An Adaptive Backstepping Sliding-Mode Control for Improving Position Tracking of a Permanent-Magnet Synchronous Motor With a Nonlinear Disturbance Observer

To provide reliable position control for a permanent-magnet synchronous motor (PMSM) under conditions of lumped disturbances such as external load torque fluctuation and system parameter variation, an adaptive backstepping sliding-mode control (ABSMC) with a nonlinear disturbance observer (NDO) is proposed. An ABSMC is a non-cascade technique that employs a position-current single-loop control structure rather than a conventional cascade control structure for vector control of the PMSM. This method uses Lyapunov theory to design a control law that ensures the motor position reaches a desired value in a finite period of time and consequently achieves a rapid transient response. The proposed ABSMC incorporates adaptive convergence gain to avoid a large overshoot under point-to-point position command. The NDO with nonlinear observer gain is proposed to estimate unknown disturbances and provide feed-forward compensation for the ABSMC, improving robustness and reducing steady-state position error. The proposed methods are applied to an industrial motor drive to confirm their validity in real-world operating environments.


I. INTRODUCTION
Permanent-magnet synchronous motors (PMSMs) are widely employed in industrial applications, such as robotics, transportation, elevators, and machine tools, due to their compact construction, high power density, high performance, and low losses [1], [2]. Field-oriented control is a popular technique for high-performance motor drive control. Field-oriented control-based cascade proportional integral derivative (PID) control is a commonly used and effective linear technique for The associate editor coordinating the review of this manuscript and approving it for publication was Jinquan Xu . regulating PMSMs [3] thanks to its simplicity and overall performance at certain operating points. However, as a nonlinear system, PMSM is often distorted by nonlinearities, parameter uncertainties, and external disturbances when operating under various conditions that can cause undesirable dynamic responses in the motor [4]. In such cases, it may not be possible to guarantee the overall performance of the motor system using conventional linear feedback control. An alternative method is therefore needed to maintain the high dynamic performance of the motor drive. Several nonlinear techniques have been developed to improve the control performance of motor systems, including backstepping control [5], [6], [7], VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ adaptive control [8], [9], fuzzy-logic control [10], [11], sliding mode control (SMC) [12], [13], [14], [15], [16], [17], and model predictive control [18], [19], [20]. Among these methods, backstepping control is a systematic and recursive approach to nonlinear systems based on Lyapunov stability theory, which ensures global asymptotic stability [21]. This method can transform complex nonlinear control systems into simple reduced-order subsystems [6]. A control law for motor position tracking or speed regulation can be derived through recursive design and virtual control variables. However, backstepping is model-based and cannot ensure the robustness of PMSM drives in the presence of uncertainties and external disturbances [5]. To overcome this limitation, a robust backstepping tracking controller with a load torque observer has been developed using PMSM machine parameter data [22] to provide precise control of the reference trajectory. Tan et al. [23] has also developed an adaptive integrated backstepping approach with a friction compensator to compensate for unknown system parameters and disturbances to a certain extent.
Extensive research has shown SMCs to be useful in industrial applications [24], [25] because of their robustness to disturbances, rapid response time, and ease of implementation. However, to guarantee the robustness of an SMC under parameter uncertainties and external disturbances, switching gain larger than the upper bound of the lumped disturbance is often required, leading to undesirable chattering phenomena [26]. Several approaches have been proposed to overcome the chattering, including a boundary layer approach [27], high-order SMC [28], fuzzy slidingmode method [10], terminal SMC [29], and a reaching law method [15], [26]. Among these approaches, the reaching law has proven to be an effective strategy for reducing chattering. It directly affects the reaching process by modifying the control gain functions that satisfy the requirements for robustness and reduction in chattering. The control gain has a potentially large value during the reaching phase, while its value will be small during the sliding phase. Therefore, robustness near the sliding-mode surface decreases when an external disturbance occurs [12]. The combination of an SMC with a disturbance observer is an attractive method to improve the robustness of the SMC-based reaching law method. A disturbance observer can estimate an unknown disturbance (including parameter variations and external disturbances) and feed it forward to the SMC. Therefore, the switching gain only needs to be greater than the estimation error [30], which is much smaller than the upper bound of the lumped disturbance, to reduce the chattering.
