# Collaborative Control of Multimotor Systems for Fixed-Time Optimisation Based on Virtual Main-Axis Speed Compensation Structure

Authors: Changfan Zhang; Mingjie Xiao; Jing He; Zhitian Liu; Xingxing Yang; Qian Zhang; Hongrun Chen
Journal: Complexity
 (2021)

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## Abstract

In response to the high-speed and high-precision collaborative control requirements of the multimotor system for filling, a new type of virtual master-axis control structure is proposed and a multimotor fixed-time optimized collaborative control algorithm is designed. Firstly, coupling relationship between virtual and slave motors is effectively established by designing a velocity compensation module for the virtual motor. Secondly, the sliding mode observer (SMO) is used to reconstruct the composite disturbance composed of motor parameter perturbation and load disturbance. Finally, the variable gain terminal sliding mode controller (SMC) is designed to ensure that each slave motor can track the given value within a fixed time. The fast convergence of the system can be proved by the fixed-time convergence theorem and Lyapunov’s stability theorem. The simulation results show that, compared with the traditional virtual main-axis control strategy, the proposed method is more effective for the tracking control of each slave motor in the initial stage.


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## Body

## 1. Introduction

High-performance collaborative control of filling multimachine is one of the key technologies in the research and development of thick sauce and viscous food packaging equipment. It is also an important embodiment of high-speed and high-precision filling operation. At present, studies on filling multimachine collaborative control technologies usually focus on synchronous control of multimotor systems [1, 2]. In the multimotor synchronous control, the synchronous topology and control algorithm are two aspects worthy of attention. Therefore, the following is a summary of domestic and foreign research status from these two aspects.Main multimotor synchronous control strategies for different synchronous topology structures are master-slave synchronous control [3], adjacent coupling synchronous control [4], relative coupling synchronous control [5], and virtual main-axis synchronous controls [6]. Yao et al. redefined adjacent cross-coupling error and designed auto disturbance rejection controller by extracting the Laplacian matrix [7]. Xu et al. proposed an optimal control strategy based on combined cross-coupling errors against various coupling errors and disturbance in multilayer and multiaxis control systems [8]. Huang et al. combined adjacent cross-coupling control strategy with adaptive algorithm to ease jitters when the speed is stable by considering phase and speed synchronisation control of multiactuators in the oscillation system [9]. Sun et al. combined fuzzy control technology with ring coupling synchronous structure to achieve the reliable operation of multimotor driving system [10]. Tao et al. proposed an improved adjacent coupling synchronous control strategy to simplify the structure of velocity synchronous control by introducing a new coupling coefficient in the traditional adjacent coupling method [11]. Shi et al. put forward an improved multimotor relative coupling synchronous control, which also decoupled tracking and synchronisation errors of the system [12]. Zhang et al. mainly improved the relative coupling synchronization structure and then realized the smooth operation of the intelligent filling equipment for sticky food under complex working conditions [13]. Xu et al. investigated the adaptive synchronisation of coupled harmonic oscillator with switching topology and proposed an edge-based servo synchronous adaptive control protocol [14]. These control strategies promote the development of multimotor synchronous control technologies. However, the virtual main-axis control structure exerts a strong control effect on the application of multimotor synchronous technology with a unique feedback mechanism and characteristics of virtual motor as the navigator. Zhang and He explored a multimotor virtual main-axis control strategy based on the observer to reflect the dynamic relationship between virtual and slave motors accurately [15]. Lin and Cai estimated the composite disturbance of the motor using sliding mode observer to ensure that slave motors reach high synchronisation accuracy [16]. He and Chen focused on the total consensus control in multimotor cooperation to ensure that the total power of the system remains constant under power loss of a single motor [17].The synchronous control algorithm mainly includes multiagent consensus [18, 19] and sliding mode variable structure control [20, 21] algorithms due to different design ideas of multimotor synchronous control. Xu and Li evaluated the multiagent consensus algorithm and proposed a semiglobal consensus control protocol based on low-gain state feedback with consideration for the intermittent semiglobal consensus problem of high-order multiagent system under input saturation [22]. Wang et al. analysed the structure of distributed observer of continuous linear time-invariant system and reduced the dependence of the system on topology structure information using adaptive rule [23]. Wang and Su investigated the collaborative control of linear time-invariant network system with uncertainty and constrained states of all agents by constructing an interval observer [24]. Liu and Su explored containment control of multiagent system based on data information of intermittent sampling position and obtained necessary and sufficient conditions for containment in the case of time delay and delay free [25]. Liu and Su further examined necessary and sufficient conditions, including control protocol under the condition of time delay, and effectively solved the problem of using the second-order multiagent system to control with only sampling position data [26]. Zheng et al. proposed an optimal operation model for microgrid based on the multiagent system to save the entire energy cost of locally observable convex function [27]. Fan and Wang assessed problems on following the consensus of a second-order multiagent system with input saturation and proved the stable consistency of the multiagent system using adaptive control technology [28]. Zhai et al. proposed a new pulse delay control method based on sampled data to realise the pulse synchronisation of multiagent systems [29]. Liu et al. examined the group controllability of the multiagent system, proposed a general definition of group controllability, and established a correlation criterion of group controllability [30]. In the research of sliding mode variable structure control algorithm, Shen et al. developed a trajectory tracking control scheme for the Mars orbiter and proposed a method of adaptive fixed-time control [31]. Shao et al. proposed a state observer with two polynomial feedback terms to identify unmeasurable speed states and unknown uncertain disturbances. The designed speedless fixed-time controller integrates the obtained observations [32]. Sun and Wang designed a sliding mode controller for angle tracking control of wearable exoskeleton based on model-free fixed-time control algorithm [33].Although the multimachine cooperative control technology has been extensively investigated [34, 35], studies on multimachine high-performance collaborative control for thick sauce and viscous food filling are limited. In particular, the viscous characteristics of the filling material greatly hinder the speed and efficiency of the filling, resulting in two technical problems in the high-speed and high-precision operation of the filling multimotor system. (1) Virtual main-axis synchronous control is a common control strategy for filling multimotor production that can effectively improve the synchronisation accuracy of multimachine collaborative operation. The rotation speed of the slave filling conveying motor cannot be fed back to the virtual motor and the insufficient coupling correlation between them are limitations of the current feedback mechanism of the virtual main-axis control structure in terms of the multimotor collaborative control strategy. (2) As one of the effective controllers of the servo system, the sliding mode controller is widely used in various filling production occasions. Approaching speed of the existing sliding mode controller must be improved because its convergence speed demonstrates difficulty in meeting the requirements of high-speed operation of the servo system, especially when the sliding mode operation reaches the sliding surface from any initial position, in terms of the multimachine collaborative control algorithm.Hence, a new multimotor collaborative control method for fixed-time optimisation based on the virtual main-axis structure is proposed in this study to solve these problems and meet the demand of the filling multimotor system for high-performance collaborative control. This study provides the following contributions:(1)
A speed compensator is added to the virtual motor to improve the existing virtual main-axis control structure, and the coupling relationship between virtual and slave motors in the speed is established given that the information on the velocity of slave motors and rotor position cannot be fed back to the virtual motor in the traditional virtual main-axis structure.(2)
A new type of variable gain terminal SMC is designed to realize the high-speed and high-precision synchronous control of the filling multimotor system. It mainly solves the problem of slow convergence speed when the system error reaches the terminal sliding surface from the initial state.This paper is organized as follows: by taking advantage of the existing virtual main shaft synchronization control structure, a speed compensator is added to the virtual motor and the synchronization topology is improved in Section2. The multimotor system model is established in Section 3. The sliding mode observer is used to overcome the parameter perturbation and load changes of the multimotor system in Section 4. The system controller is designed to ensure that the speed of each slave motor can quickly track the given value in Section 5. The experimental simulation is presented in Section 6. Finally, the conclusion is given in Section 7.


