PID Sliding Mode Control of PMSM Based on Improved Terminal Sliding Mode Reaching Law

Abstract

In order to enhance the dynamic performance and anti-disturbance ability of speed control for a permanent magnet synchronous motor (PMSM), a sliding mode control method based on a PID sliding surface and an improved terminal sliding mode reaching law (ITSMRL) is proposed. Firstly, an ITSMRL is proposed to increase the reaching speed and reduce chattering; moreover, it has been verified that the reaching law (RL) can achieve a sliding mode surface in finite time. Then, based on the dynamic model of PMSMs with uncertainties, an extended state observer (ESO) is used to estimate the lumped disturbance, and it is proven that the estimated error is bounded. Finally, on the basis of the observed feedforward disturbance, to enhance the disturbance rejection ability of PMSMs, a controller that combines the PID sliding mode surface and the ITSMRL is proposed. Moreover, the stability of the closed-loop system is proven. The composite method has the characteristics of a fast reaching speed, small chattering and strong robustness, and is verified by experiments.

1. Introduction

PMSMs have the advantages of small size, high power density, and large output torque. Owing to these merits, they are widely utilized in industrial automation, electric vehicles, wind power generation, household appliances and other fields [1,2,3]. For the control of PMSMs, the traditional methods include proportional–integral (PI) control [4], variable structure control (VSC) [5], model predictive control (MPC) [6], active disturbance rejection control (ADRC) [7], fuzzy control (FC) [8], etc. In recent years, with the development of artificial intelligence, edge computing and advanced control theory, a variety of new control strategies have emerged in PMSM control systems, such as data-driven control [9], mode-switching control [10], reinforcement learning optimization control [11], event-triggered control [12], etc. Among these, variable structure control has been widely used in the field of PMSM control because of its robustness to external perturbation, fast response and low requirements for system parameter accuracy [13,14,15,16,17]. Sliding mode control (SMC) is synonymous with VSC. Due to the existence of the signum function, chattering exists in SMC. Moreover, this chattering can only be weakened to a certain extent, or the disturbance rejection capacity of the sliding mode will be eliminated. Serious chattering will stimulate the dynamic properties of the omitted high-frequency characteristics, affect the control accuracy of the controller, enhance the energy consumption of the system, and potentially result in system vibrations and damage to the hardware equipment of the control system. Therefore, reducing chattering is the primary research direction related to SMC. In recent years, various methods have been proposed to solve the problem of chattering in PMSM systems, such as the reaching law (RL) method [18], disturbance observer composite method [19], filter method [20], boundary layer method [21], sliding mode control with fractional order characteristics [22], etc.

The disturbance observer composite method can effectively reduce the gain coefficient of a sliding mode controller and further reduce chattering by feeding the observed disturbance forward. Given that extended state observers (ESOs) estimate both system states and disturbance dynamics not based on a system model, and the ease of adjusting the parameters, they are often used to design composite controllers combined with the sliding mode control method [23,24,25,26]. Ma et al. designed a compound integral terminal sliding mode control (ITSMC) in discrete form with an ESO for the purpose of PMSM speed control [23]. A combined approach which integrated continuous fast terminal sliding mode control (FTSMC) and an ESO was presented in [24]. A speed-regulating controller based on proportional integral sliding mode control (PISMC) and an ESO was developed in [25]. A composite speed control method has been put forward by Liu et al., featuring a modified super-twisting SMC and an ESO [26]. However, the above composite methods of SMC and ESO are all based on conventional RLs.

The RL should give consideration to not only fast reaching speed but also small chatterings; that is, the RL is large when the condition of the system shows that its state is distant from a sliding mode surface, and the RL is small when the system state is near a sliding mode surface. The exponent RL was first designed by Gao et al., which is faster than the constant rate RL [27]. Given that the system state is far from the sliding mode surface, the power RL is greater than the constant rate RL. Conversely, when the system state approaches a sliding mode surface, the power RL is smaller than the constant rate RL [28]. Xu et al. designed a terminal sliding mode RL (TSMRL), which showed better performance than the power RL in terms of fast reaching speed and small chattering [24]. Wang et al. advanced the conventional power RL by integrating system state variables and an exponential function, which alleviates chattering and reduces the time required to reach the desired state [29]. However, the law contains six parameters. Therefore, an effective RL should not only balance a reduction in chattering with the acceleration of convergence but also give consideration to the complexity inherent in the algorithm structure.