The disturbance observer is an essential part of the motor controller, as uncertainties and external disturbances are inevitable in motors. Several disturbance observer approaches, such as the Luenberger observer [10], [15], the sliding-mode observer [26], [31], the finite-time disturbance compensator [32], and the extended-state observer [22], [24] have been proposed to estimate and compensate for unknown disturbances in practical applications. Experimental results indicate that the disturbance can be mitigated from the output by feeding the estimated disturbance forward to the controller without affecting system performance. Nonlinear disturbance observer (NDO) control methods [33], [34], [35] have recently been studied because they are easy to implement and produce reliable disturbance estimates. This technique has been tested on robots, converters, and motor drives, with the results demonstrating that highly robust performance is achievable [4]. However, such research used a fixed observer gain for the NDO, which may cause the observer to provide a slower convergence rate in certain critical scenarios, such as a sudden change in external load torque [36].
The main goal of this paper is to propose a controller for position tracking to enhance the overall performance of the PMSM control system through rapid transient response, small steady-state position error, reduced overshoot, and robustness against unknown disturbances. Inspired by a composite method that combines the advantages of backstepping control and SMC for PMSM speed regulation [5], this paper extends the concept to a backstepping sliding-mode control (BSMC) for position tracking. However, a BSMC can cause large overshoots in position step profiles due to the fixed convergence gain of the backstepping method. To address this problem, an adaptive BSMC (ABSMC) that contains an adaptive convergence gain is proposed. Using the proposed method, it is possible to both accelerate the actual position toward the target position and reduce overshoot. In addition, by improving the nonlinear observer gain function of the NDO, parameter uncertainties and external load torque disturbances can be observed and provided with feed-forward compensation to the ABSMC, preserving the robustness of the ABSMC and reducing steady-state error. The primary contributions to this article are: 1) A position-current single-loop control structure simplifies conventional cascade position controller structures by incorporating both backstepping control and SMC. The composite ABSMC offers the asymptotic stability of backstepping control and the robustness and rapid convergence of SMC. The ABSMC also contains an adaptive law for adjusting backstepping convergence gain, which can be used to avoid large overshoots under point-to-point position command. 2) To improve the disturbance rejection and robustness of the ABSMC, an NDO with an improved nonlinear observer gain function is used to enhance robustness and reduce chattering and steady-state position error.
3) The proposed method was implemented on an industrial motor driver and compared with PID [37], SMC [13], a super-twisting-like fractional (STLF) controller [16], and a non-singular fast terminal sliding mode controller (NFTSMC) [17]. Comparative and extensive experiments demonstrate that the proposed method has a rapid transient response, low steady-state position error and reduced overshoot, and is robust to external load and parameter disturbances. The rest of this article is constructed as follows. The mathematical formulation of a PMSM is introduced in Section II. Position control based on a BSMC scheme is detailed in Section III. Section IV addresses disturbance rejection based on the NDO and presents a combination of the BSMC and NDO. In Section V, an adaptive rule for convergence gain for the BSMC that eliminates the large overshoot when operating with step-position commands is discussed. Section VI explores the experimental results, and a conclusion is presented in Section VII.

II. MATHEMATICAL MODEL OF THE PMSM
In this paper, a non-cascade control method is proposed to improve the position tracking performance of PMSM drivers (Fig. 1). This method employs an ABSMC to design the current controller for the q and d axes, providing voltage supplies (v q and v d ) for the motor. The ABSMC is a combination of backstepping and SMC control, where the backstepping component includes an adaptive convergence gain to avoid large overshoots. Moreover, to mitigate the impact of uncertainty parameters and external disturbances, the NDO [38], [39] is utilized with some modifications to the observer gain design. This helps estimate the unknown disturbance and feed it forward to the ABSMC, ensuring the robustness of the motor system. The mathematical model of the PMSM system in the rotor d − q reference frame can be expressed as [3]: where i d and v d are the stator current and voltage in the d−axis, while i q and v q are the stator current and voltage in the q−axis, respectively. R is the stator resistance, L d and L q are the stator inductances in the d − q frame, P is the number of rotor pole pairs, is the flux linkage of the rotor permanent magnets, ω is the angular velocity of the rotor, and T e is the electromagnetic torque. To provide a constant flux condition in the PMSM, the d−axis reference was set to zero and the inductances in d− and q−axes were considered equal L d = L q [40]. The electromagnetic torque, T e , is dependent solely on the q−axis current and can be expressed as follows: where K t is the torque constant. The motion equation of the motor can be expressed as: where θ represents the position angle of the rotor, J refers to the total moment of inertia (which includes both the rotor and load), B represents the frictional coefficient, and T L represents the external load torque, which is assumed to be unknown but varies slowly within a small sampling interval.