## 2. Improved Virtual Main Shaft Control Structure Design

A virtual main-axis control structure with speed compensator is designed for the multimotor synchronous technology in this section on the basis of fully absorbing the characteristics and principles of the existing virtual main-axis strategy. Specific details of each module of this control structure designed are shown in Figure1.Figure 1
The schematic diagram of specific details of each module of this control structure.Figure1 shows the permanent magnet synchronous motor (PMSM) model, virtual motor, tracking error of permanent magnet synchronous motor, sliding mode function, controller module, virtual controller module, observer module, and virtual motor speed compensator as well as detailed formulas, logical relations, and related variables of each module. Compared with the diagrams of various modules shown in Figure 1, the golden area is the proposed virtual motor speed compensator designed in this study. The compensation signal generated by the virtual motor compensator can be directly fed back to the virtual motor and can also be adjusted by selecting a suitable virtual motor’s moment of inertia. Although the compensation signal is known and available, ωvir−ωj are bounded variables. Therefore, the output value of the virtual motor speed compensator is regarded as a variable with a known upper bound that can reduce the design difficulty and calculation of the virtual motor controller. The blue area denotes the designed variable-gain terminal SMC in this study. A dynamic switching gain term, including the sliding mode function, is designed to improve the convergence speed of the sliding mode movement because the tracking error of each slave motors is large in the initial state.


## 3. Establishment of the Multimotor System Model

The PMSM of the driving system for filling production is the research object of this study. The second-order kinetic model of motorj in the multimotor system can be expressed as follows [13]:(1)θ˙j=ωj,ω˙j=−ajωj+bjuj−pjTLj,j=1,2,…,n,where aj=RΩj/Jj; bj=3/2⋅npj2ψfj/Jj; pj=npj/Jj; uj=iqj; θj and wjare the position and electrical angular velocity of the motor rotor, respectively; TLj is the load torque on the shaft; RΩj is the rotational resistance coefficient; nPj is the number of pole pairs; Jj is the moment of inertia; Ψfj is the rotor flux; iqj is the q-axis components of the stator current; and uj is the controller to be designed.The state equation of virtual motor based on formula (1) is expressed as follows:(2)θ˙vir=ωvir,ω˙vir=−avirωvir+bviruvir−pvirTLvir,where avir=RΩvir/Jvir; bvir=3/2⋅npvir2ψfvir/Jvir; pvir=npvir/Jvir; uvir=iqvir; θvir and ωvir are the virtual motor rotor position and electrical angular velocity, respectively; TLvir is the virtual motor load torque; RΩvir is the virtual motor rotational resistance coefficient of motor; npvir is the number of pole pairs of virtual motor; Jvir is the virtual motor moment of inertia; ψfvir is the virtual motor rotor flux; iqvir is the virtual motor q-axis components of the stator current; and uviris the virtual controller to be designed.Motor parameters will be subjected to uncertain perturbation during long-time operation of the motor, and the motor load torque will change uncertainly due to the change of filling materials in the filling process. Hence, formula (1) can be transformed into the following:(3)θ˙j=ωj,ω˙j=−a¯j+Δajωj+b¯j+Δbjuj−pjΔTLj,j=1,2,…,n,where a¯j,b¯j refer to nominal values of system parameters, Δaj,Δbj refer to perturbation values of system parameters, and ΔTLj refers to the variation of load torque of the filling motor. These variables are considered unknown composite disturbances dj. Hence, formula (3) can be modified as follows:(4)θ˙j=ωj,ω˙j=−a¯jωj+b¯juj+dj,j=1,2,…,n,where dj=−Δajωj+Δbjuj−pjΔTLj.Define the vector form of each state variable asθ=θ1,…,θj,…,θnT; ω=ω1,…,ωj,…,ωnT; u=u1,…,uj,…,unT; and d=d1,…,dj,…,dnT.Formula (4) is modified in the following matrix form:(5)θ˙=ω,ω˙=−aω+bu+d,where a=diaga¯1,…,a¯j,…,a¯n and b=diagb¯1,…,b¯j,…,b¯n are coefficient matrices.