What is more, in order to improve tracking performance and enhance disturbance rejection ability, PID sliding mode surfaces have been designed for some application situations [30,31,32], especially PMSMs [33,34]. A composite fractional-order PID sliding mode controller with an ESO was designed for the speed control of a surface-mounted permanent magnet synchronous motor (SPMSM) [33]. A composite adaptive super-twisting controller with a nonlinear fractional-order PID sliding mode surface and ESO for PMSM speed control was proposed in [34]. In recent years, control methods involving fractional-order sliding mode composite disturbance observers, particularly terminal sliding mode control (TSMC) and non-singular terminal sliding mode control (NTSMC), have been applied to the speed control systems of permanent magnet synchronous motors (PMSMs). However, the observers used are always asymptotically stable and cannot guarantee the finite-time stability of a closed-loop system [14,24,35,36]. Moreover, due to the large number of parameters involved, a fractional-order sliding mode surface will face complexities when adjusting the parameters.

Motivated by the aforementioned papers, in order to balance the reduction in chattering with the acceleration of convergence and enhance the disturbance rejection ability, we propose a novel controller of PMSMs based on a PID sliding mode surface with an improved terminal sliding mode RL (ITSMRL) and an ESO. In particular, the ITSMRL is better than the TSMRL for achieving fast reaching speed and reducing chattering.

Compared with previous relevant studies, the major contributions of this paper are as follows:

(1)

An ITSMRL is proposed, which not only balances the reduction in chattering with the acceleration of convergence but also considers the complexity of the structure. It has been verified that the RL can reach a sliding mode surface within a finite time.

(2)

A novel speed composite controller based on a PID sliding mode surface with an ITSMRL and ESO is designed. The boundedness of the estimated error of the ESO and the stability of the closed-loop system are proven.

(3)

The validity and robustness of the proposed algorithm are evidenced by simulation and experimental tests.

The remainder of this paper is presented as follows. Section 2 presents a mathematical model with the uncertainty of an SPMSM. In Section 3, an ITSMRL is designed and its finite reaching time to the sliding mode surface is proven. Moreover, a PID sliding mode controller for the SPMSM based on an ITSMRL and ESO is also presented. The simulated and experimental results are presented in Section 4. Finally, in Section 5, the conclusions are presented.

2. Preliminaries

In this section, a dynamic equation of a PMSM in a speed loop with uncertainty is given. The following statement elaborates the mathematical model of the PMSM [37,38,39,40]:

$$\left\{\begin{array}{c}{L}_d{\displaystyle \frac{d{i}_d}{dt}}=-R_s{i}_d+n_p\omega L_q{i}_q+u_d,\hfill \\ L_q{\displaystyle \frac{d{i}_q}{dt}}=-R_s{i}_q-n_p\omega L_d{i}_d-n_p\omega \mathrm{\Phi }+u_q,\hfill \\ J_m{\displaystyle \frac{d\omega }{dt}}=\tau -{\tau }_L-B\omega ,\hfill \end{array}\right.$$

(1)

where $i_d$, $i_q$, $u_d$, $u_q$ and $L_d$, $L_q$ are the d-q axis current, d-q axis voltage, and d-q axis stator inductances, respectively. $n_p$ is the number of pole pairs, $\mathrm{\Phi }$ is the rotor flux linkage, $R_s$ is the stator resistance, $J_m$ is the inertia, $\omega$ is the mechanical angular velocity, ${\tau }_L$ is the load torque and B is the viscous friction coefficient. The formula of electromagnetic torque is $\tau ={\displaystyle \frac{3}{2}}{n}_p\left[(L_d-L_q)i_d{i}_q+\mathrm{\Phi }{i}_q\right]$, where $L_d=L_q$ in the SPMSM.

