Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control

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Research Article

Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control Lebao Li a,b, Lingling Sun b,n, Shengzhou Zhang a,b a b

College of Electrical Engineering, Zhejiang University, Hangzhou 310027, China Key Laboratory of RF Circuits and Systems, Ministry of Education, Hangzhou Dianzi University, Hangzhou 310018, China

art ic l e i nf o

a b s t r a c t

Article history: Received 8 November 2015 Received in revised form 3 January 2016 Accepted 20 January 2016

A new mean deviation coupling synchronization control strategy is developed for multiple motor control systems, which can guarantee the synchronization performance of multiple motor control systems and reduce complexity of the control structure with the increasing number of motors. The mean deviation coupling synchronization control architecture combining second-order adaptive sliding mode control (SOASMC) approach is proposed, which can improve synchronization control precision of multiple motor control systems and make speed tracking errors, mean speed errors of each motor and speed synchronization errors converge to zero rapidly. The proposed control scheme is robustness to parameter variations and random external disturbances and can alleviate the chattering phenomena. Moreover, an adaptive law is employed to estimate the unknown bound of uncertainty, which is obtained in the sense of Lyapunov stability theorem to minimize the control effort. Performance comparisons with master– slave control, relative coupling control, ring coupling control, conventional PI control and SMC are investigated on a four-motor synchronization control system. Extensive comparative results are given to shown the good performance of the proposed control scheme. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Mean deviation coupling Synchronization control Permanent magnet synchronous motor Second-order adaptive sliding mode control Synchronization error

1. Introduction In many industrial applications, such as distributed papermaking, continuous rolling mills and manufacturing assembly [1], the load is often driven by two or more motors simultaneously. The motors can be designed to track the given trajectory and keep their speed the same during the running process [2]. It has been recognized that the synchronization performance of system may be degraded by some factors such as system parameter variations and external disturbances in the system, and the synchronization error will affect the quality of the work pieces and even lead to stop of the working process [3,4]. So, good synchronization performance of multiple motor control systems can be obtained while there are various uncertainties and perturbations, which has become a challenge due to the increasing demand for rapid response and high accuracy manufacture and inspection. Over the past few decades, several different synchronization control strategies for multiple motor drive systems have been proposed, which mainly consist of the master–slave control, the virtual-shaft control, the cross-coupling control, the relative coupling control, n

Corresponding author. E-mail addresses: [email protected] (L. Li), [email protected] (L. Sun), [email protected] (S. Zhang).

the adjacent cross-coupling control , ring coupling control and so on [1,5–13]. The master–slave control strategy has a simple control structure [14], which sets one motor as master and the other motors as slaves, and makes the slaves track the response of the master. The disturbances on the master will be conducted to the slaves, but the reverse is impossible [15]. In the virtual-shaft control strategy, the system input needs passing through the virtual axis to get the reference signal of motor, which results in the unequal between motor’s reference signal and the system input signal [16,17]. Crosscoupling control strategy [6,10,18–20] utilizes the difference in speed response between two motors as an additional feedback tracking signal. But it is difficult to extend the method for more than two motors. To overcome this limitation, some improved synchronization control strategies have been presented, which include the relative coupling control, adjacent cross-coupling control and ring coupling control. The relative coupling control has indeed better synchronization control performance , but when the number of motor is n (n 4 2), n2-controller needs to be designed [21]. Thus the complexity of system control structure is also increasing with the increasing number of motors [22]. In order to reduce complexity of multiple motor synchronization control systems, adjacent cross-coupling control strategy [1] and ring coupling control strategy [9] are proposed by some researchers. When the number of motor is n,

http://dx.doi.org/10.1016/j.isatra.2016.01.015 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i

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3n-controller needs to be devised in adjacent cross-coupling and 2n-controller in ring coupling control system [23]; hence, these control methods can reduce the control complexity. However, they just use the speed of adjacent motor as feedback compensation, which may result in the unequal response of all motors due to the conduction delay of the speed change [15]. Therefore, to overcome the disadvantages of the aforementioned multiple motor synchronization control strategies, a new multiple motors synchronization control strategy is developed, which can ensure the synchronization performance of multiple motors and reduce complexity of the control structure with the increasing number of motors. A typical multiple motors synchronization control scheme consists of a synchronization control strategy to calculate the errors and a control algorithm to improve synchronization control precision. Many control algorithms have been proposed, such as traditional PID control [24], adaptive feedforward control [14,25], H1 control [6], iterative learning control [26], sliding mode control (SMC) [2,8,11,27,28], fuzzy control [29], neural network control [30] and so on. When the system subjects to random disturbances, mismatched drive dynamics and parameters variation [31], the synchronization performance and stability of multiple motor control systems become poor. It is well known that SMC system is robust to parameter variations and model uncertainties and insensitivity to external disturbance once the system trajectory reaches and stays on the sliding surface [32–34]. Therefore, SMC is suitable for multiple motor synchronization control systems to improve synchronization control precision. In general, to guarantee the robustness of the SMC, a large switching control gain will be used. But, it often leads to chattering phenomena which are caused by the switching (sign) function [35]. To attenuate the chattering, the sign function in traditional SMC is often replaced by the saturation function. Although the chattering can be mitigated by using saturation function, good control performance cannot be guaranteed and an indefinite steady-state error is also caused by using the selection of the boundary layer [36,37]. Another method to eliminate chattering is to diminish the switching control gain; however, the robustness of the SMC becomes poor because it is not strong enough to cope with the uncertainties. To overcome the disadvantages of the traditional SMC, high-order sliding mode controls including the second-order sliding mode control (SOSMC), which not only maintain the advantages of the traditional SMC such as robustness and simplicity, but also alleviate the chattering phenomena, have been proposed in [33,36,38–41]. In addition, the selection of the upper bound of the uncertainty, which contains mechanical parameter variations and external disturbances, has a significant effect on the control performance. Unfortunately, the upper bound of uncertainty is difficult to know in advance in practical applications, and it is very difficult to implement the sliding mode control law practically [36]. Therefore, the assumption of known uncertainty bounds is necessary in the design the SOSMC. In this paper, an adaptive law is developed to estimate the unknown bound of uncertainty to design the SOSMC law, which can minimize the control effort. Thus, to improve the synchronization control precision of multiple motor control systems and ensure the robustness of the SMC to various system uncertainties and external disturbances, a new mean deviation coupling SOASMC scheme is presented for multiple motor synchronization control systems. Considering the defects of many synchronization control strategy, the mean deviation coupling synchronization control structure of multiple motors is proposed, which can guarantee the synchronization performance of multiple motors and reduce complexity of the control structure with the increasing number of motors. Moreover, SOASMC is applied into mean deviation coupling control structure to improve speed tracking and

synchronization control precision of multiple motor control systems. The organization of the present paper is as follows. The mathematical model of permanent magnet synchronous motor (PMSM) is introduced in Section 2. The second-order adaptive sliding mode speed controller based on mean deviation coupling control structure is designed and the associated stability analysis is presented in Section 3. Comparative studies conducted on a four-motor system are given in Section 4. Section 5 gives the conclusion.

2. The mathematical model of PMSM PMSM has been chosen as plant. The dynamic model of PMSM under rotor field synchronous rotating d–q reference frame can be described as the following differential equations [42–44] x_ d;q ¼ Axd;q þ Bud;q

ð1Þ

where 2 3 21 3 pω 0 0  Rs 0 Ld Ld 6 Ld 7 6 7 6 0 6 0 L1 7  RLqs  pLωq 0 7 6 7 q 7 A¼6 7; B ¼ 6 6 7 6 Rs 41 05 0 0 pω 7 4 5 0  Rs  pω 0 0 1 h iT h iT xd;q ¼ id iq ψ d ψ q ; ud;q ¼ ud uq ud and uq are d-axis and q-axis stator voltages, respectively; id and iq are d-axis and q-axis stator currents, respectively; RS is stator resistance. Ld and Lq are d-axis and q-axis inductances, respectively;, ψd and ψq are d-axis and q-axis stator flux linkages, respectively; p is the number of pole pairs. ω is the rotor mechanical angular velocity. The equation of electromagnetic torque is stated as 3 T e ¼ p½ψ f iq þ ðLd  Lq Þid iq  2

ð2Þ

Motor dynamics is presented as B J

1 J

ω_ ðt Þ ¼  ωðt Þ þ ðT e T L Þ

ð3Þ

where Te is the electromagnetic torque of motor; TL is the load torque; J is moment of inertia; B is the viscous friction coefficient; ψf is rotor flux. By using the field-oriented mechanism with id ¼ 0 [43], we have T e ¼ ke iq ðt Þ