## 4. Sliding Mode Observer Design and Stability Analysis

The composite disturbancedj of the motor system included in formula (4) will exert a strong influence on the synchronous performance. Therefore, the SMO will be introduced in this section to explore dj.The observer is designed for system (4) as follows [36]:(6)θ^˙j=ω^j+k1sgnθj−θ^j,ω^˙j=−a¯jωj+b¯juj+k2sgnωj−ω^j,j=1,2,…,n,where θ^j and ω^j refer to observed values of rotor position and speed of motor j, respectively, and k1,k2 refer to the parameters to be designed.The observational error vector is defined ase˜j=θ˜jω˜jT=θj−θ^jωj−ω^jT, and θ˜j,ω˜j refer to observational errors of the rotor position and speed of motor j, respectively.Hypothesis 1.
A normal numberdjmax and dj≤djmax exist given that perturbation values of motor parameters Δaj,Δbj and load torque variation ΔTLj are bounded in practice.Theorem 1.
Observer (6) corresponding to motor j is designed for filling multimotor system (4). If k1>ω˜jmax+η1,k2>djmax+η2, j=1,2,…,n is satisfied, then the observational error can converge to 0 in a limited time Te˜j. The composite disturbance dj can be expressed as follows:(7)limx⟶Te˜jd^j=dj=k2sgnk1sgnθj−θ^j.Proof.
The error equation of the observer system can be obtained when formula (6) is subtracted from formula (4) as follows:(8)θ˜˙j=ω˜j−k1sgnθ˜j,ω˜˙j=dj−k2sgnω˜j.
The selected sliding mode surface of the SMO is expressed as follows:(9)se˜j=e˜j.
Take the derivative of the above formula and substitute formula (8) into it to get(10)s˙e˜j=e˜˙j=θ˜˙jω˜˙jT=ω˜j−k1sgnθ˜jdj−k2sgnω˜jT.
The Lyapunov function of the observer is expressed as follows:(11)Ve˜j=12e˜jTe˜j.
The following equation can be derived from formula (11):(12)V˙e˜j=e˜jTe˜˙j=θ˜jω˜j−k1sgnθ˜j+ω˜jdj−k2sgnω˜j=θ˜jω˜j−k1θ˜j+ω˜jdj−k2ω˜j≤θ˜jω˜j−k1+ω˜jdj−k2≤−mink1−ω˜j,k2−dje˜j.
If the designed parameters meetk1>ω˜jmax+η1,k2>djmax+η2 and η1,η2>0, then the following equation can be obtained:(13)V˙e˜j≤−minη1,η2e˜j=−ηe˜j,where η=minη1,η2.
Therefore, ifθ˜j⟶0,ω˜j⟶0, then V˙e˜j≤0 and Ve˜j⟶0. The observer system (6) at this moment demonstrates asymptotic stability. According to Hypothesis 1, the time when the observer system reaches the sliding surface is a limited value Te˜j. The following conclusions can be obtained from formulas (9) and (10) and the principle [37] of sliding mode equivalence when t>Te˜j:(14)se˜j=0s˙e˜j=0⇒θj=θ^j,ω˜j=k1sgnθ˜j=k1sgnθj−θ^j,ωj=ω^j,dj=k2sgnω˜j=k2sgnωj−ω^j.
The unknown composite disturbancedj can then be restructured as follows:(15)limx⟶Te˜jd^j=dj=k2sgnk1sgnθj−θ^j.
Ift>t1 when ∀δ>0,∃t1<Te˜j, then(16)d^j−dj<δ.
Hence, the stability of the SMO has been proven.