The mathematical model of the uncertainty of the SPMSM in the speed loop can be described as

$$\dot{\omega }=a_e+d_{\omega },$$

(2)

where $d_{\omega }$ is the lumped disturbance, including parameter variations, unknown load torques, unmodeled dynamics and external disturbances:

$$\left\{\begin{array}{c}{a}_e={\displaystyle \frac{3n_p\mathrm{\Phi }}{2J_m}}{i}_q,\hfill \\ d_{\omega }=-{\displaystyle \frac{{\tau }_L}{J_m}}-{\displaystyle \frac{B\omega }{J_m}}+{\displaystyle \frac{{\displaystyle \frac{3}{2}}{n}_p\mathrm{\Delta }\mathrm{\Phi }{i}_q-\mathrm{\Delta }{J}_m\dot{\omega }}{J_m}}+w_{\omega },\hfill \end{array}\right.$$

(3)

where $\mathrm{\Delta }\mathrm{\Phi }$ and $\mathrm{\Delta }{J}_m$ are variations in the $\mathrm{\Phi }$ and $J_m$, respectively. In addition, $w_{\omega }$ is the external disturbance.

Assumption A1.

The lumped disturbance of (2) is bounded and continuously differentiable [24,29,37,41,42].

3. Design of Sliding Mode Controller Based on ESO and ITSMRL

Figure 1 illustrates the proposed control scheme for the PMSM system. The whole control block diagram includes three parts, which are the speed loop controller, current loop controller and PMSM controlled-object. In the speed loop, firstly, the ITSMRL is proposed. Then, the ESO is developed with the purpose of estimating the total lumped disturbance. Finally, the observed lumped disturbance is fed forward; based on this, a composite sliding mode control method based on the ITSMRL is designed.

3.1. Improved TSMRL Design

In this section, the TSMRL is first presented, and then the, ITSMRL is proposed.

3.1.1. TSMRL

The RL should give consideration to fast approach and small chattering; that is, the reaching speed adapts based on distance to the sliding surface: higher when far away, lower when near.

The TSMRL is

$$\dot{s}=-k_1{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)-k_{2}s,$$

(4)

where s is the sliding mode surface, $\underset{s\to 0}{lim}\dot{s}s\le 0$, $k_1>0$, $k_2>0$ and $0<\beta <1$. The TSMRL can be thought of as the combination of the power RL $\dot{s}=-k_1{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)$ and the exponent RL $\dot{s}=-k_1\mathrm{sign}\left(s\right)-k_{2}s$. During the reaching phase, the system demonstrates accelerated convergence relative to the power RL.

3.1.2. Proposed ITSMRL

In order to further accelerate convergence when the state of the system is significantly distant from the sliding mode surface and reduce chattering when the system state is near the sliding mode surface, two power functions are introduced. The first power function is ${\left|s\right|}^{1+\beta }$, which enables the system to converge more rapidly when its state is substantially distant from the sliding mode surface. The second power function is $|x_1{|}^{1+\beta }$, where $x_1$ is the system state error and $\underset{t\to \infty }{lim}{x}_1=0$. When the state of the system approaches the sliding mode surface, it becomes small.

In this paper, an improved TSMRL (ITSMRL) is proposed, which has faster reach and reduced chattering compared to the above TSMRL. The ITSMRL can be represented as

$$\dot{s}=-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)-k_2{\left|s\right|}^{1+\beta }\mathrm{sign}\left(s\right),$$

(5)

where $k_1>0$, $k_2>0$, $0<\beta <1$. When $\left|s\right|>1$ and $|x_1|>1$, the ITSMRL $|-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)-k_2{\left|s\right|}^{1+\beta }\mathrm{sign}\left(s\right)|$ is bigger than the TSMRL $|-k_1{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)-k_{2}s|$. When $\left|s\right|<1$ and $|x_1|<1$, the ITSMRL is smaller than the TSMRL. Therefore, the ITSMRL has faster reaching speed and less chattering than the TSMRL.

Theorem 1.

The proposed ITSMRL (5) will reach convergence in finite time.

Proof. 

Define $V={\displaystyle \frac{s^2}{2}}$; according to (5), the derivative of V can be obtained as

$$\begin{array}{c}\dot{V}=s\dot{s}=-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{2-\beta }-k_2{\left|s\right|}^{2+\beta }\le 0\hfill \end{array}$$

(6)

Because of $s\dot{s}\le 0$, the existence and reachability conditions of the continuous sliding mode systems are satisfied, and there is a RL and an equilibrium point $s=0$, $\dot{s}=0$.

