Disturbance-Observer-Based Sliding-Mode Speed Control for Synchronous Reluctance Motor Drives via Generalized Super-Twisting Algorithm

Abstract

In this study, a novel composite speed controller combining a sliding-mode speed controller with a disturbance observer is proposed for the vector-controlled synchronous reluctance motor (SynRM) drive system. The proposed composite speed controller employs the generalized super-twisting sliding-mode (GSTSM) algorithm to construct both the speed controller and the disturbance observer. The GSTSM speed controller is utilized to stabilize the speed tracking error dynamics in finite time, while the GSTSM disturbance observer compensates for the total disturbance in the speed tracking error dynamics, which includes external disturbances and parametric uncertainties. Under the framework of the constant direct-axis current component vector control strategy for the SynRM drive system, comparative simulation studies are conducted among the standard STSM speed controller, the GSTSM speed controller, the composite speed controller using a GSTSM speed controller and a standard STSM disturbance observer, and the proposed composite speed controller. The effectiveness and superiority of the proposed composite speed controller are verified through simulation results.

1. Introduction

The synchronous reluctance motor (SynRM) distinguishes itself in comparison with other AC motors through its attractive advantages: relative to the induction motor (IM), it exhibits lower losses and a higher power factor; compared to the permanent-magnet synchronous motor (PMSM), it offers the benefit of lower costs by avoiding the need for expensive permanent-magnet materials; and against the switched reluctance motor, it features reduced torque ripple and noise [1,2,3,4,5]. Therefore, a century after its first introduction [6], the SynRM has gained increasing attention in the present day. In recent years, researchers worldwide have explored the application of SynRM across various fields, such as in electric and hybrid vehicles [7,8,9], water pumps [10,11,12], wind energy conversion systems [13,14,15], cold rolling mills [16,17], and uninterruptible power supplies [18]. Within the industrial sector, over the past two decades, the SynRM has been regarded as a promising alternative to IMs [5]. Some companies have recently launched a range of commercial SynRMs for adjustable-speed AC motor drive applications [19,20].

Regarding advanced control strategies for the adjustable-speed SynRM drive system, the vector control strategy is widely favored [21,22,23,24,25,26,27]. This control strategy employs a dual-loop structure, comprising speed and current control loops, to facilitate effective speed tracking. The proportional–integral (PI) speed controller, designed by treating the speed tracking error dynamics of the SynRM drive system as a linear system, is the most commonly utilized speed controller in the speed control loop. In theory, this linear speed controller can achieve asymptotic stabilization of the speed tracking error dynamics under constant disturbances, leveraging its integral term [28]. Nevertheless, the speed tracking error dynamics of the SynRM drive system represent a nonlinear system that is subject to multiple time-varying external disturbances and parametric uncertainties [29]. Theoretically, the PI speed controller cannot stabilize this system, leading to a degradation in speed tracking performance in practice [28]. To date, various nonlinear control theories have been proposed to stabilize the nonlinear system, either asymptotically or in finite time, under time-varying disturbances. Consequently, various nonlinear speed controllers based on these control theories have been developed for the vector-controlled SynRM drive system, including adaptive speed controllers [30,31,32], backstepping speed controllers [33,34], predictive speed controllers [29,35,36], neural network speed controllers [37,38], and sliding-mode speed controllers [39,40,41,42,43,44,45].

The sliding-mode control theory is highly regarded within the nonlinear control community for its compelling attributes, such as robustness against disturbances, simplicity of implementation, and finite-time convergence [46,47]. Based on this theory, sliding-mode speed controllers have been extensively employed in robust vector control strategies for AC motor drives [47,48,49]. Generally, designing a sliding-mode speed controller involves two steps. The first step involves selecting a sliding variable related to the speed tracking error. The second step entails designing a sliding-mode speed control law to drive the selected sliding variable to zero in finite time, thereby achieving the speed tracking. The conventional sliding-mode speed controller is designed using the first-order sliding-mode algorithm. This speed controller directly employs a high-gain discontinuous signum function to address the total disturbance encompassing external disturbances and parametric uncertainties in the speed tracking error dynamics, ensuring the finite-time convergence of the selected sliding variable to zero in theory. This type of speed controller has been successfully integrated into the vector-controlled SynRM drive system [39,40,41]. However, directly using the high-gain discontinuous signum function in the speed controller results in the well-known chattering phenomenon, subsequently degrading speed tracking performance in practice.

The quasi-sliding-mode algorithm and the second-order sliding-mode (SOSM) algorithm are two widely used improvements on the conventional first-order sliding-mode algorithm for chattering alleviation. The former employs a continuous function to replace the discontinuous signum function. However, this continuous function-based speed control law can only ensure the finite-time convergence of the selected sliding variable to a selected small vicinity around zero in theory [46,47]. In contrast, the latter can maintain the finite-time convergence of the selected sliding variable directly to zero theoretically [46]. Consequently, the SOSM algorithm has found wide applications in recent years [50,51,52,53,54,55,56]. Among existing SOSM algorithms, the standard super-twisting sliding-mode (STSM) algorithm is the most widely used. Its distinguishing features include the elimination of the need for the time derivative of the selected sliding variable and its applicability to both controller and observer designs for systems with a relative degree of one. Consequently, standard STSM controllers and observers have been integrated into numerous high-performance control strategies for adjustable-speed AC motor drive systems in the past few years [43,57,58,59,60]. The standard STSM speed controller has been designed for the vector-controlled SynRM drive system [43]. The standard STSM algorithm, however, still relies on bounded signum function-based nonlinear feedback terms of the tracking error, which results in relatively slow convergence of the sliding variable when it is far from zero [61,62]. Increasing the gains of the standard STSM algorithm can enhance the convergence rate of the sliding variable. However, since the standard STSM algorithm can only alleviate, not eliminate, the chattering phenomenon, increasing its gains will make this issue more severe. To address this challenge, the modified STSM algorithm and the generalized STSM (GSTSM) algorithm were proposed in [61,63], respectively. In these two improved STSM algorithms, extra linear feedback terms of the tracking error, which have relatively fast convergence of the sliding variable when it is far from zero, are added, thus improving the convergence rate of the sliding variable. To date, these two improved STSM algorithms, especially the GSTSM algorithm, have been applied to several advanced control systems, including those for adjustable-speed IM and PMSM drive systems [64,65], servo systems [66,67,68], aircraft [69,70,71,72], and robots [73,74]. However, the GSTSM speed controller for the vector-controlled SynRM drive system has yet to be explored.

Similar to their standard counterpart, the aforementioned improved STSM algorithms employ only a feedback regulation mechanism. Thus, tuning their gains involves compromises among various performance indices, which may degrade their performance in certain scenarios [28]. The disturbance-observer-based control is an effective approach to address this challenge [28,75]. It integrates a disturbance observer with the feedback controller to form a composite controller. This integration introduces a feedforward compensation mechanism specifically designed to deal with disturbances. Consequently, it enhances the control system’s degree of freedom, significantly improving the control performance. Until now, the standard STSM disturbance observer (STSM-DO) has been integrated with several nonlinear controllers, such as the predictive controller [76], the backstepping controller [77], the adaptive controller [78], the complementary sliding-mode controller [79], and the standard STSM controller [80], to construct composite controllers for vector-controlled AC motor drive systems. Nevertheless, to the best of the author’s knowledge, no published literature investigates the standard STSM-DO-based composite controller for vector-controlled SynRM drive systems, let alone their GSTSM-DO-based composite controller.

