Research on PMSM Speed Performance Based on Fractional Order Adaptive Fuzzy Backstepping Control

Abstract

A permanent magnet synchronous motor (PMSM) is a nonlinear, strongly coupled, controlled object with time-varying, fractional-order characteristics. It is difficult to achieve the ideal control effect by using the traditional control method when motor parameter changes and load perturbations occur during the operation of the PMSM, so a fractional-order adaptive fuzzy backstepping control method is proposed to improve the system’s fast response and anti-jamming ability in the case of sudden changes in rotational speed, load perturbations and other conditions. Initially, the fractional order theory is introduced, backstepping control is utilized to decompose the system into multiple subsystems, and a fractional order-based Lyapunov function is designed for each subsystem to ensure the system’s stability. Suitable control laws, as well as parameter adaptive laws, are derived through rigorous mathematical derivation. Finally, a fractional order adaptive fuzzy backstepping controller (FOAB-FPID) is designed by combining the advantages of fuzzy control. Then a mechanical simulation model of the PMSM is established to verify the validity of the designed controller, followed by three sets of comparative experiments: PID, fuzzy PID (F-PID), and integer-order adaptive fuzzy backstepping (IOAB-FPID), which are selected to simulate the PMSM under the control of the four controllers. Finally, it is validated on the constructed PMSM experimental platform. Simulation and experimental results show that FOAB-FPID can adaptively adjust system parameters during sudden speed changes, achieve real-time speed tracking, and maintain speed stability under load perturbations and internal parameter uptake. Compared with the three control strategies, reached PMSM system has better acceleration, fast response performance, and better anti-disturbance ability, which proves the rationality and effectiveness of the FOAB-FPID control method.

1. Introduction

Permanent Magnet Synchronous Motor (PMSM) is a high-performance motor type widely used because of its high efficiency, low noise, long life, etc. [1,2,3]. Conventional PMSM speed controllers usually use a cascaded PID control strategy, where an outer-loop speed controller and an inner-loop current controller act together to stabilize the desired dynamic error [4,5]. However, the PMSM is a nonlinear and strongly coupled controlled object [6,7]. It makes the design of the speed controller have some limitations and challenges.

Benefiting from the rapid development of the mathematics of differential geometry, the related theories and applications of nonlinear control systems are becoming increasingly mature. Many nonlinear control strategies have been proposed by domestic and foreign researchers, such as sliding mode control [8], fuzzy control [9], model predictive control [10], feedback linearization control [11], and backstepping control [12,13]. The literature [14] proposes an iterative learning control strategy for both performance improvement and simple controller structure. In the literature [15], an observer-based adaptive neural network finite-time dynamic surface control method is proposed to improve the robustness of the system; the literature [16] presents a novel sensor-less speed tracking control scheme to improve the robustness of the system. However, the above papers need to consider more internal parameter variations and load perturbations in the system. More seriously, they all focus on integer order, ignoring the fractional order characteristics of the PMSM system [17,18].

The fractional order theory has been developed and enriched recently and has begun to be widely applied to nonlinear systems [19,20]. During their early exploration, scholars found that fractional algorithms were usually more accurate than integer algorithms when modelling some complex nonlinear systems [21]. Some scholars have proposed some analytical and numerical methods for fractional differential equations [22] and discussed the stability and performance analysis of fractional order systems [23,24]. Then, fractional-order control strategies are more widely used in nonlinear system control [25]. Some new control algorithms and strategies are proposed [26,27,28,29], such as fractional order PID control, fractional order sliding mode control, fractional order differential equation-based control, adaptive fractional order control, etc. Some scholars and experts demonstrated that inductance and capacitance are essentially fractional orders [30], so existing PMSM systems suffer from non-integer orders in kinetic processes with memory and hereditary mass diffusion or heat conduction [31], and fractional order theory has been increasingly applied to PMSM control with good results in recent years. In the literature [32], a fractional-order sliding mode control method was proposed to solve the shortcomings of the traditional control method, such as weak immunity and significant jitter. The literature [33] improved a fractional-order controller to improve speed tracking and immunity performance. This paper explores how to ensure the system’s rapid response and excellent robustness in the case of sudden speed change, external disturbance, fractional influence, etc. Firstly, the mathematical model and structure of PMSM are studied. Then, the system is decomposed into several subsystems by using the advantage of fractional-order calculus theory combined with the mathematical derivation of the backstepping control method of multivariate adaptive change. The Lyapunov function is designed for each subsystem based on fractional-order calculus to ensure stability. A FOAB-FPID controller is designed by combining the advantages of fuzzy control with strong adaptability to parameter variation, nonlinearity, and model inaccuracy of the controlled object. The simulation and experimental results show that the method has a better control effect.

