Novel Composite Speed Control of Permanent Magnet Synchronous Motor Using Integral Sliding Mode Approach

Abstract

Permanent magnet synchronous motors (PMSMs) are widely applied in industry, and proportional integral (PI) controllers are often used to control PMSMs. Aiming at the characteristics of the poor anti-disturbance ability and speed ripple of traditional PI controllers, a novel composite speed controller for PMSMs is proposed in this paper that uses a novel sliding mode control (SMC). To improve the chattering problem of traditional SMC, a high-order approaching law super-twisting algorithm (STA) is applied. Considering the internal and external disturbance of motor driver systems, such as motor parameter drifts and load torque changes, a disturbance estimator based on an extended state observer (ESO) is proposed, and it is used for the feed-forward compensation of the current. The composite super-twisting integral sliding mode controller (ST-ISMC) with a nonlinear ESO is tested by simulations and experiments, and the comparative results verify that the proposed controller has the higher control accuracy, smaller speed ripple and stronger robustness.

1. Introduction

PMSMs are widely used in aerospace, rail transit, port ships, electric vehicles, energy storage frequency regulation and other industries due to their simple structure, high power density, high energy conversion efficiency and small installation volume [1,2,3]. However, due to the nonlinear and strongly coupled characteristics of PMSMs, reasonable control methods are usually required to exert their optimal performance [4].

The most commonly used control methods in industry are field-oriented control (FOC) and direct torque control (DTC) [5]. FOC is also known as vector control, and it decouples the three-phase alternating current of the motor stator into a straight-axis current ($i_d$) to control the excitation and a cross-axis current ($i_q$) to control the torque through vector transformation, while the PMSM control is approximated as direct current (DC) motor control. Its inner loop control strategy is usually $i_d=0$ and flux-weakening control. Its structure diagram is shown in Figure 1a [6,7]. DTC is different from the current inner loop control of FOC, and this method performs inner loop control by directly calculating the modulus of the stator flux linkage and the torque of the motor; its structure diagram is shown in Figure 1b [8,9]. Comparing FOC with DTC, the calculation of FOC is more complex, but its output torque ripple is smaller [10]. Different methods should be selected according to the applications.

FOC or DTC both require a speed controller. Speed loop controllers usually use PI control, and are simple and efficient [11], but do not react quickly enough to motor parameter drift and sudden load changes caused by temperature rise, so they cannot be used in scenarios with high performance requirements [12]. In order to improve the speed control performance of the motor in various occasions, many scholars have proposed a variety of control methods, such as backstepping control, adaptive control, sliding mode control, predictive control, etc. [13,14,15,16,17]. Backstepping control relies on the design of virtual control quantities and Lyapunov functions, which are not easily solved in real systems; adaptive control reduces the control accuracy and dynamic quality of the system because of the simple obfuscation of information; predictive control is always applied to the current loop and it is computationally intensive.

SMC, a variable structure control of a nonlinear system, has a fast response and robustness, which is well known as it is insensitive to parameter changes and other disturbances. It has been successfully applied in many fields, including motor speed control. However, conventional SMC has a significant drawback where, when the motion point of the system approaches the sliding surface $s=0$, it will shuttle at high frequency, resulting in chattering. Chattering always affects the control accuracy of the system, resulting in mechanical wear and energy losses. In SMC, the switching function is always the sign function, which is discontinuous, and its boundary is fixed. Many papers solve the chattering problem through improving the discontinuity of the switching function or using a dynamic boundary. High-order sliding mode controllers and composite sliding mode controllers are also used to suppress chattering [18,19]. To improve response speed and robustness, the integral sliding surface is selected and improved in this paper, and to minimize chattering, a high-order reaching law is used.

In the actual control process, internal disturbance such as parameter drifts when the temperature of the motor changes and external load disturbance will occur. The lumped disturbance, which degrades the performance of the controller, can be estimated and used for the feed-forward compensation control to further improve the robustness and reduce the chatting of SMC [20,21]. Several articles have designed disturbance observers such as extended state observers (ESOs) and super-twisting sliding mode observers [20,21,22,23,24]. In order to improve the performance of the ESO and adapt it to more complex system initial states and disturbances, a novel estimator based on a nonlinear ESO is designed in this paper. The main contributions of this paper mainly include:

• To reduce the steady-state error and improve the robustness of the speed controller, a novel estimator based on a nonlinear ESO is designed to estimate the lumped disturbance, including load torque and parameter changes, in real time, and is used for the feed-forward compensation;
• A novel integral sliding surface that is more sensitive to input error is designed to improve response speed and robustness. Moreover, an STA is used as a high-order reaching law to suppress chattering;
• The simulation and experimental results show that the proposed novel ST-ISMC is better at controlling than the PI controller and traditional ST-ISMC in scenarios where load and speed change frequently.

