Synchronous reluctance motor speed control based on multivariable Super-Twisting

Abstract

In order to address issues such as high chattering and insufficient anti-disturbance performance in Synchronous Reluctance Motor (SynRM) speed control systems, this paper proposes an Adaptive Multi-variable Super-Twisting Sliding Mode (AM-STSM) control strategy. First, based on the mathematical model in d-q coordinates, the torque generation principle of SynRM is analyzed. Based on this, an AM-STSM speed controller is designed by introducing time-varying gain terms, achieving rapid convergence when far from the sliding surface and effective suppression of chatter when approaching it. Furthermore, an adaptive Luenberger observer (ALDO) is constructed to observe and compensate for lumped disturbances, significantly enhancing system robustness. In “torque realization” mode, a “Maximum Torque Per Amperage control strategy” is adopted to improve stator current utilization. Simulation and experimental results demonstrate that compared to traditional STSM control methods, the proposed ALDO-AM-STSM approach achieves faster convergence during no-load startup (21.21% reduction). Under sudden load and unload conditions, the speed drop ( decreased by 82.67%), speed overshoot ( decreased by 88.95%), and adjustment time (sudden load: decreased by 47.97%; sudden unload: decreased by 50.36%) are all significantly decreased. Meanwhile, torque and speed fluctuations during steady-state operation are minimized. This control strategy effectively enhances the dynamic performance and anti-disturbance capability of the SynRM speed control system.

Introduction

Synchronous Reluctance Motors (SynRM) are a type of motor that operates using reluctance torque, representing one of the current research hotspots¹. The rotors of these motors are all iron cores, lacking permanent magnets or excitation windings, offering distinct advantages particularly under high-speed operation, high-temperature, and humid conditions. Additionally, this motor features simple manufacturing processes, low cost, and minimal rotor losses^2,3. In applications such as electric vehicles and traction machines, SynRM has emerged as a strong competitor to high-efficiency Induction Motors (IM) or partially Permanent Magnet Synchronous Motors (PMSM)^4,5,6. In recent years, structural optimizations have significantly improved the electrical performance of SynRM motors, including power factor, efficiency, and torque quality⁷. However, the inherent complexity of these motors, coupled with nonlinear and uncertain parameters and external disturbances, makes their effective control highly challenging⁸.

Linear Proportional-Integral (PI) control, due to its simple structure and ease of implementation, is widely used in speed regulation systems. However, when confronted with time-varying system parameters and external disturbances, PI control struggles to meet high-performance control requirements⁹. Consequently, nonlinear control algorithms have emerged. In recent years, driven by rapid advancements in microprocessor chip computing power, nonlinear control algorithms such as Active Disturbance Rejection Control¹⁰, Feedback Linearization¹¹, Model Predictive Control¹², and Sliding Mode Control¹³ have been applied to SynRM speed control. Today, Sliding Mode Control (SMC) stands out among numerous nonlinear control strategies due to its characteristics of low sensitivity to parameter variations, strong anti-disturbance capability, and high tracking accuracy¹⁴. Among numerous nonlinear control strategies, SMC stands out and is widely applied in AC motor control.

Traditional first-order SMC suffers from significant chattering issues due to its use of discontinuous switching functions¹⁵. To address this chattering problem, higher-order sliding mode algorithms emerged, among which Super-Twisting Sliding Mode (STSM) gained significant attention. It was shown in¹⁶ that among four second-order sliding mode algorithms, Twisting and STSM demonstrated outstanding dynamic performance improvements. Furthermore, STSM control exhibited superior convergence speed and response speed compared to Twisting. In¹⁷, STSM was applied to the control of doubly-fed induction generators to reduce instantaneous power error. However, the convergence speed of traditional STSM strategies remains suboptimal. To further enhance convergence speed and robustness, Ref.¹⁸ incorporates a linear term into STSM, proposing a multivariable STSM control strategy. This approach effectively improves convergence speed and robustness. Yet, due to the difficulty in determining disturbance upper bounds, larger sliding mode control gains must be designed to ensure control performance, which exacerbates chattering phenomena. To improve the response speed and stability of systems using multivariable STSM under different operating conditions, Ref.¹⁹ replaced the sign function with a sigmoid function. However, the use of the sigmoid function slows down the convergence speed to some extent.

To relieve the chattering problem in sliding mode control, another effective approach is to employ disturbance observers²⁰. Integrating disturbance observers as feedforward compensation terms in SMC can effectively anticipate and counteract disturbances, thereby enhancing the anti-disturbance capability of the control system²¹. Reference²² designed quasi-sliding mode observers and nonlinear disturbance observers for load disturbance estimation, noting the superior estimation speed of the quasi-sliding mode observer. Reference²³ combined a novel disturbance observer with STSM technology to reduce sliding-mode control gains, thereby enhancing the control system’s disturbance rejection capability. However, the designed observer employed constant gains, making it difficult to achieve a balance between dynamic response and steady-state error. Reference²⁴ employs a sliding mode disturbance observer with feedforward compensation for multivariable STSM. By incorporating an adaptive term into the multivariable STSM, it further reduces the chattering issues caused by excessively large gain values in the design. Reference²⁵ similarly combines a multivariable STSM with adaptive terms and a disturbance observer for PMSM speed control. Nevertheless, the observers in both²⁴ and ²⁵ feature numerous adjustable parameters, increasing parameter tuning complexity and controller development time.

It should be noted that the above literature primarily discusses the application of SMC strategies and disturbance observers in PMSM control. Compared to PMSM control, control research for SynRM is still in its infancy. Regarding SynRM, Ref.²⁶ experimentally analyzed for the first time the performance of STSM applied to SynRM speed control. Compared to traditional SMC, STSM exhibits superior response performance and reduced chattering under parameter variations and external disturbances. To enhance system robustness and control accuracy, Ref.²⁷ fully replaces the PI controller with a third-order sliding mode strategy for both speed and dq-axis current control. However, neither Ref.²⁶ nor Ref.²⁷ addresses disturbance compensation. Reference²⁸ employs a generalized STSM algorithm to construct a speed controller and disturbance observer for SynRM speed control. Simulation results demonstrate that this strategy effectively improves the system’s disturbance rejection capability. Reference²⁹ similarly employs a disturbance observer-based feedforward compensation SMC strategy, combining an extended state observer with high-order sliding mode control incorporating adaptive gains. This approach effectively reduces chattering issues and accelerates convergence. However, neither the strategy proposed in²⁸ nor that in ²⁹ has been validated for practical engineering applications. Reference³⁰ experimentally investigates the application of adaptive nonsingular terminal sliding mode control and nonlinear disturbance observers in SynRM speed control. Results demonstrate that this strategy effectively enhances system dynamic performance and robustness. However, this control strategy involves numerous adjustable parameters and is relatively complex, placing high demands on both the system’s software and hardware.

This paper investigates a dual-loop vector control system (current loop and speed loop) for SynRM. The current loop employs PI control and implements current sharing based on the Maximum Torque Per Ampere (MTPA) control strategy to enhance stator current utilization. The speed loop innovatively adopts a combined approach of a sliding mode control algorithm (STSM) and feedforward compensation. The STSM unit incorporates an adaptive term, forming an adaptive multi-variable STSM (AM-STSM), while the feedforward compensation unit employs an adaptive Luenberger disturbance observer (ALDO) to enhance the system’s disturbance rejection capability. Combining ALDO with AM-STSM yields the ALDO-AM-STSM control strategy. Finally, the performance of this speed control system is analyzed through both simulation and experimental methods, validating the feasibility of the proposed approach. The main contributions of this paper can be summarized in the following three aspects:

• 1) A AM-STSM control scheme has been designed, which automatically adjusts sliding mode gain values based on error states. This effectively accelerates settling time while reducing oscillations near the sliding surface. Stability analysis was performed using Lyapunov functions.This approach effectively accelerates the adjustment time while reducing chattering near the sliding surface. Stability analysis was performed using the Lyapunov function.