This paper is organized as follows: in Section 2, the PMSM mathematical model is derived, followed by the derivation of the control law, and the adaptive law, and the design of the FOAB-FPID controller. In Section 3, the simulation model is built and simulated. Section 4 is experimental platform validation. Section 5 compares the simulation and experimental results. Section 6 concludes.

2. Controller Design

2.1. PMSM Mathematical Model

The PMSM can be broadly classified into two forms: surface-mounted and built-in, as the permanent magnets mounted on the motor’s rotor are in different structural positions. The surface-mounted type is relatively cheaper and has a simpler structure than the built-in type. Since this paper is not a high-performance study of the motor, the surface-mounted permanent magnet synchronous motor, i_d = 0 control, is chosen. Assuming smooth rotor and stator surfaces, neglecting core saturation, ignoring eddy currents and hysteresis losses, and current triple symmetry, the voltage equation of the PMSM in the d-q axis rotating coordinate system is as follows [34].

$$\left\{\begin{array}{l}{u}_d=R_s{i}_d+\frac{d{\varphi }_d}{dt}-n_p{\omega }_r{\varphi }_q\\ u_q=R_s{i}_q+\frac{d{\varphi }_q}{dt}+n_p{\omega }_r{\varphi }_d\end{array}\right.$$

(1)

where u_d and u_q are the corresponding voltage components, i_d, and i_q are the current components, w_r is the rotor mechanical angular velocity, and R_S is the stator resistance, n_P is the number of motor pole pairs, ${\varphi }_d$ and ${\varphi }_q$ are the magnetic chain components in the d-q axis rotational coordinate system. The mechanical equation of motion and the electromagnetic torque equation are as follows [35]:

$$\left\{\begin{array}{l}{T}_e=\frac{3}{2}{n}_p{\varphi }_f{i}_q\\ T_e=T_L+J\frac{d{\omega }_r}{dt}\end{array}\right.$$

(2)

where the ${\varphi }_f$ rotor magnetic chain. Neglecting the inhomogeneity of the air gap of the surface-mounted PMSM, where $L_d=L_q=L$.$T_e$ is the electromagnetic torque, $T_L$ is the load torque, B is the damping factor, and J is the rotational inertia. Equation (3) can be deduced from Equations (1) and (2).

$$\left\{\begin{array}{l}\frac{d{\omega }_r}{dt}=\frac{3}{2}\frac{n_p}{J}{\varphi }_f{i}_q-\frac{1}{J}{T}_L\\ \frac{d{i}_d}{dt}=-\frac{R_s}{L}{i}_d+n_p{\omega }_r{i}_q+\frac{1}{L}{u}_d\\ \frac{d{i}_q}{dt}=-\frac{R_s}{L}{i}_q-n_p{\omega }_r{i}_d-\frac{n_p}{L}{\varphi }_f{\omega }_r+\frac{1}{L}{u}_q\end{array}\right.$$

(3)

2.2. FOAB-FPID Controller Design

The fractional order calculus is a non-integer order calculus with three common forms of definition, and this paper selects the Caputo form [36], defined as:

$${}_0{}^{C}D{}_t^{\alpha }f(t)=\frac{1}{\Gamma (n-\alpha )}{\displaystyle {\int }_0^t{(t-\tau )}^{n-\alpha -1}{f}^{(n)}(\tau )}d\tau$$

(4)

where $\Gamma (\alpha )={\displaystyle {\int }_0^{\infty }{e}^{-t}{t}^{\alpha -1}dt}$ is the gamma function, $\alpha$ is the order, $n-1\le \alpha <n$, and $f^{(n)}(\tau )$ is the integer order calculus of the function.

