**1. Introduction**

Three-phase induction motor drives have become a mature technology in the last years, but investigations into concepts of multiphase induction motor drives are still taking place. Multiphase drive systems have a nearly 40-year history of research and study due to their promising advantages against the conventional three-phase systems. The phase redundancy of the multiphase drives provides extra merits such as fault-tolerant operation, series-connected multimotor drive systems, asymmetry and braking systems. Six-phase induction motors (6PIMs) are known for its fault-tolerant capability, low rate of inverter switches, and low DC-link voltage utilization compared with its three-phase one [1–3]. On the other hand, the modular three-phase structure of the 6PIM allows the use of well-known three-phase technologies. The 6PIM is successfully used in special applications, such as electric ships, electric aircrafts, electric vehicles, and melt pumps, where the high reliability and continuity of the operation are critical factors for the system [4]. The phase redundancy of the 6PIM provides the ability of the open-phase fault-tolerant operation without any extra electronic components [5,6].

Among different structures of the 6PIM [4], the asymmetrical 6PIM with double isolated neutral points, which consists of two sets of three-phase windings spatially shifted by 30 electrical degrees, has attracted the interest of many researchers [7–10]. The traditional three-phase control strategies, including switching table-based direct torque control (ST-DTC) [7], modulation-based DTC [8], the field-oriented control (FOC) [9], and finite control set-model predictive control (FCS-MPC) [10], can be extended to 6PIM (or other multi-phase machines) with some modifications to use more freedom degrees that exist in multi-phase machines. DTC is a well-accepted technique due to its simplicity, quick dynamics, and robustness [11]. The modulation-based DTC strategy offers better phase current, torque, and flux response. On the contrary, this method has more complexity against conventional ST-DTC. The ST-DTC approach has straightforward and simple structure, but it is completely overshadowed by low-order harmonics due to unused voltage vectors in the losses subspaces. To overcome this restriction, the idea of duty cycle control is introduced by several researchers [12,13].

The rapid development of intelligent and high-performance control technologies has also brought about changes in the adjustable speed drive system for different industrial applications [14,15]. To operate safely and reliably under different conditions, there is a lot of debate nowadays about the main control strategy of the system [16,17]. Among different high-performance control strategies of drive systems, the DTC strategy has a straightforward algorithm. The DTC technique is inherently speed sensorless. Nevertheless, if an outer speed loop is added to the DTC, the speed value is also necessary. Sensorless three/multi-phase induction machine drives are widely addressed in the technical literature due to multiple shortcomings of shaft encoders [18–23]. To investigate the instability problem of the traditional rotor flux-based model reference adaptive system (MRAS) speed estimators in the regenerating-mode low-speed operation, a stator current-based and back electromotive force-based MRASs are addressed in [19,20], respectively. In [21], two modified adaptation mechanisms are proposed to replace the classical proportional-integral (PI) regulator. The full-order Luenberger and Kalman filter observers are discussed in [22,23], respectively. Providing a DTC drive system with parallel identification of the rotor speed and the stator resistance is a challenging task because the operation of the DTC scheme is severely dependent on the stator resistance. This problem is sporadically reported for three-phase induction machines (3PIMs) [24,25], where the rotor speed and the stator resistance estimators encounter an overlap due to limited freedom degrees of 3PIM. In this paper, the problem of parallel estimation is investigated using more freedom degrees of 6PIM.

The outer speed control loop of the DTC scheme conventionally contains the PI regulator to obtain torque command from speed error. In general, the control law of a PID regulator is a linear combination of proportional-integral-derivative terms, which is suitable for linear systems. For nonlinear systems, such as the 6PIM drive system, the PI regulator has been given a lot of attention due to its simplicity. However, it suffers from multiple problems including: (1) tuning of its parameters; (2) high sensitivity against noise and external disturbances; and (3) loss of efficiency due to oversimplified control law [26,27]. One promising technique to relatively get rid of the drawbacks of PI regulator is active disturbance rejection controller (ADRC) [26,28]. The ADRC is a nonlinear control scheme, which provides a robust control against noises, external disturbances, and parameter uncertainties. For these reasons, the ADRC technique has recently attracted more attention for electric drive systems. To address this issue, a modified FOC scheme based on first-order ADRCs for current and speed control loops is proposed in [29]. A combined active disturbance rejection and sliding-mode controller for an induction motor is presented to achieve total robustness [30].