## 5. System Controller Design and Stability Analysis

Definition 1 (see [38]).
diagx refers to the diagonal matrix of Vector x for any vector x=x1,x2,…,xnT∈Rn and expressed as follows:(17)xa=x1a,x2a,…,xnaT,xp=x1a,x2a,…,xnaT,sgnx=sgnx1,sgnx2,…,sgnxnT,sigxa=sgnx1x1a,sgnx2x2a,…,sgnxnxnaT,where sgn· represents the standard symbolic function and a refers to an arbitrary real number.Lemma 1 (see [39]).
Stability theorem of fixed-time) For the system is expressed as follows:(18)x˙t=fxt,f0=0,t0=0,x0≜x0,where xt∈Rn represents the state of the system, f:U⟶Rn refers to a continuous function from the domain containing the original point U to n-dimensional space Rn, 0∈Rn represents the zero vector, and x0 is defined as the initial state.If a positive definite and continuous functionVx exists in the domain U, then(19)V˙x≤−αVxp−βVxg,x∈U\0,where α,β,γ,p, and g are normal numbers that meet p<1 and g>1, respectively, and the system is stable in the fixed time. Hence, the system state can converge to the equilibrium point in a fixed time independent of the initial value.(20)T≤1α1−p+1βg−1.


### 5.1. Design and Stability Analysis of the Virtual Sliding Mode Controller

The virtual controlleruvir can be used to control the velocity of the virtual motor in this study given that the upper bound of the virtual motor speed compensator signal is known. The tracking error eωvir∗ of the speed of the virtual motor is expressed as follows:(21)eωvir∗=ω∗−ωvir.The following design of the virtual motor speed compensator module is added:(22)Δωvir=∑j=1nJvirJjωvir−ωj.This signal can reduce the compensation by appropriately increasing the inertiaJj of the virtual motor to improve the convenience in parameter selection of the virtual controller. Meanwhile, Δωvir is also a bounded variable, that is, Δωvir≤Δωvirmax, because ωvir and ωj are both bounded variables.The second-order system model of the improved virtual main-axis structure can be obtained from formulas (2) and (22) as follows:(23)θ˙vir=ωvir,ω˙vir=−avirωvir+bviruvir−pvirTLvir−Δωvir.The following sliding mode surface of the virtual controller is selected:(24)svir=ceωvir∗,where c refers to the normal number to be designed.The virtual controller is designed as follows:(25)uvir=1bvirω˙∗+avirωvir+pvirTLvir+k3sgnsvir.Theorem 2.
Virtual controller (2) is shown in formula (25). If the coefficient of the control can meet k3>Δωvirmax+η3, then eωvir∗ will gradually converge to 0.Proof.
Define the Lyapunov function of the virtual controller as(26)Vvir=12svir2.
The following equation can be derived from formula (26):(27)V˙vir=svirs˙vir=svirce˙vir∗=svircω˙∗+avirωvir−bviruvir+pvirTLvir+Δωvir.
The following equation can be obtained by substituting formula (25) into formula (27):(28)V˙vir=svirc−k3sgnsvir+Δωvir=−ck3svir−svirΔωvir.
The following can be obtained whenη3 is any small normal number that only guarantees k3>Δωvirmax+η3:(29)V˙vir<0.
Formula (29) and the equivalent principle of sliding mode [37] showed that eωvir∗ converges to 0 when ω∗=ωvir.
Hence, the stability of the virtual motor controller has been proven.


