Abstract
In this paper, a speed tracking and synchronization control approach is proposed for a multimotor system based on fuzzy active disturbance rejection control (FADRC) and enhanced adjacent coupling scheme. By employing fuzzy logic rules to adjust the coefficients of the extended state observer (ESO), FADRC is presented to guarantee the speed tracking performance and enhance the system robustness against external disturbance and parametric variations. Moreover, an enhanced adjacent coupling synchronization control strategy is proposed to simplify the structure of the speed synchronization controller through introducing coupling coefficients into the conventional adjacent coupling approach. Based on the proposed synchronization control scheme, an adaptive integral sliding mode control (AISMC) is investigated such that the chattering problem in conventional sliding mode control can be weakened by designing an adaptive estimation law of the control gain. Comparative simulations are carried out to prove the superiorities of the proposed method.
1. Introduction
The demands of control systems with large inertia and high power load are increasing in modern engineering systems [1]. However, due to the limited output torque, single-motor drive often fails to meet the control performance requirements when driving large inertia load and further affects the production efficiency. Recently, research on multimotor synchronous drive systems provides an effective way to solve the problem of large inertia load control and multimotor synchronous control has been applied in many manufactures [2]. However, there is a mutual coupling between motors in the multimotor system, which makes it difficult to ensure the system’s trajectory tracking and synchronous control accuracy at the same time. Therefore, a superior speed synchronization control design is significant to achieve better synchronization performance of multimotor systems [3].
Some effective synchronous control strategies have been investigated in recent decades, such as master-slave synchronization strategy [4], adjacent coupling control [5], ring coupling synchronization technique [6], cross-coupling synchronization strategy [7, 8], virtual line-shaft synchronization scheme [9], and relative coupling synchronization method [10]. Among all the above mentioned synchronization schemes, adjacent coupling control is designed based on minimum relative axial thought, which means that the torque of each motor should be able to make the tracking error of itself converge to zero and lead the synchronization error to converge to zero stably between any motor and adjacent motors. An adjacent cross coupling strategy combining with sliding mode control [5] is designed for a multiple induction motor synchronization control system. The speed tracking error and synchronize error are both guaranteed to converge to zero by Lyapunov stability theory. A total sliding mode control method based on adjacent cross coupling structure in [11] is proposed for the multiple induction motors. The speed tracking of each motor is stabilized by synchronizing with other motors, such that the synchronized errors can be converged to zero. However, the two methods cannot solve the chattering problem in the controller. Compared with other synchronization approaches, the adjacent coupling control has better synchronization performance with respect to faster convergence rate and smaller steady-state error. However, there is an obvious disadvantage which is that conventional adjacent coupling control has a complex structure and difficult stability proof [12]. An enhanced adjacent coupling synchronization method with coupling coefficients is developed in this paper so that the controller structure is simplified and stability is proved more easily.
To further improve the tracking control performance, various modern control algorithms combined with aforementioned synchronization schemes are investigated and have acquired good results. Reference [13] combined an adaptive sliding mode technique with a ring coupling scheme to achieve rapid and accurate tracking and synchronization performance. Reference [14] designed a master-slave consensus tracking algorithm via observer for coordination control of multimotor systems, which could guarantee the synchronization precision. Reference [15] constructed a synchronization controller using a fuzzy neural network to deal with the nonlinear problem effectively. Reference [16], investigated a fast terminal sliding mode synchronization scheme of a dual-motor driving system, which accelerated the convergence rates of tracking error and synchronization error. Moreover, more advanced control methods are also employed for multimotor systems, such as control [17], fuzzy control [18], and integral terminal sliding mode [19].
Compared to other control methods, active disturbance rejection control (ADRC) is a new-fashioned application algorithm and has acquired wide application in various fields, such as aerospace and vehicle manufacture [20–23]. There are some excellent merits of ADRC including error-based character, strong robustness for external disturbance and parametric variations, and superior response rate [24–28]. A nonlinear ADRC [24] is investigated for induction motors to guarantee the robust control and high performance by employing the ESO to observe the derivative signals and precise decoupling of the motor without accurate prior knowledge. An enhanced linear ADRC [25] is designed for an interior permanent magnet synchronous motor on the basis of a HF pulse voltage signal injection scheme. The cascaded ESO is proposed to ensure the timely and accurate estimation of the lumped disturbance. A novel control scheme is proposed for ball screw feed drives in [26] to improve the tracking performance and robustness of the system by combining ADRC and the proportional integral method. The ESO is employed to estimate and compensate the unmodeled dynamics, disturbances, and cutting load. However, as one of the most important parts in ADRC, the parameters of ESO mentioned in above literatures are chosen in advance, which may restrict its flexibility and cause poor dynamic system performance [29].
