- Email: [email protected]
Bidirectional synchronization control for an electrohydraulic servo loading system
Mechatronics 62 (2019) 102254
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Bidirectional synchronization control for an electrohydraulic servo loading system ✩ Yong Sang∗, Weiqi Sun, Fuhai Duan, Jianlong Zhao School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Keywords: Electrohydraulic servo loading system Synchronism Coupling BSILC
a b s t r a c t Synchronism is one of the most important factors for electrohydraulic servo loading systems with multi-axis. As electrohydraulic servo loading systems have a large output force/torque, the coupling between each axis can cause great nonlinearities and uncertainties, which seriously deteriorates the performance of the systems. In this paper, an electrohydraulic servo loading system is used for studying its bidirectional synchronization control. The mechanism of the coupling between two cylinders from two directions is discussed and the experiment results of coupling are obtained. A bidirectional synchronization iterative learning control (BSILC) method is designed to improve the tracking performance, robustness and compensate the synchronization error. The simulation and the experiment have been conducted. The results indicate that the control strategy proposed in this paper is valid. The tracking performance, robustness and synchronism of the electrohydraulic servo loading system have been remarkably improved.
1. Introduction Electrohydraulic servo loading systems have the advantages of a large output of force/torque, a high power to weight ratio, quick response and excellent anti-disturbance performance, and they have been widely applied in many important fields for control and power transmission [1,2]. With the development of electrohydraulic servo loading systems, there has been a considerable and growing demand for the synchronism of bidirectional outputs [3]. Traditional synchronous drive systems can be divided into three categories based on their control structures: the master–slave structure, the cross-coupling structure, and the model reference structure. In a master–slave structure system, the output of the master system is used as the input of the slave system [4,5]. In this case, the synchronization error can only be reduced by the slave system itself. In a cross-coupling structure system, the synchronization error is fed back to each input simultaneously to find a globally optimal solution for all of the outputs [6,7]. In a model reference structure system, each output is connected to a reference model in parallel with an individual independent synchronization controller [8,9]. In this paper, a bidirectional synchronization control for an electrohydraulic servo loading system is investigated for its tracking performance and robustness for the periodic input signal. Synchronization control methods have been widely applied in controllers to solve the synchronization problems between systems. A synchronization control
✩ ∗
scheme and system modeling for a single-axis stage driven by dual mechanically coupled parallel ball screws have been presented [10]. A precise position synchronization control of a multi-axis rotating system has been described, and the acceleration and speed controllers have been readily designed via the proportional-integral (PI) control law [11]. Motion control techniques have been proposed to improve the dynamic synchronization accuracy between translational and rotary axes in five-axis machining centers with the servo gain tuning method, feed-forward controller, signal delay and backlash compensator, rotational fluctuation compensator, and jerk limited acceleration process [12]. A two-layer sliding mode synchro-system based on friction compensation has been applied to electro-pneumatic cylinders, and a synchro-PID controller has been utilized for position tracking [13]. Modern control theories have provided many excellent control methods for improving the synchronism of bidirectional output systems, such as adaptive control, robust control, fuzzy control, and neural network control [14–17]. For the robustness of the algorithm with respect to faults, a proportional integral observer has been presented for the actuator and sensor faults estimation based on a TS fuzzy model [18]. A fault-tolerant sliding mode control has been proposed for Takagi–Sugeno fuzzy systems with time-varying delay and actuator saturation [19]. Many types of research have shown good results in improving electrohydraulic servo loading systems, but there are still limitations. A robust H∞ sliding mode control with pole placement has adequately
This paper was recommended for publication by Associate Editor Kim A. Stelson. Corresponding author. E-mail address: [email protected] (Y. Sang).
https://doi.org/10.1016/j.mechatronics.2019.102254 Received 25 December 2018; Received in revised form 2 July 2019; Accepted 20 July 2019 Available online 21 July 2019 0957-4158/© 2019 Elsevier Ltd. All rights reserved.
Y. Sang, W. Sun and F. Duan et al.
compensated for the nonlinear friction of a fluid power electrohydraulic actuator (EHA) system, but it remains in the simulation stage and has not been used in actual experiments [20]. The optimal design has effectively improved the dynamic characteristics of an electrohydraulic hydraulic valve for heavy-duty vehicle clutch actuators with certain constraints, but it cannot optimize the entire electrohydraulic servo loading system [21]. Control strategies applied together in hydraulic hybrid excavators have shown good results in reducing fuel consumption, but their optimal results cannot be optimized in unknown working cycles [22]. In this paper, a bidirectional synchronization iterative learning control (BSILC) method is designed to improve tracking performance and robustness and to reduce the synchronization error between the two cylinders from two directions in an electrohydraulic servo loading system. The traditional synchronization control structures generally show a weak ability to compensate for the synchronization error, and the cross-coupling structure system performs better in reducing the synchronization error than the other two structures. However, these control structures cannot improve the tracking performance of the entire system, which becomes even worse as the synchronism improves. Therefore, a master–slave learning structure is used for the controller in this paper to improve both tracking performance and synchronism together. The BSILC method does not rely on a mathematical model of the system. It has an excellent tracking performance for the repetitive mode executions, which can adjust the slave output to the master output and reduce the synchronization error quickly. Compared with the other control methods mentioned above, it has a smaller operation cycle and is more likely to be realized in a practical system. The simulations and experiments for the BSILC methods have been conducted. The results indicate that the control method applied in this paper is valid, and the tracking performance and robustness and the synchronization error between the two cylinders from two directions in the electrohydraulic servo loading system have been remarkably improved. The paper is organized as follows: Section 2 introduces the components of the electrohydraulic servo loading system and constructs its mathematic model. Section 3 explains the mechanism of the coupling between two cylinders from two directions and shows its experiment results. Section 4 demonstrates the design of the BSILC controller. Section 5 conducts the simulation and performances analysis for the BSILC controller. Section 6 carries out the experiment to test the BSILC controller. Section 7 comes to the conclusions.
Mechatronics 62 (2019) 102254
Fig. 1. The electrohydraulic servo loading system experiment platform.
Fig. 2. The deformation of the sample.
2. Electrohydraulic servo loading system In this paper, a dynamic triaxial apparatus is used as the experiment platform of the electrohydraulic servo loading system. A photo of the experiment platform is shown in Fig. 1. The dynamic triaxial apparatus is the most important device for studying complex geotechnical dynamics in the laboratory. The device can be used to study the instability, failure, and deformation of the foundation caused by waves from an earthquake, wind, or the ocean. It has been widely applied in the aseismic geotechnical engineering design of high-speed railways, expressways, subways, large bridges, and huge dams. In the laboratory, the dynamic triaxial apparatus usually exerts periodic sine wave loads for studying stress–strain properties, strength, and other mechanical properties of the soil. The performance indices of the electrohydraulic servo loading system shown in Fig. 1 are as follows. For the hydraulic flow power, the rated flow is <70 L/min, and the rated pressure is <21 MPa. The hydraulic gear pump has a certified capacity of 50 mL/r (PH-5B-50, Nachi, Japan). For the driving motor (Y-Type, Dalian Motor Group Co. Ltd, China), the rated speed is 1450 r/min, and the rated power is 37 kW. The stroke of the axial loading cylinder is 200 mm (CK-63/36-200, Atos, Italy). The stroke of the confining loading cylinder is 200 mm (CK50/22-200, Atos, Italy). The two electrohydraulic servo valves are the same type (D633, Moog, USA), the rated flow is 20 L/min, the hysteresis is <0.2%, and the maximum operating pressure is <35 MPa. Watercooling is used, and the heat-exchange area is >1.5 m2 . The operating
pressure of the hydraulic flow power is regulated with a pilot proportional relief valve (AGMZO, Atos, Italy), and its regulation range is 0.7– 21 MPa. The axial loading range is 0–10,000 N, and the confining pressure range is 0–4000 kPa. The pore pressure of the high precision water pressure sensor used is 0–4000 kPa. The stroke of the displacement sensor is 0–75 mm, and its precision is <0.1%. High acquisition precision can be achieved by a 16-bit AD converter. An LVDT type sensor is used as the displacement sensor, and its precision is < 0.1%. High precision pressure sensors are used for the left water discharge, right water discharge, upper pore pressure, lower pore pressure, and confining pressure, and each has a precision of <0.1%. All output signals are 0–10 V voltage signals. The measurement of water discharge is indirectly obtained by measuring the pressure of the water head. The internal force sensor and the external force sensor are the same. A VISHAY tension sensor is used as the force sensor. It ranges from 0 to 2000 kg, and its linearity is <0.02%. Fig. 2 shows the loading process of the sample. The principle diagram of the electrohydraulic servo loading system is shown in Fig. 3, including a hydraulic flow power, two servo amplifiers, two hydraulic servo valves, two loading cylinders, a water cylinder, a force sensor, a pressure sensor, a displacement sensor, a high speed AD/DA card and computers. The computers are mainly used to generate loading waveforms, operate various control algorithms, display real-time data, generate and store the data report.
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 3. The principle diagram of the electrohydraulic servo loading system.
