Sensorless Speed Control for SPMSM Using a Nonlinear Observer and Enhanced Super-Twisting ADRC

Abstract

In this article, a novel strategy called enhanced super-twisting active disturbance rejection control (ESTADRC), as well as a nonlinear observer (NOB), is used to implement a speed control scheme for permanent-magnet synchronous motors with intricate internal dynamics, and it exhibits nonlinearity and variable parameters. A new reaching law is formulated within a super-twisting sliding mode control (STSMC) framework, and a comprehensive procedure for finite convergence time analysis is provided. The convergence region of the state variables of the system is obtained using a Lyapunov function. ESTADRC is developed by integrating STSMC and linear active disturbance rejection control (LADRC), whereas the NOB is employed to estimate the motor’s position or angle value. Simulations demonstrated that the proposed approach is valid and effective compared with super-twisting active disturbance rejection control and LADRC.

1. Introduction

Sensorless velocity regulation is a widely used approach for managing permanent-magnet synchronous motors (PMSMs). In contrast to traditional induction motors, PMSMs provide superior efficiency, reduced noise levels, and better accuracy and dependability [1]. In conventional PMSMs, sensors are used to identify the rotor position and speed for precise control. However, sensorless PMSMs utilize motor controllers and algorithms to infer the rotor position and speed, thus not requiring sensors [2]. Despite the numerous benefits of sensorless velocity control, its practical application poses challenges. For example, a motor controller must precisely infer the rotor position and speed for high-precision control, and sensorless PMSMs require more intricate algorithms and controllers to guarantee motor stability and reliability [3].

To address these challenges, researchers have designed various controllers, including adaptive, predictive, robust, and optimal controllers. Among these, sliding mode control (SMC) has emerged as an important and robust technique for sensorless control [4]. Specifically, super-twisting sliding mode control (STSMC) is a type of SMC that not only ensures finite-time convergence but also effectively suppresses chattering [5]. STSMC was introduced to replace the proportional–integral (PI) adaptive law, which is not robust in sensorless speed control [6]. With these advantages, the control strategy of the super-twisting algorithm has been used in many motor speed control applications, such as 3-phase and 6-phase induction motors [7,8]. Additionally, the application of an extended state observer in active disturbance rejection control (ADRC) with proportional–integral–derivative enhancement has attracted significant interest owing to its ability to promptly estimate the “total disturbance” in a system, which comprises unmodeled internal dynamics and external disturbances [9,10]. Although ADRC has the advantage of being robust to disturbances and model parameter uncertainties, it has a complex structure and involves nonlinear terms and many adjustable coefficients, which limit its practical applications. In the early 2000s, Gao proposed the concept of linear ADRC (LADRC) with simplified and normalized parameters [11]. LADRC has been widely employed in the control of PMSMs [12,13,14,15,16], including sensorless PMSM control [17,18,19], owing to its advantages, such as strong robustness, simple parameter tuning, high adaptability, easy implementation, and excellent control performance.

In sensorless control, the accurate estimation of the rotor angle or position is critical for effective speed-loop control. The challenges of parameter variations and external disturbances have been addressed by the development of various improved sliding mode observers (SMOs) for rotor-angle estimation in the sensorless control of electric motors. These SMO variants, such as discrete-time SMO [20], new adaptive SMO [21], enhanced SMO [22], and full-order SMO [23], have several advantages, such as robustness to system uncertainties, fast response, and simplicity in design. Moreover, SMOs can provide effective control solutions without requiring precise system modeling or complex parameter tuning.

Lee et al. [24] proposed a nonlinear observer (NOB) for a surface-mounted PMSM (SPMSM) that uses the flux linkage as the state variable and eliminates its dependence on speed. The observer is characterized by simplicity and is therefore a suitable candidate for practical implementation. The NOB estimates the position ${\widehat{\theta }}_e$ by estimating $sin{\theta }_e$ and $cos{\theta }_e$ without requiring speed information, which eliminates the complexity caused by speed estimation errors. Moreover, it is easy to implement. Unlike Luenberger-type observers, it does not rely on speed information, thereby eliminating the complexities associated with speed estimation errors. Experimental results demonstrating the performance of the NOB are presented in [25,26].

Integrating the above advantages of LADRC, STSMC, and an NOB, this paper presents the implementation of speed control for sensorless PMSMs. Instead of sensors, the NOB is used to estimate the position, whereas STSMC and ADRC are combined to form a new control scheme called enhanced super-twisting active disturbance rejection control (ESTADRC) for speed control. The main contributions of this study are as follows:

• ESTADRC is designed via a new type of reaching law that enables PMSMs to achieve efficient sensorless speed control.
• The finite convergence time is derived from the new proposed reaching law.
• The convergence region of state variables “$\sigma$” and “$\dot{\sigma }$” of the uncertain system is derived via a Lyapunov function.
• Through a comparison with STADRC and LADRC, the theoretical results of the proposed system were verified by simulating a PMSM model.

