témavezető: Tar József
helyszín (magyar oldal): Óbudai Egyetem helyszín rövidítés: ÓE
A kutatási téma leírása:
In the control of "essentially nonlinear systems", i.e. in the cases in which the controlled system’s dynamic model cannot be satisfactorily replaced by an affine approximation or "linearization" in the close vicinity of a working point, the traditional control design approaches that were based on the properties of the "Linear Time Invariant" (LTI) systems by using the frequency domain, linear integral transformations as the Fourier or Laplace transforms, cannot be satisfactorily used in "Model Predictive Control" (MPC). The traditional and widely used nonlinear controllers even in our days are designed on the theoretical basis of "Lyapunov’s nd" or "Direct Method" [1] that concentrated on the problem of the stability of the motion of nonlinear systems. Taking into account the fact that in the most of the practical problems the equations of motion do not have closed analytical solutions, by giving up the need of obtaining information on the subtle details of the realized motion it was satisfied by determining its global stability or instability. With the extension of the "Lyapunov function" into the "Lyapunov–Krasovskii functional" [2] its use has been extended to systems with time delay, too.
In spite of its excellent advantages, the Lyapunov function-based approach suffers from certain disadvantages as typically prescribing rather "satisfactory" than "necessary and satisfactory" conditions in the proofs, allowing the application of a huge set of possible control parameters that maintain the stability and in the same time crucially influence the transients of the controlled motion without any "optimization" [3].
To improve this situation a novel approach was suggested in [4] in which the need for the global stability of the controlled motion was given up and the subtle details of the controlled motion were placed into the center of attention. In this approach the control task at first was transformed into a Fixed Point Problem by the use of a contractive mapping then it was solved via iteration according to Banach’s Fixed Point Theorem [5]. In this program at first the “Robust Fixed Point Transformation” (RFPT) was found to be the best method for transforming the the control task into a fixed point problem. It had only a few free parameters for the (casually necessary) setting of which various tuning possibilities were suggested. Later different fixed point transformations were applied and investigated. In [6] a possible application for solving the general task of inverse kinematics was suggested, too.
In the preceding investigations the main emphasis of the research was finding possible applications of the Fixed Point Transformations based approach. The missing aspects and elements could form the subject area of the present program. Rather “ad hoc” than systematic research was carried out on finding appropriate nonlinear and contractive maps for this purpose. In the present research, on the basis of Differen-tial Geometry or Group Theory more systematic generation of possible mappings could be carried out. The convergence proofs were based on low order Taylor series expansions and related approximations. The question is whether “exact proofs of convergence” could be generated for the suggested/found/constructed maps. Further question is the research on the possible extension of the applicability of the novel adaptive method.
References
[1] A. Lyapunov, A general task about the stability of motion. (in Russian), University of Kazan, Tatarstan (Russia), 1892.
[2] V. Kolmanovskii, S.-I. Niculescu, and D. Richard, “On the Lyapunov-Krasovskii functionals for stability analysis of linear delay systems,” International Journal of Control, vol. 72, no. 4, pp. 374–384.
[3] J. Tar, J. Bitó, and I. Rudas, “Replacement of Lyapunov’s direct method in model reference adaptive control with robust fixed point transformations,” In Proc. of the th IEEE Intl. Conf. on Intelligent Engineering Systems, Las Palmas of Gran Canaria, Spain, pp. 231–235, 2010.
[4] J. Tar, J. Bitó, L. Nádai, and J. Tenreiro Machado, “Robust Fixed Point Transformations in adaptive control using local basin of attraction,” Acta Polytechnica Hungarica, vol. 6, no. 1, pp. 21–37, 2009.
[5] S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (About the Operations in the Abstract Sets and Their Application to Integral Equations),” Fund. Math., vol. 3, pp. 133–181, 1922.
[6] B. Csanádi, J. Tar, and J. Bitó, “Matrix inversion-free quasi-differential approach in solving the inverse kinematic task,” Proc.of the th IEEE International Symposium on Computational Intelligence and Informatics, 17-19 November 2016, Budapest, Hungary, pp. 61–66, 2016.
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