Research on Control Algorithm of Electric Linear Loading System

This paper mainly focused on the problems of low loading accuracy in electric linear loading system, Firstly, the mathematical model is done on loading motor, loading motor driver and ball screw in the system. Then, the current loop proportional control is introduced ,which improves the response speed of the load motor; In order to improve the loading accuracy and restrain excess force, a parallel algorithm based on fuzzy PID and repetitive control is designed in the force loop .The fuzzy controller improves the dynamic performance and anti-interference ability of the system. The repetitive controller periodically adjusts the deviation, which reduces the steady-state error of the system. Combination of the two controller results in good dynamic and static characteristics. The simulation results show that the proposed control algorithm is feasible, which has a certain engineering reference value.


Introduction
Electric Linear Loading System (ELLS) is an important test device used to test the performance of linear servo. The successful development of ELLS can not only shorten the development cycle of the tested mechanism and reduce the development cost, but also improve test reliability and success rate. At present, most of the systems under study are electric rotary loading systems, that is, the loading of the rotary servos by the rotary electric machines, and the ELLS is mainly divided into the linear servomotors, the rotary servomotors with the ball screw. Considering the shortcomings in control difficulty and high cost of linear motor, this paper adopts the loading form of rotating motor with ball screw, but the nonlinear factors such as mechanical friction, gaps and servo position disturbance have some problems such as large excess force and signal hysteresis [1]. Therefore, on the basis of establishing the correct mathematical model of the system, how to improve the system's high response and high-precision control performance is the core issue of the study.
At present, domestic and foreign scholars have proposed a large number of control methods to improve the performance of ELLS. Ni [2] proposed a method based on dynamic fuzzy neural network to suppress the extra torque and improve the system response and tracking accuracy by combining with feedforward feedback and direct inverse control of the composite control strategy; Wang [3] proposed a nonlinear robust control algorithm of the electro-hydraulic load simulator, which address the actuator's disturbance and flow nonlinearity. Wang [4] proposed fuzzy adaptive torque control (Adaptive Fuzzy Torque Control, AFTC), which effectively restrain the extra torque and improve the stability of the closed-loop system by using the small gain theorem. ND Manring [5] proposed a feedback linearization method which reduces and nearly eliminates the load dependence of the tracking response. R Ghazali [6] adopts a robust controller design using discrete-time sliding mode control, and a two-degree-of-freedom(2-DOF) is used to turn the DSMC, which reduce the phase lag trajectories and significantly show enhancement in tracking control performance.
The above literature, mainly from the servo motion compensation and anti-jamming controller design. Based on the linear servo high-precision mechanical properties testing, in order to overcome the mechanical nonlinear factors, steering disturbance and other issues, this paper proposes parallel structure which include fuzzy PID control and the repetitive controller. It not only improves the self-adaptive anti-jamming ability of the system, but also reduces the cumulative error in sinusoidal loading. The simulation results show that the composite control can significantly reduce the extra force and improve the loading accuracy.

System structure
ELLS is a position disturbance type torque servo system, which mainly consists of load motor, motor driver, real-time controller, sensor, multi-channel data acquisition card, and the rudder system constitutes a complete loading system.
As shown in Fig.1, the host computer PC sends a sine current signal through the EtherCAT communication protocol to control the output torque of the motor. The loading motor converts the rotational force into a linear force through the ball screw to complete the linear load of the steering gear. The loading force is detected by high precision pressure sensor, the data acquisition card feedback to the controller, the formation of force closed-loop control to track load force command to complete the static and dynamic force loading.

System mathematical model
The accuracy of the ELLS mathematical model largely determines the control performance of the system. Combined with this system, the model of the motor and the ball screw model are fully considered in the process of modeling. This paper assumes that the mechanical connection parts are both rigidly connected and ignoring the influence of the friction torque.