ð4Þ

3 ke ¼ pψ f 2

ð5Þ

Substituting (4) and (5) into (3), the following result can be obtained: B J

1 J

1 J

ω_ ðt Þ ¼  ωðt Þ þ ke iq ðt Þ  T L

ð6Þ

The rotor mechanical motion equation of the ith-motor can be rewritten as B Ji

1 Ji

1 Ji

ω_ i ðt Þ ¼  i ωi ðt Þ þ keðiÞ iqðiÞ ðt Þ  T LðiÞ Set xi ðt Þ ¼ ωi ðt Þ, uqðiÞ ðt Þ ¼ iqðiÞ ðt Þ, ai ¼ keðiÞ =J i , ci ¼ Bi =J i , Eq. (7) can be presented as follows x_ i ðt Þ ¼ ai uqðiÞ ðt Þ þ bi T LðiÞ þ ci xi ðt Þ

ð7Þ bi ¼  1=J i , ð8Þ

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i

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When considering the uncertainties and time-varying parameters, the dynamic (8) can be modified as       x_ i ðt Þ ¼ ai0 þ Δai0 uqðiÞ ðt Þ þ bi0 þ Δbi0 T LðiÞ þ ci0 þ Δci0 xi ðt Þ ¼ ai0 uqðiÞ ðt Þ þ ci0 xi ðt Þ þ hi ðt Þ

ð9Þ

where ai0 is the nominal value of ai , bi0 is the nominal value of bi , and ci0 is the nominal value of ci ; Δai0 , Δbi0 and Δci0 denote the uncertainties; and hi is named as the lumped uncertainty, which is defined as   ð10Þ hi ðt Þ ¼ Δai0 uqðiÞ ðt Þ þ bi0 þ Δbi0 T LðiÞ þ Δci0 xi ðt Þ       Here, hi ðt Þ o ρi0 and h_ i ðt Þ o ρi0 are assumed, for all t A ð0; 1Þ, ρi0 is a given positive constant.

where xm ðt Þ ¼

Σ nj ¼ 1 xj ðt Þ n

¼ ωm ðt Þ.

The control objective is to design the speed controller i, which can guarantee the convergence of speed tracking error   eref ;i ðt Þ and mean speed error ei;Σ ðt Þ, that is lim eref ;i ðt Þ ¼ 0 and t-1   lim ei;Σ ðt Þ ¼ 0, in spite of the parameter uncertainties and exter-

t-1

3. Speed controller design and stability analysis In the following sections, the symbols with subscript i and j denote corresponding variables and parameters of motor i and j, respectively. 3.1. Mean deviation coupling control strategy The mean deviation coupling control strategy is proposed, which is based on the thought of the same given control and error compensation. The schematic diagram of mean deviation coupling control strategy is shown in Fig. 1. All motors can follow the given speed ωref in the proposed control approach. The idea of this control strategy is averaging all motors’ speed and getting mean value, then calculating the speed error between a motor’s speed and the mean value as the compensation signal for this one.where the given speed ωref is identical for all motors; n is motor number; eref,i (i ¼1…n) is tracking error; ωm is mean speed of all motors; ei,Σ is the mean speed error between the ith motor’s speed and ωm; ωi is the mechanical angular velocity of motor i. The speed tracking error of motor i is defined as eref ;i ðt Þ ¼ xref ðt Þ  xi ðt Þ

Fig. 2. Structure diagram of the speed controller i.

ð11Þ

nal perturbations. The structure of speed controller of motor i is shown in Fig. 2, which includes two sub controllers. One is speed tracking controller Cref,i, which is used to track the desired speed signal accurately; the other is mean speed deviation controller Ci,Σ, which is used to compensate the output speed of the controlled motor i. Considering a n-motor system with the proposed control strategy, 2n-controller will be designed, therefore, this significantly diminish the complexity of control system, especially when the number of motor is large. Therefore, the complete q-axis control effort of motor i is presented as iq;i ðt Þ ¼ uq;i ðt Þ ¼ uref ;i ðt Þ þ ui;Σ ðt Þ

ð14Þ

where uref ;i ðt Þ is the control effort for speed tracking of motor i, ui;Σ ðt Þ is the control effort for mean speed deviation of motor i. Remark 1. The speed synchronization error of en;n þ 1 ðt Þ is presented as ð15Þ en;n þ 1 ðt Þ ¼ en;1 ðt Þ ¼ xn ðt Þ x1 ðt Þ 3.2. First-order sliding mode control

where xref ðt Þ ¼ ωref ðt Þ. The speed synchronization error of motor i with motor iþ1 is defined as

3.2.1. Speed tracking controller The first derivative of the speed tracking error eref ;i ðt Þ is expressed as

ei;i þ 1 ðt Þ ¼ xi ðt Þ  xi þ 1 ðt Þ

e_ ref ;i ðt Þ ¼ x_ ref ðt Þ  x_ i ðt Þ ¼ x_ ref ðt Þ  ai0 uqðiÞ ðt Þ  ci0 xi ðt Þ  hi ðt Þ

ð12Þ

Define the mean speed error of motor i in the following way n

ei;Σ ðt Þ ¼ xi ðt Þ  xm ðt Þ ¼ xi ðt Þ 

Σ xj ðt Þ

j¼1

n

ð13Þ

ð16Þ

In order to satisfy the design requirement in the traditional SMC, a generalized sliding surface is defined as follows: Z t δref ;i ðt Þ ¼ ceref ;i ðt Þ þ eref ;i ðτ Þdτ ð17Þ 0

where c is a given positive constant. If a proper control law is developed, which can guaranteeδref ;i ðt Þδ_ ref ;i ðt Þ o 0, and then it can be obtained that Z t δref ;i ðt Þ ¼ ceref ;i ðt Þ þ eref ;i ðτ Þdτ ¼ 0 ð18Þ 0

eref ;i ðt Þ ¼ eref ;i ð0Þe

 ct

ð19Þ

Thus, it can be seen that the parameter c determines the convergence rate of tracking error. Now, taking the derivative of (17) and using (16), the following equation can be obtained   δ_ ref ;i ðt Þ ¼ ce_ ref ;i ðt Þ þ eref ;i ðt Þ ¼ c x_ ref ðt Þ  ai0 uref ;i ðt Þ  ci0 xi ðt Þ hi ðt Þ þ eref ;i ðt Þ Fig. 1. Schematic diagram of mean deviation coupling control strategy.

ð20Þ

where uref ;i ðt Þ ¼ uqðiÞ ðt Þ.

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According to (17) and (20), the speed tracking SMC law uref ;i ðt Þ is designed as follows

According to (26) and (27), the mean speed deviation SMC law ui;Σ ðt Þ is designed as follows

uref ;i ðt Þ ¼ uref ;eqi ðt Þ þuref ;hiti ðt Þ

ui;Σ ðt Þ ¼ ueqi;Σ ðt Þ þ uhiti;Σ ðt Þ

ð21Þ

The equivalent control uref ;eqi ðt Þ can be given as  1   cx_ ref ðt Þ  cci0 xi ðt Þ þ eref ;i ðt Þ uref ;eqi ðt Þ ¼ cai0

The equivalent control ueqi;Σ ðt Þ can be given as ð22Þ

The reaching control part of control effort uref ;i ðt Þ is given as 

 1 uref ;hiti ðt Þ ¼ ρref ;i sgn δref ;i ðt Þ ai0

ð23Þ

where sgn(  ) is the sign function, ρref ;i denotes the switch gain. Substituting (21)–(23) into (20), one obtains   _δ ðt Þ ¼ c x_  ðt Þ  a u ðt Þ  c x ðt Þ  h ðt Þ þ e ðt Þ i0 ref ;i i0 i i ref ;i ref ;i ref    ¼ cx_ ref ðt Þ  cai0 uref ;eqi ðt Þ þ uref ;hiti ðt Þ  cci0 xi ðt Þ  chi ðt Þ þ eref ;i ðt Þ   1   ¼ cx_ ref ðt Þ  cai0 cx_ ref ðt Þ  cci0 xi ðt Þ þ eref ;i ðt Þ cai0    1 þ ρref ;i sgn δref ;i ðt Þ ai0

ueqi;Σ ðt Þ ¼ 

 1  cc0 ei;Σ ðt Þ þei;Σ ðt Þ ca0

ð29Þ

The reaching control part of control effort ui;Σ ðt Þ is stated as uhiti;Σ ðt Þ ¼ 

