# Speed Tracking and Synchronization of a Multimotor System Based on Fuzzy ADRC and Enhanced Adjacent Coupling Scheme

Authors: Liang Tao; Qiang Chen; Yurong Nan; Fang Dong; Yan Jin
Journal: Complexity
 (2018)

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## 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.


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## Body

## 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 H∞ 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 Section2. 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 thed-q axis can be described as
(1)ud=Rid+ψ̇dt−ωψq,uq=Riq+ψ̇qt−ωψd,ψd=ψf+Ldid,ψq=Lqiq,where ud, uq are the stator voltages of d-axis and q-axis, respectively; id, iq denote the stator currents of d-axis and q-axis; ψdt, ψqt are the stator flux linkages of d-axis and q-axis; Ld, Lq are the inductances of d-axis and q-axis; ψf is the rotor flux linkage; ω is the angular speed of the the motor. It can be easily concluded that the model is strongly coupled between id and iq according to (1) and id=0 is often used to achieve decoupling for id and iq. The control schematic diagram of each PMSM in a multimotor control system is given in Figure 1.Figure 1
The control schematic diagram of each PMSM.The electromagnetic torque of PMSM can be represented as(2)Te=1.5mψdiq−ψqid=1.5mψfiq+Ld−Lqidiq,where m 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 Ld−Lq=0. Thus, (2) is written as
(3)Te=1.5mψfiq.The motion dynamics of PMSM can be shown as(4)Jω̇t=Te−TL−bωt,where J denotes the rotary inertia; TL 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:
(5)ω̇t=1.5mψfJiq−bJωt−1JTL.


### 2.2. Mathematical Model of the Multimotor System

The mathematical model of the multimotor system is(6)ω̇it=1.5mψf,iJiiq,i−biJiωit−1JiTL,where ωit,i=1,…,n, represents the angular speed of the ith motor and n denotes the motor number.In order to facilitate the controller design, defining equivalent variablesxit=ωit, Ai=1.5mψf,i/Ji, Bi=−bi/Ji, Ci=−1/Ji, and iq,i=ui. Then (6) can be rewritten as
(7)ẋit=Aiui+Bixi+CiTL.Considering the external disturbance and parameter variations, (7) can be transformed into
(8)ẋit=Aiui+Bixi+Di,where Di denotes the sum of external disturbance and parameter variations, which satisfies the boundness condition that Di≤Υ, and Υ is the upper bound. The expression of Di is
(9)Di=ΔAiui+ΔBixi+Ci+ΔCiTL,where ΔAi,ΔBi, and ΔCi denote the parameter variations.


### 2.3. Adjacent Coupling Scheme

The speed tracking error of the conventional adjacent coupling scheme can be defined as follows:(10)eit=xd−xi,where xd represents the identical speed reference signal of each motor for the multimotor system.The speed synchronization errors of adjacent motors are defined as(11)ε1t=e1t−e2t,⋮εit=eit−ei+1t,⋮εnt=ent−e1t.In general, the larger synchronization errors will be generated at the initial stage of the system or when the external disturbances are encountered. Onceεit→0, 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(12)e1εt=ε1t−εnt,⋮eiεt=εit−εi−1t,⋮enεt=εnt−εn−1t.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,ei→0,i=1,…,n;
(2)
each motor can synchronize other motors, that is,εi→0,i=1,…,n.


## 2.1. Mathematical Model of the Single-Motor System

The mathematical model of a surface-mounted permanent magnet synchronous motor (PMSM) under thed-q axis can be described as
(1)ud=Rid+ψ̇dt−ωψq,uq=Riq+ψ̇qt−ωψd,ψd=ψf+Ldid,ψq=Lqiq,where ud, uq are the stator voltages of d-axis and q-axis, respectively; id, iq denote the stator currents of d-axis and q-axis; ψdt, ψqt are the stator flux linkages of d-axis and q-axis; Ld, Lq are the inductances of d-axis and q-axis; ψf is the rotor flux linkage; ω is the angular speed of the the motor. It can be easily concluded that the model is strongly coupled between id and iq according to (1) and id=0 is often used to achieve decoupling for id and iq. The control schematic diagram of each PMSM in a multimotor control system is given in Figure 1.Figure 1
The control schematic diagram of each PMSM.The electromagnetic torque of PMSM can be represented as(2)Te=1.5mψdiq−ψqid=1.5mψfiq+Ld−Lqidiq,where m 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 Ld−Lq=0. Thus, (2) is written as
(3)Te=1.5mψfiq.The motion dynamics of PMSM can be shown as(4)Jω̇t=Te−TL−bωt,where J denotes the rotary inertia; TL 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:
(5)ω̇t=1.5mψfJiq−bJωt−1JTL.