This paper is organized as follows. Section 2 presents a model of the PMSM with parameter uncertainty. Section 3 describes the novel composite controller design and stability analysis. Comparative simulations and experimental verification are shown in Section 4, and Section 5 summarizes this study’s findings.

2. Mathematical Model of PMSM

For further research, this paper takes the surface-mount PMSM whose $L_d=L_q$ as an example. And, the idealized assumptions of the PMSM are as follows [25]:

• The three-phase winding of the motor is completely symmetrical, the air gap is uniform and the output is a standard three-phase wave current.
• The stator and rotor cores are coaxial and smooth.
• The saturation of the magnetic circuit, the influences of the magnetic hysteresis losses and the eddy currents are ignored.

The mathematical model of a PMSM in the rotating coordinate ($d-q$) reference frame can be described as follows:

$$\left\{\begin{array}{cc}{u}_d\hfill & =(R+\Delta R)i_d+(L_d+\Delta L_d)\frac{d}{dt}{i}_d-(L_q+\Delta L_q)i_q{\omega }_e\hfill \\ u_q\hfill & =(R+\Delta R)i_q+(L_q+\Delta L_q)\frac{d}{dt}{i}_q+(L_d+\Delta L_d)i_d{\omega }_e\hfill \\ \hfill & +({\phi }_f+\Delta {\phi }_f){\omega }_e\hfill \\ T_L\hfill & =T_e-(J+\Delta J)\frac{d}{dt}{\omega }_m-(B+\Delta B){\omega }_m\hfill \\ T_e\hfill & =\frac{3}{2}{P}_n({\phi }_f+\Delta {\phi }_f)i_q\hfill \end{array}\right.$$

(1)

where $u_d$ and $u_q$ are the stator voltage component on the d axis and q axis, respectively; $i_d$ and $i_q$ are the stator current component on the d axis and q axis, respectively; $L_d$ and $L_q$ are inductance on the d axis and q axis, respectively; $\Delta L_d$ and $\Delta L_q$ are measurement errors of $L_d$ and $L_q$, respectively; $R_s$ is the stator resistance; ${\phi }_f$ is the flux linkage of the permanent magnet; $\Delta R$ and $\Delta {\phi }_f$ are measurement errors of R and ${\phi }_f$, respectively; ${\omega }_e$ is the electrical angular velocity; ${\omega }_m$ is the mechanical angular velocity; $T_e$ is electromagnetic torque; $T_L$ is load torque; $Pn$ is the number of pole pairs; J is the inertia; B is the coefficient of friction; and $\Delta J$ and $\Delta B$ are errors of J and B, respectively.

Considering that $T_L$ is a limited value that varies with load, ${\omega }_m$ is a value below the maximum speed, $\frac{d}{dt}{\omega }_m$ is limited by mechanical structure and $i_q$ is bounded by the output limit. Through introducing a new bounded variable f, which is the lumped disturbance including load torque changes and parameter changes (and the derivative of f is bounded), the mathematical model can be rewritten as follows:

$$J\frac{d}{dt}{\omega }_m=\frac{3}{2}{P}_n{\phi }_f{i}_q-B{\omega }_m-f$$

(2)

where $f=T_L+\Delta J\frac{d}{dt}{\omega }_m+\Delta B{\omega }_m-\frac{3}{2}{P}_n\Delta {\phi }_f{i}_q$.

And, the $i_q$ in Equation (2) can be presented as follows:

$$\left\{\begin{array}{c}{i}_q=i_{qc}+i_{qf}\hfill \\ i_{qc}=\frac{2}{3p_n{\phi }_f}(J\frac{d}{dt}{\omega }_m+B{\omega }_m)\hfill \\ i_{qf}=\frac{2f}{3P_n{\phi }_f}\hfill \end{array}\right.$$

(3)

Then, an observer can be used to estimate the value of f and a compensation for f can be added to $i_q$.

3. Controller Design

SMC excels at addressing the aforementioned issues. By allowing the system to accurately follow the sliding surface, which directly impacts the controller’s ultimate performance, SMC enhances the overall control efficiency. Additionally, a specific approaching law is employed to guarantee the smoothness and quality of the dynamic process as the system nears the sliding surface. There is a speed loop controller and two current loop controllers in the FOC structure, and this paper concerns the speed loop controller. In this section, a novel sliding surface is proposed and an STA approaching law is applied.