### 5.2. Design and Stability Analysis of the Filling Motor Controller

The tracking error of each slave motor is defined as follows:(30)eθ=γθ−θvir=eθ1,…,eθj,…,eθn1×nT=γ1,virθ1−θvir,…,γj,virθj−θvir,…,γn,virθn−θvir1×nT,(31)eω=e˙θ=γω−ωvir=eω1,…,eωj,…,eωn1×nT=γ1,virω1−ωvir,…,γj,virωj−ωvir,…,γn,virωn−ωvir1×nT,where γ1,vir=J1/Jvir,…,γj,vir=Jj/Jvir,…,γn,vir=Jn/Jvir;γ=diagγ1,vir,⋯,γj,vir,⋯,γn,vir; θvir=θvir,…,θvir,…,θvir1×nT; and ωvir=ωvir,…,ωvir,…,ωvir1×nT.The simplification of this error equation obtained from formulas (30) and (31) can be expressed as follows:(32)E=eθ=γθ−θvir,E˙=e˙θ=eω=γω−ωvir,E¨=e˙ω=γω˙−ω˙vir.The selected terminal sliding surface is expressed as follows [38, 40]:(33)S=E˙+Sau,where(34)Sau=α1sigEq1+β1sigEg1,ifS¯=0orS¯≠0,E>ε,l1sigE+l2sigE2,ifS¯≠0,E≤ε,where S¯=E˙+α1sigEq1+β1sigEg1. α1,β1 refer to the coefficients to be designed that meet α1>1,β1>1. q1 and g1 satisfy q1∈0.5,1 and g1>1, respectively. ε refers to the switching threshold. Sau and S˙au are continuous functions in E=ε. l1=0.5α1εq1−1+0.5β1εg1−1; l2=0.5α1εq1−2+0.5β1εg1−2.This switching rule (34) is inspired by [41, 42] to avoid the singularity.The optimized variable gain terminal SMC is designed as follows:(35)u=−α2S+1sigSq2−β2S+1sigSg2+b−1aω−d^+ω˙vir−γb−1S˙au,where α2,β2 refer to coefficients to be designed that meet α2>1,β2>1, respectively, and q2 and g2 satisfy q2∈0.5,1 and g2>1, respectively.Theorem 3.
The system controller is designed in formula (35). If the coefficient of the controller satisfies α2>1,β2>1, q2∈0.5,1, and g2>1, then the system error (32) in formula (5) will converge to 0 within the fixed-time T2+T3. The upper bound on this time of convergence is(36)T2≤1α2S+1λminI+γb1−θq11−q2+121−g2β2S+1λminI+γb1−θg1g2−1,T3≤1α11−θq21−q1+121−g1β11−θg2g1−1.Proof.
Define the Lyapunov function of the optimized variable gain terminal SMC as(37)VS=STS.
Substituting formulas (32) and (33) into formula (37) gives the following:(38)V˙S=2STS˙=2STγω˙−ω˙vir+S˙au.
Substituting formulas (5) and (35) into formula (38) gives the following:(39)V˙S=2ST−α2S+1γbsigSq2−β2S+1γbsigSg2≤−2α2S+1λminγbs12q2+1/2+s22q2+1/2+⋯+sn2q2+1/2−2β2S+1λminγbs12g2+1/2+s22g2+1/2+⋯+sn2g2+1/2.
The following can be obtained on the basis of the conclusion in [43]:(40)V˙S≤−2α2S+1λminγbVSq2+1/2−22−g2β2S+1λminγbVSg2+1/2.
According to Lemma 1, formula (40) satisfies the fixed-time convergence condition, and its time upper bound is expressed as(41)T2≤1α2S+1λminI+γb1−θq11−q2+121−g2β2S+1λminI+γb1−θg1g2−1.Remark 1.
Formula (41) shows that nonlinear coefficients α1S+1 and β1S+1 satisfy the conditions of fixed-time convergence in the convergence of S.
Assuming thatS=φ exists at the moment Tφ within T2, formula (41) shows the following upper bound of the time when the sliding mode movement moves from an arbitrary initial position to the sliding surface:(42)Tφ=1α2φ+1λminI+γb1−θq11−q2+121−g2β2φ+1λminI+γb1−θg1g2−1.
The comparison of formula (42) with the method proposed in [43] shows that(43)Tφ<1α2λminI+γb1−θq11−q2+121−g2β2λminI+γb1−θg1g2−1.Remark 2.
The comprehensive analysis of formulas (41) and (42) showed that the large gain coefficient of the sig function in the controller accelerates the speed to the sliding surface when the sliding mode movement moves away from S=0. The gain coefficient of the sig function will continuously approach α1 and β1 when the SM movement is close to S=0. Therefore, the actual convergence time will be shortened. In the process of sliding mode movement from the initial state to the sliding surface, the optimized SMC has better convergence performance than the original terminal SMC.
The following can be obtained from the above conclusion and formula (33) when the sliding mode movement reaches the sliding surface:(44)E˙=−Sau.
In the process that the system moves to the equilibrium point on the sliding surface, based on sliding mode substitution method designed in formula (34), the classification should be discussed here.
WhenE>ε, we can get(45)E˙=−α1sigEq1−β1sigEg1.
The Lyapunov function is defined, and the system error is analysed as follows:(46)VE=ETE.
The following can be obtained on the basis of formula (46):(47)V˙E=2ETE˙.
The following can be obtained after substituting formula (45) into formula (47):(48)V˙E=2ET−α1sigEq1−β1sigEg1≤−2α1VEq1+1/2−22−g1β1VEg1+1/2.
According to Lemma1, formula (48) satisfies the fixed-time convergence condition, and the upper bound of time is expressed as follows:(49)T3≤1α11−θq21−q1+121−g1β11−θg2g1−1,where θq2 and θg2 refer to arbitrary normal numbers in the interval 0,1.
Formulas (41) and (49) demonstrate that the system error E will converge to 0 when the sliding mode moves for t>T2+T3.
WhenE>ε, we can get(50)E˙=−l1sigE−l2sigE2.
Similarly, the system error is also analysed as follows:(51)V˙E=2ET−l1sigE−l2sigE2≤−2l1VE.
Hence, formula (51) shows that VE will converge to 0 exponentially given that l1 is a positive real number.Remark 3.
The comprehensive analysis on formulas (45)–(51) shows that the convergence time of the linear sliding surface of Sau=l1sigE+l2sigE2 instead of the terminal sliding mode can be ignored if the value of ε is sufficiently small. The convergence time is approximately equal to T2 in this case. Therefore, system errors E and E˙ can converge to 0 within T2+T3; that is, the velocity of all motors in the multimotor system can achieve ω=ωvir in a fixed time. Moreover, filling motors can track the given value in real time because the velocity of the virtual motor is ωvir=ω∗. Hence, the stability of the filling motor controller has been proven.
The overall control block of the improved virtual main-axis system is shown in Figure2 to understand the content of Figure 1 and design details of each module further.
Figure2 shows that the most important characteristic of the control structure is the addition of the speed compensator to the virtual motor and the use of the virtual controller to provide feedback and adjust the tracking error and compensation signal of the virtual motor. Meanwhile, a variable-gain terminal SMC is designed to realise the high-speed and high-precision tracking of each filling motor speed to the given signal.Figure 2
The control block diagram of the improved virtual main shaft structure.