Lemma 1.

Let $x(t)\in R^n$ be a continuous differentiable function, then:

$$\frac{1}{2}{}_0{}^{C}D{}_t^{\alpha }{x}^T(t)x(t)\le x^T(t){}_0{}^{C}D{}_t^{\alpha }x(t)$$

(5)

Converting Equation (3) to fractional order form:

$$\left\{\begin{array}{l}{}_0{}^{C}D{}_t^{\alpha }{\omega }_r=\frac{3}{2}\frac{n_p}{J}{\varphi }_f{i}_q-\frac{1}{J}{T}_L\\ {}_0{}^{C}D{}_t^{\alpha }{i}_d=-\frac{R_s}{L}{i}_d+n_p{\omega }_r{i}_q+\frac{1}{L}{u}_d\\ {}_0{}^{C}D{}_t^{\alpha }{i}_q=-\frac{R_s}{L}{i}_q-n_p{\omega }_r{i}_d-\frac{n_p}{L}{\varphi }_f{\omega }_r+\frac{1}{L}{u}_q\end{array}\right.$$

(6)

Define the speed following error $e_{\omega }$, q-axis current following error $e_q$, d-axis current following error $e_d$, stator resistance error $e_{R_s}$, load torque error $e_{T_L}$. As shown in Equation (7).

$$\left\{\begin{array}{l}{e}_{\omega }={\omega }_d-{\omega }_r\\ e_q=i_{qe}-i_q\\ e_d=i_{de}-i_d\\ e_{R_s}={\widehat{R}}_s-R_s\\ e_{T_L}={\widehat{T}}_L-T_L\end{array}\right.$$

(7)

where ${\omega }_d$ is the desired motor speed, $i_{qe}$ is the desired q-axis current, $i_{de}$ is the desired d-axis current, ${\widehat{R}}_s$ is the estimated stator resistance, and ${\widehat{T}}_L$ is the estimated load torque. In this paper, we study the case where the speed varies occasionally, ignoring the effect of magnetic circuit saturation. According to the above parameter settings, the mathematical design of the PMSM system is carried out using the backstepping method. Based on the fractional order differential definitions and Formulas (6) and (7), get the fractional order derivative speed to follow the error.

$$\begin{array}{ccc}{}_0{}^{C}D{}_t^{\alpha }{e}_{\omega }& =& {}_0{}^{C}D{}_t^{\alpha }{\omega }_d-{}_0{}^{C}D{}_t^{\alpha }{\omega }_r\hfill \\ & =& {}_0{}^{C}D{}_t^{\alpha }{\omega }_d-\frac{3}{2}\frac{n_p}{J}{\varphi }_f{i}_q+\frac{1}{J}{T}_L\hfill \end{array}$$

(8)

Select the Lyapunov function: $Y_1=\frac{1}{2}{k}_{\omega }{e}_{\omega }^2$. Where $k_{\omega }$ is the speed error feedback gain and is greater than 0, the fractional order derivative of the Lyapunov function is obtained from Equations (4), (5) and (8).

$$\begin{array}{l}{}_0{}^{C}D{}_t^{\alpha }{Y}_1\le k_{\omega }{e}_{\omega }{}_0{}^{C}D{}_t^{\alpha }{e}_{\omega }\\ =-k_{\omega }{e}_{\omega }^2+e_{\omega }({}_0{}^{C}D{}_t^{\alpha }{\omega }_d-\frac{3}{2}\frac{n_p}{J}{\varphi }_f{i}_q+\frac{1}{J}{T}_L+k_{\omega }{e}_{\omega })\end{array}$$

(9)

