1. Introduction
Recently, multi-motor coordinated synchronization control has been extensively employed in various industrial sectors, such as wind energy generation, industrial robotics, electric vehicles, and material handling [
1]. Demands for the coordinated synchronization control precision of motors are rising, together with the repeated upgrading of industrial output [
2]. Thus, the study of the synchronous control of multi-motor synchronous motor systems with complicated uncertainties and disturbances is urgently needed.
Dual-motor synchronous control requires that the two motors must maintain the same speed in the working state. In order to improve the synchronization performance of dual-motor system, a parallel control strategy was proposed. However, without feedback of the speed difference of the two motors, its synchronization control is poor [
3]. The master–slave control strategy takes the output speed of the master motor as the reference speed of the slave motor. However, without feedback, the synchronization performance of master–slave control is not satisfactory [
4]. The structure of deviation coupling control and circular coupling control is complex, and the amount of calculation is large, which is not conducive to practical application [
5,
6]. A cross-coupling control strategy was proposed by Professor Koren to realize the synchronous and coordinated control of dual motors in 1980, and the cross-coupling structure became a common dual-motor coupling structure [
7,
8].
With the advantages of simple structure and easy implementation, PI control is widely used for motor control. Han first proposed the active disturbance rejection controller (ADRC) [
9]; with its advantages of strong stability and high precision, it became a better alternative to PI control [
10]. Nevertheless, the complexity of parameter selection in ADRCs is quite substantial, and these selections inevitably affect the control performance of the system. Therefore, the choice of ADRC parameters is an important problem. At present, methods for tuning ADRC parameters mainly include [
11] empirical methods, bandwidth methods (BMs), time-scale tuning methods, and intelligent algorithms.
Chen et al. proposed a data-driven iterative tuning method for a delayed ADRC, which balanced system performance and robustness through empirical relationships. The simulation results confirmed the effectiveness of this approach [
12]. Wang et al. developed adjustment equations for second-order linear active interference suppression control parameters using internal model control. Their approach achieved satisfactory control performance for oscillation systems by minimizing the integral of the mean square error index [
13]. Lu et al. introduced a parameterized fuzzy self-tuning approach for a load-adaptive double-loop drive system based on an enhanced position velocity-integrated ADRC. They performed experimental tests using a selective compliant assembly robot arm, which demonstrated exceptional positioning accuracy, rapid response time, and robust adaptability to changes in the applied load [
14]. However, empirical methods are overly reliant on expert experience and may not yield optimal solutions.
Gao et al. proposed a controller parameterization approach by analyzing Han’s ADRC controller [
9], wherein the controller parameters were expressed as a function of loop gain bandwidth. This methodology simplified the parameter adjustment process [
15]. Qu et al. introduced a linear–nonlinear switching ADRC strategy, in which the extended state observer (ESO) and the state error feedback (SEF) control law adopted this linear–nonlinear switch. The experimental results demonstrated that following parameter adjustment using the BM, this approach exhibited enhanced resistance to disturbances and held promising potential for practical applications in engineering [
16]. Zhang et al. built upon the foundation established by Gao [
15] in summarizing the effects of various parameters on system dynamics and adjusted gain parameters of ESO using a BM. The experimental results under different operating conditions demonstrated that the anti-interference performance of grid-connected inverters can be effectively enhanced by nonlinear ADRCs (NLADRCs) with adjusted parameters [
17]. Nevertheless, the BM is highly sensitive to small changes in control system parameters, which may cause instability in some cases. Moreover, it is usually suitable for linear or nearly linear systems but may not provide satisfactory control performance for highly nonlinear systems with complex coupling and cross-effects.
Li et al. proposed a time-scale-based parameter adjustment method for ADRCs, which facilitated the parameter tuning process [
18]. Similarly, Shao et al. developed an effective approach to adjust the parameters of motion-induced ADRC by considering the relationship between time scale and multiple parameters [
19]. However, this method relies on accurate interference prediction and may not perform well if the interference characteristics change. It also involves relatively complex calculations.