• 2) A ALDO was constructed to implement feedforward compensation for the speed controller. The observer gain matrix, incorporating an adaptive term, accelerates convergence speed and reduces observation fluctuations, thereby enhancing the system’s anti-disturbance capability.

• 3) Using a combined simulation and experimental approach, the characteristics of a speed control system for SynRM based on ALDO-AM-STSM were investigated, with comparative analysis conducted against STSM and AM-STSM.

The structure of this paper is as follows: Sect. SynRM speed controller design first establishes the mathematical model of SynRM, details the design process of the AM-STSM controller and ALDO, constructs a Lyapunov function to analyze the stability conditions of the controller, and concludes with a discussion of the MTPA current control algorithm. Section Simulation verification and analysis simulates and compares the performance of STSM, AM-STSM, and the proposed ALDO-AM-STSM controllers. Section Experimental validation and analysis presents experimental investigations of the proposed control scheme. Section Conclusions gives the conclusions.

SynRM speed controller design

SynRM and its mathematical model

The rotor of SynRM is shown in Fig. 1. It can be seen that the rotor is composed only of an iron core and an air magnetic barrier. The quadrant axis (i.e., the q-axis direction) of this motor primarily comprises the air magnetic gap, which has low magnetic permeability. The direct axis (i.e., the d-axis direction) primarily consists of the iron core material, which has high magnetic permeability. This unique rotor magnetic circuit structure results in unequal d-axis and q-axis inductance parameters L_d and L_q, where L_d > L_q. According to the motor’s torque generation principle, when ${L_d} \ne {L_q}$, reluctance torque can be formed.

Fig. 1

Internal structure of SynRM. (a) SynRM rotor, (b) Rotor model.

The stator windings of the SynRM are identical to those of conventional AC motors. The three-phase windings A-B-C can be equivalently represented as shown in Fig. 2. The electrical angle θ_e is the angle between the d-axis and the axis of phase A, and ω_e is the electrical angular velocity. The voltage equation obtained in the three-phase coordinate system based on Fig. 2 is given by Eq. (1) ³¹. In the equation, where u_ABC, i_ABC, R_s, M_ABC, and L_ABC represent the voltage, current, resistance, mutual inductance, and self-inductance of the A-B-C three-phase stator winding, respectively.

$${u_{ABC}}={R_s}{i_{ABC}}+\frac{d}{{dt}}\left( {{M_{ABC}}+{L_{ABC}}} \right){i_{ABC}}$$

(1)

Fig. 2

SynRM A-B-C Coordinate System Equivalent Model.

In the analysis of vector control systems for SynRM, the d-q axis rotating coordinate system model is widely adopted. After applying Clark and Park transformations to Eq. (1), the d-q coordinate system mathematical model for SynRM is obtained, as shown in Eq. (2) ³². Since SynRM lacks a rotor magnetic field excitation source, the voltage equation contains no rotor magnetic flux term. This represents a significant distinction from conventional synchronous motors.

$$\left\{ \begin{gathered} {u_d}={R_s}{i_d}+{L_d}\frac{{d{i_d}}}{{dt}} - {\omega _e}{L_q}{i_q} \hfill \\ {u_q}={R_s}{i_q}+{L_q}\frac{{d{i_q}}}{{dt}}+{\omega _e}{L_d}{i_d} \hfill \\ \end{gathered} \right.$$

(2)

In Eq. (2), u_d and u_q represent the stator voltage components in the d-q axes, i_d and i_q denote the stator current components in the d-q axes, R_s is the stator resistance, and ω_e is the electrical angular velocity.

The electromagnetic torque T_e and motion equations of SynRM, as shown in Eq. (3) ³³.

$$\left\{ \begin{gathered} \frac{{d{\omega _m}}}{{dt}}=\frac{{{T_e}}}{J} - \frac{{{T_L}}}{J} - \frac{B}{J}{\omega _m} \hfill \\ {T_e}=\frac{3}{2}{p_n}\left( {Ld - Lq} \right){i_d}{i_q} \hfill \\ \end{gathered} \right.$$

(3)

In the equation, ω_m is the mechanical angular velocity, T_L is the load torque, B is the friction coefficient, J is the moment of inertia, and p_n is the number of pole pairs.

Improved STSM speed controller design

The control system studied in this paper comprises a speed loop and a current loop. The speed control loop employs an improved SMC algorithm, while the current control loop utilizes an MTPA control algorithm. To discuss the speed loop design, first define the sliding surface $s={e_\omega }={\omega _m} - \omega _{m}^{*}$, where ω_m^* represents the desired mechanical angular velocity. Within the control cycle, ω_m^* can be considered constant, allowing Eq. (3) to be simplified to Eq. (4). Here, h(t) is the lumped disturbance.

$$\left\{ \begin{gathered} \frac{{ds}}{{dt}}=\frac{{d{e_\omega }}}{{dt}}=\frac{{d\left( {{\omega _m} - \omega _{m}^{*}} \right)}}{{dt}}=\frac{{T_{e}^{*}}}{J}+\frac{{({T_e} - {T_e}^{*})}}{J} - \frac{{{T_L}}}{J} - \frac{{B{\omega _m}}}{J}=\frac{{T_{e}^{*}}}{J}+h\left( t \right) \hfill \\ h(t)=\frac{1}{J}\left( {({T_e} - {T_e}^{*}) - {T_L} - B{\omega _m}} \right) \hfill \\ \end{gathered} \right.$$

(4)

To further enhance convergence speed and mitigate chattering caused by excessively large SMC gain values, an adaptive multi-variable STSM (AM-STSM) algorithm was developed, as shown in Eq. (5).

$$\left\{ \begin{gathered} \frac{{ds}}{{dt}}= - {k_1}{\left| s \right|^{\frac{1}{2}}}sign(s) - {k_2}{\varepsilon _1}\left( t \right)s+{u_1} \hfill \\ \frac{{d{u_1}}}{{dt}}= - {k_3}{\varepsilon _2}\left( t \right)sign(s) - {k_4}s \hfill \\ \end{gathered} \right.$$

(5)

In the equation, k₁ > 0, k₂ > 0, k₃ > 0, and k₄ > 0 are the gain values to be designed. ${\varepsilon _1}\left( t \right)$ and ${\varepsilon _2}\left( t \right)$ are adaptive terms, as shown in Eq. (6).

$$\left\{ \begin{gathered} {\varepsilon _1}\left( t \right)=\frac{1}{{{\eta _1}+\left( {1+{{\left| s \right|}^{ - 1}} - {\eta _1}} \right){e^{ - \left| s \right|}}}} \hfill \\ {\varepsilon _2}\left( t \right)=\frac{1}{{{\eta _1}+\left( {1 - {\eta _1}} \right){e^{ - \left| s \right|}}}} \hfill \\ \end{gathered} \right.$$

(6)

In Eq. (6), $0<{\eta _1}<1$. When the system trajectory is far from the sliding surface (i.e., |s| is large), both ${\varepsilon _1}\left( t \right)$ and ${\varepsilon _2}\left( t \right)$ converge to ${\eta _1}^{{ - 1}}$ (greater than 1). As the distance to the sliding surface decreases (i.e., |s| is small), ${\varepsilon _1}\left( t \right)$ and ${\varepsilon _2}\left( t \right)$ gradually approaches 0 and 1, respectively.

From Eq. (4), it can be seen that s is influenced by both T_e^* and h(t). Therefore, two cases can be considered during design.

• (1) When neglecting h(t).

