^{2} College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha 410073, China
Highprecision synchronization control technology is extensively used in manufacturing industries such as printing, aerospace, textile, and steel rolling^{[13]}. The higher precision design of it for system with nonlinearity and disturbances is the key to its application in manufacturing.
Traditional synchronization control strategies include masterslave control, crosscoupling control, relativecoupling control, and electronic lineshafting (ELS) control^{[4]}. As a typical one among them, ELS is extensively used in engineering application due to its excellent synchronization performance^{[5, 6]}. Lorenz^{[7]} conducted extensive research in this area, and Payette^{[8, 9]} also made contributions in ELS development. Valenzuela and Lorenz^{[10]} described the physical meaning of ELS and applied it to printing machines. Based on the previous achievements, Perez^{[11]} extended the study of ELS control strategy and perfected the ELS algorithm, Wolf and Lorenz^{[12]} proposed compensationbased ELS control to eliminate the effect of tension on printing paper. Furthermore, improved trackingcontrol algorithms were developed to increase the synchronization precision of multiaxis systems^{[13, 14]}.
In a multiaxis motion system, synchronization is required in the case of load variation. Because the real load torque is timevarying and not measurable, existing ELSbased control strategies take the virtual load torque (i.e., the coupling torque) of each axis as their load torque "feedback". These feedback signals act together on the virtual shaft to form a coordinated signal for the motion synchronization of all the axes. Under steadystate conditions, this feedback control can provide excellent synchronization performance. However, if a severe unknown load disturbance occurs in the system, loss of synchronization among the axes probably happens. To address this problem, a novel ELS control strategy is proposed in this paper. By constructing an observer, the equivalent load torque is observed, and the observed value is directly fed back to the virtual shaft.
The paper is organized as follows: Section 2 describes a direct current (DC) motor system. Section 3 introduces the traditional ELS control method. Section 4 presents a novel ELS structure. Section 5 describes the control design. Section 6 presents experimental results and conclusions in Section 7.
2 Mathematical model of the systemSynchronous coordinated running of each printing roller of a shaftlessdriven printing press is accomplished mainly by independentdrive servo motors. Register control is carried out if chromatic aberration occurs. This control scheme results in fast, accurate, and steady control of the system with respect to object position and velocity. There are many models of servo drive motors. But to reflect a practical implementation more precisely, it is desirable to take parameter variation, friction, and load torque into consideration in the model. Therefore, a DC motor is selected as the servo motor type. The dynamic system can be described by
$ \begin{eqnarray} \left\{ \begin{array}{l} \dot x_1 {\rm{ = }}x_2 \\ \dot x{}_2{\rm{ =  }}\dfrac{{k_m c_e }}{{JR}}x_2 {\rm{ + }}k_u \dfrac{{k_m }}{{JR}}u(t){\rm{  }}\dfrac{1}{J}F_f {\rm{(}}t{\rm{)  }}\dfrac{1}{J}{\rm{T}}_L \\ \end{array} \right. \end{eqnarray} $  (1) 
where
When
$ \begin{eqnarray} F_f {\rm{(}}t{\rm{) = }}\left\{ \begin{array}{l} F_m \begin{array}{*{20}c} {},&{\rm if}\quad {F_t > F_m } \\ \end{array} \\ F_t^{} \begin{array}{*{20}c} {},&{\rm if}\quad {{\rm{  }}F_m {\rm{ < }}F < F_m } \\ \end{array} \\ {\rm{  }}F_m \begin{array}{*{20}c} {},&{\rm if}\quad {F_t {\rm{ <  }}F_m }. \\ \end{array} \\ \end{array} \right. \end{eqnarray} $  (2) 
When
$ \begin{eqnarray} F_f {\rm{(}}t{) = [F}_c {\rm{ + (}}F_m {\rm{  }}F_c {\rm{)}}{\rm e}^{{\rm{  }}\sigma _2 \left {\dot \theta } \right} {\rm{]}}{\mathop{\rm sgn}} {\rm{(}}\dot \theta {\rm{) + }}k_v \dot \theta. \end{eqnarray} $  (3) 
While the driving force
$ \begin{eqnarray} F(t) = J \ddot \theta \end{eqnarray} $  (4) 
where