The electrohydraulic servo loading system is a force closed loop system, which utilizes two electrohydraulic servo valves to control two hydraulic servo cylinders for the axial loading and confining pressure, respectively. The voltage signals are used as its input and feedback signals. The mathematic model of the electrohydraulic servo loading system is described as follows: 𝑥̇ 1 = 𝑥2 1 𝑥̇ 2 = (−𝑘𝑎 𝑥1 − 𝑏𝑎 𝑥2 + 𝐴𝑎 𝑥3 ) 𝑚𝑎 √ 𝑥̇ 3 = −𝛼𝑎 𝑥2 − 𝛽𝑎 𝑥3 + (𝛾𝑎 𝑃𝑎 − sgn(𝑥4 )𝑥3 )𝑥4 𝐾 1 𝑥̇ 4 = − 𝑥4 + 𝑎 𝑢1 𝜏𝑎 𝜏𝑎 𝑥̇ 5 = 𝑥6 1 𝑥̇ 6 = (−𝑘𝑐 𝑥5 − 𝑏𝑐 𝑥6 + 𝐴𝑐 𝑥7 ) 𝑚𝑐 √ 𝑥̇ 3 = −𝛼𝑐 𝑥6 − 𝛽𝑐 𝑥7 + (𝛾𝑐 𝑃𝑐 − sgn(𝑥8 )𝑥7 )𝑥8 𝐾 1 𝑥̇ 8 = − 𝑥8 + 𝑐 𝑢2 𝜏𝑐 𝜏𝑐 𝑦1 = 𝑘𝑎 𝑥1 𝑘 𝑥 𝑦2 = 𝑐 5 𝐴𝑤
pressure piston; y1 is the axial load; and y2 is the confining pressure. The corner mark of the axial loading variable is a and the corner mark of the confining pressure variable is c. m is the mass of the actuator piston; k is the stiffness of the external load; b is the friction damping coefficient of the piston; A is the cross-sectional area of the piston; P is the supply pressure of the electrohydraulic servo valve; 𝜏 is the time constant of the electrohydraulic valve; K is the input gaining of the electrohydraulic valve; and Aw is the cross-sectional area of the water piston. 𝛼 = 4 𝐴 𝛽𝑒 ∕ 𝑉 𝑡
(2)
𝛽 = 4𝐶𝑡𝑚 𝛽𝑒 ∕𝑉𝑡
(3)
√ 𝛾 = 4𝐶𝑑 𝛽𝑒 𝑤∕(𝑉𝑡 𝜌)
(4)
where 𝛽 e is the effective bulk modulus; Vt is the total actuator volume; Ctm is the total coefficient of the internal leakage; Cd is the discharge coefficient; w is the spool valve area gradient; and 𝜌 is the fluid mass density. 3. The mechanism of the coupling between two cylinders from two directions (1)
where x1 is the axial loading piston displacement; x2 is the axial loading piston velocity; x3 is the loading pressure in the axial loading piston; x4 is the electrohydraulic valve displacement controlling the axial loading piston; x5 is the confining pressure piston displacement; x6 is the confining pressure piston velocity; x7 is the loading pressure in the confining pressure piston; x8 is the electrohydraulic valve displacement controlling the confining pressure piston; u1 is the input current to electrohydraulic servo valve controlling the axial loading piston; u2 is the input current to electro-pneumatic servo valve controlling the confining
The coupling between two cylinders from two directions is a significant practical phenomenon in electrohydraulic servo loading systems that greatly impacts tracking performance, robustness, and synchronism. The process of the coupling is shown in Figs. 4 and 5, where the symbols in red represent the inputs of the system, and the symbols in green represent the outputs of the system. Fig. 4 shows the compression coupling process. In Fig. 4(a), when the water piston controlling the confining pressure stays still, it can be seen that the confining pressure in the vessel is supposed to remain unchanged. However, as the axial loading piston compresses the sample, a part of the piston rod enters
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 4. The compression process of the electrohydraulic servo loading system. (a) Coupling from axial loading to confining pressure. (b) Coupling from confining pressure to axial loading.
Fig. 5. The decompression process of the electrohydraulic servo loading system. (a) Coupling from axial loading to confining pressure. (b) Coupling from confining pressure to axial loading.
into the closed space of the vessel and squeezes the water in it. Water has a very large bulk modulus of approximately 2.2 GPa, which directly results in a great pressure increase. The larger the displacement of the axial loading piston is, the higher the confining pressure of the water will be. Moreover, as the confining pressure grows, the sample is compressed in the radial direction and consequently expands in the axial direction according to the Poisson effect, which increases the axial load in turn. In Fig. 4(b), when the axial loading piston stays still, it can be seen that the axial loading is supposed to remain unchanged. However, the sample is compressed by the water as the water piston squeezes the water in the closed space of the vessel, increasing the confining pressure, and
thereby increasing the axial load in the same way as that mentioned in Fig. 4(a). The higher the confining pressure is, the larger the axial load will be. Fig. 5 shows the decompression coupling process. In Fig. 5(a), when the water piston controlling the confining pressure stays still and the axial loading piston is drawn back from the vessel, a part of the piston rod exits from the closed space of the vessel, resulting in a decrease in the water pressure. The larger the displacement of the axial loading piston is, the lower the confining pressure of the water will be. Moreover, as the confining pressure drops, the sample expands in the radial direction and consequently shrinks in the axial direction according to the Poisson
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 7. The pressure coupled from axial loading to confining loading.
Fig. 6. The force coupled from confining loading to axial loading.
effect, which decreases the axial load in turn. In Fig. 5(b), when the axial loading piston stays still and the water piston pumps out the water in the closed space of the vessel, thus reducing the confining pressure, the sample is recovered, which decreases the axial load in the same way as that mentioned in Fig. 5(a). The lower the confining pressure is, the smaller the axial load will be. The coupling process mentioned above indicates that the mechanism of the coupling can cause great nonlinearities and uncertainties for the electrohydraulic servo loading system. The deformation of a sample is a very complicated process. To illustrate the effect of the coupling in a simulation, four assumptions are defined as follows [23]: 1. Continuity assumption: The volume of the sample is full of the sample’s matter without any void space. 2. Homogenization assumption: The mechanical characteristics of any part of the sample are the same. 3. Isotropy assumption: The mechanical characteristics of the sample are the same in all directions. 4. The assumption of small deformation: The deformation of the sample caused by external forces is far less than its original size. In this paper, the ideal outputs of the dynamic triaxial apparatus are 𝑦∗ = 4000 sin (2𝜋 ∗ 0.5𝑡) (N)
(5)
𝑦∗ = −500 sin(2𝜋 ∗ 0.5𝑡) + 2000 (kPa)
(6)
1
2
When the confining loading pressure is set as Eq. (5), the input of the axial loading force is set as a constant. The experiment results of the axial coupling force are obtained from Fig. 6. In Fig. 6, the amplitude of the axial coupling force is approximately 600 N, which is 15% of the ideal axial loading force, and the coupling waveform is no longer a standard sine wave, which shows great distortion. When the axial loading force is set as Eq. (6), the input of the confining pressure is a constant. The experiment results of the confining coupling pressure can be obtained from Fig. 7. In Fig. 7, the amplitude of the confining coupling pressure is approximately 150 kPa, which is 30% of the ideal confining loading pressure. The experiment results demonstrate that the effect of the coupling is significant enough to seriously deteriorate the tracking performance, robustness, and synchronism of the dynamic triaxial apparatus. Therefore, an effective control method is needed to compensate for the coupling between two cylinders from two directions.