  1 kδi;Σ ðt Þ þ ρi;Σ sgn δi;Σ ðt Þ a0

ð30Þ

where k is a positive constant, ρi;Σ denotes the switch gain. Substituting (28)–(30) into (27), we have   1 n δ_ i;Σ ðt Þ ¼ c a0 ui;Σ ðt Þ þ c0 ei;Σ ðt Þ þ hi ðt Þ  Σ hj ðt Þ þ ei;Σ ðt Þ nj¼1 c n ¼ ca0 ui;Σ ðt Þ þcc0 ei;Σ ðt Þ þ chi ðt Þ  Σ hj ðt Þ þ ei;Σ ðt Þ nj¼1   ¼ ca0 ueqi;Σ ðt Þ þ uhiti;Σ ðt Þ þ cc0 ei;Σ ðt Þ þ chi ðt Þ c n Σ h ðt Þ þ ei;Σ ðt Þ nj¼1 j   1 1  cc0 ei;Σ ðt Þ þ ei;Σ ðt Þ  kδi;Σ ðt Þ ¼ ca0  ca0 a0 !   þ ρi;Σ sgn δi;Σ ðt Þ 

 cci0 xi ðt Þ  chi ðt Þ þ eref ;i ðt Þ   ¼ cx_ ref ðt Þ  cx_ ref ðt Þ þcci0 xi ðt Þ eref ;i ðt Þ  cρref ;i sgn δref ;i ðt Þ  cci0 xi ðt Þ  chi ðt Þ þ eref ;i ðt Þ   ¼  chi ðt Þ  cρref ;i sgn δref ;i ðt Þ

ð28Þ

ð24Þ

c n Σ h ðt Þ þ ei;Σ ðt Þ nj¼1 j   c n ¼ ckδi;Σ ðt Þ  cρi;Σ sgn δi;Σ ðt Þ þ chi ðt Þ  Σ hj ðt Þ nj¼1 þcc0 ei;Σ ðt Þ þ chi ðt Þ 

3.2.2. Mean speed deviation controller The following assumption for system parameters is satisfied to design the mean speed deviation controller. Assumption 1. There are two known constants a0 , c0 and the nominal values ai0 ,ci0 ,aj0 and cj0 satisfy the following equalities: a0 ¼ ai0 ¼ aj0 and c0 ¼ ci0 ¼ cj0 . The first derivative of mean speed error ei;Σ ðt Þ is expressed as

ð31Þ

3.2.3. Stability analysis Theorem 1. Considering the system dynamic equation represented by (9) and (10), if the proposed mean deviation coupling SMC law is designed via (21)–(23) and (28)–(30), and with

n

e_ i;Σ ðt Þ ¼ x_ i ðt Þ  

Σ x_ j ðt Þ

j¼1

n

¼ ai0 uqðiÞ ðt Þ þ ci0 xi ðt Þ þ hi ðt Þ

1 n Σ a u ðt Þ þcj0 xj ðt Þ þ hj ðt Þ n j ¼ 1 j0 qðjÞ

¼ a0 uqðiÞ ðt Þ þ c0 xi ðt Þ þ hi ðt Þ  n

a0 n Σ u ðt Þ n j ¼ 1 qðjÞ

ð32Þ

ρi;Σ Z 2ρi0 þ α

ð33Þ

 n  n 1   Σ hj ðt Þ o 1 Σ ρ ¼ ρ i0 n j ¼ 1  n j ¼ 1 j0

ð34Þ

n

c0 1 Σ x ðt Þ  Σ h j ðt Þ nj¼1 n j¼1 j     1 n 1 n ¼ a0 uqðiÞ ðt Þ  Σ uqðjÞ ðt Þ þ c0 xi ðt Þ  Σ xj ðt Þ nj¼1 nj¼1 

ρref ;i Z ρi0 þ α

then, the proposed mean deviation coupling SMC system guarantees the convergence of the speed tracking error and    mean  speed error simultaneously; that is lim eref ;i ðt Þ ¼ 0 and lim ei;Σ ðt Þ ¼ 0. t-1

1 n þ hi ðt Þ  Σ hj ðt Þ nj¼1 ¼ a0 ui;Σ ðt Þ þ c0 ei;Σ ðt Þ þ hi ðt Þ 

1 n Σ h ðt Þ nj¼1 j

ð25Þ

where ui;Σ ðt Þ ¼ uqðiÞ ðt Þ  1nΣ j ¼ 1 uqðjÞ ðt Þ. Time-varying surface of sliding mode of ei;Σ ðt Þ in the state R2 can be introduced as Z t δi;Σ ðt Þ ¼ cei;Σ ðt Þ þ ei;Σ ðτÞdτ ð26Þ

t-1

The stability analysis of the speed control system is illustrated by Lyapunov theory as follows. We define a two-dimension manifold as h iT δi ðt Þ ¼ δref ;i ðt Þ δi;Σ ðt Þ ¼ 0 ð35Þ

n

0

Taking the derivative of (26) and using (25), the following equation can be obtained   1 n δ_ i;Σ ðt Þ ¼ ce_ i;Σ ðt Þ þ ei;Σ ðt Þ ¼ c a0 ui;Σ ðt Þ þc0 ei;Σ ðt Þ þ hi ðt Þ  Σ hj ðt Þ nj¼1 þ ei;Σ ðt Þ

ð27Þ

Proof 1. Consider the following Lyapunov function: V ðt Þ ¼

1 n T 1 n 1 n Σ δ ðt Þδi ðt Þ ¼ Σ δ2ref ;i ðt Þ þ Σ δ2i;Σ ðt Þ 2i¼1 i 2i¼1 2i¼1

ð36Þ

Stability condition can be obtained from Lyapunov stability theorem as n

n

V_ ðt Þ ¼ Σ δref ;i ðt Þδ_ ref ;i ðt Þ þ Σ δi;Σ ðt Þδ_ i;Σ ðt Þ i¼1 i¼1  n  n  n    2 r cα Σ δref ;i ðt Þ þ Σ δi;Σ ðt Þ  ck Σ δi;Σ ðt Þ i¼1 i¼1 i¼1

ð37Þ

where αis a small positive constant.

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Substituting (24) and (31) into (37), it can be concluded that n

5

3.3. Second-order sliding mode control

n

V_ ðt Þ ¼ Σ δref ;i ðt Þδ_ ref ;i ðt Þ þ Σ δi;Σ ðt Þδ_ i;Σ ðt Þ i¼1 n

In order to design the mean deviation coupling SOSMC law, the sliding variable is defined, and its derivative can be obtained as follows: Z t δðt Þ ¼ ceðt Þ þ eðτÞdτ ð42Þ

i¼1



  ¼ Σ δref ;i ðt Þ  chi ðt Þ  cρref ;i sgn δref ;i ðt Þ i¼1

n   þ Σ δi;Σ ðt Þ  ckδi;Σ ðt Þ  cρi;Σ sgn δi;Σ ðt Þ i¼1

c n þchi ðt Þ  Σ hj ðt Þ nj¼1

0

 ð38Þ

Then, we have  n    V_ ðt Þ ¼ c Σ  ρref ;i δref ;i ðt Þ  hi ðt Þδref ;i ðt Þ þc Σ

i¼1

  2  kδi;Σ ðt Þ  ρi;Σ δi;Σ ðt Þ þ hi ðt Þδi;Σ ðt Þ

 1 n Σ hj ðt Þδi;Σ ðt Þ nj¼1 n       rc Σ  ρ δref ;i ðt Þ þ hi ðt Þδref ;i ðt Þ

  e€ ref ;i ðt Þ ¼ x€ ref ðt Þ  x€ i ðt Þ ¼ x€ ref ðt Þ  ai0 u_ ref ;i ðt Þ  ci0 x_ i ðt Þ  h_ i ðt Þ



i¼1 n

þc Σ

i¼1

δSOSMCðref ;iÞ ðt Þ ¼ δref ;i ðt Þ þ δ_ ref ;i ðt Þ ¼ ðc þ 1Þeref ;i ðt Þ

  2  kδi;Σ ðt Þ  ρi;Σ δi;Σ ðt Þ

Z

þ

      1 n þ hi ðt Þ  Σ hj ðt Þδi;Σ ðt Þ nj¼1  n  n   2  hi ðt Þ  ρref ;i δref ;i ðt Þ  ck Σ δi;Σ ðt Þ rc Σ i¼1 i¼1     n   1 n    þc Σ hi ðt Þ þ  Σ hj ðt Þ  ρi;Σ δi;Σ ðt Þ nj¼1 i¼1  n  n  2 oc Σ ρi0  ρref ;i δref ;i ðt Þ  ck Σ δi;Σ ðt Þ   δi;Σ ðt Þ

þc Σ

0

eref ;i ðτÞdτ þ ce_ ref ;i ðt Þ

ð45Þ

δ_ SOSMCðref ;iÞ ðt Þ ¼ ðc þ 1Þe_ ref ;i ðt Þ þ eref ;i ðt Þ þ ce€ ref ;i ðt Þ ¼ cx€ ref ðt Þ  cai0 u_ ref ;i ðt Þ cci0 x_ i ðt Þ ch_ i ðt Þ 