The aim of this paper is to present an ADRC-based DTC scheme for sensorless 6PIM drives. The speed estimator is based on adaptive full-order observer, and its control law is designed using Lyapunov stability theorem. Besides the speed estimation system, a stator resistance estimator is proposed using additional degrees of freedom of the 6PIM to enhance the robustness of the sensorless DTC strategy against stator resistance uncertainties. The adaptation law for the stator resistance estimator is derived using the Lyapunov stability theorem to ensure its overall convergence.

The rest of this paper is organized as follows. Section 2 introduces the mathematical model of the 6PIM. Section 3 presents the design procedure of the adaptive full-order observer, the speed estimator, and the stator resistance estimator. The DTC scheme of the 6PIM is discussed in Section 4, which includes the ST-DTC scheme, and ADRC in DTC. The experimental results are presented in Section 5. Finally, Section 6 summarizes the findings and concludes the paper.

#### **2. Dynamic Model of 6PIM**

There are two popular approaches for modeling of the multi-phase machines: (1) multiple d–q approach [9]; (2) vector space decomposition (VSD) approach [31]. The first method is exclusively used for modular three-phase structures-based multi-phase machines such as six-phase and nine-phase machines. However, the second method can be used for all types of multi-phase machines. In this research, the VSD approach is used, where a 6PIM with distributed windings is modeled in the three orthogonal subspaces, i.e., the *α* − *β*, *z*<sup>1</sup> − *z*<sup>2</sup> and *o*<sup>1</sup> − *o*2. Among them, only the *α* − *β* variables are in relation with electromechanical energy conversion, while *z*<sup>1</sup> − *z*<sup>2</sup> and *o*<sup>1</sup> − *o*<sup>2</sup> variables do not actively contribute to the torque production.

The schematic diagram of a six-phase voltage source inverter (VSI)-fed an 6PIM with two isolated neutral points is shown in Figure 1. The transfer between the normal *a* − *x* − *b* − *y* − *c* − *z* variables and *α* − *β* − *z*<sup>1</sup> − *z*<sup>2</sup> − *o*<sup>1</sup> − *o*<sup>2</sup> variables is performed by *T*<sup>6</sup> transformation matrix as follows [31]:

$$T\_6 = \frac{1}{3} \begin{bmatrix} 1 & \frac{\sqrt{3}}{2} & -\frac{1}{2} & -\frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & \frac{1}{2} & \frac{\sqrt{3}}{2} & \frac{1}{2} & -\frac{\sqrt{3}}{2} & -1 \\ 1 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} & \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} & \frac{1}{2} & \frac{\sqrt{3}}{2} & -1 \\ 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \end{bmatrix} \tag{1}$$

By applying *T*<sup>6</sup> matrix to the voltage equations in the original six-dimensional system, the 6PIM model can be represented in the three orthogonal submodels, identified as *α* − *β*, *z*<sup>1</sup> − *z*2, and *o*<sup>1</sup> − *o*2. The voltage space vector equations of the 6PIM in the *α* − *β* subspace are written as follows:

$$
\sigma\_s = R\_s \mathbf{i}\_s + p\mathbf{Y}\_s \tag{2}
$$

$$0 = R\_r \mathbf{i}\_r + p\mathbf{\varPsi}\_r - j\omega\_r \mathbf{\varPsi}\_r \tag{3}$$

The flux linkages are

$$\mathbf{Y}\_s = L\_s \mathbf{i}\_s + L\_m \mathbf{i}\_r \tag{4}$$

$$\mathbf{Y}\_r = L\_m \mathbf{i}\_s + L\_r \mathbf{i}\_r \tag{5}$$

where *v*, *i*, **Ψ**, *R*, and *L* represent voltage, current, flux linkage, resistance, and inductance, respectively, for stator (s subscript) and rotor (r subscript) quantities, and *p* denotes derivative operator. The electromagnetic torque produced by the 6PIM is expressed as

$$T\_{\mathfrak{e}} = \mathfrak{Z}P\mathbf{\bar{Y}}\_{\mathfrak{s}} \otimes \mathfrak{i}\_{\mathfrak{s}} \tag{6}$$

where *P* is pole pairs and ⊗ denotes the cross product.