When designing a speed controller, if h(t) is neglected and only T_e^* is considered, then based on Eq. (4) and Eq. (5), the AM-STSM speed controller shown in Eq. (7) can be obtained.

$$\left\{ \begin{gathered} {T_e}^{*}=J\left[ { - {k_1}{{\left| {{e_\omega }} \right|}^{\frac{1}{2}}}sign({e_\omega }) - {k_2}{\varepsilon _1}\left( t \right){e_\omega }+{u_1}} \right] \hfill \\ \frac{{d{u_1}}}{{dt}}= - {k_3}{\varepsilon _2}\left( t \right)sign({e_\omega }) - {k_4}{e_\omega } \hfill \\ \end{gathered} \right.$$

(7)

Considering the characteristics of ε₁(t) and ε₂(t), it can be seen that when the system trajectory deviates significantly from the sliding surface, k₂ε₁(t) and k₃ε₂(t) increase to k₂η₁⁻¹ and k₃η₁⁻¹, respectively. When approaching the sliding mode surface, the k₂ε₁(t) and k₃ε₂(t) terms decrease to 0 and k₃, respectively. It is evident that the AM-STSM automatically adjusts the gain coefficients based on the magnitude of s. This demonstrates that the control scheme not only enhances the system’s convergence speed but also suppresses chattering near the stable phase.

During motor startup, excessive rotational speed deviation can cause the integral term to saturate. This leads to severe overshoot in the speed control system and even oscillation. To this end, a saturation limiter module Sat with a saturation value of ${T_{es}}^{*}$ is incorporated into the speed control loop, and an anti-windup coefficient ξ is introduced. When the integration term is unsaturated, ξ = 1; when the integration is saturated (i.e., $\left| {{T_e}^{*}} \right|>\left| {T_{{es}}^{*}} \right|$), ξ = -1, causing the integration term to exit saturation. After incorporating the anti-windup coefficient, the AM-STSM speed controller is expressed as Eq. (8). Based on Eq. (8), the constructed AM-STSM speed controller is shown in Fig. 3.

$$\left\{ \begin{gathered} {T_e}^{*}=J\left[ { - {k_1}{{\left| {{e_\omega }} \right|}^{\frac{1}{2}}}sign({e_\omega }) - {k_2}{\varepsilon _1}\left( t \right){e_\omega }+{u_1}} \right] \hfill \\ \frac{{d{u_1}}}{{dt}}= - {k_3}{\varepsilon _2}\left( t \right)sign({e_\omega }) - {k_4}\xi {e_\omega } \hfill \\ \end{gathered} \right.$$

(8)

Fig. 3

AM-STSM Speed Controller Block Diagram.

• (2) When considering h(t).

To enhance the system’s dynamic response, the influence of h(t) is further considered during speed control design. An LDO is employed to obtain disturbance estimates $\hat {h}(t)$ and perform feedforward compensation. To facilitate observer design, the mathematical model of the SynRM speed control system is reformulated into state-space form as shown in Eq. (9). In the equation, the variation of h(t) is relatively slow compared to the sampling period of the speed loop, allowing the derivative of h(t) to be considered equal to zero.

$$\left\{ \begin{gathered} \frac{{dx}}{{dt}}=ax+bu \hfill \\ y=cx \hfill \\ \end{gathered} \right.$$

(9)

In the equation, the state variable $x={\left[ {\begin{array}{*{20}{c}} {{\omega _m}}&{h\left( t \right)} \end{array}} \right]^{\text{T}}}$, the system output $y={\omega _m}$, the control input $u=T_{e}^{*}$, the input matrix $b={\left[ {\begin{array}{*{20}{c}} {\frac{1}{J}}&0 \end{array}} \right]^{\text{T}}}$, the output matrix $c=\left[ {\begin{array}{*{20}{c}} 1&0 \end{array}} \right]$, and the state transition matrix $a=\left[ {\begin{array}{*{20}{c}} 0&1 \\ 0&0 \end{array}} \right]$.

According to Eq. (9), the designed LDO is given by Eq. (10) ¹⁴. In this equation, $\hat {y}$ and $\hat {x}$ represent the estimated values of y and x, respectively, while $l={\left[ {\begin{array}{*{20}{c}} {{l_1}}&{{l_2}} \end{array}} \right]^{\text{T}}}$ denotes the gain value of the observer, which is an undetermined parameter.

$$\left\{ \begin{gathered} \frac{{d\hat {x}}}{{dt}}=a\hat {x}+bu+l\left( {y - \hat {y}} \right) \hfill \\ y=c\hat {x} \hfill \\ \end{gathered} \right.$$

(10)

To determine the value of l, the error equation for x is further provided as in Eq. (11), where $\tilde {x}=x - \hat {x}$ represents the estimation error of x.

$$\frac{{d\tilde {x}}}{{dt}}=a\left( {x - \hat {x}} \right)+l\left( {y - \hat {y}} \right)=\left( {a+lc} \right)\tilde {x}$$

(11)

The characteristic equation of Eq. (11) is clearly given by Eq. (12). Here, I denotes the identity matrix and λ represents the eigenvalues.

$$\left| {\lambda I - \left( {a - lc} \right)} \right|={\lambda ^2}+{\alpha _1}\lambda +{\alpha _2}=0$$

(12)

Solving Eq. (12) reveals that selecting ${l_1}=2{\alpha _1}, {l_2}=\alpha _{1}^{2}$ and ${\alpha _1}>0$ enables $\lambda <0$ (i.e., LDO convergence). Clearly, the magnitude of α₁ affects the convergence speed of LDO: a larger α₁ results in faster convergence of the observer. However, an excessively large α₁ leads to significant overshoot and poor steady-state performance. To endow l with adaptive characteristics, $l={\left[ {\begin{array}{*{20}{c}} {2{\varepsilon _3}\left( t \right){\alpha _1}}&{{{\left( {{\varepsilon _3}\left( t \right){\alpha _1}} \right)}^2}} \end{array}} \right]^{\text{T}}}$ is adopted, where ${\varepsilon _3}\left( t \right)$ is defined as in Eq. (13).

$${\varepsilon _3}\left( t \right)=\frac{1}{{{\eta _2}+k\left( {1 - {{\left( {1+{e^{ - k\left| {{e_\omega }} \right|}}} \right)}^{ - 1}}} \right)}}$$

(13)

From the above analysis, it is evident that during operation, the observer requires both the actual rotational speed value ${\omega _m}$ and the estimated rotational speed value ${\hat {\omega }_m}$. Therefore, combining Eq. (10) and Eq. (13) yields an adaptive LDO (ALDO) as expressed in Eq. (14) and Eq. (15).

Disturbance estimation:

$$\frac{{d\hat {h}\left( t \right)}}{{dt}}={l_2}({\omega _m} - {\hat {\omega }_m})={\left[ {{\varepsilon _3}(t){\alpha _1}} \right]^2}({\omega _m} - {\hat {\omega }_m})$$

(14)

Speed estimation:

$$\frac{{d{{\hat {\omega }}_m}}}{{dt}}=\hat {h}\left( t \right)+\frac{{T_{e}^{*}}}{J}+{l_1}({\omega _m} - {\hat {\omega }_m})=\hat {h}\left( t \right)+\frac{{T_{e}^{*}}}{J}+2{\varepsilon _3}(t){\alpha _1}({\omega _m} - {\hat {\omega }_m})$$

(15)

It should be noted that the values of ${\eta _2}$ and k in ${\varepsilon _3}\left( t \right)$ are within the range: $0<{\eta _2}<1$, k > 1. When e_ω is small, ${\varepsilon _3}\left( t \right) \to {\left( {{\eta _2}+0.5k} \right)^{ - 1}}$, thereby reducing the observer’s overcompensation for h(t). When e_ω is large (e.g., during sudden loadings), ${\varepsilon _3}\left( t \right) \to {\eta _2}^{{ - 1}}$, accelerates the observer’s convergence speed.

