#### 1. Introduction

Permanent magnet synchronous motors (PMSMs) have significant advantages in structure, operational reliability, volume, power density, etc., and are therefore used in electric vehicle motor drive control system, machining center servo systems, flywheel energy storage systems, and other fields [1]-[7]. A PMSM control system with strong anti-interference ability and low parameter dependence is a performance indicator for achieving high-performance electric drive systems. At present, there are methods such as field oriented control (FOC) [8], [9], model predictive control (MPC) [10]-[13], and direct torque control (DTC) [14]-[17] to achieve high-performance control of PMSMs. Among them, the MPC can replace vector control of the current inner loop, solving the problem of high parameter dependence. Therefore, the study of high-performance MPC can effectively improve the performance of the control system.

The issues of parameter mismatch and delay in MPC have been extensively studied by numerous academics. In [18], the issue of depending too much on the motor’s mathematical model in the deadbeat MPC algorithm is addressed with a novel approach to current predictive control that uses a fuzzy algorithm. In [19], a proposed approach to improve model predictive current control is based on an incremental model. In this method, an inductance disturbance observer and inductance elimination traction algorithm are used to update the inductance of the control system in real time. In [20], a new MPC method is proposed, which uses an extended voltage vector set to improve current control accuracy. In [21], a super local model-based model-free predictive current control method (MFPCC) is proposed to address the issue of model parameter mismatch with motor parameter actualization. In view of the delay problem in MPC, Gao et al. (2020) propose a new direct compensation method that solves the time delay problem caused by a large number of calculations by predicting the current changes within the delay time. [22]. In order to improve the speed and current control performance of the MPC method, Li et al. (2021) propose a robust continuous model prediction speed and current control method based on adaptive integral sliding mode. However, the chattering problem in sliding mode control cannot be completely eliminated [23].

In the MPC algorithm, the performance of the speed loop controller also has a significant impact on the performance of the entire control system [24], [25]. The ADRC introduces an equivalent disturbance model to counteract the internal and external disturbances of the system and improve the anti-interference ability of the control system. Many scholars have conducted in-depth research on ADRCs. Currently, ADRCs can be mainly divided into linear ADRCs (LADRCs) and nonlinear ADRCs (NLADRCs), each with corresponding advantages and disadvantages. ADRC mainly consists of extended state observer, state error feedback (SEF), and tracking differentiator (TD), and the performance of each part has a significant impact on the entire system. In [26], a new ADRC-based sliding mode current control (SMCC) scheme is proposed. This method designed an extended state observer (ESO) to estimate internal disturbances in real-time, thereby improving steady-state and transient current tracking performance. The performance of the observer also has a significant impact on the control system. In [27], [28], an ADRC method using a phase-locked loop observer (PLLO) is proposed, which uses two types of ADRCs based on PLLs to improve the anti-interference performance of the speed control system. In order to solve the problem of insufficient interference observation in a single ESO, A sensorless FOC method for rotor position based on an enhanced LADRC (ELADRC) is proposed in [29]. This method uses two linear ESOs (LESOs) and a proportional current controller, thereby enhancing the control system’s overall performance. Zhu et al. (2022) propose a new NLADRC method, which replaces traditional linear functions with nonlinear functions and constructs a cascaded nonlinear ESO to ensure relatively fast and accurate disturbance estimation and compensation [30].

A cascaded LADRC multi-stage MPC (CasLADRC-MSMPC) method is proposed to meet the requirements of fast current tracking and strong anti-interference performance for PMSMs. This method proposes a finite set model predictive control algorithm based on the discrete mathematical model of PMSM. The advantages of the cascaded LADRC algorithm are verified through the design and analysis of the traditional LADRC algorithm and cascaded LADRC (CasLADRC) algorithm. Subsequently, the CasLADRC algorithm and the multi-stage finite set model predictive control algorithm are integrated to enhance the overall control system’s anti-interference capability and tracking performance.