The remainder of this paper is structured as follows: In Section 2, we introduce a mathematical model of the PMSM and present an NOB for estimating the rotor speed and position. Section 3 outlines the design of the enhanced reaching law, and Section 4 details the ESTADRC approach. In Section 5, we present numerical simulations of the speed control system and compare them with two other control strategies. Finally, the paper concludes with a summary of the main findings in Section 6.

2. Deisgn of the Nonlinear Observer

Typically, the mathematical model of an SPMSM in the $d-q$ reference frame can be expressed as [27]:

$$\begin{array}{c}{V}_d=R_s{i}_d+L_s\frac{d{i}_d}{dt}-p_n{\omega }_m{L}_s{i}_q,\hfill \\ V_q=R_s{i}_q+L_s\frac{d{i}_q}{dt}+p_n{\omega }_m{L}_s{i}_d+p_n{\omega }_m{\psi }_f,\hfill \\ J\frac{d{\omega }_m}{dt}=\frac{3p_n{\psi }_f}{2}{i}_q-T_L-B{\omega }_m,\hfill \end{array}$$

(1)

where the parameters are listed in

Table 1. Parameters of the dynamical model of PMSM.

System Parameters	Unit	Description
$V_d,V_q$	V	d-axis and q-axis stator voltages
$i_d,i_q$	A	d-axis and q-axis stator currents
${\omega }_m$	rad/s	electrical rotor angular velocity
$R_s$	$\mathrm{\Omega }$	stator resistance
$L_s$	$\mathrm{H}$	stator inductance
J	$\mathrm{kg}·{\mathrm{m}}^2$	rotor equivalent inertia
B	$\mathrm{N}·\mathrm{m}·\mathrm{s}/\mathrm{rad}$	viscous friction coefficient
${\psi }_f$	$\mathrm{V}·\mathrm{s}/\mathrm{rad}$	magnetic flux
$p_n$		number of poles
$T_L$	$\mathrm{N}·\mathrm{m}$	load torque

.

The motor equations can be reformulated in the $\alpha$–$\beta$ reference frame according to references [25,26] as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill V_{\alpha }=R_s{i}_{\alpha }+L_s{\dot{i}}_{\alpha }-{\omega }_e{\psi }_{f}sin{\theta }_e,\hfill \\ \hfill V_{\beta }=R_s{i}_{\beta }+L_s{\dot{i}}_{\beta }+{\omega }_e{\psi }_{f}cos{\theta }_e.\hfill \end{array}\right.\end{array}$$

(2)

Here, ${\theta }_e$ represents the electric angle. The state variable x can be defined as $x=\left[\begin{array}{c}{x}_1{x}_2\end{array}\right]$, where x represents the motor flux along the $\alpha$–$\beta$ axis. This enables us to expand the motor equations in matrix form.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill x_1=L_s{i}_{\alpha }+{\psi }_{f}cos{\theta }_e,\hfill \\ \hfill x_2=L_s{i}_{\beta }+{\psi }_{f}sin{\theta }_e.\hfill \end{array}\right.\end{array}$$

(3)

According to [27], the flux in the d–q axis can be represented as ${\psi }_d=L_s{i}_d+{\psi }_f$, ${\psi }_q=L_s{i}_q$. Transforming the d–q axis flux to the $\alpha$–$\beta$ reference frame yields the following expression for the SPMSM flux:

$$\left[\begin{array}{c}{\psi }_{\alpha }\\ {\psi }_{\beta }\end{array}\right]=\left[\begin{array}{cc}cos{\theta }_e& -sin{\theta }_e\\ sin{\theta }_e& cos{\theta }_e\end{array}\right]\left[\begin{array}{c}{L}_s{i}_d+{\psi }_f\\ L_s{i}_q\end{array}\right].$$

(4)

Expanding the matrices, we obtain the following expressions:

$$\begin{array}{cc}& \left\{\begin{array}{c}{\psi }_{\alpha }=L_s(\stackrel{i_{\alpha }}{\overbrace{i_{d}cos{\theta }_e-i_{q}sin{\theta }_e}})+{\psi }_{f}cos{\theta }_e\\ \hfill {\psi }_{\beta }=L_s(\underset{i_{\beta }}{\underbrace{i_{d}sin{\theta }_e+i_{q}cos{\theta }_e}})+{\psi }_{f}sin{\theta }_e\hfill \end{array}\right.\hfill \\ & \Rightarrow \left\{\begin{array}{c}{\psi }_{\alpha }=L_s{i}_{\alpha }+{\psi }_{f}cos{\theta }_e,\\ {\psi }_{\beta }=L_s{i}_{\beta }+{\psi }_{f}sin{\theta }_e.\end{array}\right.\hfill \end{array}$$