From Eq. (14), it can be seen that by integrating the current $({\omega _m} - {\hat {\omega }_m})$, the next moment’s $\hat {h}\left( t \right)$ can be estimated. From Eq. (15), it can be seen that using the current $({\omega _m} - {\hat {\omega }_m})$, $\hat {h}\left( t \right)$, and $T_{e}^{*}$, the next moment’s speed ${\hat {\omega }_m}$ can be estimated. Therefore, ALDO can be represented as shown in Fig. 4.

Fig. 4

ALDO Block Diagram.

Combining Eq. (8) and Eq. (10), the proposed controller ALDO-AM-STSM is given by Eq. (16).

$$\left\{ \begin{gathered} {T_{eo}}^{*}=J\left[ { - {k_1}{{\left| {{e_\omega }} \right|}^{\frac{1}{2}}}sign({e_\omega }) - {k_2}{\varepsilon _1}\left( t \right){e_\omega }+{u_1} - \hat {h}\left( t \right)} \right] \hfill \\ \frac{{d{u_1}}}{{dt}}= - {k_3}{\varepsilon _2}\left( t \right)sign({e_\omega }) - {k_4}\xi {e_\omega } \hfill \\ \end{gathered} \right.$$

(16)

Stability proof and analysis

When $\left| {{T_e}^{*}} \right|>\left| {{T_{es}}^{*}} \right|$, the SynRM speed control system reaches saturation, the output of the speed loop is a constant value. At this point, the stability of the speed control system depends on the stability of the current loop. Since the current loop employs a PI controller, the system remains stable during saturation. When $\left| {{T_e}^{*}} \right|<\left| {{T_{es}}^{*}} \right|$, the system is not saturated, and system stability is determined by the speed controller. Since the adaptive terms ${\varepsilon _1}\left( t \right)$ and ${\varepsilon _2}\left( t \right)$ are never negative and are less than or equal to $\eta _{1}^{{ - 1}}$ under all conditions. Therefore, when the system is unsaturated, it is only necessary to prove the stability of the speed controller Eq. (17).

$$\left\{ \begin{gathered} {T_e}^{*}=J\left[ { - {k_1}{{\left| {{e_\omega }} \right|}^{\frac{1}{2}}}sign({e_\omega }) - {k_2}{e_\omega }+{u_1}} \right] \hfill \\ \frac{{d{u_1}}}{{dt}}= - {k_3}sign({e_\omega }) - {k_4}{e_\omega }+\frac{{d\phi }}{{dt}} \hfill \\ \end{gathered} \right.$$

(17)

Assumption

When the disturbance term $\phi$ satisfies $\left| {\frac{{d\phi }}{{dt}}} \right| \leqslant \theta$ and θ > 0, i.e., $\phi$ is continuously differentiable with bounded first derivatives, and k₁, k₂, k₃, and k₄ satisfy the relationship in Eq. (18), then the speed controller in Eq. (17) meets the Lyapunov conditions for asymptotic stability and converges within a finite time.

$$\begin{gathered} {k_1}>{\left( {2\theta } \right)^{0.5}},{k_2}>0,{k_3}>\theta \hfill \\ {k_4}>\frac{{8k_{2}^{2}{k_3}+9k_{1}^{2}k_{2}^{2}+9{\theta ^2}k_{2}^{2}k_{1}^{{ - 2}} - 24\theta k_{2}^{2}}}{{4{k_3}+10\theta k_{2}^{2} - 8{\theta ^2}k_{1}^{{ - 2}}}} \hfill \\ \end{gathered}$$

(18)

Proof

Construct the Lyapunov function shown in Eq. (19).

$$V=2{k_2}\left| {{e_\omega }} \right|+{k_4}e_{\omega }^{2}+0.5u_{1}^{2}+0.5{({k_1}{\left| {{e_\omega }} \right|^{0.5}}sign\left( {{e_\omega }} \right)+{k_2}{e_\omega } - {u_1})^2}$$

(19)

Rewrite Eq. (19) in the form of Eq. (20). Since k₁, k₂, k₃, and k₄ are all larger than zero, V is a positive definite matrix.

$$V={X^{\text{T}}}QX$$

(20)

In Eq. (20), the vector X and matrix Q can be expressed as:

$$\begin{array}{*{20}{c}} {X=\left[ {\begin{array}{*{20}{c}} {{{\left| {{e_\omega }} \right|}^{0.5}}sign\left( {{e_\omega }} \right)} \\ {{e_\omega }} \\ {{u_1}} \end{array}} \right]=\left[ {\begin{array}{*{20}{c}} {{z_1}} \\ {{z_2}} \\ {{z_3}} \end{array}} \right]}&,&{Q=\frac{1}{2}} \end{array}\left[ {\begin{array}{*{20}{c}} {k_{1}^{2}+4{k_2}}&{{k_1}{k_2}}&{ - {k_1}} \\ {{k_1}{k_2}}&{k_{2}^{2}+2{k_4}}&{ - {k_2}} \\ { - {k_1}}&{ - {k_2}}&2 \end{array}} \right]$$

(21)

Equation (20) satisfies the relationship in Eq. (22). Here, ${\left\| X \right\|_2}$denotes the norm of X, while ${\lambda _{1\hbox{min} }}\left\{ Q \right\}$ and ${\lambda _{1\hbox{max} }}\left\{ Q \right\}$ represent the minimum and maximum eigenvalues of matrix Q, respectively, with both ${\lambda _{1\hbox{min} }}\left\{ Q \right\}$ and ${\lambda _{1\hbox{max} }}\left\{ Q \right\}$ being greater than zero.

$$\left\{ \begin{gathered} {\lambda _{1\hbox{min} }}\left\{ Q \right\}\left\| X \right\|_{2}^{2} \leqslant V \leqslant {\lambda _{1\hbox{max} }}\left\{ Q \right\}\left\| X \right\|_{2}^{2} \hfill \\ \left\| X \right\|_{2}^{2}=\left| {{z_2}} \right|+{z_2}^{2}+{z_3}^{2} \hfill \\ \end{gathered} \right.$$

(22)

There are ${z_1}={z_2}{\left| {{z_1}} \right|^{ - 1}}$, ${z_1}^{2}={z_2}^{2}$, and ${\left| {{z_1}} \right|^{ - 1}}={\left| {{z_2}} \right|^{ - 0.5}}$ relationships between z₁ and z₂ in vector X. Differentiate vector X and rewrite it in the form of Eq. (23).

$$\left\{ \begin{gathered} \frac{{dX}}{{dt}}=\frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}AX+CX+E\frac{{d\phi }}{{dt}} \hfill \\ \frac{{d{X^{\text{T}}}}}{{dt}}=\frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}{X^{\text{T}}}{A^{\text{T}}}+{X^{\text{T}}}{C^{\text{T}}}+{E^{\text{T}}}\frac{{d\phi }}{{dt}} \hfill \\ \end{gathered} \right.$$

(23)

In Eq. (23), the matrices A and C and the vector E can be expressed as:

$$\begin{array}{*{20}{c}} {A=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} { - {k_1}}&0&1 \\ 0&{ - 2{k_1}}&0 \\ { - 2{k_3}}&0&0 \end{array}} \right]}&,&{C=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} { - {k_2}}&0&0 \\ 0&{ - 2{k_2}}&2 \\ 0&{ - 2{k_4}}&0 \end{array}} \right]}&,&{E=\left[ {\begin{array}{*{20}{c}} 0 \\ 0 \\ 1 \end{array}} \right]} \end{array}$$

(24)