## 2.2. Mathematical Model of the Multimotor System

The mathematical model of the multimotor system is(6)ω̇it=1.5mψf,iJiiq,i−biJiωit−1JiTL,where ωit,i=1,…,n, represents the angular speed of the ith motor and n denotes the motor number.In order to facilitate the controller design, defining equivalent variablesxit=ωit, Ai=1.5mψf,i/Ji, Bi=−bi/Ji, Ci=−1/Ji, and iq,i=ui. Then (6) can be rewritten as
(7)ẋit=Aiui+Bixi+CiTL.Considering the external disturbance and parameter variations, (7) can be transformed into
(8)ẋit=Aiui+Bixi+Di,where Di denotes the sum of external disturbance and parameter variations, which satisfies the boundness condition that Di≤Υ, and Υ is the upper bound. The expression of Di is
(9)Di=ΔAiui+ΔBixi+Ci+ΔCiTL,where ΔAi,ΔBi, and ΔCi denote the parameter variations.


## 2.3. Adjacent Coupling Scheme

The speed tracking error of the conventional adjacent coupling scheme can be defined as follows:(10)eit=xd−xi,where xd represents the identical speed reference signal of each motor for the multimotor system.The speed synchronization errors of adjacent motors are defined as(11)ε1t=e1t−e2t,⋮εit=eit−ei+1t,⋮εnt=ent−e1t.In general, the larger synchronization errors will be generated at the initial stage of the system or when the external disturbances are encountered. Onceεit→0, 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(12)e1εt=ε1t−εnt,⋮eiεt=εit−εi−1t,⋮enεt=εnt−εn−1t.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,ei→0,i=1,…,n;
(2)
each motor can synchronize other motors, that is,εi→0,i=1,…,n.


## 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 Figure2.Figure 2
Block diagram of the controller for each motor.


