Research on self-coupling PID for multi-driven synchronization control with ring adjacent compensation

Multi-motor synchronous drive system is increasingly widely used in industry and manufacturing, where its control structure and control strategy affect the quality and efficiency of production. In order to solve the contradiction between fastness and overshoot, and the difficulty in determining the compensation law in the conventional PID, cross-coupling control, and master-slave control strategies used in multi-motor control, this paper proposes a self-coupling PID control strategy based on ring adjacent compensation to reduce the complexity of the control structure. Furthermore, this paper analyzes the self-coupling PID parameter tuning rules and establishes the control structure of the ring coupling strategy, and proves its validity mathematically. The simulation results verify that the proposed strategy provides a fast response speed, high control precision, good disturbance rejection, and synchronization performance.


Introduction
With the development of modern industrial technology, the using of multi-motor synchronous control is increasing, such as electric cars, 1,2 aerospace, and so on, 3,4 especially for the high-precision high-speed systems, and cooperative control is the central issue. 5,6In China and abroad, the number of studies that focus on multi-motor synchronous control technology is growing more and more.However, the early synchronous control is primarily based on the mechanical connection, which is simple, but the accuracy of synchronization is low. 7ith the development of motor transmission technology, information technology, and automatic control technology, the synchronous mode of mechanical connection transmission has been replaced by an electrical servomotor. 8,9Currently, multi-motor cooperative control mainly comprises non-coupling control and coupling control.Where, the Non-coupling control includes master command control, master-slave control, and virtual axis control.Literatures 10,11 is using master-slave control, and this scheme also has the problem of largesignal transmission delay; However, in literatures 12,13 proposed virtual axis control, which realizes the synchronous control of multiple motors, but this control strategy has some problems such as given signal delay.Coupling control includes cross-coupling, adjacent coupling, deviation coupling, etc.In literatures 14,15 the cross-coupling control was proposed.However, this scheme is suitable for dual-motor systems, but for multimotor systems, the structure is complex and the compensation effect is not ideal.While, adjacent coupling control to overcome the defect that the cross-coupling structure is only suitable for double motors, but there are many synchronous error controllers and a large amount of computation was occurred; the deviation coupling control has high coupling degree, low control delay, and good synchronization ability, but the compensator is complex.Literature 16 proposes a data comparison method and a compensation algorithm, but the calculated amount is significant.Literature 17 proposes a synchronous control strategy that combines adjacent cross-coupled structures with sliding mode variable structure control, but each compensator requires many speed signals and has sliding mode jitter.
In order to improve the precision, stability, and robustness of cooperative control, a number of researchers have proposed different methods by combining modern control methods with existing control strategies. 18In literatures 19,20 the author presents sliding mode control, but there is sliding mode jitter; Literature 21 proposes adaptive neural network.However, there is a contradiction between control accuracy and parameter estimation; Literature 22 presents active disturbance rejection control, but there are many parameter tuning; Literature 23 proposes iterative learning control, but there is a problem with initial state selection; Literature 24 uses the fuzzy control algorithm to adjust the motor torque given value in realtime to improve the synchronization performance, but due to the simple fuzzy processing method, the useful information of the system is lost, and the dynamic tracking ability is very weak; Literature 25 proposes a variable domain fuzzy PID control based on a masterslave control structure.However, it involves multiple parameters, and the effectiveness relies on fuzzy rules.In literature, 26 a synchronization controller based on the mean relative coupling structure is proposed to address the coupling issue between synchronization and tracking.However, the synchronization compensation relies on the mean value.
In order to realize the cooperative control of multimotors and ensure that each motor has good tracking, disturbance rejection, and synchronization in the operation process, based on the self-coupling PID control theory and deviation coupling control structure, this paper proposes self-coupling PID for multi-motor cooperative control based on circular adjacent compensation.The tracking and disturbance rejection of the system is improved by the self-coupling PID control, and the synchronization performance is improved by using the ring adjacent compensator.The self-coupling PID parameter tuning rules are analyzed, and the Lyapunov equation is established to analyze the stability of the compensation system.Finally, the numerical simulation test is used to verify the effectiveness of the control strategy.
The main contribution of this paper is to propose a self-coupled PID control strategy based on ring adjacent compensation for a multi-motor synchronous control system, as well as to develop a Romberg torque observer with minimal dependence on input information.The proposed ring compensation structure outperforms master-slave control, cross-coupling control, and virtual spindle control, providing new ideas for synchronous control problems.