The main contribution of this study is the first-time proposal of a novel composite speed controller based on the GSTSM algorithm for the vector-controlled SynRM drive system. The proposed composite speed controller comprises the GSTSM speed controller and the GSTSM-DO. The former is employed for the speed tracking, while the latter compensates for the total disturbance in the speed tracking error dynamics, which lumps together external disturbances and parametric uncertainties. Using the constant direct-axis current component vector control strategy for the SynRM drive system as a framework, simulation tests are performed to compare the performance of the proposed composite speed controller with that of the standard STSM speed controller, the GSTSM speed controller, and the composite speed controller using a GSTSM speed controller and a standard STSM-DO.

This rest of this article is structured as follows: Section 2 gives the dynamics of the SynRM drive system, along with the fundamentals of the vector control strategy adopted for this drive system. The design of the proposed composite speed controller is elaborated on in Section 3. Comparative simulation results are presented and analyzed in Section 4. The conclusion and future work are delineated in Section 5.

2. Preliminaries

2.1. Dynamics of the SynRM Drive System

In this study, the two-level voltage-source inverter-fed SynRM drive system is taken into account, which is depicted in Figure 1. Here, $U_{dc}$ represents the DC-link voltage, C denotes the DC-link capacitor, $Q_{ij}$ and $D_{ij}$ (i $=$a, b, c, j$=$1,2) represent the power switching device and the freewheeling diode, respectively, and $i_{sa}$, $i_{sb}$, and $i_{sc}$ denote three-phase stator currents. In this study, the adopted model for the two-level voltage-source inverter accounts for nonlinearities, comprising dead time, turn-on/-off time, saturation voltage of the power switching device, and diode forward voltage.

The switching function for each phase $S_i$ is defined as

$$S_i=\left\{\begin{array}{l}1, \mathrm{if} Q_{i1} \mathrm{on}, Q_{i2} \mathrm{off}\\ 0, \mathrm{if} Q_{i1} \mathrm{off}, Q_{i2} \mathrm{on}\end{array}\right.$$

(1)

For each phase of the two-level voltage-source inverter, over a switching period $T_s$, the relation of the actual conducting time to the applied conducting time for $Q_{i1}$ is articulated by (2), while the high-frequency pole voltage model is described as (3) [81,82,83,84,85].

$$T_i=T_i^{*}+\left(T_{off}-T_{on}-T_{dead}\right)\mathrm{sgn}\left(i_{si}\right)$$

(2)

$$u_{io}=\left(U_{dc}-U_{sat}+U_{diode}\right)\left(S_i-\frac{1}{2}\right)-\frac{1}{2}\left(U_{sat}+U_{diode}\right)\mathrm{sgn}\left(i_{si}\right)$$

(3)

where $T_i$ indicates the actual conducting time for $Q_{i1}$, $T_i^{*}$ represents the applied conducting time for $Q_{i1}$, $T_{on}$ represents the turn-on time of $Q_{ij}$, $T_{off}$ denotes the turn-off time of $Q_{ij}$, $T_{dead}$ indicates the dead time of $Q_{ij}$, $u_{io}$ is the pole voltage for the phase i, $U_{sat}$ indicates the saturation voltage of $Q_{ij}$, $U_{diode}$ represents the diode forward voltage of $D_{ij}$, and sgn($i_{in}$) is the signum function, defined as follows:

$$\mathrm{sgn}\left(i_{si}\right)=\left\{\begin{array}{l}+1, \mathrm{if} i_{si}>0\\ 0, \mathrm{if} i_{si}=0\\ -1, \mathrm{if} i_{si}<0\end{array}\right.$$

For balanced three-phase loads, the three-phase stator voltages $u_{sa}$, $u_{sb}$, and $u_{sc}$ can be derived as follows [81,85].

$$\left[\begin{array}{l}{u}_{sa}\\ u_{sb}\\ u_{sc}\end{array}\right]=\frac{1}{3}\left[\begin{array}{ccc}2& -1& -1\\ -1& 2& -1\\ -1& -1& 2\end{array}\right]\left[\begin{array}{l}{u}_{ao}\\ u_{bo}\\ u_{co}\end{array}\right]$$

(4)

To reveal the impacts of $T_{on}$, $T_{off}$, and $T_{dead}$ on $u_{sa}$, $u_{sb}$, and $u_{sc}$, the duty cycle-based low-frequency model of the two-level voltage-source inverter is adopted in this study.

Based on (2), the actual duty cycle for each phase $d_i$, indicating the average value of $S_i$ in $T_s$, is defined as (5) [85].

$$d_i=\frac{T_i}{T_s}=\frac{T_i^{*}}{T_s}+\frac{T_{off}-T_{on}-T_{dead}}{T_s}\mathrm{sgn}\left(i_{si}\right)=d_i^{*}+\frac{T_{off}-T_{on}-T_{dead}}{T_s}\mathrm{sgn}\left(i_{si}\right)$$

(5)

where $d_i^{*}$ represents the applied duty cycle for each phase.

Using $d_i$ to replace $S_i$ in (3), the low-frequency pole voltage model can be derived as

$$u_{io}=\left(U_{dc}-U_{sat}+U_{diode}\right)\left(d_i^{*}-\frac{1}{2}\right)+U_{dead}\mathrm{sgn}\left(i_{si}\right)$$

(6)

with

$$U_{dead}=\left(U_{dc}-U_{sat}+U_{diode}\right)\frac{T_{off}-T_{on}-T_{dead}}{T_s}-\frac{U_{sat}+U_{diode}}{2}$$

(7)

Based on (4) and (6), the low-frequency model of the two-level voltage-source inverter adopted in this study is derived as (8). Here, $u_{sa}^{*}$, $u_{sb}^{*}$, and $u_{sc}^{*}$ denote ideal three-phase stator voltages, while $u_{sa}^d$, $u_{sb}^d$, and $u_{sc}^d$ represent the disturbance voltages resulting from the considered inverter nonlinearities.

$$\left\{\begin{array}{l}{u}_{sa}=\underset{u_{sa}^{*}}{\underbrace{\frac{2d_a^{*}-d_b^{*}-d_c^{*}}{3}{U}_{dc}}}+\underset{u_{sa}^d}{\underbrace{\frac{2d_a^{*}-d_b^{*}-d_c^{*}}{3}\left(U_{diode}-U_{sat}\right)+\frac{U_{dead}}{3}\left[2\mathrm{sgn}\left(i_{sa}\right)-\mathrm{sgn}\left(i_{sb}\right)-\mathrm{sgn}\left(i_{sc}\right)\right]}}=u_{sa}^{*}+u_{sa}^d\\ u_{sb}=\underset{u_{sb}^{*}}{\underbrace{\frac{2d_b^{*}-d_a^{*}-d_c^{*}}{3}{U}_{dc}}}+\underset{u_{sb}^d}{\underbrace{\frac{2d_b^{*}-d_a^{*}-d_c^{*}}{3}\left(U_{diode}-U_{sat}\right)+\frac{U_{dead}}{3}\left[2\mathrm{sgn}\left(i_{sb}\right)-\mathrm{sgn}\left(i_{sa}\right)-\mathrm{sgn}\left(i_{sc}\right)\right]}}=u_{sb}^{*}+u_{sb}^d\\ u_{sc}=\underset{u_{sc}^{*}}{\underbrace{\frac{2d_c^{*}-d_a^{*}-d_b^{*}}{3}{U}_{dc}}}+\underset{u_{sc}^d}{\underbrace{\frac{2d_c^{*}-d_a^{*}-d_b^{*}}{3}\left(U_{diode}-U_{sat}\right)+\frac{U_{dead}}{3}\left[2\mathrm{sgn}\left(i_{sc}\right)-\mathrm{sgn}\left(i_{sa}\right)-\mathrm{sgn}\left(i_{sb}\right)\right]}}=u_{sc}^{*}+u_{sc}^d\end{array}\right.$$

