1M.Sc. student of Control Engineering in the Department of Electrical Engineering, Faculty of Electrical, Biomedical, and Mechatronic, Qazvin Branch, Islamic Azad University, Qazvin, Iran
2Professor, Control & Intelligent Processing Center of Excellence, School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran
Friction is a nonlinear phenomenon which has destructive effects on performance of control systems. To obviate these effects, friction compensation is an effectual solution. In this paper, an adaptive technique is proposed in order to eliminate limit cycles as one of the undesired behaviors due to presence of friction in control systems which happen frequently. The proposed approach works for nonlinear dynamic and static friction models and is applicable to a wide range of different mechanical systems. It is also applied to a simple inverted pendulum on a cart as a highly nonlinear under-actuated system. A nonlinear optimal controller based on the approximate solution of Hamilton-Jacobi-Bellman partial differential equation is designed to fulfill our control objectives and achieve preferable performance compared to those of the linear optimal controllers. It causes to have more accuracy in system's response and positioning in the presence of friction. Simulation result approve the effectiveness of both the presented technique and controller.
 Morteza Nazari Monfared and Mohammad Javad Yazdanpanah, “Adaptive Compensation Technique for Nonlinear Dynamic and Static Models of Friction,” Electrical Engineering (ICEE), 23nd Iranian Conference on IEEE.  Armstrong-Hélouvry, Brian, Pierre Dupont, and Carlos Canudas De Wit,“A survey of models, analysis tools and compensation methods for the control of machines with friction,” Automatica, vol. 30, no. 7, pp.1083-1138, 1994.  De Wit, C. Canudas, and et al, “A new model for control of systems with friction”. Automatic
Control, IEEE Transactions on, vol. 40, no. 3, pp. 419-425, 1995.  Olsson, Henrik. Control systems with friction. Diss. Lund University, 1996.  Dupont, Pierre, and et al.,“Single state elastoplastic friction models,” Automatic Control, IEEE Transactions on, vol. 47, no. 5, pp. 787-792, 2002.  Al-Bender, Farid, Vincent Lampaert, and Jan Swevers, “The generalized Maxwell-slip model: a novel model for friction simulation and compensation,” Automatic Control, IEEE Transactions on, vol. 50, no. 11, pp. 1883-1887, 2005.  Kermani, Mehrdad R., Rajnikant V. Patel, and Mehrdad Moallem, “Friction identification and compensation in robotic manipulators,” Instrumentation and Measurement, IEEE Transactions on, vol. 56, no. 6, pp. 2346-2353, 2007.  Hensen, Ron HA, Marinus JG van de Molengraft, and Maarten Steinbuch, “Frequency domain identification of dynamic friction model parameters,” Control Systems Technology, IEEE Transactions on, vol. 10, no. 2, pp.191-196, 2002.  Wang, Yongfu, Dianhui Wang, and Tianyou Chai, “Extraction and adaptation of fuzzy rules for friction modeling and control compensation,” Fuzzy Systems, IEEE Transactions on, vol. 19, no. 4, pp. 682-693, 2011.  Makkar, Charu, and et al., “Lyapunov-based tracking control in the presence of uncertain nonlinear parameterizable friction”. Automatic Control, IEEE Transactions on, vol. 52, no. 10, pp. 1988-1994, 2007.  De Wit, C. Canudas, and Pablo Lischinsky, “Adaptive friction compensation with partially known dynamic friction model,”International journal of adaptive control and signal processing vol. 11, pp. 65-80, 1998.  Kelly, Rafael, Jesús Llamas, and Ricardo Campa., “A measurement procedure for viscous and coulomb friction,” Instrumentation and Measurement, IEEE Transactions on, vol. 49, no. 4, pp. 857-861, 2000.  Rizos D., and S. Fassois, “Friction identification based upon the LuGre and Maxwell slip models,” Control Systems Technology, IEEE Transactions on, vol. 17, no.1, pp. 153-160, 2009.  Campbell, Sue Ann, Stephanie Crawford, and Kirsten Morris, “Friction and the inverted pendulum stabilization problem,” Journal of Dynamic Systems, Measurement, and Control, vol. 130, no. 5, pp. 054502, 2008.  Kirk, Donald E. Optimal control theory: an introduction. Courier Corporation, 2012.  Navasca, C. L., and A. J. Krener,“Solution of hamilton jacobi bellman equations,” Decision and Control, 2000. Proceedings of the 39th IEEE Conference on., vol. 1, 2000.  Hunt, Thomas, and Arthur J. Krener,”Improved patchy solution to the Hamilton-Jacobi-Bellman equations,” CDC, 2010.  Beard, Randal W., George N. Saridis, and John T. Wen,”Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation,” Automatica, vol. 33, no. 12, pp. 2159-2177, 1997.  Sassano, Mario, and Alessandro Astolfi, “Dynamic solution of the HJB equation and the optimal control of nonlinear systems,” Decision and Control (CDC), 49th IEEE Conference on, 2010.  Milasi, Rasoul M., Mohammad‐Javad Yazdanpanah, and Caro Lucas,“Nonlinear optimal control of washing machine based on approximate solution of HJB equation,” Optimal Control Applications and Methods, vol. 29, no. 1, pp. 1-18, 2008.  P. Ioannou and Bariş Fidan. Adaptive Control Tutorial. Society for Industrial and applied mathematics, Philadelphia, USA, 2006.  Ioannou, Petros A., Elias B. Kosmatopoulos, and Alvin M. Despain, “Position error signal estimation at high sampling rates using data and servo sector measurements,” Control Systems Technology, IEEE Transactions on, vol. 11, no. 3, pp. 325-334, 2003.