4. Design of the BSILC controller for two cylinders from two directions The BSILC controller is designed based on the state-space representation (1). The general form of Eq. (1) can be represented as 𝐱𝑗 (𝑘 + 1) = 𝐀𝐱𝑗 (𝑘) + 𝐁𝐮𝑗 (𝑘) 𝐲𝑗 (𝑘) = 𝐂𝐱 𝑗 (𝑘)
(7)
where A is the state matrix; B is the input matrix; and C is the output matrix. The state-space representation mentioned above can be applied to any system with a finite iteration length. The system can be represented in the frequency domain by extending the length of the cycle T → ∞: [
] [ ][ ] 𝑌1,𝑗 (𝑧) 𝑃 (𝑧) 0 𝑈1,𝑗 (𝑧) = 1 𝑌2,𝑗 (𝑧) 0 𝑃2 (𝑧) 𝑈2,𝑗 (𝑧) [ ][ ] [ ] 𝑃 (𝑧) 0 𝑥1 (0) 𝐷1 (𝑧) + 1,0 + 0 𝑃2,0 (𝑧) 𝑥5 (0) 𝐷2 (𝑧)
(8)
𝑌1,𝑗 (𝑧) ] is the representation of yj (k) in the frequency domain; 𝑌2,𝑗 (𝑧) Y1, j (z) is the driving output; and Y2, j (z) is the following output. The transfer matrix is given by where [
[
] 𝑃1 (𝑧) 0 = 𝐂(𝑧𝐈 − 𝐀)−1 𝐁 0 𝑃2 (𝑧) [ 𝑝 𝑧−1 + 𝑝1,22 𝑧−2 + 𝑝1,3 𝑧−3 + ⋯ = 1,1 0
]
0 𝑝2,1
𝑧−1
+ 𝑝2,2
𝑧−2
+ 𝑝2,3
𝑧−3
+⋯ (9)
where 𝑝𝑘,𝑖 = 𝐂𝐀𝑖−1 𝐁, (𝑘 = 1, 2; 𝑖 = 1, 2, 3, ⋯), and pk,i is the Markov Parameter [24]. Assuming that the initial condition is zero, the impulse responses of the system can be given by these Markov Parameters. The system’s relative degree, m, corresponds to the first nonzero Markov Parameter. The synchronization ILC is applied to the two-cylinder systems with repeated finite cycles, and the input–output relation of the first order iteration can be represented with the following matrix representation: [
] [ ] [ ] 𝐲1,𝑗 𝐮 𝐝 = 𝐏 1,𝑗 + 1 𝐲2,𝑗 𝐮2,𝑗 𝐝2
(10)
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
where ⎡ 𝑦1,𝑗 (1) ⎤ ⎡ 𝑢1,𝑗 (0) ⎤ ⎡ 𝑑1 (0) ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 𝑦1,𝑗 (2) ⎥ ⎢ 𝑢1,𝑗 (1) ⎥ ⎢ 𝑑1 (1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⋮ ⎥ ⋮ ⋮ ⎥ [ ] ⎢ ⎥ [ ] ⎢ [ ] ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝐲1,𝑗 𝑦1,𝑗 (𝑁)⎥ 𝐮1,𝑗 𝑢1,𝑗 (𝑁 − 1)⎥ 𝐝1 𝑑1 (𝑁 − 1)⎥ ⎢ ⎢ ⎢ = ; = ; = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝑦 (1) 𝑢 (0) 𝑑 (0) 𝐲2,𝑗 𝐮2,𝑗 𝐝2 2,𝑗 2,𝑗 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 𝑦2,𝑗 (2) ⎥ ⎢ 𝑢2,𝑗 (1) ⎥ ⎢ 𝑑2 (1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⋮ ⋮ ⎢ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝑦 ( 𝑁) 𝑢 ( 𝑁 − 1) 𝑑 ( 𝑁 − 1) ⎣ 2,𝑗 ⎦ ⎣ 2,𝑗 ⎦ ⎣ 2 ⎦
⎡ 𝐂1 𝐁1 ⎢ 𝐂1 𝐀1 𝐁1 ⎢ ⋮ ⎢ ⎢𝐂1 𝐀1 𝐍−𝟏 𝐁1 𝐏=⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0 𝐂1 𝐁1 ⋮ 𝐂1 𝐀1 𝐍 −𝟐 𝐁1
⋯ ⋯ ⋱ ⋯
0 0 ⋮ 𝐂1 𝐁1
𝐎
(11)
𝐎 𝐂2 𝐁2 𝐂2 𝐀2 𝐁2 ⋮ 𝐂2 𝐀2 𝐍 −𝟏 𝐁2
⋯ ⋯ ⋱ ⋯
0 𝐂2 𝐁2 ⋮ 𝐂2 𝐀2 𝐍 −𝟐 𝐁2
0 0 ⋮ 𝐂2 𝐁2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (12)
⎡ 𝑝1,1 ⎢ 𝑝1,2 ⎢ ⎢ ⋮ ⎢𝑝 = ⎢ 1,𝑁 ⎢ ⎢ ⎢ ⎢ ⎣
0 𝑝1,1 ⋮ 𝑝1,𝑁−1
⋯ ⋯ ⋱ ⋯
0 0 ⋮ 𝑝1,1
𝐎
𝐎 𝑝2,1 𝑝2,2 ⋮ 𝑝2,𝑁
⋯ ⋯ ⋱ ⋯
0 𝑝2,1 ⋮ 𝑝2,𝑁−1
0 0 ⋮ 𝑝2,1
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
𝐎 𝐂1 ], 𝐁 = [𝐁1 𝐁2 ], 𝐂 = [ ] 𝐎 𝐀2 𝐂2 Due to Eq. (7), p1,1 and p2,1 are calculated as follows:
where 𝐀 = [
𝐀1
𝑝 1 , 1 = 𝐂 1 𝐁 1 =0
(13)
𝑝 2 , 1 = 𝐂 2 𝐁 2 =0
(14)
When 𝐂1 𝐁1 =0 and 𝐂2 𝐁2 =0, a small change of the sample time will yield C1 B1 ≠ 0 and C2 B2 ≠ 0, so two relative degrees of 1 can always be obtained by adapting the sample times T1 and T2 [25]. Then, the synchronization ILC law constructs the current trial input as [ ] 𝐮1,𝑗+1 (𝑘) 𝐮2,𝑗+1 (𝑘)
[ =
] 𝐮1,𝑗 (𝑘) 𝐮2,𝑗 (𝑘)
+
[ 𝐤1,𝑝 0
0 𝐤2,𝑝
][
] 𝐞1,𝑗 (𝑘) 𝐞2,𝑗 (𝑘)
+
[ 𝐤1,𝑑 0
0 𝐤2,𝑑
][ ] 𝐞1,𝑗 (𝑘 + 1) − 𝐞1,𝑗 (𝑘) 𝐞2,𝑗 (𝑘 + 1) − 𝐞2,𝑗 (𝑘)
(15)
𝐮1,𝑗 (𝑘) 𝐞 (𝑘) ] is the kth input of a period at the jth iteration, u1, j (k) is the driving input; u2, j (k) is the following input; and [ 1,𝑗 ] is the kth 𝐮2,𝑗 (𝑘) 𝐞2,𝑗 (𝑘) output force synchronization error of a period at the jth iteration. The master–slave learning structure is defined as 𝐞1,𝑗 (𝑘) = 𝐲𝑑 (𝑘) − 𝐲1,𝑗 (𝑘) and max{𝑦∗ } 𝐤 0 𝐞2,𝑗 (𝑘) = max{𝑦2∗ } 𝐲1,𝑗 (𝑘)+𝐲2,𝑗 (𝑘); yd is the ideal output; y1, j is the driving output; y2, j is the following output; [ 1,𝑝 ] is the proportion coefficient 0 𝐤2,𝑝 1 𝐤 0 matrix; and [ 1,𝑑 ] is the derivative coefficient matrix. 0 𝐤2,𝑑 The stability of the system is given as follows: where [
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Due to Eq. (10), a backward difference vector is defined:
[
] [ ] [ ] ( [ ] [ ]) ( [ ] [ ]) [ ] Δ1,𝑗 𝑦 𝑦 𝑦 𝐮 𝐝 𝐮 𝐝 Δ𝐮 = 1,𝑗 − 1,𝑗−1 = 𝐏 1,𝑗 + 1 − 𝐏 1,𝑗−1 + 1 =𝐏 𝑗 1 Δ2,𝑗 𝑦 𝑦2,𝑗 𝑦2,𝑗−1 𝐮2,𝑗 𝐝2 𝐮2,𝑗−1 𝐝2 Δ𝑗 𝐮2
(16)
Then, the backward difference vector is applied to the control vector:
[
] [ ] [ ] ([ ] [ ]) ([ ] [ ]) [ ] [([ ] [ ]) ([ ] [ ])] Δ𝑗+1 𝐮1 𝐮 𝐮 𝐮1,𝑗 𝐞 𝐮1,𝑗−1 𝐞 Δ𝑗 𝐮1 𝐲1,𝑑 𝐲 𝐲1,𝑑 𝐲 = 1,𝑗+1 − 1,𝑗 = + 𝐋 1,𝑗 − + 𝐋 1,𝑗−1 = +𝐋 − 1,𝑗 − − 1,𝑗−1 Δ𝑗+1 𝐮2 𝐮2,𝑗+1 𝐮2,𝑗 𝐮2,𝑗 𝐞2,𝑗 𝐮2,𝑗−1 𝐞2,𝑗−1 Δ𝑗 𝐮2 𝐲1,𝑗 𝐲2,𝑗 𝐲1,𝑗−1 𝐲2,𝑗−1 [ ] [ ] [ ] Δ𝑗 𝐮1 Δ1,𝑗 𝐲 Δ𝑗 𝐮1 = −𝐋 = (𝐈 − 𝐋𝐏) Δ𝑗 𝐮2 Δ2,𝑗 𝐲 Δ𝑗 𝐮2
⎡ 𝑘1,𝑑 ⎢ ⎢𝑘1,𝑝 − 𝑘1,𝑑 ⎢ 0 ⎢ ⎢ 0 where L = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0
0
0
𝑘1,𝑑
0
0
⋱
⋱
0
0
𝑘1,𝑝 − 𝑘1,𝑑
𝑘1,𝑑
𝐎
𝐎
𝑘2,𝑑
0
𝑘2,𝑝 − 𝑘2,𝑑
𝑘2,𝑑
0
0
0
⋱
⋱
0
0
0
𝑘2,𝑝 − 𝑘2,𝑑
𝑘2,𝑑
0
0
(17)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ due to Eq. (15). ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
The discrete system of dimension N converges to zero if and only if the eigenvalues of (I-LP) lie within the unit circle:
[ ] [ ] ] [ ] ‖[𝐮 Δ𝑗 𝐮1 0 𝐮 ‖ ‖ ‖ = ⇔ lim ‖ 1,𝑗+1 − 1,𝑗 ‖ = 0 𝑗→∞ Δ𝑗 𝐮2 𝑗→∞ ‖ 𝐮2,𝑗+1 0 𝐮2,𝑗 ‖ ‖ ‖
|𝜆(𝐈 − 𝐋𝐏)| < 1 ⇔ lim
(18)
Therefore, the corresponding stability matrix can be obtained from Eqs. (12), (15), and (17):
𝐇𝐒 = 𝐈 − 𝐋𝐏 ⎡ 1 − 𝑘1,𝑑 𝑝1,1 ⎢ ⎢ (𝑘1,𝑑 − 𝑘1,𝑝 )𝑝1,2 ⎢ ⋮ ⎢ ⎢ ( 𝑘 − 𝑘 1,𝑝 )𝑝1,𝑁 = ⎢ 1,𝑑 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0
0
0
1 − 𝑘1,𝑑 𝑝1,1
0
0
⋱
⋱
0
…
(𝑘1,𝑑 − 𝑘1,𝑝 )𝑝1,2
1 − 𝑘1,𝑑 𝑝1,1
𝐎
𝐎
1 − 𝑘2,𝑑 𝑝2,1
0
0
(𝑘2,𝑑 − 𝑘2,𝑝 )𝑝2
1 − 𝑘2,𝑑 𝑝2,1
0
⋮
⋱
⋱
(𝑘2,𝑑 − 𝑘2,𝑝 )𝑝𝑁
…
(𝑘2,𝑑 − 𝑘2,𝑝 )𝑝2,1
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 1 − 𝑘2,𝑑 𝑝2,1 ⎦
(19)
Due to Eq. (19), the absolute eigenvalues of Hs are |1-k1, d p1 | and |1-k2, d p1 |. Referring to Eq. (18), the synchronization ILC system is stable if and only if
|1−𝑘𝑑 𝑝1,1 | < 1 and |1−𝑘2,𝑑 𝑝2,1 | < 1 | | | |
(20)
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Table 1 Parameters of BSILC controller. Parameter
Description
Value
Unit
ma Pa Aa kpa kda mc Pc Ac Aw kpc kda k
Axial loading motion mass Axial loading source pressure Axial loading piston cross-sectional area Axial loading proportion coefficient Axial loading derivative coefficient Confining loading motion mass Confining loading source pressure Confining loading piston cross-sectional area Water piston cross-sectional area Confining loading proportion coefficient Confining loading derivative coefficient Stiffness of the sample
11 1e7 4.2223e−3 1e−4 1e−6 9 1e7 1.00217e−3 2.82743e−3 1e−3 1e−5 2e6
kg Pa m2 – – kg Pa m2 m2 – – N/m
Fig. 8. The output force of the axial loading. (1) Ideal (2) PID (3) BSILC.
