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Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach
Author(s) -
Sanchez Tonametl,
Moreno Jaime A.
Publication year - 2018
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.4274
Subject(s) - parameterized complexity , positive definiteness , mathematics , homogeneous polynomial , lyapunov function , polynomial , class (philosophy) , representation (politics) , homogeneous , set (abstract data type) , explained sum of squares , pure mathematics , matrix polynomial , mathematical analysis , positive definite matrix , combinatorics , computer science , nonlinear system , eigenvalues and eigenvectors , physics , statistics , quantum mechanics , artificial intelligence , politics , political science , law , programming language
Summary In this paper, we provide a method to design Lyapunov functions (LFs) for a class of homogeneous systems described by functions that we call generalized forms (GFs). Homogeneous polynomial systems and several high‐order sliding modes are included in the class. The LF candidate is chosen from the same class of functions and it is parameterized in its coefficients. Since the derivative of the LF candidate along the system's trajectories is also a GF, the problem is reduced to verify positive definiteness of two GFs. We establish a procedure to represent a GF with a finite set of polynomials. Thus, the problem is changed to determine positive definiteness of a set of polynomials. Such a problem can be solved by means of Pólya's theorem or the sum of squares representation.