The 6PIM voltage equations in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace are the same as a passive R-L circuit as follows:

$$w\_{sz1} = R\_s i\_{sz1} + L\_{ls} p i\_{sz1} \tag{7}$$

$$
\sigma\_{s\pi2} = R\_s i\_{s\pi2} + L\_{ls} p i\_{s\pi2} \tag{8}
$$

where *Lls* is stator leakage inductance.

On the presumption that the stator mutual leakage inductances can be neglected, the 6PIM model in the *o*<sup>1</sup> − *o*<sup>2</sup> subspace has the same form of the *z*<sup>1</sup> − *z*<sup>2</sup> subspace. However, the applied 6PIM with two isolated neutral points avoids zero-sequence currents because it contains two sets of balanced three-phase windings.

**Figure 1.** Six-phase two-level VSI-fed 6PIM.

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

The block diagram of the proposed *Rs* and *ω<sup>r</sup>* estimators based on the adaptive state observer is shown in Figure 2. It contains the stator current and rotor flux observers, the stator resistance identifier, and the rotor speed estimator, which are discussed below.

**Figure 2.** The block diagram of the proposed parallel estimation system of the stator resistance and the rotor speed based on an adaptive full-order observer.

#### *3.1. Stator Current and Rotor Flux Observers*

The general form of state-space model of the 6PIM in the *α* − *β* subspace is

$$\begin{cases} \dot{\mathbf{x}}\_1 = A\_1 \mathbf{x}\_1 + B\_1 \mathbf{u}\_1 \\ y\_1 = C\_1 \mathbf{x} + D\_1 \mathbf{u}\_1 \end{cases} \tag{9}$$

Assuming stator current and rotor flux as state variables and using Equations (2) and (3), the elements of state-space representation in *α* − *β* subspace will be

$$\mathbf{x}\_1 = \begin{bmatrix} i\_{s\alpha} & i\_{s\beta} & \psi\_{r\alpha} & \psi\_{r\beta} \end{bmatrix}^T \tag{10}$$

$$A\_{1} = \begin{bmatrix} (-\frac{R\_s}{\sigma L\_s} - \frac{1-\sigma}{\sigma T\_r})I & \frac{L\_m}{\sigma L\_s L\_r}(\frac{1}{T\_r}I - \omega\_r I) \\ \frac{L\_m}{T\_r}I & -\frac{1}{T\_r}I + \omega\_r I \end{bmatrix} \tag{11}$$

$$B\_1 = \begin{bmatrix} \frac{1}{\sigma L\_\*} I & O \end{bmatrix}^T \tag{12}$$

$$\boldsymbol{u}\_1 = \begin{bmatrix} \boldsymbol{v}\_{s\mathfrak{A}} & \boldsymbol{v}\_{s\mathfrak{F}} \end{bmatrix}^T \tag{13}$$

$$\begin{bmatrix} y\_1 \ \end{bmatrix} = \begin{bmatrix} \dot{i}\_{s\alpha} & \dot{i}\_{s\beta} \end{bmatrix}^T \tag{14}$$

$$\mathbf{C}\_{1} = \begin{bmatrix} I & \mathbf{O} \end{bmatrix} \tag{15}$$

with

$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \tag{16}$$

where *Tr* = *Lr*/*Rr* is rotor time constant and *<sup>σ</sup>* = <sup>1</sup> − *<sup>L</sup>*<sup>2</sup> *<sup>m</sup>*/*LsLr* is leakage coefficient.

The state observer of the 6PIM has a similar form of state-space representation except that an additional compensation term based on error of measurable states and observer gain matrix is added to it. The state observer can be written as

$$\begin{cases} \dot{\mathfrak{x}}\_1 = \mathring{A}\_1 \mathfrak{x}\_1 + B\_1 \mathfrak{u}\_1 + G\_1 (\mathring{\mathfrak{z}}\_s - \mathring{\mathfrak{z}}\_s) \\\\ \mathfrak{y}\_1 = C\_1 \mathfrak{x}\_1 \end{cases} \tag{17}$$

where the marker ∧ indicates the estimated values, and *G*<sup>1</sup> is the observer gain matrix. The matrix *A*<sup>1</sup> contains unknown parameters of the 6PIM such as the rotor speed and the stator resistance. These parameters can be estimated by the designing of a suitable adaptation control law with a nonlinear theorem such as a Lyapunov stability theorem. It is worth mentioning here that the matrix *A*<sup>1</sup> also contains the rotor time constant. However, simultaneous estimation of the rotor speed, the rotor time constant, and the stator resistance is challenging because of persistency of excitation conditions problem [32]. Some techniques have recently been developed based on signal injection to provide persistent excitation [33], which suffer from steady-state torque and speed ripples. In this paper, the stator resistance is estimated from additional degrees of freedom of the 6PIM, while the rotor speed is provided using the 6PIM equations in *α* − *β* subspace. This procedure provides the stator resistance independent from the rotor speed.

