0
Research Papers

A Green's Function-Based Design for Deformation Control of a Microbeam With In-Domain Actuation

[+] Author and Article Information
Amir Badkoubeh

e-mail: amir.badkoubeh@polymtl.ca

Guchuan Zhu

e-mail: guchuan.zhu@polymtl.ca
Department of Electrical Engineering,
École Polytechnique of Montreal,
2900, Boulevard Edouard-Montpetit
Montreal, Quebec, Canada H3T 1J4

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 24, 2012; final manuscript received September 21, 2013; published online October 23, 2013. Assoc. Editor: Srinivasa M. Salapaka.

J. Dyn. Sys., Meas., Control 136(1), 011018 (Oct 23, 2013) (8 pages) Paper No: DS-12-1353; doi: 10.1115/1.4025552 History: Received October 24, 2012; Revised September 21, 2013

This paper presents a Green's function-based design for deformation control of a microbeam described by an Euler-Bernoulli equation with in-domain pointwise actuation. The Green's function is first used in control design to construct the test function that enables the solvability of a map between the original nonhomogeneous partial differential equation and a target system in standard boundary control form. Then a regularized Green's function is employed in motion planning, leading to a computationally tractable implementation of the control scheme combined by a single feedback stabilizing loop and feedforward controls. The viability and the applicability of the proposed approach are demonstrated through numerical simulations of a representative microbeam.

FIGURES IN THIS ARTICLE
<>
Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.