(8)

In the synchronous reference frame, also known as the dq coordinate system, the dynamics of the SynRM drive system can be described by the following equations [21].

$$\left\{\begin{array}{l}{\dot{\lambda }}_{sd}=-R_s{i}_{sd}+n_p{\omega }_m{\lambda }_{sq}+u_{sd}\\ {\dot{\lambda }}_{sq}=-R_s{i}_{sq}-n_p{\omega }_m{\lambda }_{sd}+u_{sq}\\ {\dot{\omega }}_m=\frac{1}{J}{T}_r-\frac{1}{J}{T}_f-\frac{1}{J}{T}_l\end{array}\right.$$

(9)

where $R_s$ denotes the stator resistance, $i_{sd}$ and $i_{sq}$ represent the direct- and quadrature-axis stator current components, respectively, $u_{sd}$ and $u_{sq}$ denote the direct- and quadrature-axis stator voltage components, respectively, ${\lambda }_{sd}$ and ${\lambda }_{sq}$ represent the direct- and quadrature-axis stator flux linkage components, respectively, ${\omega }_m$ denotes the mechanical rotor speed, $n_p$ denotes the pole pairs of the SynRM, $J$ represents the moment of inertia, and $T_r$, $T_f$, and $T_l$ are the reluctance torque, the friction torque, and the load torque, respectively.

Due to the magnetic saturation effect, the SynRM experiences significant magnetic nonlinearity, resulting in a highly nonlinear relation of the stator flux linkage to the stator current. Consequently, ${\lambda }_{sd}$ and ${\lambda }_{sq}$ can be described by the following equations [86].

$$\begin{array}{cc}{\lambda }_{sd}={\lambda }_{sd}\left(i_{sd},i_{sq}\right)=L_d\left(i_{sd},i_{sq}\right)i_{sd},& {\lambda }_{sq}={\lambda }_{sq}\left(i_{sd},i_{sq}\right)=L_q\left(i_{sd},i_{sq}\right)i_{sq}\end{array}$$

(10)

where $L_d(i_{sd}, i_{sq})$ and $L_q(i_{sd}, i_{sq})$ represent the direct- and quadrature-axis apparent inductances, respectively.

In this study, the apparent inductance model illustrated in Figure 2 is adopted. This model expressed in (11) is proposed in [86], allowing for an approximation of the values of $L_d(i_{sd}, i_{sq})$ and $L_q(i_{sd}, i_{sq})$ across a wide range of $i_{sd}$ and $i_{sq}$ combinations.

$$\begin{array}{cc}{L}_d\left(i_{sd},i_{sq}\right)=L_{d0}\left(i_{sd}\right)-L_{d1}\left(i_{sd}\right)L_{q2}\left(i_{sq}\right),& L_q\left(i_{sd},i_{sq}\right)=L_{q0}\left(i_{sq}\right)-L_{d2}\left(i_{sd}\right)L_{q1}\left(i_{sq}\right)\end{array}$$

(11)

where

$$\begin{array}{l}{L}_{d0}\left(i_{sd}\right)={\alpha }_{d0}+{\alpha }_{d1}/\left(i_{sd}^4+{\alpha }_{d2}{i}_{sd}^2+{\alpha }_{d3}\right), L_{q0}\left(i_{sq}\right)={\alpha }_{q0}+{\alpha }_{q1}/\left(i_{sq}^4+{\alpha }_{q2}{i}_{sq}^2+{\alpha }_{q3}\right)\\ L_{d1}\left(i_{sd}\right)={\alpha }_{d4}/\left(i_{sd}^4+{\alpha }_{d5}{i}_{sd}^2+{\alpha }_{d6}\right), L_{q1}\left(i_{sq}\right)={\alpha }_{q4}/\left(i_{sq}^4+{\alpha }_{q5}{i}_{sq}^2+{\alpha }_{q6}\right)\\ L_{d2}\left(i_{sd}\right)=1-1/\left({\alpha }_{dq}{i}_{sd}^2+1\right), L_{q2}\left(i_{sq}\right)=1-1/\left({\alpha }_{qd}{i}_{sq}^2+1\right)\end{array}$$

where ${\alpha }_{d0}=0.0391$, ${\alpha }_{d1}=45.4$, ${\alpha }_{d2}=-12.9$, ${\alpha }_{d3}=1329$, ${\alpha }_{d4}=19.9$, ${\alpha }_{d5}=-$13, ${\alpha }_{d6}=795$, ${\alpha }_{dq}=0.0133$, ${\alpha }_{q0}=0.01$, ${\alpha }_{q1}=0.571$, ${\alpha }_{q2}=0$, ${\alpha }_{q3}=58$, ${\alpha }_{q4}=0.825$, ${\alpha }_{q5}=0$, ${\alpha }_{q6}=63.8$, and ${\alpha }_{qd}=0.0833$ [86].

$T_r$ can be written as the following equation [21].

$$T_r=\frac{3}{2}{n}_p\left({\lambda }_{sd}{i}_{sq}-{\lambda }_{sq}{i}_{sd}\right)$$

(12)

$T_f$ can be written as the following equation [21].

$$T_l=B_m{\omega }_m$$

(13)

where $B_m$ represents the viscous friction coefficient.

Substituting (12) and (13) into (9), the ${\omega }_m$-dynamics of the SynRM drive system can be expressed as

$${\dot{\omega }}_m=\frac{3n_p}{2J}\left(L_d-L_q\right)i_{sd}{i}_{sq}-\frac{B_m}{J}{\omega }_m-\frac{1}{J}{T}_l$$

(14)

Given that the constant $i_{sd}$ control strategy described in Section 2.2 is adopted in this study, and considering that the dynamics of the current control loop are significantly faster than those of the speed control loop [21], it can be deemed that $i_{sd}=i_{sdr}$ remains valid throughout the speed controller design process, where $i_{sdr}$ indicates the predefined nonzero constant reference direct-axis current component. Consequently, the ${\omega }_m$-dynamics of the SynRM drive system can be rewritten as

$${\dot{\omega }}_m=a{i}_{sq}-b{\omega }_m-c{T}_l$$

(15)

where $a=3n_p(L_d-L_q)i_{sdr}/2J$, $b=B_m/J$, and $c=1/J$.

Treating the load and friction torques as external disturbances, along with mechanical and electrical parametric uncertainties, the ${\omega }_m$-dynamics of the SynRM drive system can be derived as

$${\dot{\omega }}_m=\left(a_r+\Delta a\right)i_{sq}-\left(b_r+\Delta b\right){\omega }_m-\left(c_r+\Delta c\right)T_l=a_r{i}_{sq}-b_r{\omega }_m+D$$

(16)

where $a_r$, $b_r$, and $c_r$ respectively denote the rated values of $a$, $b$, and $c$, $\mathsf{\Delta }a$, $\mathsf{\Delta }b$, and $\mathsf{\Delta }c$ respectively represent parametric uncertainties in terms of $a$, $b$, and $c$, and $D$ denotes the total disturbance, described as

$$D=\Delta a{i}_{sq}-\Delta b{\omega }_m-\left(c_r+\Delta c\right)T_l$$

Assuming that $D$ and $\dot{D}$ are bounded.

Remark 1. 

The units of variables and parameters in the equations presented in this article are all based on the International System of Units (SI) and its derived units.