þ ðc þ 1Þe_ ref ;i ðt Þ þ eref ;i ðt Þ

ð46Þ

The speed tracking SOSMC law uref ;i ðt Þ can be given as ð39Þ

According to (39), if the following requirement is satisfied, it can be obtained that the proposed mean deviation coupling SMC system is asymptotically stable.  8   > > hi ðt Þ o ρi0 > > n n >   > > 1 Σ h ðt Þ o 1 Σ ρ ¼ ρ > < nj ¼ 1 j  nj ¼ 1 j0 i0 ð40Þ ρ Z ρ þ α > i0 > > ref ;i > > ρ Z 2ρi0 þ α > > > i;Σ : c 40; k 4 0; α 4 0 And it also can be deduced that lim jeref ;i ðt Þj ¼ 0 and t-1   lim ei;Σ ðt Þ ¼ 0. Thus, the mean deviation coupling SMC system

t-1

guarantees the convergence of the speed tracking error and mean speed error simultaneously, even if there are parametric uncertainty and external disturbances. Remark 2. Due to all motors can follow the reference speed ωref ðt Þ in the proposed control strategy, and considering Eqs. (11) and (12), the following result can be obtained      lim ei;i þ 1 ðt Þ ¼ lim xi ðt Þ  xi þ 1 ðt Þ ¼ lim xi ðt Þ t-1 t-1 t-1    xref ðt Þ þ xref ðt Þ  xi þ 1 ðt Þ         r lim xi ðt Þ  xref ðt Þþ lim xref ðt Þ  xi þ 1 ðt Þ t-1 t-1     ð41Þ r lim eref ;i ðt Þ þ lim eref ;i þ 1 ðt Þ ¼ 0

uref ;i ðt Þ ¼ uref ;eqi ðt Þ þ uref ;hiti ðt Þ uref ;eqi ðt Þ ¼

ð47Þ

  Z t 1 cx_ ref ðt Þ cci0 xi ðt Þ þ ðc þ 1Þeref ;i ðt Þ þ eref ;i ðτÞdτ cai0 0 ð48Þ

uref ;hiti ðt Þ ¼

1 ai0

Z

t 0





ςref ;i sgn δSOSMCðref ;iÞ ðτÞ dτ

ð49Þ

where ςref ;i denotes the switch gain. Substituting (47)–(49) into (46), we have

δ_ SOSMCðref ;iÞ ðt Þ ¼ cx€ ref ðt Þ  cai0 u_ ref ;i ðt Þ  cci0 x_ i ðt Þ ch_ i ðt Þ þ ðc þ 1Þe_ ref ;i ðt Þ þ eref ;i ðt Þ    ¼ cx€ ref ðt Þ  cai0 u_ ref ;eqi ðt Þ þ u_ ref ;hiti ðt Þ cci0 x_ i ðt Þ ch_ i ðt Þ þ ðc þ 1Þe_ ref ;i ðt Þ þ eref ;i ðt Þ     ¼ cx€ ref ðt Þ  cai0 ca1i0 cx€ ref ðt Þ  cci0 x_ i ðt Þ  þ ðc þ 1Þe_ ref ;i ðt Þ þ eref ;i ðt Þ

þ

ςref ;i ai0

   sgn δSOSMCðref ;iÞ ðt Þ

cci0 x_ i ðt Þ ch_ i ðt Þ þ ðc þ 1Þe_ ref ;i ðt Þ þ eref ;i ðt Þ   ¼  ch_ i ðt Þ  cςref ;i sgn δSOSMCðref ;iÞ ðt Þ

ð50Þ

t-1

Therefore, the mean deviation coupling SMC system also guarantees  the convergence of the speed synchronization error,  that is lim ei;i þ 1 ðt Þ ¼ 0. t-1

t

Then, taking the derivative of (45) and using (44), the following equation can be obtained

i¼1

2ρi0  ρi;Σ i¼1  n  n  n    2 r  cα Σ δref ;i ðt Þ þ Σ δi;Σ ðt Þ  ck Σ δi;Σ ðt Þ i¼1 i¼1 i¼1

t-1

ð44Þ

where u_ ref ;i ðt Þ ¼ u_ qðiÞ ðt Þ. The second-order sliding surface of eref ;i ðt Þ is designed as

ref ;i

i¼1 n 

ð43Þ

3.3.1. Speed tracking controller The second derivative of the speed tracking error eref ;i ðt Þ is expressed as

i¼1

n

δ_ ðt Þ ¼ ce_ ðt Þ þ eðt Þ

Remark 3. When sliding mode control techniques is used, the chattering phenomenon will be introduced around the sliding mode surface. In order to overcome this problem, a smooth

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6

   1 vδSOSMCði;Σ Þ ðt Þ þ ςi;Σ sgn δSOSMCði;Σ Þ ðt Þ a0 þ cc e_ ðt Þ þ ch_ ðt Þ

continuous function replaces the sign function in a thin boundary layer near the sliding surface [37]. That is [11,45] 8 δSOSMCðref ;iÞ ðt Þ 4 λ > > 1;   < δSOSMCðref ;iÞ ðtÞ ð51Þ sat δSOSMCðref ;iÞ ðt Þ ¼ ;  λ o δSOSMCðref ;iÞ ðt Þ o λ λ > > :  1; δSOSMCðref ;iÞ ðt Þ o  λ



0 i;Σ

where λ is defined as the boundary layer thickness, sat(  ) is the saturation function. 3.3.2. Mean speed deviation controller

The second derivative of mean speed error ei;Σ ðt Þ is expressed as n

Σ x€ j ðt Þ

¼ ai0 u_ qðiÞ ðt Þ þ ci0 x_ i ðt Þ þ h_ i ðt Þ i aj0 u_ qðjÞ ðt Þ þ cj0 x_ j ðt Þ þ h_ j ðt Þ

j¼1

n

1  Σ nj¼1

h

n

c þ ch_ i ðt Þ  Σ h_ j ðt Þ nj¼1

ð58Þ

3.3.3. Stability analysis

Assumption 2. We have the same assumption as Assumption 1.

e€ i;Σ ðt Þ ¼ x€ i ðt Þ 

i

c n  Σ h_ j ðt Þ þ ðc þ 1Þe_ i;Σ ðt Þ þ ei;Σ ðt Þ nj¼1   ¼  cvδSOSMCði;Σ Þ ðt Þ  cςi;Σ sgn δSOSMCði;Σ Þ ðt Þ

n

a0 n c n Σ u_ ðt Þ  0 Σ x_ j ðt Þ ¼ a0 u_ qðiÞ ðt Þ þ c0 x_ i ðt Þ þ h_ i ðt Þ  n j ¼ 1 qðjÞ n j¼1 1 n _  Σ hj ðt Þ nj¼1     1 n 1 n ¼ a0 u_ qðiÞ ðt Þ  Σ u_ qðjÞ ðt Þ þ c0 x_ i ðt Þ  Σ x_ j ðt Þ nj¼1 nj¼1 1 n _ _ þ hi ðt Þ  Σ hj ðt Þ nj¼1 1 n ¼ a0 u_ i;Σ ðt Þ þ c0 e_ i;Σ ðt Þ þ h_ i ðt Þ  Σ h_ j ðt Þ nj¼1

Theorem 2. Considering the system dynamic equation represented by (9), if the proposed mean deviation coupling SOSMC law is designed as (47) and (55), which is composed of (48)–(49) and (56)– (57), respectively, and with

ςref ;i Z ρi0 þ α

ð59Þ

ςi;Σ Z 2ρi0 þ α

ð60Þ

 n  1   Σ h_ j ðt Þ o ρ i0 n j ¼ 1 

ð61Þ

then, the stability of the proposed mean deviation coupling SOSMC system can be guaranteed and the speed tracking error and mean speed error will converge of zero   to a neighborhood   simultaneously; that is lim eref ;i ðt Þ ¼ 0 and lim ei;Σ ðt Þ ¼ 0. t-1

ð52Þ

where u_ i;Σ ðt Þ ¼ u_ qðiÞ ðt Þ  1nΣ j ¼ 1 u_ qðjÞ ðt Þ. Time-varying surface of second-order sliding mode of ei;Σ ðt Þ in the state R2 can be introduced as

t-1

We also need to define a two-dimension manifold as h iT δSOSMC;i ðt Þ ¼ δSOSMCðref ;iÞ ðt Þ δSOSMCði;Σ Þ ðt Þ ¼ 0

ð62Þ

n

δSOSMCði;Σ Þ ðt Þ ¼ δi;Σ ðt Þ þ δ_ i;Σ ðt Þ ¼ ðc þ 1Þei;Σ ðt Þ Z

þ

t 0

ei;Σ ðτ Þdτ þce_ i;Σ ðt Þ

ð53Þ

Now, taking the derivative of (53) and using (52), one obtains

δ_ SOSMCði;Σ Þ ðt Þ ¼ δ_ i;Σ ðt Þ þ δ€ i;Σ ðt Þ ¼ ðc þ 1Þe_ i;Σ ðt Þ þ ei;Σ ðt Þ þ ce€ i;Σ ðt Þ ¼ ca0 u_ i;Σ ðt Þ þcc0 e_ i;Σ ðt Þ þ ch_ i ðt Þ 

