#### 1. Introduction

Permanent magnet synchronous motor (PMSM) is widely used in many high-performance fields such as robotics, automotive, aerospace, and manufacturing due to its high-power density, good dynamic performance, and reliable operation [1]-[3]. In the modern motor control system, PMSM mainly adopts linear control methods such as PI control, and adopts double closed-loop structure. However, the PI controller is difficult to maintain high-performance control under complex working conditions [4]-[7]. In order to improve the performance of PMSM control system under uncertain disturbance, such as sliding mode control algorithm (SMC) [8]-[10], adaptive control algorithm [11]-[14], neural network control algorithm [15]-[18], model predictive control algorithm (MPC) [19]-[21], active disturbance rejection control algorithm (ADRC) and so on are applied to PMSM control system.

ADRC has attracted wide attention because it does not require accurate system model and excellent anti-interference ability [22], [23]. The nonlinear control algorithm based on ADRC has been applied to the PMSM control strategy. In [24], to realize the estimation and compensation of the total disturbance of the sensorless drive system and improve the dynamic performance of the system, a sensorless control scheme of IPMSM based on ADRC is proposed. However, the PI controller is still applied to the current loop. In [25], a robust control strategy using three first-order ADRC is proposed to deal with internal and external disturbances, resulting in robustness improvement and costs reduction.

Moreover, due to the high sensitivity of sensorless control to motion parameters, ADRC is also suitable for sensorless control of motors due to its high robustness to parameters. In [26], a speed estimation strategy based on ADRC is proposed, which can accurately estimate the disturbance in the case of uncertain system model. In [27], a sensorless control scheme of PMSM based on enhanced LADRC is proposed. The system takes the estimated total disturbance as the compensation in the current loop, to further improve the accuracy of rotor position estimation. In [28], a new type of NLADRC is proposed, and the NLADRC is constructed based on the cascaded nonlinear extended state observer to improve the performance of disturbance estimation and compensation. In [29], a position sensorless control strategy based on active disturbance rejection control (ADRC) and adaptive full-order observer is designed to improve the anti-interference ability of the control system as well as accuracy of position estimation.

In this paper, an active disturbance rejection sliding mode control (ADR-SMC) is proposed to improve its anti-interference ability and dynamic response performance. Firstly, the NLADRC control system of second-order SPMSM is designed. Secondly, the NLSEF module in the system is replaced by SMC to obtain strong robustness and reduce system oscillation. Then, according to the disturbance observed by ESO, the output of SMC is adjusted by feedback to improve the current tracking performance of the disturbance under steady-state and transient conditions and obtain strong robustness. Meanwhile, the stability of ADR-SMC control system is proved by Lyapunov function. Finally, the feasibility and effectiveness of the proposed control strategy are verified by experiments.

#### 2. Mathematical model of SPMSM

In the case of ignoring the hysteresis loss and permanent magnet damping, the state equation of SPMSM is established in the \(dq\)-axis as follows.

\[\begin{equation*} \left\{ \begin{aligned} & {{u}_{d}}\text{=}R{{i}_{d}}+{{L}_{s}}\frac{\text{d}{{i}_{d}}}{\text{d}t}-{{\omega }_{e}}{{L}_{s}}{{i}_{q}} \\ & {{u}_{q}}\text{=}R{{i}_{q}}+{{L}_{s}}\frac{\text{d}{{i}_{q}}}{\text{d}t}+{{\omega }_{e}}({{L}_{s}}{{i}_{q}}+{{\psi }_{f}}) \\ \end{aligned} \right. \tag{1} \end{equation*}\] |

Where \(i_d, i_q, u_d, u_q\) are stator \(d\), \(q\)-axis current and voltage respectively. The mechanical equation and electromagnetic torque equation of the SPMSM are shown in Eq. (2).

\[\begin{equation*} \left\{ \begin{aligned} & J\frac{d{{\omega }_{m}}}{dt}={{T}_{e}}-{{T}_{L}}-B{{\omega }_{m}} \\ & {{T}_{e}}\text{=}\frac{3}{2}{{p}_{n}}{{i}_{q}}{{\psi }_{f}} \\ \end{aligned} \right. \tag{2} \end{equation*}\] |

In the formula, \(J\) is the moment of inertia, \(T_e\) is the electromagnetic torque, \(T_L\) is the load torque, \(B\) is the damping coefficient, and \(P_n\) is the polar logarithm.