Combining Eq. (23) and differentiating Eq. (20) yields Eq. (25).

$$\begin{gathered} \frac{{dV}}{{dt}}=\frac{{d{X^{\text{T}}}}}{{dt}}QX+{X^{\text{T}}}Q\frac{{dX}}{{dt}} \hfill \\ \begin{array}{*{20}{c}} {}&{=\frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}\left[ {{X^{\text{T}}}{A^{\text{T}}}QX+{X^{\text{T}}}QAX} \right]} \end{array}+{X^{\text{T}}}{C^{\text{T}}}QX+{X^{\text{T}}}QCX \hfill \\ \begin{array}{*{20}{c}} {}&{+\frac{{d\phi }}{{dt}}\left( {{E^{\text{T}}}QX+{X^{\text{T}}}QE} \right)} \end{array} \hfill \\ \end{gathered}$$

(25)

Since ${E^{\text{T}}}QX={X^{\text{T}}}QE$ and $\phi$ is a bounded constant, Eq. (25) can be rewritten in the form of Eq. (26), where the vector $B={\left[ {\begin{array}{*{20}{c}} { - {k_1}}&{ - {k_2}}&2 \end{array}} \right]^{\text{T}}}.$

$$\frac{{dV}}{{dt}} \leqslant \frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}{X^{\text{T}}}\left( {{A^{\text{T}}}Q+QA} \right)X+{X^{\text{T}}}\left( {{C^{\text{T}}}Q+QC} \right)X+\theta {B^{\text{T}}}X$$

(26)

Define $P= - {A^{\text{T}}}Q - QA$ and $H= - {C^{\text{T}}}Q - QC$, and rewrite Eq. (26) in the form of Eq. (27), where $- {\left| {{z_2}} \right|^{ - 0.5}}{X^{\text{T}}}KX=\theta {B^{\text{T}}}X.$

$$\frac{{dV}}{{dt}} \leqslant - \frac{1}{{{{\left| {{z_2}} \right|}^{\frac{1}{2}}}}}{X^{\text{T}}}\left( {P+K} \right)X - {X^{\text{T}}}HX$$

(27)

Based on matrices A, C, Q, and vector B, matrices P, K, and H in Eq. (27) can be expressed as:

$P=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} {k_{1}^{3}+2{k_1}{k_3}}&{ - {k_2}{k_3}+\frac{3}{2}k_{1}^{2}{k_2}}&{ - k_{1}^{2}} \\ { - {k_2}{k_3}+\frac{3}{2}k_{1}^{2}{k_2}}&{4{k_1}{k_4}+2{k_1}k_{2}^{2}}&{ - \frac{3}{2}{k_1}{k_2}} \\ { - k_{1}^{2}}&{ - \frac{3}{2}{k_1}{k_2}}&{{k_1}} \end{array}} \right]$

$$\begin{array}{*{20}{c}} {K=\frac{{sign\left( {{z_2}} \right)}}{2}\left[ {\begin{array}{*{20}{c}} {2\theta {k_1}}&{\theta {k_2}}&{ - 2\theta } \\ {\theta {k_2}}&0&0 \\ { - 2\theta }&{\begin{array}{*{20}{c}} 0 \end{array}}&0 \end{array}} \right]}&,&{H=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} {k_{1}^{2}{k_2}+4{k_2}{k_3}}&{ - {k_1}{k_4}+\frac{3}{2}{k_1}k_{2}^{2}}&{ - \frac{3}{2}{k_1}{k_2}} \\ { - {k_1}{k_4}+\frac{3}{2}{k_1}k_{2}^{2}}&{2{k_2}{k_4}+2k_{2}^{3}}&{ - 2k_{2}^{2}} \\ { - \frac{3}{2}{k_1}{k_2}}&{ - 2k_{2}^{2}}&{2{k_2}} \end{array}} \right]} \end{array}$$

(28)

Based on the relationship between z₁ and z₂ and the positive definite matrix property of Eq. (20), Eq. (27) is transformed into the form of Eq. (29), where $- {\left| {{z_2}} \right|^{ - 0.5}}{X^{\text{T}}}MX={X^{\text{T}}}NX.$

$$- \frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}{X^{\text{T}}}\left( {P+K} \right)X - {X^{\text{T}}}HX= - \frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}{X^{\text{T}}}\left( {P+K+M} \right)X - {X^{\text{T}}}\left( {H+N} \right)X$$

(29)

In Eq. (29), matrices M and N are expressed as follows:

$$\begin{array}{*{20}{c}} {M=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} 0&{{k_2}{k_3} - \frac{3}{2}k_{1}^{2}{k_2}}&0 \\ {{k_2}{k_3} - \frac{3}{2}k_{1}^{2}{k_2}}&{ - 2{k_1}{k_4}+3{k_1}k_{2}^{2}}&{ - \frac{3}{2}{k_1}{k_2}} \\ 0&{ - \frac{3}{2}{k_1}{k_2}}&0 \end{array}} \right]}&,&{N=} \end{array}\frac{1}{2}\left[ {\begin{array}{*{20}{c}} { - 2{k_2}{k_3}+3k_{1}^{2}{k_2}}&{{k_1}{k_4} - \frac{3}{2}{k_1}k_{2}^{2}}&{\frac{3}{2}{k_1}{k_2}} \\ {{k_1}{k_4} - \frac{3}{2}{k_1}k_{2}^{2}}&0&0 \\ {\frac{3}{2}{k_1}{k_2}}&0&0 \end{array}} \right]$$

(30)

Combining matrices M and N, matrices P and H can be simplified to:

$$\begin{array}{*{20}{c}} {P=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} {k_{1}^{3}+2{k_1}{k_3}}&0&{ - k_{1}^{2}} \\ 0&{2{k_1}{k_4}+5{k_1}k_{2}^{2}}&{ - 3{k_1}{k_2}} \\ { - k_{1}^{2}}&{ - 3{k_1}{k_2}}&{{k_1}} \end{array}} \right]}&,&H \end{array}=\left[ {\begin{array}{*{20}{c}} {2k_{1}^{2}{k_2}+{k_2}{k_3}}&0&0 \\ 0&{{k_2}{k_4}+k_{2}^{3}}&{ - k_{2}^{2}} \\ 0&{ - k_{2}^{2}}&{{k_2}} \end{array}} \right]$$

(31)

When $P+K>0,H>0$, there exists $\frac{{dV}}{{dt}}<0$ satisfying Lyapunov stability condition, the stability condition for the speed control system can be given by Eq. (32).

$$\begin{gathered} {k_1}>{\left( {2\theta } \right)^{0.5}},{k_2}>0,{k_3}>\theta \hfill \\ {k_4}>\frac{{8k_{2}^{2}{k_3}+9k_{1}^{2}k_{2}^{2}+9{\theta ^2}k_{2}^{2}k_{1}^{{ - 2}} - 24\theta k_{2}^{2}}}{{4{k_3}+10\theta k_{2}^{2} - 8{\theta ^2}k_{1}^{{ - 2}}}} \hfill \\ \end{gathered}$$

(32)

When k₁, k₂, k₃, and k₄ satisfy Eq. (32), the eigenvalues of matrices $P+K$ and H are both greater than zero. Therefore, combining with Eq. (22) yields the relationship in Eq. (33). Here, ${\lambda _{1\hbox{min} }}\left\{ {P+K} \right\}$ and ${\lambda _{1\hbox{min} }}\left\{ H \right\}$ denote the minimum eigenvalues of matrices$P+K$ and H, respectively, with ${\lambda _{1\hbox{min} }}\left\{ {P+K} \right\}>0$ and ${\lambda _{1\hbox{min} }}\left\{ H \right\}>0.$

$$\frac{{dV}}{{dt}} \leqslant - \frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}{\lambda _{1\hbox{min} }}\left\{ {P+K} \right\}\left\| X \right\|_{2}^{2}-{\lambda _{1\hbox{min} }}\left\{ H \right\}\left\| X \right\|_{2}^{2}$$