The observer gain matrix *G*<sup>1</sup> must be designed to ensure stability and good dynamic response of the observer at a wide range of the speeds. Using pole-placement method, the elements of matrix *G*<sup>1</sup> is provided as [22,34]

$$\mathbf{G}\_1 = \begin{bmatrix} \mathbb{S}\_1 & \mathbb{S}\_2 & \mathbb{S}\_3 & \mathbb{S}\_4 \\ -\mathbb{S}\_2 & \mathbb{S}\_1 & -\mathbb{S}\_4 & \mathbb{S}\_3 \end{bmatrix}^T \tag{18}$$

where

$$\begin{cases} \mathcal{g}\_1 = (1 - K\_{po}) \left( R\_s L\_r^2 + R\_r L\_m^2 \right) / \sigma L\_s L\_r^2\\ \mathcal{g}\_2 = (K\_{po} - 1) \hat{\omega}\_r\\ \mathcal{g}\_3 = (K\_{po} - 1) \left( R\_s L\_s - K\_{po} R\_s L\_r \right) / L\_m\\ \mathcal{g}\_4 = (1 - K\_{po}) \hat{\omega}\_l \sigma L\_s L\_r / L\_m \end{cases} \tag{19}$$

where *Kpo* > 0 is observer constant gain.

## *3.2. Stator Resistance Identification*

In this paper, a stator resistance adaptation system is proposed using the machine model in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace. This method can be utilized for any multi-phase machines. It is completely decoupled from the rotor speed and the rotor time constant, whereas most of the conventional stator resistance estimators, developed for three-phase machines, are related to these parameters. The proposed *Rs* estimator only depends on the stator leakage inductance *Lls*, which can be approximately assumed to be constant.

The state-space model of 6PIM in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace, with consideration of *isz*<sup>1</sup> and *isz*<sup>2</sup> as the state variables, can be derived from Equations (7) and (8) as follows:

$$
\begin{bmatrix} \dot{i}\_{sz1} \\ \dot{i}\_{sz2} \end{bmatrix} = \begin{bmatrix} -\frac{R\_b}{L\_{ls}} & 0 \\ 0 & -\frac{R\_s}{L\_{ls}} \end{bmatrix} \begin{bmatrix} \dot{i}\_{sz1} \\ \dot{i}\_{sz2} \end{bmatrix} + \frac{1}{L\_{ls}} \begin{bmatrix} v\_{sz1} \\ v\_{sz2} \end{bmatrix}
$$

$$
\begin{bmatrix} \dot{i}\_{sz1} \\ \dot{i}\_{sz2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \dot{i}\_{sz1} \\ \dot{i}\_{sz2} \end{bmatrix} \tag{20}
$$

In this case, the proposed states observer is given by

$$\begin{cases} \dot{\mathfrak{x}}\_2 = \hat{A}\_2 \hat{\mathfrak{x}}\_2 + B\_2 \mathfrak{u}\_2\\ \mathfrak{y}\_2 = \mathbb{C}\_2 \mathfrak{x}\_2 \end{cases} \tag{21}$$

It should be noted that a correction term *G*2(*x*<sup>2</sup> − *x***ˆ2**) is neglected in Equation (21) due to the inherent stability of the observer.

The proposed adaptation law for the stator resistance estimation is

$$
\hat{R}\_s = K\_{pr} \epsilon\_{R\_S} + K\_{ir} \int \epsilon\_{R\_S} dt \tag{22}
$$

where *Kir* and *Kpr* are the integral and proportional gains, respectively, and *RS* is the stator resistance error signal

$$
\varepsilon\_{R\_S} = \hat{\mathfrak{i}}\_{sz1} (\mathbf{i}\_{sz1} - \hat{\mathfrak{i}}\_{sz1}) + \hat{\mathfrak{i}}\_{sz2} (\mathbf{i}\_{sz2} - \hat{\mathfrak{i}}\_{sz2}) \tag{23}
$$

The proof for the stator resistance adaptation law is presented in Appendix A.