3.1. Design of Traditional Integral Sliding Mode Controller

Setting ${\omega }_{ref}$ as the reference speed, the speed tracking error can be defined as:

$$\left\{\begin{array}{c}e={\omega }_{ref}-{\omega }_m\hfill \\ \dot{e}=-{\dot{\omega }}_m\hfill \end{array}\right.$$

(4)

A conventional sliding surface requires both speed and acceleration. Then, the derivative of the sliding surface includes acceleration and jerk; however, jerk is hard to measure due to noise. An integral sliding surface solves this issue. The conventional integral sliding surface is designed as follows:

$$s=k_{p}e+\int k_{i}ed\tau$$

(5)

where $k_p$ and $k_i$ are the parameters of the integral sliding surface, respectively. The derivative of the integral sliding surface only includes speed and acceleration and avoids degradation of the system performance.

The main drawback of the traditional ISMC is the chattering phenomenon, which is characterized by small oscillations at the output of the controller that can result in being harmful to PMSM control systems. To address the problem of chattering, the STA is introduced, which is designed as follows [26]:

$$\left\{\begin{array}{c}\dot{s}=-k_1{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+y\hfill \\ \dot{y}=-k_2\mathrm{sign}\left(s\right)\hfill \end{array}\right.$$

(6)

where $k_1,k_2>0$, ${\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)$ is a switching function with a gain related to $\left|s\right|$ that is continuous, and y is the integral of sign function and does not include a high-frequency switching term. Therefore, $\dot{s}$ becomes a continuous value.

Combining Equations (3), (5) and (6), the conventional integral sliding mode controller is given by:

$$\begin{array}{cc}\hfill {i_{qc1}}^{*}& =\frac{2}{3P_n{\phi }_f}\left(\frac{J}{k_p}(k_1{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-y+k_{i}e)+B{\omega }_m\right)\hfill \\ \hfill & =\frac{2}{3P_n{\phi }_f}(\frac{J}{k_p}(k_1{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+\int k_2\mathrm{sign}\left(s\right)d\tau +k_{i}e)\hfill \\ & +B{\omega }_m)\hfill \end{array}$$

(7)

3.2. Design of Disturbance Estimator Based on ESO

In order to suppress the influence of various parameter mismatch and external interference on the system accuracy, a disturbance estimator is presented to estimate the parameter f. The overall structure diagram is shown in Figure 2.

Equation (2) can be rewritten as:

$$\begin{array}{cc}\hfill {\dot{\omega }}_m& =-\frac{B}{J}{\omega }_m+\frac{3p_n{\phi }_f}{2J}{i}_q-\frac{f}{J}\hfill \\ \hfill & =g\left({\omega }_m\right)-\frac{f}{J}\hfill \end{array}$$

(8)

where $g\left({\omega }_m\right)$ is a function with ${\omega }_m$. Consider the elements that need to be estimated; Equation (9) is obtained as follows:

$$\begin{array}{c}\hfill {\dot{\tilde{\omega }}}_m=g\left({\tilde{\omega }}_m\right)-\frac{\tilde{f}}{J}\end{array}$$

(9)

Combining Equations (8) and (9), we have obtained Equation (10) as follows:

$$\begin{array}{c}\hfill \tilde{f}-f=J\left\{[g\left({\tilde{\omega }}_m\right)-g\left({\omega }_m\right)]-({\dot{\tilde{\omega }}}_m-{\dot{\omega }}_m)\right\}\end{array}$$

(10)

The following estimator is described by a nonlinear ESO in active disturbance rejection control (ADRC) [27,28]:

$$\left\{\begin{array}{c}e=z_1-y\hfill \\ {\dot{z}}_1=z_2-{\beta }_1{g}_1\left(e\right)\hfill \\ {\dot{z}}_2=z_3-{\beta }_2{g}_2\left(e\right)+bu\hfill \\ {\dot{z}}_3=-{\beta }_3{g}_3\left(e\right)\hfill \end{array}\right.\left\{\begin{array}{c}{g}_1\left(e\right)=e\hfill \\ g_2\left(e\right)={\left|e\right|}^{\frac{1}{2}}sign\left(e\right)\hfill \\ g_3\left(e\right)={\left|e\right|}^{\frac{1}{4}}sign\left(e\right)\hfill \end{array}\right.$$

(11)

Then, two elements need to be estimated and two estimators are used. One is designed to estimate the error between the real speed and the speed calculated from the equation, and the other one is designed to estimate the disturbance f mentioned earlier, as shown in Equation (12).