## 5.1. Design and Stability Analysis of the Virtual Sliding Mode Controller

The virtual controlleruvir can be used to control the velocity of the virtual motor in this study given that the upper bound of the virtual motor speed compensator signal is known. The tracking error eωvir∗ of the speed of the virtual motor is expressed as follows:(21)eωvir∗=ω∗−ωvir.The following design of the virtual motor speed compensator module is added:(22)Δωvir=∑j=1nJvirJjωvir−ωj.This signal can reduce the compensation by appropriately increasing the inertiaJj of the virtual motor to improve the convenience in parameter selection of the virtual controller. Meanwhile, Δωvir is also a bounded variable, that is, Δωvir≤Δωvirmax, because ωvir and ωj are both bounded variables.The second-order system model of the improved virtual main-axis structure can be obtained from formulas (2) and (22) as follows:(23)θ˙vir=ωvir,ω˙vir=−avirωvir+bviruvir−pvirTLvir−Δωvir.The following sliding mode surface of the virtual controller is selected:(24)svir=ceωvir∗,where c refers to the normal number to be designed.The virtual controller is designed as follows:(25)uvir=1bvirω˙∗+avirωvir+pvirTLvir+k3sgnsvir.Theorem 2.
Virtual controller (2) is shown in formula (25). If the coefficient of the control can meet k3>Δωvirmax+η3, then eωvir∗ will gradually converge to 0.Proof.
Define the Lyapunov function of the virtual controller as(26)Vvir=12svir2.
The following equation can be derived from formula (26):(27)V˙vir=svirs˙vir=svirce˙vir∗=svircω˙∗+avirωvir−bviruvir+pvirTLvir+Δωvir.
The following equation can be obtained by substituting formula (25) into formula (27):(28)V˙vir=svirc−k3sgnsvir+Δωvir=−ck3svir−svirΔωvir.
The following can be obtained whenη3 is any small normal number that only guarantees k3>Δωvirmax+η3:(29)V˙vir<0.
Formula (29) and the equivalent principle of sliding mode [37] showed that eωvir∗ converges to 0 when ω∗=ωvir.
Hence, the stability of the virtual motor controller has been proven.