2.2. Adopted Vector Control Strategy

In this study, a specific dual-loop vector control strategy, commonly referred to as the constant $i_{sd}$ control strategy, is adopted as the overall control strategy of the considered drive system, as shown in Figure 3. Here, ${\theta }_m$ is the measured mechanical rotor position, ${\mathit{i}}_s$ represents the measured stator current vector, and ${\mathit{d}}^{\mathit{*}}$ denotes the applied duty cycle vector for the inverter. The adopted vector control strategy is based on the dq coordinate system. As depicted in Figure 3, this control strategy includes the speed and current control loops, as well as the duty cycle calculation module based on the principle of the space-vector pulse-width modulation. In the speed control loop, the speed controller ensures that ${\omega }_m$ tracks its predefined reference ${\omega }_{mr}$, outputting the reference quadrature-axis current component $i_{sqr}$. In the current control loop, two current controllers ensure that $i_{sd}$ and $i_{sq}$, which are obtained by performing the Park transformation on the measured $i_{sa}$, $i_{sb}$, and $i_{sc}$, track their respective references $i_{sdr}$ and $i_{sqr}$, outputting the reference direct- and quadrature-axis stator voltage components $u_{sdr}$ and $u_{sqr}$, respectively. These voltage components, after undergoing the inverse Park transformation, then feed into the duty cycle calculation module, generating the applied duty cycle for the inverter.

This study focuses on the design of the speed controller. Regarding two current controllers, two robust composite current controllers based on the standard STSM algorithm, which account for the magnetic saturation effect, are utilized as proposed in [85].

3. Controller Design

3.1. Speed Tracking Error Dynamics

The speed tracking error $e_{\omega }$ can be defined as

$$e_{\omega }={\omega }_{mr}-{\omega }_m$$

(17)

Based on (16) and (17), the $e_{\omega }$-dynamics of the SynRM drive system can be obtained as

$${\dot{e}}_{\omega }={\dot{\omega }}_{mr}-{\dot{\omega }}_m={\dot{\omega }}_{mr}-a_r{i}_{sq}+b_r{\omega }_m-D$$

(18)

The control objective is to drive $e_{\omega }$ to 0.

3.2. GSTSM-DO Design

According to (16), by selecting $D$ as the second state variable, the ${\omega }_m$-dynamics of the SynRM drive system can be extended as

$${\dot{x}}_1=a_r{i}_{sq}-b_r{\omega }_m+x_2, {\dot{x}}_2=\dot{D}$$

(19)

where $x_1={\omega }_m$ and $x_2=D$.

On the basis of (19), the GSTSM-DO for the estimation of $D$ can be designed as

$${\dot{\widehat{x}}}_1=a_r{i}_{sq}-b_r{\omega }_m+{\widehat{x}}_2+k_1{\varphi }_1, {\dot{\widehat{x}}}_2= k_2{\varphi }_2$$

(20)

with

$${\varphi }_1=\sqrt{\left|e_{x1}\right|}\mathrm{sgn}\left(e_{x1}\right)+k_3{e}_{x1}, {\varphi }_2=\frac{1}{2}\mathrm{sgn}\left(e_{x1}\right)+\frac{3}{2}{k}_3\sqrt{\left|e_{x1}\right|}\mathrm{sgn}\left(e_{x1}\right)+k_3^2{e}_{x1}$$

(21)

where ${\widehat{x}}_1={\widehat{\omega }}_m$ and ${\widehat{x}}_2=\widehat{D}$ denote the estimated mechanical rotor speed and total disturbance, respectively, $e_{x1}=x_1-{\widehat{x}}_1={\omega }_m-{\widehat{\omega }}_m$ represents the estimation error for the mechanical rotor speed, and $k_1$, $k_2$, and $k_3$ are positive constant gains.

Based on (21), the following equation is established.

$$\frac{\partial {\varphi }_1}{\partial e_{x1}}{\varphi }_1={\varphi }_2$$

(22)

Assuming that there exists a positive constant $L_1$ for which $\left|\dot{D}\right|\le L_1\left|{\varphi }_2\right|$ is satisfied [87]. From these, it can be derived that $\dot{D}=H_1{\varphi }_2$ with $\left|H_1\right|\le L_1$ [87].

Based on (19) and (20), the estimation error dynamics for the GSTSM-DO can be obtained as follows:

$${\dot{e}}_{x1}=e_{x2}-k_1{\varphi }_1, {\dot{e}}_{x2}=\dot{D}-k_2{\varphi }_2$$

(23)

where $e_{x2}=x_2-{\widehat{x}}_2=D-\widehat{D}$ represents the estimation error for the total disturbance.

Theorem 1. 

Considering the system (23), if $k_1$, $k_2$ , and $k_3$ are selected as (24), $e_{x1}$ and $e_{x2}$ will converge to zero in finite time.

$$k_1>\frac{{\epsilon }_1\left(2L_1+1\right)}{{\beta }_1}+\frac{{\left(L_1+{\beta }_1+4{\epsilon }_1^2\right)}^2}{4{\epsilon }_1{\beta }_1}, k_2=2{\epsilon }_1{k}_1, k_3>0$$

(24)

where ${\beta }_1$ and ${\epsilon }_1$ are two positive constants.

Proof. 

First, the vector $\mathit{\eta }$ is specified as follows:

$$\mathit{\eta }=\left[\begin{array}{c}{\eta }_1\\ {\eta }_2\end{array}\right]=\left[\begin{array}{c}{\varphi }_1\\ e_{x2}\end{array}\right]$$

(25)

Considering $\dot{D}=H_1{\varphi }_2$ with $\left|H_1\right|\le L_1$, based on (21)–(23) and (25), the time derivative of $\mathit{\eta }$ can be calculated as follows:

$$\begin{array}{cc}\hfill \dot{\mathit{\eta }}& =\left[\begin{array}{c}{\dot{\eta }}_1\\ {\dot{\eta }}_2\end{array}\right]=\left[\begin{array}{c}{\dot{\varphi }}_1\\ {\dot{e}}_{x2}\end{array}\right]=\left[\begin{array}{c}\frac{\partial {\varphi }_1}{\partial e_{x1}}{\dot{e}}_{x1}\\ {\dot{e}}_{x2}\end{array}\right]=\left[\begin{array}{c}\frac{\partial {\varphi }_1}{\partial e_{x1}}\left(e_{x2}-k_1{\varphi }_1\right)\\ -k_2{\varphi }_2+\dot{D}\end{array}\right]=\left[\begin{array}{c}\frac{\partial {\varphi }_1}{\partial e_{x1}}\left(-k_1{\varphi }_1+e_{x2}\right)\\ -k_2\frac{\partial {\varphi }_1}{\partial e_{x1}}{\varphi }_1+H_1\frac{\partial {\varphi }_1}{\partial e_{x1}}{\varphi }_1\end{array}\right]\hfill \\ & =\frac{\partial {\varphi }_1}{\partial e_{x1}}\left[\begin{array}{c}-k_1{\varphi }_1+e_{x2}\\ -k_2{\varphi }_1+H_1{\varphi }_1\end{array}\right]=\frac{\partial {\varphi }_1}{\partial e_{x1}}\underset{\mathit{A}}{\underbrace{\left[\begin{array}{cc}-k_1& 1\\ -k_2+H_1& 0\end{array}\right]}}\left[\begin{array}{c}{\varphi }_1\\ e_{x2}\end{array}\right]=\frac{\partial {\varphi }_1}{\partial e_{x1}}\mathit{A}\mathit{\eta }\hfill \end{array}$$

(26)

Afterward, the following Lyapunov candidate function $V_1$ is selected [87].