Since the parameters of ESO directly affect the observation of disturbances and uncertainties and then affect the control performance of ADRC, the optimization design of the parameters of ESO has been studied. An ant colony optimization (ACO) algorithm [30] is investigated to optimize the six parameters of ESO for an induction motor system by the self-learning ability of ACO. Thus, the robustness of the proposed optimization scheme is better than traditional ADRC when perturbation is produced. An adaptive hybrid biogeography-based optimization and differential evolution (AHBBODE) is represented in [31] for vessels with a dynamic positioning system to achieve the accurate movement and positioning. AHBBODE is employed to optimize the parameters of ADRC, which are not easily obtained by the trial and error method. A new Levy fight-based whale optimization algorithm [32] is proposed to estimate the parameters of ADRC for an automatic carrier landing system with internal dynamics and external disturbance. The accurate tracking performance and robustness are obtained by using the proposed optimization algorithm. However, the optimizations of [30–32] are all offline optimization algorithms and the implements are more complex. When the system is suddenly influenced by the external environment, the control of offline parameter identification may not fully guarantee the system performance. The neural network observer is designed in [33] to estimate the unmeasurable system state variables, and the weight parameter is adaptive updating online. But the weight parameter is not easy to converge to the optimal value by traditional adaptive technique [34]. Among the numerous parameter optimization methods, the fuzzy logic is a powerful tool in parameter adjustment because of its simple structure, online optimization, and easy implementation [35], and thus, it is possible to use the optimization method based on fuzzy logic to adjust the ESO parameters, so as to further improve the system control performance.
Besides, for the purpose of keeping high synchronization control performance of a multimotor system, the lumped disturbances including nonlinearities and uncertainties should be compensated. In practical multimotor applications, the lumped disturbances always follow the operation of the system, which may arise internally, such as friction and parametric variations, or externally, such as change of the load. Among the numerous feedback control schemes, the sliding mode control (SMC) technique is a superior approach which can guarantee an efficient antidisturbance performance [36–41]. A total SMC is proposed to achieve speed tracking and synchronization for multimotor induction by employing adjacent cross coupling structure in [37]. Reference [38] designed an observer-based sliding mode controller for stochastic systems. By designing the observer and a product of sliding mode variable and negative definite matrix, the sliding mode variable can be stabled almost in the beginning and the sliding mode motion can be confirmed. Reference [39] managed to attenuate chattering based on a composite controller which is set by the sliding mode feedback and disturbance compensation with a new second-order model for the speed loop.
Motivated by the discussions mentioned above, the main contributions of this paper are summarized as follows: (1)A fuzzy active disturbance rejection control (FADRC) is developed for multimotor systems to guarantee tracking performance and enhance the system robustness. Since the ESO parameters are optimized by using fuzzy logic rules, the proposed FADRC can achieve a faster response than ADRC.(2)By introducing coupling coefficients into adjacent coupling approach, an improved adjacent coupling scheme is proposed so that the controller structure is simplified and stability is proved more easily. Moreover, an AISMC approach is developed and incorporated into the synchronization control scheme to attenuate the chattering phenomenon in SMC.
The rest of this paper is organized as follows. The mathematical model of the multimotor system and the adjacent coupling scheme are shown in Section 2. In Section 3, two control schemes including a FADRC and an AISMC using improved adjacent coupling are designed for the multimotor system. The stability analysis is given in Section 4. Simulations are presented in Section 5, and a concise conclusion is provided in Section 6.
2. Problem Formulation
2.1. Mathematical Model of the Single-Motor System
The mathematical model of a surface-mounted permanent magnet synchronous motor (PMSM) under the d-q axis can be described as where , are the stator voltages of d-axis and q-axis, respectively; , denote the stator currents of d-axis and q-axis; , are the stator flux linkages of d-axis and q-axis; , are the inductances of d-axis and q-axis; is the rotor flux linkage; is the angular speed of the the motor. It can be easily concluded that the model is strongly coupled between and according to (1) and is often used to achieve decoupling for and . The control schematic diagram of each PMSM in a multimotor control system is given in Figure 1.