$$\left\{\begin{array}{c}{\tilde{\omega }}_m=\int (-\frac{B}{J}{\omega }_m+\frac{3p_n{\phi }_f}{2J}{i}_q-\frac{\tilde{f}}{J})d\tau +z_1\hfill \\ \tilde{f}=z_4\hfill \end{array}\right.$$

(12)

Figure 3 illustrates the designed structure diagram of the ESO, and the equations are described as follows:

$$\left\{\begin{array}{c}{e}_1={\tilde{\omega }}_m-{\omega }_m\hfill \\ {\dot{z}}_1=z_2-{\beta }_1{g}_1\left(e_1\right)\hfill \\ {\dot{z}}_2=z_3-{\beta }_2{g}_2\left(e_1\right)+b{u}_1\hfill \\ {\dot{z}}_3=-{\beta }_3{g}_3\left(e_1\right)\hfill \\ g_1\left(e_1\right)=e_1\hfill \\ g_2\left(e_1\right)={\left|e_1\right|}^{\frac{1}{2}}sign\left(e_1\right)\hfill \\ g_3\left(e_1\right)={\left|e_1\right|}^{\frac{1}{4}}sign\left(e_1\right)\hfill \\ u_1=-{\dot{z}}_1-{\dot{z}}_2-{\dot{z}}_3\hfill \end{array}\right.\left\{\begin{array}{c}{e}_2=\tilde{f}-f\hfill \\ {\dot{z}}_4=z_5-{\beta }_4{g}_4\left(e_2\right)\hfill \\ {\dot{z}}_5=z_6-{\beta }_5{g}_5\left(e_2\right)+b{u}_2\hfill \\ {\dot{z}}_6=-{\beta }_6{g}_6\left(e_2\right)\hfill \\ g_4\left(e_2\right)=e_2\hfill \\ g_5\left(e_2\right)={\left|e_2\right|}^{\frac{1}{2}}sign\left(e_2\right)\hfill \\ g_6\left(e_2\right)={\left|e_2\right|}^{\frac{1}{4}}sign\left(e_2\right)\hfill \\ u_2=-{\dot{z}}_4-{\dot{z}}_5-{\dot{z}}_6\hfill \end{array}\right.$$

(13)

The compensation current is obtained as follows:

$$\begin{array}{c}\hfill i_{qf}^{*}=\frac{2\tilde{f}}{3P_n{\phi }_f}\end{array}$$

(14)

The complete expression of $i_q^{*}$ is given as follows:

$$\begin{array}{cc}\hfill i_q^{*}& =\frac{2}{3P_n{\phi }_f}\{\frac{J}{k_p\left|e\right|+\xi }[k_i{\left|e\right|}^{\frac{1}{2}}e+k_1|s{|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)\hfill \\ \hfill & +\int k_2\mathrm{sign}\left(s\right)d\tau ]+B{\omega }_m\}+\frac{2\tilde{f}}{3P_n{\phi }_f}\hfill \end{array}$$

(15)

3.3. Design of a Novel Integral Sliding Mode Controller

In a conventional integral sliding surface, $k_p$ and $k_i$ are constants that affect the performance of the system. A small $k_p$ makes the starting current too large, but a large $k_p$ affects the convergence speed of the system. A small $k_i$ also affects the convergence speed of the system, but a large $k_i$ causes a large fluctuation in $i_q$. To solve the above problems, a novel sliding surface is proposed as follows:

$$s=k_p\left|e\right|e+\int k_i{\left|e\right|}^{\frac{1}{2}}ed\tau$$

(16)

where $k_p\left|e\right|$ and $k_i{\left|e\right|}^{\frac{1}{2}}$ replace the original gains $k_p$ and $k_i$, and the new gains are variable when e changes. The novel sliding surface is more sensitive than the conventional integral sliding surface when e changes. Figure 4 illustrates the novel integral sliding mode controller.

To calculate the convergence time $T_s$, Equation (17) can be obtained when the derivative of sliding surface $\dot{s}=0$:

$$2k_{p}e\dot{e}\mathrm{sign}\left(e\right)=-k_i{\left|e\right|}^{\frac{1}{2}}e$$

(17)

and $T_s$ can be obtained by taking an integral from 0 to $T_s$ and from $e_0$ to $e_{T_s}$:

$$T_s=\frac{4k_p}{k_i}{e_0}^{\frac{1}{2}}$$

(18)

Combining Equations (3), (6) and (16), the novel super-twisting integral sliding mode controller (ST-ISMC) is given by:

$$\begin{array}{cc}\hfill {i_{qc2}}^{*}& =\frac{2}{3P_n{\phi }_f}\{\frac{J}{k_p\left|e\right|+\xi }[k_i{\left|e\right|}^{\frac{1}{2}}e+k_1|s{|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)\hfill \\ \hfill & +\int k_2\mathrm{sign}\left(s\right)d\tau ]+B{\omega }_m\}\hfill \end{array}$$