### 3.1. Enhanced Adjacent Coupling Scheme

An enhanced adjacent coupling scheme is developed for the multimotor system (total number of motors isn) 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.Figure 3
Simplified structure of synchronization control.Figure 4
Schematic diagram of enhanced adjacent coupling.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(13)e1∗t=pε1t−qεnt,⋮ei∗t=pεit−qεi−1t,⋮en∗t=pεnt−qεn−1t,where p>0 and q>0 and satisfying pn≠qn.Let(14)E=A∗ε,where
(15)A=p0⋯⋯0−q⋮⋱00−qp⋯⋯⋮⋮⋯−qp⋮⋮⋮⋮⋱00⋯⋯0−qp,ε=ε1t⋮εit⋮εnt,E=e1∗t⋮ei∗t⋮en∗t.The above matrixA can be transformed into the following triangular form by a reasonable equivalent transformation
(16)A=p0⋯⋯0−q⋮⋱⋮0⋯p⋯⋯⋮0⋯0p−qipi−1⋮⋮⋱⋮0⋯00p−qnpn−1..It can be obtained that (14) has a unique solution when A is a full-rank matrix, that is, pn≠qn. Once that condition E converges to a minimal domain is satisfied, then ε converges to a minimal domain, which means that the synchronization control objective can be transformed to design the speed synchronization controller to ensure that E 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. Figure5 shows the detailed schematic diagram of FADRC.Figure 5
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(17)η0i=v1i−xd,v̇1i=−rfalη0,α,δ,where v1i denotes the output of TD, which indicates the tracking of the identical speed reference signal xd; η0i represents the error between v1i and xd; a>0, δ>0 are the tuning parameters; fal. indicates a nonlinear function and is expressed as
(18)falη,α,δ=ηδ1−α,η≤δ,ηαsignη,η>δ.Secondly, the ESO is designed as(19)ηi=z1i−xi,ż1i=z2i−β1falη,a,δ+Aiuti+Biz1i,ż2i=−β2falη,a,δ,where z1i is the observation value of xi and z2i=Di is the lumped disturbance; ηi is the observation error of xi; β1>0 and β2>0 are the tuning gains.Remark 1.
According to (18), the nonlinear function fal(.) will produce a larger gain when observation error ηi is small; on the contrary, fal(.) will produce a smaller gain when ηi is large. Reference [40] pointed out that fal(.) could stabilize the observation state by choosing the appropriate size of βi (i = 1,2), which means that xi−z1i≤d1,Di−z2i≤d2, where d1 and d2 are all small positive constants.Finally, the NLSEF is designed as(20)η1i=v1i−z1i,u0i=β3falη1i,a,δ,uti=u0i−z2ib0,where u0i and uti indicate the control input without considering disturbance and speed tracking control signal with considering disturbance, respectively, of the ith motor; η1i denotes the error between the output v1i of TD and the output z1i of ESO; b0>0 is a constant, which represents the estimated value of Ai 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(21)e01=ηi,e02=η̇i.In practical applications, the coefficients of ESO,βi,i=1,2, 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 e01 and e02 are utilized by the fuzzy logic rules to determine the coefficients βi, which can be optimally adjusted online.The error variablese01 and e02 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 Δβi. And a Gaussian function and a triangle function are chosen as the input e01, e02 membership function and the output Δβi (i = 1, 2) membership function, respectively. In this subsection, the basic universe of e01 and the basic universe of e02 are [−1, +1] and [−0.5, +0.5], and Δβi 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.Table 1
Fuzzy rules ofΔβii=1,2.
e01
e02
PB
PS
ZO
NB
NS
PB
PB/NB
PS/NS
PS/NS
ZO/ZO
PS/NS
PS
PB/NS
PS/NS
PS/NS
NS/PS
ZO/ZO
ZO
PS/NS
PS/NS
ZO/ZO
NS/PS
NS/PS
NB
ZO/ZO
NS/PS
NS/PS
NB/PB
NS/PS
NS
PS/NS
ZO/ZO
NS/PS
NB/PS
NS/PSAccording to Table1 and after fuzzy inference and defuzzification, the coefficient βi is rectified as
(22)β1=β1′+Δβ1,β2=β2′+Δβ2,where βi′ and Δβi 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(23)Sit=ei∗+λ∫0tei∗τdτ,where λ>0 denotes a tuning parameter. From (8) and (23), the general sliding mode surface controller based on ESO is given as
(24)usi′=1Ai−z2i+Bixi+pp+qẋi+1+qp+qẋi−1+λp+qei+1∗−lisignSi,where li is the control gain that satisfies that li≥d2≥0. 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 ofli since the upper bounds of estimation errors are not easy to obtain. Therefore, in the following, an AISMC combined with ESO is designed as
(25)usi=1Ai−z2i−Bixi+pp+qẋi+1+qp+qẋi−1+λp+qei∗−l^isignSi,where the adaptive law of l^i is
(26)l^̇i=σmSisignSi−ε,l^i>σ,σ,l^i≤σ,where σm>0 and σ>0 are used to guarantee that l^i>0.Substituting (25) into (8), we have
(27)ẋit=−z2i+pp+qẋi+1+qp+qẋi−1+λp+qei∗−l^isignSi+Di.