Ring adjacent compensation control structure
The structure of the multi-motor synchronous control system based on ring-adjacent compensation is illustrated in Figure 1.In the system, each permanent magnet synchronous motor adopts the structure of a speed controller, current controller, and adjacent compensator.The current closed-loop adopts a vector control scheme of i d = 0.The control strategy controls the error between the actual speed of each motor and the given speed by the speed controller and then compensates by considering the speed and load of the motor and the adjacent motor.The change in speed and load of any motor will compensate for the adjacent motor.All motors are coupled to one another and ultimately form a ring.The ring adjacent compensation is only performed between two adjacent motors.When a motor is disturbed during the system's operation, the speed change will cause a synchronization error between the motor and its two adjacent motors.In the loop compensation control strategy, the synchronization error will be feedback to the adjacent motor and the disturbed motor itself through the compensation module.The speed controller of each motor is combined with the speed compensation module to improve the following performance, immunity, and steady-state performance of the system.
The expected speed of each motor and each controller link in the system is same.The compensation amount of each compensation module is only related to the speed and a load torque of the two adjacent motors.The complexity of the compensator is independent of the motors number.The ring adjacent compensation control is more suitable for a multi-motor synchronous system in compared with the crosscoupling control strategy and the deviation coupling control strategy.In the dq coordinate system, the mechanical motion equation of the permanent magnet synchronous motor rotor is When the vector control i d = 0, the electromagnetic torque of the motor is T em = n p L md i f i q , then the motion equation is Where, v is motor speed, P n is polar logarithm, L md is d-axis excitation inductance; i f is equivalent excitation current converted to the stator side, B is the coefficient of viscosity, and T L is load torque, J is moment of inertia.

Design of multi-motor speed synchronization control
This paper mainly studies the design of the speed loop and the ring adjacent compensator of the permanent magnet synchronous motor, while the current loop adopts the conventional hysteresis control structure, which will not be repeated in this paper.

Design of self-coupling PID controller
The speed-controlled object controlled by the current loop is a second-order system, and its affine nonlinear system is expressed as Where, x 2 and x 1 are measurable state variables, u is the control quantity, y is the output of the system, f x 1 , x 2 , u ð Þis a known or unknown uncertain function, d 1 is an external bounded disturbance.For a secondorder nonaffine nonlinear uncertain system expressed in equation ( 3), let the combined disturbance of the system be the unknown uncertain dynamics and external disturbance, that is where b 0 is the estimated gain of the controlled object, then equation ( 3) can be rewritten as It can be seen from equation ( 5) that a class of nonaffine nonlinear uncertain systems is transformed into a class of linear uncertain affine systems by introducing a comprehensive disturbance, and the control problem of the nonlinear uncertain system is transformed into the control problem of the linear uncertain affine system. 27,28For the system expressed by equation ( 3) or ( 5), the control error is defined as The integral of error is Then For the system expressed by equation ( 5), the autocoupled PID control model 29,30 is defined by equation ( 8) as Then, _ e 2 =2dÀk i e 0 Àk p e 1 Àk d e 2 = _ e 2 =2dÀk i Ð e 1 dt Àk p e 1 Àk d _ e 1 .The closed-loop control system based on self-coupling PID is shown in Figure 2.
The main innovation of the strategy is that there is only a one-speed factor Z c , and its main feature is that the physical links of the three different properties such as proportion, integral, and differential are closely coupled to form the control signal through Z c .