#### *3.3. Rotor Speed Estimation*

In order to design the speed adaptation law, it is considered as an unknown parameter. First, an appropriate positive definite function is chosen as the Lyapunov candidate. Then, the adaptation law is obtained using the Lyapunov criterion to ensure asymptotic stability of the system. The speed adaptation law is

$$
\hat{\omega}\_r = K\_{p\omega} \epsilon\_\omega + K\_{i\omega} \int \epsilon\_{\omega} dt \tag{24}
$$

where *Kp<sup>ω</sup>* and *Ki<sup>ω</sup>* are proportional and integral gains, respectively, and *ω* is the speed error signal as follows:

$$
\epsilon\_{\omega} = (i\_{s\alpha} - \hat{i}\_{s\alpha})\hat{\psi}\_{r\beta} - (i\_{s\beta} - \hat{i}\_{s\beta})\hat{\psi}\_{ru} \tag{25}
$$

The proof for the speed adaptation law is presented in Appendix B.

### **4. DTC of 6PIM**

#### *4.1. ST-DTC Scheme*

A six-phase VSI contains overall 26 = 64 different voltage space vectors, 60 active, and four zero vectors, where the active voltage vectors are distributed in four non-zero levels depicted in Figure 3. The electrical angle of each sectors is 30◦. The 6PIM phase-to-neutral voltages can be calculated as

$$
\begin{bmatrix} V\_a \\ V\_b \\ V\_c \\ V\_x \\ V\_y \\ V\_z \end{bmatrix} = \frac{V\_{dc}}{3} \begin{bmatrix} 2 & -1 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ -1 & -1 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & -1 & -1 \\ 0 & 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & -1 & -1 & 2 \end{bmatrix} \begin{bmatrix} S\_a \\ S\_b \\ S\_c \\ S\_x \\ S\_y \\ S\_z \end{bmatrix} \tag{26}
$$

where *Si* = {0, 1}, *i* = {*a*, *x*, *b*, *y*, *c*, *z*} is the switching state. When *Si* = 1 (*Si* = 0), the corresponding stator terminal is connected to positive (negative) DC-link rail. The voltage space vectors are given by

$$v\_s = \frac{1}{3} [V\_a + aV\_x + a^4V\_b + a^5V\_y + a^8V\_c + a^9V\_z] \tag{27}$$

$$v\_z = \frac{1}{3} [V\_d + a^5 V\_x + a^8 V\_b + a V\_\mathcal{Y} + a^4 V\_c + a^9 V\_z] \tag{28}$$

where *vz* = *vsz*<sup>1</sup> + *jvsz*<sup>2</sup> and *a* = *ej<sup>π</sup>*/6.

The flux estimator is obtained from

$$
\psi\_{\rm sa} = \int (\upsilon\_{\rm sa} - \hat{R}\_s i\_{\rm sa}) dt \tag{29}
$$

$$
\psi\_{s\S} = \int (\upsilon\_{s\S} - \hat{\mathsf{R}}\_s \dot{\imath}\_{s\S}) dt \tag{30}
$$

and the toque estimator is obtained from (6). In the traditional ST-DTC, the torque and stator flux errors are applied to hysteresis regulators to provide the sign of torque (*T*) and stator flux (*ψ*). According to gained signals and also the position of stator flux, a proper large voltage vector is selected based on Table 1 during each sampling period. From Figure 3, the corresponding voltage vectors in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace will produce large current harmonics, when only large voltage vectors are used to control the torque and flux. Hence, it can alleviate the current harmonics through reduction of the *z*<sup>1</sup> − *z*<sup>2</sup> components by applying a combined voltage vector during each sampling period. This technique is referred to as duty cycle control, where a virtual vector (synthesized by large and medium voltage space vectors) is applied to the inverter in each sampling period because the large and medium voltage vectors are in the opposite direction in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace (see Figure 3). The duration of the applied vectors is calculated in order to reduce the average volt-seconds in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace [4]. The block diagram of the proposed sensorless DTC strategy with the adaptive full-order observer is shown in Figure 4a. In this figure, the speed control loop is based on the ADRC strategy, which will be discussed in the next subsection.