In practical operating conditions, the motor can often be affected by both variations in internal parameters and external disturbances. According to equation (6), the mechanical equation can be rewritten as follows where J n and B n are the nominal values of J and B, respectively, with J = J n + J and B = B n + B. The value of d is defined as d = ( J dω dt + Bω + T L ), which contains internal parameter variations and external load torque. These are considered as lumped disturbances in the motor. The value of d is unknown, causing an undesired dynamic response.
Assumptions : 1) The lumped disturbances do not have an excessive effect on the system, and they are bound (i.e., |d| < D 1 with D 1 > 0) and vary slowly during a small sampling interval in the actual system. 2) The disturbance derivative will also be bound in the controller (i.e., there exists D 2 > 0 such that |ḋ| < D 2 ).
The main purpose of this study was to construct a position controller that improves the position tracking of the motor. First, a BSMC was designed to replace the conventional position control based on a cascade control structure. Then a NDO was used with a modified observer gain function to improve robustness and reduce steady-state error in position tracking. However, because the backstepping in the BSMC used fixed convergence gains, a large overshoot occurred in the position step profile. To address this problem, an adaptive convergence gain for the backstepping was proposed.

III. DESIGN OF THE BSMC FOR POSITION CONTROLLER
The BSMC for position control is described first in this section. The drawbacks of the BSMC are then analyzed, and the design of the proposed ABSMC is described in Section V. To facilitate the controller design, equations (5), (7), (2), VOLUME 11, 2023 and (1) are collected and rewritten as (8)- (11), respectively: where x 0 = θ, x 1 = ω, x 2 = i q and x 3 = i d are the measurable variables; x d = −d/J n is lumped disturbance and unknown; and a n = B n /J n , (10) were used to construct the q−axis current controller (Steps 1 to 3 in Fig. 1), and equation (11) was used to construct the d−axis current controller (Step 4 in Fig. 1). The main idea of backstepping control is to reduce a high-order system into several first-order subsystems by providing virtual control variables [6]. The control law is designed based on the Lyapunov stability theory.

A. Q-AXIS CURRENT CONTROLLER DESIGN
Step 1: Select a candidate Lyapunov function as V 0 = 0.5 e 2 0 , then take the derivative of V 0 with respect to time: From (12), the virtual control input for the (9) is designed as: where c 0 > 0 is a strictly positive constant for an asymptotic rate of convergence.
Step 2: Based on the value x ref From (13) and (9), the termė 1 is determined bẏ (15) From (14) and (15), the term e 0 +ė 1 is designed to equal −c 1 e 1 : The virtual control input for the (10) can then be determined as is the nominal virtual control signal without considering the uncertainties and external load torque: The virtual control signal x ref 2 contains the lumped disturbance x d , which will cause an undesirable response in the motor system. To overcome the negative effect of the disturbance on the controller, a NDO was designed to estimate and compensate for it in Section IV.
The positive gains c 0 and c 1 in (13) and (18) represent the convergence gain of the backstepping controller. If these gains are large, the system will have a faster response and superior robustness, but a large overshoot will occur in the step profile. If the value of these gains is small, the system will respond slowly, and the overshoot will be reduced. There is therefore a trade-off between settling time and overshoot. This problem will be addressed in Section V.
