Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 7064141, 13 pages
https://doi.org/10.1155/2019/7064141

Research Article

An Improved Electronic Line Shafting Control for Multimotor Drive System Based on Sliding Mode Observer

Qiang Geng, Wei Liu, Huimin Wang, Zhanqing Zhou, and Guozheng Zhang

School of Electrical Engineering and Automation, Tianjin Polytechnic University, Tianjin 300387, China

Correspondence should be addressed to Huimin Wang; wanghuimin@tju.edu.cn and Guozheng Zhang; zhanggz@tju.edu.cn

Received 21 January 2019; Accepted 16 April 2019; Published 30 April 2019

Academic Editor: José Domingo Álvarez

Copyright © 2019 Qiang Geng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

As the computation delays between the reference torques and the load torques, the speed and position synchronous errors of the multimotor drive system employed traditional electronic line shafting (ELS) control would become significant during the process of large load disturbances. Therefore, an improved ELS control strategy is proposed in this paper. In this strategy, the load torques observed by the sliding model observers are fed back to the virtual motor directly, so as to shorten the adjusting time and improve the antidisturbance performance of entire control system. Meanwhile, to reduce the chattering of the sliding mode observer, a novel exponential reaching law is designed in this paper. The experimental results show that the improved ELS control strategy could reduce the speed and position synchronous errors effectively.

1. Introduction

With the development of modern industrial technology, the multimotor drive system has been applied in various fields as paper making, textile, printing industrials, etc. [1, 2]. So far, the common control strategies of the multimotor drive system include master-slave control, cross coupling control, relative coupling control, and electronic line shafting (ELS) control [3–7].

In the master-slave control, the reference speed is fed to the master motor and then the output speed is offered to the slave motor as its reference. There is an extra time delay of signal transmission for the slave motor to reach the initial reference speed. The cross coupling control system includes two motors; the feedback signal of each motor consists of the output speed and the relative difference between the speeds of both motors. Nonetheless, it is difficult to extend this strategy for more than two motors. In the relative coupling control, speeds of all motor are fed back to every motor by the relative speed blocks, in which the importance of each motor’s speed is identical. However, the structure of the control system is complicated. The ELS strategy mimics mechanical synchronization method by utilizing a virtual electronic line shaft, and the strategy has already been applied to paper machine successfully for its advantages as flexible configuration and easy adjustment [8].

In 1989, Loren and Schmidt proposed the control strategy of ELS based on the definition of the relative stiffness of the mechanical axes connections [9]. The ELS control system is composed of a virtual motor, a virtual mechanical unit, multiple motors, and some other parts. Meanwhile, the virtual motor and the virtual mechanical unit could be implemented by software. The ELS control strategy employs some virtual parts to replace the mechanical major axes, so that the mechanical constraints could be avoided and the control parameters could be adjusted easily [10–13]. Therefore, the ELS control strategy, applied to diverse circumstances, has received wide attention with the flexible configurations of the virtual parts. Taking the paper machine as the research target, [14] replaces the real mechanical unit with a virtual mechanical unit to reduce the speed and position synchronous errors between the virtual motor and each motor during start, stop, and load disturbances of the motors. Reference [15] has designed a compensator, which solves the problem of deviations between the sampling frequencies of each motor.

In the traditional ELS control strategy, the reference torques of motors are fed back to the virtual motor. At steady state, the reference torques are approximately equal to the load torques. However, during the process of large load disturbances, the computation delay between the reference torques and the load torques will result in significant speed and position synchronous errors.

The sliding mode observer is applied successfully to the load torque observation of permanent magnet synchronous motor (PMSM) drive system with the advantages of parameter robustness and good antidisturbances ability [16]. Reference [17] analyzes the stability of the control systems based on the load torque sliding mode observer. References [18, 19] propose novel PMSM control strategies with the extended sliding mode observer which take the speed and load torque as the observation target to eliminate the load torque disturbances effectively. Although the sliding mode observer could improve the control performance and robustness of the system efficiently, the existence of chattering limited its range of application [20, 21]. Hence, [22] introduces the sigmoid function to redesign the sliding mode observer, so as to overcome the serious chattering of the tradition sliding mode observer. Besides, [23] replaces the signum function in the traditional sliding mode observer with S type function, which solved the problems such as chattering and filter delay caused by the switching characteristic of the signum function.