## 3.1. Enhanced Adjacent Coupling Scheme

An enhanced adjacent coupling scheme is developed for the multimotor system (total number of motors isn) 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.Figure 3
Simplified structure of synchronization control.Figure 4
Schematic diagram of enhanced adjacent coupling.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(13)e1∗t=pε1t−qεnt,⋮ei∗t=pεit−qεi−1t,⋮en∗t=pεnt−qεn−1t,where p>0 and q>0 and satisfying pn≠qn.Let(14)E=A∗ε,where
(15)A=p0⋯⋯0−q⋮⋱00−qp⋯⋯⋮⋮⋯−qp⋮⋮⋮⋮⋱00⋯⋯0−qp,ε=ε1t⋮εit⋮εnt,E=e1∗t⋮ei∗t⋮en∗t.The above matrixA can be transformed into the following triangular form by a reasonable equivalent transformation
(16)A=p0⋯⋯0−q⋮⋱⋮0⋯p⋯⋯⋮0⋯0p−qipi−1⋮⋮⋱⋮0⋯00p−qnpn−1..It can be obtained that (14) has a unique solution when A is a full-rank matrix, that is, pn≠qn. Once that condition E converges to a minimal domain is satisfied, then ε converges to a minimal domain, which means that the synchronization control objective can be transformed to design the speed synchronization controller to ensure that E 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. Figure5 shows the detailed schematic diagram of FADRC.Figure 5
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(17)η0i=v1i−xd,v̇1i=−rfalη0,α,δ,where v1i denotes the output of TD, which indicates the tracking of the identical speed reference signal xd; η0i represents the error between v1i and xd; a>0, δ>0 are the tuning parameters; fal. indicates a nonlinear function and is expressed as
(18)falη,α,δ=ηδ1−α,η≤δ,ηαsignη,η>δ.Secondly, the ESO is designed as(19)ηi=z1i−xi,ż1i=z2i−β1falη,a,δ+Aiuti+Biz1i,ż2i=−β2falη,a,δ,where z1i is the observation value of xi and z2i=Di is the lumped disturbance; ηi is the observation error of xi; β1>0 and β2>0 are the tuning gains.Remark 1.
According to (18), the nonlinear function fal(.) will produce a larger gain when observation error ηi is small; on the contrary, fal(.) will produce a smaller gain when ηi is large. Reference [40] pointed out that fal(.) could stabilize the observation state by choosing the appropriate size of βi (i = 1,2), which means that xi−z1i≤d1,Di−z2i≤d2, where d1 and d2 are all small positive constants.Finally, the NLSEF is designed as(20)η1i=v1i−z1i,u0i=β3falη1i,a,δ,uti=u0i−z2ib0,where u0i and uti indicate the control input without considering disturbance and speed tracking control signal with considering disturbance, respectively, of the ith motor; η1i denotes the error between the output v1i of TD and the output z1i of ESO; b0>0 is a constant, which represents the estimated value of Ai 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(21)e01=ηi,e02=η̇i.In practical applications, the coefficients of ESO,βi,i=1,2, 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 e01 and e02 are utilized by the fuzzy logic rules to determine the coefficients βi, which can be optimally adjusted online.The error variablese01 and e02 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 Δβi. And a Gaussian function and a triangle function are chosen as the input e01, e02 membership function and the output Δβi (i = 1, 2) membership function, respectively. In this subsection, the basic universe of e01 and the basic universe of e02 are [−1, +1] and [−0.5, +0.5], and Δβi 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.Table 1
Fuzzy rules ofΔβii=1,2.
e01
e02
PB
PS
ZO
NB
NS
PB
PB/NB
PS/NS
PS/NS
ZO/ZO
PS/NS
PS
PB/NS
PS/NS
PS/NS
NS/PS
ZO/ZO
ZO
PS/NS
PS/NS
ZO/ZO
NS/PS
NS/PS
NB
ZO/ZO
NS/PS
NS/PS
NB/PB
NS/PS
NS
PS/NS
ZO/ZO
NS/PS
NB/PS
NS/PSAccording to Table1 and after fuzzy inference and defuzzification, the coefficient βi is rectified as
(22)β1=β1′+Δβ1,β2=β2′+Δβ2,where βi′ and Δβi 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(23)Sit=ei∗+λ∫0tei∗τdτ,where λ>0 denotes a tuning parameter. From (8) and (23), the general sliding mode surface controller based on ESO is given as
(24)usi′=1Ai−z2i+Bixi+pp+qẋi+1+qp+qẋi−1+λp+qei+1∗−lisignSi,where li is the control gain that satisfies that li≥d2≥0. 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 ofli since the upper bounds of estimation errors are not easy to obtain. Therefore, in the following, an AISMC combined with ESO is designed as
(25)usi=1Ai−z2i−Bixi+pp+qẋi+1+qp+qẋi−1+λp+qei∗−l^isignSi,where the adaptive law of l^i is
(26)l^̇i=σmSisignSi−ε,l^i>σ,σ,l^i≤σ,where σm>0 and σ>0 are used to guarantee that l^i>0.Substituting (25) into (8), we have
(27)ẋit=−z2i+pp+qẋi+1+qp+qẋi−1+λp+qei∗−l^isignSi+Di.