Step 3: Based on the value In this step, a SMC is used to design the voltage supply v q for the motor, and the procedure is separated into two parts: designing the slidingmode surface and the reaching law. First, the sliding-mode surface is designed based on error e 2 : Considering the Lyapunov function V 2 = V 1 + 0.5 S 2 1 , the differential of V 2 with respect to time becomes: From (17) and (18), the term (ẋ Substitute (21) into (20), theV 2 can be rewritten aṡ The constant-proportional rate reaching law [13] for the sliding mode is designed as beloẇ The resulting control system is therefore asymptotically sta- The derivative with respect to time of S 1 is: Substituting (15) and (9) into (25) yields: where (23), (24) and (26) theṠ 1 is deriveḋ From the above equation, the voltage supply for the q−axis can be derived as From equation (28), it can be seen that the control signal v q contains the terms x d and its derivativeẋ d . The design of v q is therefore not complete. To overcome this problem, the lumped disturbance is considered in Section IV.

B. D-AXIS CURRENT CONTROLLER DESIGN
Step 4: This step designs the control signal v d to make the d−axis current output from the motor track the reference to provide a constant flux). To achieve this goal, the difference e 3 = 0 − x 3 should be minimized. As in Step 3, the SMC is used to design the control signal v d . The sliding surface is designed with: where α 1 > 0, and the control signal v d is derived using the reaching law [13]: From (29), taking the derivative of S 2 yields: From (30) and (31), the control signal v d can be calculated: Defining the Lyapunov function V 3 = 0.5S 2 2 . ThenV 3 = S 2Ṡ2 = −k 3 S 2 sgn(S 2 ) − k 4 S 2 2 ≤ 0 can be ensured. The value of x 3 will therefore converge to 0.  (17) and the control law for the voltage supply v q in (28), the lumped disturbance, x d , and its first derivative term,ẋ d , will cause an undesirable dynamic. Although it is possible to compensate for a lumped disturbance and retain a certain level of robustness by designing a large sliding-mode switching gain (k 1 and k 2 ), this can cause significant chattering, and fail to provide an effective dynamic response in some applications [26]. An alternative solution is to design a disturbance observer to estimate the lumped disturbance and feed it forward to the control law online.

IV. DESIGN OF THE NONLINEAR DISTURBANCE OBSERVER AND PROOF OF SYSTEM STABILITY
Considering the subsystem (9) contained uncertainty and external disturbances with the virtual control x ref 2 that was obtained from step 2 The NDO [38], [39] is designed as: where l(x 1 ) > 0 represents the observer gain. As the statė x 1 cannot be measured directly, the preceding equation was rewritten as: An internal state z is defined such thaṫ and a nonlinear function p(x 1 ) with a derivative with respect to x 1 equal to l(x 1 ), so that ∂p(x 1 ) ∂x 1 = l(x 1 ). Integrating (36) with respect to time yields z =x d − p(x 1 ), and the lumped disturbancex d can be estimated From (35), (36) and (37) theż can be calculated aṡ Considering the observer error e d = x d −x d , theė d is obtained: For a simple design [33], [34], [35], the observer gain is commonly selected to be a fixed constant value L 1 > 0, and the function p(x 1 ) = L 1 x 1 becomes a linear function to estimate the disturbance. As a result, the disturbance observer can be referred to as a linear disturbance observer (LDO) when using a constant observer gain. The derivative of observer error becomesė d =ẋ d − L 1 e d . It follows that the observer error can be obtained by [36]: where |ẋ d | < ε, with ε > 0 is the bound of derivative of the lumped disturbance x d . From equation (40), in the case when the system experiences a constant lumped disturbance,ẋ d = 0, meaning ε = 0, the observer error, e d , will converge to zero. This implies that the disturbance observer can accurately estimate the lumped disturbance in the steady state. When the lumped disturbance changes slowly and its derivative is bounded by ε, the observer error (40) is asymptotically and exponentially stable for the initial error, e d (0). However, a steady state error, ε/L 1 , still exists, and depends on the value of L 1 . Increasing the magnitude of the observer gain decreases the observer error, e d , and leads to a faster disturbance estimation response. However, a fixed observer gain may result in a slower convergence rate in critical scenarios, such as sudden changes in external load