(5)

Let y be a column vector of size 2, given by $y={\left[\begin{array}{c}\hfill y_1{y}_2\hfill \end{array}\right]}^T.$

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill y_1=-R_s{i}_{\alpha }+V_{\alpha },\hfill \\ \hfill y_2=-R_s{i}_{\beta }+V_{\beta }.\hfill \end{array}\right.\end{array}$$

(6)

Applying (3), (4), and (6), we can derive the expression for $\dot{x}={\left[\begin{array}{c}\hfill {\dot{x}}_1{\dot{x}}_2\hfill \end{array}\right]}^T$ as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill {\dot{x}}_1=y_1=L_s{\dot{i}}_{\alpha }-{\omega }_e{\psi }_{f}sin{\theta }_e,\hfill \\ \hfill {\dot{x}}_2=y_2=L_s{\dot{i}}_{\beta }+{\omega }_e{\psi }_{f}cos{\theta }_e.\hfill \end{array}\right.\end{array}$$

(7)

Given the NOB equation $\dot{\tilde{x}}=\gamma \eta (\widehat{x})\left[{\psi }_f^2-|\eta (\widehat{x}){|}^2\right]/2$, where $\dot{\tilde{x}}=\dot{\widehat{x}}-\dot{x}$, $\gamma$ represents the observer gain, $\eta (\widehat{x})=\widehat{x}-L_s{i}_{\alpha \beta }$, and $|\eta (\widehat{x}){|}^2=\eta {\left({\widehat{x}}_1\right)}^2+\eta {\left({\widehat{x}}_2\right)}^2$, the values of $\gamma$ and $L_s$ should be adjusted based on practical requirements. By substituting the variables into the observer equation, we obtain

$$\begin{array}{c}\hfill \dot{\widehat{x}}=y+\frac{\gamma }{2}\left(\widehat{x}-L_s{i}_{\alpha \beta }\right)\left[{\psi }_f^2-{\left({\widehat{x}}_1-L_s{i}_{\alpha }\right)}^2-{\left({\widehat{x}}_2-L_s{i}_{\beta }\right)}^2\right].\end{array}$$

(8)

where $i_{\alpha \beta }$ means a vector defined as $i_{\alpha \beta }={\left[i_{\alpha }{i}_{\beta }\right]}^T$.

By expanding (8) on the $\alpha$–$\beta$ axis, where $\widehat{\eta }\left(x_1\right)={\widehat{x}}_1-L_s{i}_{\alpha }$ and $\widehat{\eta }\left(x_2\right)={\widehat{x}}_2-L_s{i}_{\beta }$, we can express the NOB equation as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill {\dot{\widehat{x}}}_1=y_1+\frac{\gamma }{2}\eta \left({\widehat{x}}_1\right)\left[{\psi }_f^2-\eta {\left({\widehat{x}}_1\right)}^2-\eta {\left({\widehat{x}}_2\right)}^2\right],\hfill \\ \hfill {\dot{\widehat{x}}}_2=y_2+\frac{\gamma }{2}\eta \left({\widehat{x}}_2\right)\left[{\psi }_f^2-\eta {\left({\widehat{x}}_1\right)}^2-\eta {\left({\widehat{x}}_2\right)}^2\right].\hfill \end{array}\right.\end{array}$$

(9)

By substituting ${\widehat{x}}_1-L_s{i}_{\alpha }={\psi }_{f}cos{\theta }_e$ of (9) and ${\widehat{x}}_2-L_s{i}_{\beta }={\psi }_{f}sin{\theta }_e$ from (3), the angle $\left({\theta }_e\right)$ can be calculated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\hfill cos{\theta }_e=\frac{1}{{\psi }_f}\eta \left({\widehat{x}}_1\right)=\frac{1}{{\psi }_f}\left({\widehat{x}}_1-L_s{i}_{\alpha }\right),\hfill \\ \hfill sin{\theta }_e=\frac{1}{{\psi }_f}\eta \left({\widehat{x}}_2\right)=\frac{1}{{\psi }_f}\left({\widehat{x}}_2-L_s{i}_{\beta }\right),\hfill \end{array}\right.\end{array}$$

(10)

which implies that the electrical angle of the motor can be computed as

$$\begin{array}{c}\hfill {\widehat{\theta }}_e=atan2\left({\widehat{x}}_2-L_s{i}_{\beta },{\widehat{x}}_1-L_s{i}_{\alpha }\right).\end{array}$$