Control system performance analysis
Theorem 1: Assuming that the sum disturbance defined by equation ( 4) is bounded: jdj \ ' when the three roots are all in the left half of the S plane, the closedloop control system composed of the SC-PI controller defined by equation ( 9) is stable in a wide range, and the SC-PI control system has good disturbance rejection and robustness.
Proving: Substitute the SC-PID controller defined by equation ( 9) into the controlled error system shown in equation ( 8) to obtain a closed-loop control system as se 0 (s) À e(0) = e 1 (s) Since e 0 (0) = 0 then It can be obtained from equation ( 11) as e 1 (s)= sd(s)+s 2 e 1 (0)+se 2 (0)+k d e 1 (0) The first term is the zero-state response and the second term is the zero-input response.From equation ( 12), the transfer function of the closed-loop system can be obtained as Let equation ( 13) have three identical real roots, three different real roots (namely z 1 , z 2 , and z 3 , respectively), one real root z 1 , or two identical real roots z 1 = z 2 (or complex roots z 2 = z 3 = a + jb), the corresponding unit impulse excitation response forms are as follows In addition, according to the principle that the zeroinput response of the system has a fixed relationship with h(t), it can be obtained that if each feature root has a negative real part, when jdj \ ', the system has good robustness against total disturbance.

Parameter tuning analysis
In order to stabilize the system, let the three roots be S 1, 2, 3 = 2Z c and z c .0, then k p = 3z 2 c , k i = z 3 c , and k d = 3z c can be obtained from If the three roots of the system are s 1 = 2z c , s 2, 3 À z c 6jl, where z c and l are positive real numbers, respectively.Then k p = 3z 2 c + l 2 , k i = z 3 c + z c l 2 , and k d = 3z c can be obtained from Therefore, only by determining z c and l, the parameters of the controller can be determined.
If the three roots of the system are s 1 = 2z c , s 2, 3 = 2z c2 s 2, 3 = 2z c2 , where z c1 and z c2 are positive real numbers, respectively.Then , and k d = z c1 + 2z c2 can be obtained from Therefore, only by determining z c1 and z c2 , the parameters of the controller can be determined, that is, two parameters need to be tuned.
If the three roots of the system are s 1 = 2z c1 , s 1 = 2z c2 , and s 1 = 2z c3 where z c1 , z c2 , and z c3 are positive real numbers, respectively.Then k p = z c1 z c2 + z c2 z c3 + z c1 z c3 , k i = z c1 z c2 z c3 , and k d = z c1 + z c2 + z c3 can be obtained from The parameters of the controller can be determined only after confirming z c1 , z c2 , and z c3 .At this time, three parameters need to be tuned.

Adjacent ring compensation control strategy
Let the synchronization error between the ith motor and the next motor be Then the tracking error of the ith motor after being corrected by the compensation module is According to the Lyapunov theorem, construct the Lyapunov function as Therefore, the system is globally stable when E i b !0. When i = 1, equation ( 25) can be obtained from Assuming that all motor parameters are the same, that is, J 1 =J 2 = ÁÁÁJ, n p 1 =n p 2 = ÁÁÁn p , and B 1 =B 2 = ÁÁÁB.
The compensation current of the first motor is As can be seen from the above, the torque T 1 and T 2 are required to obtain the compensation current.

Design of load torque observer
According to formula (1), the rotational speed can be measured, while the load torque cannot be measured.Select, x = ½v T i T u = T e , y = v.When the sampling period is very small, the load torque is considered constant within the sampling period, and the state space expression is obtained as The form of the Luenberger observer is constructed as follows where L = L 1 L 2 ½ T .Derivation and finishing can get Substitute equations ( 27) and ( 28) into equation (30) to get It can be seen from equation ( 33) that the load torque can be observed only by the electromagnetic torque of the motor and the actual speed of the motor without additional hardware equipment.The system control process is shown in Figure 3.