From Equation (9) above, the desired q-axis current can be set as follows:

$$i_{q\mathrm{e}}=\frac{2}{3n_p{\varphi }_f}(J{k}_{\omega }{e}_{\omega }-{\widehat{T}}_L+J{}_0{}^{C}D{}_t^{\alpha }{\omega }_d)$$

(10)

When the q-axis current satisfies the above equation, we obtain the fractional derivative of the d-q-axis current following the error. Introduce the Lyapunov function, and then obtain the fractional order derivation. Where, ${\theta }_1$ and ${\theta }_2$ are the adaptive rates of change of $T_L$ and $R_s$,and are greater than 0.

$$Y_2=Y_1+\frac{1}{2}{e}_q^2+\frac{1}{2}{e}_d^2+\frac{1}{2{\theta }_1}{e}_{T_L}^2+\frac{1}{2{\theta }_2}{e}_{R_s}^2$$

(11)

After a series of derivations, in order to satisfy the PMSM system stabilization, the control law and adaptive law are selected as follows:

$$\left\{\begin{array}{c}\hspace{1em}\hspace{1em}{u}_d=L(-e_d+\frac{{\widehat{R}}_s}{L^2}{i}_d-\frac{1}{L}{n}_p{\omega }_r{i}_q)\\ \hspace{1em}\hspace{1em}{u}_q=L[-e_q+\frac{1}{L}(\frac{{\widehat{R}}_s}{L}-k_{\omega })i_q+\frac{1}{L}{n}_p{\omega }_r{i}_d+\frac{n_p}{L^2}{\varphi }_f{\omega }_r\\ \hspace{1em}\hspace{1em}\hspace{1em}+\frac{2}{3n_p{\varphi }_{f}L}(k_{\omega }{\widehat{T}}_L+J{k}_{\omega }{}_0{}^{C}D{}_t^{\alpha }{\omega }_d-{}_0{}^{C}D{}_t^{\alpha }{\widehat{T}}_L+J{}_0{}^{C}D{}_t^{\alpha }{}_0{}^{C}D{}_t^{\alpha }{\omega }_d)]\\ {}_0{}^{C}D{}_t^{\alpha }{e}_{T_L}={}_0{}^{C}D{}_t^{\alpha }{\widehat{T}}_L={\theta }_1(\frac{1}{J}{k}_{\omega }{e}_{\omega }+\frac{2}{3n_p{\varphi }_f}{k}_{\omega }{e}_q)\\ {}_0{}^{C}D{}_t^{\alpha }{e}_{R_s}={}_0{}^{C}D{}_t^{\alpha }{\widehat{R}}_s={\theta }_2(\frac{1}{L}{i}_q{e}_q+\frac{1}{L}{i}_d{e}_d)\end{array}\right.$$

(12)

By substituting Equation (12) into the equation, the fractional order derivative of Y₂ is less than or equal to 0. Through the above mathematical derivation, it can be seen that the system is asymptotically stable as the time ‘t’ increases.$e_{\omega }(t)$, $e_q(t)$, $e_d(t)$ converges to zero exponentially when ‘t’ tends to infinity, the speed ${\omega }_r$ of the PMSM and the d-q-axis current $i_d,i_q$ converge exponentially to its desired value. To differ from the traditional control with fixed parameters, this paper sets the adaptive adjustment feedback gain in the dynamic regulation process, sets the adaptive feedback gain $k_{\omega }$ for the speed error of fractional order derivatives, and sets the adaptive feedback gain ${\theta }_1$ for the load torque error and the adaptive feedback gain ${\theta }_2$ for the stator resistance error. It adaptively adjusts the parameters according to the real-time operation of PMSM to ensure the dynamic response performance of PMSM speed regulation.