$$V_1={\mathit{\eta }}^T{\mathit{P}}_1\mathit{\eta },\begin{array}{c} {\mathit{P}}_1=\left[\begin{array}{cc}{\beta }_1+4{\epsilon }_1^2& -2{\epsilon }_1\\ -2{\epsilon }_1& 1\end{array}\right]\end{array}$$

(27)

Based on (26) and (27), the time derivative of $V_1$ can be calculated as

$$\begin{array}{c}{\dot{V}}_1={\dot{\mathit{\eta }}}^T{\mathit{P}}_1\mathit{\eta }+{\mathit{\eta }}^T{\mathit{P}}_1\dot{\mathit{\eta }}=\frac{\partial {\varphi }_1}{\partial e_{x1}}{\mathit{\eta }}^T\left({\mathit{A}}^T{\mathit{P}}_1+{\mathit{P}}_1\mathit{A}\right)\mathit{\eta }\hfill \\ \hspace{1em}=\frac{\partial {\varphi }_1}{\partial e_{x1}}{\mathit{\eta }}^T\left(\left[\begin{array}{cc}-k_1& -k_2+H_1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}{\beta }_1+4{\epsilon }_1^2& -2{\epsilon }_1\\ -2{\epsilon }_1& 1\end{array}\right]+\left[\begin{array}{cc}{\beta }_1+4{\epsilon }_1^2& -2{\epsilon }_1\\ -2{\epsilon }_1& 1\end{array}\right]\left[\begin{array}{c}\hfill -k_1 1\\ -k_2+H_1 0\end{array}\right]\right)\mathit{\eta }\hfill \\ \hspace{1em}=\frac{\partial {\varphi }_1}{\partial e_{x1}}{\mathit{\eta }}^T\left[\begin{array}{c}-2{\beta }_1{k}_1-4{\epsilon }_1\left(2{\epsilon }_1{k}_1-k_2\right)-4{\epsilon }_1{H}_1 *\\ 2{\epsilon }_1{k}_1-k_2+{\beta }_1+4{\epsilon }_1^2+H_1 -4{\epsilon }_1\end{array}\right]\mathit{\eta }=-\frac{\partial {\varphi }_1}{\partial e_{x1}}{\mathit{\eta }}^T\mathit{Q}\mathit{\eta }\hfill \end{array}$$

(28)

with

$$\mathit{Q}=\left[\begin{array}{c}2{\beta }_1{k}_1+4{\epsilon }_1\left(2{\epsilon }_1{k}_1-k_2\right)+4{\epsilon }_1{H}_1 *\\ -2{\epsilon }_1{k}_1+k_2-{\beta }_1-4{\epsilon }_1^2-H_1 4{\epsilon }_1\end{array}\right]$$

(29)

Substituting (24) into (29), Q can be simplified as

$$\mathit{Q}=\left[\begin{array}{c}2{\beta }_1{k}_1+4{\epsilon }_1{H}_1 *\\ -{\beta }_1-4{\epsilon }_1^2-H_1 4{\epsilon }_1\end{array}\right]$$

(30)

Regarding the $2\times 2$ identity matrix ${\mathit{I}}_{2\times 2}$, $(\mathit{Q}-2{\epsilon }_1{\mathit{I}}_{2\times 2})$ can be expressed as

$$Q-2{\epsilon }_1{I}_{2\times 2}=\left[\begin{array}{cc}-2{\epsilon }_1+2{\beta }_1{k}_1+4{\epsilon }_1{H}_1& *\\ -{\beta }_1-4{\epsilon }_1^2-H_1& 2{\epsilon }_1\end{array}\right]$$

(31)

Substituting (24) into (31), $\mathit{Q}\ge 2{\epsilon }_1{\mathit{I}}_{2\times 2}$ is ensured.

Given that ${\mathit{P}}_1$ is a positive definite symmetric matrix, with ${\lambda }_{min}${${\mathit{P}}_1$} and ${\lambda }_{max}${${\mathit{P}}_1$} denoting its minimum and maximum eigenvalues, respectively, the following inequality is ensured.

$${\lambda }_{\min }\left\{{\mathit{P}}_1\right\}{‖\mathit{\eta }‖}_2^2\le V_1\le {\lambda }_{\max }\left\{{\mathit{P}}_1\right\}{‖\mathit{\eta }‖}_2^2$$

(32)

where ${‖·‖}_2$ represents the Euclidean norm of a vector.

Based on (21) and (25), ${‖\mathit{\eta }‖}_2^2$ can be calculated as

$${‖\mathit{\eta }‖}_2^2=\left|e_{x1}\right|+2k_3{\left|e_{x1}\right|}^{\frac{3}{2}}+k_3^2{e}_{x1}^2+e_{x2}^2$$

(33)

According to (32) and (33), the following inequality is ensured.

$$-\frac{1}{\sqrt{\left|e_{x1}\right|}}\le -\frac{1}{\sqrt{\left|e_{x1}\right|+2k_3{\left|e_{x1}\right|}^{\frac{3}{2}}+k_3^2{e}_{x1}^2+e_{x2}^2}}=-\frac{1}{{‖\mathit{\eta }‖}_2}\le -\sqrt{\frac{{\lambda }_{\min }\left\{{\mathit{P}}_1\right\}}{V_1}}$$

(34)

Considering $\mathit{Q}\ge 2{\epsilon }_1{\mathit{I}}_{2\times 2}$, based on (21) and (28), the following inequality is ensured.

$${\dot{V}}_1=-\frac{\partial {\varphi }_1}{\partial e_{x1}}{\mathit{\eta }}^T\mathit{Q}\mathit{\eta }\le -\frac{\partial {\varphi }_1}{\partial e_{x1}}2{\epsilon }_1{\mathit{\eta }}^T\mathit{\eta }=-2{\epsilon }_1\left(\frac{1}{2\sqrt{\left|e_{x1}\right|}}+k_3\right){‖\mathit{\eta }‖}_2^2=-\frac{{\epsilon }_1}{\sqrt{\left|e_{x1}\right|}}{‖\mathit{\eta }‖}_2^2-2{\epsilon }_1{k}_3{‖\mathit{\eta }‖}_2^2$$

(35)

Based on (32), (34) and (35), the following inequality is ensured.

$$\begin{array}{c}{\dot{V}}_1\le -\frac{{\epsilon }_1}{\sqrt{\left|e_{x1}\right|}}{‖\mathit{\eta }‖}_2^2-2{\epsilon }_1{k}_3{‖\mathit{\eta }‖}_2^2\hfill \\ \hspace{1em}\le -{\epsilon }_1\sqrt{\frac{{\lambda }_{\min }\left\{{\mathit{P}}_1\right\}}{V_1}}\frac{V_1}{{\lambda }_{\max }\left\{{\mathit{P}}_1\right\}}-2{\epsilon }_1{k}_3\frac{V_1}{{\lambda }_{\max }\left\{{\mathit{P}}_1\right\}}=-\frac{{\epsilon }_1\sqrt{{\lambda }_{\min }\left\{{\mathit{P}}_1\right\}}}{{\lambda }_{\max }\left\{{\mathit{P}}_1\right\}}\sqrt{V_1}-\frac{2{\epsilon }_1{k}_3}{{\lambda }_{\max }\left\{{\mathit{P}}_1\right\}}{V}_1\hfill \\ \hspace{1em}\le -\frac{{\epsilon }_1\sqrt{{\lambda }_{\min }\left\{{\mathit{P}}_1\right\}}}{{\lambda }_{\max }\left\{{\mathit{P}}_1\right\}}\sqrt{V_1}\hfill \end{array}$$

(36)

According to (36) and the comparison principle [88], the finite-time convergence of $e_{x1}$ and $e_{x2}$ to zero can be guaranteed. The proof is completed. □

3.3. Proposed Composite Speed Controller Design

Selecting $e_{\omega }$ as the sliding variable, and based on (18), the proposed composite speed control law is designed as follows:

$$i_{sq}=\frac{1}{a_r}\left(i_{GSTSM}+{\dot{\omega }}_{mr}+b_r{\omega }_m-\widehat{D}\right)$$

(37)

where $i_{GSTSM}$ is written as

$$i_{GSTSM}=p_1{\psi }_1+p_2{\displaystyle \int {\psi }_{2}dt}$$

(38)

with

$${\psi }_1=\sqrt{\left|e_{\omega }\right|}\mathrm{sgn}\left(e_{\omega }\right)+p_3{e}_{\omega },\hspace{1em}{\psi }_2=\frac{1}{2}\mathrm{sgn}\left(e_{\omega }\right)+\frac{3}{2}{p}_3\sqrt{\left|e_{\omega }\right|}\mathrm{sgn}\left(e_{\omega }\right)+p_3^2{e}_{\omega }$$

(39)

where $p_1$, $p_2$, and $p_3$ are positive constant gains.

