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Robust tracking control of an array of nanoparticles moving on a substrate. (English) Zbl 1260.93048

Summary: Control of nanosystems with frictional dynamics using feedback control methods is important to a wide range of applications of nanotribology. This paper studies the tracking control problem of an array of nanoparticles moving on a substrate with friction between the substrate and the particles. The focus of this study is on control design and stability analysis. The major challenges in this problem include nonlinearities and uncertainties in the frictional dynamics and limited availability of measurable states in nanosystems. The particle-substrate interaction is considered to be unknown, and the unknown effect of unmodeled particle dynamics on the dynamics of the center of mass of the array is also considered. A nonlinear identifier is first developed to identify these unmodeled dynamics. A feedback controller is then developed based on the identifier to control the center of mass of the particles to track a desired trajectory. Boundedness of the closed-loop states and semiglobal asymptotic stability of the tracking error are proven using Lyapunov theory for the case of linear inter-particle interactions. An example with more general Morse-type inter-particle interactions is included to provide some level of confidence that the results are general but not assuredness that they are. Numerical simulation results are provided to demonstrate the performance of the developed identification and control law.

MSC:

93B35 Sensitivity (robustness)
93B52 Feedback control
93D20 Asymptotic stability in control theory
93C95 Application models in control theory
93C15 Control/observation systems governed by ordinary differential equations
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[1] Braiman, Y.; Barhen, J.; Protopopescu, V., Control of friction at the nanoscale, Physical Review Letters, 90, 9, 094301 (2003), (1-4)
[2] Carpick, R. W., Physics: controlling friction, Science, 313, 5784 (2006)
[3] Edwards, C. H., Advanced calculus of several variables (1994), Courier Dover Publications · Zbl 0308.26002
[4] Godsil, C.; Royle, G., (Algebraic graph theory. Algebraic graph theory, Graduate texts in mathematics, Vol. 207 (2001), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0968.05002
[5] Guerra, R.; Vanossi, A.; Urbakh, M., Controlling microscopic friction through mechanical oscillations, Physical Review E, 78, 036110 (2008), (1-5)
[6] Guo, Y.; Qu, Z., Control of frictional dynamics of a one-dimensional particle array, Automatica, 44, 2560-2569 (2008) · Zbl 1155.93356
[7] Guo, Y.; Qu, Z.; Braiman, Y.; Zhang, Z.; Barhen, J., Nanotribology and nanoscale friction, IEEE Control Systems Magazine, 28, 92-100 (2008) · Zbl 1395.93477
[8] Guo, Y.; Qu, Z.; Zhang, Z., Lyapunov stability and precise control of the frictional dynamics of a one-dimensional nanoarray, Physical Review B, 73, 9, 094118 (2006)
[9] Khalil, H. K., Nonlinear systems (2002), Prentice-Hall, Inc: Prentice-Hall, Inc New Jersey · Zbl 0626.34052
[10] Makkar, C.; Hu, G.; Sawyer, W. G.; Dixon, W. E., Lyapunov-based tracking control in the presence of uncertain nonlinear parameterizable friction, IEEE Transactions on Automatic Control, 52, 10, 1988-1994 (2007) · Zbl 1366.93443
[11] Merris, R., Laplacian matrices of graphs: a survey, Linear Algebra and its Applications, 197-198, 143-176 (1994) · Zbl 0802.05053
[12] Olfati-Saber, R.; Murray, R. M., Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49, 9, 1520-1533 (2004) · Zbl 1365.93301
[13] Park, J.; Ogletree, D.; Salmeron, M.; Ribeiro, R.; Canfield, P.; Jenks, C.; Thiel, P., High frictional anisotropy of periodic and aperiodic directions on a quasicrystal surface, Science, 309, 1354 (2005)
[14] Park, J.; Ogletree, D.; Thiel, P.; Salmeron, M., Electronic control of friction in silicon pn junctions, Science, 313, 5784 (2006)
[15] Protopopescu, V.; Barhen, J.; Amselem, G.; Dahan, J., Non-lipschitzian control algorithms: application to a nanofriction model, Mathematical Methods in the Applied Sciences, 29, 249-266 (2006) · Zbl 1134.93347
[16] Reiter, G.; Demirel, A.; Granick, S., From static to kinetic friction in confined liquid films, Science, 263, 1741 (1994)
[17] Socoliuc, A.; Gnecco, E.; Maier, S.; Pfeiffer, O.; Baratoff, A.; Bennewitz, R.; Meyer, E., Atomic-scale control of friction by actuation of nanometersized contacts, Science, 313, 207-210 (2006)
[18] Urbakh, M.; Klafter, J.; Gourdon, D.; Israelachvili, J., The nonlinear nature of friction, Nature, 430, 525-528 (2004)
[19] Xian, B.; Dawson, D. M.; de Queiroz, M.; Chen, J., A continuous asymptotic tracking control strategy for uncertain nonlinear systems, IEEE Transactions on Automatic Control, 49, 7, 1206-1211 (2004) · Zbl 1365.93219
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