For the problems existing in the traditional ELS control strategy, an improved ELS control strategy is put forward in this paper. In this strategy, the load torques of motors, observed by the sliding mode observer, will be fed back to the virtual motor, which improves the antidisturbances ability of the control system. Besides, to avoid the serious chattering problem of the traditional sliding mode observer, the sliding mode observer based on a novel exponential reaching law algorithm is proposed in this paper.

2. The Mathematical Model of PMSM

Assume that the number of motors in the multi-PMSM drive system is n; the voltage equation of the -th ( = 1,2,…,n) PMSM under the synchronization reference frame is described by [17]

where , , , , , are the direct axis (d-axis) and quadrature axis (q-axis) components of the stator voltage, current, and inductance, respectively. , , are the stator winding resistance, electrical angular velocity, and rotor flux, respectively.

The mechanical equation of the -th PMSM is as follows:

where , , , , and represent the electromagnetic torque, load torque, the number of the pole-pairs, mechanical angular velocity, and moment of inertia of the -th PMSM, respectively.

When using the control strategy, the electromagnetic torque of the -th PMSM can be expressed as

3. The Improved ELS Control Strategy

3.1. The Traditional ELS Control Strategy

The block diagram of the traditional ELS control strategy for three PMSMs is shown in Figure 1.

Figure 1: The block diagram of the traditional ELS control.

In Figure 1, denotes the reference speed of the virtual motor. , , used as the reference signal of the speed and position of each motor, denote the output speed and position of the virtual motor, respectively. denotes the reference torque of the virtual motor. denotes the reference torque of each motor, which is fed back to the virtual motor. According to Hooke theorem, can be obtained as

where Ki is the stiffness gain of each motor; Bi is the damping gain of each motor. Meanwhile, according to the torque balance mechanism, the reference torques are approximately equal to the load torques at steady state.

From Figure 1, the traditional ELS control strategy could adjust the speed and the position of each motor to track the reference signals which are provided by the virtual motor. When operating at steady states, the motors follow the virtual motor to achieve the synchronization of the speeds and the positions. Then, the balance equation on the virtual motor can be expressed as

where is the moment of inertia of the virtual motor.

The principle of the traditional ELS control strategy is analyzed as follows. If load torque increases suddenly, of the disturbed motor would decrease. From (4), the decrease of would lead to the increase of ; then would be fed back to the virtual motor. Furthermore, according to (5), the increase of would cause the decrease of the reference signal , which also leads to the decrease of speeds of other motors. Then, the speed and position differences between the disturbed motor and other motors will be reduced, which ensures the synchronization of the speeds of all motors.

According to Figure 1, of each motor is fed back to the virtual motor simultaneously in the traditional ELS control strategy. Hence, when large load disturbances appear in some certain motors, the computation delay of (4) will cause relatively large errors between the reference torques and the load torques. The errors of the disturbed motors will increase the synchronous errors of the entire control system and degrade the system performance.

3.2. The Improved ELS Control Strategy

To reduce the speed and position synchronous errors under large load disturbances, an improved ELS control strategy is proposed in this paper. And the block diagram is shown in Figure 2.

Figure 2: The block diagram of the improved ELS control.

In Figure 2, denotes the observed value of the load torque of the -th PMSM. In the improved ELS control strategy, , constructed by the sliding mode observer of each motor, is fed back to the virtual motor instead of . In this way, the virtual motor could respondto the large load variation rapidly. The balance equation on the virtual motor could be rewritten as

In another aspect, is fed forward to the current loop of each motor in the system for the improved ELS control strategy.

The control structure of each motor is shown in Figure 3. is the torque coefficient of the PMSM; is the feedforward current; the relationship between and could be expressed as

where is the feedforward coefficient of the observed torque, and . Here, as has contained the information of speed and position, the speed loop is not necessary for the control structure of Figure 3. In short, when a large load disturbance occurs, the sliding mode observer will feed forward to the current loop of each motor directly, which shortens the adjusting time and improves the antidisturbance performance of entire control system.

