Stability analysis of the homogeneous nonlinear dynamical systems using Lyapunov function generation based on the basic functions

Abstract

In this paper, based on Lyapunov functions candidates, a new approach in the stability analysis of homogeneous nonlinear systems is proposed in which instead of concentrating on the positive definiteness of the Lyapunov candidate functions, we stress on the negative definiteness of its derivative. Having ensured of negative definiteness of the derivative function, based on sign assignment of the primitive function, the stability of the equilibrium is analyzed wherein the necessary and sufficient conditions are declared simultaneously. Selecting the trend of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions whose unknown coefficients in the candidate function structure are computed based on negative definiteness of the derivative function. Afterward, using these determined coefficients in the Lyapunov function, the sign of the primitive function in the state space is argued. Therefore, the triple sign attitudes of the candidate function can be used to deduce the stability/instability of the equilibrium point. Moreover, in the process of the negative definiteness of the derivative function, the coefficients are obtained using two independent methods. Numerical simulations support the proposed theoretical results and show their effectiveness.

Numerical Construction of Lyapunov Functions Using Homotopy Continuation Method

“…In this current study, we focus on developing an approach that extends the search of polynomial Lyapunov function to polynomials of even degree that are not necessarily quadratic nor a p − f orm. This extends the work of [9,13] by removing the requirements of the Lyapunov candidates to be quadratic or in p − f orm. We use SOS decomposition and Newton polytope to have decomposition with fewer monomials.…”

“…From our results, we can say that our algorithm for constructing Lyapunov function where the candidates are not required to be quadratic or p − f orm is successful. This has removed two important restrictions in search of Lyapunov functions reported in the literature, see for instance [13,9,20]. We included the eigenvalues of Q and H to show that the deniteness of the two matrices is satised, unlike the work reported in [9].…”

“…These polynomials are constructed using the technique of Zhikun et al reported in [20] and by constructing and solving a semi-algebraic system using cylindrical algebraic decomposition (CAD) reported by Collins in [6]. Recently an algorithm for construction of Lyapunov functions in p − f orm was developed in [13]. In that report, the authors utilize using Sum Of Square technique (SOS) and the requirement for the candidate to be in quadratic form was removed.…”

Stability Analysis of Equilibrium Point and Limit Cycle of Two-Dimensional Nonlinear Dynamical Systems—A Tutorial

“…The Lyapunov theorem can be categorized into local theorem and global theorem. The local theorem can be used to prove whether an equilibrium point is stable or asymptotically stable [1][2][3][4][5][6], whereas the global theorem can be used to prove if an equilibrium point is globally asymptotically stable and if it has been used in several applications [7][8][9][10][11][12][13][14][15][16][17]. The LaSalle theorem can also be categorized into a local version and a global version, and they are collectively called invariant set theorem [18,19].…”