TECHNICAL PAPER Integral sliding mode control for nonlinear damped model of arch microbeams Arman Rajaei 1 • Amin Vahidi-Moghaddam 2 • Moosa Ayati 2 • Mostafa Baghani 2 Received: 13 March 2018 / Accepted: 28 April 2018 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In this paper, a second order integral sliding mode controller (SMC) and a two-dimensional integral sliding mode controller are designed for a nonlinear damped model of arch microbeam with two electrodes as a bistable system. The latest model of the arch microbeam is introduced in which the squeezed film damping effect is modeled through considering nonlinear terms. The actuating voltage is considered as the control effort of the system somehow expands as a combined static DC and harmonic AC voltage. The second order integral SMC and two-dimensional integral SMC are proposed as the robust controllers to stabilize the system in the presence of the uncertain parameter due to the damping coefficient. The controller formula, stability and convergence of the closed-loop system are derived and formulated for the arch microbeam. Sim- ulation results and comparison of the proposed controllers are presented to demonstrate the performance of the designed control schemes for achieving set point tracking in the closed-loop system. 1 Introduction Recently, arch microbeams have important roles and applications as the perfect mechanisms for micro-electro- mechanical systems (MEMS) such as micro mechanical memories (Hosseini et al. 2017), micro resonators (Tajad- dodianfar et al. 2016), optical switches (Huang et al. 2011), micro actuators (Park and Hah 2008), inertial sensors (Rabanim et al. 2011) and etc. Therefore, the nonlinear dynamic behaviors of the arch microbeams have been investigated by many researchers as one of the main problems in micro-structures (Krylov and Dick 2010; Medina et al. 2012; Baghani 2012; Daneshpajooh and Zand 2015; Medina et al. 2017). Krylov and Dick (2010) investigated the dynamic stability of electrostatically actuated initially arch shallow microbeams. The dynamic behavior of an initially arch shallow microbeam loaded by a distributed electrostatic force has been studied by Medina et al. (2012). The analytical approaches for studying the vibration behavior of the arch micro/nano-beams have been employed by Daneshpajooh and Zand (2015). Electrostatic actuation as the main actuator for MEMS devices has significant applications due to high reliability, low power consumption and simple structure (Senturia 2007; Varadan et al. 2003). Over the past years, many investigations have been carried out to study the nonlinear dynamic behaviors of the micro/nano-structures in the presence of electrostatic actuation (Chuang et al. 2010; Lotfi et al. 2017; Bouchaala et al. 2017; Zhao and Trimble 2017; Kumar et al. 2017; Bouchaala 2018). Small changing of the voltage causes the snap through and bistability for arch microbeam, so they are perfect mechanisms for mechanical memories, gripers, manipulator and etc. Bouchaala et al. (2017) have conducted an analytical study for the frequency shifts of micro/nano-resonators due to an added mass. An electrostatic actuation scheme for low voltage micro switch has been investigated by Kumar et al. (2017). Bouchaala (2018) reported a theoretical study on an electrostatically actuated torsional micro sensor for bio- logical applications. To enhance the performance, resolution and efficiency of micro-structures, the vibrations control of these structures has been investigated as a significant subject (Alsaleem and Younis 2010, 2011; Vagia 2012; Zhankui and Sun 2013; Vatankhah et al. 2015; Rega and Lenci 2015). Alsaleem and Younis (2010) have designed a delayed feedback controller & Moosa Ayati m.ayati@ut.ac.ir 1 School of Mechanical Engineering, Shiraz University, Shiraz, Iran 2 School of Mechanical Engineering, College of Engineering, University of Tehran, P.O.B. 11155-4563, Tehran, Iran 123 Microsystem Technologies https://doi.org/10.1007/s00542-018-3931-1