(33)

Equation (22) satisfies the relationship ${\left| {{z_2}} \right|^{0.5}} \leqslant {\left\| X \right\|_2} \leqslant {V^{0.5}}\lambda _{{1\hbox{min} }}^{{ - 0.5}}\left\{ Q \right\}$. Combined with $V \leqslant {\lambda _{1\hbox{max} }}\left\{ Q \right\}\left\| X \right\|_{2}^{2}$, this further yields the relationship in Eq. (34).

$$\left\{ \begin{gathered} \frac{1}{{{{\left| {{z_2}} \right|}^{0.5}}}}\left\| X \right\|_{2}^{2} \geqslant \frac{{\lambda _{{1\hbox{min} }}^{{0.5}}\left\{ Q \right\}}}{{{V^{0.5}}}} \cdot \frac{V}{{{\lambda _{1\hbox{max} }}\left\{ Q \right\}}} \hfill \\ {\lambda _{1\hbox{min} }}\left\{ H \right\}\left\| X \right\|_{2}^{2} \geqslant {\lambda _{1\hbox{min} }}\left\{ H \right\}\frac{V}{{{\lambda _{1\hbox{max} }}\left\{ Q \right\}}} \hfill \\ \end{gathered} \right.$$

(34)

Combining Eq. (33) and Eq. (34) yields the relationship in Eq. (35).

$$\left\{ \begin{gathered} \frac{{dV}}{{dt}} \leqslant - {\delta _1}{V^{0.5}}-{\delta _2}V \hfill \\ {\delta _1}=\frac{{{\lambda _{1\hbox{min} }}\left\{ {P+K} \right\}\lambda _{{1\hbox{min} }}^{{0.5}}\left\{ Q \right\}}}{{{\lambda _{1\hbox{max} }}\left\{ Q \right\}}}>0,{\delta _2}=\frac{{{\lambda _{1\hbox{min} }}\left\{ H \right\}}}{{{\lambda _{1\hbox{max} }}\left\{ Q \right\}}}>0 \hfill \\ \end{gathered} \right.$$

(35)

Since both ${\delta _1}$ and ${\delta _2}$ are greater than zero in Eq. (35), it follows that Eq. (17) converges to zero in finite time. Thus, the hypothesis is proven.

Current loop design

After the speed loop calculates the desired torque ${T_{eo}}^{*}$, it serves as the input to the current loop. The current loop converts ${T_{eo}}^{*}$ into the desired stator current based on the selected control strategy. As shown by the electromagnetic torque equation in Eq. (3), an infinite number of current possibilities exist for a given ${T_{eo}}^{*}$. To obtain the minimum stator current i_s among them; this paper adopts the MTPA vector control strategy. This strategy aims to generate the required electromagnetic torque output using the smallest possible i_s, which can be equivalently formulated as solving the extremum problem of Eq. (36).

$$\left\{ \begin{gathered} {\text{constant}}:{T_e}=\frac{3}{2}{p_n}\left( {{L_d} - {L_q}} \right){i_d}{i_q} \hfill \\ \hbox{min} :i_{s}^{2}={i_d}^{2}+{i_q}^{2} \hfill \\ \end{gathered} \right.$$

(36)

To further compute i_d and i_q in Eq. (36), we construct the auxiliary function Eq. (37), where µ is the Lagrange multiplier.

$$K\left( {{i_d},{i_q},\mu } \right)={i_d}^{2}+{i_q}^{2}+\mu [T_{e}^{*} - \frac{3}{2}{p_n}\left( {{L_d} - {L_q}} \right){i_d}{i_q}]$$

(37)

For Eq. (37), find the partial derivatives of i_d, i_q, and µ, and set the partial derivatives equal to zero, yielding the relationship in Eq. (38).

$$\left\{ \begin{gathered} \frac{{\partial K}}{{\partial {i_d}}}=2{i_d} - \frac{3}{2}\mu {p_n}({L_d} - {L_q}){i_q}=0 \hfill \\ \frac{{\partial K}}{{\partial {i_q}}}=2{i_q} - \frac{3}{2}\mu {p_n}({L_d} - {L_q}){i_d}=0 \hfill \\ \frac{{\partial K}}{{\partial \mu }}={T_e} - \frac{3}{2}{p_n}({L_d} - {L_q}){i_d}{i_q}=0 \hfill \\ \end{gathered} \right.$$

(38)

Solving the first two equations of Eq. (38) yields the relationship ${i_d}= \pm {i_q}$. Substituting this into the third equation of Eq. (38) provides the desired currents i_q^* and i_d^* for the current loop, as shown in Eq. (39).

$$\left\{ \begin{gathered} i_{q}^{*}=i_{d}^{*},\;\;T_{{eo}}^{*}>0 \hfill \\ i_{d}^{*}=\sqrt {\frac{{T_{{eo}}^{*}}}{{1.5{p_n}\left( {{L_d} - {L_q}} \right)}}} \hfill \\ \end{gathered} \right.\begin{array}{*{20}{c}} {{\text{and}}}&{\left\{ \begin{gathered} i_{q}^{*}= - i_{d}^{*},\;\;T_{{eo}}^{*}<0 \hfill \\ i_{d}^{*}=\sqrt {\frac{{ - T_{{eo}}^{*}}}{{1.5{p_n}\left( {{L_d} - {L_q}} \right)}}} \hfill \\ \end{gathered} \right.} \end{array}$$

(39)

Simulation verification and analysis

To investigate the performance of the SynRM speed control system based on ALDO-AM-STSM, a corresponding simulation system was established in Matlab/Simulink, as shown in Fig. 5. In the simulation model, the solver was configured as a fixed-step discrete system without continuous states, with a step size T_s = 1e-6 s. The sampling time for current, torque, and electrical angle was set to T_pwm = 1e-5 s, while the speed sampling time was 10 times T_pwm to simulate the 10 kHz switching frequency of the experimental equipment. The speed loop comprises an AM-STSM controller and an ALDO. The AM-STSM block generates the desired torque value T_e^* (without disturbance consideration) based on speed deviation. This T_e^* undergoes feedforward compensation via the ALDO stage to yield the desired torque value ${T_{eo}}^{*}$ (with disturbance consideration). The ${T_{eo}}^{*}$ serves as the input to the current control module, where the MTPA current control unit solves for i_d^* and i_q^* according to Eq. (39). Finally, the current loop combines PI control with voltage compensation to enable the actual current to rapidly track i_d^* and i_q^*.

Fig. 5

Block Diagram of the Speed Control System for the SynRM.

The design values in AM-STSM are k₁ = 350, k₂ = 45, k₃ = 5000, k₄ = 35, and η₁ = 0.6, which clearly satisfy the stability condition of Eq. (32). The design values in ALDO are η₂ = 0.5, k = 9, and α₁ = 750. The parameters of the d-q axis current PI controller in the current loop are designed as K_pd = 226.08, K_pq = 108.6, and K_id = K_iq = 3756.6. The SynRM parameters used in the simulation are listed in Table 1.

Table 1 Parameters of SynRM.