Figure 3: Single motor control structure.

4. The Design of the Load Torque Sliding Mode Observer

To reduce the chattering phenomenon, the sliding mode observer is constructed with the exponential reaching law. According to the exponential reaching law, the movement of the system states to the original point depends on the exponential function which varies with the distance between the states and sliding surface. However, the switching gap of the traditional exponential reaching law is the ribbon pattern, which makes the system cannot reach the original point ultimately. To solve this problem, a novel exponential reaching law is designed in this section for the sliding mode observer.

4.1. The Design of the Reaching Law

The traditional exponential reaching law could be expressed as

where −ks is the exponential reaching term; , are the positive constants; is the signum function. From (8), the response speed will be increased with the increment of ; however, the chattering phenomenon will appear if the value of k is large. And, the reaching process, moving towards the sliding surface, is a gradual process and has not sliding mode for a simple exponential reaching. Thus, this process does not ensure arrivals within a limited time. So, a constant speed reaching term is added to (8). But, if the value of in this term was too large, the chatting phenomenon would become obvious; conversely, the reaching speed would become slow if the value of ε is too small, though the chatting phenomenon get weakened. Hence, there is a tradeoff between the chatting phenomenon and reaching speed of the observer.

To build the balance control mechanism, a novel exponential reaching law is proposed as [24, 25]

where , , , , .

When the system state variables are far from the sliding surface, (9) will degenerate to the ordinary exponential reaching law

Equation (10) consists of two parts. The first part is the exponential reaching term , where k is the reaching term gain. In order to ensure that the system state variables can quickly converge to the sliding surface and is required to be large enough. However, if k is too large, it will cause serious chatting phenomenon when the system state variables are close to the sliding surface. In this paper, the combination of simulation and experiment is used to select . The second part is the constant speed reaching term . The term is to ensure that there is still a significant sliding mode on the switching surface when the system state variables are close to the sliding surface. For the stability of the sliding mode observer, is generally 2~5. In order to adjust the reaching rate, an adjustment factor is added. Specifically, if , it is the traditional constant speed reaching term; if , the sliding mode gain is increased on the basis of the traditional term.

Otherwise, when the system state variables reach or even go through the sliding surface, (9) will degenerate to the power exponential reaching law:

Equation (11) also includes two parts. The first part is the exponential reaching term and the other part is the power exponential reaching term . In order to avoid the severe chattering phenomenon, the constant speed reaching term is replaced by the power exponential reaching term, and is generally 0~1.

It can be concluded from the above analysis that the novel exponential reaching law proposed in this paper can automatically degenerate to the exponential reaching law or the power exponential reaching law according to the distance of the system state variables from the sliding surface. It not only ensures that the system state variables converge quickly to the sliding surface, but also effectively eliminates the chattering phenomenon after the system state variables reach the sliding surface.

4.2. The Design of the Sliding Mode Observer

Taking and of the -th PMSM as the state variables of the control system, it has

Besides, define variables as

In (12), the derivative of the load torque is approximate to zero considering the control period is short; the following can be obtained:

From (14), taking and as the observation objects, the sliding mode observer could be built as

where is the observation value of ; is the observation value of ; is the feedback gain; is the novel exponential reaching law from (9), which can be expressed as

Define the observation errors of and as

Then, from (14) and (15), we have

Taking as the sliding surface, which means , thus, when the sliding mode observer goes through the sliding surface. In this situation, the following expression could be obtained from (18):

According to stability theory, it can be concluded that the control system would be stable if and the observation error is reaching zero. The reaching manner can be expressed as

where is the coefficient, which is decided by the initial condition of system.

4.3. The Stability Analysis of the Sliding Mode Observer

To verify the stability of the observer, the Lyapunov stability function is taken as

The derivation of (21) can be obtained as

It can be obtained from (22) that if

were satisfied, we have

which means that the observer designed with the new exponential reaching law could meet the system stability requirements, as the derivation of is a negative constant value. It is worth mentioning that could be replaced with the saturation function , so as to weaken the chattering phenomenon more obviously.

