Paper:

# Comparison of Fractional Robust- and Fixed Point Transformations- Based Adaptive Compensation of Dynamic Friction

## József K. Tar^{*}, Imre J. Rudas^{*}, and Béla Pátkai^{**}

^{*}Institute of Intelligent Engineering Systems, John von Neumann Faculty of Informatics, Budapest Tech, H-1034 Budapest, Bécsi út 96/B, Hungary

^{**}Distributed Information and Automation Lab, Department of Engineering, University of Cambridge, Mill Lane, Cambridge CB2 1RX, UK

The features of fractional order robust and fixed-point transformation based adaptive controllers of a “Ball-Beam System” are compared to each other. The speciality of this task is that the position of the ball along the beam is indirectly controlled via directly controlling the other axis, the tilting angle of the beam. It is assumed that this tilting axle suffers from considerable dynamic friction mathematically approximated by the LuGre model. By neglecting the internal physics of the tilting drive this system can be modeled as a 4^{th} order one because only the 4^{th} time-derivative of the ball’s position can directly be influenced by the tilting torque. The system also has saturation since the available acceleration of the ball is limited by the gravitation. It is shown that little reduction of the order of the differential equation controlling the decay of the error metrics in a Sliding Mode / Variable Structure controller considerably improves the robust controller. However, really precise solution can be obtained by the adaptive controller. These statements are illustrated and substantiated via simulation.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11, No.9, pp. 1062-1071, 2007.

