Super-Twisting Algorithm-Based Sensorless Sliding-Mode Control for PMSM

Abstract

To address the issues of sluggish dynamic response, significant steady-state fluctuations, and poor disturbance rejection associated with traditional proportional–integral (PI) and conventional speed control methods, a novel sensorless sliding-mode speed control strategy for permanent magnet synchronous motors (PMSMs) based on the super-twisting algorithm (STA) is proposed. First, an advanced sliding-mode speed controller is designed by integrating an integral nonsingular fast terminal sliding-mode surface with the STA, thereby enhancing the dynamic response and transient stability of the PMSM under speed variations. Subsequently, to mitigate inherent sliding-mode chattering, a novel load torque observer is developed. This observer continuously feeds forward real-time load estimates to the speed controller, which substantially improves the system’s robustness against external disturbances. Furthermore, to eliminate the reliance on mechanical sensors and ensure reliable operation across diverse scenarios, an improved sliding-mode observer (SMO) incorporating the STA is utilized to achieve more precise rotor position and speed estimation. Finally, an experimental platform is established to conduct comprehensive variable-speed and variable-load tests on the PMSM. Experimental results demonstrate that the proposed method improves the dynamic response and disturbance immunity of the PMSM by 58.33% and 71.75%, respectively, while reducing steady-state fluctuations by 33.33%. These results demonstrate the effectiveness of the proposed sensorless sliding-mode control strategy and show improved speed regulation performance for PMSM drives.

1. Introduction

Permanent magnet synchronous motors (PMSMs) have been widely employed in new-energy systems, robotics, aerospace, and other high-performance applications owing to their high efficiency, simple structure, high power density, and wide speed regulation capability [1,2]. With the continuous expansion of application scenarios, increasingly stringent requirements have been imposed on drive performance, particularly in terms of stability, transient response speed, and robustness [3].

However, the PMSM control system is inherently multivariable, strongly coupled, and nonlinear, and is subject to multiple uncertainties, including internal parameter variations and external load disturbances [4]. These uncertainties significantly affect speed-tracking performance and may deteriorate system stability under varying operating conditions. Therefore, achieving reliable speed regulation requires control strategies that explicitly account for both internal parameter uncertainties and external disturbances [5,6,7].

In practical applications, proportional-integral (PI) controllers remain widely adopted for PMSM speed regulation due to their simplicity and ease of implementation [8,9]. Nevertheless, PI controllers rely heavily on accurate system modeling and fixed parameter tuning. Their performance is easily degraded by nonlinear dynamics and time-varying operating conditions, resulting in limited robustness and reduced adaptability to complex working environments.

To address these limitations, various nonlinear control strategies have been proposed, including neural network control [10], adaptive control [11], model predictive control [12], fuzzy control [13,14], and sliding-mode control (SMC) [15,16]. These approaches enhance PMSM control performance from different perspectives. Among them, SMC has attracted considerable attention in engineering applications due to its simple structural design, reduced dependence on precise system modeling, and strong robustness against parameter variations and external disturbances. Motor speed regulation represents one of its most important application domains.

Sliding-mode control (SMC) is a representative variable-structure control strategy in which a sliding surface is designed to drive the system states toward and subsequently along a predefined manifold. Owing to its inherent robustness against parameter variations and external disturbances, SMC has been widely applied in motor drive systems. However, the discontinuous switching nature of conventional SMC introduces high-frequency control signals, which may cause severe chattering phenomena, leading to mechanical vibration, increased losses, and degradation of control accuracy.

To alleviate the chattering problem while preserving robustness, various improved SMC schemes have been developed, which can generally be classified into first-order, second-order, and higher-order sliding-mode methods according to the order of the sliding-variable derivatives.

For first-order SMC, numerous convergence-law and sliding-surface modifications have been proposed. In [17], a hyperbolic sinusoidal function with a variable boundary-layer thickness was introduced to achieve smoother switching behavior, and fuzzy logic was employed to adjust the function coefficients online, thereby attenuating chattering and improving both dynamic and steady-state performance of the PMSM drive. A fixed-time dynamic convergence law combined with a nonsingular fast terminal sliding surface was presented in [18] to enhance convergence speed while reducing oscillations. In [19], a composite convergence law integrating terminal and exponential proportional terms was developed, and an extended disturbance observer was incorporated to improve disturbance rejection capability. An integral sliding surface together with an improved convergence law introducing state variables was designed in [20], further enhancing system response performance.

Among second-order sliding-mode approaches, the super-twisting algorithm (STA) is the most widely adopted due to its relatively simple structure and effective chattering attenuation. In [21], an STA-based PMSM controller was proposed, and a parameter-tuning method grounded in high-gain theory was developed to improve robustness. An improved STA incorporating linear correction terms was introduced in [22], achieving enhanced convergence speed and stability. Furthermore, Ref. [23] combined a segmented integral terminal sliding surface with the STA to realize improved tracking performance and reduced chattering in PMSM speed control.

Higher-order sliding-mode techniques theoretically provide superior chattering suppression compared with lower-order counterparts. However, they often involve significantly increased computational complexity and typically require high-resolution encoders and high-precision sensing devices, which restricts their practical implementation in industrial motor drives. To simplify higher-order sliding-mode structures, a relay-polynomial algorithm was proposed in [24] to reduce controller complexity.

In summary, first-order SMC features structural simplicity and fast response but suffers from residual chattering and limited robustness. Higher-order SMC offers excellent theoretical performance in chattering suppression but entails substantial computational burden and demanding hardware requirements. Second-order SMC, particularly the super-twisting algorithm, provides a favorable compromise between robustness, convergence speed, and implementation complexity. Therefore, this paper adopts a second-order sliding-mode framework to design the speed controller and load observer.