torque [36]. To address this issue, we propose a nonlinear design function, p(x 1 ), referred to as the NDO: where L 1 and L 2 > 0 are constant parameters. In contrast to the fixed gain designed in [33], [34], and [35], the proposed NDO employs a nonlinear design function p(x 1 ), which combines first-and second-order state variables. Differentiating the nonlinear function p(x 1 ) with respect to x 1 yields: The observer gain l(x 1 ) is constructed to meet the condition of globally exponential stability [39] of the observer error in (39). It is clear that the proposed l(x 1 ) has a larger magnitude when compared with the constant gain L 1 , which is used in [33], [34], and [35]. As a result, the nonlinear gain l(x 1 ) can provide a faster convergence rate while also reducing observer error. Fig. 3 shows the results of the NDO and LDO when combined with the (A)BSMC under trapezoidal velocity motion profiles (Fig. 2). The term (A)BSMC means that the ABSMC and BSMC performed similarly under trapezoidal velocity motion profiles. The main advantage of the ABSMC is the reduction of overshoot under the step position profile, which will be demonstrated later. Fig. 3(a)

B. PROOF OF THE SYSTEM STABILITY
Through the design of the d−axis current controller, it can be seen that the d−axis control system is stable;V 3 ≤ 0. To design the q−axis current controller, considering the motor system (9) with the lumped disturbance x d , the NDO is employed to estimate the lumped disturbance and provide a feed-forward combination for the control law in the control signal (Fig. 1). The control signals x ref 2 and v q can be expressed as Define the Lyapunov function V = V 2 + V 3 . Taking the derivative of V and combining (20) andV 3 : The term (ẋ ref 1 −ẋ 1 ) is calculated by using (43), (13), and (18): The derivative with respect to time of S 1 is calculated by using (44), (26), and (10):  Substituting (46) and (47) into (45) yields: where k ac > 0, then c 0 + c 1 > a n . Since by Young's inequality, it follows thaṫ where M = 0.75 c 1 . As shown in Fig. 4, thė V II < 0 outside of an ellipse. Since the size of the ellipse can be made arbitrarily small by taking a sufficiently large observer gain L 1 and suitable values of c 1 and k 2 , V II (e 1 (t), e 2 (t)) tends to an arbitrarily small number as time increases. The robustness of the BSMC+NDO can therefore be maintained.

V. ADAPTIVE BACKSTEPPING SLIDING MODE CONTROL
As mentioned previously, the proposed BSMC contains convergence gains c 0 and c 1 for calculating the virtual control    (18). Based on experiments, c 0 was selected to equal c 1 to obtain reliable control performance. However, the BSMC poses a problem with respect to practical applicability. If these gains are small, the system will respond slowly, and the overshoot will be small. If the gains are large, the system will have a faster response and superior robustness, but a large overshoot will occur in position step profile.
Because of the large changes in the step profile, excessively large errors (e 0 and e 1 ) occur. As described in (18), these errors will be multiplied by large constant gains (c 0 and c 1 ), resulting in a large control signal x ref 2 . Due to limitations in the physical system, the current is always constrained within a specific range of values. As a result, control saturation occurs, leading to an overshoot in the position response. To tackle this problem, an ABSMC was proposed. The experimental results of the BSMC under the step position command are shown in Figs. 5(a) and Fig. 6. The red line in Fig. 5(a) shows the position response of the BSMC, while Fig. 6 shows the e 0 , e 1 ,ė 0 , and the virtual control  position of 10 rad in 0.18 s at the end of region I, as shown in Fig. 5(a). The values of e 0 and e 1 are small at the end of region I, but the large gains (c 0 , c 1 ), and the large value oḟ e 0 in the term (c 0 + c 1 )ė 0 , which is contained in (18), will continue to produce a large control signal x ref 2 in region II, as shown in Fig. 6(d). This results in an overshoot of 38% in the position in Fig. 5(a). Therefore, when excessively large errors occur, the gains of c 0 and c 1 should be lowered to reduce the excessive control signal. The proposed adaptive gain of c * 0 and c * 1 for the ABSMC which ensures k ac > 0 is: where δ > 0 is the experimentally determined threshold. It should be noted that the convergence gains c * 0 and c * 1 are calculated independently of the backstepping procedure (steps 1, 2, 3, and 4), and the calculated gains are considered constant values in the backstepping procedure, as shown in Fig. 8. When an excessively large error occurs (e 0 and e 1 are both large), |e 0 e 1 | will also be large. The convergence gains of c 0 and c 1 should then be lowered to reduce overshoot. Because the derivative of e 0 indicates how fast the position error e 0 changes in time, it is used to adjust the gain c * 0 . To provide a rapid transient response and reduce overshoot, the gains of c 0 and c 1 should be reasonably adjusted. When an excessively large errors occur, the degree of gain reduction is determined by the magnitude ofė 0 and the exponential function f (e 0 ) with (λ > 0 and η > 0) to appropriately decrease the gain c * 0 . When the actual position is far from the reference position, e 0 and e 1 are excessively large, and the magnitude ofė 0 tends to increase from a small to a large value, corresponding to region I (Fig. 5). If the gain of c * 0 is immediately changed to a small value at the beginning of region I, the controller will exhibit a slow transient response. As a result, when e 0 is large, f (e 0 ) should converge to a small value, gradually reducing the gain c * 0 ≈ c 0 − |ė 0 | and accelerating the actual position to the reference position. Because of the reduction of the gain of c * 0 to a small value at the end of region I, the excessively large control signal caused by the term (c 0 +c 1 )ė 0 as analyzed above can be avoided. When the actual position is close to the reference position, e 0 is small, and the magnitude ofė 0 tends to decrease to a small value, corresponding to region II; f (e 0 ) should therefore converge to λ, forcing the gain c * 0 ≈ c 0 − |ė 0 |(1 + λ) to gradually increase to avoid an excessively large control signal.
The blue line in Figs. 5(a), 5(b), and 7 show the experimental results of the proposed adaptive convergence gain for the ABSMC, demonstrating that the overshoot can be reduced. Fig. 5(b) depicts the adaptive gain, c * 0 , to reduce the overshoot of the proposed ABSMC. When excessive position errors occur due to large changes in the motion profile, the convergence gain c * 0 was adapted to reduce the overshoot. Fig. 7 shows the e 0 , e 1 ,ė 0 , and the virtual control signal x ref 2 using the proposed ABSMC+NDO. Using the adaptive convergence gain, the excessive control signal x ref 2 was reduced at region II, as shown in Fig. 7(d), resulting in the reduced overshoot in Fig. 5(a).
The following statements summarize guidance for selecting control parameters for the proposed control method: The ABSMC has parameters c 0 , c 1 , k 1 , k 2 , k 3 , k 4 , α 1 for the BSMC; and parameters δ, λ, η for the adaptive algorithms, which adjust the convergence gain (c 0 , c 1 ) to reduce overshoot under the position-step profile. For a simple design, c 0 was selected to equal c 1 , k 1 = k 3 , and k 2 = k 4 . From analysis of system stability (Section IV, k ac > 0), the resulting c 0 and c 1 were selected to be larger than 0.5 a n . If the gains c 0 and c 1 are small, the system will respond slowly, the transient position error will be large, and the overshoot will be small. If these gains are large, the system will have a faster response and superior robustness, but a large overshoot will occur in the step profile. The parameters (k 1 , k 2 ) and (k 3 , k 4 ) are related to SMC for v q and v d , under the constantproportional rates reaching law [13].
To begin, we configured the BSMC parameters by running the motor with a trapezoidal velocity motion profile but without the combination of NDO. The initial values c 0 and c 1 were chosen to be greater than 0.5 a n . The factors k 1 , k 2 , k 3 , k 4 , and α 1 for the constant-proportional rate reaching law [13] were determined according to experimental results such that system chattering would be effectively reduced. We then gradually increased the c 0 , c 1 to reduce the transient position error to an acceptable value. After pre-tuning the parameters for BSMC, the NDO was combined with BSMC to reduce the effects of disturbances and uncertainties in the motor system. The transient position error should decrease after adding the NDO. The observer gain L 2 = 0 and L 1 was first selected with a small value and then L 1 was gradually increased to obtain acceptable position error. Similarly, keeping the tuned L 1 , we configured L 2 with a small value and gradually increased L 2 to further reduce steady-state position error.