(19)

where $\xi$ is a very small positive constant to avoid when $\left|e\right|=0$, and the denominator of $i_{qc2}$ is 0. At the initial point of the system, the value of e is large, the value of $k_p\left|e\right|$ is set sightly large to reduce the fluctuation in and amplitude of $i_{qc}^{*}$ and, when e is close to 0, $\dot{s}=-k_1|s{|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+y$ is close to 0; then, $i_{qc2}^{*}$ is close to $i_{i_{qc1}^{*}}$. The $k_i{\left|e\right|}^{\frac{1}{2}}$ is designed to ensure that $\frac{k_i{\left|e\right|}^{\frac{1}{2}}e}{k_p\left|e\right|+\xi }$ is related to the value of e.

3.4. Stability Analysis for Novel ST-ISMC

To analyze the stability of the observer [29], Equation (11) can be rewritten as follows:

$$\left\{\begin{array}{c}e=z_1-y\hfill \\ {\dot{z}}_1=z_2-{\beta }_{1}e\hfill \\ {\dot{z}}_2=z_3-{\beta }_2\frac{g_2\left(e\right)}{e}e+bu\hfill \\ {\dot{z}}_3=-{\beta }_3\frac{g_3\left(e\right)}{e}e\hfill \end{array}\right.$$

(20)

Then, Equation (20) can be seen as a variable gain linear ESO, and its transfer function can be written as:

$$\left\{\begin{array}{c}{z}_1=\frac{{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}{s^3+{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}y+\frac{s}{s^3+{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}bu\hfill \\ z_2=\frac{{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}{s^3+{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}y+\frac{s+{\beta }_1}{s^3+{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}bu\hfill \\ z_3=\frac{{\beta }_3\frac{g_2\left(e\right)}{e}{s}^{2}y-{\beta }_3\frac{g_3\left(e\right)}{e}bu}{s^3+{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}-\frac{{\beta }_3\frac{g_3\left(e\right)}{e}}{s^3+{\beta }_1{s}^2+{\beta }_2\frac{g_2\left(e\right)}{e}s+{\beta }_3\frac{g_3\left(e\right)}{e}}f\left(s\right)\hfill \end{array}\right.$$

(21)

According to the Routh criterion, the necessary and sufficient condition for the stability of the ESO is

$$\begin{array}{c}\hfill {\beta }_1{\beta }_2\frac{g_2\left(e\right)}{e}>{\beta }_3\frac{g_3\left(e\right)}{e}\end{array}$$

(22)

When the range of e is known, the range of gain is determined. It is known that the root locus method does not limit the form of parameter perturbations but only needs to know the range of parameter perturbations. Therefore, the perturbation range can be used to equate at $\frac{g_i\left(e\right)}{e}$ and the time-varying gain independent of e instead, so the condition for the stability of the ESO can be described as

$$\begin{array}{c}\hfill {\beta }_1{\beta }_2>{\beta }_3,{\beta }_4{\beta }_5>{\beta }_6\end{array}$$

(23)

To analyze the stability of the sliding mode controller, firstly analyze the stability of the sliding surface. Define a Lyapunov function:

$$\begin{array}{c}\hfill V_1=\frac{1}{2}{s}^2\end{array}$$

(24)

The derivative of $V_1$ is:

$$\begin{array}{cc}\hfill {\dot{V}}_1& =\dot{s}s\hfill \\ & =s(2k_p|e|\dot{e}+k_i{|e|}^{\frac{1}{2}}e)\hfill \\ & =s(-2k_p|e|\frac{3P_n{\phi }_f}{2J}{i}_{qc}+k_i{|e|}^{\frac{1}{2}}e)\hfill \\ & =2s(-k_1|s{|}^{\frac{1}{2}}\mathrm{sign}(s)-\int k_2\mathrm{sign}(s)d\tau )\hfill \end{array}$$

(25)

Then, this is equal to analyzing the stability of the approaching law; Equation (6) can be rewritten as follows:

$$\dot{s}=-k_1{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+\int -k_2\mathrm{sign}\left(s\right)d\tau$$

(26)