#### 3. ADR-SMC model

##### 3.1 Mathematical model of speed-current compound loop based on SPMSM

The object is SPMSM based on \(L_q=L_d\). From the Eqs. (1), (2), it can be seen that there is a coupling term between \(i_d, i_q\) and \(\omega_{m}\) in SPMSM. However, the SPMSM control is \(i_d = 0\), so the coupling between \(i_d\) and \(i_q\) can be ignored. Therefore, the state equation of the composite loop can be obtained from Eq. (1) and Eq. (2).

\[\begin{equation*} {{\ddot{\omega }}_{m}}=-\frac{{{{\dot{T}}}_{L}}}{J}-\frac{B{{{\dot{\omega }}}_{m}}}{J}\text{-}\frac{3{{P}_{n}}{{\psi }_{f}}({{R}_{s}}{{i}_{q}}+{{P}_{n}}{{\omega }_{m}}{{\psi }_{f}})}{2J{{L}_{s}}}+\frac{3{{P}_{n}}{{\psi }_{f}}{{u}_{q}}}{2J{{L}_{s}}} \tag{3} \end{equation*}\] |

Let

\[\begin{equation*} \begin{aligned} & b_0=\frac{3{{P}_{n}}{{\psi }_{f}}}{2J{{L}_{s}}}, \\ & f({{\omega }_{m}},{{i}_{q}},{{T}_{L}})=-\frac{{{{\dot{T}}}_{L}}}{J}-\frac{B{{{\dot{\omega }}}_{m}}}{J}\frac{3{{P}_{n}}{{\psi }_{f}}({{R}_{s}}{{i}_{q}}+{{P}_{n}}{{\omega }_{m}}{{\psi }_{f}})}{2J{{L}_{s}}} \\ \end{aligned} \tag{4} \end{equation*}\] |

By combining Eq. (3) and Eq. (4), the mathematical model of SPMSM speed-current composite loop can be obtained as follows.

\[\begin{equation*} {{\ddot{\omega }}_{m}}=f({{\omega }_{m}},{{i}_{q}},{{T}_{L}})+{{b}_{0}}{{u}_{q}} \tag{5} \end{equation*}\] |

Where, \(f(\omega_{m}, i_q, T_L)\) denotes the total disturbance of the SPMSM composite loop.

##### 3.2 NLADRC system model

NLADRC consists of three parts: tracking differentiator (TD), extended state observer (ESO) and nonlinear state error feedback (NLSEF).

Construct second-order TD as follows.

\[\begin{equation*} \left\{ \begin{aligned} & {{{\dot{v}}}_{1}}={{v}_{2}} \\ & {{{\dot{v}}}_{2}}=fhan({{v}_{1}}-\omega_{m}^{*},{{v}_{2}},r,h) \\ \end{aligned} \right. \tag{6} \end{equation*}\] |

Where \(v_1\) and \(v_2\) are the tracking value and differential value of the given speed \(\omega_{m}^{*}\) of the motor, respectively. \(r\) is the gain; \(h\) is the integral step; \(fhan()\) is the fastest tracking function, which is expressed as follows.

\[\begin{equation*} \begin{aligned} & fsg(x,d)=(sign(x+d)-sign(x-d))/2 \\ & \left\{ \begin{aligned} & d=r{{h}^{2}},{{a}_{0}}=h{{x}_{2}},y={{x}_{1}}+{{a}_{0}},{{a}_{1}}=\sqrt{d(d+8\left| y \right|)} \\ & {{a}_{2}}={{a}_{0}}+sign(y)({{a}_{1}}-d)/2 \\ & a=({{a}_{0}}+y)fsg(y,d)+{{a}_{2}}(1-fsg(y,d)) \\ & fhan=-r(\frac{a}{d}-sign(a))fsg(a,d)-sign(a) \\ \end{aligned} \right. \\ \end{aligned} \tag{7} \end{equation*}\] |