In sliding-mode-based PMSM control systems, accurate rotor position and speed information is a prerequisite for achieving high-performance regulation. Conventionally, such information is obtained using mechanical sensors mounted on the rotor shaft. However, the use of physical sensors increases system cost, volume, and mechanical complexity, while potentially reducing reliability and robustness. Consequently, sensorless control technology has attracted considerable research interest in recent years [25].

Sensorless control strategies estimate rotor position and speed from measurable electrical quantities, such as stator voltages and currents [26]. Existing PMSM sensorless techniques mainly include high-frequency signal injection methods [27], Kalman filtering approaches [28], and sliding-mode-observer (SMO)-based schemes. Among these methods, SMO demonstrates strong potential due to its inherent robustness, disturbance rejection capability, and relatively simple algorithmic structure.

The SMO typically estimates the back electromotive force (EMF) by exploiting the mismatch between the measured stator current and the observer output, and subsequently extracts rotor position information from the estimated EMF. However, conventional SMOs often rely on low-pass filters to obtain the EMF components, which introduces phase delay and residual chattering, thereby degrading estimation accuracy. To overcome these limitations, numerous improvements have been proposed, focusing on enhanced convergence laws and reduced sensitivity to parameter variations.

For instance, a sliding-mode predictive observer was introduced in [29] to achieve fast convergence and reduced chattering without requiring manual tuning of sliding gains, while mitigating the influence of motor parameter variations. In [30], a sigmoid switching function combined with fuzzy adaptive tuning was employed to modify convergence characteristics and suppress chattering. A novel convergence-law-based SMO was developed in [31], enabling different convergence behaviors during distinct stages of the sliding process to improve dynamic response and reduce oscillations. Moreover, an adaptive phase-locked loop (PLL) was proposed in [32] to dynamically adjust bandwidth and minimize rotor position estimation error.

In summary, although existing speed control and sensorless strategies have achieved notable improvements in robustness and estimation accuracy, challenges remain in simultaneously enhancing response speed and smoothness while maintaining implementation simplicity. To address these issues, this paper proposes a nonsingular terminal sliding-mode speed control scheme based on an integral-type fast terminal sliding surface combined with a fast super-twisting algorithm. Furthermore, a corresponding load torque observer and a super-twisting-based sensorless control strategy are developed to improve disturbance rejection and estimation performance.

This paper develops an integrated STA-based sensorless sliding-mode control framework for PMSM drives by combining speed regulation, load torque observation, and sensorless estimation. Since the focus is on the speed-loop and observer design, a conventional PI current controller is retained to ensure stable current tracking and to avoid introducing additional current-loop optimization effects into the performance comparison [33].

The remainder of this paper is organized as follows. Section 2 presents the proposed sliding-mode speed controller. Section 3 describes the load torque observer design. Section 4 introduces the super-twisting-based sensorless control strategy. Experimental validation is provided in Section 5, and conclusions are drawn in Section 6.

2. Sliding-Mode Speed Controller Design

This study considers a surface-mounted permanent magnet synchronous motor (SPMSM). For analytical tractability, magnetic saturation, eddy-current losses, and hysteresis effects are neglected. It is further assumed that the stator windings generate sinusoidally distributed air-gap flux, and that rotor motion induces a corresponding sinusoidal back electromotive force (EMF).

To address the limitations of conventional first-order sliding-mode control, particularly the residual chattering and limited robustness, a second-order super-twisting sliding-mode algorithm is adopted to optimize the design of the speed controller. The super-twisting framework enhances convergence performance while reducing control discontinuity, thereby improving smoothness and robustness in PMSM speed regulation.

For the SPMSM with $i_d=0$ control, the electromagnetic torque is mainly determined by the $q$-axis current. Therefore, the mechanical dynamics can be expressed as follows:

$${\dot{\omega }}_m={\displaystyle \frac{3P_n{\psi }_f}{2J}}{i}_q-{\displaystyle \frac{B}{J}}{\omega }_m-{\displaystyle \frac{T_L}{J}}$$

(1)

Thus, the $q$-axis reference current $i_q^{*}$ is regarded as the control input of the speed loop, while the load torque $T_L$ is treated as an external disturbance. The control objective is to make ${\omega }_m$ track the reference speed ${\omega }_m^{*}$ with fast response, small steady-state fluctuation, and strong robustness against load disturbance.

2.1. Conventional Sliding-Mode Control

Sliding-mode control (SMC) is a nonlinear control method that forces the system states to reach and subsequently move along a predefined trajectory, known as the sliding surface, thereby ensuring system stability.

The conventional sliding surface is defined as a linear function given by the following expression:

$$s_1=c_1{x}_1+x_2$$

(2)

where $x_1$ and $x_2$ denote the speed-tracking error and its derivative, respectively.

The conventional exponential reaching law is given by the following:

$${\dot{s}}_1=-{\epsilon }_1\mathrm{sgn}(s_1)-q_1{s}_1$$

(3)

In the formulation, $c_1$ denotes the error weighting gain, ${\epsilon }_1$ represents the switching gain coefficient, and $q_1$ denotes the exponential reaching-law coefficient. Under the i_d = 0 control strategy, the q-axis reference current $i_q^{*}$ is given by the following:

$$i_q^{*}={\displaystyle \frac{2J}{3P_n{\psi }_f}}\left[-c_1\dot{\omega }+{\displaystyle \frac{B}{J}}\omega +{\displaystyle \frac{T_L}{J}}+{\epsilon }_1\mathrm{sgn}(s_1)+q_1{s}_1\right]$$

(4)

In the equation, ω denotes the actual rotor speed, i_q represents the quadrature-axis current, P_n is the number of pole pairs in the PMSM, ψ_f indicates the permanent magnet flux linkage, J is the rotational inertia, B is the damping coefficient, T_L corresponds to the load torque, and sgn(·) defines the signum function.