(11)

where ${\widehat{\theta }}_e$ represents the estimation of ${\theta }_e$. It should be noted that even when the denominator approaches zero, the arctangent function remains insensitive. The error dynamics can then be derived directly from (3) to (8) due to $\tilde{x}\equiv \widehat{x}-x$ as follows:

$$\begin{array}{c}\hfill \dot{\tilde{x}}=-\left\{\frac{\gamma }{2}{\parallel \tilde{x}\parallel }^2+\gamma {\psi }_f\left[{\tilde{x}}_{1}cos{\theta }_e\left(t\right)+{\tilde{x}}_{2}sin{\theta }_e\left(t\right)\right]\right\}\left\{\tilde{x}+{\psi }_f\left[\begin{array}{c}cos{\theta }_e\left(t\right)\\ sin{\theta }_e\left(t\right)\end{array}\right]\right\}.\end{array}$$

(12)

In [25], it is demonstrated that (12) exhibits the following stability properties:

• Global stability: For any given speeds, the disk defined as the set of $\{\tilde{x}\in {\mathbb{R}}^2\mid \parallel \tilde{x}\parallel \le 2{\psi }_f\}$ exhibits global attractiveness. This indicates that all trajectories of (12) will converge toward this disk.
• Local stability under persistent excitation: The zero equilibrium of (12) achieves exponential stability if certain constants T and $\Delta >0$ exist such that $\left({\int }_t^{t+T}{\omega }_e^2\left(s\right)ds\right)/T\ge \Delta .$
• Constant nonzero speed: In the case of a constant speed that meets $|{\omega }_e|>\gamma {\psi }_f^2/4,$ the origin represents the sole equilibrium point of (12), and it is globally asymptotically stable. It is essential to take note of the influence of the free adaptation gain $\gamma$ on the lower bound.

3. Design of the Enhanced Reaching Law

A novel and enhanced quick-reaching law is proposed to achieve rapid convergence to the sliding-mode surface during the entire reaching process. The improved quick-reaching law is formulated as follows:

$$\begin{array}{c}\hfill \dot{\sigma }=-k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{|\sigma |}^{b}sgn\left(\sigma \right)-k_2\left(c^{-|\sigma |}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau ,\end{array}$$

(13)

where $a>0$, $k_1>0$, $k_2>0$, $0<b<1$, and $c=1+k_1/k_2$. When the state of the system is far from the sliding surface, that is, $|\sigma |>1$, the second term in (13) becomes dominant. Its rate of change is higher than that of the power function in the first term, which accelerates the reaching process. Similarly, when the system state approaches the sliding surface, i.e., when $|\sigma |<1$, the first term in (13) becomes dominant and aids the system in arriving at the sliding surface with a slightly higher velocity than the power function in the second term. By combining the effects of both terms, the reaching law defined in (13) enhances the dynamic characteristics of the system throughout the reaching process.

Remark 1.

When b is set to 1/2, (13) is a typical super-twisting law. Because the range of b is set to be greater than 0 and less than 1, $b=1/2$ is reasonable in this paper.

3.1. Finite Convergence Time Analysis

Theorem 1.

In terms of the novel enhanced reaching law described in (12), it is worth noting that the state variables σ and $\dot{\sigma }$ can converge to the equilibrium point (0, 0) in a finite time, implying that the state variables will eventually reach $\dot{\sigma }=\sigma =0$ after a finite convergence time.

Proof. 

When the initial state $\sigma \left(0\right)>1$, the motion process from the initial state to the sliding mode can be divided into two stages: $\sigma \left(0\right)\to \sigma =1$ and $\sigma =1\to \sigma =0$. The duration of each stage can be calculated separately. In the initial stage, where $\sigma \left(0\right)>1$ and $\sigma >1$, the second term in (13) is more significant than the first term because of the conditions of $c=1+k_1/k_2$ and $k_2(c^{-|\sigma |}-1)>k_1{e}^{\left|\sigma /(\sigma +a)\right|}|\sigma {|}^b$ [28]. Therefore, (13) can be simplified by disregarding the first term as follows:

$$\begin{array}{c}\frac{d\sigma }{dt}\cong -k_2\left(c^{-|\sigma |}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau \hfill \\ \hspace{1em} =-k_2\left(c^{-\sigma }-1\right){\int }_0^{t}d\tau \hfill \end{array}$$

(14)

We integrate (14) with respect to time t on the right-hand side and $\sigma$ on the left-hand side.

$$\begin{array}{c}\underset{\sigma \left(0\right)}{\overset{1}{\int }}\frac{1}{k_2(1-c^{-\sigma })}d\sigma =\underset{0}{\overset{t_1}{\int }}\underset{0}{\overset{{\tau }_1}{\int }}d{\tau }_{2}d{\tau }_1,\hfill \\ \underset{\sigma \left(0\right)}{\overset{1}{\int }}\frac{1}{k_{2}lnc}d(ln(c^{-\sigma }-1))=\underset{0}{\overset{t_1}{\int }}\underset{0}{\overset{{\tau }_1}{\int }}d{\tau }_{2}d{\tau }_1,\hfill \end{array}$$

