Robust tracking control of an array of nanoparticles moving on a substrate

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.