A vector is defined as $Z={\left[{\zeta }_1,{\zeta }_2\right]}^T={\left[{\left|s\right|}^{\frac{1}{2}}\mathrm{sign}\left(s\right),\int -k_2\mathrm{sign}\left(s\right)d\tau \right]}^T$; $\dot{\sigma }=[-k_1,1]Z$, then, Equation (20) can be written as follows:

$$\begin{array}{cc}\hfill \dot{Z}& =\left[\begin{array}{c}{\dot{\zeta }}_1\\ {\dot{\zeta }}_2\end{array}\right]\hfill \\ & =\left[\begin{array}{c}\frac{1}{2}|s{|}^{-\frac{1}{2}}(-k_1|s{|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)-\int k_2\mathrm{sign}\left(s\right)d\tau )\\ -k_2\mathrm{sign}\left(s\right)\end{array}\right]\hfill \\ & =\frac{1}{\left|{\zeta }_1\right|}\left[\begin{array}{cc}-\frac{1}{2}{k}_1& \frac{1}{2}\\ -k_2& 0\end{array}\right]Z\hfill \end{array}$$

(27)

A candidate Lyapunov function is given by:

$$\begin{array}{cc}\hfill V_1& =ZP{Z}^T\hfill \\ \hfill & =({\lambda }_1+4{{\lambda }_2}^2){{\zeta }_1}^2+{{\zeta }_2}^2-4{\lambda }_2{\zeta }_1{\zeta }_2\hfill \end{array}$$

(28)

where $P=\left[\begin{array}{cc}{\lambda }_1+4{{\lambda }_2}^2& -2{\lambda }_2\\ -2{\lambda }_2& 1\end{array}\right]$ and P is positive definite. The derivative of the Lyapunov function is given:

$$\begin{array}{cc}\hfill {\dot{V}}_1=& 2({\lambda }_1+4{\lambda }_2^2){\zeta }_1{\dot{\zeta }}_1+2{\zeta }_2{\dot{\zeta }}_2-4{\lambda }_2{\dot{\zeta }}_1{\zeta }_2-4{\lambda }_2{\zeta }_1{\dot{\zeta }}_2\hfill \\ \hfill =& \frac{1}{\left|{\zeta }_1\right|}\left\{2({\lambda }_1+4{\lambda }_2^2){\zeta }_1(-\frac{1}{2}{k}_1{\zeta }_1+\frac{1}{2}{\zeta }_2)+2{\zeta }_2(-k_2{\zeta }_1)\right.\hfill \\ \hfill & \left.-4{\lambda }_2(-\frac{1}{2}{k}_1{\zeta }_1+\frac{1}{2}{\zeta }_2){\zeta }_2-4{\lambda }_2{\zeta }_1(-k_2{\zeta }_1)\right\}\hfill \\ \hfill =& -\frac{1}{\left|{\zeta }_1\right|}\left\{\left[k_1({\lambda }_1+4{\lambda }_2^2)-4k_2{\lambda }_2\right]{\zeta }_1^2\right.\hfill \\ \hfill & \left.-\left[({\lambda }_1+4{\lambda }_2^2)+2k_1{\lambda }_2-2k_2\right]{\zeta }_1{\zeta }_2+2{\lambda }_2{\zeta }_2^2\right\}\hfill \\ \hfill =& -\frac{1}{\left|{\zeta }_1\right|}{Z}^{T}QZ\hfill \end{array}$$

(29)

where $A^{T}P+PA=-Q$.

For every symmetric and positive definite matrix $Q=Q^T>0$, Equation $A^{T}P+PA=-Q$ has a unique symmetric and positive definite solution $P=P^T>0$ [30,31].

In this case, if the controller gains satisfy ${\lambda }_2>0$, ${\lambda }_1>-4{\lambda }_2^2$, $\frac{4k_2{\lambda }_2}{({\lambda }_1+4{\lambda }_2^2)}<k_1<\frac{k_2}{{\lambda }_2}+\frac{{\lambda }_1+4{\lambda }_2^2}{4{\lambda }_2}$, then $P>0$, $Q>0$, $V>0$, $\dot{V}<0$, and the system is asymptotically stable.

4. Simulation and Experiment

To verify the performance of the proposed controller, comparative simulations and experimental studies are introduced in this section. The conventional PI controller is given by

$$\begin{array}{c}\hfill i_q=k_{p}e+k_i\int ed\tau \end{array}$$

(30)

The ST-ISMC is given by

$$\begin{array}{c}\hfill i_{qc}=\frac{2}{3P_n{\phi }_f}\{\frac{J}{k_p}[k_{i}e+k_1|s{|}^{\frac{1}{2}}\mathrm{sign}\left(s\right)+\int k_2\mathrm{sign}\left(s\right)d\tau ]+B{\omega }_m\}\end{array}$$

(31)

The parameters of the surface-mount PMSM for the simulation and experiment are shown in

Table 1. Parameter values of PMSM.

