pwpfit/pwroaest

Toolbox for region-of-attraction estimation of piecewise-defined polynomial systems using sum-of-squares prgramming

pwroaest : Piecewise Region of Attraction Estimation

This toolbox offers sum-of-squares based algorithms for the region-of-attraction estimation for piecewise defined polynomial systems. Such systems are defined as ordinary differential equations dx = f(x) where

    f(x) = f1(x), if φ(x) <= 0
    f(x) = f2(x), else

and f1, f2, φ are polynomials in n variables x.

It is assumed throughout this work that f1(0) = 0 if φ(0) <= 0 and f2(0) = 0 if φ(0) >= 0.

Dependencies

This work is based on the Robustness Analysis of Nonlinear Systems of the University of Minnesota for the analysis of polynomial systems using sum-of-squares programming. In particular, the toolbox requires sosopt, multipoly, and the polynomial roaest routines of SOSAnalysis and their dependencies (i.e., SeDuMi) to be accessible for MATLAB.

Theoretical background

A set X ⊂ ℝ^n is an estimate of the region of attraction of the equilibrium x* = 0 for f if any trajectory starting in X stays in X as well as converges to x* when time approaches infinity; i.e., X is both forward-invariant for f and attractive to x*.

Definition: Lyapunov Functions

Let f be a system with f(0) = 0, V be a scalar field over ℝ^n and X a subset of ℝ^n; we will call V Lyapunov function of f in X if and only if

1. V(0) = 0
2. V(x) > 0 for all x ∈ R^n - {0}
3. grad V(x)*f(x) < 0 for all x ∈ X - {0},

where grad V(x) denotes the gradient of V evaluated in x.

Theorem: Piecewise Region of Attraction

Let f1, f2, φ define a piecewise system f as above, X be a subset of ℝ^n, and γ > 0; if one of the following hold,

• there is a scalar field V that is Lyapunov function of both f1 and f2 in {x| V(x) <= γ}, respectively, ("Local Lyapunov function"); and X = {x| V(x) < γ}
• there is a scalar field V that is Lyapunov function of both f1 and f2 in {x| V(x) <= γ, φ(x) <= 0} and {x| V(x) <= γ, φ(x) >= 0}, respectively, ("Common Lyapunov function"); and X = {x| V(x) < γ}
• there are scalar fields V1, V2 such that V1 is Lyapunov function of f1 in {x| V1(x) <= γ, φ(x) <= 0} and V2 is Lyapunov function of f2 in {x| V2(x) <= γ, φ(x) >= 0}; V1(x) = V2(x) for all x s.t. φ(x) = 0; and X = {x| V(x) < γ} ∩ {x| V(x) < γ}

then X is an estimate of the region of attraction for f.