#### 2. Mathematical model and analysis of multi-stage model predictive control

##### 2.1 Model predictive control mathematical model

In the d and q rotating coordinate systems, the stator voltage equation of PMSM is as follows:

\[\begin{equation*} \boldsymbol{U}=\boldsymbol{AI}+\boldsymbol{B} \tag{1} \end{equation*}\] |

where \(\boldsymbol{U}=\left[ \begin{matrix} u_d& u_q\\ \end{matrix} \right] ^{\mathrm{T}}\), \(\boldsymbol{I}=\left[ \begin{matrix} i_d& i_q\\ \end{matrix} \right] ^{\mathrm{T}}\), \(\boldsymbol{A}=\left[ \begin{matrix} R+pL_d& -\omega _{\mathrm{e}}L_q\\ \omega _{\mathrm{e}}L_d& R+pL_q\\ \end{matrix} \right]\), \(\boldsymbol{B}=\left[ \begin{matrix} 0& \omega _{\mathrm{e}}\lambda _f\\ \end{matrix} \right] ^{\mathrm{T}}\), the subscripts d and q respectively represent the coordinate axes corresponding to the rotating coordinate system. Therefore, \(i_d\), \(i_q\), \(L_d\), \(L_q\), \(u_d\) and \(u_q\) are the current, inductance, and voltage components corresponding to the coordinate axis. \(R\) represents stator resistance, \(\lambda_f\) represents the flux linkage of permanent magnet, \(\omega_e\) represents the rotor angular velocity value, and in surface mounted permanent magnet synchronous motors (SPMSM), there exists \(L_d\)=\(L_q\)=\(L_s\) because the \(d\)- and \(q\)-axis reluctance are equal.

Use the forward Euler method to discretize formula (1) and convert it into the stator current expression form. The specific form is:

\[\begin{equation*} \boldsymbol{I}\left( k+1 \right) =\boldsymbol{C}\left( k \right) \boldsymbol{I}\left( k \right) +D\boldsymbol{U}\left( k \right) -D\boldsymbol{B}\left( k \right) \tag{2} \end{equation*}\] |

where \(\boldsymbol{C}=\left[ \begin{matrix} \left( 1-\frac{R}{L_s}T_s \right)& \omega _eT_s\\ -\omega _eT_s& \left( 1-\frac{R}{L_s}T_s \right)\\ \end{matrix} \right]\), \(D=\frac{T_s}{L_s}\), \((k)\) and \((k + 1)\) represent the corresponding states of the parameters at the current time and the next time, respectively, while \(T_s\) represents the control period.

##### 2.2 Multi-stage model predictive control algorithm

Single vector MPC selects the voltage vector only once per control cycle, and the optimal voltage vector can be applied in the next cycle to achieve optimal results, but this also limits performance. To solve this problem, a multi-stage series model predictive control method is proposed. Unlike multi-step prediction, this method pushes the predicted current trajectory obtained from each voltage vector to multiple control cycles, and evaluates each voltage vector using multiple cost functions to ultimately select the optimal voltage vector. The multi-stage series MPC method considers multiple control cycles as a whole, and selects the same voltage vector as the optimal vector within multiple control cycles. The control block diagram is shown in Fig. 1.

The predicted current expression is shown in (2). According to (2), the predicted current values of the first stage prediction and evaluation process generated by different voltage vectors at time (k+1) can be calculated. The current error corresponding to different voltage vectors is shown by the arrow trajectory in Fig. 2(a). Define a cost function to calculate the corresponding value, and the calculation formula is:

\[\begin{equation*} g_x^{(k+1)}=\left[ i_{d}^{*}-i_d\left( k+1 \right) \right] ^2+\left[ i_{q}^{*}-i_q\left( k+1 \right) \right] ^2 \tag{3} \end{equation*}\] |

In order to save computational resources, the current prediction trajectory shown in Fig. 2(b) is used for the second stage prediction and evaluation. From Fig. 2(b), it can be seen that \(U_1\) and \(U_2\) are the values with the smallest current error, and there is no need to select \(U_4\) and \(U_6\). Therefore, in the second stage screening, the two values with the smallest current error can be directly selected, and then the other six values can be removed.