## 4. Stability Analysis

Lemma 1 (see [42]).
For the given nonlinear uncertainty system (8) with the sliding mode (23), the control gain l^it has an upper bound, which satisfies that l^it≤li.Lemma 2 (see [43]).
Suppose that there is a continuous and positive definite functionVt, which satisfies the follow differential equation:
(28)V̇t≤−α0Vξ,Vt0>0,where α0>0 and 0<ξ<1 are positive constants. Then for any given t0, there is a finite time t1 that satisfies
(29)V1−ξt≤V1−ξt0−α01−ξt−t0,t0≤t≤t1,Vt≡0,∀t≥t1,where t1=t0+V1−ξt0/α01−ξ.Theorem 1.
Considering multimotor system (8), integral sliding mode surface (23), and controller (25), the integral sliding mode surface Si can converge to be uniformly ultimately bounded in finite time and the synchronization error can stably converge to be uniformly ultimately bounded when ρ≤σmε/kα+ε with ϵ and ka being two positive constants.Proof 1.
Considering system (8), the Lyapunov function is designed
(30)Vi=12Si2+12ρl~i2,where l~i=l^i−li.Differentiating (30) and substituting into (13), we can obtain
(31)V̇i=Si∗Ṡi+1ρl~il~̇i=Siėi∗t+λei∗t+1ρl~il~̇i=Sipėit−ėi+1t−qėi−1t−ėit+λei∗+1ρl~il~̇i.Substituting (10) into (31), we have
(32)V̇i=Sipẋi+1t−pẋti+qẋi−1t−qẋit+λei∗+1ρl~il~̇i=Sipẋi+1t−p+qẋit+qẋi−1t+λei∗+1ρl~il~̇i.Substituting (27) into (32), then
(33)V̇i=Si−z2i−lsignSi+Dit+1ρl^i−lil^̇i=SiDit−z2i−l^isignSi+1ρl^i−lil^̇i≤SiDit−z2i−l^iSi+liSi−liSi+1ρl^i−lil^̇i≤−li−d2Si−l^i−liSi+1ρl^i−lil^̇i=−li−d2Si+l^i−li1ρl^̇i−Si.Introducing parameterka>0, (33) can be equivalently transformed into
(34)V̇i=−li−d2Si−l^i−liSi−1ρl^̇i+kal^i−li−kal^i−li.According to Lemma1, we can obtain
(35)V̇i≤kbSi−ϱ−kal^i−li,where kb=li−d2>0, ϱ=1/ρl^̇i−Si−kal^i−li. Ultimately, the derivative of Vi is
(36)V̇i≤kbSi−ϱ−kal^i−li=−2kbSi2−2ρkal^i−li2ρ−ϱ≤−kmV1/2−ϱ,where km=min2kb,2ρka.Case 1.
IfSi>ϵ, then according to (26), we have
(37)l^̇i=σmSi,ϱ=1ρσmSi−Si−kal^i−li.
When the parameterρ is selected to satisfy ρ≤σmε/kα+ε, we can conclude from (37) that ϱ>0, which leads to V̇i≤−kmVi1/2 from (36).Case 2.
Supposing thatSi≤ϵ, we have the function ϱ in (35) which is negative and l^̇i is also negative. So, l^i will decrease until less than d2. From (33), we can conclude that V̇i will be positive, which means that Si will increase until larger than ϵ. Therefore, the case becomes the same as Case 1, in which V̇i≤−kmVi1/2 can be finally guaranteed by choosing the appropriate parameter ρ.From Case1 and Case 2, we have V̇i≤−kmVi1/2. According to Lemma 2, the sliding surface Si could be guaranteed to converge to the domain Si≤ϵ in finite time. According to V̇i≤−kmVi1/2, we have SiṠi≤2/2kmSi so Ṡi≤2/2km. Differentiating both sides of (23), we have ėi∗+λei∗≤2/2km and the solution is ei∗≤e−λt+ϵ, which means that ei∗ converges to domain ei∗≤2/2km when t→+∞. According to (14), we can conclude that εi 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
(38)ui=uti+usi=u0i−z2ib0+Bixi+pẋi+1/p+q+qẋi−1/p+q+λei∗/p+q−lisignSiA.Remark 2.
To further improve the control performances, the functionsign•, which may cause sliding mode chattering, is replaced by a saturation function sat•
(39)satSi=Siξ,Si<ξ,1,Si≥ξ,−1,Si≤−ξ,where ξ 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 Table2. 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.Table 2
Parameters of four motors.
Parameter
Motor 1
Motor 2
Motor 3
Motor 4
RSΩ
2.875
2.875
2.88
2.86
LdH
0.0085
0.0085
0.0085
0.0085
LqH
0.0085
0.0085
0.0085
0.0085
ψfWb
0.067
0.072
0.076
0.068
Jkg∙m2
0.008
0.0083
0.0073
0.0065
BN∙m∙s
0.00051
0.00047
0.00056
0.00061
mkg
1
1
1
1


### 5.1.M1: The Proposed Method

The parameters in tracking controllers (17), (18), and (19) are a=0.3, δ=0.5, b0=50, β1′=5000, β2′=50000, and r=2000; the parameters in synchronization controller (25) are p=2, q=1, λ=30, l=100, and ξ=0.5; the parameters of adaptive law (26) are set as σm=0.15 and σ=0.01.


