Multi-motor servo system cooperative control method based on adaptive observer

Abstract

The invention aims to provide a cooperative control algorithm for realizing motor synchronization and load position tracking so as to effectively solve the problem of coupling control of motor synchronization and load tracking and the problems of unknown system state, unknown nonlinearity and parameter uncertainty estimation and compensation, so that a multi-motor servo system has the characteristics of rapidness, accuracy and stability. In order to achieve the purpose, the technical scheme of the invention provides a multi-motor servo system cooperative control method based on an adaptive observer. Aiming at a multi-motor servo system, through the steps of model conversion, self-adaptive observer design, self-adaptive cooperative control design and the like, the influence of an unknown system state, unknown parameters and nonlinearity is solved, and rapid motor synchronization and accurate load position tracking control are realized.

Description

Technical Field

[0001] The invention relates to a cooperative control method for a multi-motor servo system. Background Art

[0002] The motor servo system is a complex coupled nonlinear system composed of a drive motor and a load. It is widely used in industrial fields such as industrial steel rolling because of its high response speed and driving ability. With the continuous development of industry, the demand for high-driving force, high-power and high-precision motor servo systems is increasing. However, due to the limited driving capacity of a single motor and the high cost of high-power motors, the original single-motor drive system can no longer meet the requirements. Therefore, multi-motor servo control systems have gradually attracted widespread attention and become a research hotspot. Unlike single-motor servo systems, multi-motor servo systems not only have nonlinear phenomena such as friction nonlinearity and parameter uncertainty, but also have complex coupling relationships between motors, between motors and loads, and between motor synchronization and load tracking. This problem has a huge impact on the accuracy and stability of multi-motor servo systems, thereby limiting their wide application in practice. However, existing research often focuses on single load tracking control or only considers the synchronization problem between motors, without considering the relationship and control method between the two at the same time.

[0003] Therefore, how to design a control strategy to simultaneously solve the coordinated control problems of motor synchronization and load tracking in order to achieve fast, accurate and steady-state control effects has become the subject of current research. Summary of the invention

[0004] The purpose of the present invention is to provide a collaborative control algorithm for realizing motor synchronization and load position tracking, so as to effectively solve the coupling control problem of motor synchronization and load tracking, as well as the unknown system state, unknown nonlinearity, parameter uncertainty estimation and compensation problems, so that the multi-motor servo system can achieve fast, accurate and stable characteristics.

[0005] In order to achieve the above object, the technical solution of the present invention is to provide a multi-motor servo system cooperative control method based on an adaptive observer, characterized in that it includes the following steps:

[0006] Step 1: Design the system model of the multi-motor drive servo system:

[0007] The kinematic model of the design consideration system is shown as follows:

[0008]

[0009] In the formula, θ mi and θ l are the angular displacements of motor i and load, respectively, and are the angular velocities of motor i and load, J and J l is the moment of inertia of motor i and load, is the friction force at the motor end i, b l is the viscous friction coefficient at the load end, u i is the input torque of motor i, τ i is the torque transmitted from motor i to the load;

[0010] Let the position and speed of the load and motor i be the state variables of the system x1 = θ l , x 3i =θ mi , Then the state equation of the multi-motor drive servo system is:

[0011]

[0012] In the formula, f l is the unknown nonlinear term, f(x 4i ) is the friction force at the i end of the motor;

[0013] Step 2: Design an adaptive variable gain observer:

[0014] The design of the adaptive variable gain observer is shown as follows:

[0015]

[0016] In the formula, is the estimated value of state χ, χ=[x1,x2] T , is the estimated value of the unknown parameter ρ, ρ=[k / J l kα / J l -b l / J l ] T , α is the gap width, k is the torque coefficient, z i =θ mi -θ l , L=[L1,L2] T is the time-varying gain, r is the robust term, where the designed time-varying gain L is shown as follows:

[0017]

[0018] Where: l1>l0>0 is a positive constant and l0≤L≤l1; constant k t >0; is the state estimation error;

[0019] Step 3: Design an adaptive robust cooperative controller to achieve load tracking and motor synchronization:

[0020] Step 301: Design a neural network tracking controller as shown in the following equation:

[0021] u t = (-k1s - k2s / (|s| + 1) - u w ) / b t

[0022] where: s is the switching sliding surface, defined as s(t) = e2(t) + βf a (e1(t)), β is a positive constant, e2 is the speed error, e1 is the tracking error, and there is y d is the tracking signal, f t is the unknown nonlinear term, f a (e1(t)) is defined as positive constant 0 < q < 1, parameter δ > 0;

[0023] k1, k2 > 0 are control gains, and -k2s / (|s| + 1) is the robust control term used to eliminate the estimation error and external disturbance;

[0024] u w is the neural network compensation term used to compensate for the unknown nonlinearity, defined as is the estimated value of W t T W t T is the expected weight of the neural network, Φ is the basis vector;

[0025] b t = n / J l , n is the number of motors;

[0026] Step 302: Design a motor synchronization controller as shown in the following equation:

[0027]

[0028] where, k s > 0 is the synchronization control gain, is the neural network control term used to compensate for the influence of the transmitted torque and friction nonlinearity on the motor synchronization control;

[0029] Step 303: Combine the neural network tracking controller and the motor synchronization controller to obtain the cooperative controller as:

[0030]

[0031] where, is a normal number, is the time-varying gain, e si is the synchronization error.

[0032] Preferably, in step 1, the transmission torque τ from motor i to the load i The simplified model is:

[0033] τ i = k(z i -αtanh(ρz i ))+ε i

[0034] In the formula, ρ is a positive constant, ε i is the bounded approximation error of motor i.

[0035] Preferably, in step 1, Defined as:

[0036]

[0037] In the formula, β 1i , β 2i , β 3i , β 4i , β 5i and β 6i is an unknown parameter.

[0038] Preferably, in step 1, the unknown nonlinear term f l Defined as:

[0039]

[0040] Where n is the number of motors.

[0041] Preferably, in step 2, for the state estimation error, there is an error state equation as shown below:

[0042]

[0043] In the formula, is a nonlinear function of the error state variable, And there exists a positive constant γ that satisfies |Δ|≤γ.

[0044] Preferably, in step 2, design The adaptive rate is:

[0045]

[0046] Where Γ is a positive definite matrix and λ is a positive constant.

[0047] Preferably, in step 2, the robust term r is designed to be:

[0048]

[0049] Where r0 is a positive constant and satisfies r0>γ, and parameter r1>0.

[0050] Preferably, in step 301, the unknown nonlinear term f t Defined as:

[0051]

[0052] Preferably, in step 301, The adaptive rate is:

[0053]

[0054] In the formula, Γ t is a positive definite matrix, g t Is a normal number.

[0055] Preferably, in step 301, the parameter The adaptive rate is:

[0056]

[0057] Where: si is a positive definite matrix;

[0058] g si is a positive constant;

[0059] e si and e sn is the synchronization error, expressed as:

[0060] e si =x 4i -x 4i+1 ,i=1,2,…,n-1

[0061] e sn =x 4n -x 41 ,

[0062] And the synchronization error dynamic equation is:

[0063]

[0064] The technical solution disclosed in the present invention is applied to a multi-motor drive servo system. The present invention is aimed at a multi-motor servo system, and solves the influence of unknown system states, unknown parameters and nonlinearity through steps such as model conversion, adaptive observer design, and adaptive collaborative control design, so as to achieve fast motor synchronization and accurate load position tracking control. The present invention can achieve synchronization of servo motors and control the motors to push loads to achieve position tracking, and has functions such as motor synchronization control, load tracking control, speed estimation, and adaptive compensation.