The reaching time can be obtained by integrating (5) as follows:

$$\begin{array}{c}t={\int }_{\left|s\right(0\left)\right|}^0{\displaystyle \frac{1}{-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{1-\beta }-k_2{\left|s\right|}^{1+\beta }}}\mathrm{d}s\hfill \\ ={\displaystyle \frac{\arctan \left[s{\left(0\right)}^{\beta }{k}_1^{-\frac{1}{2}}{k}_2^{\frac{1}{2}}{\left|x_1\right|}^{-\frac{1+\beta }{2}}\right]}{\beta k_1^{\frac{1}{2}}{k}_2^{\frac{1}{2}}{\left|x_1\right|}^{\frac{1+\beta }{2}}}}\hfill \end{array}$$

(7)

The proof is thus completed. □

3.2. Design of ESO

Define $x_1=\omega$ and $x_2=d_{\omega }$; consequently, the system dynamics described by Equation (2) can be formulated as follows:

$$\left\{\begin{array}{c}{\dot{x}}_1=b{i}_q+x_2,\hfill \\ {\dot{x}}_2={\dot{d}}_{\omega },\hfill \end{array}\right.$$

(8)

where $b={\displaystyle \frac{3n_p\mathrm{\Phi }}{2J_m}}$.

For the above mentioned equation, the ESO can be formulated to estimate the state variables and the lumped disturbance. Its design is presented as follows:

$$\left\{\begin{array}{c}{\dot{z}}_1=b{i}_q+z_2-{\beta }_1(z_1-x_1),\hfill \\ {\dot{z}}_2=-{\beta }_2(z_1-x_1),\hfill \end{array}\right.$$

(9)

where $z_1$ and $z_2$ are the estimations of $x_1$ and $x_2$. According to the bandwidth selection method [43], ${\beta }_1=2{\omega }_0$ and ${\beta }_2={\omega }_0^2$, where ${\omega }_0>0$ is the bandwidth of the ESO.

Theorem 2.

Considering the dynamic (2) and its corresponding ESO (9), by selecting the parameters ${\beta }_1=2{\omega }_0$, ${\beta }_2={\omega }_0^2$ (${\omega }_0>0$), the estimation error $e_{\omega }$ in any time is bounded. In addition, increasing the bandwidth of the ESO ${\omega }_0$ can reduce the estimation error and increase the convergence speed.

Proof. 

From (8) and (9), the dynamic of the observation error related to the ESO can be formulated as

$${\dot{e}}_{\omega }=A{e}_{\omega }+D{\dot{d}}_{\omega },$$

(10)

where $e_{\omega }=x-z$, $A=\left[\begin{array}{cc}-{\beta }_1& 1\\ -{\beta }_2& 0\end{array}\right]$, $D=\left[\begin{array}{c}0\\ 1\end{array}\right]$. The characteristic polynomial of A can be presented in the following form:

$$\lambda \left(s\right)=\det (sI-A)=s^2+{\beta }_{1}s+{\beta }_2,$$

(11)

where ${\beta }_1=2{\omega }_0$, ${\beta }_2={\omega }_0^2$(${\omega }_0>0$). So,

$$\lambda \left(s\right)={(s+{\omega }_0)}^2.$$

(12)

Thus, the matrix A with the eigenvalues ${\lambda }_i=-{\omega }_0<0$($i=1,2$) is a Hurwitz matrix, and the observed error dynamic (10) of the ESO is stable. Additionally,

$$\exp \left\{At\right\}=\sum _{i=1}^2{W}_i\left[{\displaystyle \frac{t^{i-1}}{(i-1)!}}\right]\exp \{-{\omega }_{0}t\},$$

(13)

where $W_i\in {\mathbb{R}}^{2\times 2}$. It can be found that

$$\left|\right|\exp \left\{At\right\}\left|\right|\le \sum _{i=1}^2\left|\right|W_i\left|\right|\left[{\displaystyle \frac{t^{i-1}}{(i-1)!}}\right]\exp \{-{\omega }_{0}t\}.$$

(14)

Subsequently, for each $\delta >0$, with $\rho =({\omega }_0-\delta )>0$, a constant $P_1$ can be found that satisfies

$$\left|\right|\exp \left\{At\right\}\left|\right|\le \sum _{i=1}^2\left|\right|W_i\left|\right|\left[{\displaystyle \frac{t^{i-1}}{(i-1)!}}\right]\exp \{-{\omega }_{0}t\}\le P_1\exp \{-\rho t\}.$$