In conventional sliding-mode control, the convergence behavior of the linear sliding surface is determined by the reaching law, as shown in Equation (3). The exponential reaching law introduces the convergence coefficient $q_1$ to improve the convergence rate under the guidance of the linear sliding surface. However, due to the inherent characteristics of the exponential function, the system state cannot converge to zero within finite time.

Moreover, under the exponential reaching law, the presence of the discontinuous switching term ${\epsilon }_1\mathrm{s}\mathrm{g}\mathrm{n}\left(s_1\right)$ prevents the state trajectory from strictly remaining on the sliding surface. Instead, it oscillates in the vicinity of the surface, resulting in the well-known chattering phenomenon.

2.2. Design of a Sliding-Mode Speed Controller Based on STA

To rapidly reduce tracking errors, suppress sliding-mode chattering at its source, and accelerate both the reaching and sliding phases, this paper proposes a sliding-mode speed controller that combines an integral-type nonsingular fast terminal sliding surface with the super-twisting algorithm (STA).

Since the PMSM is a nonlinear multivariable system, it can be described by the following nonlinear state equation:

$$\dot{x}=f(t,x,u)$$

(5)

In the equation, $x\in R^n,u=R^m,t=R$.

To drive the system trajectory toward the sliding-mode surface, the sliding-mode surface is designed as follows:

$$s(x)=s(x_1,x_2,\dots ,x_n)=0$$

(6)

At this stage, the state space is divided by the sliding-mode surface into three regions, $S>0$, $S<0$, and $S=0$, as illustrated in Figure 1.

The control function is as follows:

$$u_i(x,t)=\left\{\begin{array}{c}{u}_i^{+}(x,t),s_i(x,t)>0,\\ u_i^{-}(x,t),s_i(x,t)<0,\end{array}\right.i=1,2,\dots ,m.$$

(7)

where the control function $u_i(x,t)$ should satisfy the reachability condition as well as ensure the stability of the subsequent sliding-mode motion. On this basis, the sliding-mode control function is designed to optimize the dynamic performance of the control system.

Building upon Equation (6), a fast nonsingular terminal sliding surface is designed by introducing a fractional-order term to eliminate singularity and an integral term to enhance robustness against disturbances, which is expressed as follows:

$$s_2=k_1{x}_1+k_2{\displaystyle {\int }_0^t{x}_2}(\tau )d\tau +k_3{\displaystyle {\int }_0^t|}{x}_1(\tau ){|}^{q/p}\mathrm{sgn}\left(x_1(\tau )\right)d\tau$$

(8)

In the formulation, S₂ denotes the designed sliding surface, k₁, k₂, and k₃ represent sliding-surface gains, p and q are constants satisfying p > q, and t specifies the controller sampling period.

In conventional terminal sliding-mode control, singularity may occur because negative-power error terms are generated after differentiating the sliding surface. In the proposed integral-type nonsingular terminal sliding surface, the fractional-power term is introduced in integral form; therefore, its derivative only contains the bounded term and no negative-power error term appears. Thus, the proposed method avoids singularity while maintaining fast convergence.

Differentiating Equation (8) yields the following:

$${\dot{s}}_2=k_1{x}_2+k_2{x}_2+k_3|x_1{|}^{q/p}\mathrm{sgn}(x_1)$$

(9)

The derivative of the fast nonsingular terminal sliding surface can be obtained as follows:

$${\dot{s}}_2=-{\displaystyle \frac{3k_1{P}_n{\psi }_f}{2J}}{i}_q+{\displaystyle \frac{k_{1}B}{J}}\omega +{\displaystyle \frac{k_1{T}_L}{J}}+k_{2}e+k_3|e{|}^{q/p}\mathrm{sgn}(e)$$

(10)

To effectively suppress chattering during the reaching phase of sliding-mode motion, a fast STA incorporating proportional and integral terms is designed as the reaching law. The proportional term accelerates the convergence process, while the integral term suppresses chattering on the sliding surface. The reaching law is expressed as follows:

$$\left\{\begin{array}{l}{\dot{s}}_2=-k_1\alpha {\left|s_2\right|}^{1/2}\mathrm{sgn}(s_2)-k_1\gamma s_2+k_1{s}_3\\ {\dot{s}}_3=-\beta \mathrm{sgn}(s_2)-\eta s_2\end{array}\right.$$

(11)

In the formulation, $k_1\gamma s_2$ is the proportional term, $k_1\eta s_2$ is the integral term, and α, β, γ, and η are all constants.

Combining the derivative of the sliding surface with the proposed fast STA reaching law, the q-axis reference current is derived as follows:

$$i_q^{*}={\displaystyle \frac{2J\left[k_1\left(\frac{B}{J}\omega +\frac{T_L}{J}\right)+k_{2}e+k_3|e{|}^{q/p}\mathrm{sgn}(e)+\alpha |s_2{|}^{1/2}\mathrm{sgn}(s_2)+\gamma s_2-v\right]}{3k_1{P}_n{\psi }_f}}$$

(12)

2.3. Stability and Convergence Analysis

This section first analyzes the stability of the proposed controller based on Lyapunov theory and then verifies the convergence behavior of the sliding variable through simulation.