5. Simulation and analysis To test the effect of the BSILC controller designed before the experiment, this paper utilizes MATLAB to conduct force–pressure-coupling closed-loop control simulations for the classic PID controller and the BSILC controller. The parameters of the electrohydraulic servo force loading system in this study are given in Table 1. Two 0.5-Hz standard sine waves are used as the input signals of the axial loading force and the confining loading pressure. The external load acting on the axial loading piston and the confining pressure in the water vessel are used as the outputs in the simulating process. The axial loading output comparison of the classic PID control is shown in Fig. 8. In Fig. 8, the output force amplitude of the classic PID control shows a significant overshoot. The reason can be found in Figs. 4–7. Figs. 4 and 5 show four situations of the unidirectional coupling between two cylinders from two directions. When the electrohydraulic servo loading system functions, two situations in Fig. 4 occur at the same time in the first half period, and two situations in Fig. 5 occur at the same time in the second half period. The combinations of the two unidirectional couplings create a bidirectional coupling, which is even more complex than the original and can cause significant uncertainties. In Fig. 6, the waveform of the force coupled from confining loading to axial loading shows distinct distortion. In Fig. 7, the amplitude of the pressure coupled from axial loading to confining loading is extremely large. These characteristics can lead to a significant nonlinear disturbance to the force control system. The confining loading output comparison of the classic PID control is shown in Fig. 10. In Fig. 10, the output pressure of the classic PID control also shows a significant overshoot, which is caused by the same
Fig. 9. The output force error of the axial loading. (1) PID (2) BSILC.
reason as the axial loading output. The simulation details and the errors between the ideal output and the real output are given as follows. In Figs. 8 and 9, the tracking performance of the classic PID control for the axial loading output is deteriorated by the coupling between two cylinders from two directions so that its maximum force error is approximately 20% of the amplitude of the ideal output force. The BSILC shows better trajectories one period after another, and its output force error has already been reduced by 40% in the second period and continuously converges to 0 as time passes. In Figs. 10 and 11, the tracking performance of the classic PID control for the confining loading output is seriously deteriorated by the coupling between two cylinders from two directions so that its maximum pressure error is over 30% of the amplitude of the ideal output pressure. The BSILC shows better trajectories one period after another, and its output pressure error has been reduced by 20% in the second period and continuously converges to 0 as time passes. To compare the synchronization error of the axial loading force and the confining pressure, the sine wave of the confining pressure is transformed into the same amplitude as the axial loading force. According to Eqs. (5) and (6), the ideal amplitude of the axial loading force is approximately 4000 N, and the ideal amplitude of the confining pressure is approximately 500 kPa. Therefore, the force synchronization error in Fig. 11 is described as 𝑒𝑠 = 𝑦1 +
4000 𝑦 = 𝑦 1 +8 𝑦 2 500 2
(21)
In Fig. 12, the synchronism of the classic PID is also deteriorated by the coupling between two cylinders from two directions so that
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 13. The output force of the axial loading. (1) Ideal (2) PID (3) BSILC. Fig. 10. The output pressure of the confining loading (1) Ideal (2) PID (3) BSILC.
Fig. 14. The output force error of the axial loading. (1) PID (2) BSILC.
Fig. 11. The output pressure error of the confining loading. (1) PID (2) BSILC.
its maximum force synchronization error is approximately 10% of the amplitude of the ideal output force. The BSILC shows an increasingly better synchronism one period after another, and its force synchronization error has already been reduced by 20% in the fourth period and continuously converges to 0 as time passes. From the above analysis, the BSILC shows excellent tracking performance and robustness for the ideal axial output force and ideal output pressure, and the synchronism between two cylinders from two directions has been remarkably improved. Therefore, the BSILC method is an ideal control scheme for the coupling in the electrohydraulic servo loading system. 6. Experiment results
Fig. 12. The force synchronization error between the axial loading and confining pressure. (1) PID (2) BSILC.
The experiment results are shown in Figs. 13–17. In Figs. 13 and 14, the output force of the classic PID control for the axial loading output overshoots by approximately 30%, and its waveform shows great distortions. For the BSILC, the output force error has already been reduced by 17.5% in the third period, and the distortion of the output waveform has been corrected. The tracking performance and robustness of the BSILC are increasingly better one period after another. In Figs. 15 and 16, the output pressure of the classic PID control for the confining loading output overshoots by approximately 10%. For the BSILC, the output pressure error has already been reduced by 10% in the second period. The tracking performance and robustness of the BSILC are increasingly better one period after another.
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
In Fig. 17, the synchronization error of the classic PID is approximately 50% of the amplitude of the ideal output force. For the BSILC, the force synchronization error has already been reduced by 20% in the second period. The synchronism of the BSILC is increasingly better one period after another. The experiment results prove that the BSILC proposed in this paper is valid. The tracking performance, robustness and synchronism of the electrohydraulic servo loading system have been remarkably improved. 7. Conclusion In this paper, a BSILC controller is designed to improve the tracking performance and robustness and to compensate for the synchronization error of the coupling between two cylinders from two directions in an electrohydraulic servo loading system. The conclusions are as follows:
Fig. 15. The output pressure of the confining loading (1) Ideal (2) PID (3) BSILC.
(1) The BSILC controller can improve the tracking performance for ideal axial loading and ideal confining loading and effectively restrain the overshoot of the amplitude. (2) The BSILC controller shows excellent robustness in resisting the distortion of the waveform caused by the coupling between two cylinders from two directions and good synchronism. (3) The BSILC controller designed in this paper has been successfully applied in the bidirectional synchronization control of an electrohydraulic servo loading system, with obvious improvements in the precision of the periodic loading system. This system has value in engineering applications, such as wave simulators for earthquakes, wind and ocean, dynamic tests for mechanical devices, and natural frequency measurement tests. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We gratefully acknowledge the support by the National Natural Science Foundation of China through the grant number 51275068, and the support by the Fundamental Research Funds for the Central Universities Grant no. DUT18LK22.
Fig. 16. The output pressure error of the confining loading. (1) PID (2) BSILC.
References
Fig. 17. The force synchronization error between the axial loading and confining pressure. (1) PID (2) BSILC.
[1] Jiao Z, Yao J. Nonlinear control of electrohydraulic servo system. Beijing: Science Press; 2016. p. 1–5. [2] Merritt HE. Hydraulic control systems. John Wiley & Sons; 1967. [3] Rybak AT, Temirkanov AR, Lyakhnitskaya OV. Synchronous hydromechanical drive of a mobile machine. Russ Eng Res 2018;38:212–17. [4] Karimi HR, Gao H. LMI-based H∞ synchronization of second-order neutral master-slave systems using delayed output feedback control. Int J Control Autom Syst 2009;7(3):371–80. [5] Karimi HR. Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations. Int J Control Autom Syst 2011;9(4):671. [6] Lin FJ, Chou PH, Chen CS, et al. DSP-based cross-coupled synchronous control for dual linear motors via intelligent complementary sliding mode control. IEEE Trans Ind Electron 2012;59(2):1061–73. [7] Byun JH, Choi MS. A method of synchronous control system for dual parallel motion stages. Int J Precis Eng Manuf 2012;13(6):883–9. [8] Byun JH, Kim YB. A study on construction of synchronous control system for extension and stability. Trans Korean Soc Mech Eng A 2002;26(6):1135–42. [9] Chulines E, Rodríguez MA, Duran I, et al. Simplified model of a three-phase induction motor for fault diagnostic using the synchronous reference frame DQ and parity equations. IFAC-PapersOnLine 2018;5113:662–7. [10] Hsieh MF, Yao WS, Chiang CR. Modeling and synchronous control of a single-axis stage driven by dual mechanically-coupled parallel ball screws. Int J Adv Manuf Technol 2007;34(9-10):933–43. [11] Jeong SK, You SS. Precise position synchronous control of multi-axis servo system. Mechatronics 2008;18(3):129–40. [12] Sato R, Tsutsumi M. Motion control techniques for synchronous motions of translational and rotary axes. Procedia CIRP 2012;1:265–70.
Y. Sang, W. Sun and F. Duan et al. [13] Zhao H, Ben-Tzvi P. Synchronous position control strategy for bi-cylinder electro-pneumatic systems. Int J Control Autom Syst 2016;14(6):1501–10. [14] Roozegar M, Mahjoob MJ, Ayati M. Adaptive tracking control of a nonholonomic pendulum-driven spherical robot by using a model-reference adaptive system. J Mech Sci Technol 2018;32(2):845–53. [15] Chen Z, Yao B, Wang Q. Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME Trans Mech 2013;18(3):1122–9. [16] Golnargesi S, Shariatmadar H, Razavi HM. Seismic control of buildings with active tuned mass damper through interval type-2 fuzzy logic controller including soil–structure interaction. Asian J Civ Eng 2018;19(2):177–88. [17] Artale V, Collotta M, Milazzo C, et al. An integrated system for UAV control using a neural network implemented in a prototyping board. J Intell Rob Syst 2016;84(1-4):5–19. [18] Youssef T, Chadli M, Karimi HR, et al. Actuator and sensor faults estimation based on proportional integral observer for TS fuzzy model. J Franklin Inst 2017;354(6):2524–42. [19] Selvaraj P, Kaviarasan B, Sakthivel R, et al. Fault-tolerant SMC for takagi–sugeno fuzzy systems with time-varying delay and actuator saturation. IET Control Theory Appl 2017;11(8):1112–23. [20] Zhang H, Liu X, Wang J, et al. Robust H∞ sliding mode control with pole placement for a fluid power electrohydraulic actuator (EHA) system. Int J Adv Manuf Technol 2014;73(5-8):1095–104. [21] Meng F, Shi P, Karimi HR, et al. Optimal design of an electro-hydraulic valve for heavy-duty vehicle clutch actuator with certain constraints. Mech Syst Signal Process 2016;68:491–503. [22] Shen W, Jiang J, Su X, et al. Control strategy analysis of the hydraulic hybrid excavator. J Franklin Inst 2015;352(2):541–61. [23] Fan Y, Du Y. Material mechanics. 1st ed. Beijing: Tsinghua University Press; 2017. 2–2. [24] Barton KL, Alleyne AG. A cross-coupled iterative learning control design for precision motion control. IEEE Trans Control Syst Technol 2008;16(6):1218–31. [25] Hillenbrand S, Pandit M. A discrete-time iterative learning control law with exponential rate of convergence. In: Decision and control, 1999. Proceedings of the 38th IEEE conference on, 2. IEEE; 1999. Yong Sang is an associate professor at the school of mechanical engineering, Dalian University of Technology. He received his M.S. degree in Mechatronics from Shandong University in 2004 and the PhD in Mechatronics from Beijing University of Aeronautics and Astronautics in 2007. He worked in the department of mechanical engineering as a postdoctoral researcher, University of Minnesota from Aug. 2012 to Dec. 2013. He is studying on advanced testing technique, hydraulic transmission and control.
Mechatronics 62 (2019) 102254 Weiqi Sun is a doctor at the school of mechanical engineering, Dalian University of Technology. He is studying on measurement and control technology.
Fuhai Duan is a professor at the school of mechanical engineering, Dalian University of Technology. He received his M.S. degree in vehicle guidance, navigation and control technology in 1996 and the PhD in 1999 from Northwestern Polytechnical University. He is studying on embedded computer testing technique, virtual instrument technology and precision control of servo motor.
Jianlong Zhao is a doctor at the school of mechanical engineering, Dalian University of Technology. He is studying on measurement and control technology.