## 5.2. Design and Stability Analysis of the Filling Motor Controller

The tracking error of each slave motor is defined as follows:(30)eθ=γθ−θvir=eθ1,…,eθj,…,eθn1×nT=γ1,virθ1−θvir,…,γj,virθj−θvir,…,γn,virθn−θvir1×nT,(31)eω=e˙θ=γω−ωvir=eω1,…,eωj,…,eωn1×nT=γ1,virω1−ωvir,…,γj,virωj−ωvir,…,γn,virωn−ωvir1×nT,where γ1,vir=J1/Jvir,…,γj,vir=Jj/Jvir,…,γn,vir=Jn/Jvir;γ=diagγ1,vir,⋯,γj,vir,⋯,γn,vir; θvir=θvir,…,θvir,…,θvir1×nT; and ωvir=ωvir,…,ωvir,…,ωvir1×nT.The simplification of this error equation obtained from formulas (30) and (31) can be expressed as follows:(32)E=eθ=γθ−θvir,E˙=e˙θ=eω=γω−ωvir,E¨=e˙ω=γω˙−ω˙vir.The selected terminal sliding surface is expressed as follows [38, 40]:(33)S=E˙+Sau,where(34)Sau=α1sigEq1+β1sigEg1,ifS¯=0orS¯≠0,E>ε,l1sigE+l2sigE2,ifS¯≠0,E≤ε,where S¯=E˙+α1sigEq1+β1sigEg1. α1,β1 refer to the coefficients to be designed that meet α1>1,β1>1. q1 and g1 satisfy q1∈0.5,1 and g1>1, respectively. ε refers to the switching threshold. Sau and S˙au are continuous functions in E=ε. l1=0.5α1εq1−1+0.5β1εg1−1; l2=0.5α1εq1−2+0.5β1εg1−2.This switching rule (34) is inspired by [41, 42] to avoid the singularity.The optimized variable gain terminal SMC is designed as follows:(35)u=−α2S+1sigSq2−β2S+1sigSg2+b−1aω−d^+ω˙vir−γb−1S˙au,where α2,β2 refer to coefficients to be designed that meet α2>1,β2>1, respectively, and q2 and g2 satisfy q2∈0.5,1 and g2>1, respectively.Theorem 3.
The system controller is designed in formula (35). If the coefficient of the controller satisfies α2>1,β2>1, q2∈0.5,1, and g2>1, then the system error (32) in formula (5) will converge to 0 within the fixed-time T2+T3. The upper bound on this time of convergence is(36)T2≤1α2S+1λminI+γb1−θq11−q2+121−g2β2S+1λminI+γb1−θg1g2−1,T3≤1α11−θq21−q1+121−g1β11−θg2g1−1.Proof.
Define the Lyapunov function of the optimized variable gain terminal SMC as(37)VS=STS.
Substituting formulas (32) and (33) into formula (37) gives the following:(38)V˙S=2STS˙=2STγω˙−ω˙vir+S˙au.
Substituting formulas (5) and (35) into formula (38) gives the following:(39)V˙S=2ST−α2S+1γbsigSq2−β2S+1γbsigSg2≤−2α2S+1λminγbs12q2+1/2+s22q2+1/2+⋯+sn2q2+1/2−2β2S+1λminγbs12g2+1/2+s22g2+1/2+⋯+sn2g2+1/2.
The following can be obtained on the basis of the conclusion in [43]:(40)V˙S≤−2α2S+1λminγbVSq2+1/2−22−g2β2S+1λminγbVSg2+1/2.
According to Lemma 1, formula (40) satisfies the fixed-time convergence condition, and its time upper bound is expressed as(41)T2≤1α2S+1λminI+γb1−θq11−q2+121−g2β2S+1λminI+γb1−θg1g2−1.Remark 1.
Formula (41) shows that nonlinear coefficients α1S+1 and β1S+1 satisfy the conditions of fixed-time convergence in the convergence of S.
Assuming thatS=φ exists at the moment Tφ within T2, formula (41) shows the following upper bound of the time when the sliding mode movement moves from an arbitrary initial position to the sliding surface:(42)Tφ=1α2φ+1λminI+γb1−θq11−q2+121−g2β2φ+1λminI+γb1−θg1g2−1.
The comparison of formula (42) with the method proposed in [43] shows that(43)Tφ<1α2λminI+γb1−θq11−q2+121−g2β2λminI+γb1−θg1g2−1.Remark 2.
The comprehensive analysis of formulas (41) and (42) showed that the large gain coefficient of the sig function in the controller accelerates the speed to the sliding surface when the sliding mode movement moves away from S=0. The gain coefficient of the sig function will continuously approach α1 and β1 when the SM movement is close to S=0. Therefore, the actual convergence time will be shortened. In the process of sliding mode movement from the initial state to the sliding surface, the optimized SMC has better convergence performance than the original terminal SMC.
The following can be obtained from the above conclusion and formula (33) when the sliding mode movement reaches the sliding surface:(44)E˙=−Sau.
In the process that the system moves to the equilibrium point on the sliding surface, based on sliding mode substitution method designed in formula (34), the classification should be discussed here.
WhenE>ε, we can get(45)E˙=−α1sigEq1−β1sigEg1.
The Lyapunov function is defined, and the system error is analysed as follows:(46)VE=ETE.
The following can be obtained on the basis of formula (46):(47)V˙E=2ETE˙.
The following can be obtained after substituting formula (45) into formula (47):(48)V˙E=2ET−α1sigEq1−β1sigEg1≤−2α1VEq1+1/2−22−g1β1VEg1+1/2.
According to Lemma1, formula (48) satisfies the fixed-time convergence condition, and the upper bound of time is expressed as follows:(49)T3≤1α11−θq21−q1+121−g1β11−θg2g1−1,where θq2 and θg2 refer to arbitrary normal numbers in the interval 0,1.
Formulas (41) and (49) demonstrate that the system error E will converge to 0 when the sliding mode moves for t>T2+T3.
WhenE>ε, we can get(50)E˙=−l1sigE−l2sigE2.
Similarly, the system error is also analysed as follows:(51)V˙E=2ET−l1sigE−l2sigE2≤−2l1VE.
Hence, formula (51) shows that VE will converge to 0 exponentially given that l1 is a positive real number.Remark 3.
The comprehensive analysis on formulas (45)–(51) shows that the convergence time of the linear sliding surface of Sau=l1sigE+l2sigE2 instead of the terminal sliding mode can be ignored if the value of ε is sufficiently small. The convergence time is approximately equal to T2 in this case. Therefore, system errors E and E˙ can converge to 0 within T2+T3; that is, the velocity of all motors in the multimotor system can achieve ω=ωvir in a fixed time. Moreover, filling motors can track the given value in real time because the velocity of the virtual motor is ωvir=ω∗. Hence, the stability of the filling motor controller has been proven.
The overall control block of the improved virtual main-axis system is shown in Figure2 to understand the content of Figure 1 and design details of each module further.
Figure2 shows that the most important characteristic of the control structure is the addition of the speed compensator to the virtual motor and the use of the virtual controller to provide feedback and adjust the tracking error and compensation signal of the virtual motor. Meanwhile, a variable-gain terminal SMC is designed to realise the high-speed and high-precision tracking of each filling motor speed to the given signal.Figure 2
The control block diagram of the improved virtual main shaft structure.