[0065] Compared with the prior art, the present invention adopts a variable gain adaptive observer strategy and a cooperative control strategy, which can realize multi-motor control when the load state is unmeasurable, and is conducive to realizing fast synchronization and precise tracking control of a multi-motor servo system. In the model conversion part of the present invention, the original complex dead zone model is converted into a linear parameter form by using linear parameterization technology, thereby simplifying the model and facilitating controller design. In the observer part of the present invention, a variable gain strategy is adopted to increase the convergence rate of the observation error and reduce external disturbances, so as to avoid the influence of the estimated curve oscillation on the observation accuracy. In the control part of the present invention, a switching function algorithm is used to design a switching sliding surface, which can effectively improve the convergence speed of the system tracking error, reduce the jitter of the tracking curve, and enhance the dynamic performance of the system. In the present invention, a time-varying gain compensation method based on synchronization error is adopted, which can reduce the mutual influence between the two under the premise of realizing load tracking and motor synchronization control, so as to facilitate the precise tracking control of the load. The present invention can better coordinate tracking and synchronization control, increase the steady-state and dynamic performance of the system, and provide a new research and development idea and application method for the development of other products and algorithms in the field of multi-motor servo. It can produce quite valuable economic benefits and social impact in the future. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] Figure 1 It is a schematic diagram of a multi-motor driven servo system;

[0067] Figure 2 This is the control block diagram of the servo system;

[0068] Figure 3 Estimation curve diagram for load position;

[0069] Figure 4 It is a load speed estimation curve graph;

[0070] Figure 5 It is a sine tracking curve diagram;

[0071] Figure 6 It is the sinusoidal tracking error curve;

[0072] Figure 7 It is the motor speed synchronization curve diagram;

[0073] Figure 8 This is the motor synchronization error curve. DETAILED DESCRIPTION

[0074] The present invention will be further described below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not intended to limit the scope of the present invention. In addition, it should be understood that after reading the content taught by the present invention, those skilled in the art can make various changes or modifications to the present invention, and these equivalent forms fall within the scope limited by the appended claims of the application equally.

[0075] Conventional multi-motor drive servo systems such as Figure 1 As shown in the figure, its main components are multiple servo motors, transmission mechanisms, and loads. The system has complex friction and dead zone nonlinearity, unknown system parameters (such as friction parameters, torque parameters, etc.), and complex coupling control relationships (such as motor synchronization and load tracking), so it has the characteristics of high coupling and high complexity.

[0076] Combination Figure 2 , a multi-motor servo system collaborative control method based on an adaptive observer disclosed in an embodiment of the present invention includes the following contents:

[0077] 1. System model

[0078] The multi-motor driven servo control system includes multiple drive motors, transmission links and load links. Multiple motors work through gear transmission links to achieve the purpose of jointly driving the load movement. The kinematic model of the system is considered as:

[0079]

[0080] In formula (1), θ mi and θ l are the angular displacements of motor i and load, respectively, and are the corresponding angular velocities, J and J l is the moment of inertia of motor i and load, is the friction force at the motor end i, b l is the viscous friction coefficient at the load end, u i is the input torque of motor i, τ i is the torque transferred from the motor to the load and is defined as:

[0081] τ i = kf(z i ) (2)

[0082] In formula (2):

[0083]

[0084] And zi =θ mi -θ l , α is the tooth gap width, and k is the torque coefficient.

[0085] In order to facilitate the design of the controller, a simplified model of the transmitted torque is proposed:

[0086] τ i = k(z i -αtanh(ρz i ))+ε i (4)

[0087] In formula (4), ρ is a positive constant, ε i is a bounded approximation error.

[0088] In addition, the continuous friction model is defined for:

[0089]

[0090] In formula (5), β 1i , β 2i , β 3i , β 4i , β 5i and β 6i is an unknown parameter.

[0091] Let the position and speed of the load and motor be the state variables of the system x1=θ l , x 3i =θ mi , Then the state equation of the multi-motor drive servo system is:

[0092]

[0093] In formula (6), f(x 4i ) is the friction force at the motor i end, f l is an unknown nonlinear term and is defined as:

[0094]

[0095] In formula (7), n is the number of motors.

[0096] Model conversion is the basic part of the present invention. According to the dynamic relationship of the servo system, the dynamic models of the motor and the load are established respectively. Taking into account the shortcomings of the original dead zone model of the torque transmission, which is piecewise discontinuous and non-differentiable, a linear parameterized form is proposed, which can effectively reduce the design complexity. In addition, since the Stribeck friction model is too complex containing exponential terms, the present invention designs a continuously differentiable friction model to replace the original model. After the above-mentioned modeling conversion, the original discontinuous dead zone and friction nonlinearity are converted into a continuous model. Compared with the traditional servo system model, the above modeling method simplifies the original complex system model, thereby facilitating the controller design.