Assuming that the total disturbance is differentiable, the state space expression of the SPMSM composite loop system is.

\[\begin{equation*} \left\{ \begin{aligned} & {{{\dot{x}}}_{1}}={{x}_{2}}\text{=}{{{\dot{\omega }}}_{m}} \\ & {{{\dot{x}}}_{2}}={{x}_{3}}+{{b}_{0}}u_{q} \\ & {{{\dot{x}}}_{3}}=\overset{\bullet }{\mathop{f({{\omega }_{m}},{{i}_{q}},{{T}_{L}})}}\, \\ & y={{x}_{1}}={{\omega }_{m}} \\ \end{aligned} \right. \tag{8} \end{equation*}\] |

According to Eq. (8), ESO can be expressed as.

\[\begin{equation*} \left\{ \begin{aligned} & {{e}}={{z}_{1}}-y \\ & {{{\dot{z}}}_{1}}={{z}_{2}}-{{\beta }_{0}}{fal(e,{{a}_{0}},\delta )} \\ & {{{\dot{z}}}_{2}}={{z}_{3}}-{{\beta }_{1}}{fal(e,{{a}_{1}},\delta )}+{{b}_{0}}u \\ & {{{\dot{z}}}_{3}}=-{{\beta }_{2}}{fal(e,{{a}_{2}},\delta )} \\ \end{aligned} \right.\ \tag{9} \end{equation*}\] |

Where \(z_1\), \(z_2\) and \(z_3\) are the observed values of motor speed, speed differential and total disturbance, respectively. \(\beta_i\) is the observer gain, \(b_0\) is the compensation factor, and \(fal()\) is an error processing function.

From Eq. (9), ESO follows the order of \(z_i\) tracking \(x_i\).

The \(fal()\) is defined as.

\[\begin{equation*} {{\varphi }_{i}}(e)=fal(e,{{a}_{i}},\delta )\text{=}\left\{ \begin{aligned} & \frac{e}{{{\delta }^{1-{{a}_{i}}}}},\left| e \right|\le \delta \\ & {{\left| e \right|}^{{{a}_{i}}}}sign(e),\left| e \right|>\delta \\ \end{aligned} \right. \tag{10} \end{equation*}\] |

Where \(a\) is a nonlinear factor; \(\delta\) is the length of the nonlinear interval.

The NLSEF is constructed as follows

\[\begin{equation*} \left\{ \begin{aligned} & {{e}_{1}}={{v}_{1}}-{{z}_{1}} \\ & {{e}_{2}}={{v}_{2}}-{{z}_{2}} \\ & {{u}_{0}}={{k}_{1}}fal({{e}_{1}},{{a}_{1}},\delta )+{{k}_{2}}fal({{e}_{2}},{{a}_{2}},\delta ) \\ & u_q={{u}_{0}}-\frac{{{z}_{3}}}{{{b}_{0}}} \\ \end{aligned} \right. \tag{11} \end{equation*}\] |

Where \(k_1\) and \(k_2\) are the gains of NLSEF, \(u_0\) is the output variable of NLSEF.

##### 3.3 ADR-SMC system design

Furthermore, ESO in ADRC is difficult to completely estimate the disturbance of the system. Therefore, SMC can be used to improve the performance of ADRC by replacing NLSEF with sliding mode control law because of its low requirement and strong robustness to the system model. Meanwhile, the chattering problem in SMC can be eliminated by ADRC, which also improves the observation ability of ESO. The ADR-SMC combined with the two control methods has distinct advantages.

Define the state error equation.

\[\begin{equation*} \left\{ \begin{aligned} & {{e}_{1}}={{v}_{1}}-{{z}_{1}} \\ & {{e}_{2}}={{{\dot{e}}}_{1}}={{v}_{2}}-{{z}_{2}} \\ \end{aligned} \right. \tag{12} \end{equation*}\] |

The introduction of exponential function and power function in SMC is to achieve faster convergence speed and weaken chattering simultaneously, and output smooth signal.