The gain trajectories of the adaptive terms ${\varepsilon _1}\left( t \right)$, ${\varepsilon _2}\left( t \right)$, and ${\varepsilon _3}\left( t \right)$ in Eq. (6) and Eq. (13) are shown in Fig. 6. During no-load startup and sudden load application, the angular velocity error |e_ω| is large, causing the system trajectory to deviate significantly from the sliding surface. The ${\varepsilon _1}\left( t \right)$ and ${\varepsilon _2}\left( t \right)$ terms increase and approach $\eta _{1}^{{ - 1}}$ (approximately 1.67), enabling the system to rapidly converge toward the sliding surface (s = 0). At this point, ${\varepsilon _3}\left( t \right)$ approaches 2, accelerating the convergence of the observed values. After system stabilization, ${\varepsilon _1}\left( t \right)$, ${\varepsilon _2}\left( t \right)$, and ${\varepsilon _3}\left( t \right)$ converge to 0, 1, and 0.2 respectively, effectively suppressing chattering after reaching the sliding surface. The observer gain α₁ is reduced to 150 to minimize fluctuations in the observed values after stabilization, meeting the design requirements for the adaptive terms.

Fig. 6

Adaptive term gain value trajectory.

The speed response curves for AM-STSM and STSM are shown in Fig. 7. During no-load startup, both control schemes exhibit approximately 10 r/min of overshoot, converging to the desired speed at 0.1 s and 0.13 s, respectively. However, AM-STSM demonstrates faster convergence and response speeds. To evaluate anti-disturbance capabilities, a sudden 7 N·m load was applied at 1 s. STSM dropped from 1500 r/min to 1250 r/min, exhibiting a 250 r/min drop (16.67% drop) and recovered to the setpoint within 0.4 s. However, the AM-STSM reduced the drop by 150 r/min (a decrease by 60%) and shortened the recovery time by approximately 38%.

Fig. 7

Speed Response of AM-STSM Controller and STSM Controller.

The electromagnetic torque response of the speed control system is shown in Fig. 8. During no-load startup, the torque output of both control schemes is limited to 10.5 N·m, approximately 1.5 times the rated torque. As shown in Fig. 8a, during startup, the AM-STSM system exhibits adaptive capabilities, resulting in a higher current sliding mode control gain. This causes the electromagnetic torque to reach the limit value, enabling rapid startup. When the no-load speed exceeds the desired, the AM-STSM’s sliding-mode gain rapidly decreases, significantly suppressing chattering near the sliding surface (s = 0) and quickly entering a stable state, concluding the start-up. As shown in Fig. 8b, after applying a sudden 7 N·m load at 1 s, the STSM’s torque output peaks at 7.45 N·m and stabilizes around 1.4 s. The AM-STSM achieves a maximum torque output of 7.2 N·m, stabilizing around 1.23 s. After stabilization, both controllers exhibit minor ripples of 0.1 N·m. Simulation results demonstrate that the AM-STSM exhibits smaller overshoot, faster response, and faster convergence speed.

Fig. 8

Electromagnetic Torque Response of AM-STSM Controller and STSM Controller. (a) No-load start, (b) Sudden load.

To investigate the impact of the adaptive amplification factor $\eta _{1}^{{ - 1}}$ on the performance of the speed control system, simulations were conducted under no-load start and sudden load application (7 N·m) conditions while keeping all other parameters constant and varying only ${\eta _1}$, as shown in Fig. 9. When ${\eta _1}=0.9$, the speed overshoot during no-load start-up is minimal, approximately 5 r/min (0.33% overshoot), while the speed drop during sudden load application is most significant, approximately 100 r/min (6.67% drop). When ${\eta _1}=0.2$, the speed drop during sudden load application was minimal, approximately 55 r/min (3.67% drop), but the overshoot during no-load start was maximum, approximately 30 r/min (2% overshoot). Simulation data indicates that reducing ${\eta _1}$ enhances the system’s resistance to load disturbances but increases overshoot during no-load starts. Therefore, the ${\eta _1}$ value should be selected based on actual requirements.

Fig. 9

Effect of Adaptive Item amplification factor $\eta _{1}^{{ - 1}}$ on speed. (a) η₁ = 0.2, 0.4, 0.6, (b) η₁ = 0.6, 0.8, 0.9.

A comparison of disturbance observations between ALDO and LDO is shown in Fig. 10. The initial α₁ values for the two observers were set to 750 and 1500, respectively. When disturbances are present, ALDO’s adaptive term amplifies α₁ to 1500, resulting in identical gain matrix values. Therefore, when a 7 N·m load is suddenly applied at 1 s, both observers exhibit the same overshoot of approximately 2.5 N·m (35.71% overshoot). After the rotational speed stabilizes, ALDO’s adaptive term reduces α₁ to 150, resulting in smaller fluctuations in ALDO’s observed values and significantly improving observation accuracy.

Fig. 10

Observed Values of ALDO and LDO for Lumped Disturbances.

The trajectory of the anti-windup term in Eq. (16) is shown in Fig. 11a. During the motor start-up phase, the $- {k_4}\xi {e_\omega }$ term is a large positive value, enabling the integral term to rapidly exit saturation. After exiting saturation, the $- {k_4}\xi {e_\omega }$ term becomes negative to accelerate the elimination of steady-state speed error. Rewriting the derivative term in Eq. (16) as $- {k_3}{\varepsilon _2}\left( t \right)sign({e_\omega }) - {k_4}{e_\omega }+{k_5}\left( {T_{e}^{*} - T_{{es}}^{*}} \right)$ yields the standard anti-windup method described in reference³⁴. A comparison of the speed response between this method and the proposed anti-windup strategy is shown in Fig. 11b. When k₅ is appropriately designed, both strategies exhibit comparable performance in terms of speed overshoot and recovery time.

Fig. 11

Anti-windup Strategy Validation and Comparison. (a) Anti-windup Strategy Validation, (b) Comparison of Two Anti-windup Strategies.

During motor operation, parameter shifts may occur. A comparison of speed responses under inductance-resistance parameter mismatches is shown in Fig. 12. The inductance and resistance parameters of the motor model were varied from (L_dq, R_s) to (0.7L_dq, 1.3R_s) and (0.5L_dq, 1.5R_s), respectively. Figure 12 shows that when the parameter mismatch is small, there are no significant changes in the overshoot during startup, the speed drop under sudden load, or the speed stabilization time. When the parameter mismatch is significant, the speed overshoot during startup increases by 5 r/min (increase of 50%), and the speed drop under a sudden 7 N·m load increases by 20 r/min (increase of 50%). Both increases are relatively small. This demonstrates that the ALDO-AM-STSM exhibits good parameter robustness.

Fig. 12

Impact of mismatch in resistive and inductive parameters on control performance.

To investigate the impact of ALDO on the system, the speed responses of ALDO-AM-STSM and AM-STSM (without ALDO) were compared as shown in Fig. 13. After the sudden load application (7 N·m), the speed drop of ALDO-AM-STSM was smaller, amounting to 60 r/min (a 60% reduction compared to AM-STSM), and it recovered to the desired speed within 0.35 s. At 1.3 s after load unloading, ALDO-AM-STSM exhibited a smaller overshoot (approximately 25 r/min, 1.67% overshoot) and shorter recovery time. Results indicate that compared to AM-STSM, ALDO-AM-STSM demonstrates superior performance in both speed drop and recovery time.

Fig. 13

Load Sudden Change Speed Response Curve.

Experimental validation and analysis

The SynRM controller’s hardware utilizes the TMS320F28335 as its main control chip, implementing the previously discussed control algorithms as software programs. To investigate the experimental characteristics of SynRM speed control, a hysteresis brake was employed to apply a load to the motor, capable of exerting up to 10 N·m. The constructed experimental platform is shown in Fig. 14, where the hysteresis brake, dynamic torque sensor, and motor rotate coaxially during operation. Speed information is obtained from an optoelectronic encoder with a resolution of 3600 pulses per revolution. Three-phase current information is obtained after processing by the AD7616 chip, which has a resolution of 16 bits. The dynamic torque sensor displays the real-time load value applied by the brake, along with the motor speed and power output. The inverter switch’s PWM interrupt frequency is 10 kHz. Sampling times for current, electrical angle, and speed match those set in the simulation (speed sampling interval: 1 ms; current and electrical angle sampling interval: 0.1 ms).

