Session: MSNDC-04/VIB-09-03: Nonlinear Dynamics of Systems and Structures

Paper Number: 116835

116835 - On Selecting and Reconstructing Lyapunov Functions for Stability Analysis of Nonlinear Dynamical Systems 

Lyapunov theorems did not tell us how to choose the Lyapunov function to analyze the stability of equilibrium states for nonlinear systems, and currently there is no particular way or formula that can guide us on how to choose the right Lyapunov function when using Lyapunov theorems and LaSalle theorems. Typically, a Lyapunov candidate function is chosen based on the trial and error approach. In this paper, the author presents an approach that is able to guide us to choose the right Lyapunov function based on the initial random Lyapunov candidate function when using LaSalle theorems to study the equilibrium states’ stability for the nonlinear systems. Difference case studies are given in the paper in order to show the effectiveness of the approach.

The main concept of the proposed approach in this paper is that when we randomly choose a Lyapunov function and derive the time derivative of the Lyapunov function, it turns out to be a composite variable, which contains all the state variables. Consequently, we are using all the components of the composite variable to reconstruct the new Lyapunov function. This approach works well for the first order linear and nonlinear systems, as well as for the case 1 of the second order nonlinear dynamical system.

       Further work will focus on how to further refine the Lyapunov function based on the proposal approach so that the limit cycle of the Van der Pol system (i.e. the case 2 of the second order nonlinear dynamical system) can be shown by using the reconstructed Lyapunov function, and how to extend the proposed approach so that it can be applied to a wide arrange of second order nonlinear dynamical systems.

Presenting Author: Bin Wei Texas A&M University - Kingsville

Presenting Author Biography: Dr. Wei is an Assistant Professor in the department of mechanical engineering at Texas A&M University - Kingsville.

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