5. Simulation Results

The parameters of the virtual motor (VM) and the PMSMs employed in the simulations and experiments are shown in Table 1.

Table 1: Parameters of virtual motor and PMSMs.

Parameters
Units
VM
PMSMs

1.27
1.27

mH
8.05
8.05

2
2

Wb
0.5
0.5

kg.cm²
50
27.2

The initial condition is settled as follows: = 400 r/min; = 3.00, = 0.02; = = = 4N·m. The parameters of the sliding mode observer are taken as = 0.1, = 10, = 0.1, = 2, = 100. The current inner loop controller of each PMSM adopts PI controller, and the controller's parameters tuning method is as follows [26]:

where and are the proportional coefficient and integral time constant of the current loop controller, respectively; , , and are the current loop controller bandwidth, the motor stator inductance, and the resistance, respectively. From (25), the nominal value parameters of each current loop controller are = 0.6, = 0.1.

The PI controller parameters of the virtual motor are obtained by the critical ratio method. The specific operation steps are as follows: first, a sufficiently short control period is selected which is much smaller than the mechanical time constant of the virtual motor; then, in the case that the controller only includes the proportional part, the proportional coefficient is adjusted to cause a critical oscillation of the step response of the system, and the proportional coefficient and the critical oscillation period at this time are recorded; finally, combining [26], the parameters of the PI controller of the virtual motor are = 7.0, = 0.8.

From (3), the torque coefficient and feedforward coefficient are approximately equal to . Taking into account the parameter errors of the actual system, using simulation and experimental methods, the coefficients are finally determined to be = 0.9 and = 1.05.

5.1. Simulation of Sliding Mode Observer

When increased suddenly from 4 N·m to 6 N·m at 0.2 s and decreased suddenly from 6 N·m to 4 N·m at 0.6 s, the waveforms of load torques observed by the sliding mode observer is shown in Figure 4.

(a)

(b)

Figure 4: Observed load torque waveforms during load changes: (a) actual load torque of PMSM3; (b) observed load torque of PMSM3.

From Figure 4, it can be seen that the large observation errors appear only at the starting stage of PMSM3, because the state variables do not reach the sliding surface. When the state variables reach the defined sliding surface, the observed load torque could track the actual load torque.

5.2. Simulation of Load Disturbances

The speed synchronous errors with the traditional and the improved ELS control strategies under the condition of large load torque disturbances of PMSM3 are given in Figure 5.

(a)

(b)

Figure 5: Speed synchronization errors of three PMSMs: (a) the conventional ELS; (b) the improved ELS.

As can be seen from Figure 5, when the load disturbances appear on the PMSM3, for the traditional ELS control strategy, the synchronous speed error between the PMSM1 and PMSM3 is 66 r/min, and the adjust time last for 0.2 s. For the improved ELS control strategy, the synchronous speed error between PMSM1 and PMSM3 is 17 r/min, and the adjust time last 0.1 s with the observed load torques fed back to the virtual motor and fed forward to the current loops of PMSMs. The synchronous speed error between PMSM2 and PMSM3 is equal to the error between PMSM1 and PMSM3.

6. Experimental Results

To prove the effectiveness of the improved ELS control strategy, experiments are carried on the paper transmission unit which is composed of three PMSMs and two load motors. The MCU produced by TI is employed in the experiments. The hardware experimental platform is shown in Figure 6.

Figure 6: Hardware experimental platform.

6.1. Load Disturbances in Single PMSM

The waveforms of the observed load torques when increased from 4 N·m to 6 N·m at 1 s and decreased from 6 N·m to 4 N·m at 4s are shown in Figure 7. From Figure 7, it can be seen that the observed load torques could track the actual load torques rapidly and exactly.

(a)

(b)

Figure 7: Observed load torque waveforms during load changes: (a) PMSM3 increases load; (b) PMSM3 decreases load.

The speed synchronous errors with the traditional and the improved ELS control strategies under the condition of large load torque disturbances of PMSM3 are given in Figure 8. It can be seen that the speed synchronous errors are large and the adjust time lasts for a long time for the traditional ELS control strategy. However, for the improved ELS control strategy, the synchronous speed errors between PMSMs are less affected by the load torque changes and have shorter adjust time, which is consistent with the simulation results.