The fuzzy system consists of four parts: fuzzification, knowledge base, fuzzy rule inference, and defuzzification [37]. A two-dimensional fuzzy controller is selected, and the fractional order derivatives of the speed error $e_{\omega }$ and the rate of change of speed error $\stackrel{•}{e_{\omega c}}$ are used as the fuzzy controller inputs, and $k_{\omega }$, ${\theta }_1$, and ${\theta }_2$ are used as the fuzzy controller outputs. $e_{\omega }$ theoretical domain setting needs to take into account the control accuracy, and it must be able to decay to 0 in the range of this domain, introducing the scaling factor and continuously simulating the experiment; the normalized field of $e_{\omega }$, $\stackrel{•}{e_{\omega c}}$ was determined as [–3, 3]. The variation of random variables is approximately normally distributed, and the normalized theoretical field of $k_{\omega }$ can be found as [0, 1/3, 2/3, 1, 4/3, 5/3, 2], the normalized field of ${\theta }_1$ as [−0.06, 0.06], and the normalization field of ${\theta }_2$ as [−0.06, 0.06], corresponding to the fuzzy language representation as [NB, NM, NS, ZO, PS, PM, PB], in combination with the control variable method. $k_{\omega }$, ${\theta }_1$, and ${\theta }_2$ three can be approximated as three of the traditional PID controller. From Equations (5), (14) and (18), it can be seen that with constant output electromagnetic torque and constant viscous friction coefficient, the fractional order speed error rate of change $\stackrel{•}{e_{\omega c}}$ is related to the load torque $T_L$. During the motor acceleration phase, $\stackrel{•}{e_{\omega c}}$ decreases, and it is necessary to increase the value of $k_{\omega }$, it decreases the value of ${\theta }_1$ thus increasing the influence of $\stackrel{•}{e_{\omega c}}$ feedback on the control law and let $\stackrel{•}{e_{\omega c}}$ converge to 0 asymptotically. Similarly, in the motor acceleration phase, adding ${\theta }_2$ cannot speed up the system response, and it is necessary to reduce ${\theta }_2$ to reduce the impact on $\stackrel{•}{e_{\omega c}}$. Similarly, the fuzzy rules of $k_{\omega }$, ${\theta }_1$, ${\theta }_2$ can be obtained by debugging. The fuzzy rules are shown in

Table 1. k_ω fuzzy control rules table.

$\stackrel{•}{{\mathit{e}}_{\mathit{\omega }\mathit{c}}}$	${\mathit{e}}_{\mathit{\omega }}$
	NB	NM	NS	ZO	PS	PM	PB
NB	PB	NS	ZO	PS	NM	NB	PS
NM	PM	NM	ZO	PS	NM	NB	PS
NS	PM	NM	ZO	PM	NS	NM	PM
ZO	PS	NB	NS	PB	NS	NS	PM
PS	PS	NB	NS	PM	ZO	ZO	PB
PM	PS	NB	NM	PS	ZO	PS	PB
PB	PS	NB	NM	PS	ZO	PS	PB

and

Table 2. Fuzzy control rules table.

$\stackrel{•}{{\mathit{e}}_{\mathit{\omega }\mathit{c}}}$	${\mathit{e}}_{\mathit{\omega }}$
	NB	NM	NS	ZO	PS	PM	PB
NB	NB	NS	ZO	PM	PB	ZO	NS
NM	NB	NM	ZO	PM	PM	ZO	NS
NS	NB	NS	PM	PM	PS	ZO	NM
ZO	NB	NS	PS	PB	PS	PS	NM
PS	NB	NM	PM	PB	PS	PS	NB
PM	NB	ZO	PM	PB	ZO	NS	NB
PB	NB	ZO	PM	PB	ZO	NS	NB

.

