_{\mathsf{f}\mathsf{a}\mathsf{s}} - \hat{\mathsf{i}}\_{\mathsf{a}\mathsf{s}} \right) - \stackrel{\rightarrow}{\Psi}\_{\mathsf{a}\mathsf{ar}} \left( \mathsf{i}\_{\mathsf{f}\mathsf{a}\mathsf{s}} - \hat{\mathsf{i}}\_{\mathsf{f}\mathsf{b}\mathsf{s}} \right) \right] \mathsf{d}\mathsf{t} \tag{19}$$

Like MRAs, the proposed scheme would be separated into two key components, which can be seen in Figure 3; each model is supplied with the same input signal → vs. Additionally, the adaptive model is tuned by dual closed loops. The first takes into account the error among measured current →i s and estimated → ˆi s from (17), while the other considers the adaptively calculated speed (20). The first closed-loop is being responsible for compensating the offsets that have the major source in current sensors. The controller time constant is calibrated proportionally to a speed value for proper compensation over the entire range of IM rotor velocity. Moreover, for two main reasons, the controllers work slowly: to ensure a lack of effect on transient estimation and because it integrates errors through sinusoidal signals. The second closed-loop provides the rotor velocity ω<sup>ˆ</sup> r, calculated on the basis of difference among estimated and measured currents Δ → i s multiplied by estimated stator flux conjugate vector → Ψˆ s ∗ as per as (17).

$$\begin{cases} \begin{array}{c} \mathbf{e}\_{\omega \nu} = \operatorname{Im} \left( \stackrel{\rightarrow}{\Psi}\_{s}^{\*} \, \Delta \, \stackrel{\rightarrow}{\mathbf{i}}\_{s} \right) = \Psi\_{\beta \kappa} \left( \mathbf{i}\_{\alpha \kappa} - \mathbf{f}\_{\alpha \kappa} \right) \, -\Psi\_{\alpha \kappa} \left( \mathbf{i}\_{\beta \kappa} - \mathbf{f}\_{\beta \kappa} \right) \\ \updownarrow \quad \begin{pmatrix} \mathbf{K}\_{\Gamma \omega} + \frac{\mathbf{K}\_{\Gamma \omega}}{s} \end{pmatrix} \mathbf{e}\_{\omega \nu} \end{cases} \tag{20}$$

**Figure 3.** Proposed sensorless observer.

#### *3.3. Other Parameter Estimation*

IM stator resistance will vary thanks to temperature change during operation. To provide stator resistance estimation, adaptive control observer could be extended. According to the same Lyapunov's theory, stator resistance Rs can also be estimated like rotor speed via a PI controller [39].

$$\begin{cases} \begin{array}{c} \mathbf{e\_{R\_{\mathsf{s}}}} = \hat{\mathbf{1\_{\alpha\mathsf{s}}}} \left( \mathbf{\dot{1\_{\alpha\mathsf{s}}}} - \mathbf{\dot{1\_{\alpha\mathsf{s}}}} \right) + \hat{\mathbf{1\_{\beta\mathsf{s}}}} \left( \mathbf{\dot{1\_{\beta\mathsf{s}}}} - \mathbf{\dot{1\_{\beta\mathsf{s}}}} \right) \\ \mathbf{\dot{R\_{\mathsf{s}}}} = \left( \mathbf{K\_{PR\_{\mathsf{s}}}} + \frac{\mathbf{K\_{R\_{\mathsf{s}}}}}{\mathbf{\dot{s}}} \right) . \mathbf{e\_{R\_{\mathsf{s}}}} \end{array} \end{cases} \tag{21}$$

In addition to the temperature variation effect on stator resistance, various parameters in the suggested observer will also adjust during operation. Owing to magnetic saturation, parameters *Ls*; *Lr* and *Lm* differ. Although magnetic saturation variance can be compensated for through the nonlinear magnetic model, rotor resistance variation would have a significant effect on the speed-accuracy of our adaptive observer.

It is recognized that the misestimating of Rr gives correct estimates of the rotor and stator fluxes during-steady state, but results in a speed misestimating [44]. Rotor resistance estimate can be incorporated into adaptive observer using the approach followed in [42] or IM thermal model. The influence of the core-loss on the estimation of both stator and rotor resistance and its compensation has been well defined in our previous work [45]. Stator current and rotor flux error estimation can give rotor resistance estimate utilizing Lyapunov theory via the suggested AFOO in (17) as:

$$\begin{cases} \mathbf{e}\_{\mathsf{R}\_{\mathsf{r}}} = (\boldsymbol{\Psi}\_{\mathsf{car}} - \mathbf{L}\_{\mathsf{m}}\mathbf{\hat{l}}\_{\mathsf{cas}}) \left( \mathbf{i}\_{\mathsf{cas}} - \mathbf{\hat{l}}\_{\mathsf{cas}} \right) + (\boldsymbol{\Psi}\_{\mathsf{f}\mathsf{c}\mathsf{r}} - \mathbf{L}\_{\mathsf{m}}\mathbf{\hat{l}}\_{\mathsf{f}\mathsf{s}s}) \left( \mathbf{i}\_{\mathsf{f}\mathsf{s}s} - \mathbf{\hat{l}}\_{\mathsf{f}\mathsf{s}s} \right) \\\ \mathbf{i}\_{\mathsf{r}} = \left( \mathbf{K}\_{\mathsf{PR}\_{\mathsf{r}}} + \frac{\mathbf{K}\_{\mathsf{RL}}}{\mathsf{s}} \right) \cdot \mathbf{e}\_{\mathsf{R}\_{\mathsf{r}}} \end{cases} \tag{22}$$