(15)

Because ${\zeta }_0=\underset{t>0}{sup}\left|\right|{\dot{d}}_{\omega }\left|\right|$, the observed error dynamic (10) satisfies

$$\left|\right|e_{\omega }\left(t\right)\left|\right|\le P_1\exp \{-\rho (t-t_0)\}\left[\left|\right|e_{\omega }\left(t_0\right)\left|\right|-{\displaystyle \frac{{\zeta }_0}{\rho }}\right]+{\displaystyle \frac{P_1{\zeta }_0}{\rho }},t\ge t_0.$$

(16)

And

$$\underset{t\to \infty }{lim}\left|\right|e_{\omega }\left(t\right)\left|\right|={\displaystyle \frac{P_1{\zeta }_0}{\rho }}.$$

(17)

This indicates that the estimated state z converges exponentially to a bounded region centered around the system state x. Additionally, it can be seen that an increase in the ESO bandwidth ${\omega }_0$ leads to a higher convergence speed and a reduced radius of the region within which the estimation error remains bounded. This completes the proof. □

3.3. Design of Proposed Controller

The speed error is defined as $e={\omega }^{*}-\omega$, where ${\omega }^{*}$ is the reference speed. Define the PID sliding mode surface as

$$s=\dot{e}+{\rho }_{1}e+{\rho }_2{\int }_0^{t}e\left(\tau \right)\mathrm{d}\tau +d_s,$$

(18)

where ${\rho }_1>0$, ${\rho }_2>0$ and $d_s=d_{\omega }-z_2$.

Assumption A2.

The residual disturbance and its derivative are bounded, which are defined by $\epsilon =\underset{t>0}{sup}\left|\right|d_s\left|\right|$ and $\eta =\underset{t>0}{sup}\left|\right|{\dot{d}}_s\left|\right|$.

The devirative of the global PID surface is

$$\begin{array}{c}\dot{s}=\ddot{e}+{\rho }_1\dot{e}+{\rho }_{2}e+{\dot{d}}_s\hfill \\ =({\ddot{\omega }}^{*}-\ddot{\omega })+{\rho }_1\dot{e}+{\rho }_{2}e+{\dot{d}}_s.\hfill \end{array}$$

(19)

The controller can be obtained with the ITSMRL (5) and the lumped disturbance observed by the ESO (9):

$$i_q^{*}={\displaystyle \frac{1}{b}}\left\{\int \left[{\rho }_1\dot{e}+{\rho }_{2}e+k_1|x_1{|}^{1+\beta }{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)+k_2{\left|s\right|}^{1+\beta }\mathrm{sign}\left(s\right)\right]-z_2\right\}$$

(20)

Theorem 3.

Consider the PMSM system with uncertainties (2). Utilizing the proposed PID sliding mode surface (18), in cases where the control input is configured as (20), the sliding variable s will asymptotically converge to the bounded region $min({\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2)$ in a finite time, where ${\mathrm{\Theta }}_1={\left({\displaystyle \frac{{\rho }_1\epsilon +\eta }{k_2}}\right)}^{{\displaystyle \frac{1}{1+\beta }}}$, ${\mathrm{\Theta }}_2={\left({\displaystyle \frac{{\rho }_1\epsilon +\eta }{k_1}}\right)}^{{\displaystyle \frac{1}{1-\beta }}}$.

Proof. 

Consider a Lyapunov function $V={\displaystyle \frac{1}{2}}{s}^2$. By differentiating V with respect to time under the control input (20),

$$\begin{array}{c}\dot{V}=s\dot{s}\hfill \\ =s[{\rho }_1{d}_s+{\dot{d}}_s-k_1|x_1{|}^{1+\beta }|s{|}^{1-\beta }\mathrm{sign}\left(s\right)-k_2|s{|}^{1+\beta }\mathrm{sign}\left(s\right)]\hfill \\ ={\rho }_1{d}_{s}s+{\dot{d}}_{s}s-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{2-\beta }-k_2{\left|s\right|}^{2+\beta }\hfill \\ \le {\rho }_1\epsilon s+\eta s-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{2-\beta }-k_2{\left|s\right|}^{2+\beta }\hfill \\ =-(k_2{\left|s\right|}^{\beta }-{\displaystyle \frac{{\rho }_1\epsilon +\eta }{s}})s^2-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{2-\beta }.\hfill \end{array}$$