Load motor model
The load motor(PMSM) uses 0 d i  control strategy. Under ideal conditions [7], the voltage balance equation and the electromagnetic torque equation can be written as: The back EMF coefficient and electromagnetic torque are: In the above (1) and (2),where q U is the voltage on q axis; a R is the phase resistance of the motor stator; q i is the current on q axis ; a L is the inductance on q axis; p is the motor rotor pole pairs; f  is the rotor permanent magnet flux; m W is the motor output shaft mechanical angular velocity; e T is the electromagnetic torque; L T is the load torque; a B is the motor friction coefficient; a J is the motor's moment of inertia; K e is the motor's back EMF coefficient.

Intermediate transformation model
Permanent magnet synchronous motor will rotate torque into linear force, and it needs to go through the ripple coupling, torque sensor and ball screw pair. These intermediate links should be taken into consideration because they have a certain impact on the accuracy of the system.
Torque sensor integrated error is 0.1%  FS, which meets the Hulk's law: so system torque balance equation is: In the above (3) and (4), w T is elastic torque; L J is the load moment of inertia, which includes ball screw s J and the coupling G J , 

Torque and force relationship of ball screw
Rotating torque transforms into linear load force through the ball screw. The relation between output torque and linear relationship is: Linear servo run a distance L, then the corresponding angle of screw turned, displacement and angle are as follows: 2 L LP   (6) In the above (5) and (6), F is the linear load force; r is the ball screw radius;  is the ball screw lead angle; P is the ball screw lead, L is the linear servo displacement.
When constructing the system model, the servo actuator, actuator and transmission mechanism are a closed system. Therefore, the actual displacement of the linear servo is approximated as the displacement command, and the displacement of the servo is regarded as a disturbance of the system.
Laplacian transformation of the above (1)-(6) can get the system control open loop control block diagram shown in Fig.2, where in the voltage control signal q U is the input signal, the servo linear displacement, L is the disturbance signal, the straight line force F is the output signal.

Current loop design
During the servo failure test, the torque of the PMSM needs to be precisely controlled so that it can respond quickly. Therefore, the current is controlled by the proportional control, and the control block diagram is shown in Fig.3.
Current open loop transfer function is: The above (7) is the proportional coefficient of the current loop controller. It can be seen from the bode diagram of the system, and it is a stable system whose phase does not exceed -90  . The closed-loop transfer function of the system is as shown in (8),The current closed-loop transfer function consists of a first-order lag and a proportional.
Static deviation of the control system commonly uses S=0 closed-loop gain to evaluate. That is, the closer to 0 the gain is, the smaller the static deviation is. From (9), since the proportional gain is far greater than a R , the gain is close to 0 and the static deviation of the system is small.
It can be seen from the frequency characteristics of the closed loop current loop in Fig.4 that the amplitude-frequency characteristics of the system are stable and are of a high bandwidth (3Khz),which shows fast current response and good dynamic and static characteristics.

Force controller design
Considering the problems of large load steady-state error, phase lag and low response in the system, this paper uses the closed-loop feedback of output force and proposes the structure of parallel fuzzy PID and repetitive control. The block diagram of the concrete structure is shown in Fig.5 The fuzzy PID structure adopts the gain adjustment type.
Because the controlled increment only relates to the error and the error rate of change sampling value, it has the advantages of saving the regulator memory and calculating the time [8]. Based on the premise of a stable system, the initial parameters of PID uses Z-N method of self-tuning: 0 .06.The specific formula is: ,The input deviation e and deviation of the rate of change ec fuzzy domain are defined as (-6,-4,-2,0,2,4,6) .The fuzzy universe of output variable U is (-3,-2,-1,0,1,2,3) . In actual application, the quantization factor ( K e , ec K ) and the scale factor ( 1 K , 2 K , 3 K ) can be adjusted to map to the corresponding universe of discourse range. Membership functions adopt triangular functions with online calculations and occupy smaller systems.
Fuzzy rules are summed up after a lot of people's experiments and work experience [9].The general rule of control rules are: 1. When the ELLS starts or stops running, the error e of the output force of the loading system becomes larger, so as to speed up the response of the system, p K should be greater. To avoid the effect of differential saturation caused by the excessive e, d K should be medium; To prevent loading system output force from greater overshoot. i K should be zero. 2. When the ELLS is in normal operation, the error e and the rate of change of error e are medium-sized, so as to reduce the overshoot of the system, properly take small p K , i K and take d K appropriate values.

