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Vol: 51(65) No: 2 / June 2006         On the Generalized Friction Dynamic Systems Identification by Distributions Constantin Marin Department of Automation, University of Craiova, Faculty of Automation, Computers and Electronics, Bd. Decebal, no. 5, 200440 Craiova, Romania, phone: +40.251.438198, e-mail: cmarin@automation.ucv.ro Anca Petrisor Department of Electromechanics, University of Craiova, Faculty of Electromechanics, Bd. Decebal, no. 5, 200440 Craiova, Romania, phone: +40.251.438198, e-mail: aperisorn@em.ucv.ro Virginia Finca Department of Electromechanics, University of Craiova, Faculty of Electromechanics, Bd. Decebal, no. 5, 200440 Craiova, Romania, phone: +40.251.438198 Keywords: identification, distribution theory, friction. Abstract A class of nonlinear systems is represented by smooth systems with discontinuous feedback loops depending on both the state and input vectors called generalized friction dynamic systems (GFDS) as they have been presented for the first time in [17]. The feedback loops generate some reaction vectors assimilated to friction vectors from mechanics. This paper extends some aspects of the above approach regarding identification procedures based on distributions theory to GFDS. Both GFDS with static friction models (SFM) and dynamic friction models (DFM) are considered. The identification problem is formulated as a condition of vanishing the existence relation of the system. Then, this relation is represented by functionals using techniques from distribution theory based on testing function from a finite dimensional fundamental space. The proposed method in [17] is a batch on-line identification method because identification results are obtained during the system evolution after some time intervals but not in any time moment. Even if it is based on the input-output measurements only, the method is insensitive to the initial state of any transient. In this paper, the identificability conditions and the sticky surface are presented in detail. The advantages of representing information by distributions are pointed out when special evolutions as sliding mode, or limit cycle can appear. The proposed method does not require the derivatives of measured signals for its implementation. Some experimental results are presented to illuminate further its advantages and practical use. References [1] B. Armstrong-Helouvry, Control of machines with friction, Boston MA, Kluwer, 1991 [2] C. Canudas de Wit, H. Olsson, K. J. Ã…ström and P. Lischinsky, "A New Model for Control of Systems with Friction", IEEE Trans. Automat. Contr., vol.40, pp. 419–425, March 1999. [3] H. Du and N. Satish, "Identification of Friction at Low Velocities Using Wavelets Basis Function Network", Proc. of American Control Conference, Philadelphia, June 1998, pp. 1918–1922. [4] S.J. Kim. and I.J. Ha, “A Frequency-Domain Approach to Identification of Mechanical Systems with Friction”, IEEE Trans. Automat. Contr., vol.46, pp. 888–893, June. 2001. [5] P. Dahl, “A solid friction model”, Aerospace Corp., El Segundo, CA,Tech. Rep. TOR-0158(3107-18)-1, 1968. [6] B. Armstrong-Helouvry, P. Dupont, and C. 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. [7] C. Canudas de Wit, P. Noel, A. Aubin and B. Brogliato, "Adaptive Friction Compensation in Robot Manipulators: Low Velocities", Int. Journal of Robotics Research, vol. 10, no. 3, pp. 189-199, 1991. [8] D.A. Haessig and B. Friedland, "On the Modeling and Simulation of Friction", ASME Joumal of Qynamic Systems, Measurement, and Control, vol. 113, pp. 354-362, Sept. 1991. [9] J.R. Dieterich, "Modeling of Rock Friction: 1.Experimental Results and Constitutive Equations", Joumal of Geophysical Research, vol. 84 (BS), pp. 2161-2168, 1979. [10] C. Walarth, "Adaptive Bearing Friction Compensation Based on Recent Knowledge of Dynamic Friction", Automatica, vol. 20, no. 6, pp. 717-727, 1984. [11] B. Helouvry, P. Dupont and C. 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. [12] S.J. Kim and I.J. Ha, “On the existence of Carathèodory Solutions in mechanical systems with friction”, IEEE Trans. Automat. Contr., vol. 44, pp. 2086-2089, Nov. 1999. [13] C. Marin, "System Identification Based on Distribution Theory", Proceedings of the IASTED International Conference Applied Simulation and Modelling (ASM 2002), Crete, June, 2002. [14] C. Marin, E. Petre, D. Selisteanu and D. Sendrescu, "Identification of Nonlinear Plants via Distributions. Application for Wastewater Biodegradation Process", Proc. 15th Int. Conference on Control Systems and Computer Science-CSCS15, May 25-27, 2005, Bucharest. [15] N.K.Sinh and G. P. Rao, Identification of continuous–time systems, Kluwer Academic Press, Dordreht. [16] C. Marin, “System identification by the method of weighted moments”, Proc. 12th Int. Conference on Control Systems and Computer Science, CSCS 12, Bucharest. [17] C. Marin, A. Petrişor and V. Fincă, "Identification of Systems with Friction via Distributions", CONTI’2006, Proc. 7th Int. Conference on Technical Informatics, 8-9 June 2006, Timişoara, Romania. [18] A.E. Pearson and A.E. Lee, "On the identification of polynomial input-output differential systems", IEEE Trans.Aut.Contr.,vol. AC30, no. 8, pp. 778-782. [19] A. Patra and H. Unbehauen, "Identification of a class of nonlinear continuous time systems using Hartley modulating functions", Int.J.Contr. vol. 62, no. 6, pp. 1431-1452. [20] A. Ohsumi and K. Kameiama, "Subspace identification for continuous-time systems via distribution-based approach", Automatica, vol. 38, pp. 63-79, 2002. [21] W. Kecs and P. Teodorescu, Introduction to distribution theory with technical applications, Editura Tehnica, Bucharest, 1975. [22] V. Barbu, Differential equations, Editura Junimea, Iasi, 1985. [23] V. Utkin, Sliding modes in control optimization, Springer, 1992.