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On finite‐time stability analysis of homogeneous vector fields with multiplicative perturbations
Author(s) -
Braidiz Youness,
Polyakov Andrey,
Efimov Denis,
Perruquetti Wilfrid
Publication year - 2022
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.6330
Subject(s) - mathematics , bounded function , stability (learning theory) , homogeneous , multiplicative function , nonlinear system , class (philosophy) , mathematical analysis , exponential stability , stability theory , vector field , control theory (sociology) , computer science , physics , geometry , control (management) , combinatorics , machine learning , quantum mechanics , artificial intelligence
The problem of finite‐time and fixed‐time stability analysis is considered for a class of nonlinear systemsx ˙ = H ( x ) b ( x ) , $$ \dot{x}=H(x)b(x), $$ whereH $$ H $$ is homogeneous of a degreeν ≠ 0 $$ \nu \ne 0 $$ andb $$ b $$ is bounded. It is shown that under certain conditions onb $$ b $$ the asymptotic stability of the system implies its finite‐time stability forν < 0 $$ \nu <0 $$ or nearly fixed‐time stability forν > 0 $$ \nu >0 $$ . The so‐called homogeneous extensions are utilized for the analysis. The results are generalized to a class of systemsx ˙ = ∑ i = 0 pH i ( x ) b i ( x ) , $$ \dot{x}=\sum \limits_{i=0}^p\kern.17em {H}_i(x){b}_i(x), $$ where for everyi = 0 , 1 , … , p , p ∈ ℕ , $$ i=0,1,\dots, p,\kern0.3em p\in \mathbb{N}, $$H i$$ {H}_i $$ are homogeneous of some (possibly different) degrees andb i$$ {b}_i $$ satisfy certain boundedness conditions. An example of a mechanical system is presented in the last section to illustrate the obtained results.