When ELLS outputs a constant force, e and ec
should be small at this moment. In order to make the system stable, i K and d K should be increased appropriately. In order to avoid system oscillation in setting value and strengthen the anti-interference of the system, the reasonable value d K should be taken. x , and the final output of the system multiplied 0 x by the scaling factor to get the actual control volume.

Repetitive controller design
The repetitive control is based on the internal model principle and can be used to control the repetitive trajectory of the servo system accurately. It can compensate the periodic signal and suppress the periodic disturbance of the load [10].
The (13) can be transformed into: In (13), G(S) P(S) From (14), we get the system's characteristic equation: According to the stability criterion of classical control theory, all the eigenvalues of the system are in the left half of the S plane, so the system is stable.

b) Compensator design
Dynamic compensator C(S) is designed to compensate for the amplitude and phase of the system to improve the system loading accuracy and stability. When |1 C(S)P(S) | 1   converges to zero, the repetitive controller loading system converges fast and the external disturbance rejection capability of the system increases. However, C(S) is impossible to approach the controlled object P(S) in the entire frequency band. Considering repetitive amplitude attenuation and phase compensation in the high frequency range, Therefore, C(S) with low-pass filter function, cycle delay compensation for the phase, so the design should meet the (16) , we can see that when Q(S) is close to 1, the tracking capability is good, the steady-state error is small and the system is stable.
The selection of Q(S) should ensure system stability and tracking accuracy, under the premise of On the basis of Fig.2, the current loop and the force loop are introduced into the system, therefore the block diagram of the control system is obtained and shown in Fig8.

System simulation
According to the control strategy of fuzzy PID and repetitive control presented in this paper, a system simulation model based on Matlab/Simulink is established. The system parameter

Linear load force tracking performance
Set sine load force command F=3000sin(6 ft)  as a system of incentives, compare traditional PID control and composite control load force tracking effect. Fig.9 is the PID control, there is a large deviation both in amplitude and phase, which can not meet the project "double ten indicators"; Fig.10 is a composite control tracking curve, the force command and output force curve basically coincide, greatly reducing the amplitude and phase deviation, the tracking performance of the system has also been greatly improved; Fig.11

Excess force suppression ability
ELLS endures position coupling disturbance from active motion of actuation system, and this inherent is called excess force, So suppression of excess force is an important property for electric linear loading systems, and generally the commanding force is zero. The linear actuator follows the specified sinusoidal path acted as the disturbance input. In order to compare the control effects under different controllers, traditional PID control and redundant force suppression simulation under compound control are carried out. Fig.12 shows that when the steering gear moves at 1mm/1Hz, the composite control compared with traditional PID extra force reduces from 150N to 60N. Fig.13 shows that when the steering gear moves at 1mm/3Hz, the composite control reduces 380N to 110N compared with the traditional PID, and as time goes by, the effect of restrain force is getting better and better.

Conclusion
ELLS is a key part in the design of electric load simulator. Compared with the torque servo system, ELLS has strong coupling and many nonlinear factors. In this paper, proposes a two-loop control structure of current loop and force loop closed-loop feedback. Current loop is controlled by Proportional control, which improves the current response and steady-state error. The force loop uses the parallel method of fuzzy PID and repetitive control, which not only improves the accuracy of the loading force, but also reduces the steady-state error. The simulation shows that the control method is very good at tracking servo capability and suppressing excess force in low frequency.