Based on the Lyapunov theorem, the stability of the proposed controller is proven by selecting the state as follows:

$$Z={\left[\begin{array}{ccc}{Z}_1& Z_2& Z_3\end{array}\right]}^T={\left[\begin{array}{c}|s_2{|}^{1/2}\mathrm{sgn}(s_2)\hspace{1em}{s}_2\hspace{1em}{s}_3\end{array}\right]}^T$$

(13)

A positive-definite Lyapunov function is chosen as follows:

$$V(Z)=Z^{T}PZ$$

(14)

where $P=P^T>0$ is a real symmetric positive-definite matrix.

Taking the time derivative of $V\left(Z\right)$ along the trajectories,

$$\dot{V}\le -{\displaystyle \frac{1}{|s_2{|}^{1/2}}}{Z}^{T}AZ-Z^{T}BZ$$

(15)

Since $A>0$, $B>0$, and $P>0$, Equation (17) further yields

$$\dot{V}\le -c_1{V}^{1/2}-c_{2}V<0$$

(16)

where

$$c_1={\displaystyle \frac{{\lambda }_{\min }(A)}{{\lambda }_{{\max }^{1/2}}(P)}},c_2={\displaystyle \frac{{\lambda }_{\min }(B)}{{\lambda }_{\max }(P)}}$$

(17)

${\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}(\cdot )$ and ${\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}(\cdot )$ denote the minimum and maximum eigenvalues of the corresponding matrix, respectively.

From Equation (16), $V\left(Z\right)$ is strictly decreasing for $Z\ne 0$, and the closed-loop sliding dynamics are stable. Moreover, since $\dot{V}\le -c_1{V}^{1/2}$, the state vector $Z$ converges to the origin in finite time. The reaching time satisfies

$$T_r\le {\displaystyle \frac{2}{c_2}}\ln \left(1+{\displaystyle \frac{c_2}{c_1}}{V}^{1/2}(0)\right)$$

(18)

where $V\left(0\right)$ is the initial value of the Lyapunov function. Therefore, $s_2$, $\xi$, and $s_3$ converge to zero in finite time. This proves that the proposed fast-STA-based sliding-mode speed controller ensures finite-time convergence of the sliding variable and improves the robustness of the PMSM speed control system against bounded disturbances.

To further verify the stability of the designed sliding-mode controller, a corresponding simulation model was established in Simulink. A comparative analysis of the sliding variables with those of the conventional sliding-mode controller was conducted, and the PMSM parameters used in the simulation are listed in

Table 1. List of main symbols and parameters.

Symbol	Description
$i_d,i_q,i_q^{*}$	d-axis current, q-axis current, and q-axis reference current
$i_{\alpha },i_{\beta },{\widehat{i}}_{\alpha },{\widehat{i}}_{\beta }$	Actual and estimated stator currents in the $\alpha \beta$ frame
${\omega }_m,{\omega }_e,{\widehat{\omega }}_e$	Mechanical speed, electrical speed, and estimated electrical speed
${\theta }_e,{\widehat{\theta }}_e$	Actual and estimated electrical rotor positions
$T_L,{\widehat{T}}_L$	Actual and estimated load torques
$J,B,P_n,\psi$	Inertia, viscous damping coefficient, pole pairs, and PM flux linkage
$R_s,L_s$	Stator resistance and stator inductance
$s_1,s_2,s_5$	Sliding surfaces used in the controller and observer
$s_3$	Auxiliary variable of the super-twisting algorithm
$k_1,k_2,k_3$	Sliding-surface gains
$\alpha ,\beta ,\gamma ,\eta$	Super-twisting algorithm gains
$K_1,K_2$	Gains of the STA-based sliding-mode observer
$E_{\alpha },E_{\beta },{\widehat{E}}_{\alpha },{\widehat{E}}_{\beta }$	Actual and estimated back-EMF components
$V,A,B,P$	Lyapunov function and matrices used in stability analysis

.

The comparison results of the sliding variables are shown in Figure 2. The traditional sliding-mode controller requires approximately 0.019 s to reach the sliding surface, whereas the proposed method converges within about 0.01 s. In addition, the sliding variable under the proposed approach exhibits improved smoothness compared with the conventional scheme, and the chattering phenomenon is significantly reduced.

In summary, the proposed fast-STA-based controller is stable and outperforms the conventional sliding-mode controller in terms of chattering suppression and convergence time, thereby improving the dynamic performance of the system.

3. Load Torque Disturbance Observer Design

During motor operation, load variations are inevitable [34]. To ensure that the system robustly reaches the sliding surface in the presence of disturbances, the sliding-mode controller gains are typically selected according to the upper bound of the disturbances. However, these gains directly influence the chattering amplitude of the system states. By feeding forward accurate disturbance estimates to the controller, the required sliding gains can be reduced, thereby mitigating chattering.

In conventional sliding-mode load torque observers, where the sliding surface is defined solely by the observation error, the feedback gain plays a critical role in determining observer performance. This leads to an inherent trade-off between convergence speed and chattering amplitude. To simultaneously reduce chattering and accelerate convergence, this work improves the observer design by incorporating a fast terminal sliding surface combined with the STA.