Simulation and analysis
In order to further prove the correctness, stability, and effectiveness of the cooperative control scheme proposed in this study, a multi-motor synchronous control system composed of three motors is constructed and simulated on the MATLAB/Simulink platform.The three selected motors have the same parameters, as summarized in Table 1.

Simulation of self-coupling PID and conventional PID
In order to verify the effect of the self-coupling PID control, compared with the conventional PID control, two control methods use the hysteresis controller in the current loop and the ring adjacent compensation.The controller parameters are summarized in Table 2.The expected speed value is n Ã = 1000r=min, when t = 3:5s, the same pulse load disturbance is added to motor 1 of each control structure, and the other motor loads remain unchanged.Figure 4 shows the response curve of self-coupling PID and PID control.Figure 5 depicts the comparison curve of synchronous error between motor 1 and motor 2. Figure 6 shows the comparison curve of synchronous error between motor 2 and motor 3. Figure 7 depicts the comparison curve of synchronous error between motor 3 and motor 1. Figure 8 illustrates each self-coupled PID controller output.Figure 9 illustrates each PID controller output.Comparative analysis demonstrate that the self-coupling PID controller has fast output response and small fluctuations; The PID control output has slow adjustment time, large fluctuations, and multiple oscillations.
It can be seen from Figure 4 that the overshoot of PID control is 10% and the adjustment time is 0.7 s during the startup process, while the overshoot of selfcoupling PID is 1% and the adjustment time is 0.1 s.
When the load disturbance occurs, the dynamic speed drop of PID control is 20r=min, and the recovery time is 0.05 s.The dynamic speed drop of self-coupling PID control is 10r=min, and the recovery time is 0.025 s.It can be seen from Figures 5 and 6 that the motor load disturbance has a great impact on the motor synchronization performance in the system with the PID control strategy.The maximum speed synchronization deviation is 25r=min, the convergence time is 0.07-0.4s, the maximum speed synchronization deviation of selfcoupling PID control is 5r=min, and the convergence time is 0.01 s.

The simulation of master-slave control, cross-coupling control, and proposed control strategy
In order to verify the effectiveness of the adjacent ring compensation control strategy proposed in this paper, it is compared with the master-slave control and crosscoupling control.In the master-slave control strategy system, motor 1 is set as the main motor, and motor 2 and motor 3 are the slave motors.The expected speed of the three control strategies is n Ã = 1000r=min, when t = 3:5s, the same pulse load disturbance is added to motor 2 of each control structure, and the other motor loads are unchanged.Figure 10 shows the synchronous error comparison curve between motor 1 and motor 2. Figure 11 illustrates the synchronous error comparison curve between motor 2 and motor 3. Figure 12 depicts the synchronous error comparison curve between motor 3 and motor 1. Figure 13 illustrates the output of each self-coupled PID controller, Figure 14 illustrates the output of each controller in master-slave mode, and Figure 15 illustrates the output of each controller in cross-coupling mode.Through comparative analysis, the output response speed of the self-coupling PID controller is the fastest and the fluctuation is small; The output response speed of the master-slave control mode controller is slow, and the maximum output value is smaller than that of the self-coupling PID controller; The output response of the cross coupling control method controller is slow, but the output fluctuates greatly.
It can be seen from Figures 10-12 that the synchronization errors of the three control strategies in the startup process finally tend to 0, and the synchronization errors of the adjacent ring compensation control and the cross-coupling control are 0, while the synchronization error of the master-slave control strategy is large, and the maximum speed synchronization error             can reach 578r=min, which needs about 0.015 s to converge to 0. The synchronization performance and crosscoupling control strategy in the startup stage is basically the same, which are significantly better than the master-slave control strategy.The load disturbance of the motor has a significant influence on the synchronous performance of the motor in the system using the master-slave control strategy, which needs 0.06-0.09s to gradually converge to 0, and the cross-coupling control needs 0.035-0.04s to gradually converge to 0; with the use of adjacent ring compensation control, the synchronization error between the disturbance motor and the adjacent two motors is less than that of the master-slave control and crosscoupling control strategy, and the synchronization error converges to 0 in 0.01-0.037s and runs stably.Among them, the relative maximum synchronous speed deviation is 40r=min when the master-slave control strategy is adopted; the relative maximum dynamic speed drop of the cross-coupling control strategy and the adjacent ring compensation control strategy is about 10r=min.Thus, the adjacent ring compensation control can not only effectively suppress the synchronization error caused by a motor disturbance in the system, but also accelerate its convergence speed.
According to the simulation results of motor starting, when the system adopts the adjacent ring compensation control strategy, the motor synchronization error always approaches 0 and converges rapidly in the motor starting process, consequently, it has a good starting synchronization performance.The synchronous error caused by motor load disturbance is relatively small and converges rapidly, and its disturbance rejection is significantly better than that of the masterslave control strategy and cross-compensation control strategy.Based on the above analysis, it can be seen that the adjacent ring compensation control strategy significantly improves the dynamic and steady-state characteristics of the system, which shortens the disturbance recovery time, weakens the oscillation phenomenon, and offers good synchronization.In addition, the proposed control strategy has fewer controller parameters, simple parameter setting and small amount of system computation.