(21)

If $k_2{\left|s\right|}^{\beta }-{\displaystyle \frac{{\rho }_1\epsilon +\eta }{s}}>0$, there exist ${\lambda }_1>0$ and ${\lambda }_2>0$, which satisfy

$$\begin{array}{c}\dot{V}\le -{\lambda }_1{s}^2-{\lambda }_2{\left|s\right|}^{2-\beta }\hfill \\ =-2{\lambda }_{1}V-2^{{\displaystyle \frac{2-\beta }{2}}}{\lambda }_2{V}^{{\displaystyle \frac{2-\beta }{2}}}.\hfill \end{array}$$

(22)

The region of s needs to meet $k_2{\left|s\right|}^{\beta }-{\displaystyle \frac{{\rho }_1\epsilon +\eta }{s}}>0$, as

$$\left|s\right|>{\left({\displaystyle \frac{{\rho }_1\epsilon +\eta }{k_2}}\right)}^{{\displaystyle \frac{1}{1+\beta }}}={\mathrm{\Theta }}_1.$$

(23)

Therefore, the system reaches the following region within a finite time

$$\left|s\right|\le {\mathrm{\Theta }}_1.$$

(24)

$\dot{V}$ also can be written as

$$\begin{array}{c}\dot{V}=s\dot{s}\hfill \\ \le {\rho }_1\epsilon s+\eta s-k_1|x_1{|}^{1+\beta }{\left|s\right|}^{2-\beta }-k_2{\left|s\right|}^{2+\beta }\hfill \\ =-k_2{\left|s\right|}^{\beta }{\left|s\right|}^2-(k_1|x_1{|}^{1+\beta }-{\displaystyle \frac{{\rho }_1\epsilon s+\eta s}{{\left|s\right|}^{2-\beta }}}{\left)\right|s|}^{2-\beta }.\hfill \end{array}$$

(25)

If $k_1{\left|x_1\right|}^{1+\beta }-{\displaystyle \frac{{\rho }_1\epsilon s+\eta s}{{\left|s\right|}^{2-\beta }}}>0$, there exist ${\lambda }_3>0$ and ${\lambda }_4>0$, which satisfy

$$\begin{array}{c}\dot{V}\le -{\lambda }_3{s}^2-{\lambda }_4{\left|s\right|}^{2-\beta }\hfill \\ =-2{\lambda }_{3}V-2^{{\displaystyle \frac{2-\beta }{2}}}{\lambda }_4{V}^{{\displaystyle \frac{2-\beta }{2}}}.\hfill \end{array}$$

(26)

The region of s needs to meet $k_1{\left|x_1\right|}^{1+\beta }-{\displaystyle \frac{{\rho }_1\epsilon s+\eta s}{{\left|s\right|}^{2-\beta }}}>0$, as

$$\left|s\right|>{\left({\displaystyle \frac{{\rho }_1\epsilon +\eta }{k_1}}\right)}^{{\displaystyle \frac{1}{1-\beta }}}={\mathrm{\Theta }}_2.$$

(27)

Therefore, the system approaches the following region in a finite time:

$$\left|s\right|\le {\mathrm{\Theta }}_2.$$

(28)

In conclusion, the sliding variable s will asymptotically converge to the bounded region $min({\mathrm{\Theta }}_1,{\mathrm{\Theta }}_2)$ in a finite time, where ${\mathrm{\Theta }}_1={\left({\displaystyle \frac{{\rho }_1\epsilon +\eta }{k_2}}\right)}^{{\displaystyle \frac{1}{1+\beta }}}$, ${\mathrm{\Theta }}_2={\left({\displaystyle \frac{{\rho }_1\epsilon +\eta }{k_1}}\right)}^{{\displaystyle \frac{1}{1-\beta }}}$. This completes the proof. □

4. Simulations and Experiments

In this section, simulations and experiments are implemented to evaluate the performance of this method. The parameters of the PMSM are shown in

Table 1. The parameters of the PMSM.