Remark 2. 

By eliminating $\widehat{D}$ in (37), the GSTSM speed control law is derived.

Remark 3. 

By setting $k_3=0$ in (21), the composite speed control law combining a GSTSM speed control law with a standard STSM-DO is derived.

Remark 4. 

By eliminating $\widehat{D}$ in (37) and setting $p_3=0$ in (39), the standard STSM speed control law is derived.

The block diagram of the proposed composite speed controller is depicted in Figure 4.

Based on (39), the following equation is established.

$$\frac{\partial {\psi }_1}{\partial e_{\omega }}{\psi }_1={\psi }_2$$

(40)

Substituting (37) into (18), the controlled $e_{\omega }$-dynamics with the proposed composite speed control law are expressed as

$${\dot{e}}_{\omega }=-p_1{\psi }_1-p_2{\displaystyle \int {\psi }_{2}dt}-e_{x2}$$

(41)

Since $e_{x2}$ will converge to zero in finite time based on (24), it can be assumed that there exists a positive constant $L_2$ for which $\left|-{\dot{e}}_{x2}\right|\le L_2\left|{\psi }_2\right|$ is satisfied. From these, it can be derived that $-{\dot{e}}_{x2}=H_2{\psi }_2$ with $\left|H_2\right|\le L_2$.

Theorem 2. 

Considering the system (41), if $p_1$ ,  $p_2$, and $p_3$are selected as (42), $e_{\omega }$ will converge to zero in finite time.

$$p_1>\frac{{\epsilon }_2\left(2L_2+1\right)}{{\beta }_2}+\frac{{\left(L_2+{\beta }_2+4{\epsilon }_2^2\right)}^2}{4{\epsilon }_2{\beta }_2},p_2=2{\epsilon }_2{p}_1, p_3>0$$

(42)

 where ${\beta }_2$ and ${\epsilon }_2$ are two positive constants.

Proof. 

First, the system (41) is extended as (43) and the vector $\mathit{\xi }$ is specified as (44).

$${\dot{e}}_{\omega }=-p_1{\psi }_1+\phi , \dot{\phi }=-p_2{\psi }_2-{\dot{e}}_{x2}$$

(43)

$$\xi =\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\end{array}\right]=\left[\begin{array}{c}{\psi }_1\\ \phi \end{array}\right]$$

(44)

Considering $-{\dot{e}}_{x2}=H_2{\psi }_2$ with $\left|H_2\right|\le L_2$, based on (39)–(41) and (44), the time derivative of $\mathit{\xi }$ can be calculated as follows:

$$\begin{array}{ll}\dot{\xi }& =\left[\begin{array}{c}{\dot{\xi }}_1\\ {\dot{\xi }}_2\end{array}\right]=\left[\begin{array}{c}{\dot{\psi }}_1\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{c}\frac{\partial {\psi }_1}{\partial e_{\omega }}{\dot{e}}_{\omega }\\ \dot{\phi }\end{array}\right]=\left[\begin{array}{c}\frac{\partial {\psi }_1}{\partial e_{\omega }}\left(-p_1{\psi }_1+\phi \right)\\ -p_2{\psi }_2-{\dot{e}}_{x2}\end{array}\right]=\left[\begin{array}{c}\frac{\partial {\psi }_1}{\partial e_{\omega }}\left(-p_1{\psi }_1+\phi \right)\\ -p_2\frac{\partial {\psi }_1}{\partial e_{\omega }}{\psi }_1+H_2\frac{\partial {\psi }_1}{\partial e_{\omega }}{\psi }_1\end{array}\right]\\ & =\frac{\partial {\psi }_1}{\partial e_{\omega }}\left[\begin{array}{c}-p_1{\psi }_1+\phi \\ -p_2{\psi }_1+H_2{\psi }_1\end{array}\right]=\frac{\partial {\psi }_1}{\partial e_{\omega }}\underset{\Xi }{\underbrace{\left[\begin{array}{cc}-p_1& 1\\ -p_2+H_2& 0\end{array}\right]}}\left[\begin{array}{c}{\psi }_1\\ e_{\omega }\end{array}\right]=\frac{\partial {\psi }_1}{\partial e_{\omega }}\Xi \xi \end{array}$$

(45)

Afterward, the following Lyapunov candidate function $V_2$ is selected [87].

$$V_2={\xi }^T{\mathit{P}}_2\xi \begin{array}{c}, {\mathit{P}}_2=\left[\begin{array}{cc}{\beta }_2+4{\epsilon }_2^2& -2{\epsilon }_2\\ -2{\epsilon }_2& 1\end{array}\right]\end{array}$$

(46)

Based on (45) and (46), the time derivative of $V_2$ can be calculated as

$$\begin{array}{c}{\dot{V}}_2={\dot{\xi }}^T{\mathit{P}}_2\xi +{\xi }^T{\mathit{P}}_2\dot{\xi }=\frac{\partial {\psi }_1}{\partial e_{\omega }}{\xi }^T\left({\Xi }^T{\mathit{P}}_2+{\mathit{P}}_2\Xi \right)\xi \hfill \\ \hspace{1em}=\frac{\partial {\psi }_1}{\partial e_{\omega }}{\xi }^T\left(\left[\begin{array}{cc}-p_1& -p_2+H_2\\ 1& 0\end{array}\right]\left[\begin{array}{cc}{\beta }_2+4{\epsilon }_2^2& -2{\epsilon }_2\\ -2{\epsilon }_2& 1\end{array}\right]+\left[\begin{array}{cc}{\beta }_2+4{\epsilon }_2^2& -2{\epsilon }_2\\ -2{\epsilon }_2& 1\end{array}\right]\left[\begin{array}{c}-p_1 1\\ -p_2+H_2 0\end{array}\right]\right)\xi \hfill \\ \hspace{1em}=\frac{\partial {\psi }_1}{\partial e_{\omega }}{\xi }^T\left[\begin{array}{c}-2{\beta }_2{p}_1-4{\epsilon }_2\left(2{\epsilon }_2{p}_1-p_2\right)-4{\epsilon }_2{H}_2 *\\ 2{\epsilon }_2{p}_1-p_2+{\beta }_2+4{\epsilon }_2^2+H_2 -4{\epsilon }_2\end{array}\right]\xi =-\frac{\partial {\psi }_1}{\partial e_{\omega }}{\xi }^T\mathit{\Omega }\xi \hfill \end{array}$$