By sorting, the two quantities with the smallest current error and the 6 quantities excluded are obtained. The two smallest quantities will be selected as candidate voltage vectors in the second stage prediction, as shown in the two voltage vectors retained in the second control cycle in Fig. 2(b).

In order to apply the filtered voltage vector to the motor, the predicted current at time (k+2) and a new cost function needs to be provided, with the expression:

\[\begin{equation*} \begin{cases} \boldsymbol{I}\left( k+2 \right)\!=\!\boldsymbol{C}\left( k+1 \right) \boldsymbol{I}\left( k+1 \right) +D\boldsymbol{U}\left( k+1 \right) -D\boldsymbol{B}\left( k+1 \right)\\ g_x^{(k+2)}=\left[ i_{d}^{*}-i_d\left( k+2 \right) \right] ^2+\left[ i_{q}^{*}-i_q\left( k+2 \right) \right] ^2\\ \end{cases} \tag{4} \end{equation*}\] |

#### 3. Cascaded active disturbance rejection control algorithm and analysis

##### 3.1 Traditional LADRC algorithm

The following formulas can be used to express the electromagnetic torque and mechanical motion of SPMSMs:

\[\begin{equation*} \begin{cases} T_e=1.5n_p\lambda _fi_q\\ J\frac{\text{d}\omega _m}{\text{d}t}=T_e-T_L-B\omega _m\\ \end{cases} \tag{5} \end{equation*}\] |

where \(n_p\) is the number of pole pairs of the motor, \(\omega _m\) is the mechanical angular velocity of the motor, \(J\) is the moment of inertia of the rotor, and \(B\) is the viscous damping of the motor.

According to the above two equations, it can be rewritten as:

\[\begin{equation*} \frac{\text{d}\omega _e}{\text{d}t}=\frac{1.5n_{p}^{2}\lambda _f}{J} i_q-\frac{\left( n_pT_L+B\omega _e \right)}{J}=bu+f_w \tag{6} \end{equation*}\] |

where \(b={1.5n_{p}^{2}\lambda _f}/{J}, u=i_q, f_w=-{\left( n_pT_L+B\omega _e \right)}/{J}\), and \(f_w\) is the total perturbation of the system.

The form of the above equation conforms to the general form of ADRC, so it can be used for controlling the speed loop. The control block diagram is shown in Fig. 3.

The expression of the LESO shown in Fig. 3 is:

\[\begin{equation*} \begin{cases} \dot{z}_{21}=z_{22}-\beta _1\left( z_{21}-\omega _e \right) +bu\\ \dot{z}_{22}=-\beta _2\left( z_{21}-\omega _e \right)\\ \end{cases} \tag{7} \end{equation*}\] |

where \(z_\mathrm{21}\) is the rotor electrical angular velocity observation value, \(z_\mathrm{22}\) is the total disturbance observation value, \(\beta_ 1\) and \(\beta_ 2\) are the observer gains.

The LSEF equation and total control output part shown in Fig. 3 can be represented as:

\[\begin{equation*} \begin{cases} u_0=\beta _3\left( \omega _{e}^{*}-z_{21} \right)\\ u=u_0-\frac{z_{22}}{b}\\ \end{cases} \tag{8} \end{equation*}\] |

where \(\omega_\mathrm{e}^*\) is the given value of rotor electrical angular velocity, \(\beta_ 3\) is the gain of LSEF, \(u_0\) is the LSEF output value.

TLADRC requires high control bandwidth to meet the fast tracking performance requirements of the control system, but high bandwidth will lead to a decrease in system stability. Therefore, this article designs a cascaded form of LADRC to enhance the anti-interference ability and stability of the control system.

##### 3.2 Cascaded LADRC algorithm

The cascaded LADRC (CasLADRC) algorithm is based on the TLADRC algorithm, using two LESO cascades. LESO1 estimates the total disturbance preliminarily, and LESO2 estimates the unknown disturbance, thereby reducing the burden of single LESO estimation disturbance. Its structure is shown in Fig. 4.