ð54Þ

  Z t 1 ueqi;Σ ðt Þ ¼  cc0 ei;Σ ðt Þ þ ðc þ 1Þei;Σ ðt Þ þ ei;Σ ðτÞdτ ca0 0

ð56Þ

t 0

   vδSOSMCði;Σ Þ ðτÞ þ ςi;Σ sgn δSOSMCði;Σ Þ ðτÞ dτ

where v is a positive constant, ςi;Σ denotes the switch gain. Substituting (55)–(57) into (54), we have

δ_

_ i;Σ ðt Þ þcc0 e_ i;Σ ðt Þ þ ch_ i ðt Þ SOSMCði;Σ Þ ðt Þ ¼ ca0 u c n  Σ h_ j ðt Þ þ ðc þ 1Þe_ i;Σ ðt Þ þ ei;Σ ðt Þ nj¼1   ¼ ca0 u_ eqi;Σ ðt Þ þ u_ hiti;Σ ðt Þ þ cc0 e_ i;Σ ðt Þ þ ch_ i ðt Þ n

c Σ h_ ðt Þ þ ðc þ 1Þe_ i;Σ ðt Þ þ ei;Σ ðt Þ nj¼1 j    1 ¼ ca0  ca cc0 e_ i;Σ ðt Þ þ ðc þ 1Þe_ i;Σ ðt Þ þ ei;Σ ðt Þ 

0

Stability condition can be obtained from Lyapunov stability theorem as

n

ð55Þ

1 a0

ð63Þ

n

ui;Σ ðt Þ ¼ ueqi;Σ ðt Þ þ uhiti;Σ ðt Þ

uhiti;Σ ðt Þ ¼ 

1 n T 1 n 2 Σ δ ðt ÞδSOSMC;i ðt Þ ¼ Σ δSOSMCðref ;iÞ ðt Þ 2 i ¼ 1 SOSMC;i 2i¼1 1 n 2 þ Σ δSOSMCði;Σ Þ ðt Þ 2i¼1

i¼1

According to (53) and (54), the mean speed deviation SOSMC law ui;Σ ðt Þ is designed as follows

Z

V ðt Þ ¼

V_ ðt Þ ¼ Σ δSOSMCðref ;iÞ ðt Þδ_ SOSMCðref ;iÞ ðt Þ

c n _ Σ h ðt Þ nj¼1 j

þ ðc þ 1Þe_ i;Σ ðt Þ þ ei;Σ ðt Þ

Proof 2. Consider the following Lyapunov function:

þ Σ δSOSMCði;Σ Þ ðt Þδ_ SOSMCði;Σ Þ ðt Þ i¼1  n  n     r  cα Σ δSOSMCðref ;iÞ ðt Þ þ Σ δSOSMCði;Σ Þ ðt Þ i¼1

n

 cv Σ δ i¼1

i¼1

2 SOSMCði;Σ Þ ðt Þ

ð64Þ

where α is a small positive constant. Substituting (50) and (58) into (64), it can be concluded that n

V_ ðt Þ ¼ Σ δSOSMCðref ;iÞ ðt Þδ_ SOSMCðref ;iÞ ðt Þ i¼1

ð57Þ

n

þ Σ δSOSMCði;Σ Þ ðt Þδ_ SOSMCði;Σ Þ ðt Þ i¼1 n

   ¼ Σ δSOSMCðref ;iÞ ðt Þ  ch_ i ðt Þ  cςref ;i sgn δSOSMCðref ;iÞ ðt Þ i¼1 n

þ Σ δSOSMCði;Σ Þ ðt Þ i¼1    cvδSOSMCði;Σ Þ ðt Þ  cςi;Σ sgn δSOSMCði;Σ Þ ðt Þ  c n þ ch_ i ðt Þ  Σ h_ j ðt Þ nj¼1

ð65Þ

Then, one can obtain n    V_ ðt Þ ¼ c Σ  h_ i ðt ÞδSOSMCðref ;iÞ ðt Þ  ςref ;i δSOSMCðref ;iÞ ðt Þ i¼1

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  2  vδSOSMCði;Σ Þ ðt Þ  ςi;Σ δSOSMCði;Σ Þ ðt Þ  1 n þ h_ i ðt ÞδSOSMCði;Σ Þ ðt Þ  Σ h_ j ðt ÞδSOSMCði;Σ Þ ðt Þ nj ¼ 1  n      _  r c Σ hi ðt Þ δSOSMCðref ;iÞ ðt Þ  ςref ;i δSOSMCðref ;iÞ ðt Þ n

þc Σ

Table 1 Nominal parameters of four motors.

i¼1

i¼1

n    2 þc Σ  vδSOSMCði;Σ Þ ðt Þ  ςi;Σ δSOSMCði;Σ Þ ðt Þ i¼1       1 n þ h_ i ðt Þ  Σ h_ j ðt ÞδSOSMCði;Σ Þ ðt Þ nj¼1   n  n  _  2 r c Σ hi ðt Þ ςref ;i δSOSMCðref ;iÞ ðt Þ  cv Σ δSOSMCði;Σ Þ ðt Þ i¼1 i¼1     1 n n      þc Σ h_ i ðt Þþ  Σ h_ j ðt Þ  ςi;Σ δSOSMCði;Σ Þ ðt Þ nj¼1 i¼1  n   δSOSMCðref ;iÞ ðt Þ oc Σ ρ ς i¼1

n   2 þc Σ 2ρi0  ςi;Σ δSOSMCði;Σ Þ ðt Þ  cv Σ δSOSMCði;Σ Þ ðt Þ i¼1 i¼1  n  n     r  cα Σ δSOSMCðref ;iÞ ðt Þ þ Σ δSOSMCði;Σ Þ ðt Þ



n

i¼1 n

i¼1

cv Σ δSOSMCði;Σ Þ ðt Þ 2

i¼1

ð66Þ

According to (66), if the following requirement is satisfied, it can be obtained that the proposed mean deviation coupling SOSMC system is asymptotically stable.  8   > > h_ i ðt Þ o ρi0 > > >   > >  n  > > < 1n Σ h_ j ðt Þ o ρi0 j¼1 ð67Þ > ςref ;i Z ρi0 þ α > > > > > ςi;Σ Z 2ρi0 þ α > > > : c 40; v 4 0; α 4 0 Thus, the mean deviation coupling SOSMC system guarantees the stability of the speed tracking error eref ;i ðt Þ and mean speed error ei;Σ ðt Þ simultaneously, even if parametric uncertainty and external disturbances. 3.4. Second-order adaptive sliding mode control The designed SOSMC not just maintain the advantages of the traditional SMC such as robustness and simplicity, but alleviate the chattering phenomena. From the Theorem 2, the switch gain ςref ;i and ςi;Σ are related to the magnitude of lumped uncertainty to keep the trajectory on the sliding surface. The selection of the upper bound of the uncertainty has a significant effect on the control performance. However, the upper bound of lumped uncertainty is difficult to know in advance in practical applications. The value of the upper bound can be chosen by the trial and error method to realize the requirement of convergence of speed tracking errors and synchronization errors in the mean deviation coupling SOSMC law, but it is time consuming and cannot give enough robustness practically [36]. Therefore, an adaption law is developed to estimate the upper bound of lumped uncertainty in the following section, which can alleviate the above drawbacks and minimize the control effort (47) and (55), so that the stability condition (64) can also be guaranteed via Lyapunov stability theorem.   So, the adaptive algorithm for the bound of hi ðt Þ is presented as   ς_ ref ;i ¼ cϖ ref ;i δSOSMCðref ;iÞ ðt Þ ð68Þ 



ς_ i;Σ ¼ cϖ i;Σ δSOSMCði;Σ Þ ðt Þ

Parameters

Motor 1

Motor 2

Motor 3

Motor 4

RS (Ω) Ld (H) Lq (H) ψf (Wb) J (kg m2) B (N m s) p

2.875 0.0085 0.0085 0.175 0.008 0.00051 1

2.875 0.0085 0.0085 0.172 0.0076 0.00047 1

2.88 0.0085 0.0085 0.177 0.0083 0.00056 1

2.88 0.0085 0.0085 0.177 0.0083 0.00056 1

Table 2 The load torques.

ref ;i

i0

7

ð69Þ

where the position constant ϖ ref ;i and ϖ i;Σ act like adaptive filters to minimize the control effort.