## 6. Simulation

The simulation of the multimotor system composed of three motors in this section is conducted using MATLAB/Simulink to verify the proposed technical solution on the basis of theoretical studies. The comparative experimental simulation is added in this section to reflect the advantages of the improved synchronization topology and the optimised fixed-time controller. The terminal SMC proposed in [43] is used as the reference and modified according to the PMSM model in this study, but the control concept is consistent with the design idea of quotations. The given curve for multimotor system simulation is as follows:(52)ω∗=500t,300,0≤t<0.6,0.6≤t<1.4,−500t+1000,1.4≤t<2.Some typical motor parameters are shown in Table1.Table 1
Some typical motor parameters.
ParameterPMSM 1PMSM 2PMSM 3RΩN⋅m⋅s4.831×10−54.846×10−54.827×10−5ψfWb0.1750.1730.178np222Jkg⋅m28.5×10−48.47×10−48.51×10−4The simulation time in the entire experiment is 2 s. The experimental simulation of the improved virtual main-axis multimotor synchronous control structure is carried out for the optimised variable-gain terminal SMC based on fixed-time convergence and the traditional terminal SMC. The tracking error of each motor is compared with the simulation result of the synchronisation error under the enabling conditions of two controllers. Parameters of the observer are set tok1=4000 and k2=10. Virtual moment of inertia in the virtual motor is set to Jvir=0.0001. The gain of controller in the virtual controller is set to k3=200 and c=50. Coefficients of the sliding mode surface in the trial design of the variable-gain terminal SMC are set to α1=β1=2.1, q1=0.833, and g1=2. Coefficients of the controller are set to α2=β2=2.1, q2=0.833, and g2=2. The consistency between coefficients of the controller in the designed comparative experiment simulation in [43] and parameters of the observer ensures the validity and preciseness of the comparative experiment. The parameter perturbation value and motor torque variation of motors 1, 2, and 3 are Δa1=Δb1=ΔTL1=sinπt,Δa2=Δb2=ΔTL2=sin0.5πt, and Δa3=Δb3=ΔTL3=−sinπt, respectively. The following will show the simulation results of this experiment in turn.As shown in Figure3, motors of the filling multimotor system can successfully track the given value. The designed three-stage given signal includes the following processes: (1) starting and accelerating, (2) running at a uniform velocity, and (3) slowing down to stop. The experimental results showed that virtual and slave motors can rapidly track the given value in the three stages and run stably without evident overshooting. The actual simulation effect of the virtual motor can verify that the selection of the virtual motor’s moment of inertia and the design of the virtual controller are reasonable.Figure 3
Schematic diagram of speed simulation results of multimotor system.Simulation comparison charts in Figures4(a)–4(c) that present the tracking error of the velocity of motors 1, 2, and 3, respectively, are further representations of the simulation results in Figure 3. New_TSMC and TSMC represent the simulation results of the tracking error between each slave motor and virtual motor under the enabling conditions of the improved variable-gain terminal SMC and controller in [43], respectively. The tracking error of motors 1, 2, and 3 refers to the error between the actual velocity of each motor and the velocity of the virtual motor. The analysis of the simulation results showed that the tracking performance of the improved variable-gain terminal SMC is significantly improved compared with that of the traditional terminal SMC. Both the amplitude of overshooting and the convergence time of all slave motors are reduced by approximately one-third, especially in the initial stage of the simulation.Figure 4
Simulation comparison of tracking error between (a) motor 1 and virtual motor, (b) motor 2 and virtual motor, and (c) motor 3 and virtual motor.
(a)(b)(c)Figure5(a) represents the synchronization error between motor 1 and motor 2. Figure 5(b) represents the synchronization error between motor 1 and motor 3. Figure 5(c) represents the synchronization error between motor 2 and motor 3. New_TSMC and TSMC, respectively, represent the simulation results of the synchronisation error between adjacent motors under the enabling condition of the variable-gain terminal SMC and the controller enable condition proposed in [43]. The simulation results showed that the synchronisation performance of the improved controller significantly enhances compared with that of the traditional terminal SMC. Convergence amplitudes of all slave motors are reduced significantly, especially in the initial stage of the simulation.Figure 5
Comparison of simulation diagram of synchronization errors between (a) motors 1 and 2, (b) motors 1 and 3, and (c) motors 2 and 3.
(a)(b)(c)Figure6 shows that the tracking error between the real-time speed of the virtual motor and the given value is constantly controlled in a small range and vibrates in a small amplitude of 0.4–1.4 s during start-up, respectively. This phenomenon is mainly caused by the high change rate of the given value signal that causes fluctuation in the velocity of the virtual motor whilst the tracking error of the virtual motor can gradually converge within 0 to 0.001 s. Hence, the convergence is fast and the parameter selection for the virtual motor and the virtual controller is reasonable.Figure 6
Simulation results of virtual motor speed tracking error.As shown in Figure7, the velocity compensation signal constantly maintained in a small range is conducive to the design of the gain coefficient of the virtual SMC. This signal is mainly affected by the velocity of all slave motors. Therefore, the velocity compensator signal can only generate a small amplitude of vibration after selecting a small moment of inertia of the virtual motor. The velocity compensation signal gradually converges to zero when the velocity of each motor follows the given value. The result can be used to verify the effectiveness of the design of the velocity compensation signal.Figure 7
Speed compensator signal simulation results.Figures8(a)–8(c) show the simulation results of disturbances and observed values of motors 1, 2, and 3, respectively. The effect of the estimation of the sliding mode observer of each motor that tracks the actual disturbance is better for different unknown composite disturbances of the three slave motors. The observed value of each motor will evidently jitter in the initial stage of the simulation but can be controlled to follow the actual value of the disturbance within 0.2 s, which exerts a minimal impact on the entire system. This result is consistent with the expectation of the simulation experiment.Figure 8
The disturbance of (a) motor 1, (b) motor 2, and (c) motor 3 and the simulation results of their observations.
(a)(b)(c)


## 7. Conclusions

A novel synchronous control method for the filling multimotor system is put forward in this study. On the one hand, the velocity compensator added to the virtual motor to establish the coupling relationship between virtual and slave motors in the output velocity improves the correlation between systems on the basis of the existing virtual main-axis control structure. On the other hand, the variable-gain terminal sliding mode controller is designed on the basis of the tracking error between each slave motor and the virtual motor to ensure that the multimotor system can be stable within a fixed time. The proposed method can effectively weaken the adverse effects of the system parameter perturbation and motor load variation and improve the convergence robustness performance of the system. Notably, the convergence of each slave motor in the initial stage of the start-up phase significantly improves. The simulation results verify that the method has a relatively prominent effect on the high-performance cooperative control of filling multimotor system.



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*Source: 4113022-2021-07-28.xml*