# Optimal Design and Hybrid Control for the Electro-Hydraulic Dual-Shaking Table System
## 1 Introduction  
The electro-hydraulic earthquake simulation shaking table is the most important piece of equipment for replicating actual earthquake vibration environments, and can be used for the earthquake resistance test in the field of civil and architectural engineering [1]. For the large-span structures, such as bridges, dams, railways and pipelines, etc., the shaking tables system composed of more than one shaking tables is proposed to carry out the vibration test simultaneously [2]. Mechanical test and simulation (MTS) corporation built the two horizontal directions shaking tables system for the University of Nevada, Reno in 2003 [3] and built the Six-degree of freedom (DOF) shaking tables system for University of Buffalo, state university of New York (SUNY) in 2003 [4]. Many large-span structure earthquake simulation experiments have been successfully performed, which indicates the superiority of the shaking tables system in the experimental research [5, 6]. Although some shaking tables systems have been built, the design and control technique are still strictly secret and undocumented by a few commercial companies out of commercial benefit, which is a hindrance to the development of the shaking tables system research. Especially, there is no academic reference describing how to design the parameters of the hydraulic cylinders, servo valvesand hydraulic power supply aiming at the shaking table system, let alone how to optimize these parameters. For the shaking tables system, to realize vibration environment replication accurately, not only should the synchronization precision of all shaking tables be ensured but also the tracking precision of the desired signal. It is because the control precision directly determines the reality of earthquake replication and in turn influences the feasibility of the performance research of the specimen, which is explained in [7,8]. However, due to the eccentricity of the load mass, cross-coupled characteristic, system nonlinearity and disturbance, the control system has a poor tracking and synchronization precision [9,10]. The cross-coupled controller (CCC) proposed by [PERSON] [11] to reduce contour error of the computer numerical control (CNC) machine tools has a satisfactory effect. Nowadays, the CCC and many modified schemes have been researched to improve the system synchronization precision such as variable-gain, adaptive controller, linear quadratic gauss (LQG) optimal control, fuzzy logic control and network control, etc. [12-14]. Recently, SMC has attracted much attention due to strong robustness, fast dynamic response and simplicity of design and implementation [15-17]. The SMC is introduced to the synchronization control of multiple robotic manipulators in [15]. [PERSON] [16] developed a robust control strategy for the dual-motor system by incorporating a second order SMC. In [17], a sliding mode synchronization controller was designed for the fixture clamp equipment. However, one drawback of the traditional SMC is the large control chatting phenomenon, which can wear coupled mechanisms and degrade the system performance. For the control of single multi-axis shaking table, [PERSON] [18] proposed a DOF control method to realize the transformation between actuators space and DOF space. [PERSON] et al. [19] studied the three-variable control (TVC) technique to improve the stability and tracking performance of the system. [PERSON] et al. [20] proposed the inner force suppression (IFS) control strategy to solve the problem of inner force coupling. To improve the tracking performance, many researchers have studied advanced control strategies, such as least mean square (LMS)-based, robust control, adaptive control, disturbance observer, internal model control and iterative learning control, etc. [21-26]. Feed-forward inverse control has been widely used in high precision tracking systems due to its perfect control performance without affecting system stability [27].