It is assumed that
$ \begin{eqnarray} \left\{ \begin{array}{l} \dot x_1 = x_2 \\ \dot x_2 = a_1 x_2 { + b}u(t){\rm{ + }}a_2 F_f {\rm{(}}t{\rm{) + }}a_2 {\rm{T}}_L \\ \end{array} \right. \end{eqnarray} $  (5) 
where
Define
$ \begin{eqnarray} \left\{ \begin{array}{l} \dot x_1 = x_2 \\ \dot x_2 = \bar a_1 x_2 + \bar bu(t) + \bar a_2 F_f (t) + T_L^ * .\\ \end{array} \right. \end{eqnarray} $  (6) 
The mechanical shaft driven system provides power to each individual servo drive unit by means of a mechanical shaft and maintains their motion synchronization simultaneously. In the ELS control strategy, a virtual shaft is used to replace the mechanical shaft by realistically simulating its physical characteristics. Each axis follows the motion of the virtual shaft and couples with it by torque integration and feedback. When the signal from the virtual shaft control system is applied to the mechanical shaft, a reference signal for each unit controller is obtained. In other words, the reference input signal, instead of the system input signal, is synchronized among all unit controllers. Because it is a filtered signal applied to the shaft, it can be tracked more easily by the unit controllers, and can improve synchronization performance. The ELS control structure is shown in Fig. 1.
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Fig. 1. Structure of traditional ELS system 
Fig. 1 shows that the virtual electric line shaft, known as the master, involves the adjustment of shaft velocity or position and provides the velocity or position reference value to its slaves. Under steadystate conditions, every axis follows the virtual shaft, and excellent synchronization performance can be achieved. If one or more axes deviate from the reference value due to a disturbance, torque integration and feedback enables the virtual shaft to sense this variation and produce an adjusted reference value for the other slaves. In this manner, synchronization is always achieved among the axes. The torque balancing equation is
$ \begin{eqnarray} T{\rm{  }}\sum T_i {\rm{ = }}J_m \ddot \theta_m \end{eqnarray} $  (7) 
where
$ \begin{eqnarray} T_i = b_r \Delta \omega + k_r \Delta \theta + k_{ir} \int {\Delta \theta {\kern 1pt} {\kern 1pt} {\rm d}t} \end{eqnarray} $  (8) 
where
Because the load torque of the servo drive
$ \begin{eqnarray} T{\rm{  }}\sum \hat T_{Li}^ * {\rm{ = }}J_m \ddot \theta _m \end{eqnarray} $  (9) 
where
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Fig. 2. Structure of improved ELS system 
5 Observer design for synchronization system
Fig. 2 shows that every servo drive unit under the control of virtual shaft has symmetric structure. Therefore, the same control algorithm can be applied to each of them. Taking axis 1 for example, the design of a tracking controller and an equivalent loadtorque observer for servo motor No. 1 is described in this section.
5.1 Design of the tracking controllerFor a shaftless driven printing machine, each paper roller is driven by an independent servo motor, and in case of chromatic aberration, register control is carried out. These lead to higher requirements on multiaxis synchronization control in which the driving motor is considered as a complex object with nonlinear and variable parameters. With common PID control, satisfactory speed regulation and positioning are hard to achieve due to the weak robustness to disturbances and parameter variations.
In contrast, with the slidingmode variable structure control, a type of nonlinear and discontinuous control, the system is capable to obtain excellent robustness which is actually called invariability through keeping motion within sliding surface^{[16]}. Together with its simplicity, high realtime performance, and easy implementation, this method applied to highprecision tracking control has attracted significant attention^{[17]}. This section describes the design of a slidingmode controller for a singleaxis system which guarantees not only the excellent tracking performance but also the high performance of observer. Moreover, it is also applicable to a multiaxis system with symmetrical motors.
