# Noninteracting Control Design for 6-DoF Active Vibration Isolation Table with LMI Approach
## 1 Introduction  
In recent years, the performance requirements of high-precision machinery and measuring instruments have increased at a rapid rate. Microscopes, 3D surface measurement systems, and nano analysis instruments can be named among them. These systems are very sensitive to ambient vibrations, such as ground vibration produced by other machines in the same workspace. A slight vibration may lead to undesirable system movements and the performance of the system deteriorates, which may result in product defects. Therefore, there is a need for mechanisms that isolate these systems from six degrees-of-freedom (DoF) vibrations on a wide range of frequencies. The vibration isolation capability determines the work performance in high-precision operations, which leads to a high level of manufacturing and measurement yields. In general, there are two methods for vibration isolation: passive and active. The passive vibration isolation systems (PVIS), such as springs and dampers, are widely used for their low cost, simple installation, wide load range, and no external energy usage. The vibration isolation can be achieved with the appropriate dynamic structural design of a PVIS, such as in [1]. Although the PVIS can effectively suppress high-frequency vibrations, at low-frequency ranges, it is difficult to achieve satisfying vibration isolation performances. Moreover, the vibrations are amplified whenever their frequencies are near the system resonance frequency. In order to overcome the PVIS limitations, active actuators andadvanced control techniques were introduced with the development of active vibration isolation systems (AVIS). A typical vibration isolation system consists of two plates: a bottom plate fixed on the ground and a top plate carrying the machinery that needs isolating. Springs, dampers, and/or active actuators are situated between the two plates such that the weight of the machinery is supported, the vibration from the ground is suppressed, and the top plate is stabilized. Several types of actuators were used in AVIS systems; for instance, pneumatic actuators [2], hybrid actuators comprising pneumatic actuators and giant magneto-strictive actuators [3], piezoelectric stack actuators [4], and voice coil motors (VCMs) [5]. Each actuator has various advantages and disadvantages, as described in [6], which makes them suitable for diverse applications. VCMs, especially, are widely used because of their advantages, such as good linearity, low noise, and high acceleration and deceleration. In AVISs, the number of actuators must be equal to or greater than the controlled DoFs. Hence, vibrations in every direction can be suppressed; however, the resulting system is often over-actuated. Moreover, motion sensors, such as accelerometers, velocity sensors, and vibration sensors, are mounted directly to the top plate to sense the presence of movements of the machinery and a controller regulates the actuators accordingly. Additional sensors can also be mounted on the bottom plate and the ground vibration can be fed directly to the control systems. Moreover, there is little difference in the mechanical structure and the operating principle of vibration isolation systems. Therefore, recent studies have focused on the design of the control schemes rather than the system structure. One of the greatest difficulties in controlling an AVIS is the large number of DoFs as well as the interactions between them. The presence of these interactions was clear in the system dynamics of the 6-DoFs model in [5] or even the simplified 2-DoFs suggested in [7]. Many authors have tried to propose control methods so that a controller for each DoF can be individually designed. For example, in [8], the author used the modal decomposition technique to transform the system model into a diagonal structure, and then proposed a proportional-integral-derivative (PID) to regulate the performance in each DoF.