Parameters	Label	Value
rated power	$P_{rated}$	64 W
rated voltage	$V_{rated}$	24 V
rated current	$I_{rated}$	4 A
rated speed	${\omega }_{rated}$	3000 rpm
rated torque	${\tau }_{rated}$	$0.2\mathrm{N}·\mathrm{m}$
pole pairs	$n_p$	4
stator inductance	L	$0.295$ mH
stator resistance	$R_s$	$0.51\mathrm{\Omega }$
inertia constant	$J_m$	$0.000028\mathrm{Kg}·{\mathrm{m}}^2$

.

To validate the effectiveness and robustness of the proposed approach, three other speed controllers were tested. The first one is a composite controller with a PID sliding surface (18), TSMRL (4) and ESO (9) (PIDSMC+ TSMRL), where the PID sliding mode surface and the ESO are consistent with the proposed algorithm; only the RL is different. The PIDSMC+TSMRL can be expressed as

$$i_q^{*}={\displaystyle \frac{2J_m}{3n_p\mathrm{\Phi }}}\left\{{\int }_0^t\left[{\rho }_1\dot{e}+{\rho }_{2}e+k_1{\left|s\right|}^{1-\beta }\mathrm{sign}\left(s\right)+k_{2}s\right]-z_2\right\}.$$

(29)

The second one is the TSMC described in [36] with an ESO (9), represented as

$$i_q^{*}={\displaystyle \frac{2J_m}{3n_p\mathrm{\Phi }}}\left\{{c\left|e\right|}^{\alpha }\mathrm{sat}\left(e\right)+p{\int }_0^t\mathrm{sign}\left[\dot{e}+c{\left|e\right|}^{\alpha }\mathrm{sat}\left(e\right)\right]-z_2\right\},$$

(30)

where $c>0$, $0<\alpha <1$, $p>0$, $\mathrm{sat}\left(e\right)=1$ when $e>▵e$, $\mathrm{sat}\left(e\right)=e$ when $e\le ▵e$ and $\mathrm{sat}\left(e\right)=-1$ when $e<-▵e$. The third tested controller is a conventional SMC(CSMC) [44], represented as

$$i_q^{*}={\displaystyle \frac{2J_m}{3n_p\mathrm{\Phi }}}\left\{{\int }_0^t\lambda (-\dot{\omega })+\eta \mathrm{sign}\left[\lambda ({\omega }^{*}-\omega )-\dot{\omega })\right]\right\},$$

(31)

where $\lambda >0$ and $\eta >0$.

4.1. Simulations

The relevant control parameters used in the experiments are presented in

Table 2. The parameters of controllers.

Algorithms	Parameters
The proposed method	$k_1=3.5$, $k_2=160$, ${\rho }_1=6000$, ${\rho }_2=0.01$, $\beta =0.08$, ${\omega }_0=10$
PIDSMC+TSMRL	$k_1=3.5$, $k_2=380$, ${\rho }_1=6000$, ${\rho }_2=0.01$, $\beta =0.08$, ${\omega }_0=10$
TSMC	$c=1020$, p = 25,000,000, $\alpha =0.6$, $▵e=1$, ${\omega }_0=10$
CSMC	$\lambda =200$, $\eta$ = 35,000,000

. To ensure fairness, the controller parameters were selected as the relative optimum after numerous repeated tests, incorporating both dynamic and static performance characteristics. For the surface-mounted PMSM, the parameters of the PI controller in the current loop are designed as $k_{cp}=\alpha L$ and $k_{ci}=\alpha R_s$; here, the bandwidth of the current loop is $\alpha =2\pi /\tau$. $\tau =L/R_s$ represents the time constant of the PMSM [44].

During the simulation process, the reference speed of the PMSM was 800 rpm when powered on, and a load torque disturbance of $0.2\mathrm{N}·\mathrm{m}$ was added at $0.1$ s. The speed curve and q-axis current curve of the PMSM during acceleration and loading are shown in Figure 2 and Figure 3, respectively. It can be seen that because of the ITSMRL, compared to the other two controllers, the proposed method has the fastest acceleration, the fastest recovery and the smallest speed fluctuation.

For quantitative comparison, the settling time (ST) and q-axis current root mean square error (RMSEA) were calculated in acceleration simulation, and the recovery time, speed fluctuation and q-axis current root mean square error (RMSEL) were calculated in the loading simulation. The quantitative comparison values are shown in

Table 3. Quantitative comparisons of simulation.