(47)

with

$$\mathit{\Omega }=\left[\begin{array}{c}2{\beta }_2{p}_1+4{\epsilon }_2\left(2{\epsilon }_2{p}_1-p_2\right)+4{\epsilon }_2{H}_2 *\\ -2{\epsilon }_2{p}_1+p_2-{\beta }_2-4{\epsilon }_2^2-H_2 4{\epsilon }_2\end{array}\right]$$

(48)

Substituting (42) into (48), $\mathit{\Omega }$ can be simplified as

$$\mathit{\Omega }=\left[\begin{array}{c}2{\beta }_2{p}_1+4{\epsilon }_2{H}_2 *\\ -{\beta }_2-4{\epsilon }_2^2-H_2 4{\epsilon }_2\end{array}\right]$$

(49)

$(\mathit{\Omega }-2{\epsilon }_2{\mathit{I}}_{2\times 2})$ can be expressed as

$$\mathit{\Omega }-2{\epsilon }_2{\mathit{I}}_{2\times 2}=\left[\begin{array}{c}-2{\epsilon }_2+2{\beta }_2{p}_1+4{\epsilon }_2{H}_2 *\\ -{\beta }_2-4{\epsilon }_2^2-H_2 2{\epsilon }_2\end{array}\right]$$

(50)

Substituting (42) into (50), $\mathit{\Omega }\ge 2{\epsilon }_2{\mathit{I}}_{2\times 2}$ is ensured.

Given that ${\mathit{P}}_2$ is a positive definite symmetric matrix, the following inequality is ensured.

$${\lambda }_{\min }\left\{{\mathit{P}}_2\right\}{‖\xi ‖}_2^2\le V_2\le {\lambda }_{\max }\left\{{\mathit{P}}_2\right\}{‖\xi ‖}_2^2$$

(51)

Based on (39) and (44), ${‖\mathit{\xi }‖}_2^2$ can be calculated as

$${‖\xi ‖}_2^2=\left|e_{\omega }\right|+2p_3{\left|e_{\omega }\right|}^{\frac{3}{2}}+p_3^2{e}_{\omega }^2+{\phi }^2$$

(52)

According to (51) and (52), the following inequality is ensured.

$$-\frac{1}{\sqrt{\left|e_{\omega }\right|}}\le -\frac{1}{\sqrt{\left|e_{\omega }\right|+2p_3{\left|e_{\omega }\right|}^{\frac{3}{2}}+p_3^2{e}_{\omega }^2+{\phi }^2}}=-\frac{1}{{‖\xi ‖}_2}\le -\sqrt{\frac{{\lambda }_{\min }\left\{{\mathit{P}}_2\right\}}{V_2}}$$

(53)

Considering $\mathit{\Omega }\ge 2{\epsilon }_2{\mathit{I}}_{2\times 2}$, based on (39) and (47), the following inequality is ensured.

$${\dot{V}}_2=-\frac{\partial {\psi }_1}{\partial e_{\mathit{\omega }}}{\xi }^T\mathit{\Omega }\xi \le -\frac{\partial {\psi }_1}{\partial e_{\omega }}2{\epsilon }_2{\xi }^T\xi =-2{\epsilon }_2\left(\frac{1}{2\sqrt{\left|e_{\omega }\right|}}+p_3\right){‖\xi ‖}_2^2=-\frac{{\epsilon }_2}{\sqrt{\left|e_{\omega }\right|}}{‖\xi ‖}_2^2-2{\epsilon }_2{p}_3{‖\xi ‖}_2^2$$

(54)

Based on (51), (53), and (54), the following inequality is ensured.

$$\begin{array}{c}{\dot{V}}_2\le -\frac{{\epsilon }_2}{\sqrt{\left|e_{\omega }\right|}}{‖\xi ‖}_2^2-2{\epsilon }_2{p}_3{‖\xi ‖}_2^2\hfill \\ \hspace{1em}\le -{\epsilon }_2\sqrt{\frac{{\lambda }_{\min }\left\{{\mathit{P}}_2\right\}}{V_2}}\frac{V_2}{{\lambda }_{\max }\left\{{\mathit{P}}_2\right\}}-2{\epsilon }_2{p}_3\frac{V_2}{{\lambda }_{\max }\left\{{\mathit{P}}_2\right\}}=-\frac{{\epsilon }_2\sqrt{{\lambda }_{\min }\left\{{\mathit{P}}_2\right\}}}{{\lambda }_{\max }\left\{{\mathit{P}}_2\right\}}\sqrt{V_2}-\frac{2{\epsilon }_2{p}_3}{{\lambda }_{\max }\left\{{\mathit{P}}_2\right\}}{V}_2\\ \hspace{1em}\le -\frac{{\epsilon }_2\sqrt{{\lambda }_{\min }\left\{{\mathit{P}}_2\right\}}}{{\lambda }_{\max }\left\{{\mathit{P}}_2\right\}}\sqrt{V_2}\hfill \end{array}$$

(55)

According to (55) and the comparison principle [88], the finite-time convergence of $e_{\omega }$ to zero can be guaranteed. The proof is completed. □

4. Simulation Results

The two-level voltage-source inverter-fed SynRM drive system is built in the software MATLAB/Simulink. The sampling period is selected as $10 \mathsf{\mu }\mathrm{s}$. The inverter model, based on (8), incorporates the following parameters: $T_s=100 \mathsf{\mu }\mathrm{s}$, $T_{on}=1.3 \mathsf{\mu }\mathrm{s}$, $T_{off}=1.3 \mathsf{\mu }\mathrm{s}$, $T_{dead}=2.0 \mathsf{\mu }\mathrm{s}$, $U_{sat}=1.6 \mathrm{V}$, and $U_{diode}=1.5 \mathrm{V}$ [84]. Additionally, the SynRM model depends on (9)–(11), with parameters set as follows: $U_{dc}$ at $250\mathrm{V}$, the rated mechanical rotor speed at $1500 \mathrm{r}\mathrm{p}\mathrm{m}$, the rated value of $R_s$ at $1.05 \mathsf{\Omega }$, the rated values of $L_d$ and $L_d$ selected as $L_d\left(\mathrm{0,0}\right)$ and $L_q\left(\mathrm{0,0}\right)$ in (11), respectively, the rated value of $T_r$ at $4.8 \mathrm{N}\mathbf{⸱}\mathrm{m}$, the rated value of $J$ at $0.0208 \mathrm{k}\mathrm{g}\mathbf{⸱}{\mathrm{m}}^2$, and the rated value of $B_m$ at $0.00268 \mathrm{N}\mathbf{⸱}\mathrm{m}\mathbf{⸱}\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$, $n_p$ at 2 [86]. It is important to note that, since the apparent inductance model (11) is incorporated into the SynRM model, the magnetic saturation effect is always included in the simulation tests.

To verify the effectiveness and superiority of the proposed composite speed controller, three comparative simulation tests are performed under the framework of the constant $i_{sd}$ control strategy illustrated in Figure 3. These tests compare the standard STSM speed controller, the GSTSM speed controller, the composite speed controller using a GSTSM speed controller and a standard STSM-DO, and the proposed composite speed controller. The parameters of the four tested speed controllers are presented in

Table 1. Parameters of tested speed controllers.

Speed Controller	Controller Parameters	Observer Parameters
Standard STSM Speed Controller	$p_1=60$, $p_2=200$, $p_3=0$	-
GSTSM Speed Controller	$p_1=60$, $p_2=200$, $p_3=0.03$	-
Composite Speed Controller Using GSTSM Speed Controller and Standard STSM-DO	$p_1=60$, $p_2=200$, $p_3=0.03$	$k_1=30$, $k_2=80$, $k_3=0$
Proposed Composite Speed Controller	$p_1=60$, $p_2=200$, $p_3=0.03$	$k_1=30$, $k_2=80$, $k_3=0.05$

.