References

Tyson, R. K., 2000, Adaptive Optic Engineering Handbook, Marcel Dekker, NY.
Fraanje, R., Massioni, P., and Verhaejen, M., 2010, “A Decomposition Approach to Distributed Control of Dynamic Deformable Mirrors,” Int. J. Optomech., 4(3), pp. 269–284. [CrossRef]
Vogel, C. R., and Yang, Q., 2007, “Modelling, Simulation, and Open-Loop Control of a Continuous Facesheet MEMS Deformable Mirror,” J. Opt. Soc. Am., 24(12), pp. 3827–3833. [CrossRef]
Bifano, T. G., Mali, R. K., Dorton, J. K., Perreault, J., Vandelli, N., Horenstein, M. N., and Castanon, D. A., 1997, “Continuous-Membrane Surface-Micromachined Silicon Deformable Mirror,” J. Opt. Eng., 36(5), pp. 1354–1360. [CrossRef]
Morzinski, K., Harpsoe, K., Gavel, D., and Ammons, S., 2007, “The Open-Loop Control of MEMS: Modeling and Experimental Results,” Proc. SPIE6467, pp. 645–654.
Stewart, J. B., Diouf, A., Zhou, Y., and Bifano, T. G., 2007, “Open-Loop Control of a MEMS Deformable Mirror for Large-Amplitude Wavefront Control,” J. Opt. Soc. Am., 24(12), pp. 3827–3833. [CrossRef]
Preumont, A., 2004, Vibration Control of Active Structures: An Introduction, Kluwer Academic, New York.
Timoshenko, S., and Woinowsky, S., 1959, Theory of Plates and Shells, 2nd ed., MacGraw-Hill Book Company, New York.
Ventsel, E., and Krauthammer, T., 2001, Thin Plate and Shells: Theory, Analysis, and Applications, Marcel Dekker, New York.
Ammari, K., Mercier, D., Regnier, V., and Valein, J., 2012, “Spectral Analysis and Stabilization of a Chain of Serially Connected Euler-Bernoulli Beam and Strings,” Commun. Pure Appl. Anal., 11(2), pp. 785–807. [CrossRef]
Aoustin, Y., Fliess, M., Mounier, H., Rouchon, P., and Rudolph, J., 1997, “Theory and Practice in the Motion Planning and Control of a Flexible Robot Arm Using Mikusinski Operators,” Proceedings of the Fifth IFAC Symposium on Robot Control, pp. 287–293.
Bensoussan, A., Da Prato, G., Delfor, M., and Mitter, S. K., 2006, Representation and Control of Infinite-Dimensional Systems, Birkhauser, Boston, MA.
Krstić, M., and Smyshyaev, A., 2008, Boundary Control of PDEs: A Course on Backstepping Designs, SIAM, New York.
Lasiecka, I., and Triggiani, R., 2000, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge University, Cambridge, UK.
Meurer, T., Thull, D., and Kugi, A., 2008, “Flatness-Based Tracking Control of a Piezoactuated Euler-Bernoulli Beam With Non-Collocated Output Feedback: Theory and Experiments,” Int. J. Control, 81(3), pp. 473–491. [CrossRef]
Rudolph, J., 2003, Flatness Based Control of Distributed Parameter Systems, Shaker-Verlag, Aachen, Germany.
Ammari, K., and Tucsnak, M., 2000, “Stabilization of Bernoulli-Euler Beams by Means of a Pointwise Feedback Force,” SIAM J. Control Optim., 39(4), pp. 1160–1181. [CrossRef]
Conrad, F., 1990, “Stabilization of Beam by Pointwise Feedback Control,” SIAM J. Control Optim., 28(2), pp. 423–437. [CrossRef]
Chen, G., Delfour, M. C., Krall, A., and Payre, G., 1987, “Modeling, Stabilization and Control of Serially Connected Beams,” SIAM J. Control Optim., 25(3), pp. 526–546. [CrossRef]
Le Gall, P., Prieur, C., and Rosier, L., 2007, “Output Feedback Stabilization of a Clamped-Free Beam,” Int. J. Control, 80(8), pp. 1201–1216. [CrossRef]
Liu, K., Huang, F., and Chen, G., 1989, “Exponential Stability Analysis of a Chain of Coupled Vibrating Strings With Dissipative Linkage,” SIAM J. Appl. Math., 33(1), pp. 1–28.
Rebarber, R., 1995, “Exponential Stability of Coupled Beam With Dissipative Joints: A Frequency Domain Approach,” SIAM J. Control Optim., 33(1), pp. 1–28. [CrossRef]
Fliess, M., Lévine, J., Martin, P., and Rouchon, P., 1995, “Flatness and Defect of Nonlinear System Introductory Theory and Examples,” Int. J. Control, 61, pp. 1327–1361. [CrossRef]
Lévine, J., 2009, Analysis and Control of Nonlinear Systems: A Flatness-Based Approach, Springer-Verlag, Berlin.
Laroche, B., Martin, P., and Rouchon, P., 2000, “Motion Planning for the Heat Equation,” Int. J. Robust Nonlinear Control, 10, pp. 629–643. [CrossRef]
Lynch, A. F., and Rudolph, J., 2002, “Flatness-Based Boundary Control of a Class of Quasilinear Parabolic Distributed Parameter Systems,” Int. J. Control, 75(15), pp. 1219–1230. [CrossRef]
Meurer, T., and Kugi, A., 2009, “Tracking Control for Boundary Controlled Parabolic PDEs With Varying Parameters: Combining Backstepping and Differential Flatness,” Automatica, 45, pp. 1182–1194. [CrossRef]
Petit, N., Rouchon, P., Boueih, J. M., Guerin, F., and Pinvidic, P., 2002, “Control of an Industrial Polymerization Reactor Using Flatness,” Int. J. Control, 12(5), pp. 659–665.
Badkoubeh, A., and Zhu, G., 2012, “Flatness-Based Deformation Control of a 1-Dimensional Microbeam With in-Domain Actuation,” IECON’12, Montreal, Que, pp. 2301–2306.
Badkoubeh, A., and Zhu, G., 2012, “Deformation Control of a 1-Dimensional Microbeam With in-Domain Actuation,” CDC’12, Maui, Hawaii, pp. 3145–3150.
Polyanin, A. D., 2002, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, New York.
Kovacs, G. T. A., 1998, Micromachined Transducers Sourcebook, McGraw-Hill, New York.
Diouf, A., Legendre, A. P., Stewart, J. B., Bifano, T. G., and Lu, Y., 2010, “Open-Loop Shape Control for Continuous Microelectromechanical System Deformable Mirror,” Appl. Opt., 49(31), pp. 148–154. [CrossRef]
Maithripala, D. H. S., Berg, J. M., and Dayawansa, W. P., 2005, “Control of an Electrostatic MEMS Using Static and Dynamic Output Feedback,” ASME J. Dyn. Syst., Meas., Control, 127, pp. 443–450. [CrossRef]
Zhu, G., 2011, “Electrostatic MEMS: Modelling, Control, and Applications,” Advances in the Theory of Control, Signals and Systems With Physical Modeling, Vol. 407 (Lecture Notes in Control and Information Sciences), J.Lévine and P.Mullhaupt, eds., Springer-Verlag, Berlin, pp. 113–123.
Senturia, S. D., 2002, Microsystem Design, Kluwer Academic Publishers, Norwell, MA.
Lasiecka, I., 2002, Mathematical Control Theory of Coupled PDEs, SIAM, Philadelphia, PA.
Cortez, R., and Minion, M., 2000, “The Blob Projection Method for Immersed Boundary Problems,” J. Comput. Phys, 161, pp. 428–453. [CrossRef]
Pederson, M., 2000, Functional Analysis in Applied Mathematics and Engineering, Chapman and Hall/CRC, Boca Raton, FL.
Rodino, L., 1993, Linear Partial Differential Operators in Gevrey Spaces, World Scientific, River Edge, NJ.
Trefethen, L. N., 2000, Spectral Method in MATLAB, SIAM, Philadelphia, PA.

Figures

Grahic Jump Location
Fig. 1

Schematic of the deformable microbeam

Grahic Jump Location
Fig. 2

Stabilizing feedback control

Grahic Jump Location
Fig. 3

Green's function of the beam for ξ={0.1,0.2,…,0.9}

Grahic Jump Location
Fig. 4

Deformation control: (a) desired shape; (b) beam deflection; and (c) tracking error

Grahic Jump Location
Fig. 5

Control signals: (a) α1α5 and (b) α6α10

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In