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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

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.

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