**Figure 3.** The *α* − *β* (top side) and the *z*<sup>1</sup> − *z*<sup>2</sup> (down side) vector subspaces for a six-phase VSI.

**Table 1.** Switching table of DTC strategy.


**Figure 4.** *Cont*.

**Figure 4.** Block diagram of (**a**) the proposed sensorless DTC strategy; (**b**) ADRC.

### *4.2. ADRC in DTC*

To enhance the robustness of the DTC technique against external disturbances and measurement noises, the ADRC is proposed to replace with the conventional PI regulator in the outer speed control loop. The block diagram of ADRC is shown in Figure 4b. It consists of three main elements: (1) nonlinear differentiator; (2) extended state observer; (3) nonlinear control law.

In some industrial applications, the command values are changed as step function, which is not suitable for the control system because of a sudden jump of output and control signals. To solve this problem, the nonlinear differentiator is used, which makes a reasonable transient profile from command signals for tracking [26]. The nonlinear differentiator can be expressed by

$$\begin{cases} \upsilon\_1(k+1) = \upsilon\_1(k) + h\upsilon\_2(k) \\ \upsilon\_2(k+1) = \upsilon\_2(k) + h f\_1(\upsilon\_1(k) - \upsilon(k), \upsilon\_2(k), r\_0, h\_0) \end{cases} \tag{31}$$

where *f*<sup>1</sup> is a nonlinear function as

$$f\_1(v\_1(k), v\_2(k), r\_0, h\_0) = -\begin{cases} a(k)/h\_0 & |a(k)| \le r\_0 h\_0 \\ r\_0 \text{sign}(a(k)) & |a(k)| > r\_0 h\_0 \end{cases} \tag{32}$$

with

$$a(k) = \begin{cases} v\_2(k) + y\_0(k)/h\_0 & |a(k)| \le r\_0 h\_0^2\\ v\_2(k) + (a\_0(k) - r\_0 h\_0)/2 & |a(k)| > r\_0 h\_0^2 \end{cases}$$

$$y\_0(k) = v\_1(k) + h\_0 v\_2(k)$$

$$a\_0(k) = \sqrt{(r\_0 h\_0)^2 + 8r\_0|y\_0(k)|}$$

where *r*<sup>0</sup> and *h*<sup>0</sup> are the parameters of the nonlinear differentiator, and *h* is sampling period.

The extended state observer is an enhanced version of feedback linearization method to compensate the total disturbances of the system. Using this observer, the state feedback term can be estimated online; hence, it is an adaptive robust observer against model uncertainties and external disturbances. The extended state observer is represented as follows:

$$\begin{cases} z\_1(k+1) = z\_1(k) + h[z\_2(k) - \beta\_1 f\_2(e(k), a\_1, \delta\_1) + b\_0 u(k)] \\ z\_2(k+1) = z\_2(k) - h\beta\_2 f\_2(e(k), a\_1, \delta\_1) \\ e(k) = z\_1(k) - y(k) \end{cases} \tag{33}$$

where the nonlinear function *f*<sup>2</sup> is defined as

$$f\_2(\varepsilon(k), a, \delta) = \begin{cases} \varepsilon(k)/\delta^{1-a} & |\varepsilon(k)| \le \delta \\ |\varepsilon(k)|^a \text{sign}(\varepsilon(k)) & |\varepsilon(k)| > \delta \end{cases} \tag{34}$$

where *α*1, *δ*1, *β*1, *β*2, and *b*<sup>0</sup> are the parameters of the extended state observer.

The conventional PI controller is based on the linear combination of proportional and integral terms of error, which may degrade the performance of the DTC scheme. Different nonlinear combination of error can be presented to overcome this problem. In this paper, the following nonlinear control law is used:

$$\begin{cases} \boldsymbol{e}\_{1}(k) = \boldsymbol{v}\_{1}(k) - \boldsymbol{z}\_{1}(k) \\ \boldsymbol{u}\_{0}(k) = \beta\_{3} f\_{2}(\boldsymbol{e}\_{1}(k), \boldsymbol{u}\_{2}, \delta\_{2}) \\ \boldsymbol{u}(k) = \boldsymbol{u}\_{0}(k) - \boldsymbol{z}\_{2}(k) / b\_{0} \end{cases} \tag{35}$$

where *α*2, *β*3, and *δ*<sup>2</sup> are the parameters of nonlinear control law.

### **5. Experimental Validation**