The designed integral-type fast nonsingular terminal sliding manifold is expressed as follows:

$$s_5=x_1+{\alpha }_1{\displaystyle {\int }_0^t{x}_1}(\tau )d\tau +{\beta }_1{\displaystyle {\int }_0^t|}{x}_1(\tau ){|}^{q/p}\mathrm{sgn}\left(x_1(\tau )\right)d\tau$$

(19)

Based on the speed-loop model in Equation (1), the load torque is further introduced as an augmented state with ${\dot{T}}_L=0$, and the observer-oriented model is written as follows:

$$\left\{\begin{array}{l}{\dot{\omega }}_m={\displaystyle \frac{3P_n{\psi }_f}{2J}}{i}_q-{\displaystyle \frac{B}{J}}{\omega }_m-{\displaystyle \frac{T_L}{J}}\hfill \\ {\dot{T}}_L=0.\hfill \end{array}\right.$$

(20)

According to Equation (20) and substituting the two estimated state variables and the designed sliding surface into the fast STA with linear terms yields the observer expression as follows:

$$\left\{\begin{array}{l}{\dot{\widehat{\omega }}}_m={\displaystyle \frac{3P_n{\psi }_f}{2J}}{i}_q-{\displaystyle \frac{B}{J}}{\widehat{\omega }}_m-{\displaystyle \frac{{\widehat{T}}_L}{J}}-\alpha |s_5{|}^{1/2}\mathrm{sgn}(s_5)-\gamma s_5\hfill \\ {\dot{\widehat{T}}}_L=-\beta \mathrm{sgn}(s_5)-\eta s_5.\hfill \end{array}\right.$$

(21)

The block diagram of the improved load observer design is shown in Figure 3, and the stability proof is the same as that of the improved sliding-mode speed controller described above, which will not be repeated here.

4. Sensorless Control Based on STA

The implementation of speed control in the aforementioned methods requires accurate rotor position and velocity information of the PMSM. Although sensors are traditionally used for this purpose, they increase system size and may suffer from reduced measurement accuracy under harsh operating conditions, thereby degrading speed regulation performance. To overcome these limitations, a sliding-mode observer is designed to estimate rotor position and speed using a sensorless control strategy.

4.1. Sliding-Mode Observer Design

Although a conventional sliding-mode observer (SMO) can estimate rotor position with strong robustness, it usually employs discontinuous switching functions to reconstruct the back electromotive force (back-EMF). As a result, high-frequency chattering is introduced into the estimated back-EMF. A low-pass filter is commonly used to suppress these high-frequency components, but it inevitably causes phase delay and degrades rotor position estimation accuracy.

To overcome this problem, STA is introduced into the sliding-mode observer. Compared with the conventional first-order SMO, the STA-based SMO generates a continuous control signal and can effectively reduce chattering without relying on an additional low-pass filter. Therefore, the phase delay caused by filtering can be alleviated, and the accuracy of sensorless rotor speed and position estimation can be improved.

For the SPMSM considered in this study, $L_d=L_q=L_s$. In the stationary $\alpha \beta$ reference frame, the current dynamics of the motor can be expressed as follows:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]=-{\displaystyle \frac{R_s}{L_s}}\left[\begin{array}{c}{i}_{\alpha }\\ i_{\beta }\end{array}\right]+{\displaystyle \frac{1}{L_s}}\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]-{\displaystyle \frac{1}{L_s}}\left[\begin{array}{c}{E}_{\alpha }\\ E_{\beta }\end{array}\right]$$

(22)

where $i_{\alpha }$ and $i_{\beta }$ are the stator currents, $u_{\alpha }$ and $u_{\beta }$ are the stator voltages, $R_s$ is the stator resistance, $L_s$ is the stator inductance, and $E_{\alpha }$ and $E_{\beta }$ are the $\alpha$- and $\beta$-axis back-EMF components, respectively.

Based on Equation (22), the STA-based sliding-mode observer is designed as follows:

$${\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{\widehat{i}}_{\alpha }\\ {\widehat{i}}_{\beta }\end{array}\right]=-{\displaystyle \frac{R_s}{L_s}}\left[\begin{array}{c}{\widehat{i}}_{\alpha }\\ {\widehat{i}}_{\beta }\end{array}\right]+{\displaystyle \frac{1}{L_s}}\left[\begin{array}{c}{u}_{\alpha }\\ u_{\beta }\end{array}\right]-\left[\begin{array}{c}{S}_{\alpha }\\ S_{\beta }\end{array}\right]$$

(23)

where ${\widehat{i}}_{\alpha }$ and ${\widehat{i}}_{\beta }$ are the estimated stator currents, and $S_{\alpha }$ and $S_{\beta }$ are the STA-based sliding-mode injection terms.

The STA-based injection terms are given by

$$\left\{\begin{array}{l}{S}_{\alpha }=K_1|{\tilde{i}}_{\alpha }{|}^{1/2}\mathrm{sgn}({\tilde{i}}_{\alpha })+S_{1\alpha }\hfill \\ S_{\beta }=K_1|{\tilde{i}}_{\beta }{|}^{1/2}\mathrm{sgn}({\tilde{i}}_{\beta })+S_{1\beta }\hfill \\ {\dot{S}}_{1\alpha }=K_2\mathrm{sgn}({\tilde{i}}_{\alpha })\hfill \\ {\dot{S}}_{1\beta }=K_2\mathrm{sgn}({\tilde{i}}_{\beta })\hfill \end{array}\right.$$

(24)

where $K_1>0$ and $K_2>0$ are observer gains, and $S_{1\alpha }$ and $S_{1\beta }$ are the internal integral states of the STA.

When the observer reaches the sliding mode, the current estimation errors satisfy ${\tilde{i}}_{\alpha }=0$, ${\tilde{i}}_{\beta }=0$, ${\dot{\tilde{i}}}_{\alpha }=0$, and ${\dot{\tilde{i}}}_{\beta }=0$. According to the equivalent control principle, the back-EMF components can be reconstructed as follows:

$$\left[\begin{array}{c}{\widehat{E}}_{\alpha }\\ {\widehat{E}}_{\beta }\end{array}\right]=L_s\left[\begin{array}{c}{S}_{\alpha }\\ S_{\beta }\end{array}\right]$$

(25)

where ${\widehat{E}}_{\alpha }$ and ${\widehat{E}}_{\beta }$ denote the estimated $\alpha$- and $\beta$-axis back-EMF components. Since the STA output is continuous, the proposed observer can reduce chattering in the estimated back-EMF and avoid the phase delay introduced by a conventional low-pass filter.