Parameter	Value	Unit
Rated power	$0.2$	[$\mathrm{kW}$]
Rated line voltage	48	[V]
Rated line current	$5.7$	[A]
Rated speed	3000	[r/min]
Rated torque	$0.64$	[N·m]
Rotor inertia	$0.0000175$	[kg·m${}^2$]
Winding resistance	$0.3$	[$\Omega$]
Winding inductance	$1.378$	[mH]
Rotor flux linkage	$0.0181$	[Wb]

.

4.1. Simulation Results

Performance comparison tests were carried out to prove the effectiveness of the proposed novel controller with compensation through simulations. We compared the performance of the conventional PI controller, ST-ISMC and the novel ST-ISMC, as shown in Figure 5 and Figure 6. The optimal controller parameters are determined by the trial and error method and are given in

Table 2. Parameter values of three tested speed controllers (simulation).

Controller	${\mathit{k}}_{\mathit{p}}$	${\mathit{k}}_{\mathit{i}}$	${\mathit{k}}_1$	${\mathit{k}}_2$
PI controller	1	50	*	*
ST-ISMC	$0.08$	5000	$0.8$	$0.2$
Novel ST-ISMC	$0.08$	5000	$0.8$	$0.2$

* means this parameter does not exist.

, and the performance comparison of the three tested speed controllers is shown in

Table 3. Performance comparison of three tested speed controllers (simulation).

Controller	PI Controller	ST-ISMC	Novel ST-ISMC
Settling time at initial speed (s)	$0.0041$	$0.0039$	$0.0034$
Overshoot at initial speed (r/min)	108	95	86
Maximum speed drop at initial speed (r/min)	60	62	41
Settling time when load changes (s)	$0.003$	$0.0027$	$0.0025$
Overshoot when load changes (r/min)	7	7	17
Maximum speed drop when load changes (r/min)	31	31	43
Settling time when speed changes (s)	$0.0055$	$0.0053$	$0.0052$
Overshoot when speed changes (r/min)	35	33	32
Maximum speed drop when speed changes (r/min)	36	48	21

. The parameters for the current loop PI controllers are the same: $k_p=2.756$, $k_i=2.756$. The switching frequency of the inverter is 10 kHz and the sampling time is $T_s=0.00001$ s. The parameters for the nonlinear ESO are chosen as ${\beta }_1=2000, {\beta }_2=1, {\beta }_3=1, {\beta }_4=130, {\beta }_5=1,$${\beta }_6=1$.

The motor starting load torque was set to 0.1 N·m. When the simulation time was 0.4 $\mathrm{s}$, the load torque suddenly increased to 0.5 N·m. The motor starting speed was set to 1000 $\mathrm{r}/\min$. When the simulation time was 0.6 $\mathrm{s}$, the reference speed suddenly increased to 2000 $\mathrm{r}/\min$.

It can be seen in Figure 5 that all controllers have a good step response. When the reference speed changes, the overshoot of the PI controller is the largest of them and that of the novel ST-ISMC is the smallest. When the load changes, the novel ST-ISMC reaches stability the earliest. In terms of control accuracy, both the ST-ISMC and novel ST-ISMC perform better than the PI controller. In Figure 6, it can be obviously seen that the novel ST-ISMC experienced smaller current ($i_q$) fluctuations during speed changes and sudden load torque changes.

The performance of the estimator designed based on a nonlinear ESO is shown in Figure 7, including a comparison with the estimated value of f at the original value of PMSM controller parameters, changed inertia and changed flux linkage.

It can be observed that the value of f at the original value of PMSM parameters tracks the load torque very well. When we change the J in the controller from 0.0000175 kg·m${}^2$ to 0.0000275 kg·m${}^2$, the estimated value of f changes because of $\Delta J\frac{d}{dt}{\omega }_m$, which means that the fluctuation is greater in the presence of acceleration. When the ${\phi }_f$ is changed in the controller from 0.0181 Wb to 0.0281 Wb, the estimated value of f changes because of $\frac{3}{2}{P}_n\Delta {\phi }_f{i}_q$, which means that the convergence value changes with ${\phi }_f$.

4.2. Experimental Results

Experimental tests were conducted for the validation. Figure 8 shows the experimental setup, where a DSP controller board with TMS320f28335 and an inverter drive circuit board with IR2136 are used to implement various control algorithms.

The program was designed and values were observed using communication software with an upper monitor. According to the official examples of Texas Instruments (TI), the sampling frequency of the current loop is 10 $\mathrm{kHz}$ and the sampling time is $T_s=0.001$ s. The load motor is connected to a rectifier bridge and a resistance of 20 $\Omega$ is used to consume energy. The optimal controller parameters are determined by the trial and error method and given in

Table 4. Parameter values of three tested speed controllers (experiment).