(15)

Therefore, the convergence time of the first stage ($t_1$) is calculated as follows:

$$\begin{array}{c}\hfill t_1=\sqrt{2\frac{ln\left(c^{-1}-1\right)-ln\left(c^{\sigma \left(0\right)}-1\right)}{k_{2}lnc}}.\end{array}$$

(16)

In the second stage of the motion process, when $\sigma$ shifts from 1 to 0, the conditions $c=1+k_1/k_2$ and $k_2(c^{-|\sigma |}-1)<k_1{e}^{\left|\sigma /(\sigma +a)\right|}|\sigma {|}^b$ [28] suggest that the first term in (1) is more significant than the second term. Therefore, the second term can be ignored. Consequently, (13) can be approximated as follows:

$$\begin{array}{c}\dot{\sigma }\cong -k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{|\sigma |}^{b}sgn\left(\sigma \right)\hfill \\ \hspace{1em}=-k_1{e}^{\frac{\sigma }{\sigma +a}}{\sigma }^b.\hfill \end{array}$$

(17)

By applying the same integration method used in (14)–(17), the results of (18) can be determined as follows.

$$\begin{array}{c}\hfill \underset{0}{\overset{1}{\int }}\frac{1}{k_1}{e}^{-\frac{\sigma }{\sigma +a}}{\sigma }^{-b}d\sigma <\underset{0}{\overset{1}{\int }}\frac{1}{k_1}{\sigma }^{-b}d\sigma =\underset{0}{\overset{t_2}{\int }}d\tau .\end{array}$$

(18)

Therefore, the convergence time of the second stage can be obtained using the following expression.

$$\begin{array}{c}\hfill t_2=\frac{1}{k_1(1-b)}.\end{array}$$

(19)

The convergence time of the two stages is calculated under the assumption that the minor term can be neglected. Thus, the total convergence time ${t_{\sigma }}_{\left(0\right)>1}$ of the reaching law can be computed as follows:

$$\begin{array}{cc}\hfill & \hfill {t_{\sigma }}_{\left(0\right)>1}<t_1+t_2=\sqrt{2\frac{ln\left(c^{-1}-1\right)-ln\left(c^{\sigma \left(0\right)}-1\right)}{k_{2}lnc}}+\frac{1}{k_1(1-b)}.\hfill \end{array}$$

(20)

Similarly, when the initial state is $\sigma \left(0\right)<-1$, the motion process from the initial state to the sliding mode can also be divided into two stages: $\sigma \left(0\right)\to \sigma =-1$ and $\sigma =-1\to \sigma =0$. The motion times for these two stages can also be calculated separately. During the first stage, $\sigma \left(0\right)\to \sigma =-1$, we observe that the second term in (13) dominates over the first term, and thus, the first term can be ignored. Therefore, we can write

$$\begin{array}{c}\frac{d\sigma }{dt}\cong -k_2\left(c^{-|\sigma |}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau \hfill \\ \hspace{1em} =-k_2\left(c^{\sigma }-1\right){\int }_0^{t}d\tau .\hfill \end{array}$$

(21)

By integrating both sides of (21) and (22), we obtain

$$\begin{array}{c}\underset{\sigma \left(0\right)}{\overset{-1}{\int }}\frac{1}{k_2(1-c^{\sigma })}d\sigma =\underset{0}{\overset{t_1^{{}^{\prime }}}{\int }}\underset{0}{\overset{{\tau }_1}{\int }}d{\tau }_{2}d{\tau }_1,\hfill \\ \underset{\sigma \left(0\right)}{\overset{-1}{\int }}\frac{1}{k_{2}lnc}d(ln(1-c^{-\sigma }))d\sigma =\underset{0}{\overset{t_1^{{}^{\prime }}}{\int }}\underset{0}{\overset{{\tau }_1}{\int }}d{\tau }_{2}d{\tau }_1,.\hfill \end{array}$$

(22)

Thus, the corresponding convergence time can be obtained using the following expression:

$$\begin{array}{c}\hfill t_1^{{}^{\prime }}=\sqrt{2\frac{ln\left(1-c\right)-ln\left(1-c^{\sigma \left(0\right)}\right)}{k_{2}lnc}}\end{array}$$

(23)

In the interval in which $\sigma$ ranges from −1 to 0, the significance of the first term in (13) exceeds that of the second term. Therefore, the contribution of the second term can be neglected, which results in a simplified equation.

$$\begin{array}{c}\hfill \dot{\sigma }\cong -k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{|\sigma |}^{b}sgn\left(\sigma \right)=k_1{e}^{-\frac{\sigma }{\sigma +a}}{(-\sigma )}^b.\end{array}$$