TL (N m)

Motor 1

Motor 2

Motor 3

Motor 4

0.08 r to 0.12 s 0.12r tr 0.2 s

2 5

4 3

3 2

1 6

Then, the adaptive switch gain ςref ;i and ςi;Σ can be rewritten as Z t   δSOSMCðref ;iÞ ðτÞdτ ςref ;i ¼ cϖ ref ;i ð70Þ 0

Z

ςi;Σ ¼ cϖ i;Σ

t 0

  δSOSMCði;Σ Þ ðτÞdτ

ð71Þ

Thus, the reaching control parts (49) and (57) of control effort are changed as

uref ;hiti ðt Þ ¼

Z

cϖ ref ;i ai0

t 0

Z

t 0

     δSOSMCðref ;iÞ ξ sgn δSOSMCðref ;iÞ ðτÞ dξdτ ð72Þ

Z

 vδSOSMCði;Σ Þ ðτÞ  Z t      δSOSMCði;Σ Þ ξ dξsgn δSOSMCði;Σ Þ ðτÞ dτ þ cϖ i;Σ

uhiti;Σ ðt Þ ¼ 

1 a0

t

0

0

ð73Þ

We select ς^ ref ;i and ς^ i;Σ as the estimation values of ςref ;i and ςi;Σ , respectively. The estimated errors can be defined as

ς~ ref ;i ¼ ςref ;i  ς^ ref ;i

ð74Þ

ς~ i;Σ ¼ ςi;Σ  ς^ i;Σ

ð75Þ

Proof 3. Choose the Lyapunov function candidate for the SOASMC system as V ðt Þ ¼

1 n 2 1 n 2 Σ δ ðt Þ þ Σ δSOSMCði;Σ Þ ðt Þ 2 i ¼ 1 SOSMCðref ;iÞ 2i¼1 1 n 1 1 n 1 2 2 þ Σ ς~ ðt Þ þ Σ ς~ ðt Þ 2 i ¼ 1 ϖ ref ;i ref ;i 2 i ¼ 1 ϖ i;Σ i;Σ

ð76Þ

Taking the time derivative of the Lyapunov function and using (65), one can obtain n

V_ ðt Þ ¼ Σ δSOSMCðref ;iÞ ðt Þδ_ SOSMCðref ;iÞ ðt Þ i¼1

n

þ Σ δSOSMCði;Σ Þ ðt Þδ_ SOSMCði;Σ Þ ðt Þ i¼1 n

þ Σ

i¼1

1

ϖ ref ;i

n

1

i¼1

ϖ i;Σ

ς~ ref ;i ς_~ ref ;i þ Σ

ς~ i;Σ ς_~ i;Σ

 n   ¼ Σ δSOSMCðref ;iÞ ðt Þ  ch_ i ðt Þ  cςref ;i sgn δSOSMCðref ;iÞ ðt Þ i¼1

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8

Fig. 3. System structure diagram of motor i.

mean deviation coupling control 2500

2000

2000 motor 1 motor 2 motor 3 motor 4

1500

1000

n/(r/min)

n/(r/min)

master-slave control 2500

500

0

motor 1 motor 2 motor 3 motor 4

1500

1000

500

0

0.05

0.1 Time (s)

0.15

0

0.2

0

0.05

2500

2000

2000 motor 1 motor 2 motor 3 motor 4

1500

1000

500

0

0.15

0.2

ring coupling control

2500

n/(r/min)

n/(r/min)

relative coupling control

0.1 Time (s)

1500

motor 1 motor 2 motor 3 motor 4

1000

500

0

0.05

0.1

0.15

0.2

0

0

0.05

Time (s)

0.1

0.15

0.2

Time (s)

Fig. 4. Speed curves of four motors with PI control in Case 1.



  þ Σ δSOSMCði;Σ Þ ðt Þ  cvδSOSMCði;Σ Þ ðt Þ  cςi;Σ sgn δSOSMCði;Σ Þ ðt Þ n

i¼1

  n c n 1  ςref ;i  ς^ ref ;i ς_~ ref ;i þ ch_ i ðt Þ  Σ h_ j ðt Þ þ Σ nj¼1 i ¼ 1 ϖ ref ;i n

þ Σ

i¼1

1 

ϖ i;Σ

 ςi;Σ  ς^ i;Σ ς~_ i;Σ

2

i¼1



   ςi;Σ δSOSMCði;Σ Þ ðt Þ þ h_ i ðt ÞδSOSMCði;Σ Þ ðt Þ  1 n  Σ h_ j ðt ÞδSOSMCði;Σ Þ ðt Þ nj¼1  n   þ c Σ ςref ;i  ς^ ref ;i δSOSMCðref ;iÞ ðt Þ n

þc Σ

i¼1

ð77Þ

Substituting (68) and (69) into (77) and considering (64) yields n    V_ ðt Þ ¼ c Σ  h_ i ðt ÞδSOSMCðref ;iÞ ðt Þ  ςref ;i δSOSMCðref ;iÞ ðt Þ i¼1

n

 cv Σ δSOSMCði;Σ Þ ðt Þ

i¼1 n

þc Σ

i¼1

  ςi;Σ  ς^ i;Σ δSOSMCði;Σ Þ ðt Þ



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9

e1,2

e2,3 100

master-slave control mean deviation coupling control relative coupling control ring coupling control

100

synchronization error (r/min)

synchronization error (r/min)

150

50

0

-50

-100

0

0.05

0.1

0.15

50

0

-50

-100

-150

0.2

master-slave control mean deviation coupling control relative coupling control ring coupling control

0

0.05

Time (s) e3,4 100

synchronization error (r/min)

synchronization error (r/min)

80 master-slave control mean deviation coupling control relative coupling control ring coupling control

40

0.15

0.2

e4,1

100

60

0.1 Time (s)

20 0

master-slave control mean deviation coupling control relative coupling control ring coupling control

50

0

-50

-100

-20 -40

0

0.05

0.1 Time (s)

0.15

0.2

-150

0

0.05

0.1 Time (s)

0.15

0.2

Fig. 5. Speed synchronization errors with PI control in Case 1.

   n  _  2 hi ðt Þ ςref ;i δSOSMCðref ;iÞ ðt Þ  cv Σ δSOSMCði;Σ Þ ðt Þ i¼1 i¼1       1 n n      þc Σ h_ i ðt Þþ  Σ h_ j ðt Þ  ςi;Σ δSOSMCði;Σ Þ ðt Þ n i¼1 j¼1  n   þc Σ ςref ;i  ς^ ref ;i δSOSMCðref ;iÞ ðt Þ n

rc Σ

i¼1 n 

þc Σ

i¼1

the stability and stability condition (64) of multiple motor synchronization control systems via the proposed mean deviation coupling SOASMC speed controller are guaranteed by selecting ς^ ref ;i Z ρi0 þ α and ς^ i;Σ Z 2ρi0 þ α.

  ςi;Σ  ς^ i;Σ δSOSMCði;Σ Þ ðt Þ

4. An example

   n    2 ¼ c Σ h_ i ðt Þ  ς^ ref ;i δSOSMCðref ;iÞ ðt Þ  cv Σ δSOSMCði;Σ Þ ðt Þ i¼1 i¼1       1 n n      þc Σ h_ i ðt Þþ  Σ h_ j ðt Þ  ς^ i;Σ δSOSMCði;Σ Þ ðt Þ nj¼1 i¼1  n  n  2 o c Σ ρi0  ς^ ref ;i δSOSMCðref ;iÞ ðt Þ  cv Σ δSOSMCði;Σ Þ ðt Þ n

i¼1

n

þc Σ i¼1  n r  cα Σ



i¼1 n

2ρi0  ς^ i;Σ

i¼1

 n     δSOSMCðref ;iÞ ðt Þ þ Σ δSOSMCði;Σ Þ ðt Þ

cv Σ δ i¼1

  δSOSMCði;Σ Þ ðt Þ i¼1

2 SOSMCði;Σ Þ ðt Þ

ð78Þ

Therefore, ς^ ref ;i and ς^ i;Σ can be chose so that the values of  ς^ ref ;i þ ρi0 þ α and  ς^ i;Σ þ 2ρi0 þ α remain negative. That is to say,

Comparative studies of four-motor synchronization control driving system are performed to demonstrate the effectiveness of the proposed control approach. The system control structure diagram of motor i is shown in Fig. 3. The nominal parameters of four motors used in the experiment are listed in Table 1. The parameters of the controllers are given as follows. k¼ 0.03, c¼4.5, v¼ 0.02, λ ¼0.45, ϖ ref ;i ¼460 and ϖ i;Σ ¼ 400 (i¼1,…, 4) are adopted for speed tracking controllers and mean speed deviation controllers. ρref ;i ¼370, ρi;Σ ¼230, ςref ;i ¼ 260, and ςi;Σ ¼ 200 are selected as the switching gain for the conventional SMC and SOSMC, respectively. The coefficients of PI speed tracking controllers are selected as KP(ref,1) ¼ KP(ref,2) ¼ KP (ref,3) ¼ KP(ref,4) ¼11.5; KI(ref,1) ¼ KI(ref,2) ¼ KI(ref,3) ¼ KI(ref,4) ¼138.9. The coefficients of PI mean speed deviation controllers are