\[\begin{equation*} \left\{ \begin{aligned} & \dot{s}=-{{\chi }_{1}}{{\left| s \right|}^{\mu }}H(s)-{{\chi }_{2}}\left( {{e}^{|s|}}-1 \right)H(s) \\ & H(s)=\frac{{{e}^{as}}-{{e}^{-as}}}{{{e}^{as}}+{{e}^{-as}}} \\ \end{aligned} \right. \tag{13} \end{equation*}\] |

where, \(\chi_{1},\chi_{2}\) are positive constants, \(H(s)\) is hyperbolic tangent function, \(a\) is boundary layer, \(\mu\) is positive coefficient, \(0<\mu<1\). Select the sliding surface \(s\).

\[\begin{equation*} s=c{{e}_{1}}\text{+}{{e}_{2}} \tag{14} \end{equation*}\] |

Where \(c>0\). The derivative of the sliding surface is formulated as.

\[\begin{equation*} \begin{aligned} & \dot{s}=c{{{\dot{e}}}_{1}}+{{{\dot{e}}}_{2}}=c({{v}_{2}}-{{z}_{2}})+({{{\dot{v}}}_{2}}-{{{\dot{z}}}_{2}}) \\ & =c({{v}_{2}}-{{z}_{2}})+({{{\dot{v}}}_{2}}-f-b\centerdot {{u}_{q}}) \\ & \text{=-}{{\chi }_{1}}{{\left| s \right|}^{\mu }}H(s)-{{\chi }_{2}}\left( {{e}^{|s|}}-1 \right)H(s) \\ \end{aligned} \tag{15} \end{equation*}\] |

According to Eq. (15), the sliding mode error feedback control law can be obtained as.

\[\begin{equation*} {{u}_{q}}\text{=-}\frac{{{\chi }_{1}}{{\left| s \right|}^{\mu }}H(s)-{{\chi }_{2}}\left( {{e}^{|s|}}-1 \right)H(s)+c{{e}_{2}}-f+{{{\dot{v}}}_{2}}}{{{b}_{0}}} \tag{16} \end{equation*}\] |

According to the Lyapunov function \(V=0.5s^2\), it is proved that the stability is expressed as follows.

\[\begin{equation*} \dot{V}=-\frac{1}{2}s({{\chi }_{1}}{{\left| s \right|}^{\mu }}H(s)+{{\chi }_{2}}({{e}^{\left| s \right|}}-1)H(s))\le 0 \tag{17} \end{equation*}\] |

When \(\chi_{1}>0,\chi_{2}>0\), then \(\dot{V} \le 0\). The improved SMC satisfies the Lyapunov stability, so it can be concluded that the state variable error of the system converge to 0 in a finite time. The block diagram of the ADR-SMC system based on the new sliding mode error feedback law is shown in Fig. 1.

The block diagram of SPMSM speed-current composite loop control system based on ADR-SMC is shown in Fig. 2. The ADR-SMC controller is designed. The TD preprocessing of \(\omega_{ref}\) ensures fast response and small overshoot of the system. In addition, in order to make correspording compensation, ESO can not only observe the change of speed, but also observe the disturbance. Finally, the control variable is output by the sliding mode error feedback control law. In this paper, the improvement of typical NLADRC reduces the system oscillation and improves the robustness and accuracy of SPMSM control.

#### 4. Experimental verification and analysis

##### 4.1 Experimental platform

On the experimental platform of SPMSM system, the ADR-SMC algorithm proposed in this paper is verified by experiments, and compared with PI algorithm and NLADRC algorithm, which proves the advantages of the algorithm. In addition, the SPMSM control test system of the platform is shown in Fig. 3, and the SPMSM parameters listed in Table I are used in the test.

In addition, the parameters of TD module and ESO module in NLADRC and ADR-SMC are the same. The \(r\) in TD is set to 10000, and the gain in ESO is set to \(\beta_{1}=1000, \beta_{12}=10000, \beta_{3}= 12500, b_0=1500\). The parameters of SMC module in ADR-SMC were set as follows: \(c=15, \chi_{1}=150, \chi_{2}=100, \mu=0.5, a=10\). The parameters of NLSEF module in NLADRC are set as follows: \(k_1=8.5, k_2=9.3,\delta=0.05\). PI control speed loop parameters: \(k_{sp}=20, k_{si}=0.5\). Current loop parameters: \(k_{cp}=25, k_{ci}=1.2\).

