Generalized Homogeneity in Systems and Control by Andrey Polyakov

Author:Andrey Polyakov
Language: eng
Format: epub, pdf
ISBN: 9783030384494
Publisher: Springer International Publishing

and a strictly increasing continuous function can be constructed similarly to . For we have

Hence,

and

Example 7.13

Let the operator is given by with and the dilation is defined as follows then , then the -homogeneous approximation of f at the 0-limit is defined as and the -homogeneous approximation of f at the -limit is given .

In the case of smooth nonlinear systems of ODEs

the so-called first-order approximation (or the first-order Taylor expansion)

is utilized for stability analysis and control design (see e.g. [18] and references therein). If the system is globally asymptotically stable then is, at least, locally asymptotically stable.

However, the first-order approximation is not informative in some cases. For example, the first-order approximation of the asymptotically stable system is the system , which is not asymptotically stable. The reasonable question in this case: is it possible to introduce a homogeneity-based analog of the first-order approximation? The main beauty of the classical first-order approximation is a reduction of the local stability analysis of a nonlinear system to the same problem for a linear system. The homogeneous approximations introduced by Definition 7.5 do not allow this in the general case. That is why, we specify a special subclass of nonlinear systems which admit a similar property.

Definition 7.7

A -homogeneous approximation at L-limit is said to be the first-order -homogeneous approximation at L-limit if there exists an operator such that

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