### Abstract

The mathematical foundations of the modern Soft Computing (SC) techniques go back to Kolmogorov's approximation theorem stating that each multi-variable continuous function on a compact domain can be approximated with arbitrary accuracy by the composition of single-variable continuous functions [1]. Since the late eighties several authors have proved that different types of neural networks possess the universal approximation property (e.g. [2]). Similar results have been published since the early nineties in fuzzy theory claiming that different fuzzy reasoning methods are related to universal approximators (e.g. in [3]). Due to the fact that Kolmogorov's theorem aims at the approximation of the very wide class of continuous functions, the functions to be constructed are often very complicated and highly non-smooth, therefore their construction is difficult. As is well known, continuity allows very extreme behavior even in the case of single-variable functions. The first example of a function that is everywhere continuous but nowhere differentiable was given by Weierstraß in 1872 [4]. At that time mathematicians believed that such functions are only rare extreme examples, but nowadays it has become clear that the great majority of the continuous functions have extreme properties. The seemingly antagonistic contradiction between the complicated nature of the universal approximators and their successful practical applications makes one arrive at the conclusion that if we restrict our models to the far better behaving "everywhere differentiable" functions, these problems ab ovo can be evaded or at least reduced.

Original language | English |
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Title of host publication | Lecture Notes in Control and Information Sciences |

Pages | 157-166 |

Number of pages | 10 |

Volume | 360 |

DOIs | |

Publication status | Published - 2007 |

### Publication series

Name | Lecture Notes in Control and Information Sciences |
---|---|

Volume | 360 |

ISSN (Print) | 01708643 |

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### ASJC Scopus subject areas

- Library and Information Sciences

### Cite this

*Lecture Notes in Control and Information Sciences*(Vol. 360, pp. 157-166). (Lecture Notes in Control and Information Sciences; Vol. 360). https://doi.org/10.1007/978-1-84628-974-3_14

**Fixed point transformations-based approach in adaptive control of smooth systems.** / Tar, J.; Rudas, I.; Kozłowski, Krzysztof R.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Control and Information Sciences.*vol. 360, Lecture Notes in Control and Information Sciences, vol. 360, pp. 157-166. https://doi.org/10.1007/978-1-84628-974-3_14

}

TY - GEN

T1 - Fixed point transformations-based approach in adaptive control of smooth systems

AU - Tar, J.

AU - Rudas, I.

AU - Kozłowski, Krzysztof R.

PY - 2007

Y1 - 2007

N2 - The mathematical foundations of the modern Soft Computing (SC) techniques go back to Kolmogorov's approximation theorem stating that each multi-variable continuous function on a compact domain can be approximated with arbitrary accuracy by the composition of single-variable continuous functions [1]. Since the late eighties several authors have proved that different types of neural networks possess the universal approximation property (e.g. [2]). Similar results have been published since the early nineties in fuzzy theory claiming that different fuzzy reasoning methods are related to universal approximators (e.g. in [3]). Due to the fact that Kolmogorov's theorem aims at the approximation of the very wide class of continuous functions, the functions to be constructed are often very complicated and highly non-smooth, therefore their construction is difficult. As is well known, continuity allows very extreme behavior even in the case of single-variable functions. The first example of a function that is everywhere continuous but nowhere differentiable was given by Weierstraß in 1872 [4]. At that time mathematicians believed that such functions are only rare extreme examples, but nowadays it has become clear that the great majority of the continuous functions have extreme properties. The seemingly antagonistic contradiction between the complicated nature of the universal approximators and their successful practical applications makes one arrive at the conclusion that if we restrict our models to the far better behaving "everywhere differentiable" functions, these problems ab ovo can be evaded or at least reduced.

AB - The mathematical foundations of the modern Soft Computing (SC) techniques go back to Kolmogorov's approximation theorem stating that each multi-variable continuous function on a compact domain can be approximated with arbitrary accuracy by the composition of single-variable continuous functions [1]. Since the late eighties several authors have proved that different types of neural networks possess the universal approximation property (e.g. [2]). Similar results have been published since the early nineties in fuzzy theory claiming that different fuzzy reasoning methods are related to universal approximators (e.g. in [3]). Due to the fact that Kolmogorov's theorem aims at the approximation of the very wide class of continuous functions, the functions to be constructed are often very complicated and highly non-smooth, therefore their construction is difficult. As is well known, continuity allows very extreme behavior even in the case of single-variable functions. The first example of a function that is everywhere continuous but nowhere differentiable was given by Weierstraß in 1872 [4]. At that time mathematicians believed that such functions are only rare extreme examples, but nowadays it has become clear that the great majority of the continuous functions have extreme properties. The seemingly antagonistic contradiction between the complicated nature of the universal approximators and their successful practical applications makes one arrive at the conclusion that if we restrict our models to the far better behaving "everywhere differentiable" functions, these problems ab ovo can be evaded or at least reduced.

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U2 - 10.1007/978-1-84628-974-3_14

DO - 10.1007/978-1-84628-974-3_14

M3 - Conference contribution

SN - 9781846289736

VL - 360

T3 - Lecture Notes in Control and Information Sciences

SP - 157

EP - 166

BT - Lecture Notes in Control and Information Sciences

ER -