Paper:

# Replacement of Lyapunov Function by Locally Convergent Robust Fixed Point Transformations in Model-Based Control a Brief Summary

## József K. Tar

Transportation Informatics and Telematics Knowledge Centre, Óbuda University H-1034 Budapest, Bécsi út 96/B, Hungary

^{nd}Method is a popular approach in the model-based control of nonlinear systems since normally it can guarantee

*global asymptotic stability*. However, the cost of the application of Lyapunov function frequently can be inefficient and complicated parameter tuning process containing unnecessary number of almost arbitrary control parameters, and vulnerability of the tuning process against not modeled, unknown external disturbances. Improved versions of the original, model-based tuning are especially sensitive to the external perturbations. All these disadvantages can simply be avoided by the application of robust fixed point transformations at the cost of giving up the guarantee of global asymptotic stability. Instead of that simple,

*stable iterative control*of local basin of attraction can be constructed on the basis of an approximate system model that can well compensate the effects of unknown external disturbances, too. Since the basin of convergence of the method in principle can be left, setting the three adaptive parameters of this controller needs preliminary simulation investigations. These statements are illustrated and substantiated via simulation results obtained for the adaptive control of various physical systems.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.14 No.2, pp. 224-236, 2010.

- [1] J. K. Tar, I. J. Rudas, and J. Gáti, “Improvements of the Adaptive Slotine & Li Controller - Comparative Analysis with Solutions Using Local Robust Fixed Point Transformations,” invited lecture and paper at The 14th WSEAS Int. Conf. on APPLIED MATHEMATICS (MATH’09), Puerto De La Cruz, Canary Islands, Spain, December 14-16, 2009, pp. 305-311, 2009.
- [2] J. J. E. Slotine and W. Li, “Applied Nonlinear Control,” Prentice Hall Int., Inc., Englewood Cliffs, New Jersey, 1991.
- [3] G. W. Stewart, “On the early history of singular value decomposition,” Technical Report TR-92-31, Institute for Advanced Computer Studies, University of Mariland, March 1992.
- [4] G. H. Golub and W. Kahan, “Calculating the Singular Values and Pseudoinverse of a Matrix,” SIAM J. on Numerical Analysis, Vol.2, pp. 205-224, 1965.
- [5] J. K. Tar, J. F. Bitó, I. J. Rudas, S. Preitl, and R.-E. Precup, “An SVD Based Modification of the Adaptive Inverse Dynamics Controller,” Proc. of 5th Int. Symposium on Applied Computational Intelligence and Informatics, Timişoara, Romania, pp. 193-198, 2009.
- [6] 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.
- [7] S. Lacroix, “Traité du calcul differentiel et du calcul intégral,” Courciel, Paris, France, 1819.
- [8] J. Liouville, “Mémoire sur le calcul des différentielles a indices quelconcues,” J. Ecole Polytechn., Vol.13, pp. 71-162, 1832.
- [9] A. K. Grünwald, “Über ’begrenzte’ Derivationen und deren Anwendung,” Zeitshrift für angewandte Mathematik und Physik, Vol.12, pp. 41-480, 1867.
- [10] J. K. Tar, I. J. Rudas, L. Nádai, K. R. KozBlowski, and J. A. Tenreiro Machado, “Fixed Point Transformations in the Adaptive Control of Fractional Order MIMO Systems”, Lecture Notes in Control and Information Sciences 396 (Eds: M. Thoma, F. Allgöver, M. Morari) – Robot Motion and Control 2009 (Ed. K. R. KozBlowski), Springer-Verlag Berlin Heidelberg, Chapter 10, pp. 103-112, 2009.
- [11] J. K. Tar, I. J. Rudas, and K. R. KozBlowski, “Fixed Point Transformations-Based Approach in Adaptive Control of Smooth Systems,” Lecture Notes in Control and Information Sciences 360 (Eds. M. Thoma and M. Morari), Robot Motion and Control 2007 (Ed. K. R. KozBlowski), Springer Verlag London Ltd., pp. 157-166, 2007.
- [12] L. Madarász, R. Andoga, L. Fozo, and T. Lazar, “Situational control, modeling and diagnostics of large scale systems,” Springer: Towards Intelligent Engineering and Information Technology, No.143, pp. 153-164. 2009.
- [13] J. K. Tar, J. F. Bitó, I. J. Rudas, K. R. Kozlowski, and J. A. Tenreiro Machado, “Possible Adaptive Control by Tangent Hyperbolic Fixed Point Transformations Used for Controlling the Φ
^{6}-type Van der Pol oscillator,” Proc. of the 6^{th}IEEE Int. Conf. on Computational Cybernetics (ICCC 2008), November 27-29, 2008, Stará Lesná, Slovakia, pp. 15-20, 2008. - [14] 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.
- [15] J. K. Tar, J. F. Bitó, I. Gergely, and L. Nádai, “Possible Improvement of the Operation of Vehicles Driven by Omnidirectional Wheels,” Proc. of the 4th Int. Symposium on Computational Intelligence and Intelligent Informatics, 21-25 October 2009 Egypt (ISCIII 2009), pp. 63-68, 2009.
- [16] J. K. Tar, I. J. Rudas, I. Nagy, K. R Kozlowski, and J. A. Tenreiro Machado, “Simple Adaptive Dynamical Control of Vehicles Driven by Omnidirectional Wheels,” Proc. of the 7th IEEE Int. Conf. on Computational Cybernetics (ICCC 2009), Palma de Mallorca, Spain, November 26-29, 2009, pp. 91-95, 2009.
- [17] J. K. Tar, I. J. Rudas, J. F. Bitó, S. Preitl, and R–E. Precup, “Adaptive Control of a 3rd Order Electromechanical System Using Robust Sigmoidal Fixed Point Transformation,” Proc. of the RAAD 2009 18th Int. Workshop on Robotics in Alpe-Adria-Danube Region, May 25-27, 2009, Brasov, Romania, CD issue, file: 87.pdf, 2009.
- [18] R. Molero, J. Roca, D. Separovich, J. Rojas, M. Montes, and N. Cuellar, “Nonlinear Control of an Electromagnetic System Based on Exact Linearization and Sliding Mode Control,” Mechanics Based Design of Structures and Machines, Vol.36, Issue 4, pp. 426-445, October 2008.
- [19] J. K. Tar and J. F. Bitó, “Adaptive Control Using Fixed Point Transformations for Nonlinear Integer and Fractional Order Dynamic Systems,” in the series “Studies in Computational Intelligence 241 – Aspects of Soft Computing, Intelligent Robotics and Control,” Eds. J. Fodor and J. Kacprzyk, Springer-Verlag Berlin Heidelberg, pp. 253-267, 2009.
- [20] B. Van der Pol, “VII. Forced oscillations in a circuit with non-linear resistance (Reception with reactive triode),” Philosophical Magazine Series 7, Vol.3, Issue 13, pp. 65-80, January 1927.

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