In test 1, the value of ${\omega }_{mr}$ increases from $0 \mathrm{r}\mathrm{p}\mathrm{m}$ to $1500 \mathrm{r}\mathrm{p}\mathrm{m}$, starting at $2.0 \mathrm{s}$ and reaching $1500 \mathrm{r}\mathrm{p}\mathrm{m}$ by $4.0 \mathrm{s}$, with the value of $T_l$ set to $0 \mathrm{N}\mathbf{⸱}\mathrm{m}$. Regarding $i_{sdr}$, its value is set to $5 \mathrm{A}$ for the current control loop and $6 \mathrm{A}$ for the speed controller design, accounting for the parametric uncertainty of $i_{sdr}$. Furthermore, in the SynRM model, the value of $J$ is adjusted to twice its rated value to account for the parametric uncertainty of $J$, while $B_m$ remains at its rated value. The waveforms of ${\omega }_{mr}$, ${\omega }_m$, and $e_{\omega }$ or four tested speed controllers in this test are presented in Figure 5. The dynamic performance of these speed controllers is shown in

Table 2. Dynamic performance of speed controllers in the first test.

Speed Controller	Overshoot (rpm)	Settling Time (s)
Standard STSM Speed Controller	63.98	3.74
GSTSM Speed Controller	55.69	3.52
Composite Speed Controller Using GSTSM Speed Controller and Standard STSM-DO	51.54	3.14
Proposed Composite Speed Controller	49.48	2.96

, where the settling time is calculated using a tolerance band of $\pm 1 \mathrm{r}\mathrm{p}\mathrm{m}$. It is clear that, among the tested speed controllers, the standard STSM speed controller exhibits the highest overshoot and the longest settling time. In comparison, the GSTSM speed controller achieves lower overshoot and shorter settling time than the standard STSM speed controller. The composite speed controller, combining a GSTSM speed controller with a standard STSM-DO, further reduces both overshoot and settling time relative to the two aforementioned speed controllers. Specifically, this controller achieves a reduction of 19.44% in overshoot and 16.04% in settling time compared with the standard STSM speed controller, and 7.45% in overshoot and 10.80% in settling time compared with the GSTSM speed controller. Among the tested speed controllers, the lowest overshoot and shortest settling time are achieved with the proposed composite speed controller.

In test 2, the value of ${\omega }_{mr}$ is maintained at $1500 \mathrm{r}\mathrm{p}\mathrm{m}$, and the value of $T_l$ suddenly changes from $0 \mathrm{N}\mathbf{⸱}\mathrm{m}$ to $4.0 \mathrm{N}\mathbf{⸱}\mathrm{m}$ at $2.0 \mathrm{s}$. Regarding $i_{sdr}$, its value is set to $5 \mathrm{A}$ for the current control loop and $6 \mathrm{A}$ for the speed controller design, accounting for the parametric uncertainty of $i_{sdr}$. Moreover, in the SynRM model, the value of $J$ is adjusted to twice its rated value to account for the parametric uncertainty of $J$, while $B_m$ remains at its rated value. The waveforms of ${\omega }_{mr}$, ${\omega }_m$, and $e_{\omega }$ for the four tested speed controllers in this test are depicted in Figure 6. The dynamic performance of these speed controllers is illustrated in

Table 3. Dynamic performance of speed controllers in test 2.

Speed Controller	Maximum Tracking Error (rpm)	Settling Time (s)
Standard STSM Speed Controller	96.59	2.27
GSTSM Speed Controller	81.65	1.91
Composite Speed Controller Using GSTSM Speed Controller and Standard STSM-DO	75.74	1.45
Proposed Composite Speed Controller	72.29	1.21

, where the settling time is calculated using a tolerance band of $\pm 1 \mathrm{r}\mathrm{p}\mathrm{m}$. The test results show that the standard STSM speed controller has the highest maximum tracking error and the longest settling time among the tested speed controllers. In contrast, the GSTSM speed controller demonstrates improvements with a lower maximum tracking error and a shorter settling time than its standard STSM counterpart. The composite speed controller, which combines a GSTSM speed controller with a standard STSM-DO, achieves further reductions in both maximum tracking error and settling time compared to the previously mentioned speed controllers. Specifically, it reduces the maximum tracking error by 21.59% and the settling time by 36.12% relative to the standard STSM speed controller, and by 7.24% and 24.08%, respectively, in comparison to the GSTSM speed controller. Among all the speed controllers tested, the proposed composite speed controller stands out by exhibiting the lowest maximum tracking error and the shortest settling time.

In test 3, the values of ${\omega }_{mr}$ and $T_l$ are maintained at $1500 \mathrm{r}\mathrm{p}\mathrm{m}$ and $0 \mathrm{N}\mathbf{⸱}\mathrm{m}$, respectively. Regarding $i_{sdr}$, its value is set to $5 \mathrm{A}$ for the current control loop and $6 \mathrm{A}$ for the speed controller design, accounting for the parametric uncertainty of $i_{sdr}$. Moreover, in the SynRM model, the value of $B_m$ is suddenly changed to ten times its rated value at $2.0 \mathrm{s}$, while $J$ remains at its rated value. The waveforms of ${\omega }_{mr}$, ${\omega }_m$, and $e_{\omega }$ for the four tested speed controllers in this test are depicted in Figure 7. The dynamic performance of these speed controllers is illustrated in

Table 4. Dynamic performance of speed controllers in test 3.

Speed Controller	Maximum Tracking Error (rpm)	Settling Time (s)
Standard STSM Speed Controller	88.53	2.09
GSTSM Speed Controller	76.49	1.79
Composite Speed Controller Using GSTSM Speed Controller and Standard STSM-DO	72.99	1.34
Proposed Composite Speed Controller	70.87	1.12

, where the settling time is calculated using a tolerance band of $\pm 1 \mathrm{r}\mathrm{p}\mathrm{m}$. The test results indicate that, relative to other tested speed controllers, the standard STSM speed controller records the maximum value in tracking error as well as the longest settling time. On the other hand, the GSTSM speed controller marks a notable reduction in both maximum tracking error and settling time when compared to the standard STSM version. The composite speed controller, which combines a GSTSM speed controller with a standard STSM-DO, further diminishes the tracking error and accelerates the settling process beyond the improvements noted with the individual GSTSM and standard STSM controllers. The proposed composite speed controller distinguishes itself by securing the lowest maximum tracking error and the quickest convergence among the lineup of tested controllers.

5. Conclusions and Future Work

In this study, a novel composite speed controller is proposed for the vector-controlled SynRM drive system. Through the DO-based control principle, it utilizes the GSTSM algorithm to construct both the speed controller and the DO. This approach aims to enhance the control system’s robustness against external disturbances and parametric uncertainties in the speed tracking error dynamics. The finite-time convergence of the speed tracking error is ensured by the GSTSM speed controller, whereas the external disturbances and parametric uncertainties in the speed tracking error dynamics are compensated for by the GSTSM-DO, thereby improving control performance. Comparative simulation tests among the standard STSM speed controller, the GSTSM speed controller, the composite speed controller that combines a GSTSM speed controller with a standard STSM-DO, and the proposed composite speed controller were carried out in the software MATLAB/Simulink. The simulation results confirm the effectiveness and superiority of the proposed composite speed controller.

For future work, experiments are planned to further verify the effectiveness and superiority of the proposed composite speed controller. Additionally, the GSTSM algorithm will be extended to other components of the vector control strategy for the SynRM drive system to further enhance its performance.