Controller	${\mathit{k}}_{\mathit{p}}$	${\mathit{k}}_{\mathit{i}}$	${\mathit{k}}_1$	${\mathit{k}}_2$
PI controller	$1.5$	$0.8$	*	*
ST-ISMC	$0.08$	5000	$1.5$	$1.5$
Novel ST-ISMC	$0.08$	5000	$1.5$	$1.5$

* means this parameter does not exist.

, and the performance comparison of the three tested speed controllers is shown in

Table 5. Performance comparison of three tested speed controllers (experiment).

Controller	PI Controller	ST-ISMC	Novel ST-ISMC
Settling time at initial speed (s)	$1.35$	$1.33$	$1.91$
Overshoot at initial speed (r/min)	42	3	64
Maximum speed drop at initial speed (r/min)	0	13	1
Settling time when load changes (s)	$1.84$	$0.46$	$0.28$
Overshoot when load changes (r/min)	0	0	1
Maximum speed drop when load changes (r/min)	191	59	29
Settling time when speed changes (s)	$2.58$	$1.41$	$1.57$
Overshoot when speed changes (r/min)	0	0	2
Maximum speed drop when speed changes (r/min)	3	58	2

. The parameters for the current loop PI controllers are the same: $k_p=2$, $k_i=0.4$. The parameters for the nonlinear ESO are chosen as ${\beta }_1=200, {\beta }_2=0.1, {\beta }_3=0.1,$${\beta }_4=13, {\beta }_5=0.1, {\beta }_6=0.1$.

As depicted in Figure 9, we conducted a performance comparison between the conventional PI controller, ST-ISMC and the novel ST-ISMC. The motor’s initial speed was set at 1000 $\mathrm{r}/\min$. At 4 $\mathrm{s}$ into the experiment, a sudden load was introduced to the motor. Subsequently, at 8 $\mathrm{s}$, the reference speed was increased to 2000 $\mathrm{r}/\min$.

From Figure 9, we can observe that the PI controller exhibits the fastest response at low speed when there is no load. However, both the ST-ISMC and novel ST-ISMC demonstrate faster responses at high speeds compared to the PI controller. When subjected to load changes, the novel ST-ISMC achieves stability before the others. In terms of control accuracy, the PI controller and novel ST-ISMC perform similarly, while the ST-ISMC exhibits a notable steady-state error. Figure 10 displays the q-axis current for the PI controller, ST-ISMC and novel ST-ISMC. In Figure 10c, the issue of current glitches can be observed. The sudden current change at low speed raises the speed and reduces the steady-state error, but it may have adverse effects on the hardware. However, once the permanent magnet synchronous motor (PMSM) reaches stable operation, the novel ST-ISMC performs exceptionally well.

The performance of the estimator, designed based on a nonlinear ESO, is showcased in Figure 11. This figure includes a comparison of the estimated value of f with the original PMSM controller parameters, as well as variations in inertia and flux linkage. When we change the inertia J in the controller from 0.0000175 kg·m${}^2$ to 0.0000275 kg·m${}^2$, the estimated value of f changes due to the presence of acceleration. However, it is important to note that the experimental sampling frequency is much lower than the simulation frequency, which may result in the sampled acceleration leading to a larger value of f. Similarly, when we modify the flux linkage ${\phi }_f$ in the controller from 0.0181 Wb to 0.0281 Wb, the estimated value of f varies accordingly with ${\phi }_f$.

5. Conclusions

This paper introduced a novel composite speed loop controller based on the FOC structure. To enhance controller accuracy, a novel disturbance estimator utilizing a nonlinear ESO has been proposed. This estimator effectively estimated disturbances such as load torque, allowing for compensation to be applied to $i_q$. Additionally, a novel ST-ISMC has been developed, which incorporates adaptive sliding surface gains that adjust according to the speed error. This feature significantly reduces overshoot and fluctuations in both speed and current. Simulation and experimental results confirmed the effectiveness of the proposed controller. Compared to the PI controller and traditional ST-ISMC, the proposed controller demonstrates superior performance in scenarios involving frequent load and speed changes, such as blenders, cranes, flywheel energy storage systems, etc.

At present, the current glitches problem at low speed for the designed controller has not been solved. We hope for the current $i_q$ to be smoother and have a smaller amplitude of fluctuations, and this will be the focus of further research. We will concentrate on optimizing the current loop controllers in future work, and a more responsive current loop controller may make the control system perform better.