Fig. 14

SynRM Speed Control Experimental Platform.

During the experiment, the relevant parameters in the AM-STSM were set as follows: k₁ = 20, k₂ = 26, k₃ = 0.65, k₄ = 0.0016, and ${\eta _1}=0.6$. The corresponding parameters in ALDO were set as: ${\eta _2}=0.5$, k = 9, α₁ = 0.075. Among them, k₃, k₄, and α₁ are all parameter values after integral discretization multiplied by 10 times T_pwm. The integral and proportional terms in STSM were identical to those in AM-STSM. The speed was uniformly set to 1000 r/min during the experiments.

At a load value of 5 N·m, the three-phase stator currents after motor stabilization are shown in Fig. 15. The current waveforms are essentially sinusoidal with an amplitude of approximately 3.95 A. The disturbance observation values of the control system are depicted in Fig. 16. A disturbance of 5.5 N·m was applied at 5.5 s, and the stabilized ALDO observation value was approximately 5.6 N·m, slightly higher than the actual value. Experiments demonstrate that ALDO can effectively track lumped disturbances.

Fig. 15

Three-phase stator current during stable operation.

Fig. 16

Lumped disturbance observation value.

A comparison of stator current waveforms using MTPA and non-MTPA ($i_{d}^{*}=0.4\left| {{T_{eo}}^{*}} \right|,{\text{ }}i_{q}^{*}={T_{eo}}^{*}$) current control is shown in Fig. 17. After no-load start-up speed stabilization, the current consumption of both current allocation strategies is essentially identical. Under a 5 N·m load, the average stator current of the system employing the MTPA current allocation strategy is lower than that of the non-MTPA control (reduced by 0.33 A), consistent with theoretical analysis.

Fig. 17

Stator Current Waveforms for Different Current Distribution Strategies. (a) MTPA Control, (b) Non-MTPA control.

During no-load startup, the speed and current response curves for the three speed control schemes (Scheme 1: ALDO-AM-STSM; Scheme 2: AM-STSM; Scheme 3: STSM) are shown in Fig. 18, with corresponding data listed in Table 2. Compared to the other two schemes, Scheme 3 exhibits the smallest peak starting current. After stabilization, all three control schemes exhibit good current sinusoidal quality. Regarding overshoot, Scheme 2 and Scheme 3 are comparable, with overshoots of approximately 10 r/min and 9 r/min, respectively. Scheme 1 exhibits a larger starting current due to the ${T_{eo}}^{*}$ supplied to the MTPA during the initial startup phase, which results from the superposition of the disturbance observer and sliding mode controller components. Consequently, Scheme 1 demonstrates a slightly larger overshoot of approximately 34 r/min. Schemes 1 and 2 stabilized at the set speed in 0.52 s and 0.56 s, respectively, both converging faster than Scheme 3 (Scheme 1 is 7.14% faster than Scheme 2 and 21.21% faster than Scheme 3.).

Fig. 18

No-Load Start-Up Experiment. (a) ALDO-AM-STSM, (b) AM-STSM, (c) STSM.

The response curves for rotational speed, electromagnetic torque, and stator current during 2 N·m load startup are shown in Fig. 19. During load startup, no overshoot was observed in any of the three control schemes. After speed stabilization, all schemes exhibited minor fluctuations of approximately ± 1.5 r/min, with good sinusoidal phase current waveforms. Among the three control schemes, Scheme 1 achieved the shortest convergence time of approximately 0.63 s. Scheme 3 produced the smallest torque peak. Following speed stabilization, all three control methods exhibited torque ripples of approximately 0.3 N·m.

Fig. 19

Load Start-up Experiment. (a) ALDO-AM-STSM, (b) AM-STSM, (c) STSM.

The response to sudden load changes during operation is shown in Fig. 20, with corresponding data listed in Table 3. After applying a sudden load (5 N·m), Schemes 1, 2, and 3 exhibited speed reductions of approximately 43 r/min, 102 r/min, and 248 r/min, respectively. Scheme 1 demonstrated the smallest reduction and the shortest time to recover to the desired speed. Compared to scheme 3, scheme 1 reduces speed drop by 82.67% and adjustment time by 47.96%. Compared to scheme 2, it reduces speed drop by 57.84% and adjustment time by 17.95%. After speed stabilization, all three control schemes exhibited approximately 0.8 N·m (15%) of torque ripple. After sudden load removal, Scheme 1 demonstrated the smallest speed overshoot and fastest convergence speed. Compared to Scheme 2 and Scheme 3, Scheme 1 reduced overshoot by 61.09 r/min (decreased by 66.49%) and 247.74 r/min (decreased by 88.95%), respectively, while shortening convergence time by 0.17 s and 0.69 s, respectively. Scheme 1 demonstrated superior anti-disturbance capability.

Fig. 20

Anti-disturbance Performance Experiment. (a) ALDO-AM-STSM, (b) AM-STSM, (c) STSM.

Table 2 No-load startup experimental data.

Table 3 Anti-disturbance experimental data.

Based on the combined results of no-load start, load start, and anti-disturbance tests, it is evident that applying the ALDO-AM-STSM controller to SynRM speed control delivers superior recovery characteristics and anti-disturbance capability. These properties are particularly crucial for electric vehicle applications. For instance, during overtaking maneuvers or hill climbs, the drive motor faces sudden torque demands. This solution ensures smooth speed transitions, eliminating jerky sensations to enhance ride comfort and safety. Simultaneously, when robotic arms grasp workpieces, it guarantees stable operation, preventing vibrations or positioning errors caused by abrupt load changes.

Conclusions

This paper proposes a composite control scheme combining adaptive multivariable super-twisting sliding mode (AM-STSM) control with an adaptive Luenberger disturbance observer (ALDO) to enhance the dynamic and anti-disturbance performance of the SynRM speed control system. Through theoretical analysis, simulation studies, and experimental validation, the following conclusions are obtained:

• 1) The proposed AM-STSM controller effectively balances the trade-off between system response speed and steady-state accuracy through an adaptive gain scheme. The adaptive terms ${\varepsilon _1}\left( t \right)$ and ${\varepsilon _2}\left( t \right)$ dynamically adjust gains based on rotational speed error, enabling strong convergence during startup and large disturbance phases while smoothly suppressing sliding mode chattering as steady-state approaches.

• 2) The designed ALDO achieves high-precision observation and real-time compensation for lumped disturbances. Its adaptive gain ${\varepsilon _3}\left( t \right)$ ensures rapid response during dynamic processes and smooth decay in steady-state phases, thereby enhancing system robustness while avoiding excessive observation noise.

• 3) Simulation and experimental results demonstrate that compared to traditional STSM, ALDO-AM-STSM achieves shorter adjustment times during no-load startup, but with a slight increase in speed overshoot. Under sudden load additions, it reduces speed drop by 82.67% and shortens adjustment time by approximately 47.96%. During sudden load shedding, it reduces speed overshoot by 88.95% and shortens regulation time by approximately 50.36%, while significantly reducing steady-state torque and speed fluctuations.

ALDO-AM-STSM provides an effective solution for addressing chattering and anti-disturbance issues in SynRM speed control systems, significantly enhancing the system’s dynamic and static performance as well as control quality. Compared to nonlinear control strategies such as model predictive control, feedback linearization, and nonlinear optimal control, ALDO-AM-STSM requires lower model accuracy, imposes a lighter computational burden, and is simpler to implement. However, further research will be conducted on the parameter system adjustment of this strategy, the mismatch perturbation caused parameter mismatches, and control adaptability issues during wide speed range operation.