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Bidirectional synchronization control for an electrohydraulic servo loading system ✩ Yong Sang∗, Weiqi Sun, Fuhai Duan, Jianlong Zhao School of Mechanical Engineering, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Keywords: Electrohydraulic servo loading system Synchronism Coupling BSILC
a b s t r a c t Synchronism is one of the most important factors for electrohydraulic servo loading systems with multi-axis. As electrohydraulic servo loading systems have a large output force/torque, the coupling between each axis can cause great nonlinearities and uncertainties, which seriously deteriorates the performance of the systems. In this paper, an electrohydraulic servo loading system is used for studying its bidirectional synchronization control. The mechanism of the coupling between two cylinders from two directions is discussed and the experiment results of coupling are obtained. A bidirectional synchronization iterative learning control (BSILC) method is designed to improve the tracking performance, robustness and compensate the synchronization error. The simulation and the experiment have been conducted. The results indicate that the control strategy proposed in this paper is valid. The tracking performance, robustness and synchronism of the electrohydraulic servo loading system have been remarkably improved.
1. Introduction Electrohydraulic servo loading systems have the advantages of a large output of force/torque, a high power to weight ratio, quick response and excellent anti-disturbance performance, and they have been widely applied in many important fields for control and power transmission [1,2]. With the development of electrohydraulic servo loading systems, there has been a considerable and growing demand for the synchronism of bidirectional outputs [3]. Traditional synchronous drive systems can be divided into three categories based on their control structures: the master–slave structure, the cross-coupling structure, and the model reference structure. In a master–slave structure system, the output of the master system is used as the input of the slave system [4,5]. In this case, the synchronization error can only be reduced by the slave system itself. In a cross-coupling structure system, the synchronization error is fed back to each input simultaneously to find a globally optimal solution for all of the outputs [6,7]. In a model reference structure system, each output is connected to a reference model in parallel with an individual independent synchronization controller [8,9]. In this paper, a bidirectional synchronization control for an electrohydraulic servo loading system is investigated for its tracking performance and robustness for the periodic input signal. Synchronization control methods have been widely applied in controllers to solve the synchronization problems between systems. A synchronization control
✩ ∗
scheme and system modeling for a single-axis stage driven by dual mechanically coupled parallel ball screws have been presented [10]. A precise position synchronization control of a multi-axis rotating system has been described, and the acceleration and speed controllers have been readily designed via the proportional-integral (PI) control law [11]. Motion control techniques have been proposed to improve the dynamic synchronization accuracy between translational and rotary axes in five-axis machining centers with the servo gain tuning method, feed-forward controller, signal delay and backlash compensator, rotational fluctuation compensator, and jerk limited acceleration process [12]. A two-layer sliding mode synchro-system based on friction compensation has been applied to electro-pneumatic cylinders, and a synchro-PID controller has been utilized for position tracking [13]. Modern control theories have provided many excellent control methods for improving the synchronism of bidirectional output systems, such as adaptive control, robust control, fuzzy control, and neural network control [14–17]. For the robustness of the algorithm with respect to faults, a proportional integral observer has been presented for the actuator and sensor faults estimation based on a TS fuzzy model [18]. A fault-tolerant sliding mode control has been proposed for Takagi–Sugeno fuzzy systems with time-varying delay and actuator saturation [19]. Many types of research have shown good results in improving electrohydraulic servo loading systems, but there are still limitations. A robust H∞ sliding mode control with pole placement has adequately
This paper was recommended for publication by Associate Editor Kim A. Stelson. Corresponding author. E-mail address: [email protected] (Y. Sang).
https://doi.org/10.1016/j.mechatronics.2019.102254 Received 25 December 2018; Received in revised form 2 July 2019; Accepted 20 July 2019 Available online 21 July 2019 0957-4158/© 2019 Elsevier Ltd. All rights reserved.
Y. Sang, W. Sun and F. Duan et al.
compensated for the nonlinear friction of a fluid power electrohydraulic actuator (EHA) system, but it remains in the simulation stage and has not been used in actual experiments [20]. The optimal design has effectively improved the dynamic characteristics of an electrohydraulic hydraulic valve for heavy-duty vehicle clutch actuators with certain constraints, but it cannot optimize the entire electrohydraulic servo loading system [21]. Control strategies applied together in hydraulic hybrid excavators have shown good results in reducing fuel consumption, but their optimal results cannot be optimized in unknown working cycles [22]. In this paper, a bidirectional synchronization iterative learning control (BSILC) method is designed to improve tracking performance and robustness and to reduce the synchronization error between the two cylinders from two directions in an electrohydraulic servo loading system. The traditional synchronization control structures generally show a weak ability to compensate for the synchronization error, and the cross-coupling structure system performs better in reducing the synchronization error than the other two structures. However, these control structures cannot improve the tracking performance of the entire system, which becomes even worse as the synchronism improves. Therefore, a master–slave learning structure is used for the controller in this paper to improve both tracking performance and synchronism together. The BSILC method does not rely on a mathematical model of the system. It has an excellent tracking performance for the repetitive mode executions, which can adjust the slave output to the master output and reduce the synchronization error quickly. Compared with the other control methods mentioned above, it has a smaller operation cycle and is more likely to be realized in a practical system. The simulations and experiments for the BSILC methods have been conducted. The results indicate that the control method applied in this paper is valid, and the tracking performance and robustness and the synchronization error between the two cylinders from two directions in the electrohydraulic servo loading system have been remarkably improved. The paper is organized as follows: Section 2 introduces the components of the electrohydraulic servo loading system and constructs its mathematic model. Section 3 explains the mechanism of the coupling between two cylinders from two directions and shows its experiment results. Section 4 demonstrates the design of the BSILC controller. Section 5 conducts the simulation and performances analysis for the BSILC controller. Section 6 carries out the experiment to test the BSILC controller. Section 7 comes to the conclusions.
Mechatronics 62 (2019) 102254
Fig. 1. The electrohydraulic servo loading system experiment platform.
Fig. 2. The deformation of the sample.
2. Electrohydraulic servo loading system In this paper, a dynamic triaxial apparatus is used as the experiment platform of the electrohydraulic servo loading system. A photo of the experiment platform is shown in Fig. 1. The dynamic triaxial apparatus is the most important device for studying complex geotechnical dynamics in the laboratory. The device can be used to study the instability, failure, and deformation of the foundation caused by waves from an earthquake, wind, or the ocean. It has been widely applied in the aseismic geotechnical engineering design of high-speed railways, expressways, subways, large bridges, and huge dams. In the laboratory, the dynamic triaxial apparatus usually exerts periodic sine wave loads for studying stress–strain properties, strength, and other mechanical properties of the soil. The performance indices of the electrohydraulic servo loading system shown in Fig. 1 are as follows. For the hydraulic flow power, the rated flow is <70 L/min, and the rated pressure is <21 MPa. The hydraulic gear pump has a certified capacity of 50 mL/r (PH-5B-50, Nachi, Japan). For the driving motor (Y-Type, Dalian Motor Group Co. Ltd, China), the rated speed is 1450 r/min, and the rated power is 37 kW. The stroke of the axial loading cylinder is 200 mm (CK-63/36-200, Atos, Italy). The stroke of the confining loading cylinder is 200 mm (CK50/22-200, Atos, Italy). The two electrohydraulic servo valves are the same type (D633, Moog, USA), the rated flow is 20 L/min, the hysteresis is <0.2%, and the maximum operating pressure is <35 MPa. Watercooling is used, and the heat-exchange area is >1.5 m2 . The operating
pressure of the hydraulic flow power is regulated with a pilot proportional relief valve (AGMZO, Atos, Italy), and its regulation range is 0.7– 21 MPa. The axial loading range is 0–10,000 N, and the confining pressure range is 0–4000 kPa. The pore pressure of the high precision water pressure sensor used is 0–4000 kPa. The stroke of the displacement sensor is 0–75 mm, and its precision is <0.1%. High acquisition precision can be achieved by a 16-bit AD converter. An LVDT type sensor is used as the displacement sensor, and its precision is < 0.1%. High precision pressure sensors are used for the left water discharge, right water discharge, upper pore pressure, lower pore pressure, and confining pressure, and each has a precision of <0.1%. All output signals are 0–10 V voltage signals. The measurement of water discharge is indirectly obtained by measuring the pressure of the water head. The internal force sensor and the external force sensor are the same. A VISHAY tension sensor is used as the force sensor. It ranges from 0 to 2000 kg, and its linearity is <0.02%. Fig. 2 shows the loading process of the sample. The principle diagram of the electrohydraulic servo loading system is shown in Fig. 3, including a hydraulic flow power, two servo amplifiers, two hydraulic servo valves, two loading cylinders, a water cylinder, a force sensor, a pressure sensor, a displacement sensor, a high speed AD/DA card and computers. The computers are mainly used to generate loading waveforms, operate various control algorithms, display real-time data, generate and store the data report.
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 3. The principle diagram of the electrohydraulic servo loading system.