[0097] 2. Design of Adaptive Variable Gain Observer

[0098] Considering the problem that the load position can be measured but the speed is difficult to measure in actual engineering, an adaptive observer is used to realize the speed estimation of the system with unknown nonlinear load. Define χ=[x1,x2] T , then the state equation of the load can be rewritten as:

[0099]

[0100] In formula (8), ρ=[k / J l kα / J l -b l / J l ] T ,

[0101] To achieve speed estimation, the following adaptive observer is designed

[0102]

[0103] In formula (9), is the estimated value of state χ, is the estimated value of the unknown parameter ρ, r is the robust control term used to eliminate external disturbances, L = [L1, L2] T is a time-varying gain, which is used to ensure the convergence of the estimated state, and there exists a positive definite matrix P l and Q l satisfy:

[0104] (A-Lc T ) T P l +P l (A-Lc T )=-Q l (10)

[0105] The state estimation error is defined as Then, by combining equations (8) and (9), we can get the error state equation:

[0106]

[0107] In formula (11), is a nonlinear function of the error state variable, And there exists a positive constant γ that satisfies |Δ|≤γ.

[0108] To ensure the estimation error The convergence of , the designed observer gain L is:

[0109]

[0110] In formula (12), L1 and L2 are the time-varying gains of the observer, l1>l0>0 is a positive constant and l0≤L≤l1, and the constant k t >0 is used to control the decay rate of the time-varying gain. The time-varying gain can be gradually reduced as the estimation error converges. Therefore, compared with the traditional controller, the observer can use a high gain when the error is large, increase the convergence rate and reduce external disturbances. When the error is small, a small gain is used to avoid the oscillation of the estimation curve affecting the observation accuracy.

[0111] Considering the influence of unknown parameters on state estimation, the design parameter adaptation rate is

[0112]

[0113] In formula (13), Γ is a positive definite matrix and λ is a positive constant.

[0114] In addition, in order to further reduce the influence of parameter adaptation error and external disturbance, the robust term is designed as:

[0115]

[0116] In formula (14), r0 is a positive constant and satisfies r0>γ, and parameter r1>0.

[0117] 3. Collaborative control

[0118] Different from single motor control, the control of multi-motor servo system needs to consider both load position tracking and motor synchronization to ensure fast and smooth operation of the system. To this end, an adaptive robust cooperative controller will be designed to achieve load tracking and motor synchronization.

[0119] To achieve load following control, the state equation is transformed into

[0120]

[0121] In formula (15), u t is the tracking controller, b t= n / J l , f t is an unknown non - linear term and is defined as:

[0122]

[0123] Define the tracking error as e1 = x1 - y d , then the dynamic equation of the tracking error of the servo system is:

[0124]

[0125] In equation (17), y d is the tracking signal, and e2 is the velocity error.

[0126] Design a control strategy for the tracking error. First, propose the switching sliding surface as:

[0127] s(t) = e2(t) + βf a (e1(t)) (18)

[0128] In equation (18), β is a positive constant, and f a (e1(t)) is defined as:

[0129]

[0130] In equation (19), the positive constant 0 < q < 1 affects the convergence rate of the system, δ > 0 is a small parameter to achieve function switching. The function (19) is a continuously differentiable function. When |e1| > δ, it achieves fast finite - time convergence. When |e1| ≤ δ, it achieves linear convergence to reduce curve fluctuations.

[0131] Combining equations (18) and (19) and differentiating equation (18) gives

[0132]

[0133] In equation (20),

[0134] Use a neural network to approximate the non - linear f m to obtain:

[0135]

[0136] In equation (21), W t T is the expected weight of the neural network, F(×) is the basis vector, ε is the approximation error, and there exist positive constants W N , ε N satisfying ||W t T || ≤ W Nand ||ε||≤ε N .