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i

L. Li et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎

10

eref,1

eref,2

140

140 with PI control with SOASMC

120

100

tracking error (r/min)

tracking error (r/min)

100 80 60 40 20 0

80 60 40 20

-20

0

-40

-20

-60 0.05

with PI control with SOASMC

120

0.1

0.15

-40 0.05

0.2

0.1

Time (s) eref,3 160 with PI control with SOASMC

140

with PI control with SOASMC

140 120

tracking error (r/min)

120

tracking error (r/min)

0.2

eref,4

160

100 80 60 40 20

100 80 60 40 20

0

0

-20

-20

-40 0.05

0.15 Time (s)

0.1

0.15

0.2

-40 0.05

0.1

Time (s)

0.15

0.2

Time (s)

Fig. 6. Speed tracking errors with mean deviation coupling control in Case 2.

selected as KP(1,Σ) ¼ KP(2,Σ) ¼ KP(3,Σ) ¼ KP(4,Σ) ¼2.1; KI(1,Σ) ¼ KI(2,Σ) ¼ KI(3,Σ) ¼ KI(4,Σ) ¼10.2. The load torques are listed in Table 2. For simulation tests, the following cases are considered. The comparison study with master–slave control, relative coupling control, ring coupling control, conventional PI control and SMC is investigated. Case 1. The parameters of four motors are set as the nominal values, whereas the load torques TL of the four motors changed at t¼0.08 s and t¼ 0.12 s. The load torques are shown in Table 2. The speed command is 2000 r/min. Case 2. The moments of inertia J of the four motors changed as half of the nominal values at time t ¼0.08 simultaneously, whereas the other parameters are shown in Table 1. The load torques are set as TL1 ¼ 2 N  m, TL2 ¼4 N m, TL3 ¼3 N m and TL4 ¼ 1 N m at t¼0.12 s. The speed command is 2000 r/min. Case 3. The parameters are set as the nominal values, load torques of four motors are the same as that in Case 2, whereas the speed command changed from 2000 r/min to 1000 r/min at t ¼0.08 s. The simulation results in Case 1 are shown in Figs. 4 and 5.

Figs. 4 and 5 show the speed curves of four motors and speed synchronization error curves with master–slave control, mean deviation coupling control, relative coupling control and ring coupling control in Case 1, respectively. It can be seen that synchronization performance with mean deviation coupling control has little difference with relative coupling control, but much better than with master–slave control and ring coupling control. The speed synchronization errors with mean deviation coupling control converge to zero fast. These results show that mean deviation coupling control strategy is effective to synchronize the output speed of multiple motors. The simulation results in Case 2 are shown in Figs. 6–8. Figs. 6–8 illustrate the speed tracking error curves, speed synchronization error curves and mean speed error curves with PI control and SOASMC in Case 2, respectively. From the performance comparison, it can be seen that the actual speeds with the proposed mean deviation coupling SOASMC can track given value accurately, the precision of speed synchronization with mean deviation coupling SOASMC is much higher than with PI control and the mean speed errors with mean deviation coupling SOASMC are significantly less than with PI control despite parameters variation and external load disturbance. At t ¼0.08 s

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i

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11

e1,2

e2,3

30

20

synchronization error (r/min)

synchronization error (r/min)

15 20 10 0 -10 -20 -30

with PI control with SOASMC 0

0.05

10 5 0 -5 -10 -15 -20 with PI control with SOASMC

-25 0.1

0.15

-30

0.2

0

0.05

Time (s) e3,4

0.2

0.15

0.2

30

synchronization error (r/min)

10

synchronization error (r/min)

0.15

e4,1

15

5 0 -5 -10 -15 -20

0.1 Time (s)

0.05

10 0 -10 -20

with PI control with SOASMC 0

20

0.1 Time (s)

0.15

0.2

-30

with PI control with SOASMC 0

0.05

0.1 Time (s)

Fig. 7. Speed synchronization errors with mean deviation coupling control in Case 2.

and t ¼0.12 s, the speed tracking errors, synchronization errors and mean speed errors with SOASMC converge to zero rapidly, which shows the high robustness of the proposed control approach against parameters change and external load disturbance. The simulation results in Case 3 are shown in Figs. 9–11. Fig. 9 describes the control effort uq;2 and uq;4 with PI control, SMC and SOASMC, respectively. Fig. 10 shows the speed tracking curves of four motors with mean deviation coupling synchronization control strategy in Case 3. Fig. 11 shows the speed synchronization error curves with PI control, SMC and SOASMC, when the speed command changed from 2000 r/min to 1000 r/min at t¼ 0.08 s. It can be observed that the tracking precision with the proposed mean deviation coupling SOASMC is much higher than with PI control when the speed command changes from 2000 r/ min to 1000 r/min at t¼ 0.08 s. The speed synchronization error curves show that the synchronization performance with the proposed mean deviation coupling SOASMC has little effect due to the change of the speed command. Therefore, the high accuracy in speed tracking and speed synchronization of the proposed control scheme is verified in simulation.

5. Conclusions This study has successfully demonstrated the effectiveness of the proposed mean deviation coupling SOASMC scheme for multiple motor control systems. The dynamics of PMSM drive system with lumped uncertainty are described first. Then, a SMC is designed to stabilize multiple motor control systems. In order to better cope with the uncertainties, improve synchronization control precision of multiple motor control systems and make speed tracking errors, mean speed errors of each motor and speed synchronization errors converge to zero rapidly, the SOASMC is proposed for the synchronization control of multiple motors. In SOASMC, an adaptive law is employed to estimate the unknown bound of uncertainty, which is derived using the Lyapunov stability theorem to minimize the control effort and guarantee the asymptotical stability of the closed-loop system. Finally, the comparative studies with master–slave control, relative coupling control, ring coupling control, PI control and SMC have been conducted on a four-motor synchronization control system under various test conditions to verify the effectiveness of the proposed synchronization control scheme. Through simulation analysis and comparison, the proposed SOASMC has accurate synchronization control performance with robustness for multiple motor control systems.

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i

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12

e2, Σ 10

15

5

mean speed error (r/min)

mean speed error (r/min)

e1,Σ 20

10 5 0 -5 -10

-20

0

0.05

-5 -10 -15

with PI control with SOASMC

-20

with PI control with SOASMC

-15

0

0.1

0.15

-25

0.2

0

0.05

Time (s)

0.2

0.15

0.2

15 10

5

mean speed error (r/min)

mean speed error (r/min)

0.15

e4, Σ

e3, Σ 10

0

-5

-10

-15

0.1 Time (s)

0.05

0 -5 -10

with PI control with SOASMC 0

5

0.1

0.15

-15

0.2

with PI control with SOASMC 0

0.05

0.1 Time (s)

Time (s) Fig. 8. Mean speed errors with mean deviation coupling control in Case 2.

uq,4

25

25

20

20

15

15

10

10

current iq,4 (A)

current iq,2 (A)

uq,2

5 0 -5 -10

0 -5 -10

-15

with PI control with SMC with SOASMC

-20 -25

5

0

0.05

0.1

0.15

0.2

-15

with PI control with SMC with SOASMC

-20 -25

0

0.05

Time (s)

0.1

0.15

0.2

Time (s)

Fig. 9. Control effort with mean deviation coupling control in Case 3.

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i

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13

PI control

SOASMC

2500

2500 reference motor 1 motor 2 motor 3 motor 4

1500

2000

n/(r/min)

n/(r/min)

2000

1000

500

0

reference motor 1 motor 2 motor 3 motor 4

1500

1000

500

0

0.05

0.1

0.15

0

0.2

Time (s)

0

0.05

0.1

0.15

0.2

Time (s)

Fig. 10. Speed tracking curves of four motors with mean deviation coupling control in Case 3.

e1,2

e2,3 with PI control with SMC with SOASMC

20 10 0 -10

30

synchronization error (r/min)

synchronization error (r/min)

30

-20 -30

with PI control with SMC with SOASMC

20 10 0 -10 -20

0

0.05

0.1

0.15

-30

0.2

0

0.05

e3,4

0.15

0.2

e4,1

10

20

6 4 2 0 -2 -4 -6 -8

with PI control with SMC with SOASMC

15

synchronization error (r/min)

with PI control with SMC with SOASMC

8

synchronization error (r/min)

0.1 Time (s)

Time (s)

10 5 0 -5 -10

-10 -12

0

0.05

0.1 Time (s)

0.15

0.2

-15

0

0.05

0.1

0.15

0.2

Time (s)

Fig. 11. Speed synchronization errors with mean deviation coupling control in Case 3.