The electrohydraulic servo loading system is a force closed loop system, which utilizes two electrohydraulic servo valves to control two hydraulic servo cylinders for the axial loading and confining pressure, respectively. The voltage signals are used as its input and feedback signals. The mathematic model of the electrohydraulic servo loading system is described as follows: 𝑥̇ 1 = 𝑥2 1 𝑥̇ 2 = (−𝑘𝑎 𝑥1 − 𝑏𝑎 𝑥2 + 𝐴𝑎 𝑥3 ) 𝑚𝑎 √ 𝑥̇ 3 = −𝛼𝑎 𝑥2 − 𝛽𝑎 𝑥3 + (𝛾𝑎 𝑃𝑎 − sgn(𝑥4 )𝑥3 )𝑥4 𝐾 1 𝑥̇ 4 = − 𝑥4 + 𝑎 𝑢1 𝜏𝑎 𝜏𝑎 𝑥̇ 5 = 𝑥6 1 𝑥̇ 6 = (−𝑘𝑐 𝑥5 − 𝑏𝑐 𝑥6 + 𝐴𝑐 𝑥7 ) 𝑚𝑐 √ 𝑥̇ 3 = −𝛼𝑐 𝑥6 − 𝛽𝑐 𝑥7 + (𝛾𝑐 𝑃𝑐 − sgn(𝑥8 )𝑥7 )𝑥8 𝐾 1 𝑥̇ 8 = − 𝑥8 + 𝑐 𝑢2 𝜏𝑐 𝜏𝑐 𝑦1 = 𝑘𝑎 𝑥1 𝑘 𝑥 𝑦2 = 𝑐 5 𝐴𝑤
pressure piston; y1 is the axial load; and y2 is the confining pressure. The corner mark of the axial loading variable is a and the corner mark of the confining pressure variable is c. m is the mass of the actuator piston; k is the stiffness of the external load; b is the friction damping coefficient of the piston; A is the cross-sectional area of the piston; P is the supply pressure of the electrohydraulic servo valve; 𝜏 is the time constant of the electrohydraulic valve; K is the input gaining of the electrohydraulic valve; and Aw is the cross-sectional area of the water piston. 𝛼 = 4 𝐴 𝛽𝑒 ∕ 𝑉 𝑡
(2)
𝛽 = 4𝐶𝑡𝑚 𝛽𝑒 ∕𝑉𝑡
(3)
√ 𝛾 = 4𝐶𝑑 𝛽𝑒 𝑤∕(𝑉𝑡 𝜌)
(4)
where 𝛽 e is the effective bulk modulus; Vt is the total actuator volume; Ctm is the total coefficient of the internal leakage; Cd is the discharge coefficient; w is the spool valve area gradient; and 𝜌 is the fluid mass density. 3. The mechanism of the coupling between two cylinders from two directions (1)
where x1 is the axial loading piston displacement; x2 is the axial loading piston velocity; x3 is the loading pressure in the axial loading piston; x4 is the electrohydraulic valve displacement controlling the axial loading piston; x5 is the confining pressure piston displacement; x6 is the confining pressure piston velocity; x7 is the loading pressure in the confining pressure piston; x8 is the electrohydraulic valve displacement controlling the confining pressure piston; u1 is the input current to electrohydraulic servo valve controlling the axial loading piston; u2 is the input current to electro-pneumatic servo valve controlling the confining
The coupling between two cylinders from two directions is a significant practical phenomenon in electrohydraulic servo loading systems that greatly impacts tracking performance, robustness, and synchronism. The process of the coupling is shown in Figs. 4 and 5, where the symbols in red represent the inputs of the system, and the symbols in green represent the outputs of the system. Fig. 4 shows the compression coupling process. In Fig. 4(a), when the water piston controlling the confining pressure stays still, it can be seen that the confining pressure in the vessel is supposed to remain unchanged. However, as the axial loading piston compresses the sample, a part of the piston rod enters
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 4. The compression process of the electrohydraulic servo loading system. (a) Coupling from axial loading to confining pressure. (b) Coupling from confining pressure to axial loading.
Fig. 5. The decompression process of the electrohydraulic servo loading system. (a) Coupling from axial loading to confining pressure. (b) Coupling from confining pressure to axial loading.
into the closed space of the vessel and squeezes the water in it. Water has a very large bulk modulus of approximately 2.2 GPa, which directly results in a great pressure increase. The larger the displacement of the axial loading piston is, the higher the confining pressure of the water will be. Moreover, as the confining pressure grows, the sample is compressed in the radial direction and consequently expands in the axial direction according to the Poisson effect, which increases the axial load in turn. In Fig. 4(b), when the axial loading piston stays still, it can be seen that the axial loading is supposed to remain unchanged. However, the sample is compressed by the water as the water piston squeezes the water in the closed space of the vessel, increasing the confining pressure, and
thereby increasing the axial load in the same way as that mentioned in Fig. 4(a). The higher the confining pressure is, the larger the axial load will be. Fig. 5 shows the decompression coupling process. In Fig. 5(a), when the water piston controlling the confining pressure stays still and the axial loading piston is drawn back from the vessel, a part of the piston rod exits from the closed space of the vessel, resulting in a decrease in the water pressure. The larger the displacement of the axial loading piston is, the lower the confining pressure of the water will be. Moreover, as the confining pressure drops, the sample expands in the radial direction and consequently shrinks in the axial direction according to the Poisson
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 7. The pressure coupled from axial loading to confining loading.
Fig. 6. The force coupled from confining loading to axial loading.
effect, which decreases the axial load in turn. In Fig. 5(b), when the axial loading piston stays still and the water piston pumps out the water in the closed space of the vessel, thus reducing the confining pressure, the sample is recovered, which decreases the axial load in the same way as that mentioned in Fig. 5(a). The lower the confining pressure is, the smaller the axial load will be. The coupling process mentioned above indicates that the mechanism of the coupling can cause great nonlinearities and uncertainties for the electrohydraulic servo loading system. The deformation of a sample is a very complicated process. To illustrate the effect of the coupling in a simulation, four assumptions are defined as follows [23]: 1. Continuity assumption: The volume of the sample is full of the sample’s matter without any void space. 2. Homogenization assumption: The mechanical characteristics of any part of the sample are the same. 3. Isotropy assumption: The mechanical characteristics of the sample are the same in all directions. 4. The assumption of small deformation: The deformation of the sample caused by external forces is far less than its original size. In this paper, the ideal outputs of the dynamic triaxial apparatus are 𝑦∗ = 4000 sin (2𝜋 ∗ 0.5𝑡) (N)
(5)
𝑦∗ = −500 sin(2𝜋 ∗ 0.5𝑡) + 2000 (kPa)
(6)
1
2
When the confining loading pressure is set as Eq. (5), the input of the axial loading force is set as a constant. The experiment results of the axial coupling force are obtained from Fig. 6. In Fig. 6, the amplitude of the axial coupling force is approximately 600 N, which is 15% of the ideal axial loading force, and the coupling waveform is no longer a standard sine wave, which shows great distortion. When the axial loading force is set as Eq. (6), the input of the confining pressure is a constant. The experiment results of the confining coupling pressure can be obtained from Fig. 7. In Fig. 7, the amplitude of the confining coupling pressure is approximately 150 kPa, which is 30% of the ideal confining loading pressure. The experiment results demonstrate that the effect of the coupling is significant enough to seriously deteriorate the tracking performance, robustness, and synchronism of the dynamic triaxial apparatus. Therefore, an effective control method is needed to compensate for the coupling between two cylinders from two directions.
