Region of Attraction Estimation Using Invariant Sets and Rational Lyapunov Functions

This work addresses the problem of estimating the region of attraction (RA) of equilibrium points of nonlinear dynamical systems. The estimates we provide are given by positively invariant sets which are not necessarily defined by level sets of a Lyapunov function. Moreover, we present conditions for the existence of Lyapunov functions linked to the positively invariant set formulation we propose. Connections to fundamental results on estimates of the RA are presented and support the search of Lyapunov functions of a rational nature. We then restrict our attention to systems governed by polynomial vector fields and provide an algorithm that is guaranteed to enlarge the estimate of the RA at each iteration

A Semi-Definite Programming Approach to Stability Analysis of Linear Partial Differential Equations

We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled PDEs. Our approach is Lyapunov based which allows us to reduce the stability problem to the verification of integral inequalities on the subspaces of Hilbert spaces. Then, using fundamental theorem of calculus and Green's theorem, we construct a polynomial problem to verify the integral inequalities. Constraining the solution of the polynomial problem to belong to the set of sum-of-squares polynomials subject to affine constraints allows us to use semi-definite programming to algorithmically construct Lyapunov certificates of stability for the systems under consideration. We also provide numerical results of the application of the proposed method on different types of PDEs

A Convex Approach to Hydrodynamic Analysis

We study stability and input-state analysis of three dimensional (3D) incompressible, viscous flows with invariance in one direction. By taking advantage of this invariance property, we propose a class of Lyapunov and storage functionals. We then consider exponential stability, induced L2-norms, and input-to-state stability (ISS). For streamwise constant flows, we formulate conditions based on matrix inequalities. We show that in the case of polynomial laminar flow profiles the matrix inequalities can be checked via convex optimization. The proposed method is illustrated by an example of rotating Couette flow.Comment: Preliminary version submitted to 54rd IEEE Conference on Decision and Control, Dec. 15-18, 2015, Osaka, Japa

Barrier Functionals for Output Functional Estimation of PDEs

We propose a method for computing bounds on output functionals of a class of time-dependent PDEs. To this end, we introduce barrier functionals for PDE systems. By defining appropriate unsafe sets and optimization problems, we formulate an output functional bound estimation approach based on barrier functionals. In the case of polynomial data, sum of squares (SOS) programming is used to construct the barrier functionals and thus to compute bounds on the output functionals via semidefinite programs (SDPs). An example is given to illustrate the results.Comment: 8 pages, 1 figure, preprint submitted to 2015 American Control Conferenc

A geometric stabilization of planar switched systems

International audienceIn this paper, we investigate a particular class of switching functions between two linear systems in the plan. The considered functions are defined in terms of geometric constructions. More precisely, we introduce two criteria for proving uniform stability of such functions, both criteria are based on the construction of a Lyapunov function. The first criterion is constructed in terms of an algebraic reformulation of the problem and linear matrix inequalities. The second one is purely geometric. Finally, we illustrate the second method with a numerical example

Finite Gain L p Stability for Hybrid Dynamical Systems ⋆

Abstract We characterize finite gain Lp stability properties for hybrid dynamical systems. By defining a suitable concept of hybrid Lp norm, we introduce hybrid storage functions and provide sufficient Lyapunov conditions for Lp stability of hybrid systems, which cover the well-known continuous-time and discrete-time Lp stability notions as special cases. We then focus on homogeneous hybrid systems and prove a result stating the equivalence among local asymptotic stability of the origin, global exponential stability, existence of a homogeneous Lyapunov function with suitable properties for the hybrid system with no inputs, and input-to-state stability, and we show how these properties all imply Lp stability. Finally we characterize systems with direct and reverse average dwell time properties and establish parallel results for this class of systems. We also make several connections to the existing results on dissipativity properties of hybrid dynamical systems

Stability analysis and controller synthesis for saturating polynomial systems

La classe des systèmes non-linéaires dont la dynamique est définie par un champ de vecteurs polynomial est étudié. Des modèles polynomiaux peuvent représenter différents systèmes réels ou bien définir des approximations plus riches que des modèles linéaires pour des systèmes non-linéaires différentiables. Des techniques de programmation semi-définie développées récemment ont rendu possible l'étude de cette classe de systèmes avec des outils numériques. Le problème d'analyse en stabilité locale est résolu via des conditions basées sur la positivité de polynomes. Dans le cadre de la synthèse de lois de commande nous proposons un changement de variables linéaire pour traiter la synthèse de lois de commande non-linéaire qui garantissent la stabilité locale. Les ensembles définissant des estimations de la région d'attraction, définis par des courbes de niveau de la fonction de Lyapunov pour le système, sont également donnés par des fonctions polynomialesWe study the class of nonlinear dynamical systems which vector field is defined by polynomial functions. A large set of systems can be modeled using such class of functions. Tests for stability are formulated as semidefinite programming problems by considering positive polinomials to belong to the class of Sum of Squares polynomials. Polynomial control law gains are computed based on a linear change of coordinates and guarantee the local stability of the closed-loop system. Lyapunov theory is then applied in order to obtain estimates of the region of attraction for stable equilibrium points. Such estimates are given by level sets of polynomial positive function