Nano-precision control of micromirrors using output feedback D. H. S. Maithripala * , R. O. Gale ** , M. W. Holtz † , J. M. Berg * , W. P. Dayawansa ‡ Depts. of * Mechanical Engineering, ** Electrical Engineering, † Physics, ‡ Mathematics & Statistics Texas Tech University Lubbock, TX 79409 USA daya@math.ttu.edu Abstract— Micromirror arrays may be used as spatial light modulators, to vary the amplitude and phase of an incident image. The difficulty of controlling analog micromirrors to within a fraction of a wavelength of the incident light—that is, to within a few nanometers—has led to the development and successful commercialization of digital devices, in which the components take one of only two possible configurations. In this paper we revisit true analog operation, in light of recent advances in nonlinear control. Our focus is electrostatically actuated mirrors, for which control objectives include extending the range of motion, improving transient performance, im- proving positioning accuracy and preventing electrode contact. We consider two models of individual micromirrors, suitable for controller design. The first treats each micromirror as a rigid body, and allows arbitrary rotation and translation. This model is developed in the context of dynamics on the Lie group SE(3). We simulate the performance of an observer-based controller for this system. The second model, based on work by Pelesko, is developed directly from a PDE representation of an electrostatically forced membrane. We show via simulation that an unstable equilibrium point of this model may be stabilized by proportional output feedback. I. I NTRODUCTION Micro electromechanical systems and micro optoelec- tromechanical, MEMS and MOEMS, respectively, are minia- ture devices with potentially revolutionary impact. An exam- ple of a MEMS device which has had a major commercial impact is the accelerometer used to trigger air bag de- ployment in automobiles. A hallmark commercial MOEMS device is the Texas Instruments digital micromirror device (DMD) used in projection technology. Additional areas where MOEMS technology is promising include holographic data storage, wavelength division multiplexing (WDM), and adaptive optics. MEMS and MOEMS devices are gener- ally fabricated using silicon micromachining and integrated circuit manufacturing techniques, and typically include at least one moving component. Several methods are in use to actuate these devices, including piezoelectric [1], magnetic and magnetic/electrostatic [2], and electrostatic [7] – [14]. The primary subject of this paper is non-linear control of electrostatically actuated MOEMS devices. We first motivate the need for such control with a brief overview of spatial light modulators (SLMs) based on micromirror arrays. Micromirror arrays function either diffractively or reflec- tively, depending on their size. These devices include com- mercial digital devices such as the DMD [3] and grating light Fig. 1. Optical Image Correlator with Reflective Phase and Amplitude SLM. valve (GLV) [7], and laboratory analog devices such as are described in [8], [9]. The function of a SLM array is to take nominally planar incoming light and modulate its amplitude and/or phase according to some array function. The array function can be simple, such as alternating stripes to produce a diffraction grating, or complex, such as a known image function to serve as an optical comparator. Potential uses include holographic data storage and manipulation, matched filter correlation for rapid search of holographically stored data, joint transform correlators for “lock-and-key” security systems, and adaptive optics for improved spatial resolution. Associated with these capabilities are numerous applications in the areas of human identification, genomic data bases, and security. Figure 1 shows an implementation of an all- optical image recognition system due to Florence and Gale [6], that exploits a reflective SLM. The use of such systems to perform complex computational tasks has grown from the Vander Lugt image recognition system [4], through analysis of synthetic aperture radar for terrain mapping, to the present day, when miniature optical correlators, small enough to fit into a modest personal computer chassis can be bought off the shelf [5]. In the system depicted, the array function of the SLM is chosen to match the Fourier transform of a target pattern. An intensity peak in the image correlation plane indicates the presence of the target pattern in the input image. Conceptually, the use of an array of reflective micromirrors as a phase and amplitude SLM is straightforward. Translation of a mirror above or below the device reference plane by distances of a fraction of a wavelength cause the reflection