4. Design of the BSILC controller for two cylinders from two directions The BSILC controller is designed based on the state-space representation (1). The general form of Eq. (1) can be represented as 𝐱𝑗 (𝑘 + 1) = 𝐀𝐱𝑗 (𝑘) + 𝐁𝐮𝑗 (𝑘) 𝐲𝑗 (𝑘) = 𝐂𝐱 𝑗 (𝑘)
(7)
where A is the state matrix; B is the input matrix; and C is the output matrix. The state-space representation mentioned above can be applied to any system with a finite iteration length. The system can be represented in the frequency domain by extending the length of the cycle T → ∞: [
] [ ][ ] 𝑌1,𝑗 (𝑧) 𝑃 (𝑧) 0 𝑈1,𝑗 (𝑧) = 1 𝑌2,𝑗 (𝑧) 0 𝑃2 (𝑧) 𝑈2,𝑗 (𝑧) [ ][ ] [ ] 𝑃 (𝑧) 0 𝑥1 (0) 𝐷1 (𝑧) + 1,0 + 0 𝑃2,0 (𝑧) 𝑥5 (0) 𝐷2 (𝑧)
(8)
𝑌1,𝑗 (𝑧) ] is the representation of yj (k) in the frequency domain; 𝑌2,𝑗 (𝑧) Y1, j (z) is the driving output; and Y2, j (z) is the following output. The transfer matrix is given by where [
[
] 𝑃1 (𝑧) 0 = 𝐂(𝑧𝐈 − 𝐀)−1 𝐁 0 𝑃2 (𝑧) [ 𝑝 𝑧−1 + 𝑝1,22 𝑧−2 + 𝑝1,3 𝑧−3 + ⋯ = 1,1 0
]
0 𝑝2,1
𝑧−1
+ 𝑝2,2
𝑧−2
+ 𝑝2,3
𝑧−3
+⋯ (9)
where 𝑝𝑘,𝑖 = 𝐂𝐀𝑖−1 𝐁, (𝑘 = 1, 2; 𝑖 = 1, 2, 3, ⋯), and pk,i is the Markov Parameter [24]. Assuming that the initial condition is zero, the impulse responses of the system can be given by these Markov Parameters. The system’s relative degree, m, corresponds to the first nonzero Markov Parameter. The synchronization ILC is applied to the two-cylinder systems with repeated finite cycles, and the input–output relation of the first order iteration can be represented with the following matrix representation: [
] [ ] [ ] 𝐲1,𝑗 𝐮 𝐝 = 𝐏 1,𝑗 + 1 𝐲2,𝑗 𝐮2,𝑗 𝐝2
(10)
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
where ⎡ 𝑦1,𝑗 (1) ⎤ ⎡ 𝑢1,𝑗 (0) ⎤ ⎡ 𝑑1 (0) ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 𝑦1,𝑗 (2) ⎥ ⎢ 𝑢1,𝑗 (1) ⎥ ⎢ 𝑑1 (1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⋮ ⎥ ⋮ ⋮ ⎥ [ ] ⎢ ⎥ [ ] ⎢ [ ] ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝐲1,𝑗 𝑦1,𝑗 (𝑁)⎥ 𝐮1,𝑗 𝑢1,𝑗 (𝑁 − 1)⎥ 𝐝1 𝑑1 (𝑁 − 1)⎥ ⎢ ⎢ ⎢ = ; = ; = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝑦 (1) 𝑢 (0) 𝑑 (0) 𝐲2,𝑗 𝐮2,𝑗 𝐝2 2,𝑗 2,𝑗 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 𝑦2,𝑗 (2) ⎥ ⎢ 𝑢2,𝑗 (1) ⎥ ⎢ 𝑑2 (1) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⋮ ⋮ ⎢ ⋮ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 𝑦 ( 𝑁) 𝑢 ( 𝑁 − 1) 𝑑 ( 𝑁 − 1) ⎣ 2,𝑗 ⎦ ⎣ 2,𝑗 ⎦ ⎣ 2 ⎦
⎡ 𝐂1 𝐁1 ⎢ 𝐂1 𝐀1 𝐁1 ⎢ ⋮ ⎢ ⎢𝐂1 𝐀1 𝐍−𝟏 𝐁1 𝐏=⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0 𝐂1 𝐁1 ⋮ 𝐂1 𝐀1 𝐍 −𝟐 𝐁1
⋯ ⋯ ⋱ ⋯
0 0 ⋮ 𝐂1 𝐁1
𝐎
(11)
𝐎 𝐂2 𝐁2 𝐂2 𝐀2 𝐁2 ⋮ 𝐂2 𝐀2 𝐍 −𝟏 𝐁2
⋯ ⋯ ⋱ ⋯
0 𝐂2 𝐁2 ⋮ 𝐂2 𝐀2 𝐍 −𝟐 𝐁2
0 0 ⋮ 𝐂2 𝐁2
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (12)
⎡ 𝑝1,1 ⎢ 𝑝1,2 ⎢ ⎢ ⋮ ⎢𝑝 = ⎢ 1,𝑁 ⎢ ⎢ ⎢ ⎢ ⎣
0 𝑝1,1 ⋮ 𝑝1,𝑁−1
⋯ ⋯ ⋱ ⋯
0 0 ⋮ 𝑝1,1
𝐎
𝐎 𝑝2,1 𝑝2,2 ⋮ 𝑝2,𝑁
⋯ ⋯ ⋱ ⋯
0 𝑝2,1 ⋮ 𝑝2,𝑁−1
0 0 ⋮ 𝑝2,1
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
𝐎 𝐂1 ], 𝐁 = [𝐁1 𝐁2 ], 𝐂 = [ ] 𝐎 𝐀2 𝐂2 Due to Eq. (7), p1,1 and p2,1 are calculated as follows:
where 𝐀 = [
𝐀1
𝑝 1 , 1 = 𝐂 1 𝐁 1 =0
(13)
𝑝 2 , 1 = 𝐂 2 𝐁 2 =0
(14)
When 𝐂1 𝐁1 =0 and 𝐂2 𝐁2 =0, a small change of the sample time will yield C1 B1 ≠ 0 and C2 B2 ≠ 0, so two relative degrees of 1 can always be obtained by adapting the sample times T1 and T2 [25]. Then, the synchronization ILC law constructs the current trial input as [ ] 𝐮1,𝑗+1 (𝑘) 𝐮2,𝑗+1 (𝑘)
[ =
] 𝐮1,𝑗 (𝑘) 𝐮2,𝑗 (𝑘)
+
[ 𝐤1,𝑝 0
0 𝐤2,𝑝
][
] 𝐞1,𝑗 (𝑘) 𝐞2,𝑗 (𝑘)
+
[ 𝐤1,𝑑 0
0 𝐤2,𝑑
][ ] 𝐞1,𝑗 (𝑘 + 1) − 𝐞1,𝑗 (𝑘) 𝐞2,𝑗 (𝑘 + 1) − 𝐞2,𝑗 (𝑘)
(15)
𝐮1,𝑗 (𝑘) 𝐞 (𝑘) ] is the kth input of a period at the jth iteration, u1, j (k) is the driving input; u2, j (k) is the following input; and [ 1,𝑗 ] is the kth 𝐮2,𝑗 (𝑘) 𝐞2,𝑗 (𝑘) output force synchronization error of a period at the jth iteration. The master–slave learning structure is defined as 𝐞1,𝑗 (𝑘) = 𝐲𝑑 (𝑘) − 𝐲1,𝑗 (𝑘) and max{𝑦∗ } 𝐤 0 𝐞2,𝑗 (𝑘) = max{𝑦2∗ } 𝐲1,𝑗 (𝑘)+𝐲2,𝑗 (𝑘); yd is the ideal output; y1, j is the driving output; y2, j is the following output; [ 1,𝑝 ] is the proportion coefficient 0 𝐤2,𝑝 1 𝐤 0 matrix; and [ 1,𝑑 ] is the derivative coefficient matrix. 0 𝐤2,𝑑 The stability of the system is given as follows: where [
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Due to Eq. (10), a backward difference vector is defined:
[
] [ ] [ ] ( [ ] [ ]) ( [ ] [ ]) [ ] Δ1,𝑗 𝑦 𝑦 𝑦 𝐮 𝐝 𝐮 𝐝 Δ𝐮 = 1,𝑗 − 1,𝑗−1 = 𝐏 1,𝑗 + 1 − 𝐏 1,𝑗−1 + 1 =𝐏 𝑗 1 Δ2,𝑗 𝑦 𝑦2,𝑗 𝑦2,𝑗−1 𝐮2,𝑗 𝐝2 𝐮2,𝑗−1 𝐝2 Δ𝑗 𝐮2
(16)
Then, the backward difference vector is applied to the control vector:
[
] [ ] [ ] ([ ] [ ]) ([ ] [ ]) [ ] [([ ] [ ]) ([ ] [ ])] Δ𝑗+1 𝐮1 𝐮 𝐮 𝐮1,𝑗 𝐞 𝐮1,𝑗−1 𝐞 Δ𝑗 𝐮1 𝐲1,𝑑 𝐲 𝐲1,𝑑 𝐲 = 1,𝑗+1 − 1,𝑗 = + 𝐋 1,𝑗 − + 𝐋 1,𝑗−1 = +𝐋 − 1,𝑗 − − 1,𝑗−1 Δ𝑗+1 𝐮2 𝐮2,𝑗+1 𝐮2,𝑗 𝐮2,𝑗 𝐞2,𝑗 𝐮2,𝑗−1 𝐞2,𝑗−1 Δ𝑗 𝐮2 𝐲1,𝑗 𝐲2,𝑗 𝐲1,𝑗−1 𝐲2,𝑗−1 [ ] [ ] [ ] Δ𝑗 𝐮1 Δ1,𝑗 𝐲 Δ𝑗 𝐮1 = −𝐋 = (𝐈 − 𝐋𝐏) Δ𝑗 𝐮2 Δ2,𝑗 𝐲 Δ𝑗 𝐮2
⎡ 𝑘1,𝑑 ⎢ ⎢𝑘1,𝑝 − 𝑘1,𝑑 ⎢ 0 ⎢ ⎢ 0 where L = ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0
0
0
𝑘1,𝑑
0
0
⋱
⋱
0
0
𝑘1,𝑝 − 𝑘1,𝑑
𝑘1,𝑑
𝐎
𝐎
𝑘2,𝑑
0
𝑘2,𝑝 − 𝑘2,𝑑
𝑘2,𝑑
0
0
0
⋱
⋱
0
0
0
𝑘2,𝑝 − 𝑘2,𝑑
𝑘2,𝑑
0
0
(17)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ due to Eq. (15). ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
The discrete system of dimension N converges to zero if and only if the eigenvalues of (I-LP) lie within the unit circle:
[ ] [ ] ] [ ] ‖[𝐮 Δ𝑗 𝐮1 0 𝐮 ‖ ‖ ‖ = ⇔ lim ‖ 1,𝑗+1 − 1,𝑗 ‖ = 0 𝑗→∞ Δ𝑗 𝐮2 𝑗→∞ ‖ 𝐮2,𝑗+1 0 𝐮2,𝑗 ‖ ‖ ‖
|𝜆(𝐈 − 𝐋𝐏)| < 1 ⇔ lim
(18)
Therefore, the corresponding stability matrix can be obtained from Eqs. (12), (15), and (17):
𝐇𝐒 = 𝐈 − 𝐋𝐏 ⎡ 1 − 𝑘1,𝑑 𝑝1,1 ⎢ ⎢ (𝑘1,𝑑 − 𝑘1,𝑝 )𝑝1,2 ⎢ ⋮ ⎢ ⎢ ( 𝑘 − 𝑘 1,𝑝 )𝑝1,𝑁 = ⎢ 1,𝑑 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
0
0
0
1 − 𝑘1,𝑑 𝑝1,1
0
0
⋱
⋱
0
…
(𝑘1,𝑑 − 𝑘1,𝑝 )𝑝1,2
1 − 𝑘1,𝑑 𝑝1,1
𝐎
𝐎
1 − 𝑘2,𝑑 𝑝2,1
0
0
(𝑘2,𝑑 − 𝑘2,𝑝 )𝑝2
1 − 𝑘2,𝑑 𝑝2,1
0
⋮
⋱
⋱
(𝑘2,𝑑 − 𝑘2,𝑝 )𝑝𝑁
…
(𝑘2,𝑑 − 𝑘2,𝑝 )𝑝2,1
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 1 − 𝑘2,𝑑 𝑝2,1 ⎦
(19)
Due to Eq. (19), the absolute eigenvalues of Hs are |1-k1, d p1 | and |1-k2, d p1 |. Referring to Eq. (18), the synchronization ILC system is stable if and only if
|1−𝑘𝑑 𝑝1,1 | < 1 and |1−𝑘2,𝑑 𝑝2,1 | < 1 | | | |
(20)
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Table 1 Parameters of BSILC controller. Parameter
Description
Value
Unit
ma Pa Aa kpa kda mc Pc Ac Aw kpc kda k
Axial loading motion mass Axial loading source pressure Axial loading piston cross-sectional area Axial loading proportion coefficient Axial loading derivative coefficient Confining loading motion mass Confining loading source pressure Confining loading piston cross-sectional area Water piston cross-sectional area Confining loading proportion coefficient Confining loading derivative coefficient Stiffness of the sample
11 1e7 4.2223e−3 1e−4 1e−6 9 1e7 1.00217e−3 2.82743e−3 1e−3 1e−5 2e6
kg Pa m2 – – kg Pa m2 m2 – – N/m
Fig. 8. The output force of the axial loading. (1) Ideal (2) PID (3) BSILC.
