Using SOS for optimal semialgebraic representation of sets: Finding minimal representations of limit cycles, chaotic attractors and unions

In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as in the case of attractors of nonlinear systems. We define optimality in the sense of minimum volume, while satisfying constraints that can include set containment, convexity, or Lyapunov stability conditions. Our admittedly heuristic approach to volume minimization is based on the use of a determinant-like objective function. We provide numerical examples for the Lorenz attractor and the Van der Pol limit cycle.