Lyapunov-based stability analysis automated by genetic programming

doi:10.1016/j.automatica.2008.07.014

Automatica

Volume 45, Issue 1, January 2009, Pages 252-256

https://doi.org/10.1016/j.automatica.2008.07.014 Get rights and content

Abstract

This contribution describes an automatic technique to detect suitable Lyapunov functions for nonlinear systems. The theoretical basis for the work is Lyapunov’s Direct Method, which provides sufficient conditions for stability of equilibrium points. In our proposed approach, genetic programming (GP) is used to search for suitable Lyapunov functions, that is, those that best predict the true domain of attraction. In the work presented here, our GP approach has been extended by defining a target function accounting for the Lyapunov function level sets.

Introduction

The formal definition of system stability is at the focus of differential and integral analysis, having engaged the attention of leading mathematicians and physicists including Torricelli, Laplace, Lagrange and others. However, it was only in 1892 that a clear criterion was established, with the publication of the work of the Russian mathematician, Aleksandr Mikhailovich Lyapunov (Lyapunov, 1949). He defined a scalar function inspired by a classical energy function (Lyapunov’s direct method), which has three important properties that are sufficient for establishing the asymptotic stability of an equilibrium point: (a) it must be local positive definite, (b) it must have continuous partial derivatives, and (c) its time derivative along any state trajectory must be local negative definite (Slotine & Li, 1991). While Lyapunov theory provides powerful guarantees concerning a system’s stability once an appropriate function is identified, it regrettably provides no guidance on how this function should be selected.

Section snippets

Lyapunov stability principles

This study concerns stability analysis of autonomous systems, of the general form: $\underline{\dot{x}} = \underline{f} (\underline{x})$ where $\underline{x} = R^{n}$ . A system is said to be autonomous if $\underline{f}$ does not depend explicitly on time. An equilibrium point of the system of Eq. (1), $\underline{x} = {\underline{x}}^{*}$ , is one that satisfies: $\underline{f} ({\underline{x}}^{*}) = 0 .$

An equilibrium point is said to be stable in the sense of Lyapunov if for any $n$ -dimensional ball of radius $ε > 0$ there exists an $n$ -dimensional ball of radius $δ (ε)$ , such that for any trajectory $\underline{x} (t_{0}, {\underline{x}}_{0})$ , starting in $δ$ , $‖ \underline{x} (t, x_{0}) ‖ < ε$

Selection of Lyapunov functions using genetic programming

Genetic programming (GP) is an optimization method inspired by the principles of Darwinian evolution (Koza, 1992). Unlike conventional optimization techniques that manipulate the parameters of an initial estimate of the solution, GPs maintain a population of potential models, each structured in a tree-like fashion, with basis functions linking nodes of inputs and constants. The probability of a given model surviving into the next generation depends on its performance, which is evaluated using a

Results

Our approach is driven by the maximization of predicted domains of attraction. These abilities were compared with some well-known published results, one of which was described in Vannelli and Vidyasagar (1985). Their method is based on an iterative procedure for finding what they call maximal Lyapunov functions, which are rational by their nature and are the result of the division of two multivariable polynomials: $V (\underline{x}) = N (\underline{x}) / D (\underline{x})$ . In the following, we compare our results with those of

Conclusions

This paper has presented a novel approach for the automatic generation of Lyapunov functions suitable for stability analysis of nonlinear systems. The results demonstrate the ability of the GP to detect complex structures for Lyapunov functions. The approach was tested on four examples from Vannelli and Vidyasagar (1985), whose aim was to define the true domain of attraction. Our original motivation was to determine acceptable Lyapunov functions for nonlinear system analysis rather than to

Benyamin Grosman received his B.Sc. degree in 1997, his M.Sc. degree in 2002, and his Ph.D. degree in 2008, all at the Technion Institute of Technology, Haifa, Israel. His research interests are in process systems engineering, with a particular emphasis on evolutionary computational methods.

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Daniel R. Lewin holds the Churchill Family Chair at the Department of Chemical Engineering and is the director of the PSE Research Group at the Technion, the Israel Institute of Technology. His research focuses on the interaction of process design and process control and operations, with emphasis on model-based methods. He has authored or co-authored over 100 technical publications in the area of process systems engineering, as well as the textbook: Product and Process Design Principles: Synthesis, Analysis and Evaluation, published in 2003 by John Wiley and Sons. He is a member of International Federation of Automatic Control (IFAC) Committee on Process Control.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henri Huijberts, under the direction of Editor Hassan K. Khalil.

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Brief paperLyapunov-based stability analysis automated by genetic programming☆