5. Simulation and analysis To test the effect of the BSILC controller designed before the experiment, this paper utilizes MATLAB to conduct force–pressure-coupling closed-loop control simulations for the classic PID controller and the BSILC controller. The parameters of the electrohydraulic servo force loading system in this study are given in Table 1. Two 0.5-Hz standard sine waves are used as the input signals of the axial loading force and the confining loading pressure. The external load acting on the axial loading piston and the confining pressure in the water vessel are used as the outputs in the simulating process. The axial loading output comparison of the classic PID control is shown in Fig. 8. In Fig. 8, the output force amplitude of the classic PID control shows a significant overshoot. The reason can be found in Figs. 4–7. Figs. 4 and 5 show four situations of the unidirectional coupling between two cylinders from two directions. When the electrohydraulic servo loading system functions, two situations in Fig. 4 occur at the same time in the first half period, and two situations in Fig. 5 occur at the same time in the second half period. The combinations of the two unidirectional couplings create a bidirectional coupling, which is even more complex than the original and can cause significant uncertainties. In Fig. 6, the waveform of the force coupled from confining loading to axial loading shows distinct distortion. In Fig. 7, the amplitude of the pressure coupled from axial loading to confining loading is extremely large. These characteristics can lead to a significant nonlinear disturbance to the force control system. The confining loading output comparison of the classic PID control is shown in Fig. 10. In Fig. 10, the output pressure of the classic PID control also shows a significant overshoot, which is caused by the same
Fig. 9. The output force error of the axial loading. (1) PID (2) BSILC.
reason as the axial loading output. The simulation details and the errors between the ideal output and the real output are given as follows. In Figs. 8 and 9, the tracking performance of the classic PID control for the axial loading output is deteriorated by the coupling between two cylinders from two directions so that its maximum force error is approximately 20% of the amplitude of the ideal output force. The BSILC shows better trajectories one period after another, and its output force error has already been reduced by 40% in the second period and continuously converges to 0 as time passes. In Figs. 10 and 11, the tracking performance of the classic PID control for the confining loading output is seriously deteriorated by the coupling between two cylinders from two directions so that its maximum pressure error is over 30% of the amplitude of the ideal output pressure. The BSILC shows better trajectories one period after another, and its output pressure error has been reduced by 20% in the second period and continuously converges to 0 as time passes. To compare the synchronization error of the axial loading force and the confining pressure, the sine wave of the confining pressure is transformed into the same amplitude as the axial loading force. According to Eqs. (5) and (6), the ideal amplitude of the axial loading force is approximately 4000 N, and the ideal amplitude of the confining pressure is approximately 500 kPa. Therefore, the force synchronization error in Fig. 11 is described as 𝑒𝑠 = 𝑦1 +
4000 𝑦 = 𝑦 1 +8 𝑦 2 500 2
(21)
In Fig. 12, the synchronism of the classic PID is also deteriorated by the coupling between two cylinders from two directions so that
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
Fig. 13. The output force of the axial loading. (1) Ideal (2) PID (3) BSILC. Fig. 10. The output pressure of the confining loading (1) Ideal (2) PID (3) BSILC.
Fig. 14. The output force error of the axial loading. (1) PID (2) BSILC.
Fig. 11. The output pressure error of the confining loading. (1) PID (2) BSILC.
its maximum force synchronization error is approximately 10% of the amplitude of the ideal output force. The BSILC shows an increasingly better synchronism one period after another, and its force synchronization error has already been reduced by 20% in the fourth period and continuously converges to 0 as time passes. From the above analysis, the BSILC shows excellent tracking performance and robustness for the ideal axial output force and ideal output pressure, and the synchronism between two cylinders from two directions has been remarkably improved. Therefore, the BSILC method is an ideal control scheme for the coupling in the electrohydraulic servo loading system. 6. Experiment results
Fig. 12. The force synchronization error between the axial loading and confining pressure. (1) PID (2) BSILC.
The experiment results are shown in Figs. 13–17. In Figs. 13 and 14, the output force of the classic PID control for the axial loading output overshoots by approximately 30%, and its waveform shows great distortions. For the BSILC, the output force error has already been reduced by 17.5% in the third period, and the distortion of the output waveform has been corrected. The tracking performance and robustness of the BSILC are increasingly better one period after another. In Figs. 15 and 16, the output pressure of the classic PID control for the confining loading output overshoots by approximately 10%. For the BSILC, the output pressure error has already been reduced by 10% in the second period. The tracking performance and robustness of the BSILC are increasingly better one period after another.
Y. Sang, W. Sun and F. Duan et al.
Mechatronics 62 (2019) 102254
In Fig. 17, the synchronization error of the classic PID is approximately 50% of the amplitude of the ideal output force. For the BSILC, the force synchronization error has already been reduced by 20% in the second period. The synchronism of the BSILC is increasingly better one period after another. The experiment results prove that the BSILC proposed in this paper is valid. The tracking performance, robustness and synchronism of the electrohydraulic servo loading system have been remarkably improved. 7. Conclusion In this paper, a BSILC controller is designed to improve the tracking performance and robustness and to compensate for the synchronization error of the coupling between two cylinders from two directions in an electrohydraulic servo loading system. The conclusions are as follows:
Fig. 15. The output pressure of the confining loading (1) Ideal (2) PID (3) BSILC.
(1) The BSILC controller can improve the tracking performance for ideal axial loading and ideal confining loading and effectively restrain the overshoot of the amplitude. (2) The BSILC controller shows excellent robustness in resisting the distortion of the waveform caused by the coupling between two cylinders from two directions and good synchronism. (3) The BSILC controller designed in this paper has been successfully applied in the bidirectional synchronization control of an electrohydraulic servo loading system, with obvious improvements in the precision of the periodic loading system. This system has value in engineering applications, such as wave simulators for earthquakes, wind and ocean, dynamic tests for mechanical devices, and natural frequency measurement tests. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We gratefully acknowledge the support by the National Natural Science Foundation of China through the grant number 51275068, and the support by the Fundamental Research Funds for the Central Universities Grant no. DUT18LK22.
Fig. 16. The output pressure error of the confining loading. (1) PID (2) BSILC.
References
Fig. 17. The force synchronization error between the axial loading and confining pressure. (1) PID (2) BSILC.
[1] Jiao Z, Yao J. Nonlinear control of electrohydraulic servo system. Beijing: Science Press; 2016. p. 1–5. [2] Merritt HE. Hydraulic control systems. John Wiley & Sons; 1967. [3] Rybak AT, Temirkanov AR, Lyakhnitskaya OV. Synchronous hydromechanical drive of a mobile machine. Russ Eng Res 2018;38:212–17. [4] Karimi HR, Gao H. LMI-based H∞ synchronization of second-order neutral master-slave systems using delayed output feedback control. Int J Control Autom Syst 2009;7(3):371–80. [5] Karimi HR. Robust synchronization and fault detection of uncertain master-slave systems with mixed time-varying delays and nonlinear perturbations. Int J Control Autom Syst 2011;9(4):671. [6] Lin FJ, Chou PH, Chen CS, et al. DSP-based cross-coupled synchronous control for dual linear motors via intelligent complementary sliding mode control. IEEE Trans Ind Electron 2012;59(2):1061–73. [7] Byun JH, Choi MS. A method of synchronous control system for dual parallel motion stages. Int J Precis Eng Manuf 2012;13(6):883–9. [8] Byun JH, Kim YB. A study on construction of synchronous control system for extension and stability. Trans Korean Soc Mech Eng A 2002;26(6):1135–42. [9] Chulines E, Rodríguez MA, Duran I, et al. Simplified model of a three-phase induction motor for fault diagnostic using the synchronous reference frame DQ and parity equations. IFAC-PapersOnLine 2018;5113:662–7. [10] Hsieh MF, Yao WS, Chiang CR. Modeling and synchronous control of a single-axis stage driven by dual mechanically-coupled parallel ball screws. Int J Adv Manuf Technol 2007;34(9-10):933–43. [11] Jeong SK, You SS. Precise position synchronous control of multi-axis servo system. Mechatronics 2008;18(3):129–40. [12] Sato R, Tsutsumi M. Motion control techniques for synchronous motions of translational and rotary axes. Procedia CIRP 2012;1:265–70.
Y. Sang, W. Sun and F. Duan et al. [13] Zhao H, Ben-Tzvi P. Synchronous position control strategy for bi-cylinder electro-pneumatic systems. Int J Control Autom Syst 2016;14(6):1501–10. [14] Roozegar M, Mahjoob MJ, Ayati M. Adaptive tracking control of a nonholonomic pendulum-driven spherical robot by using a model-reference adaptive system. J Mech Sci Technol 2018;32(2):845–53. [15] Chen Z, Yao B, Wang Q. Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect. IEEE/ASME Trans Mech 2013;18(3):1122–9. [16] Golnargesi S, Shariatmadar H, Razavi HM. Seismic control of buildings with active tuned mass damper through interval type-2 fuzzy logic controller including soil–structure interaction. Asian J Civ Eng 2018;19(2):177–88. [17] Artale V, Collotta M, Milazzo C, et al. An integrated system for UAV control using a neural network implemented in a prototyping board. J Intell Rob Syst 2016;84(1-4):5–19. [18] Youssef T, Chadli M, Karimi HR, et al. Actuator and sensor faults estimation based on proportional integral observer for TS fuzzy model. J Franklin Inst 2017;354(6):2524–42. [19] Selvaraj P, Kaviarasan B, Sakthivel R, et al. Fault-tolerant SMC for takagi–sugeno fuzzy systems with time-varying delay and actuator saturation. IET Control Theory Appl 2017;11(8):1112–23. [20] Zhang H, Liu X, Wang J, et al. Robust H∞ sliding mode control with pole placement for a fluid power electrohydraulic actuator (EHA) system. Int J Adv Manuf Technol 2014;73(5-8):1095–104. [21] Meng F, Shi P, Karimi HR, et al. Optimal design of an electro-hydraulic valve for heavy-duty vehicle clutch actuator with certain constraints. Mech Syst Signal Process 2016;68:491–503. [22] Shen W, Jiang J, Su X, et al. Control strategy analysis of the hydraulic hybrid excavator. J Franklin Inst 2015;352(2):541–61. [23] Fan Y, Du Y. Material mechanics. 1st ed. Beijing: Tsinghua University Press; 2017. 2–2. [24] Barton KL, Alleyne AG. A cross-coupled iterative learning control design for precision motion control. IEEE Trans Control Syst Technol 2008;16(6):1218–31. [25] Hillenbrand S, Pandit M. A discrete-time iterative learning control law with exponential rate of convergence. In: Decision and control, 1999. Proceedings of the 38th IEEE conference on, 2. IEEE; 1999. Yong Sang is an associate professor at the school of mechanical engineering, Dalian University of Technology. He received his M.S. degree in Mechatronics from Shandong University in 2004 and the PhD in Mechatronics from Beijing University of Aeronautics and Astronautics in 2007. He worked in the department of mechanical engineering as a postdoctoral researcher, University of Minnesota from Aug. 2012 to Dec. 2013. He is studying on advanced testing technique, hydraulic transmission and control.
Mechatronics 62 (2019) 102254 Weiqi Sun is a doctor at the school of mechanical engineering, Dalian University of Technology. He is studying on measurement and control technology.
Fuhai Duan is a professor at the school of mechanical engineering, Dalian University of Technology. He received his M.S. degree in vehicle guidance, navigation and control technology in 1996 and the PhD in 1999 from Northwestern Polytechnical University. He is studying on embedded computer testing technique, virtual instrument technology and precision control of servo motor.
Jianlong Zhao is a doctor at the school of mechanical engineering, Dalian University of Technology. He is studying on measurement and control technology.