(24)

Integrating both sides of (24), we obtain the following equation:

$$\begin{array}{c}\hfill \underset{-1}{\overset{0}{\int }}\frac{1}{k_1}{e}^{\frac{\sigma }{\sigma +a}}{(-\sigma )}^{-b}d\sigma <\underset{-1}{\overset{0}{\int }}\frac{1}{k_1}{(-\sigma )}^{-b}d\sigma =\underset{0}{\overset{t_2^{{}^{\prime }}}{\int }}d\tau \end{array}$$

(25)

Thus, the convergence time of the second stage can be calculated as

$$\begin{array}{c}\hfill t_2^{{}^{\prime }}=\frac{1}{k_1(1-b)}.\end{array}$$

(26)

Similarly, when estimating the convergence time for the two stages mentioned above, insignificant terms are neglected. Therefore, the overall convergence time for the reaching law, denoted as ${t_{\sigma }}_{\left(0\right)<-1}$, can be expressed as

$$\begin{array}{c}\hfill {t_{\sigma }}_{\left(0\right)<-1}<t_1^{{}^{\prime }}+t_2^{{}^{\prime }}=\sqrt{2\frac{ln\left(1-c\right)-ln\left(1-c^{\sigma \left(0\right)}\right)}{k_{2}lnc}}+\frac{1}{k_1(1-b)}.\end{array}$$

(27)

In summary, the system can reach the sliding mode within a finite time. Furthermore, (13) indicates that when $\sigma$ equals 0, the derivative of $\sigma$ becomes 0. This implies that the system reaches sliding mode with zero velocity, which can effectively reduce chattering. Theorem 1 is completed. □

3.2. Stability Analysis via Lyapunov Function

If the system is subject to uncertain and bounded disturbances, the state variables $\sigma$ and $\dot{\sigma }$ defined by (13) can converge to a region near the equilibrium point $(0,0)$ within a finite amount of time.

Lemma 1

([29]). Consider a continuous function $g:{\mathbb{R}}^n\to {\mathbb{R}}^n$, where n is a positive integer defined on an open neighborhood N of the origin such that $g\left(0\right)=0$, and g is locally Lipschitz on $N\setminus \left\{0\right\}$. Let x be a vector of N and $\dot{x}=g\left(x\right)$. Suppose a continuous function $S:N\to \mathbb{R}$ exists that satisfies the following conditions:

• S is positive definite;
• $\dot{S}$is negative on$N\setminus \left\{0\right\}$;
• Positive constants k and $\beta \in (0,1)$, as well as a neighborhood $D\subset N$ of the origin, exist such as $\dot{S}+k{S}^{\beta }\le 0$ on $D\setminus \left\{0\right\}$. Thus, the origin is a finite-time-stable equilibrium of $\dot{x}=g\left(x\right)$.

Theorem 2.

The novel enhanced reaching law with disturbance is expressed as follows:

$$\begin{array}{c}\hfill \dot{\sigma }=-k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{\left|\sigma \right|}^{b}sgn\left(\sigma \right)-k_2\left(c^{-\left|\sigma \right|}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau +d\left(t\right),\end{array}$$

(28)

where a bounded disturbance $d\left(t\right)$ exists such that $|d(t)|\le \delta$, with $\delta >0$ being a positive constant. The state variables σ and $\dot{\sigma }$ of the uncertain system converge separately to the following regions:

$$\begin{array}{c}\hfill \left|\sigma \right|\le {log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right)\end{array}$$

(29)

$$\begin{array}{c}\hfill \left|\dot{\sigma }\right|\le k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{\left({log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right)\right)}^b+2\delta .\end{array}$$

(30)

Proof. 

The candidate Lyapunov function can be set as follows:

$$\begin{array}{c}\hfill S=\frac{1}{2}{\sigma }^2.\end{array}$$

(31)

The time derivative of S can be obtained as follows:

$$\begin{array}{c} \dot{S}=\sigma ·\dot{\sigma }\hfill \\ \hspace{1em}=\sigma \left[-k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{|\sigma |}^{b}sgn\left(\sigma \right)-k_2\left(c^{-|\sigma |}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau +d\left(t\right)\right]\hfill \\ \hspace{1em}=-k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{|\sigma |}^{b+1}-k_2\left(c^{-|\sigma |}-1\right)|\sigma |·t+|\sigma |d\left(t\right)\hfill \\ \hspace{1em}\le -k_1{e}^{\frac{\sigma }{\sigma +a}\mid }{|\sigma |}^{b+1}-k_2\left(c^{-|\sigma |}-1\right)|\sigma |·t+|\sigma \parallel d\left(t\right)|,\hfill \end{array}$$