##### 4.2 Start-up performance

The first comparative experiment is the start-up performance test of SPMSM. The comparison of the start-up performance of ADR-SMC, NLADRC and PI control strategies is shown in Fig. 4. The experimental results are shown in Table II, where \(\delta\) is the speed overshoot and \(t_s\) is the time required to achieve stability. It can be clearly seen that the speed overshoot and the time required to achieve stability with PI control strategy are relatively large, while the start-up performance with ADR-SMC and NLADRC strategy is much better. Specifically, in the case of large speed difference, the step speed of NLADRC is slow, but NLADRC can effectively alleviate the speed overshoot. ADR-SMC avoids the shortcomings of both, and its starting performance is better than the one with NLADRC and PI, resulting in small speed overshoot and short stability time.

##### 4.3 Steady-state performance

The second comparison experiment is the steady-state performance test of SPMSM at a given speed based on ADR-SMC, NLADRC and PI control strategies. The given speed is set to the reference value 3000 rpm. Fig. 5 and Fig. 6 show the speed, A-phase current and current harmonics of SPMSM. From Fig. 5, it can be seen that the speed fluctuation with PI control is the largest, the speed fluctuation of NLADRC is between the two, and the speed fluctuation of ADR-SMC is the smallest. Meanwhile, from the current harmonic analysis shown in Fig. 6, the current harmonics of ADR-SMC, NLADRC and PI control strategies are 10.59%, 9.40% and 7.96%, respectively, which further indicates that the proposed ADR-SMC has excellent steady-state performance.

##### 4.4 Dynamic performance

The third comparative experiment is the dynamic response performance test of SPMSM, which mainly compares the dynamic response ability of motor speed under sudden load. When the given speed is 3000 rpm and the sudden load torque is 0.5 N\(\cdot\)m, the results of the speed and phase current waveforms are shown in Fig. 7. The experimental results of dynamic performance comparison of SPMSM are shown in Table III. This clearly shows that when the sudden load is 0.5 N\(\cdot\)m, the dynamic response performance of NLADRC is better than that of SPMSM with PI control, and the response of ADR-SMC to disturbance is faster than that of NLADRC.Therefore, the dynamic performance with ADR-SMC proposed in this paper is better than the one with NLADRC and PI control strategies. In addition, the current waveform of ADR-SMC is more stable, and the pulsation is smaller, which further indicates that ADR-SMC can maintain good dynamic performance.

#### 5. Conclusion

In order to improve the control performance of SPMSM, this paper proposes a new SPMSM control strategy ADR-SMC. Firstly, the NLSEF module in the NLADRC speed-current compound loop is replaced by a SMC-based sliding mode error feedback control law. By introducing exponential function and power function into the sliding mode error feedback control law, the sliding mode chattering is effectively suppressed and the convergence time is shortened. The ADR-SMC control strategy has faster dynamic response, smaller steady-state error and stronger robustness. Finally, the comparative experimental results of ADR-SMC, NLADRC and PI control strategies show that the ADR-SMC control strategy has the best start-up performance and steady-state performance, as well as faster dynamic response capability. When the sudden load is 0.5 N\(\cdot\)m, the maximum speed fluctuation of ADR-SMC is about 27 rpm smaller than that of NLADRC, and the stability time is about 0.067 s smaller than that of NLADRC, which verifies the effectiveness and feasibility of ADR-SMC control strategy for SPMSM control.

#### Acknowledgments

National Natural Science Foundation of China: Grant No.51975526; 51505425. Technical service contract: KYY-HX-20231005; KYY-HX-20231005.

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#### Authors

Xu Zhang

Institute of Process Equipment and Control Engineering, Zhejiang University of Technology

Jianfeng Mao

Institute of Process Equipment and Control Engineering, Zhejiang University of Technology

Fujiong Zhao

Institute of Process Equipment and Control Engineering, Zhejiang University of Technology

Weigang Wang

Provincial Research Design Institute, Hangzhou Fusheng Electrical Appliances Co. Ltd.

Rongsheng Jia

Provincial Research and Design Institute, Hangzhou Xinhengli Motor Manufacturing Co. Ltd.