The electromagnetic torque of PMSM can be represented as where is the number of motor pole pairs. As a surface-mounted PMSM is employed, the permeability of permanent magnets and air permeability are almost the same. Hence, it is reasonable to think that . Thus, (2) is written as
The motion dynamics of PMSM can be shown as where denotes the rotary inertia; and b represent the load torque and viscous coefficient of friction, respectively. Due to the great influences of load torque, it is regarded as an external system disturbance.
Substituting (3) into (4), the following equation is obtained:
2.2. Mathematical Model of the Multimotor System
The mathematical model of the multimotor system is where represents the angular speed of the ith motor and denotes the motor number.
In order to facilitate the controller design, defining equivalent variables , , , and . Then (6) can be rewritten as
Considering the external disturbance and parameter variations, (7) can be transformed into where denotes the sum of external disturbance and parameter variations, which satisfies the boundness condition that , and is the upper bound. The expression of is where , and denote the parameter variations.
2.3. Adjacent Coupling Scheme
The speed tracking error of the conventional adjacent coupling scheme can be defined as follows: where represents the identical speed reference signal of each motor for the multimotor system.
The speed synchronization errors of adjacent motors are defined as
In general, the larger synchronization errors will be generated at the initial stage of the system or when the external disturbances are encountered. Once , all the motors have achieved a good synchronization performance.
For the purpose of keeping good synchronization performance for the multimotor system, the adjacent coupling errors are defined for the conventional adjacent-coupling synchronization technique
It should be noted that in order to enhance the synchronization performance of the multimotor control system, both synchronization error and adjacent coupling error should be considered simultaneously in the controller design. However, the approach may greatly increase the complexity of controller and lead to the difficulty of the system stability analysis.
In this paper, the control objective is to design the speed tracking and synchronization controllers for the multimotor system such that (1)the rotor speeds of all the motors can track the command speed signal, that is, (2)each motor can synchronize other motors, that is,
3. Controller Design
In this section, two controllers are proposed for the speed tracking and synchronous control of the multimotor system based on ADRC and AISMC, respectively. The design objectives of speed tracking and synchronization controllers are to ensure that the speed tracking errors and synchronization errors, respectively, can effectively converge to zero. The block diagram of the controller for each motor is given in Figure 2.
3.1. Enhanced Adjacent Coupling Scheme
An enhanced adjacent coupling scheme is developed for the multimotor system (total number of motors is ) to reduce the synchronization errors between the adjacent motors. The speed signal of each motor and adjacent motor are used as the input of the controller. Simultaneously, coupling coefficients are added to ensure that the synchronization errors converge to zero and simplify the design of the synchronous controller. The simplified structure of synchronization control is given in Figure 3, and Figure 4 shows the control schematic diagram of the enhanced adjacent coupling scheme.
In the enhanced adjacent coupling synchronization scheme, coupling coefficients are introduced into adjacent coupling errors to ensure the stability of the multimotor system and simplify the controller design. Consequently, the adjacent coupling errors are defined as where and and satisfying .
Let where
The above matrix can be transformed into the following triangular form by a reasonable equivalent transformation
It can be obtained that (14) has a unique solution when is a full-rank matrix, that is, . Once that condition converges to a minimal domain is satisfied, then converges to a minimal domain, which means that the synchronization control objective can be transformed to design the speed synchronization controller to ensure that converges to a minimal domain.
3.2. Speed Tracking Controller
ADRC is a powerful antidisturbance technique, including tracking differentiator (TD), ESO, and nonlinear state error feedback (NLSEF) controller. Smoother input signals and smaller overshoot can be obtained by using TD when the system starts. And the multimotor states and lumped disturbances including nonlinearities and uncertainties can be effectively observed by designing the ESO. The NLSEF is employed to realize lumped disturbance compensation with the feedback state errors. Figure 5 shows the detailed schematic diagram of FADRC.
From (7), the ith motor’s tracking FADRC is designed as (17), (18), (19), and (20).
Firstly, the TD is given as where denotes the output of TD, which indicates the tracking of the identical speed reference signal ; represents the error between and ; , are the tuning parameters; indicates a nonlinear function and is expressed as
Secondly, the ESO is designed as where is the observation value of and is the lumped disturbance; is the observation error of ; and are the tuning gains.
Remark 1. According to (18), the nonlinear function fal(.) will produce a larger gain when observation error is small; on the contrary, fal(.) will produce a smaller gain when is large. Reference [40] pointed out that fal(.) could stabilize the observation state by choosing the appropriate size of (i = 1,2), which means that ,, where and are all small positive constants.