As shown in Fig. 2,
Choose the linear sliding surface
$ \begin{eqnarray} s = ce + \dot e \end{eqnarray} $  (10) 
where
The output of the sliding mode controller can be obtained from
$ \begin{align} u(t) = \frac{1}{{\bar b}}_{_{} } \left[ {c\dot e + \ddot \theta _m + \beta (x, t){\mathop{\rm sgn}} (s) + ks  \bar a_1 x_2  \bar a_2 F_f (t)} \right] \end{align} $  (11) 
where
Assume that
If design
$ \begin{eqnarray} \beta (x, t) = M + \eta \end{eqnarray} $  (12) 
then
A sigmoid function
$ \begin{eqnarray} \lambda {\rm{(}}s{\rm{) = }}\frac{2}{{1 + {\rm e}^{  as} }}  1 \end{eqnarray} $  (13) 
where
From (6), the sliding mode observer is designed as
$ \begin{eqnarray} \left\{ \begin{array}{l} \dot {\hat x}_1 = \hat x_2 + v_1 \\ v_1 = L_1 {\mathop{\rm sgn}} (x_1  \hat x_1 ) \\ \end{array} \right. \end{eqnarray} $  (14) 
$ \begin{eqnarray} \left\{ \begin{array}{l} \dot{\hat x}_2 = \bar a_1 \hat x_2 + \bar bu(t) + \bar a_2 F_f (t) + v_2 \\ v_2 = L_2 {\mathop{\rm sgn}} (x_2  \hat x_2 ) \\ \end{array} \right. \end{eqnarray} $  (15) 
where
Define the estimation errors of the observer as
$ \begin{eqnarray} \left\{ \begin{array}{l} e_1 = x_1  \hat x_1 \\ e_2 = x_2  \hat x_2. \\ \end{array} \right. \end{eqnarray} $  (16) 
From
$ \begin{eqnarray} \left\{ \begin{array}{l} \dot e_1 = e_2  v_1 \\ \dot e_2 = \bar a_1 e_2 + T_L^ *  v_2 . \\ \end{array} \right. \end{eqnarray} $  (17) 
Proposition 1. Consider the system described by (6) and its observers described by (14) and (15). If
Proof.
The first step: Define the sliding surface
$ \begin{eqnarray} s_1 = e_1 \end{eqnarray} $  (18) 
The second step: Choose the Lyapunov function
Along the trajectory of system (17), the derivative of the Lyapunov function with respect to time is
$ \begin{eqnarray} &\dot V_1 = e_1 \dot e_1 = e_1 e_2  e_1 v_1 =\notag\\ &\qquad e_1 e_2  e_1 L_1 {\mathop{\rm sgn}} e_1 =\notag\\ &\;\; e_1 e_2  L_1 \left {e_1 } \right \le\notag\\ &\qquad\left {e_1 } \right\left {e_2 } \right  L_1 \left {e_1 } \right =\notag\\ &\left {e_1 } \right\left[ {\left {e_2 } \right  L_1 } \right]. \end{eqnarray} $  (19) 
When
$ \begin{eqnarray} s_1 = \dot s_1 = 0. \end{eqnarray} $  (20) 
Then, from (14)
$ \begin{eqnarray} e_2 = L_1 {\mathop{\rm sgn}} e_1. \end{eqnarray} $  (21) 
Now
The third step: Define the sliding surface
$ \begin{eqnarray} s_2 = e_2. \end{eqnarray} $  (22) 
The fourth step: Choose the Lyapunov function
$ \begin{eqnarray} \dot V_2 = e_1 \dot e_1 + e_2 \dot e_2 = \dot V_1 + e_2 \dot e_2. \end{eqnarray} $  (23) 
Applying the same reasoning as in the proof of the second step, we obtain
Then, from (17)
$ \begin{eqnarray} \begin{array}{l} \dot V_2 \le e_2 \dot e_2 = \bar a_1 e_2^2 + \left {e_2 } \right\left[ {T_L^ *  L_2 } \right]= \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \;  \left {\bar a_1 } \righte_2^2 + \left {e_2 } \right\left[ {T_L^ *  L_2 } \right]. \\ \end{array} \end{eqnarray} $  (24) 
When
$ \begin{eqnarray} s_2 = \dot s_2 = 0. \end{eqnarray} $  (25) 
The estimated value of the equivalent load torque can be obtained from (14)(18), (23) and (25):
$ \begin{eqnarray} \hat T_L^ * = v_2 = L_2 {\mathop{\rm sgn}} e_2 = L_2 {\mathop{\rm sgn}} \left[ {L_1 {\mathop{\rm sgn}} e_1 } \right]. \end{eqnarray} $  (26) 
Now the load torque
Based on the above analysis, the procedure of observer design for system synchronization can be summarized as follows:
Step 1. Design the system control algorithm of a singleaxis system according to (11) in order to maintain the tracking performance of the system.