Conclusions
In order to solve the performance of multi-motor coopcontrol, a self-coupling PID control structure with ring adjacent compensation is proposed to improve the synchronization, dynamic and static performance of the multi-motor system.The following conclusions are obtained through experiments: 1) In order to improve the synchronization of the multi-motor system, a new ring-adjacent compensation structure is proposed based on the coupling compensation principle.Where the complexity of the control structure can't be affected by the number of motors; 2) In order to improve the system tracking and disturbance rejection, a self-coupled PID control strategy is used to analyze the system performance by pulse excitation response.The tuning rules of controller parameters are summarized by applying the relationship between roots and coefficients; 3) In order to achieve adjacent compensation, by constructing a Luenberger observer, the load torque can be observed only by the electromagnetic torque of the motor and the actual speed of the motor.
The convergence of the control algorithm is demonstrated by the use of the Lyapunov function.The proposed control strategy is applied to the synchronous control system of three motors and compared to the system using master-slave control and cross-coupling control strategy.The simulation results show that the proposed control strategy makes the system have good starting performance, dynamic performance, and disturbance rejection, and ensures the synchronization accuracy of the system, which is suitable for the synchronous control of multiple motors.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 1 .
Figure 1.The structure of multi-motor synchronous control system with circular adjacent compensation.

Figure 2 .
Figure 2. Closed-loop control system model based on self-coupled PID.

Figure 5 .
Figure 5. Speed synchronization difference between motor 1 and motor 2 under the separate control of sc-pid and pid.

Figure 6 .
Figure 6.Speed synchronization difference between motor 2 and motor 3 under the separate control of sc-pid and pid.

Figure 7 .
Figure 7. Speed synchronization difference between motor 1 and motor 3 under the separate control of sc-pid and pid.

Figure 10 .
Figure 10.Speed synchronization difference between motor 1 and motor 2 under the separate control of sc-pid, m-s and c-c.

Figure 11 .
Figure 11.Speed synchronization difference between motor 2 and motor 3 under the separate control of sc-pid m-s and c-c.

Figure 12 .
Figure 12.Speed synchronization difference between motor 1 and motor 3under the separate control of sc-pid, m-s and c-c.

Figure 14 .
Figure 14.Output of each controller in master-slave mode.

Figure 15 .
Figure 15.Output of each controller in cross-coupling mode.

Figure 13 .
Figure 13.Output of each self-coupled PID controller.