After the back-EMF components are obtained, a phase-locked loop (PLL) is used to extract the rotor position and speed. For an SPMSM, the back-EMF components can be written as follows:

$$E_{\alpha }=-{\omega }_e{\psi }_f\sin {\theta }_e,E_{\beta }={\omega }_e{\psi }_f\cos {\theta }_e$$

(26)

where ${\theta }_e$ and ${\omega }_e$ are the electrical rotor position and angular velocity, respectively. The PLL error signal is defined as follows:

$$\Delta E=-{\widehat{E}}_{\alpha }\cos {\widehat{\theta }}_e-{\widehat{E}}_{\beta }\sin {\widehat{\theta }}_e$$

(27)

Substituting the back-EMF expressions into the above equation gives

$$\Delta E=k_E\sin ({\theta }_e-{\widehat{\theta }}_e)\approx k_E({\theta }_e-{\widehat{\theta }}_e),k_E={\omega }_e{\psi }_f$$

(28)

where the approximation holds when the rotor position estimation error is small. Therefore, the proposed STA-based SMO can provide smoother back-EMF estimation for the PLL, thereby improving the accuracy of sensorless rotor speed and position estimation.

4.2. Stability Analysis

To analyze the stability of the proposed STA-based sliding-mode observer, the current estimation errors in the stationary $\alpha \beta$ reference frame are defined as

$${\tilde{i}}_{\alpha }=i_{\alpha }-{\widehat{i}}_{\alpha },{\tilde{i}}_{\beta }=i_{\beta }-{\widehat{i}}_{\beta }.$$

(29)

For each axis $r\in \{\alpha ,\beta \}$, the current estimation error dynamics of the STA-based observer can be expressed in the following standard super-twisting form:

$$\left\{\begin{array}{l}{\dot{\tilde{i}}}_r=-K|{\tilde{i}}_r{|}^{1/2}\mathrm{sgn}({\tilde{i}}_r)+z_r+{\rho }_r(t),\hfill \\ {\dot{z}}_r=-\delta \mathrm{sgn}({\tilde{i}}_r),\hfill \end{array}\right.$$

(30)

where $K>0$ and $\delta >0$ are observer gains, $z_r$ is the internal state of the super-twisting observer, and ${\rho }_r\left(t\right)$ denotes the lumped disturbance term caused by parameter uncertainty, voltage error, current measurement error, and back-EMF variation.

For the observer stability analysis, it is assumed that the lumped disturbance is differentiable and its derivative is bounded, namely

$$|{\dot{\rho }}_r(t)|\le L_r,r\in \{\alpha ,\beta \}$$

(31)

where $L_r$ is a positive constant.

$$L=\max \{L_{\alpha },L_{\beta }\}$$

(32)

Define

$${\zeta }_r=z_r+{\rho }_r(t)$$

(33)

Then Equation (31) can be rewritten as

$$\left\{\begin{array}{l}{\dot{\tilde{i}}}_r=-K|{\tilde{i}}_r{|}^{1/2}\mathrm{sgn}({\tilde{i}}_r)+{\zeta }_r\hfill \\ {\dot{\zeta }}_r=-\delta \mathrm{sgn}({\tilde{i}}_r)+{\dot{\rho }}_r(t)\hfill \end{array}\right.$$

(34)

According to the finite-time convergence condition of the STA, if the observer gains are selected to satisfy

$$\delta >L$$

(35)

$$K^2>{\displaystyle \frac{4L(\delta +L)}{\delta -L}}$$

(36)

then the current estimation error ${\tilde{i}}_r$ and the auxiliary variable ${\zeta }_r$ converge to zero in finite time. This indicates that the estimated stator currents ${\widehat{i}}_{\alpha }$ and ${\widehat{i}}_{\beta }$ converge to the measured stator currents $i_{\alpha }$ and $i_{\beta }$, respectively.

Therefore, under the assumptions of bounded motor parameters, bounded disturbance derivatives, and properly selected observer gains satisfying Equations (35) and (36), the proposed STA-based sliding-mode observer guarantees finite-time convergence of the current estimation error and provides accurate back-EMF estimation.

5. Simulation and Experimental Verification

To improve reproducibility, the main simulation and implementation settings are further specified. The PMSM drive system was implemented in MATLAB/Simulink R2023b using the $i_d=0$ vector control strategy and SVPWM. A fixed-step solver with the ode3 integration method was adopted, and the fixed-step size was set to $T_s=10 \mathsf{\mu }\mathrm{s}$. The PWM carrier period was $2\times {10}^{-4}$ s, corresponding to a switching frequency of 5 kHz. The DC-link voltage was set to 560 V, and the total simulation time was 0.6 s. In the comparative simulations, speed and load changes were applied at 0.2 s and 0.4 s, respectively. The motor parameters, current-loop controller, sampling time, PWM frequency, and operating conditions were kept identical for all compared methods.

The main gains used in the reported results were specified as follows. The current-loop PI gains were set as $K_p=5.9077$ and $K_i=860.9610$. The baseline speed-loop PI gains were $K_{p\omega }=0.1362$ and $K_{i\omega }=6.81$. For the proposed STA-based speed control module, the main implementation gains were $k_{q1}=100$ and $k_{q2}=80$. The load torque observer parameters were ${\alpha }_1=15$, ${\alpha }_2=9$, and ${\epsilon }_o=0.0005$.