3. Analysis of Simulation Results

To verify the effectiveness of the design scheme in this paper, the control system of PMSM is built in MATLAB/Simulink, and the parameters of the PMSM system are set in the simulation: stator inductance $L=0.0085H$, motor rotational inertia $J=0.008 \mathrm{K}\mathrm{g}·{\mathrm{m}}^2$, magnetic chain ${\varphi }_{\mathrm{f}}=0.175 \mathrm{W}\mathrm{b}$, and the number of pole pairs is 4. Considering the influence of the gain on the control accuracy and convergence of the system, the feedback gains $k_1$, $k_2$ were finally taken as 550 and 950 through repeated tests. The simulation sampling rate is 10 kHz, the simulation period is 0.1 s, and the initial reference speed of the motor is set to 900 rpm. The simulation results are compared for the PMSM system with four controllers: PID, FPID, IOAB-FPID, and FOAB-FPID. The speed is adjusted to 600 rpm at 0.05 s and 930 rpm at 0.07 s. The speed tracking of the system under the four control strategies is shown in Figure 1. When the speed rises in the start-up phase, the PID control responds the slowest, the FPID control responds fast but with a significant overshoot, and the FOAB-FPID control curve is smooth with fast response and no overshoot. The reaction of PID and IOAB-FPID control is slow when the command speed decreases, and the reaction by FPID control and FOAB-FPID control is fast. The adjustment time is short, but the FPID overshoot is still significant, and the result is the same as above when the command speed increases. It can be concluded that the FOAB-FPID control performs best and has the best control performance in the three-time states of the start-up phase, commanded speed decrease, and oversaw speed increase.

The load torque T_L is changed when the speed is constant, and then the speed immunity of the PMSM system is analyzed. The load torque varies: 3.2 N-m is added at 0.05 s, and 2.4 N-m is subtracted at 0.07 s. Figure 2 shows the speed comparison after adding the load torque. The figure shows that the FOAB-FPID control speed is the least affected than the other three controls when the load torque is added.

When the PMSM runs at no load, the stator resistance is changed for simulation analysis, and the initial stator resistance value is set to 2.875 $\Omega$. When the PMSM runs at no load, the stator resistance Rs changes as follows: at 0.05 s, add 5 $\Omega$, and at 0.07 s, subtract 3 $\Omega$. A comparison of speed change when stator resistance changes is shown in Figure 3. The figure shows that the FOAB-FPID control has a shorter speed rise time and faster response.

4. Experimental Verification

The algorithm implementation of four controllers, PID, FPID, IOAB-FPID, and FOAB-FPID, is completed on the experimental platform to verify the speed regulation performance of PMSM under the four controllers. To ensure consistency, the experimental motor parameters are selected in the same way as in the simulation. The experimental platform and the principle of PMSM vector control are shown in Figure 4. The control core is chosen as TI’s DSP28335 chip with a carrier frequency of 10 KHz; the speed measurement module is used to collect the motor speed display (the feedback speed of the speed loop is calculated from the encoder data collected by EQEP); the magnetic powder brake module is used to adjust the external load torque input.

When the speed changes abruptly, the tracking and response ability of the system speed under different control schemes are tested. Firstly, the instruction speed is changed and compared, with speed changes from 900 rpm to 840 rpm and 900 rpm to 960 rpm, to verify the speed tracking response of the PMSM under different controls during speed drop and rise.

The three-phase currents and rotational speeds of the speed reduction experimental results are shown in Figure 5, which can be seen: With the PID controller, the motor rotational speed showed an overshoot of about 20% when the speed reduction command was received, and the adjustment time was about 904.6 ms, whereas the FPID control rotational speed was overshooting by approximately 35.5%, and the adjustment time was about 787.7 ms. The IOAB-FPID control of the three-phase current shows apparent fluctuation, and the adjustment time is about 585.7 ms. At the same time, the FOAB-FPID still responds smoothly and quickly when the speed command changes in the same way, and the current does not fluctuate obviously, which shortens the adjustment period to about 488.9 ms. The comprehensive analysis shows that when the rotational speed of the PMSM decreases, the FOAB-FPID control of the rotational speed response is the fastest, and the tracking performance is the best. It is the quickest and has the best tracking performance.

The results of the speed-up experiment are shown in Figure 6. The speed overshoot of the PID control is about 27.5%, and the adjustment time is about 987 ms. The speed overshoot of the FPID control is about 30%, and the adjustment time is about 786.7 ms. The IOAB-FPID control has a slight jitter when the speed rises, no noticeable overshoot, and the adjustment time is about 701.9 ms. While the FOAB-FPID control can still respond smoothly and quickly when the speed command changes, no evident overshoot occurs, and the adjustment time is about 368.8 ms. It can be seen that the FOAB-FPID control performance is better than the other three when the motor speed rises. The controller is better than the other three.