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References [1] Zhao DZ, Li CW, Ren J. Speed synchronisation of multiple induction motors with adjacent cross-coupling control. IET Control Theory A 2010;4:119–28. [2] Chen W, Wu YF, Du RH, Chen QW, Wu XB. Speed tracking and synchronization of a dual-motor system via second order sliding mode control. Math Probl Eng 2013. [3] Chou PH, Chen CS, Lin FJ. DSP-based synchronous control of dual linear motors via Sugeno type fuzzy neural network compensator. J Frankl Inst – Eng Appl Math 2012;349:792–812. [4] Xiao Y, Zhu KY. Optimal synchronization control of high-precision motion systems. IEEE Trans Ind Electron 2006;53:1160–9. [5] Rafique MA, Rehan M, Siddique M. Adaptive mechanism for synchronization of chaotic oscillators with interval time-delays. Nonlinear Dyn 2015;81:495–509. [6] Chen CS, Chen LY. Robust cross-coupling synchronous control by shaping position commands in multiaxes system. IEEE Trans Ind Electron 2012;59:4761–73. [7] Zhao DZ, Ding QQ, Zhang SM, Li CW, Stobart R. Integrated feedback scheduling and control codesign for motion coordination of networked induction motor systems. Math Probl Eng 2014;2014. [8] Zhan LX, Zhou K. Adaptive fuzzy sliding mode control for a robotic aircraft flexible tooling system. Int J Adv Manuf Technol 2013;69:1469–81. [9] Liu R, Sun JZ, Luo YQ, Sun W, Li WD. Research on multi-motor synchronization control based on the ring coupling strategy for cutterhead driving system of shield machines. Adv Mech Eng 2011;52–54:65–72. [10] Sun D, Shao X, Feng G. A model-free cross-coupled control for position synchronization of multi-axis motions: theory and experiments. IEEE Trans Control Syste Technol 2007;15:306–14. [11] Lin FJ, Chou PH, Chen CS, Lin YS. DSP-based cross-coupled synchronous control for dual linear motors via intelligent complementary sliding mode control. IEEE Trans Ind Electron 2012;59:1061–73. [12] Zhang Z, Chau KT, Wang Z. Chaotic speed synchronization control of multiple induction motors using stator flux regulation. IEEE Trans Magn 2012;48:4487– 90. [13] He WL, Qian F, Han QL, Cao JD. Synchronization error estimation and controller design for delayed lur'e systems with parameter mismatches. IEEE Trans Neur Net Learn 2012;23:1551–63. [14] Tomizuka M, Kamano T, Hu J-S, Chiu T-C. Synchronization of two motion control axes under adaptive feedforward control. ASME J Dyn Syst, Meas, Control 1992;114:196–203. [15] Bin Z, Feng G, Tian L. Normalized coupling method for speed synchronization of multi-axis driving vehicles. Int J Adv Robot Syst 2012:9. [16] Anderson RG, Meyer AJ, Valenzuela MA, Lorenz RD. Web machine coordinated motion control via electronic line-shafting. IEEE Trans Ind Appl 2001;37:247– 54. [17] Valenzuela MA, Lorenz RD. Electronic line-shafting control for paper machine drives. IEEE Trans Ind Appl 2001;37:158–64. [18] Koren Y. Cross-coupled biaxial computer control for manufacturing systems. J Dyn Syste Meas Control-Trans ASME 1980;102:265–72. [19] Sun D, Mills JK. Adaptive synchronized control for coordination of multirobot assembly tasks. IEEE Trans Robot Autom 2002;18:498–510. [20] Cheng MH, Li YJ, Bakhoum EG. Controller synthesis of tracking and synchronization for multiaxis motion system. IEEE Trans Control Syst Technol 2014;22:378–86. [21] Cao LZ, Li CW, Niu C, Zhao DZ, Wei SB. Synchronized sliding-mode control for multi-induction motors based on adjacent cross-coupling. Dianji yu Kongzhi Xuebao/Electric Mach Control 2008;12:586–92. [22] Perez-Pinal FJ, Calderon G, Araujo-Vargas I. Relative coupling strategy. In: Proceedings of the electric machines and drives conference, 2003 IEMDC'03 IEEE International. IEEE, Madison; 2003. p. 1162–6.

[23] Sun J, Liu R, Luo Y, Sun W. Research on multi-motor synchronization control for cutter head of shield machine based on the ring coupled control strategy. Intell Robot Appl Springer 2009:345–54. [24] Ouyang PR, Dam T, Pano V. Cross-coupled PID control in position domain for contour tracking. Robotica 2015;33:1351–74. [25] Zhao DY, Li SY, Gao F. Fully adaptive feedforward feedback synchronized tracking control for Stewart Platform systems. Int J Control Autom 2008;6:689–701. [26] Barton KL, Alleyne AG. A cross-coupled iterative learning control design for precision motion control. IEEE Trans Control Syste Technol 2008;16:1218–31. [27] Zhao D, Li S, Gao F, Zhu Q. Robust adaptive terminal sliding mode-based synchronised position control for multiple motion axes systems. IET Control Theory A 2009;3:136–50. [28] Zhao DY, Zhu QM. Position synchronised control of multiple robotic manipulators based on integral sliding mode. Int J Syst Sci 2014;45:556–70. [29] Shao J, Tan P, Xu Y, Jin XJ, Malekian R. An improved synchronous control strategy based on fuzzy controller for PMSM. Elektron Elektrotech 2014;20:17–23. [30] Lin FJ, Hsieh HJ, Chou PH, Lin YS. Digital signal processor-based cross-coupled synchronous control of dual linear motors via functional link radial basis function network. IET Control Theory A 2011;5:552–64. [31] Lee HC, Jeon GJ. A neuro-controller for synchronization of two motion axes. Int J Intell Syst 1998;13:571–86. [32] Veselic B, Perunicic-Drazenovic B, Milosavljevic C. Improved discrete-time sliding-mode position control using euler velocity estimation. IEEE Trans Ind Electron 2010;57:3840–7. [33] Loukianov AG, Canedo JM, Fridman LM, Soto-Cota A. High-order block slidingmode controller for a synchronous generator with an exciter system. IEEE Trans Ind Electron 2011;58:337–47. [34] Shao XL, Wang HL. Sliding mode based trajectory linearization control for hypersonic reentry vehicle via extended disturbance observer. ISA Trans 2014;53:1771–86. [35] Beltran B, Ahmed-Ali T, Benbouzid ME. High-order sliding-mode control of variable-speed wind turbines. IEEE Trans Ind Electron 2009;56:3314–21. [36] Lin FJ, Hung YC, Ruan KC. An intelligent second-order sliding-mode control for an electric power steering system using a wavelet fuzzy neural network. IEEE Trans Fuzzy Syst 2014;22:1598–611. [37] Saghafinia A, Ping HW, Uddin MN, Gaeid KS. Adaptive fuzzy sliding-mode control into chattering-free IM drive. IEEE Trans Ind Appl 2015;51:692–701. [38] Manceur M, Essounbouli N, Hamzaoui A. Second-order sliding fuzzy interval type-2 control for an uncertain system with real application. IEEE Trans Fuzzy Syst 2012;20:262–75. [39] Girin A, Plestan F, Brun X, Glumineau A. High-order sliding-mode controllers of an electropneumatic actuator: application to an aeronautic benchmark. IEEE Trans Control Syst Technol 2009;17:633–45. [40] Amodeo M, Ferrara A, Terzaghi R, Vecchio C. Wheel slip control via secondorder sliding-mode generation. IEEE Trans Intell Transp 2010;11:122–31. [41] Na J, Ren XM, Zheng DD. Adaptive control for nonlinear pure-feedback systems with high-order sliding mode observer. IEEE Trans Neur Net Learn 2013;24:370–82. [42] Lin FJ, Shen PH. Robust fuzzy neural network sliding-mode control for twoaxis motion control system. IEEE Trans Ind Electron 2006;53:1209–25. [43] Zhang BT, Pi YG, Luo Y. Fractional order sliding-mode control based on parameters auto-tuning for velocity control of permanent magnet synchronous motor. ISA Trans 2012;51:649–56. [44] Qi L, Shi HB. Adaptive position tracking control of permanent magnet synchronous motor based on RBF fast terminal sliding mode control. Neurocomputing 2013;115:23–30. [45] Foo G, Rahman MF. Sensorless sliding-mode MTPA control of an IPM synchronous motor drive using a sliding-mode observer and HF signal injection. IEEE Trans Ind Electron 2010;57:1270–8.

Please cite this article as: Li L, et al. Mean deviation coupling synchronous control for multiple motors via second-order adaptive sliding mode control. ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.01.015i