- [1] D. Hilbert, “Mathematische Probleme,” 2
^{nd}Int. Congress of Mathematicians, Paris, France, 1900. - [2] V. I. Arnold, “Doklady Akademii Nauk USSR,” 114, pp. 679-681 (in Russian).
- [3] A. N. Kolmogorov, “Doklady Akademii Nauk USSR,” 114, pp. 953-956, (in Russian).
- [4] J. P. De Figueiredo, IEEE Tr. Autom. Control, Vol.25, Issue 6, pp. 1227-1231, 1980.
- [5] K. Hornik, M. Stinchcombe, and H. White, Neural Networks, Vol.2, pp. 359-366.
- [6] E. K. Blum and L. K. Li, Neural Networks, Vol.4, No.4, pp. 511-515, 1991.
- [7] V. Kurková, Neural Networks, Vol.5, pp. 501-506, 1992.
- [8] L. X. Wang, “Fuzzy systems are universal approximators,” Proc. of the IEEE Int. Conf. on Fuzzy Systems, San Diego, USA, California, Mar 8-12, 1992, pp. 1163-1169.
- [9] B. Kosko, IEEE Trans. on Computers, Vol.43, Issue 11, pp. 1329-1333, 1994.
- [10] J. L. Castro, IEEE Trans. on SMC, Vol.25, pp. 629-635, 1995.
- [11] B. Moser, Fuzzy Sets and Systems, Vol.104, No.2, pp. 269-277, 1999.
- [12] D. Tikk, Tatra Mountains Math. Publ., Vol.16, pp. 369-377, 1999.
- [13] E. P. Klement, L. T. Kóczy, and B. Moser, Int. J. General Systems, Vol.28, No.2, pp. 259-282, 1999.
- [14] D. Tikk, P. Baranyi, and J. Patton, “Polytopic and TS model are nowhere dense in the approximation model space,” Proc. of IEEE Int. Conf. on Systems, Man and Cybernetics SMC 2002, Hammamet, Tunisia, October 6-9, pp. 150-153, 2002.
- [15] Gy. Schuster, “Fuzzy Approach of Backward Identification of Quasi-linear and Quasi-time-invariant Systems,” Proc. of the 11th Int. Workshop on Robotics in Alpe-Adria-Danube Region (RAAD 2002), June 30-July 2, 2002, Balatonfüred, Hungary, pp. 43-50.
- [16] Gy. Schuster, “Adaptive Fuzzy Control of Thread Testing Furnace,” Proc. of the ICCC 2003 IEEE Int. Conf. on Computational Cybernetics, August 29-31, Gold Coast, Lake Balaton, Siófok, Hungary, pp. 299-304.
- [17] Gy. Schuster, “Improved Method of Adaptive Fuzzy Control of a Thread Testing Furnace,” Proc. of the 2006 IEEE Int. Conf. on Computational Cybernetics (ICCC 2006), Tallinn, Estonia, August 20-22, 2006, pp. 125-129.
- [18] Gy. Hermann, “Application of Neural Network Based Sensor Fusion in Drill Monitoring,” Proc. of Symposium on Applied Machine Intelligence (SAMI 2003), Herl’any, Slovakia, pp. 11-24, February 12-14, 2003.
- [19] J. Tick and J. Fodor, “Some classes of binary operations in approximate reasoning,” Proc. of the 2005 IEEE Int. Conf. on Intelligent Engineering Systems (INES’05), September 16-19, 2005, pp. 123-128.
- [20] J. Tick and J. Fodor, “Fuzzy Implications and Inference Processes,” Computing and Informatics, Vol.24, No.6, pp. 591-602, 2005.
- [21] I. J. Rudas and M. O. Kaynak, Fuzzy Sets and Systems, Vol.98, No.1, pp. 83-94, 1998.
- [22] I. J. Rudas, Intl. Journal of Fuzzy Systems, Vol.2, Part 4, pp. 236-243, 2000.
- [23] K. Weierstraß, “Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen,” A paper presented to the ‘Königliche Akademie der Wissenschaften’ on 18 of July 1872. English translation available in: On continuous functions of a real argument that do not have a well-defined differential quotient, in: G.A. Edgar, Classics on Fractals, Addison-Wesley Publishing Company, 1993, 3-9.
- [24] J. K. Tar, “Dynamic Nonlinear Control of Mechanical and Analogous Devices/Processes,” Invited plenary lecture at the IEEE Int. Conf. on Computational Cybernetics (ICCC 2006), Tallinn, Estonia, August 20-22, 2006.
- [25] J. K. Tar, I. J. Rudas, Á. Szeghegyi, and K. Kozłowski, “Novel Adaptive Control of Partially Modeled Dynamic Systems,” In “Lecture Notes in Control and Information Sciences,” Springer Berlin/Heidelberg, Robot Motion and Control: Recent Development, Part II – Control and Mechanical Systems, Krzysztof KozBłowski (Ed.), 335, pp. 99-111, 2006.
- [26] J. K. Tar, “Extension of the Modified Renormalization Transformation for the Adaptive Control of Negative Definite SISO Systems,” In the Proc. of the 2nd Romanian-Hungarian Joint Symposium on Applied Computational Intelligence (SACI 2005), May 12-14, 2005, Timişoara, Romania, pp. 447-457.
- [27] B. Armstrong-Hèlouvry, “Stick Slip and Control in Low Speed Motion,” IEEE Trans. On Automatic Control, Vol.38(10), pp. 1483-1496, October, 1990.
- [28] C. Caundas de Wit, H. Ollson, K. J. Åstrom, and P. Lischinsky, “A New Model for Control of Systems with Friction,” IEEE Trans. On Automatic Control, Vol.40(3), pp. 419-425, March, 1995.
- [29] C. Caundas de Wit, “Comments on “A New Model for Control of Systems with Friction”,” IEEE Trans. On Automatic Control, Vol.43(8), pp. 1189-1190, August, 1998.
- [30] J. A. Tenreiro Machado, “Fractional Calculus and Dynamical Systems,” invited plenary lecture at the IEEE Int. Conf. on Computational Cybernetics (ICCC 2006), Tallinn, Estonia, August 20-22, 2006.
- [31] S. Lacroix, “Traité du calcul differentiel et du calcul intégral,” Courciel, Paris, France, 1819.
- [32] J. Liouville, “Mémoire sur le calcul des différentielles a indices quelconcues,” J. Ecole Polytechn., Vol.13, pp. 71-162, 1832.
- [33] A. K. Grünwald, “Über ‘begrenzte’ Derivationen und deren Anwendung,” Zeitshrift für angewandte Mathematik und Physik, Vol.12, pp. 441-480, 1867.
- [34] A. Gemant, “Method of Analyzing Experimental Results Obtained from Elasto-Viscous Bodies,” Physics, Vol.7, pp. 311-317, 1936.
- [35] A. Gemant, “On Fractional Differentials,” The Phylosophical Magzine, Vol.25, pp. 540-549, 1938.
- [36] K. B. Oldham and J. Spanier, “The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order,” Academic Press, 1974.
- [37] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” John Wiley and Sons, 1993.
- [38] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, 1999.
- [39] P. J. Torvik and R. L. Bagley, “On the Appearance of the Fractional Derivative in the Behaviour of Real Materials,” ASME Journal of Applied Mechanics, Vol.51, pp. 294-298, 1984.
- [40] C. G. Koh and J. M. Kelly, “Application of Fractional Derivatives to Seismic Analysis of Base-isolated Models,” Earthquake Engineering and Structural Dynamics, Vol.19, pp. 229-241, 1990.
- [41] J. Machado and A. Azenha, “Fractional-order hybrid control of robot manipulators,” IEEE Int. Conf. on Systems, Man and Cybernetics, pp. 788-793, 1998.
- [42] O. P. Agrawal, “Solution for a Fractional Diffusion-wave Equation in a Bounded Domain,” Nonlinear Dynamics, Vol.29, pp. 145-155, 2002.
- [43] T. Roska, “Development of Kilo Real-time Frame Rate TeraOPS Computational Capacity Topographic Microprocessors,” Plenary Lecture at 10th Int. Conf. on Advanced Robotics (ICAR 2001), Budapest, Hungary, August 22-25, 2001.