After selecting the parameters for the BSMC and NDO, we operated the motor under the position-step profile to tune the adaptive gain (λ and η) for adjusting the convergence gains c 0 and c 1 . By conducting experiments, we realize that the value of λ is in the range [1; 4], η is in the range [0.1; 2] the proposed control method will provide a better result. The threshold δ was configured to begin at a small value and gradually increase to twice the previous value until the overshoot was reduced.

VI. EXPERIMENTAL RESULTS
This section presents experimental results that demonstrate the practicability of the proposed method. The experimental setup and nominal parameters are shown in Fig. 9 and Table 1.
The experimental system consists of a PMSM attached to an incremental encoder with 5000 lines, or 20000 pulses per revolution to measure the position and speed of the motor shaft. To test the performance under an external load torque, a ZKG-20AN powder clutch was mounted coaxially to the motor shaft, and a CTA3200 brake controller was used to regulate the current applied to the brake powder to adjust the value of the load torque. An STM32F446VC processor (Cortex-M4, 168Mhz) was integrated into the driver board to implement the proposed method. RS232 communication software was connected to the driver module to record experimental data for debugging and plotting results using the MATLAB software.
The effectiveness of the proposed ABSMC+NDO in this research was compared with the PID [37]+NDO, SMC [13]+NDO, super-twisting-like fractional (STLF) controller [16], and non-singular fast terminal sliding mode controller (NFTSMC) [17]. To assure a fair comparison of the proposed method and the other methods, the experiments were conducted under the same conditions. To obtain the best possible system response for each technique, the controller parameters for the PID, SMC, STLF, NFTSMC, and proposed methods were iteratively tuned to select the appropriate parameters by trial and error. Relevant parameters of the five control methods are given below. The parameters for the nonlinear observer gain were L 1 = 900 and L 2 = 1.6. The PID, SMC, STLF, and NFTSMC position controls were based on the cascade control structure. They consisted of the d and q inner PI current controllers. The control gains of the two PI controllers for the d− and q−axis current loops were constructed using a cancellation approach [3]. The cutoff frequency of the current controls was 1000 Hz, and the current control gains were k p = 7 and k i = 8796. The parameters for the PID control were k p,pos = 64.5, k i,pos = 738, and k d,pos = 1. The SMC position control [13] with sliding surface S 3 =ė 0 + c 3 e 0 and the reaching laẇ S 3 = −k 5 sgn(S 3 ) − k 6 S 3 was designed with the parameters  c 3 = 300, k 5 = 15, and k 6 = 30. The (STLF) controller [16] was configured such that α STLF = 0.95, β STLF = 30, ε STLF = −0.05. The (NFTSMC) [17] with l NFTSMC = 191, h NFTSMC = 101, p NFTSMC = 111, q NFTSMC = 101, α NFTSMC = 0.1, β NFTSMC = 0.01, ε NFTSMC = 25, k 1,NFTSMC = 70, k 2,NFTSMC = 0.3, δ NFTSMC = 0.9. The sampling frequency of the current loops was 20 kHz, and the position control loop was 2 kHz. The common parameters for the BSMC and ABSMC were c 0 = c 1 = 180, α 1 = 800, k 1 = k 3 = 700, k 2 = k 4 = 1500. The adaptive parameters for ABSMC were λ = 2.5, η = 0.5, and the threshold δ = 2.
The following results show the performance of the proposed ABSMC+NDO for position control when uncertainties and external load torque are taken into account. The real inertia of the system is J r = J n + J = 111 × 10 −6 kg.m 2 . In this study, the proposed method was tested under the trapezoidal velocity motion profile as shown in Fig. 2, the step profile, and the sinusoidal profile to illustrate the effectiveness of the proposed method. Fig. 10 shows the results of the PID+NDO, SMC+NDO, STLF, NFTSMC+NDO, and ABSMC+NDO. From Fig. 10(a) to 10(e), we can see that the ABSMC+NDO produced the best results, with a maximum position error of 0.008 rad, while the PID+NDO had a maximum error of 0.017 rad, the SMC+NDO had a maximum error of 0.021 rad, the STLF had a maximum error of 0.022 rad, and the NFTSMC had a maximum error of 0.022 rad. As mentioned above in Fig. 3, the ABSMC+NDO and BSMC+NDO performed similarly under trapezoidal velocity motion profiles. The main advantage of the ABSMC is the reduction of overshoot under the step position profile.