The parameters were tuned sequentially. The current-loop PI controller was first adjusted to ensure stable current tracking, and the baseline speed-loop controller was then tuned for comparison. Based on the stability and finite-time convergence requirements, the STA-related gains were selected and further refined through simulation by balancing response speed, chattering suppression, disturbance rejection, and steady-state fluctuation [35]. Finally, the load torque observer and STA-based sliding-mode observer parameters were adjusted to improve load-disturbance estimation, back-EMF estimation, and sensorless estimation accuracy, while avoiding excessively large gains that may cause high-frequency oscillations or current command fluctuations.

The block diagram of the speed control system based on the integral fast terminal sliding-mode surface combined with STA is shown in Figure 4.

In the proposed sensorless vector control system, the speed controller generates the $q$-axis reference current $i_q^{*}$, the load torque observer provides the estimated load torque ${\widehat{T}}_L$ for feed-forward disturbance compensation, and the STA-based sliding-mode observer supplies the estimated rotor position and speed for coordinate transformation and speed feedback. Thus, these three modules are coordinated within a unified PMSM vector control framework.

5.1. Simulation Verification of Speed Regulation and Sensorless Estimation

This section evaluates the proposed control strategy through simulations under speed-variation and load-disturbance conditions, including speed response, disturbance rejection, and sensorless estimation performance.

To evaluate the load observation performance of the conventional load torque observer and the proposed observer, a 10 Nm load disturbance was applied to the motor at 0.2 s and removed at 0.4 s in the Matlab/Simulink environment, as shown in Figure 5.

As illustrated in Figure 5, the conventional observer requires approximately 0.005 s to reach the reference value following the sudden load change, with a steady-state observation fluctuation of about 0.048 Nm. In contrast, the proposed observer converges within 0.001 s, and the steady-state fluctuation is reduced to approximately 0.000006 Nm. The proposed method improves the load observation response by about 80% and significantly suppresses steady-state fluctuation.

To evaluate the optimization performance of the proposed controller in PMSM speed regulation, simulations were conducted under identical parameter settings. The reference speed was set to 1000 r/min. The motor was started, and the speed was increased to 1200 r/min at 0.2 s and decreased to 800 r/min at 0.4 s, as shown in Figure 6.

As illustrated in Figure 6, compared with the conventional double closed-loop PI control, the proposed controller significantly reduces the overshoot from 28.5% to 0.4%, corresponding to a reduction of 98.60%. Moreover, the acceleration and deceleration response time is shortened from 2 s to 0.5 s, representing a 75% improvement.

In comparison with the traditional sliding-mode controller, the proposed method decreases the overshoot from 14.8% to 0.4%, achieving a reduction of 97.30%. The acceleration and deceleration response time is reduced from 0.18 s to 0.015 s, which corresponds to a 91.67% decrease.

To evaluate the disturbance rejection capability and steady-state performance of the PMSM under different control strategies, the reference speed was set to 1000 r/min. The motor was started, and a load of 10 Nm was applied at 0.2 s and removed at 0.4 s, as shown in Figure 7.

As illustrated in Figure 7, compared with the conventional double closed-loop PI control, the proposed controller significantly reduces the speed deviation caused by the sudden load disturbance from 70.20 r/min to 21.11 r/min, corresponding to a reduction of 69.93%. In addition, the steady-state speed fluctuation is reduced from approximately 8 r/min to about 1 r/min, representing an 87.5% decrease.

Compared with the traditional sliding-mode controller, the proposed method reduces the speed variation induced by the load disturbance from 93.12 r/min to 21.11 r/min, achieving a 77.33% improvement. Furthermore, the steady-state speed fluctuation is decreased from about 7 r/min to approximately 1 r/min, corresponding to a reduction of 85.71%.

Based on the above simulation results, the proposed method also shows improved performance over conventional control strategies in three key aspects of PMSM speed regulation: dynamic speed response, load-disturbance rejection, and steady-state fluctuation. The dynamic performance of the PMSM is therefore significantly improved.

To further evaluate the position and speed estimation accuracy of the proposed STA-based sliding-mode observer, simulations were conducted in Simulink. The speed estimation and tracking error curves are shown in Figure 8 and Figure 9.

Under the proposed sensorless control strategy, the PMSM rapidly reaches the reference speed, and the estimated speed closely tracks the actual value. Compared with the conventional sliding-mode observer, the steady-state speed estimation error is reduced from approximately 15 r/min to about 2 r/min, corresponding to an improvement of 86.67%.

The rotor position estimation and tracking error curves are shown in Figure 10 and Figure 11. The proposed STA-based sliding-mode observer enables the estimated rotor position angle to rapidly track the actual value. By employing the STA, chattering is significantly reduced, and the reliance on a low-pass filter is minimized, thereby alleviating the phase-delay issue.

Compared with the conventional sliding-mode observer, the position estimation error is reduced from 0.039 rad (~2.24°) to approximately 0.002 rad (~0.11°), corresponding to a 94.87% reduction. These results demonstrate that the optimized sliding-mode observer substantially improves estimation accuracy and enhances the performance of the sensorless control system.

In summary, the above simulation results preliminarily verify that the proposed sensorless control strategy for the PMSM vector control system effectively reduces the estimation errors of rotor speed and position, thereby improving overall sensorless control accuracy.

5.2. Analysis of Experimental Results

To further validate the effectiveness of the proposed sensorless speed optimization strategy, experimental verification was conducted to compare theoretical and simulation results with practical performance.