The LESO1 shown in Fig. 4 can be represented as:

\[\begin{equation*} \begin{cases} \dot{v}_{21}=v_{22}-\beta _1\left( v_{21}-\omega _e \right) +bu\\ \dot{v}_{22}=-\beta _2\left( v_{21}-\omega _e \right)\\ \end{cases} \tag{9} \end{equation*}\] |

where \(v_{21}\) is the rotor’s estimated angular velocity and \(v_{22}\) is the initial estimate of the entire perturbation.

The LESO2 shown in Fig. 4 is an estimate of the remaining unknown perturbations based on \(v_{22}\) and is expressed as:

\[\begin{equation*} \begin{cases} \dot{s}_{21}=s_{22}+v_{22}-\beta _3\left( s_{21}-\omega _e \right) +bu\\ \dot{s}_{22}=-\beta _4\left( s_{21}-\omega _e \right)\\ \end{cases} \tag{10} \end{equation*}\] |

The \(s_{22}\) in LESO2 can complete the estimation of remaining unknown disturbances. Finally, the total disturbance \(v_{22}\) estimated in LESO1 and the remaining disturbance \(s_{22}\) estimated in LESO2 are added to form the estimation function for all disturbances.

##### 3.3 Time domain analysis

According to (7), the transfer function of the disturbance tracking error of LESO in the frequency domain can be expressed as:

\[\begin{equation*} \begin{cases} Z_{21}\left( s \right) =\frac{\Delta _2-s^2}{\Delta _2}\omega _{e}^{*}+\frac{s}{\Delta _2}bU\left( s \right)\\ Z_{22}\left( s \right) =\frac{\beta _2}{\Delta _2}\left[ s\omega _{e}^{*}+bU\left( s \right) \right] =\frac{\beta _2}{\Delta _2}X_2\left( s \right)\\ \end{cases} \tag{11} \end{equation*}\] |

where \(\Delta _2=s^2+\beta _1s+\beta _2\).

\[\begin{equation*} \frac{Z_{22}-X_2\left( s \right)}{X_2\left( s \right)}=-\frac{s\left( s+\beta _1 \right)}{s^2+\beta _1s+\beta _2} \tag{12} \end{equation*}\] |

For LSEF, the general output form is shown in the following equation. After performing Laplace transform, it can be represented as:

\[\begin{equation*} \frac{y\left( s \right)}{r\left( s \right)}=\frac{k_p}{s+k_p}= \frac{\omega _{c}}{s+\omega _{c}} \tag{13} \end{equation*}\] |

Therefore, the coefficient \(k_p = \omega _{c}\).

For LESO in (11), its general form can be expressed as:

\[\begin{equation*} \left\{ \begin{array}{l} \boldsymbol{\dot{r}}=\boldsymbol{Ar}+\boldsymbol{B}u+\boldsymbol{E}h\\ \boldsymbol{o}=\boldsymbol{Cr}\\ \end{array} \right. \tag{14} \end{equation*}\] |

where \(\boldsymbol{A}=\left[ \begin{matrix} 0& 1\\ 0& 0\\ \end{matrix} \right] ,\boldsymbol{B}=\left[ \begin{array}{c} b\\ 0\\ \end{array} \right] ,\boldsymbol{C}=\left[ \begin{array}{c} 1\\ 0\\ \end{array} \right] ^{\text{T}},\boldsymbol{E}=\left[ \begin{array}{c} 0\\ 1\\ \end{array} \right]\).