Step 2. Replace the sgn(s) in (11) with the sigmoid function in (13) in order to minimize slidingmodel chattering. This chattering reduction method will be used again in the subsequent design.
Step 3. Construct the observer according to (14) and (15) and calculate the estimated value of equivalent load torque according to (26).
Step 4. Carry out ELS synchronization control according to Fig. 2 and (9).
6 Experiments 6.1 Setting experimental parametersTaking a fouraxis printing machine with shaftless drive for example, a Matlab simulation model was built. Considering that in printing machine, the strong system nonlinearity only happens during the low speed preregister stage, a lowamplitude, lowfrequency sinusoidal signal
6.2 Analysis of the experimental results 6.2.1 Case 1. Tracking performance
Use the sinusoidal signal sin(4
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Fig. 3. Tracking performance of singleaxis system 
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Fig. 4. Estimated value of the designed observer 
6.2.2 Case 2. Multiaxis synchronization coordination performance
To verify the effectiveness of the method proposed in this paper, a comparative experiment was performed based on the traditional ELS control method shown in Fig. 1 and the improved ELS control method shown in Fig. 2.
1) Synchronization performance without disturbance.
2) Synchronization performance with disturbance.
At 2 s, a 3 N step signal was added on axis 1 to simulate an abrupt load change.
As Fig. 5 shows, if there is no any external disturbance, both control schemes achieve excellent tracking performance by means of welldesigned controllers. However, by examining the details shown in Figs. 6 and 7, it apparently shows that taking coupling torque as the torque feedback leads to a constant deviation between the main reference value and each axis, so that the tracking error of each axis deviates from zero to the balance point. It implies that the traditional ELS achieves high tracking performance with the cost of tracking performance loss in each axis.
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Fig. 5. Tracking performance of multiaxis system without disturbance 
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Fig. 6. Synchronization errors of multiaxis system without disturbance 
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Fig. 7. Synchronization errors of multiaxis system with disturbance 
In contrast, the improved ELS introduces a solution based on the equivalent load torque observer. By taking the observed value of the equivalent load torque as the torque feedback, the control system can guarantee not only high synchronization performance, but also excellent tracking performance on each axis. In addition, slidingmode variable structure keeps system operation status stable in case of disturbance. Furthermore, it plays its role on improving overall robustness, steadystate performance, and synchronization performance.
7 ConclusionsFocusing on improving the synchronization performance of traditional ELS control, an equivalent load observer based control strategy is proposed in this paper. First, slidingmode controllers are designed for each axis in a multiaxis synchronization system to achieve high tracking performance and improve system resistance to disturbances. Second, the equivalent loadtorque observer is designed to get the observer equivalent torque value of each axis for eliminating the constant deviation between the main reference value and each axis. This control strategy is especially helpful in maintaining synchronized motion control when severe load disturbances occur. Experimental results have demonstrated the effectiveness of the proposed control strategy for multiaxis systems when taking unexpected load disturbances into account during normal operations.
AcknowledgementsThis work was supported by Natural Science Foundation of China (Nos. 61773159 and 61473117), Hunan Provincial Natural Science Foundation of China (No. 13JJ8020 and 14JJ5024), and Hunan Province Education Department (No. 12A040).
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