Function of several real variables

Variables, functions and mappings

Published in Alan Jeffrey, Mathematics, 2004

The chapter ends with a brief discussion of the extension of the notion of a function of a single real variable to a function of several real variables which is essential for the development of mathematics and also for its applications. It is shown how, when a function of two real variables needs to be visualized, the use of level curves or of sectioning planes is often helpful, as these interpret a three-dimensional problem in terms of a set of two-dimensional curves.

A note on the ‘task-space control of robots using an adaptive Taylor series uncertainty estimator’

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Journal Information

Published in International Journal of Control, 2021

Alireza Izadbakhsh, Payam Kheirkhahan

Ahmadi and Fateh (2018a) proposed a new control scheme utilising Taylor series as universal approximator. The joint-space form of this work has been presented in Ahmadi and Fateh (2018b). They assumed that the ith row of the lumped uncertainty (a real-valued function of several real-variables) can be approximated by a Taylor series estimator represented for functions with only one of those variables. This approximation, however, is not technically true and produces an incredible approximation error. The respectable authors may claim that this error can be eliminated utilising the existing robustifying term in the proposed control law. However, other control strategies such as PID control (Qu & Dorsey, 1991) can give good results with a simpler structure and less tuning parameters. In this paper, it will be shown that the Stone–Weierstrass theorem will not be satisfied for such an approximation.

Function of several real variables

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Variables, functions and mappings

A note on the ‘task-space control of robots using an adaptive Taylor series uncertainty estimator’