## 1 Introduction
ERFs comprise micrometer-size dielectric particles in insulating liquid, so that their apparent viscosity is varying in response to the applied electric field. Based on electrorheological properties of ERs, many researchers developed different applications and systems using ERF. For a hollow cantilevered beam filled with ER fluid, in the absence of the electric field, the ERF produces the shear force only caused by the fluid resistance. However, if a certain level of electric field is supplied to the ERF, a shear stiffness arises owing to the pre-yielding behavior of the ERF. Therefore, the stiffness of the beam can be tuned by controlling the intensity of the electric field applied to the ERF. [PERSON] et al. investigated the vibration properties for hollow cantilevered beams filled with ER fluid and derived an empirical model for predicting the vibration characteristics in responding to electric field [48]. Many studies focus on the progress of ER materials in modeling the mechanisms and rheology of electrorheological fluids. There are several literature reviews about material properties including water-containing system such as silica gel, polymer, cellulose, and water-free system such as aluminosilicate, carbonaceous, semiconducting polymers [50; 51; 52; 53; 54; 55; 56; 57]. During the past decades, effective semi-active suspension and absorber systems featuring ERF has been proposed by many investigators. [PERSON] et al. proposed a semi-active vibration control for ER adaptive beams and illustrated their capabilities by experiments [58]. [PERSON] et al. designed a feedback vibration controller for the cantilevered ER beam [59]. [PERSON] et al. studied the vibration control of rotating flexible beams and investigated the feasibility of ER fluids by experiments [60]. [PERSON] et al. proposed an ER damper for cylindrical suspension system of small-sized vehicles [61]. However, because ER fluids change from Newtonian flow to Bingham plastic with response to the applied electric field, the damping and stiffness of the ATVA is very nonlinear. Therefore, accurate parameters of ERF models are generally difficult to obtain so that there are modeling uncertainties for the ATVAsystems. The vibration controller must have the adaptive capability and the robustness to the modeling uncertainties and the external disturbance. The damping force of ERF consists a viscous damping without an electric field. With an electric field, The ERF can provide a Coulomb friction damping force for vibration control. Generally, the Coulomb friction damping force is in proportion to the square of voltage applied to the ERF [62]. The relation between the applied voltage and the Coulomb friction is intensively nonlinear. The controller based on the linear control theory is difficult to design to achieve the required performance. Therefore, these systems were mostly modeled by highly nonlinear differential equations, thus making it difficult when considering controller design. Various controller design schemes were proposed to manipulate ERF absorber systems, such as the intelligent control, adaptive control, and robust control. [PERSON] et al. designed a fuzzy controller to deal with the nonlinearity for a hybrid smart composite beam actuated by piezoceramics and ERF [62]. [PERSON] et al. proposed a robust control of electrorheological suspension system subjected to parameter uncertainties associated with sprung mass of the vehicle and the ER damper [63]. Because the membership functions of type-1 fuzzy sets do not have uncertainties associated with them, the modeling uncertainties of ERF cannot be described in the membership function of the type-1 fuzzy set. Recently, more and more researchers around the world are studying about type-2 fuzzy sets and systems for various applications [64; 65; 66]. Membership functions of type-2 fuzzy set are different from the one of type-1 fuzzy set. For type-2 fuzzy sets, the membership functions are three-dimensional.