[0137] For the load tracking control problem, the following neural network tracking controller is designed based on the switching sliding surface shown in formula (18):

[0138] u t =(-k1s-k2s / (|s|+1)-u w ) / b t (twenty two)

[0139] In formula (22), k1, k2>0 is the control gain, and -k2s / (|s|+1) is the robust control term used to eliminate estimation errors and external disturbances, while the neural network compensation term u w Used to compensate for unknown nonlinearity, defined as W t T The estimated value of and the adaptive rate is:

[0140]

[0141] In formula (23), Γ t is a positive definite matrix, g t Is a normal number.

[0142] For the motor synchronous control problem, the synchronization error is defined as

[0143] e si =x 4i -x 4i+1 ,i=1,2,…,n-1

[0144] e sn =x 4n -x 41 , (twenty four)

[0145] Combined with the motor state equation shown in equation (3), the dynamic equation of synchronization error can be obtained as follows:

[0146]

[0147] According to formula (25), since the same tracking controller is used, the tracking control does not affect the synchronous control effect. Based on the error dynamic equation, the following motor synchronous controller is designed:

[0148]

[0149] In formula (26), k s >0 is synchronous control gain, is a neural network control item, which is used to compensate for the influence of transmission torque and friction nonlinearity on motor synchronous control, and the parameter The adaptive rate is

[0150]

[0151] In formula (27), Γ si is a positive definite matrix, g si Is a normal number.

[0152] Combining the tracking controller and the synchronization controller, the cooperative controller can be obtained as:

[0153]

[0154] In formula (28), is a positive constant, which is used to determine the synchronization control weight. In order to further reduce the impact of synchronization control on tracking control, a time-varying gain based on synchronization error is introduced into the controller. This function will continue to decrease as the synchronization error converges, thereby weakening the impact of synchronization control on tracking.

[0155] This strategy greatly reduces the interaction between synchronization control and tracking control while ensuring rapid synchronization of the motor speed, thereby achieving coordinated control of load tracking and motor synchronization at the same time.

[0156] The simulation is carried out by taking a dual-motor servo system as an example. The state estimation effect of the invention is as follows: Figure 3 and Figure 4 As shown in the figure, it can be seen from the simulation results that the estimation algorithm disclosed in the present invention can achieve accurate estimation of load position and speed. Although there is a certain estimation error in the initial stage, the variable gain function used in the estimation algorithm disclosed in the present invention can ensure that the error converges quickly. In addition, the adaptive strategy used in the estimation algorithm disclosed in the present invention can better compensate for unknown nonlinear interference.

[0157] Figure 5 and Figure 6 The sine signal tracking curve and tracking error curve of the present invention, wherein the reference signal is y d =2sin(2πt / 5). Figure 5 It can be seen that the estimation algorithm disclosed in the present invention can achieve accurate tracking of the load. There is no peaking phenomenon or hysteresis phenomenon at the maximum value of the tracking curve, which shows that the control algorithm can effectively eliminate the influence of friction nonlinearity and dead zone nonlinearity on the system control performance. In addition, the tracking error curve Figure 6It can be seen that the estimation algorithm disclosed in the present invention can ensure that the load accurately tracks the expected signal and has a small steady-state error. Although there is a small overshoot in the initial stage, the estimation algorithm disclosed in the present invention can achieve rapid convergence of the tracking error.

[0158] Figure 7 and Figure 8 The motor synchronization curve and synchronization error curve of the present invention. In order to clearly show the synchronization effect of the algorithm, the initial speed values ​​of the two motors are set to 0.8rad / s and 0rad / s respectively in the simulation. It can be seen from the simulation diagram that the estimation algorithm disclosed in the present invention can ensure that motors with different speeds reach the synchronous state, and has the advantages of small overshoot, fast convergence speed and low steady-state synchronization error. Therefore, the estimation algorithm disclosed in the present invention has good steady-state and transient performance in synchronization control.

[0159] It can be seen from the above simulation that the present invention can achieve accurate load tracking and rapid motor synchronization when the load state is unknown. Although the synchronous control has a certain impact on the tracking control in the initial stage, resulting in overshoot and oscillation in the tracking curve, the collaborative control strategy adopted by the present invention can effectively weaken this impact and make the tracking error converge quickly. Therefore, the present invention can effectively solve the problem of multi-motor control and realize the collaborative control of load tracking and motor synchronization.