(32)

Subsequently,

$$\begin{array}{c}\hfill \dot{S}\le -k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{\left|\sigma \right|}^{b+1}-|\sigma |\left[k_2(c^{-|\sigma |}-1)·t-\delta \right].\end{array}$$

(33)

When $k_2(c^{-|\sigma |}-1)·t-\delta \ge 0,$ then $\dot{S}\le 0$, which indicates that

$$\begin{array}{c}\hfill \left|\sigma \right|\le {log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right).\end{array}$$

(34)

Moreover, the state variable $\dot{\sigma }$ of the system can converge to the following region within a finite time.

$$\begin{array}{c}|\dot{\sigma }|={\left.\left|-k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}\right|\sigma \right|}^{b}sgn\left(\sigma \right)|+|-k_2\left(c^{-|\sigma |}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau |+|d\left(t\right)|\hfill \\ \hspace{1em}\le k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{\left({log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right)\right)}^b+k_2\left(c^{-{log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right)}-1\right)·t+\delta \hfill \\ \hspace{1em}\le k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{\left({log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right)\right)}^b+k_2\left(\frac{\delta +t·k_2}{t·k_2}-1\right)·t+\delta \hfill \\ \hspace{1em}=k_1{e}^{\left|\frac{\sigma }{\sigma +a}\right|}{\left({log}_c\left(\frac{t·k_2}{\delta +t·k_2}\right)\right)}^b+2\delta .\hfill \end{array}$$

(35)

Therefore, Theorem 2 is proved. □

4. Design of ESTADRC

In an SPMSM, as the d-axis inductance and q-axis inductance are equal, the electromagnetic torque equation of the motor is $T_e=1.5p_n{\psi }_f{i}_q$, where $T_e$ is the electromagnetic torque output of the motor.

$$\begin{array}{c}J\frac{\mathrm{d}{\omega }_m}{\mathrm{d}t}=T_e-T_L-B{\omega }_m,\hfill \\ \hspace{1em}\hspace{1em}=1.5p_n{\psi }_f{i}_q-T_L-B{\omega }_m.\hfill \end{array}$$

(36)

where ${\omega }_m=N_r\pi /30$, and $N_r$ is the reference rotation speed (Figure 1). Figure 1 contains an entire control block diagram of this study. For convenience in designing the disturbance observer, the motion equation in (36) is transformed into the following equation:

$$\begin{array}{c}\hfill {\dot{\omega }}_m=\frac{1.5p_n{\psi }_f}{J}{i}_q-\frac{B}{J}{\omega }_m-\frac{T_L}{J}.\end{array}$$

(37)

The inertia and damping of the motor’s own bearings do not change. Instead, the inertia and damping of the load driven by the bearings vary randomly with changes in the load environment. Therefore, by adding a disturbance to (37), where the disturbance f is a variable that changes with the load, we can rewrite (37) as follows:

$$\begin{array}{c}\hfill {\dot{\omega }}_m=b_0{i}_q^{*}+f.\end{array}$$

(38)

where $b_0$ and f denote the gain and disturbance, respectively. Let us design an LESO in Figure 2 with adjustable coefficients ${\omega }_o$ and $b_0$. The $sat(.)$ means a saturation function in Figure 2.

$$\begin{array}{c}\left\{\begin{array}{c}{\dot{\widehat{\omega }}}_m=\widehat{f}-2{\omega }_o\left({\widehat{\omega }}_m-{\omega }_m\right)+b_0{i}_q^{*},\hfill \\ \dot{\widehat{f}}=-{\omega }_0^2\left({\widehat{\omega }}_m-{\omega }_m\right).\hfill \end{array}\right.\hfill \end{array}$$

(39)

To track the reference rotation speed, the sliding surface is designed as follows:

$$\sigma ={\omega }_m^{ref}-{\widehat{\omega }}_m.$$

(40)

In addition, the proposed reaching law is provided in (13). This output is used as the reference current $i_q^{*}$ for the current loop using (13) and (38)–(40) as follows:

$$\begin{array}{c}\hfill i_q^{*}=\frac{{\omega }_c}{b_0}\left(-k_1{e}^{\left|\frac{\sigma }{\sigma +a\mid }\right|}{|\sigma |}^{b}sgn\left(\sigma \right)-k_2\left(c^{-|\sigma |}-1\right){\int }_0^{t}sgn\left(\sigma \right)d\tau -{\widehat{\omega }}_m-\widehat{f}\right),\end{array}$$

(41)

where a, ${\omega }_c$, $b_0$, b, $k_1$, and $k_2$ are positive constants, and $c=1+k_1/k_2$. To demonstrate the good performance of our designed ESTADRC, LADRC and STADRC were selected as comparative controllers. The outputs for LADRC and STADRC are as follows [30,31]:

$$i_{q\_LADRC}^{*}=\frac{{\omega }_c}{b_0}\left({\omega }_m^{ref}-\widehat{f}\right),$$

(42)

and

$$i_{q\_STADRC}^{*}=\frac{{\omega }_c}{b_0}\left(-k_1{\left|\sigma \right|}^b\mathrm{sgn}\left(\sigma \right)-k_2\underset{0}{\overset{t}{\int }}\mathrm{sgn}\left(\sigma \right)d\tau -{\widehat{\omega }}_m-\widehat{f}\right).$$

(43)

Remark 2.