The motor control experimental platform, constructed according to the configuration shown in Figure 4, is presented in Figure 12. The platform is based on the SP1000 controller (Nanjing Yanxu Company, China) and the TMS320F28335 processor (Nanjing Yanxu Company, China), and includes a torque sensor, speed sensor, magnetic powder brake, and host computer. Experimental data are recorded, stored, and analyzed via the host computer.

The experimental motor is a surface-mounted permanent magnet synchronous motor (PMSM). The motor parameters are listed in

Table 2. Motor parameters.

Parameter	Value
Rated Voltage (V)	24
Rated Power (W)	100
Rated Current (A)	5.5
Rated Speed (r/min)	3000
Rated Torque (Nm)	0.32
Peak Torque (Nm)	0.96
Torque Constant (Nm/A)	0.05
Pole Pairs	5
Rotor Inertia (kg·m2)	0.000034
EMF (V/krpm)	4.0

, and the specifications of the magnetic powder brake are provided in

Table 3. Magnetic powder brake parameters.

Parameter	Value
Peak Torque (Nm)	1
Rated Power (W)	10
Rated Voltage (V)	24
Mass (kg)	0.54
Shaft End Diameter (mm)	7
Shaft End Length (mm)	12.2

. Experimental tests were performed under sensorless operation to evaluate start-up, acceleration, deceleration, and sudden load disturbances. The results were compared with those obtained using the conventional control method for validation.

5.3. Experiments on Speed Control for Variable-Speed Conditions

The reference speed was set to 1000 r/min via the host computer. The motor was started, and the speed was increased to 1200 r/min at 4 s and decreased to 800 r/min at 8 s, as shown in Figure 13.

As illustrated in Figure 13, compared with the conventional double closed-loop PI control, the proposed controller reduces the overshoot from 6.5% to 0.55%, corresponding to a 91.54% reduction. Moreover, the acceleration and deceleration response time is shortened from 0.67 s to 0.5 s, representing a 25.37% reduction.

Compared with the traditional sliding-mode controller, the proposed method decreases the overshoot from 12.10% to 0.55%. Furthermore, the acceleration and deceleration response time is reduced from 1.2 s to 0.5 s, indicating a significant enhancement in system responsiveness.

5.4. Experiments on Speed Control Under Variable-Load Conditions

The reference speed was set to 1000 r/min via the host computer. After the motor was started, a load of 0.2 Nm was applied at 4 s and removed at 8 s, as shown in Figure 14.

As illustrated in Figure 14, compared with the conventional double closed-loop PI control, the proposed controller significantly improves disturbance rejection performance. During sudden load changes, the speed deviation is reduced from 176 r/min to 25 r/min, corresponding to an 85.80% reduction. Meanwhile, the speed recovery time is shortened from 3 s to 0.24 s, representing a 92% improvement. In steady-state operation, the speed fluctuation decreases from 31 r/min to 12 r/min, a reduction of 61.29%.

Compared with the traditional sliding-mode controller, the proposed method also shows improved performance. During load disturbances, the speed deviation is reduced from 88.5 r/min to 25 r/min, corresponding to a 71.75% decrease, and the recovery time is shortened from 0.6 s to 0.24 s, achieving a 60.00% improvement. In steady state, the speed fluctuation is reduced from 18 r/min to 12 r/min, representing a 33.33% decrease.

To ensure a consistent quantitative comparison, the performance indices are defined as follows. The overshoot is the maximum speed exceeding the reference value after a speed command change. The response time is the time required for the speed to enter and remain within a ±2% band of the reference speed. Under load-disturbance conditions, the speed deviation is defined as the maximum absolute speed error caused by the load step, and the recovery time is the time required for the speed to return to the ±2% band after the disturbance. The steady-state fluctuation is calculated as the peak-to-peak speed variation after the transient process. The same definitions are used for PI control, traditional SMC, and the proposed method.

To provide a clearer quantitative comparison, the main performance indices of PI control, traditional SMC, and the proposed method are summarized in

Table 4. Quantitative comparison of control performance.

Condition	Index	PI	SMC	Proposed
Speed change (Sim.)	Overshoot (%)	28.5	14.8	0.4
Load change (Sim.)	Speed deviation	70.20	93.12	21.11
Load change (Sim.)	Speed fluctuation	8	7	1
Speed change (Exp.)	Overshoot (%)	6.50	12.10	0.55
Load change (Exp.)	Speed deviation	176	88.5	25
Load change (Exp.)	Speed fluctuation	31	18	12

.

The experimental results show that the proposed sensorless sliding-mode speed control strategy achieves improved dynamic response, load-disturbance rejection, and steady-state performance compared with the conventional control methods.

6. Conclusions

To address the limited control accuracy of conventional sliding-mode methods caused by the trade-off between chattering and dynamic response, this paper proposes a sensorless speed control strategy that integrates an integral-type fast terminal sliding-mode surface with the STA. A sliding-mode controller with load compensation is developed, and stability analysis is provided to ensure robustness and convergence.

An STA-based sliding-mode observer is further designed to reduce chattering and eliminate phase delay without relying on a low-pass filter, thereby improving rotor position and speed estimation accuracy.

Experimental validation is conducted on a dedicated motor control platform. Simulation results demonstrate improvements of 91.67% in dynamic response, 77.33% in disturbance rejection, and an 85.71% reduction in steady-state fluctuation. Experimental results further confirm enhancements of 58.33% in dynamic response, 71.75% in disturbance rejection, and a 33.33% reduction in steady-state fluctuation.

These results demonstrate that the proposed sensorless sliding-mode speed control strategy can improve the dynamic response, disturbance rejection capability, and steady-state performance of PMSM drives under the tested operating conditions.