(a)

(b)

(c)

(d)

Figure 8: Speed synchronization errors of three PMSMs: (a) PMSM3 increases load of the conventional ELS; (b) PMSM3 increases load of the improved ELS; (c) PMSM3 decreases load of the conventional ELS; (d) PMSM3 decreases load of the improved ELS.

The position synchronous errors with the traditional and the improved ELS control strategies under the condition of large load torque disturbances of PMSM3 are given in Figure 9. It can be seen that the improved ELS control strategy could restrain the position synchronous errors brought by the load variation effectively, which means the system has a good antidisturbances performance.

(a)

(b)

Figure 9: Position synchronization errors of three PMSMs: (a) the conventional ELS; (b) the improved ELS.

6.2. Load Disturbances in Two PMSMs

The experimental waveforms of the observed load torques, when increased from 4 N·m to 6 N·m and decreased from 7 N·m to 4 N·m at 1 s and decreased from 6 N·m to 4 N·m and increased from 4 N.m to 7 N.m at 3 s, are shown in Figure 10. It can be seen that the observed load torques could track the actual load torques rapidly and exactly under the condition of different load disturbances of two PMSMs.

(a)

(b)

Figure 10: Observed load torque waveforms during load change in PMSM1 and PMSM3: (a) PMSM1 increases load and PMSM3 decreases load; (b) PMSM1 decreases load and PMSM3 increases load.

The speed synchronous errors with the traditional and the improved ELS control strategies under the condition of different load disturbances of PMSM1 and PMSM3 are given in Figure 11. Among Figures 11(a) and 11(b), there are the speed synchronous errors of the traditional and the improved ELS control strategies, respectively, when increased from 4N·m to 6N·m and decreased from 7 N·m to 4 N·m; Figures 11(c) and 11(d) are the speed synchronous errors of the traditional and the improved ELS control strategies, respectively, when decreased from 6 N·m to 4 N·m and increased from 4 N·m to 7 N·m.

(a)

(b)

(c)

(d)

Figure 11: Speed synchronization errors of three PMSMs: (a) PMSM1 increases load and PMSM3 decreases load of the conventional ELS; (b) PMSM1 increases load and PMSM3 decreases load of the improved ELS; (c) PMSM1 decreases load and PMSM3 increases load of the conventional ELS; (d) PMSM1 decreases load and PMSM3 increases load of the improved ELS.

It can be seen that the speed synchronous errors with the traditional ELS control strategy are relatively large. However, the speed synchronous errors with the improved ELS control strategy are obviously smaller, and the adjust time is relatively short.

The position synchronous errors with the traditional and the improved ELS control strategies under the condition of different load disturbances of PMSM1 and PMSM3 are given in Figure 12. It can be seen that the improved ELS control strategy could decrease the position synchronous errors under the condition of different load disturbances of two PMSMs.

(a)

(b)

Figure 12: Position synchronization errors of three PMSMs: (a) the conventional ELS; (b) the improved ELS.

7. Conclusion

Taking the multi-PMSM drive system as the study object, the traditional ELS control strategy is analyzed in this paper. On this basis, it can be found that the speed and position errors are relatively large under large load disturbances as the computation delay between the reference torques and the load torques for the traditional strategy. To solve this problem, an improved ELS control strategy based on the load torque sliding mode observer is designed.

(1) For the improved ELS control strategy, the observed load torques are fed back to the virtual motor directly and fed forward to the motors, which promotes the antidisturbances of control system.

(2) The load torques are observed by the sliding mode observer with a novel exponential reaching law, which could reduce the chatting phenomenon effectively.

Compared with the traditional ELS control strategy, the improved ELS control strategy could effectively improve the antidisturbances capability of the system under the prerequisite of ensuring the dynamic performance of the system.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was funded by the National Natural Science Foundation of China (no. 51577134 and no. 51507111). The research of this manuscript has also been funded by the Science & Technology Development Fund of Tianjin Education Commission for Higher Education (no. 2018KJ207).

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