To depict the robustness of the proposed method under a sudden external load torque, the brake was controlled to produce an external load torque of 0.12 Nm at 1 s and release the load at 2 s. The position errors of the motor with the PID+NDO, SMC+NDO, STLF, NFTSMC+NDO, and the proposed ABSMC+NDO are shown in Fig. 11. The proposed ABSMC+NDO can produce superior disturbance rejection compared with the other methods in the event of a sudden external load torque. Fig. 5 shows the results of the ABSMC+NDO and BSMC+NDO under the position step profile. As analyzed in Section V, the proposed ABSMC+NDO can reduce overshoot when compared with the BSMC+NDO. Fig. 12  To test the performance of the ABSMC+NDO in a position step profile under an external load torque, an external load torque of 0.12 Nm was added during the step position profile, and the results are shown in Fig. 13. The ABSMC+NDO also had a shorter settling time (0.28 s) compared with the PID+NDO (0.72 s), SMC+NDO (0.67 s), the STLF (0.33 s),   NFTSMC+NDO (0.42 s), and a smaller overshoot compared with the BSMC+NDO (30%). Fig. 14 supplies the experimental results under the sinusoidal position profile of 10sin(1.5πt) with an external load torque of 0.12 Nm. The PID+NDO had the maximum error, settling time, and maximum steady state error results of 2.08 rad, 0.6 s, and 0.015 rad, respectively. The results for the SMC+NDO are 2.14 rad, 0.6 s, and 0.02 rad, respectively. The results for the STLF are 2.33 rad, 0.75 s, and 0.022 rad, respectively. The results for the NFTSMC+NDO are 2.1 rad, 0.31 s, and 0.017 rad, respectively. While the proposed ABSMC+NDO had the best results (0.43 rad, 0.16 s, and 0.009 rad, respectively). Table 2 summarizes the experimental results of the proposed ABSMC+NDO when compared with the PID+NDO, SMC+NDO, STLF, and NFTSMC+NDO. According to these results, the ABSMC+NDO method performed better than the PID+NDO, SMC+NDO, STLF, and NFTSMC+NDO.

VII. CONCLUSION
In this study, an effective method for improving the position response of a PMSM system was proposed based on an ABSMC and a NDO. The ABSMC position controller includes the BSMC and an adaptive convergence gain algorithm. The BSMC is a combination of backstepping and SMC that simplifies the control structure and improves the dynamic response of position tracking. An adaptive convergence gain algorithm was used to reduce the large overshoot position response when the motor operated under a point-topoint position profile. The NDO with a nonlinear observer gain function was proposed to estimate the uncertainty of parameters and external load torque and was used as feedforward compensation for the ABSMC. The stability of the composite control, ABSMC+NDO, is guaranteed by Lyapunov theory. The effectiveness of the proposed method was confirmed by experiments that demonstrated the proposed method can improve the overall performance of the PMSM control system in terms of fast transient response, small steady-state position error, reduced overshoot, and robustness against unknown disturbances when compared with PID, SMC, STLF, and NFTSMC position control methods. From 2020 to 2022, he was with the Department of Electrical and Computer Engineering, Sungkyunkwan University, as a Postdoctoral Researcher. Since 2022, he has been a Lecturer and a Researcher with the Department of Information Technology, FPT University, Can Tho, Vietnam. His research interests include signal processing, motion control, embedded systems, autonomous robotics, and machine learning. From 1990 to 1994, he was a Senior Researcher with Samsung Electronics, Suwon, South Korea. Since 1994, he has been working as an Assistant Professor with the School of Electrical and Computer Engineering, Sungkyunkwan University, Suwon, where he is currently a Professor with the School of Information and Communication Engineering. His research interests include robotics, embedded systems, and factory automation. VOLUME 11, 2023