Design of Lyapunov functions for a class of homogeneous systems: Generalized forms approach

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.

Digital implementation of sliding‐mode control via the implicit method: A tutorial

“…where B P 3 is the unit ball in the metric defined by P . [110] Lyapunov functions of arbitrarily large degree of differentiability are derived for similar controllers as (97). Their use to derive consistent numerical schemes is not trivial, however, as they may be lacking the central symmetry property.…”

“…Under homogeneity and asymptotic stability, [ 111, theorem 6.2] states the existence of a homogeneous Lyapunov function, and [ 34, theorem 3.8 (1)] states the existence of a homogeneous and infinitely differentiable Lyapunov function. In Reference 124 Lyapunov functions of arbitrarily large degree of differentiability are derived for similar controllers as (89). Their use to derive consistent numerical schemes is not trivial, however, as they may be lacking the central symmetry property.…”

Homogeneous Lp Stability for Homogeneous Systems

“…The dissipation inequality in Definition 6 depends on the construction of storage function. Although there is no general method to obtain storage functions, for certain classes of homogeneous systems the recent work [39] provides a methodology to construct them, using generalized homogeneous forms and the sums of squares technique.…”

Numerical design of Lyapunov functions for a class of homogeneous discontinuous systems

“…Moreover, Efimov et al 11 give a methodology for numerical construction of such functions. With a different approach, Sanchez and Moreno 12,13 uses generalized homogeneous polynomials as Lyapunov functions for a class of homogeneous systems including negative homogeneity degree and discontinuous ones.…”

“…For instance, generalized polynomial Lyapunov functions for homogeneous HOSM and continuous HOSM described by generalized forms 1 are presented by Sanchez and Moreno 12,13 . A family of strict Lyapunov functions for the Super-Twisting Algorithm is provided by Moreno and Osorio 17 by means of quadratic forms.…”