An interval arithmetic approach for the estimation of the domain of attraction

Abstract

Since the analysis of asymptotic stability is not sufficient for safety-critical nonlinear systems, the analysis of the domain of attraction is the focus of current research. While several approaches for polynomial systems were presented in the last years, based on sums of squares/linear matrix inequalities (SOS/LMIs) techniques or using a sampling method, only a few approaches are available for non-polynomial systems. In this paper we present a new branch-and-bound-method using the Lyapunov stability theory. It is based on interval arithmetic and delivers lower and upper bounds for the maximum contour line of a given Lyapunov function, which bounds a subset of the domain of attraction.Our approach can applied for polynomial systems as well as for non-polynomial systems.

Estimation of the domain of attraction for non-polynomial systems: A novel method

“…The effectiveness of our approach is shown by means of three benchmark examples. These examples were presented in Chesi [2005Chesi [ , 2009 and were also used in Warthenpfuhl et al [2010]. Our results were computed with MATLAB 2008b on a standard PC and are not only consistent, but tighter than the inclusions given by Warthenpfuhl et al [2010].…”

“…The method was tested on three benchmark examples from literature. On these, it yielded better results than those presented in Chesi [2005Chesi [ , 2009 and Warthenpfuhl et al [2010]. Future research will concentrate on adapting the method to work with higherorder and rational Lyapunov functions and on using it in controller design to maximize the domain of attraction of the controlled system.…”

“…Techniques for non-polynomial systems are rare: In Chesi [2005Chesi [ , 2009, an LMI technique which substitutes the non-polynomial terms with their Taylor expansions, was proposed. An interval arithmetic approach was presented in Warthenpfuhl et al [2010].…”

A numerical algebraic geometry approach to regional stability analysis of polynomial systems

“…In addition, [7] assumes finitely many local optimizers and relies on generally expensive computer algebra routines. Interval arithmetic methods are considered in [15] and [17]. These methods employ numerical procedures for solving polynomial systems and are therefore similar in spirit to the technique we describe.…”

Nonlinear static state feedback controller design to enlarge the domain of attraction for a class of nonlinear systems

“…In [14] a computational methodology of estimating the DOA for non-polynomial systems by a descriptor system approach is presented. An interval arithmetic approach was presented in [11]. Two different multidimensional gridding approaches for non-polynomial systems were presented in [1] and [2], in which a theorem of Ehlich and Zeller [12] for polynomial systems was modified and adjusted to deal with non-polynomial systems using the interpolation error of the non-polynomial terms.…”