Yang et al. utilized a particle swarm optimization (PSO) algorithm to refine the pertinent parameters within the initial ADRC parameter configuration approach, consequently attaining accurate motion control of the antenna servo system. The simulation result indicated that the enhanced ADRC exhibited advantages such as minimal overshoot, rapid response, strong anti-interference capability, high reliability, and robustness [
20]. Ma et al. proposed integrating the ADRC with intelligent algorithms to enhance its controller performance by introducing fuzzy inference and radial basis function neural network algorithms for self-adjustment of controller parameters. The experimental results indicated that this approach provided a reliable guarantee for interventional surgical robot stability [
21]. Liu et al. introduced a novel ADRC design based on an improved meme algorithm (IMA) for permanent magnet synchronous motors (PMSMs). The experimental findings revealed significant optimization effects of the IMA-based ADRC [
22]. Moreover, various methods exist for tuning ADRC parameters, including differential evolution algorithms [
23,
24], artificial bee colonies [
25], genetic algorithms [
26], whale optimization algorithms [
27,
28], gray wolves [
29,
30,
31,
32], neural network algorithms [
21,
33,
34,
35,
36], and other sophisticated approaches. Nevertheless, intelligent algorithms may be prone to premature convergence, leading to suboptimal parameter selection. The ADRC parameter adjustment methods are summarized in
Table 1.
In this work, an improved PSO method (IPM) is proposed to solve the parameter-tuning problem in ADRCs. The key findings of this article can be succinctly summarized as follows:
- (i)
The proposed method effectively addresses the issue of ADRC parameter setting without local premature convergence and does not rely on dataset training.
- (ii)
The ADRC is added into the dual-motor coupling control system to replace the PI control in the traditional dual-motor coupling control system. The new dual-motor coupling control system uses the ADRC speed loop and current loop as feedback control.
- (iii)
The experimental results demonstrate the efficient parameter selection capability of the IPM for the ADRC, leading to reduced overshoot steady-state errors (REs) and enhanced immunity performance in dual-motor ADRC systems. The evaluation function (EF) designed in this study is only 1.139, which is 1.298 lower than that of the state-of-the-art method.
The remainder of this paper is structured as follows. A mathematical model of PMSM is provided in
Section 2. A dual-motor system is integrated with the IPM to form a driver in
Section 3, and each part of the drive is introduced in detail, including the motor control mode, ADRC selection, and IPM process and parameter selection. Comparison experiment is used to verify that the IPM has better performance in ADRC parameter setting in
Section 4. The content of the paper is summarized in
Section 5.
2. Mathematical Model of SPMSM
The subject of investigation in this paper is the surface-mounted PMSM (SPMSM), ignoring the nonlinear properties of the PMSM, the air gap permeance, the internal permeance of the permanent magnet, and the winding damping on the rotor. Moreover, the three-phase stator winding of the motor is symmetrically distributed in space, and the air gap magnetic field is sinusoidally distributed. The mathematical model of the SPMSM can be acquired in accordance with the motor control theory in [
37].
The stator voltage equation of the PMSM in the
d–q synchronous rotating coordinate system is
where
and
are the stator voltage components of
d- and
q-axes, respectively;
and
are the stator flux components of
d- and
q-axes, respectively;
and
are the inductance components of
d- and
q-axes, respectively;
is the electric angular velocity; and
is permanent-magnet flux linkage.
The PMSM electromagnetic torque equation is
The SPMSM’s permanent magnet has a homogeneous air gap, and its relative permeability is essentially 1 when
. The electromagnetic torque may be expressed as follows:
where
is the PMSM electromagnetic torque and
is the number of pole pairs.
According to the force’s equilibrium conditions, the mechanical motion equation of the SPMSM is as follows:
where
is the moment of inertia,
is the damping coefficient,
is the load torque, and
is the mechanical angular velocity of the motor.