Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

Abstract

The notion of Lyapunov function plays a key role in the design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives. Furthermore, we present a method for automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is relatively complete in the sense that it is able to discover all polynomial RLFs with a given predefined template for any PDS. Therefore it can also generate all polynomial RLFs for the PDS by enumerating all polynomial templates.

Exact Asymptotic Stability Analysis and Region-of-Attraction Estimation for Nonlinear Systems

“…All the computations have been performed on an Intel Core 2 Duo 2.0 GHz processor with 2 GB of memory. In Table 1 Examples 1-3 correspond to [9,Examples 2,3,9] and Examples 4-6 correspond to [39,Example 7], [6,Example 22], and [40, page 1341]. For all these examples, let the degree bound of the SOSes be 4, and set = 10 −2 , = 100.…”

“…A function (x) satisfying the conditions (5) and (6) in Theorem 2 is commonly known as a Lyapunov function. And we can verify globally asymptotic stability of system (1) by using Lyapunov functions stated as follows.…”

“…Lyapunov functions can be used to verify the asymptotic stability, including locally asymptotic stability and globally asymptotic stability. In the literature, there has been lots of work on constructing Lyapunov functions [1][2][3][4][5][6][7][8][9][10][11][12]. For example, [4,5] used the linear matrix inequality (LMI) method to compute quadratic Lyapunov functions.…”

Adaptive Backstepping Control for a 2-DOF Robot Manipulator: A State Augmentation Approach

“…In order to design the backstepping controller, the equation system (3) is separated into subsystems. The first equation subsystem is, [10]. By defining the Lyapunov function for (6) as (7).…”

“…The designed controller should satisfy the stability of the controlled system in existence of model nonlinearity and the system variable structure. Lyapunov theorem is a very strong tool proving the stability of the controlled manipulator by proposing a positive decreasing energy function [10]. The controller should adapt its structure to confront the parameter variations and despite its adaptive structure it should prove the stability of the controlled system.…”