Domain of Attraction: Analysis and Control via SOS Programming

Citation
G. Chesi. Domain of Attraction: Analysis and Control via SOS Programming. Springer, 2011

Other
Series: Lecture Notes in Control and Information Sciences
Volume: 415
Pages: 286
ISBN: 978-0-85729-958-1

Abstract
Stability of equilibrium points plays a fundamental role in dynamical systems. For nonlinear dynamical systems, investigating stability requires characterizing the domain of attraction (DA) of an equilibrium point, i.e. the set of initial conditions from which the trajectory of the system converges to such a point. Unfortunately, it is well-known that estimating the DA, or even more attempting to control it, are very difficult problems due to the complex relationship of this set with the model of the system. This book addresses the estimation and control of the DA via the newborn SOS programming, i.e. optimization techniques that have been recently developed based on polynomials that are sum of squares of polynomials (denoted hereafter as SOS polynomials) and that amount to solving convex optimization problems with linear matrix inequality (LMI) constraints, also known as semidefinite programs (SDPs). It is shown for the first time in the literature how these issues can be addressed under a unified framework in various cases depending on the nature of the considered nonlinear systems, including the cases of polynomial systems, uncertain polynomial systems, and nonlinear non-polynomial systems (possibly uncertain). The methodology proposed in this book is illustrated through various examples with fictitious and real systems, such as chemical reactors, electric circuits, mechanical devices, and social models. The book also offers a concise and simple description of the main features of SOS programming which can be used for general purpose in research and teaching, in particular introducing various classes of SOS polynomials and their characterization via LMIs, and addressing typical problems such as establishing positivity or non-positivity of polynomials and matrix polynomials, determining the minimum of rational functions, and solving systems of polynomial equations, in both cases of unconstrained variables and constrained ones.

Software
The techniques presented in this book are available in the Matlab toolbox SMRSOFT

Keywords
Control systems
Domain of attraction
Hilbert's 17th problem
Nonlinear systems
Optimization over polynomials
SOS
Stability analysis
Uncertain systems

Errata
List of corrections

Organization of the Book

The book is organized in two parts: Part I, which addresses SOS programming, and Part II, which addresses the estimation and control of the DA.

For Part I, Chapter 1 introduces the representation of polynomials via power vectors and via the square matrix representation (SMR), the concepts of SOS polynomials, SOS parameter-dependent polynomials and SOS matrix polynomials, their characterization via the SMR and LMIs, the concept of SOS index (which measures how SOS is a polynomial), the extraction of power vectors from linear subspaces, and the gap between positive polynomials and SOS polynomials.

Chapter 2 describes the use of SOS polynomials for unconstrained optimization (in particular, establishing positivity or non-positivity of a polynomial and determining the minimum of a rational function through the introduction of the generalized SOS index, which measures how ratio of two SOS polynomials is a polynomial), constrained optimization (in particular, determining the minimum of a rational function, solving systems of polynomial equations and establishing positivity of a matrix polynomial over semialgebraic sets through the Positivstellensatz), and optimization over special sets (in particular, establishing positivity or non-positivity of a polynomial over ellipsoids and over the simplex). This chapter also addresses the search for a positive semidefinite SMR matrix of a SOS polynomial with the largest rank.

For Part II, Chapter 3 introduces some preliminaries of nonlinear systems, in particular the concepts of equilibria, stability, DA, and their robustness properties in the presence of model uncertainties.

Chapter 4 investigates the DA in polynomial systems, in particular considering the problems of establishing whether a sublevel set of a Lyapunov function (LF) is an inner estimate of the DA, estimating the largest estimate of the DA (LEDA) for such a function and verifying the tightness of the found estimate, presenting an alternative technique based on homogeneous polynomials for the case of quadratic LFs, searching for optimal estimates and establishing global asymptotical stability by considering variable LFs, and designing static output controllers for enlarging the DA.

Chapter 5 addresses the robust DA (RDA) in uncertain polynomial systems, in particular considering the problems of establishing whether a sublevel set of a common LF or a parameter-dependent sublevel set of a parameter-dependent LF is an inner estimate of the RDA, computing the largest estimate of the RDA (LERDA) for such functions according to various criteria, estimating the intersection of parameter-dependent estimates, searching for optimal estimates and establishing robust global asymptotical stability by considering variable LFs, and designing static output controllers for enlarging the RDA.

Chapter 6 addresses the case of nonlinear non-polynomial systems by introducing truncated Taylor expansions of the non-polynomial terms and by taking into account their worst-case remainders over the considered estimate. This is done in both cases of certain dynamics and uncertain ones, by considering the problems of establishing whether a sublevel set of a LF is an inner estimate of the DA or the RDA, estimating the LEDA or the LERDA for such a function, searching for optimal estimates and establishing global asymptotical stability by considering variable LFs, and designing static output controllers for enlarging the DA or the RDA.

Lastly, Chapter 7 describes the use of SOS programming in some miscellaneous problems related to the DA, in particular the estimation of the DA via union of a continuous family of estimates for nonlinear systems, and the computation and the minimization via constrained control design of the extremes of the trajectories of nonlinear systems over a set of initial conditions. This chapter also provides some remarks on the estimation of the DA and the construction of global LFs for degenerate polynomial systems.