Algorithms	ST (s)	RMSEA (A)	RT (s)	SF (rpm)	RMSEL (A)
The proposed method	$0.009$	$0.0853$	$0.007$	$12.27$	$0.7039$
PIDSMC+TSMRL	$0.014$	$0.0879$	$0.011$	$12.61$	$0.7142$
TSMC	$0.013$	$0.0935$	$0.011$	$15.21$	$0.7326$
CSMC	$0.025$	$0.1574$	$0.015$	$16.17$	$0.8057$

. It can be seen that the proposed method has the smallest settling time, recovery time, speed fluctuation and q-axis current root mean square error compared to PIDSMC+TSMRL, TSMC and CSMC.

4.2. Experiments

The acceleration experiments and loading experiments were carried out on the platform shown below to verify the effectiveness and robustness of the proposed method. The hardware platform of the PMSM experiments is shown in Figure 4. The main controller is a Digital Signal Processor of the C2000 series which is manufactured by Texas Instruments (TI) in Dallas, Texas, USA. The load is added by another PMSM.

To ensure fairness, the controller parameters were optimized through numerous repeated tests, considering both dynamic and static performance. The parameters of the controllers used in the experiments are shown in

Table 4. The parameters of the controllers.

Algorithms	Parameters
The proposed method	$k_1=0.001$, $k_2=2.5$, ${\rho }_1=6000$, ${\rho }_2=0.01$, $\beta =0.08$, ${\omega }_0=50$
PIDSMC+TSMRL	$k_1=0.001$, $k_2=5.5$, ${\rho }_1=6000$, ${\rho }_2=0.01$, $\beta =0.08$, ${\omega }_0=50$
TSMC	$c=2500$, $p=8000$, $\alpha =0.6$, $▵e=1$, ${\omega }_0=50$
CSMC	$\lambda =500$, $\eta =5000$

. For the surface-mounted PMSM, the PI parameters within the current loop were configured as $k_{cp}=0.1$ and $k_{ci}=0.001$.

4.2.1. Acceleration Experiments

In the acceleration experiments, the reference speed of the PMSM was 2000rpm when the PMSM ran at 800 rpm. Figure 5a,b, Figure 6a,b, Figure 7a,b and Figure 8a,b show the speed and $i_q$ response curves under different controllers. The quantitative comparison values are shown in

Table 5. Quantitative comparison of acceleration experiment.

Algorithms	ST (s)	RMSEA (A)
The proposed method	$0.42$	$0.06$
PIDSMC+TSMRL	$0.57$	$0.07$
TSMC	$0.65$	$0.09$
CSMC	$0.73$	$0.09$

. It can be seen that the proposed method has the fastest tracking performance compared to PIDSMC+TSMRL, TSMC and CSMC, in addition to reduced chattering, due to the use of the ITSMRL.

4.2.2. Loading Experiments

In the loading experiments, a sudden load torque disturbance was added when the PMSM was running at the speed of $2000\mathrm{rpm}$. The response curves for speed and $i_q$ under different controllers are shown in Figure 9a,b, Figure 10a,b, Figure 11a,b and Figure 12a,b. The quantitative comparison values are shown in

Table 6. Quantitative comparison of loading experiment.

Algorithms	SF (rpm)	RMSEL (A)
The proposed method	$4.5$	$0.15$
PIDSMC+TSMRL	$5.9$	$0.17$
TSMC	$12.1$	$0.25$
CSMC	$69.8$	$0.23$

. It can be seen that the smallest speed fluctuation and smallest chattering occurred under the proposed method compared to PIDSMC+TSMRL, TSMC and CSMC when loading, as shown in Table 6. Due to the use of the ITSMRL, the proposed method showed reduced chattering and strong robustness.

5. Conclusions

A composite controller based on a PID sliding mode surface and the ITSMRL has been proposed in this paper. The ITSMRL has the advantages of fast reaching speed and reduced chattering. Moreover, it is verified that the RL can drive the system to reach the sliding mode surface in a finite time. On the basis of the observed disturbance of the ESO feedforward, a composite controller based on the PID sliding mode surface and ITSMRL is designed to enhance the disturbance rejection ability of the PMSM system. The composite method has the characteristics of fast reaching speed, reduced chattering and strong disturbance rejection ability, and is verified by simulation and experiments.