\(\boldsymbol{r}\), \(\boldsymbol{u}\) and \(\boldsymbol{o}\) in the formula (14) correspond to reference vector, input and output value of controlled object respectively. \(h\) is considered as the differential of the total disturbance. These variables have different meanings according to different design processes. The corresponding LESO is

\[\begin{equation*} \left\{ \begin{array}{l} \boldsymbol{\dot{z}}=\boldsymbol{Az}+\boldsymbol{B}u+\boldsymbol{L}\left( \boldsymbol{o}-\boldsymbol{\hat{o}} \right)\\ \boldsymbol{\hat{o}}=\boldsymbol{Cz}\\ \end{array} \right. \tag{15} \end{equation*}\] |

where the observer gain is \(\boldsymbol{L}\). The configuration of \(\boldsymbol{L}\) allows for the stabilization of the observation system, and varies the control performance of the system with varied gain coefficients. In order to prove the stability of the observation system, subtract the formula (15) from formula (14) to obtain

\[\begin{equation*} \boldsymbol{\dot{r}}-\boldsymbol{\dot{z}}=\left( \boldsymbol{A}-\boldsymbol{LC} \right) \left( \boldsymbol{r}-\boldsymbol{z} \right) =\boldsymbol{He}_{\mathbf{s}} \tag{16} \end{equation*}\] |

where \(\boldsymbol{H}=(\boldsymbol{A}-\boldsymbol{LC}) , \boldsymbol{e}_{\mathbf{s}}= (\boldsymbol{{r}}-\boldsymbol{{z}})\).

The characteristic equation of (16) can be described as:

\[\begin{equation*} |\lambda \boldsymbol{I}-\left( \boldsymbol{A}-\boldsymbol{LC} \right) |=0 \tag{17} \end{equation*}\] |

where \(\boldsymbol{I}\) is the second-order unit matrix.

The formula (17) expand to

\[\begin{equation*} \lambda ^2+\beta _1\lambda +\beta _2=0 \tag{18} \end{equation*}\] |

where \(\lambda\) is the characteristic value of \(\boldsymbol{H}\), \(\beta_1\) and \(\beta_2\) are the observer gains.

For the purpose of linking the bandwidth and control system parameters, let the eigenvalue \(\lambda _1=\lambda _2=-\omega _0\), therefore, the formula (18) can be rewritten as

\[\begin{equation*} \left( \lambda +\omega _0 \right) ^2=\lambda ^2+2\omega _0\lambda +\omega _{0}^{2}=0 \tag{19} \end{equation*}\] |

When the gain matrix \(\boldsymbol{L}=\left[ \begin{matrix} \beta _1& \beta _2\\ \end{matrix} \right] =\left[ \begin{matrix} 2\omega _0& \omega _{0}^{2}\\ \end{matrix} \right]\), the error matrix corresponding to the formula (19) will approach 0, and the whole observer system will remain stable.

By incorporating the calculated gain coefficient into the above equation, it can be obtained that:

\[\begin{equation*} \frac{Z_{22}-X_2\left( s \right)}{X_2\left( s \right)}= \frac{s\left( s+2\omega _0 \right)}{\left( s+\omega _0 \right) ^2} \tag{20} \end{equation*}\] |

By setting different bandwidths, the error tracking effect in (20) can be analyzed. The disturbance tracking error of slope signals with different bandwidth settings is shown in Fig. 5(a). From the figure, it can be seen that the higher the bandwidth \(\omega_{0}\), the smaller the tracking error, but the error cannot be eliminated.

The transfer function of the disturbance tracking error of LESO1 in the frequency domain can be expressed as:

\[\begin{equation*} \frac{V_{22}\left( s \right)}{X_2\left( s \right) -V_{22}\left( s \right)}=\frac{\beta _4}{s^2+s\beta _3+\beta _4} \tag{21} \end{equation*}\] |

The calculation process of LESO2 is similar to that of LESO.

Similar to the traditional LESO parameter analysis process, \(\beta_3\) and \(\beta_4\) are replaced with terms related to \(\omega_{0}\), respectively. By setting different bandwidths, the error tracking effect can be analyzed. The disturbance tracking error of slope signals with different bandwidth settings is shown in Fig. 5(b). From the figure, it can be seen that the higher the bandwidth \(\omega_{0}\), the faster the error convergence speed, the smaller the amplitude of the error, and can completely eliminate the error of slope signals.

