Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach. (18th July 2018)

• Record Type: Journal Article
• Authors: Sanchez, Tonametl; Moreno, Jaime A.
• Other Names: Fridman Leonid guestEditor.; Poznyak Alex guestEditor.
• Is Part Of: International journal of robust and nonlinear control. Volume 29:Number 3(2019)
• Page Start: 661
• Page End: 681
• Publication Date: 2018-07-18
• Subjects: homogeneous systems -- Lyapunov function -- sliding modes; Automatic control -- Periodicals; Control theory -- Periodicals; Nonlinear systems -- Periodicals; 629.836
• DOI: 10.1002/rnc.4274
• Languages: English
• ISSNs: 1049-8923

Abstract

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.