|
Vol: 58(72) No: 2 / June 2013 Application of Truncated Linear Sigmoid Functions in Adaptive Controllers Based on Robust Fixed Point Transformations Krisztián Kósi Doctoral School of Applied Informatics, Óbuda University, Bécsi út 96/B, H-1034 Budapest, Hungary, phone: +36 (1) 666-5543, e-mail: kosi.krisztian@phd.uni-obuda.hu János F. Bitó Institute of Applied Mathematics, Óbuda University, Bécsi út 96/B, H-1034 Budapest, Hungary, e-mail: bito@uni-obuda.hu József K. Tar Institute of Applied Mathematics, Óbuda University , Bécsi út 96/B, H-1034 Budapest, Hungary, e-mail: tar.jozsef@uni-obuda.hu Keywords: adaptive control, robust fixed point transformation, contractive map, Banach’s fixed point theorem Abstract The application of Robust Fixed Point Transformations (RFPT) in the design of adaptive controllers was invented as an alternative of Lyapunov’s 2nd method with the aim of evading the mathematical difficulties related to the construction of a Lyapunov function. It is based on Banach’s Fixed Point Theorem since it applies sigmoid functions to construct a contractive map that generates an iterative sequence of digital control signals that converge to the solution of the control task. The cost of mathematical simplicity is the existence of only a local basin of convergence in contrast to the globally stable solutions that can be obtained by the use of Lyapunov’s method. This deficiency was amended by the use of complementary tuning of one of the altogether three parameters of this map. The details of convergence of the control signal also depend on the particular sigmoid function used for bringing about the contractive map. In the present paper the use of the simplest solution i.e. the truncated linear function is investigated in comparison with the latest results of parameter tuning. It can be stated that this simple solution works well, too. References [1] A.M. Lyapunov, A general task about the stability of motion (in Russian), PhD Thesis, 1892. [2] A.M. Lyapunov, Stability of Motion, Academic Press, New-York and London, 1966. [3] J.-J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall International, Inc., Englewood Cliffs, NJ, 1991. [4] R. Isermann, K.H. Lachmann, and D. Matko, Adaptive Control Systems, New York DC, Prentice-Hall, USA, 1992. [5] K. Hosseini–Suny, H. Momeni, and F. Janabi-Sharifi, “Model reference adaptive control design for a teleoperation system with output prediction,” Journal of Intelligent and Robotic Systems, vol. 59, no. 3-4, pp. 319-339, 2010. [6] J.K. Tar, “Towards replacing Lyapunov’s “direct” method in adaptive control of nonlinear systems,” in Proceedings of 3rd Conference in Mathematical Methods in Engineering, Coimbra, Portugal, 2010, Paper 11 (CD issue). [7] S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fund. Math., vol, 3, pp. 133-181, 1922. [8] J.K. Tar and K. Eredics, “Simulation studies on various tuning methods for convergence stabilization in a novel approach of model reference adaptive control based on robust fixed point transformations,” Acta Technica Jaurinensis, vol. 4, no. 1, pp. 37-57, 2010. [9] J.K. Tar, J.F. Bitó, I.J. Rudas, K. Eredics K, and J.A. Tenreiro Machado, “Comparative analysis of a traditional and a novel approach to model reference adaptive control,” in Proceedings of 11th IEEE International Symposium on Computational Intelligence and Informatics, Budapest, Hungary, 2010, pp. 93-98. [10] J.K. Tar, L. Nádai, I.J. Rudas, and T.A. Várkonyi, “RFPT-based adaptive control stabilized by fuzzy parameter tuning,” in Proceedings of 9th European Workshop on Advanced Control and Diagnosis (ACD 2011), Budapest, Hungary, 2011, paper 6, 8 pp. [11] K. Kósi, Sz. Hajdu, J.F. Bitó, and J.K. Tar, “Chaos formation and reduction in robust fixed point transformations based adaptive control,” in Proceedings of 4th IEEE International Conference on Nonlinear Science and Complexity, Budapest, Hungary, 2012, pp. 211-216. [12] T.A. Várkonyi, J.K. Tar, I.J. Rudas, and I. Krómer, “VS-type stabilization of MRAC controllers using robust fixed point transformations,” in Proceedings of 7th IEEE International Symposium on Applied Computational Intelligence and Informatics, Timisoara, Romania, 2012, pp. 389-394. [13] K. Kósi, J.K. Tar, and I.J. Rudas, “Improvement of the stability of rfpt-based adaptive controllers by observing precursor oscillations,” in Proceedings of IEEE 9th International Conference on Computational Cybernetics, Tihany, Hungary, 2013, pp. 267-272. [14] K. Kósi, J.F. Bitó, and J.K. Tar, “Fine tuning with sigmoid functions in robust fixed point transformation,” in Proceedings of 8th IEEE International Symposium on Applied Computational Intelligence and Informatics, Timisoara, Romania, 2013, pp. 411-416. [15] R.T. Bupp, D.S Bernstein, and V.T. Coppola, “A benchmark problem for nonlinear control design: Problem statement, experiment testbed and passive nonlinear compensation,” in Proceedings of American Control Conference, Seattle, USA, 1995, pp. 4363-4376. [16] M. Jankovic, D. Fontanie, and P.V. Kokotovic, “TORA example: Cascade- and passivity based control designs,” IEEE Transactions on Control Systems Technology, vol. 4, pp. 292-297, 2006. [17] P. Baranyi, Z. Petres, P. Várlaki, and P. Michelberger, “Observer and control law design to the TORA system via TPDC framework,” Transactions on Systems, vol. 1, no. 5, pp. 156-163, 2006. |