From the above analysis, it can be seen that the proposed CasLADRC method can achieve better slope signal tracking ability. Therefore, when the total disturbance action undergoes a slope change, the improved LADRC has no steady-state error in estimating the disturbance. Compared to TLADRC, it improves the estimation accuracy of disturbances and enhances the robustness of the system to disturbances.

#### 4. Experimental analysis

To verify the effectiveness of the proposed CasLADRC-MSMPC method, a total control block diagram is established as shown in Fig. 6, and the TLADRC-MPC, CasLADRC-MPC, and CasLADRC-MSMPC experiments are conducted on the experimental platform shown in Fig. 7. The parameters of the prototype and controller in the experimental platform are shown in Table I.

The first experiment is the TLADRC-MPC experiment, in which the speed is increased from 0 r/min to 1500 r/min, and then increased to 3000 r/min after 0.2 s. At this time, the motor remained in an unloaded state, and at 0.5 s, a sudden load of 0.64 N\(\cdot\)m is applied. Fig. 8(a) shows the waveform of the corresponding speed, d-axis current, and q-axis current under the above working conditions. From these figures, it can be seen that the TLADRC-MPC method takes 0.18 s to reach the given speed again after a sudden change in speed, with a speed fluctuation amplitude of 55 r/min. The fluctuation amplitude of the d-axis current when the speed increases from 1500 r/min to 3000 r/min is 1.25 A, and the q-axis current also reaches the limit amplitude of 6.5 A, and quickly recovers to a stable state.

Then the experiment of CasLADRC-MPC method is carried out, and the experimental working conditions are the same as TLADRC-MPC. Figure 8(b) is a waveform diagram of rotational speed, d-axis current and q-axis current of CasLADRC-MPC method. It can be seen from these figures that the control method of CasLADRC-MPC will have a certain degree of overshoot after the sudden change of speed, the overshoot amplitude is 95 r/min, and the time to restore stability is 0.08 s. After sudden loading, the fluctuation amplitude of speed is 93 r/min, and the time to restore stability is 0.06 s. After the sudden change of speed and load, the time for q-axis current to regain stability is longer than that of TLADRC-MPC method, but the time for speed recovery is 0.1 s shorter than that of TLADRC-MPC method. The d-axis current changes suddenly when the speed changes suddenly, and the amplitude of the change is 1.13 A.

Finally, the experiment of proposed control method is carried out, and the experimental conditions are the same as TLADRC-MPC. Figure 9 is a waveform diagram of rotational speed, d-axis current and q-axis current of proposed control method. As can be seen from these figures, the control method of CasLADRC-MSMPC will have a certain degree of overshoot after the sudden change of speed. The overshoot amplitude is 38 r/min, and the time to restore stability is 0.08 s. After sudden load, the fluctuation amplitude of speed is 40 r/min, and the time to restore stability is 0.08 s, which is 0.1 s shorter than that of TLADRC-MPC. After the sudden change of speed and load, the time for q-axis current to regain stability is similar to that of TLADRC-MPC method, but the overshoot is smaller than that of TLADRC-MPC and CasLADRC-MPC. The d-axis current changes suddenly when the speed changes suddenly, and the amplitude of the change is 0.82 A, which is smaller than that of TLADRC-MPC and CasLADRC-MPC.

#### 5. Conclusion

This paper proposes a CasLADRC-MSMPC method for the fast response and strong anti-interference capabilities required by PMSMs to improve the response speed and anti-interference ability of the entire control system. Through theoretical analysis and experimental verification, the proposed method has good control performance and strong disturbance suppression ability in the case of sudden changes in speed and load. Compared with TLADRC-MPC and CasLADRC-MPC, the proposed control method has better speed and current anti-interference ability.

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#### Authors

Mingxiang Zhu

Nanjing Normal University Taizhou College

Hongjun Ni

Nanjing Normal University Taizhou College

Hongyan Sun

Nanjing Normal University Taizhou College

Jue Wang

Nanjing Normal University Taizhou College