## 3 Control Synchronization  
From Equation (1), it can be remarked that the vibration system excited by three ERs is a typical underactuated system, which is also a multiple-input-multiple-output system [15]. The backstepping control is usually selected to design the controller of an underactuated system. However, it is not necessary for the vibration system. Since the object of this underactuated system in this paper is the motion trajectory of the mass center of the vibration system, Equation (1) can be uncoupled from a dynamic perspective. That is, the motion of the mass center of the system depends on the phases of three ERs. If the phase difference is zero, the resultant force acting on the vibration system is maximum, and its driving power is also maximum. To achieve the desired motion of the maximum force and the maximum driving power, the object is converted into controlling the same phase motion (synchronization) of three ERs (\(\varphi_{1}=\varphi_{2}=\varphi_{3}\)). Due to the ER being driven directly by the motor, the synchronous motion converts to the motion control of three motors. However, control synchronization of three motors in the vibration system is different from that in the general system [15; 16]. Because of the self-adjusting of the vibration system, the phases of three ERs are usually different in the case of asymmetric structural parameters [33; 34]. Figure 2 shows the self-adjusting that the motor drives ER to excite the rigid body and the motion of the rigid body affects the load torque of the motor, which is expressed in (ii) and (iii) using \(\varphi_{i}\) and \(T_{Li}(i=1,2,3)\). Based on the above analysis, the complex motion control of the vibration system converts the tracking control of three motors operating in the disturbance of the self-adjusting of the vibration system. Although control synchronization may satisfy the process requirements of the vibration machines, it still increases the manufacturing cost of the system. Considering this engineering application, this paper will make a comparative study on the control schemes of two motors or three motors. Based on this purpose, the master-slave control is selected as the control strategy of this paper. For the general system driven by three motors, velocity is usually the control object. However, the phase of ER is the control object of the vibration system in this paper. Since the phases of ERs are increasing with time and unbounded, the phase difference between ERs is selected as the control object. Hence, the master motor is used only for open-loop and closed-loop velocity control, and the slave motor is used for closed-loop phase tracking control as shown in (i) of Figure 2.  
To sum up, Figure 2 shows the electromechanical coupling of the vibration system driven by three ERs. The vibration system using control synchronization can be divided into three parts The key part is (ii) which acts as a bridge between (i) and (iii). The coupling process is that controller outputs voltage \(u_{i}(i=1,2,3)\) to the motor, the motor drives ER to produce the exciting force, the rigid body is forced to move and affects the velocity and phase of ER, and the controller detects the signal to adjust the output voltages of the motors. Where, \(2\kappa_{1}=\varphi_{1}-\varphi_{2}\) and \(2\kappa_{2}=\varphi_{2}-\varphi_{3}\) denote the phase difference. Figure 2: Electromechanical coupling of the vibration system driven by three ERs. Because the AC motor has a strong mechanical characteristic, it is not easy to observe the change of velocity with the change of load torque. Thus, a DC motor is used as the driving source to study the control synchronization of three ERs. At present, the pulse width modulation (PWM) of voltage is the most common technique of motor control in engineering.