To ensure fairness in comparison, except for the unique parameters of these three controllers (LADRC, STADRC, and ESTADRC), all other parameters adopt the same parameter values as shown in

Table 2. Parameter values of LADRC, STADRC and ESTADRC.

Parameters	ESTADRC	STADRC	LADRC
$\eta$	30	30	30
$\gamma$	500,000	500,000	500,000
${\omega }_0$	5	5	5
${\omega }_c$	150	150	150
$b_0$	8000	8000	8000
a	40	-	-
b	1/2	-	-
$k_1$	20	20	-
$k_2$	10	10	-

.

5. Simulation

In this section, we present simulation results to demonstrate the effectiveness of the proposed control strategy. The simulation parameters of the model and controllers used in this study are summarized in Table 2 and

Table 3. Parameter values of the PMSM.

Parameters	Value	Unit
Resistance   $R_s$	2.875	$\mathrm{\Omega }$
Inductance   $L_s$	8.5	$\mathrm{mH}$
Flux linkage   ${\phi }_f$	0.175	$\mathrm{Wb}$
Inertia   J	0.0003	$\mathrm{kg}·{\mathrm{m}}^2$
Damping coefficient   $B$	0.0008	$\mathrm{N}·\mathrm{m}·\mathrm{s}/\mathrm{r}$
Pole pairs   $p_n$	4
Reference speed   $N_r$	1000	$\mathrm{r}/\min$

, respectively. The pulse width modulation (PWM) switching frequency is set to be 10 kHz and the sampling period is 100 $\mathsf{\mu }$s in this simulation. In this paper, we engaged in an iterative process of trial and error to adjust the control parameters within the simulation. With meticulous consideration of factors such as steady-state error, convergence speed, and speed response stability, we systematically arrived at the selection of the final control parameters. Figure 3 shows a comparison of the times required for the LADRC, STADRC, and ESTADRC systems to achieve the desired rotational speed.

Table 4. Performance index about settling time for speed regulation.

Controllers	Settling Time
LADRC	0.15 s
STADRC	0.9 s
ESTADRC	0.075 s

demonstrates the performance index about settling time for speed regulation. As depicted in Figure 3a, ESTADRC achieved the desired speed of 220 r/min in the shortest time of 0.075 s, whereas the LADRC and STADRC required 0.15 and 0.09 s, respectively. Hence, this comparison indicated that ESTADRC outperformed LADRC and STADRC. Figure 3b shows the corresponding current variations, which represent the changes in the control inputs.

Figure 4 presents a comparison of the speed tracking performances of LADRC, STADRC, and ESTADRC. The desired rotational speed changed from 200 to −500 r/min at 0.55 s; then, it changes from −500 to 500 r/min at 1.45 s, from 500 to −490 r/min at 1.90 s, and finally from −490 to 1200 r/min at 24 s before decreasing to 20 r/min. Figure 4a depicts the tracking results of the three controllers, and Figure 4(a-1–a-3) provide zoomed-in views over time. These figures demonstrate that ESTADRC outperformed LADRC and STADRC in terms of tracking performance. Figure 4b shows the corresponding changes in the current, which represents the control input.

Figure 5 shows the speed response of the motor to a sudden load of 3 $\mathrm{N}·\mathrm{m}$ applied at 1 s. As shown in Figure 5a, ESTADRC achieved a smaller speed deviation from the desired speed than LADRC and STADRC. Figure 5(a-1) shows a zoomed-in view of Figure 5a, which indicates that ESTADRC could return to the desired speed of 200 r/min at 1.036 s, whereas STADRC and LADRC required 1.04 and 1.1 s, respectively. The corresponding changes in the current are shown in Figure 5b.

6. Conclusions

In this article, we investigate the use of NOB and ESTADRC for the sensorless speed control of an SPMSM. We propose a novel reaching approach based on the conventional super-twisting reaching law to enhance the speed control performance and provide a detailed procedure for finite convergence time analysis. A Lyapunov function is used to derive the convergence region of the state variables and of the uncertain system. We implement the ESTADRC and NOB for speed control and angle estimation, respectively. The results demonstrate the superiority of the ESTADRC over LADRC and STADRC.