The speed is kept constant, and the load torque is added to test the anti-interference capability of the PMSM system. A load torque of 3.2 N-m is added at 900 rpm. The magnetic powder braking module provides the load torque to analyze the variation of PMSM speed and three-phase current under different control conditions. The experimental results are shown in Figure 7.

As shown in Figure 7, the motor speed fluctuated under all four controls when the load torque was added, and the percentage speed drop and the specific value could be calculated based on the marker in the upper left corner of the oscilloscope. The speed fluctuated more under PID and FPID control, and overshoot occurred when the rate picked up. The speed fluctuation of IOAB-FPID control is slight, and the response time is fast but fails to return to the command speed. The speed fluctuation of FOAB-FPID control is slight, and the adjustment is fast and returns to the command speed after a particular adjustment time. The performance of the system parameters under the four control schemes is shown in

Table 3. Performance indicators under different controls.

Performance Indicators	PID	FPID	IOAB-FPID	FOAB-FPID
Decreasing speed/rpm	44.8	33.6	21.6	17.5
Decrease in percentage/%	4.98	3.73	2.4	1.94
Adjustment time/ms	771.4	747.3	745.5	715.6

.

5. Discussion

In simulations and experiments, we conducted several sets of experiments with sudden changes in speed and the addition of load to analyze the controller performance regarding response speed, regulation time, and overshoot. Comparative analysis of the simulation and experimental results shows that there is almost no overshooting in the PID control in simulation, but the response is slow with the longest regulation time; overshooting occurs in the experiment with the longest regulation time. The FPID control was more quickly regulated in simulation, but the overshoot was very large; in the experiment, the overshoot was large, and the regulation time was long. The IOAB-FPID control at simulation did not almost overshoot, and the regulation time was shorter; at the experiment, there was almost no overshoot, and the regulation time was shorter, but the speed did not return to the original value. The FOAB-FPID control has no overshoot and rapid regulation during simulation; it has no overshoot and the fastest response and shortest regulation time during the experiment. Through comparative analysis, we found that the experimental results and simulation results are the same, but there are subtle differences in the reasons that are manifold, such as debugging too many parameters that failed to adjust to the optimal matching parameters of the system, the experimental data acquisition by the time of the impact of the larger cannot be continuously collected, the software program algorithm failed to capture the speed change, and so on. By analyzing the results of several experiments on the command speed down, up, and adding load torque, both in the simulation and experimental FOAB-FPID controller is the best; in the speed of the sudden change and input load torque after the speed fluctuation is smaller and can return to the command faster; the responsiveness and tracking performance are the best; to a certain extent, it has a better ability to resist interference; and the parameter adjustment after matching the experimental The effect will be better after parameter adjustment and matching. Both simulation and experiment show that it has some practical significance for engineering applications.

6. Conclusions

The problem of speed regulation performance for a PMSM based on a fractional-order adaptive fuzzy backstepping control method has been studied. First, the fractional order theory is introduced in the mathematical modelling of PMSM to overcome the influence of non-integer order on the system speed regulation, and the backstepping method is used to derive the system’s adaptive and control laws. Second, a FOAB-FPID controller is designed in conjunction with fuzzy logic, which ensures that the system is able to adaptively regulate the parameters to overcome the load disturbance problem better. Third, the control group was designed under the same conditions. For the proposed method, this paper draws the following conclusions through simulation and experiments: The FOAB-FPID control has the fastest response and the best following adaptation to reach the commanded rotational speed at the earliest, with the smallest fluctuation in rotational speed, the fastest dynamic response, and the best resistance to the load disturbance. It is proved by analysis that, compared with PID control, FPID control, and IOAB-FPID control, this control strategy not only improves the response speed and reduces the overshooting amount of speed but also ensures the speed control accuracy and dynamic performance under perturbation and enhances the robustness of the system. It verified the rationality and effectiveness of our proposed control method, which can be applied to practical engineering.