### 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 a=0.3, δ=0.5, b0=50, β1′=5000, β2′=50000, β3′=500, and r=2000.


### 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.
(40)uti∗=1Aiẋd−Bixi+lisignSi+λei,usi∗=1Ai−Bixi+pp+qẋi+1+qp+qẋi−1+λp+qei∗−lisignSi,where li=500, i=1,…,4 and p,q,λ,Ai,Bi 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 Figure6. 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 xd=1000r/min. 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, ei,j,i,j=1,…,4 and i≠j, indicates the synchronization error between the ith motor and jth 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.Figure 6
Speed synchronization performance comparison.
(a)
Synchronization performances of M1
(b)
Synchronization performances of M2
(c)
Synchronization performances of M3Figure 7
Synchronization error comparison.
(a)
Synchronization errorse1,2
(b)
Synchronization errorse2,3
(c)
Synchronization errorse3,4
(d)
Synchronization errorse4,1Figure 8
Tracking error comparison.
(a)
Tracking errorse1
(b)
Tracking errorse2
(c)
Tracking errorse3
(d)
Tracking errorse4Figure 9
Control inputs.
(a)
Control inputu1
(b)
Control inputu2All the parameters are chosen the same as those in the former case. The speed synchronization performance comparison of four motors is shown in Figure10. 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.Figure 10
Speed synchronization and tracking performances.
(a)
Synchronization performances of M1
(b)
Synchronization performances of M2
(c)
Synchronization performances of M3Figure 11
Synchronization error comparison.
(a)
Synchronization errorse1,2
(b)
Synchronization errorse2,3
(c)
Synchronization errorse3,4
(d)
Synchronization errorse4,1Figure 12
Tracking error comparison.
(a)
Tracking errorse1
(b)
Tracking errorse2
(c)
Tracking errorse3
(d)
Tracking errorse4Figure 13
Control inputs.
(a)
Control inputu1
(b)
Control inputu2From 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.


## 5.1.M1: The Proposed Method

The parameters in tracking controllers (17), (18), and (19) are a=0.3, δ=0.5, b0=50, β1′=5000, β2′=50000, and r=2000; the parameters in synchronization controller (25) are p=2, q=1, λ=30, l=100, and ξ=0.5; the parameters of adaptive law (26) are set as σm=0.15 and σ=0.01.


## 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 a=0.3, δ=0.5, b0=50, β1′=5000, β2′=50000, β3′=500, and r=2000.


## 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.
(40)uti∗=1Aiẋd−Bixi+lisignSi+λei,usi∗=1Ai−Bixi+pp+qẋi+1+qp+qẋi−1+λp+qei∗−lisignSi,where li=500, i=1,…,4 and p,q,λ,Ai,Bi 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 Figure6. 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 xd=1000r/min. 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, ei,j,i,j=1,…,4 and i≠j, indicates the synchronization error between the ith motor and jth 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.Figure 6
Speed synchronization performance comparison.
(a)
Synchronization performances of M1
(b)
Synchronization performances of M2
(c)
Synchronization performances of M3Figure 7
Synchronization error comparison.
(a)
Synchronization errorse1,2
(b)
Synchronization errorse2,3
(c)
Synchronization errorse3,4
(d)
Synchronization errorse4,1Figure 8
Tracking error comparison.
(a)
Tracking errorse1
(b)
Tracking errorse2
(c)
Tracking errorse3
(d)
Tracking errorse4Figure 9
Control inputs.
(a)
Control inputu1
(b)
Control inputu2All the parameters are chosen the same as those in the former case. The speed synchronization performance comparison of four motors is shown in Figure10. 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.Figure 10
Speed synchronization and tracking performances.
(a)
Synchronization performances of M1
(b)
Synchronization performances of M2
(c)
Synchronization performances of M3Figure 11
Synchronization error comparison.
(a)
Synchronization errorse1,2
(b)
Synchronization errorse2,3
(c)
Synchronization errorse3,4
(d)
Synchronization errorse4,1Figure 12
Tracking error comparison.
(a)
Tracking errorse1
(b)
Tracking errorse2
(c)
Tracking errorse3
(d)
Tracking errorse4Figure 13
Control inputs.
(a)
Control inputu1
(b)
Control inputu2From 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.



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*Source: 5632939-2018-09-09.xml*