Finally, the NLSEF is designed as where and indicate the control input without considering disturbance and speed tracking control signal with considering disturbance, respectively, of the ith motor; denotes the error between the output of TD and the output of ESO; is a constant, which represents the estimated value of and is given by experience.
It is pointed out in [20] that a ADRC-based single-input and single-output system is absolutely stable, and thus, the speed tracking errors are ensured to converge to zero through the speed tracking controller.
3.3. Coefficient Determination via Fuzzy Logic Rules
In the following, we define that
In practical applications, the coefficients of ESO, , are usually adjusted based on prior knowledge to acquire a good estimation performance. However, it may restrict its flexibility and cause poor dynamic system performance. In the following, the error variables and are utilized by the fuzzy logic rules to determine the coefficients , which can be optimally adjusted online.
The error variables and are employed as the fuzzy inputs, and five linguistic rules, that is, positive big (PB), positive small (PS), zero (ZO), negative big (NB), and negative small (NS), are given as the membership functions. Table 1 shows the fuzzy rules of . And a Gaussian function and a triangle function are chosen as the input , membership function and the output (i = 1, 2) membership function, respectively. In this subsection, the basic universe of and the basic universe of are [−1, +1] and [−0.5, +0.5], and are chosen within [−0.1, +0.1] and [−0.5, +0.5], respectively. The Mamdani type is adopted as the fuzzy inference, and the weighted average method is utilized for the defuzzification.
According to Table 1 and after fuzzy inference and defuzzification, the coefficient is rectified as where and are the initial value and rectified value, respectively, of the ESO coefficients.
3.4. Speed Synchronization Controller
In this subsection, we employ AISMC to design the speed synchronization controller.
The integral sliding mode surface can be designed as where denotes a tuning parameter. From (8) and (23), the general sliding mode surface controller based on ESO is given as where is the control gain that satisfies that . The state variables can be stabilized to the sliding surface by the designing of controller (24).
However, it is difficult to determine the accurate value of since the upper bounds of estimation errors are not easy to obtain. Therefore, in the following, an AISMC combined with ESO is designed as where the adaptive law of is where and are used to guarantee that .
Substituting (25) into (8), we have
4. Stability Analysis
Lemma 1 (see [42]). For the given nonlinear uncertainty system (8) with the sliding mode (23), the control gain has an upper bound, which satisfies that .
Lemma 2 (see [43]). Suppose that there is a continuous and positive definite function, which satisfies the follow differential equation: where and are positive constants. Then for any given , there is a finite time that satisfies where .
Theorem 1. Considering multimotor system (8), integral sliding mode surface (23), and controller (25), the integral sliding mode surface can converge to be uniformly ultimately bounded in finite time and the synchronization error can stably converge to be uniformly ultimately bounded when with and being two positive constants.
Proof 1. Considering system (8), the Lyapunov function is designed where .
Differentiating (30) and substituting into (13), we can obtain
Substituting (10) into (31), we have
Substituting (27) into (32), then
Introducing parameter , (33) can be equivalently transformed into
According to Lemma 1, we can obtain where , . Ultimately, the derivative of is where .
Case 1. If , then according to (26), we have When the parameter is selected to satisfy , we can conclude from (37) that , which leads to from (36).
Case 2. Supposing that , we have the function in (35) which is negative and is also negative. So, will decrease until less than . From (33), we can conclude that will be positive, which means that will increase until larger than . Therefore, the case becomes the same as Case 1, in which can be finally guaranteed by choosing the appropriate parameter .
From Case 1 and Case 2, we have . According to Lemma 2, the sliding surface could be guaranteed to converge to the domain in finite time. According to , we have so . Differentiating both sides of (23), we have and the solution is , which means that converges to domain when . According to (14), we can conclude that converges to be uniformly ultimately bounded. This completes the theorem proof.
From (20) and (25), the complete speed controller for the ith motor can be obtained as
Remark 2. To further improve the control performances, the function , which may cause sliding mode chattering, is replaced by a saturation function where is a small positive constant, which indicates the thickness of the boundary layer.
5. Simulation Results
In this section, numerical simulations are performed on a four-motor synchronization control system to evaluate the effectiveness of the proposed control scheme. In order to show the superior synchronization and tracking performance of the proposed method, two other different approaches and two different reference signals are provided for the fair comparison. For notation convenience, all the three methods are denoted as (M1) the proposed method, (M2) master-slave scheme with ADRC [44], and (M3) adjacent coupling scheme with SMC [12].