Claims

1. A cooperative control method for multi-motor servo system based on adaptive observer, It is characterized in that The following steps are involved: Step 1: Design the system model of the multi-motor drive servo system: The kinematic model of the design consideration system is shown as follows: In the formula, θ mi and θ l are the angular displacements of motor i and load, respectively, and are the angular velocities of motor i and load, J and J l is the moment of inertia of motor i and load, is the friction force at the motor end i, b l is the viscous friction coefficient at the load end, u i is the input torque of motor i, τ i is the torque transmitted from motor i to the load; Let the position and speed of the load and motor i be the state variables x of the system 1 =θ l , x 3i =θ mi , Then the state equation of the multi-motor drive servo system is: In the formula, f l is the unknown nonlinear term, f(x 4i ) is the friction force at the i end of the motor; Step 2: Design an adaptive variable gain observer: The design of the adaptive variable gain observer is shown as follows: In the formula, is the estimated value of state χ, χ=[x 1 ,x 2 ] T , is the estimated value of the unknown parameter ρ, ρ=[k / J l kα / J l -b l / J l ] $ , α is the gap width, k is the torque coefficient, z i =θ mi -θ l , L=[L 1 ,L 2 ] T is the time-varying gain, r is the robust term, where the designed time-varying gain L is shown as follows: Where: l 1 > 0 >0 is a positive constant and l 0 ≤L≤l 1 ; Constant k t >0; is the state estimation error; Step 3: Design an adaptive robust cooperative controller to achieve load tracking and motor synchronization: Step 301, design a neural network tracking controller as shown below: u t =(-k 1 s-k 2 s / (|s|+1)-u w ) / b t where: s is the switching sliding mode surface, defined as s(t) = e 2 (t) + βf a (e 1 (t)), β is a positive constant, e 2 is the velocity error, e 1 is the tracking error, and there is y d is the tracking signal, f t is the unknown nonlinear term, f a (e 1 (t)) is defined as a positive constant 0 < q < 1, parameter δ > 0; k 1 ,k 2 >0 is the control gain, and -k 2 s / (|s|+1) is a robust control term used to eliminate estimation errors and external disturbances; u w The neural network compensation term is used to compensate for unknown nonlinearity and is defined as W t T The estimated value of W t T is the expected weight of the neural network, Φ is the basis vector; b t =n / J l , n is the number of motors; Step 302: Design a motor synchronous controller as shown below: In the formula, k s >0 is synchronous control gain, It is a neural network control item, which is used to compensate for the influence of transmission torque and friction nonlinearity on motor synchronous control; Step 303: Combine the neural network tracking controller and the motor synchronization controller to obtain a coordinated controller: In the formula, is a positive constant, is the time-varying gain, e si is the synchronization error.

2. A method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 1, the torque τ transmitted from motor i to the load is i The simplified model is: t i =k(z i -atanh(ρz i ))+e i In the formula, ρ is a positive constant, ε i is the bounded approximation error of motor i.

3. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 1, Defined as: In the formula, β 1i , β 2i , β 3i , β 4i , β 5i and β 6i is an unknown parameter.

4. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 1, the unknown nonlinear term f l Defined as: Where n is the number of motors.

5. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 2, for the state estimation error, there is an error state equation as shown below: In the formula, is a nonlinear function of the error state variable, And there exists a positive constant γ that satisfies |Δ|≤γ.

6. A method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 2, design The adaptive rate is: Where Γ is a positive definite matrix and λ is a positive constant.

7. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 2, the robust term r is designed as: In the formula, r 0 is a positive constant and satisfies r 0 >γ, parameter r 1 >0.

8. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 301, the unknown nonlinear term f t Defined as:

9. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 301, The adaptive rate is: In the formula, Γ t is a positive definite matrix, g t Is a normal number.

10. The method for cooperative control of a multi-motor servo system based on an adaptive observer as claimed in claim 1, It is characterized in that In step 301, the parameters The adaptive rate is: Where: Γ si is a positive definite matrix; g si is a positive constant; e si and e sn is the synchronization error, expressed as: e si =x 4i -x 4i+1 ,i=1,2,…,n-1 e sn =x 4n -x 41 , And the synchronization error dynamic equation is:

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