Design and Experimental Validation of a Nonlinear Tracking Control Law for an Electrostatic Micromirror Carlos G. Agudelo, Muthukumaran Packirisamy, Guchuan Zhu, and Lahcen Saydy Abstract— This work aims at demonstrating potential ben- efits of applying nonlinear control techniques to electrostatic micromirrors in order to extend their stable operational range and enhance the system’s performance. A nonlinear tracking control based on feedback linearization and trajectory planning has been developed. Aspects essential to the implementation of such a system, such as prevention of the device from its destruction on contact, modeling and sensing schemes allowing for the removal of on-chip sensors, influence of the dynamics of the driving circuit on the performance, and characterization of the device, have been thoroughly addressed and practical solutions have been proposed. The experimentation is per- formed on a set-up built with low cost, commercial off-the-shelf (COTS) instruments and components in an ordinary laboratory environment. The experimental results show that the developed control system can achieve a stable operation beyond the pull-in position for both set-point and scanning controls. I. I NTRODUCTION The electrostatic micromirror is one of the most popular microelectromechanical systems (MEMS), which is used in a variety of scientific, commercial, and defense applications, such as adaptive optics [15], optical network switching [4], projection systems [16], and resonant microsensors [1], among others. However, electrostatic actuation results in highly nonlinear dynamics, giving rise to a saddle-node bi- furcation, called “pull-in,” which limits the stable open-loop operation to a small portion of the whole physically avail- able range [12]. Extending the stable operation range and enhancing the performance of electrostatic MEMS constitute the main motivation of the majority of the works in the appli- cation of closed-loop control in this area. The work reported in the literature has addressed the application of a variety of techniques, in particular nonlinear control methods, to the control of diverse electrostatic micro-actuators [9], [10], [21], [19], [13], [11], as well as experimental implementations of closed-loop control algorithms for micromirrors [4], [3], [18], [6]. The present work deals with the design of nonlinear tracking control of a one-degree of freedom (1DOF) scanning micromirror. It emphasizes particularly issues related to the experimental implementation. More specifically, in order to ensure safe and repeatable operations, the micromirror is designed in such a way as to prevent the destruction of This work was supported in part by the Natural Science and Engineering Research Council of Canada (NSERC). C. G. Agudelo, G. Zhu, and L. Saydy are with the Department of Electrical Engineering, ´ Ecole Polytechnique de Montr´ eal, P.O. Box 6079, Station Centre-Ville, Montr´ eal, QC, Canada H3C 3A7. M. Packirisamy is with the Department of Mechanical and Industrial Engineering, Concordia University, 1455 de Maisonneuve Blvd. West, Montr´ eal, QC, Canada H3G 1M8. the structure due to, e.g., short-circuit when the moveable and fixed electrodes come into contact (intentionally or ac- cidently). To this end, stopping mechanisms are considered in the design of micromirror. We will see later that this structure will also allow avoiding a singularity in the dynamics of the device associated to short-circuit, making the system model more reliable. Another important issue that affects both micro-actuator design of and control algorithm development is the choice of state variable in the dynamic model. In the literature, it is common to take the charge as a state variable for describing the dynamics of electrostatic MEMS (see, e.g., [12], [10], and [20]). However, the implementation of on-chip charge measurement requires additional structures (see, e.g., [2]), besides electrical interference between the actuation and the sensing during the operation. To circumvent this problem, the actuation voltage across the device is taken instead as a state variable in the dynamic model. Consequently, on-chip charge sensing is no longer required. As voltage measurement is instantaneous and trivial to implement, we can expect to achieve a simple but reliable implementation of control systems. A position sensitive detector (PSD) will be used as angular deflection sensor. This makes it possible to implement new control systems with existing devices that are not equipped with on-chip position and charge sensors. The model used in controller design incorporates also the dynamics of the driving circuit, including output impedance of the high-voltage amplifier used in the experimental sys- tem. Note that the output capacitance of the high voltage amplifier is usually of several orders of magnitude greater than that of the device. Therefore, it has an important impact on the performance of the system and cannot be ignored in practice. Finally, to obtain the angular velocity required for con- troller implementation, we use the discrete derivative in order to respect the capability offered by the embedded computa- tion platform. Closed-loop stability and system performance will be assessed by numerical simulation and experimental test. The rest of the paper is organized as follows. Section II presents the micromirror and its dynamic model. Section III is dedicated to the control synthesis. Section IV introduces the experimental setup and the characterization of the mi- cromirror. The results of the simulation study and the exper- imental test are reported in Section V. Finally, Section VI contains some concluding remarks. 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 FrA08.2 978-1-4244-4524-0/09/$25.00 ©2009 AACC 4202