In the simulation, all the parameters of the four motors are shown in Table 2. The simulation time and step size are set as 0.5 s and 0.001 s, respectively. The load torque is given as 2 N and changes to 11.8 N at 0.2 s. The detailed parameters are set as follows.
5.1. M1: The Proposed Method
The parameters in tracking controllers (17), (18), and (19) are , , , , , and ; the parameters in synchronization controller (25) are , , , , and ; the parameters of adaptive law (26) are set as and .
5.2. M2: Master-Slave Scheme with ADRC
The synchronization controller can be avoided to design for this method, because the command signals of other slave motors are the output signals of the master motor. The speed tracking controller has been shown in (20), and the relevant parameters are given as , , , , , , and .
5.3. M3: Adjacent Coupling Scheme with SMC
In this control approach, the speed tracking and synchronization controller are designed according to the procedures of [10], which can be expressed as follows, respectively. where , and are set the same as M1.
In the following, two different command signals are given for comparison. (1)xd = 1000 r/min.
The speed synchronization performance comparison of four motors is shown in Figure 6. From Figure 6, we can see that the output speed signals of four motors with M1 can achieve the best synchronization performance when command signal is . Moreover, M1 has a faster convergent speed than the other two methods when motors suffer to load torque variations. The corresponding synchronization errors and tracking errors are shown in Figures 7 and 8, respectively. In Figure 7, and , indicates the synchronization error between the motor and motor and we can see that the maximum synchronization error with M2 is about 40 r/min and the synchronization error with M3 is about 10 r/min. However, the maximum synchronization error with M1 is no more than 5 r/min. Consequently, we can conclude that compared with M2 and M3, M1 has a good performance with respect to small synchronization errors. As seen in Figure 8, it is clear to see that tracking error using M1 has a faster convergence rate when suffering to load variation. The control input signals with M1 and M3 are compared in Figure 9, which shows that the chattering problem with M1 is much smaller than M3. (2)xd = 100sin (8πt) r/min.
(a) Synchronization performances of M1
(b) Synchronization performances of M2
(c) Synchronization performances of M3
(a) Synchronization errors
(b) Synchronization errors
(c) Synchronization errors
(d) Synchronization errors
(a) Tracking errors
(b) Tracking errors
(c) Tracking errors
(d) Tracking errors
(a) Control input
(b) Control input
All the parameters are chosen the same as those in the former case. The speed synchronization performance comparison of four motors is shown in Figure 10. From Figure 10, we can see that speed synchronization performance with M1 is superior to that with the other two methods. The corresponding synchronization errors and tracking errors are provided in Figures 11 and 12, respectively. It is clear to see from Figure 11 that the synchronization errors of M1 have faster convergence rate and smaller steady-state values than those of M2 and M3. Figure 12 shows that M1 has smaller tracking errors compared to M2 and M3. The control input signals with M1 and M3 are compared in Figure 13. From Figure 13, we can see that M1 can attenuate the chattering problem compared to M3, which verified the effectiveness of the proposed method.
(a) Synchronization performances of M1
(b) Synchronization performances of M2
(c) Synchronization performances of M3
(a) Synchronization errors
(b) Synchronization errors
(c) Synchronization errors
(d) Synchronization errors
(a) Tracking errors
(b) Tracking errors
(c) Tracking errors
(d) Tracking errors
(a) Control input
(b) Control input
From all the simulations above, we can conclude that (1)Compared with M2, M1 has faster convergence rate when suffering to load variation;(2)Compared with M3, M1 has a much smaller chattering problem;(3)For different command signals, M1 can achieve better synchronization and tracking performance compared with M2 and M3.
6. Conclusion
Two kinds of controllers are investigated in this paper to improve the synchronization and tracking performances of the multimotor system. The synchronous controller is designed to ensure the speed synchronous performances and reduce the SMC chattering of the adjacent motor by utilizing an enhanced adjacent coupling approach with an adaptive integral sliding mode control. For the second tracking controller, the fuzzy logic rules are employed to design the active disturbance rejection control technique, and thus, a fast performance response and better robustness can be achieved. Simulation results prove the superior synchronization and tracking performance of the proposed method. The future work is aimed at simplifying the controller design and applying the proposed method to a practical system, such as a straight wire drawing machine.
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 they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant no. 61403343, and Zhejiang Provincial Natural Science Foundation under Grant no. LY17F030018 and no. LY17F010015.