Modeling aspects and Gain scheduled H ∞ Controller Design for an Electrostatic micro-Actuator with Squeezed Gas Film Damping Effects Marialena Vagia and Anthony Tzes Abstract— In this article the modeling and control design aspects of an electrostatic microactuator (EmA) with squeezed thin film damping effects are presented. The modeling analysis of the squeezed film damping effect is investigated in the case of an EmA composed by a set of two plates. The bottom plate is clamped to the ground, while the moving plate is driven by an electrically induced force which is opposed by the force exerted by a spring element. The nonlinear model of the EmA is linearized at various operating points, and the feedforward compensator provides the nominal voltage. Subsequently a gain scheduled H ∞ controller is used to tune the controller- parameters depending on the EmA’s operating conditions. The controller is designed at various operating points based on the distance between its plates. The parameters of the controller are tuned in an optimal manner and computed via the use of the Linear Matrix Inequalities. Special attention is paid in order to examine the stability issue in the intervals between the operating points. Simulation results investigate the efficacy of the suggested modeling and control techniques. I. INTRODUCTION With the rapid progress of micro and nano fabrication processes in the recent years it is now possible to fabricate miniaturized devices whose size varies from micro to nano scales [1]. The focus is on creating high performance devices which are sensitive and have high quality factor [2], [3]. The dynamic behavior of movable parts in MEMS is largely affected by the supporting environmental conditions such as the air pressure, temperature et. al. This gas-structure inter- action [4] has been encountered in certain devices such as accelerometers, gyroscopes and RF-switches [5], [6] which are designed to operate in rarified air condition, whereas other devices, such as microphones, ultrasonic transducers and micro mirrors, generally work with ambient air surround- ing them. Therefore, the effects of the surrounding air and most notably the damping force which can be neglected in structures of conventional dimensions, play a critical role with micro-structures with diminutive size [7]. Squeezed film damping is a term used to describe the most common fluid-structure interaction that impacts the performance of MEMS devices. Squeezed film damping occurs when a thin film layer of air or some other fluid separating the free structure from the substrate is “squeezed” due to any possible movement of the free structure normal to the substrate. Silicon microstructures (sensors and actuators) that make use of the capacitive measurement principles [8], or electrostatic driving forces [9], are characterized by very small gaps between their moving surfaces [5] and so the dynamic behavior of movable parts in these electrostatic actuators is largely affected by the air’s presence (i.e. low The authors are with the Electrical and Computer Engineering De- partment, University of Patras, Rio, Achaia-26500, Greece. Corresponding author’s e-mail: mvagia@ece.upatras.gr vacuum conditions for micro-accelerometers [10], ultra thin gas film in magnetic/disk interfaces [11] and tilting micro- mirrors in DLP type projectors [12], [13]). The understanding of the squeeze film damping mechanism in such electrostatic micro-actuator (EmA) devices [14] is necessary in order to optimize the controller designs. The inclusion of the squeeze film damping effects in- creases the complexity of the dynamics of the micro actuator plant [15], [16], and appropriate controllers should deal with it [17]. Accordingly, since these systems are highly non- linear and have a large order, the tendency is to design controllers that are based on linearized models of the system [13]. In the present article a Gain Scheduled H ∞ controller [18], [19] is designed for a class of Linear Parameter Varying (LPV) plants characterizing the EmA. The main idea is to separate the control design process into two steps. Firstly the local linear controllers are designed based on the linearizations of the nonlinear system at several operating points [20]. In the sequel, a global controller for the nonlinear plant is obtained by interpolating or scheduling the gains of the local operating points design. The linearized plants’ state space matrices are assumed to depend on a vector of spatial varying parameters. The measured parameters, are fed to the controller to optimize the performance and the robustness of the closed loop system. The resulting controller is auto- matically “gain scheduled” along parameter trajectories. The synthesis problem of the controller is fulfilled with the use of the Linear Matrix Inequalities (LMIs). In the rest of this article the modeling of the EmA with squeezed film damping effects is presented in Section II. In Section III the design of the Gain scheduled H ∞ con- trol scheme is presented. Simulation results that prove the efficacy of the proposed control architecture are presented in Section IV, while the conclusions are drawn in the last Section V. II. MODELING OF PARALLEL PLATE ACTUATORS WITH SQUEEZE AIR DAMPING The EmA from a structural point of view corresponds to a micro–capacitor whose one plate is attached to the ground while its other moving plate is floating on the air [21], [22], [23] with the aid of an additional external spring. Figure 1 presents the structure of the EmA. The dynamic nonlinear governing equation of the system [2] is: m ¨ η + F d + kη = ε ℓ U 2 2(η max − η ) 2 = F el (1) where η is the displacement of the plates from the relaxed position, m is the plate’s mass, k is the spring’s stiffness, ℓ is the length of the square plate, U is the applied voltage 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 FrB06.6 978-1-4244-4524-0/09/$25.00 ©2009 AACC 4805