• ciclo limite
• limit cycle
• region of attraction
• regione di asintotica stabilità
• stability analysis
• sum of squares

Limit cycles represent the natural and energy-efficient oscillatory behaviour of nonlinear dynamical systems. Their identification, appearance and stability properties have a long history of study, with some unresolved theoretical questions, such as Hilbert's sixteenth problem. In particular, stability analysis plays a crucial role in engineering applications: the knowledge of the stability properties of a limit cycle makes it possible to exploit the system evolution in a control perspective, considerably reducing energy consumption; and represents a metric of the robustness to disturbances. Examples can be found from aerospace to energy systems, as well as in robotics, where the legged robot locomotion is one of the clearest examples of a system to be kept oscillating on a limit cycle.
The problem of numerically estimating the region of attraction of a limit cycle, i.e. the set of all initial conditions for which the system evolves in the limit cycle, has been addressed in several ways. The classic technique involves constructing a Poincarè map through various simulations. Still, more recently methods have been introduced that formulate the estimation as a reachability problem or Sum of Squares optimization problem. The latter are considered in this thesis, with their advantage of leading to semidefinite programming problems, reducing computational complexity.
The presented work proposes a novel approach to exploit Sum of Squares optimization for the region of attraction estimation of limit cycles: current methods manipulate the system to use algorithms for the stability analysis on equilibrium points; rather, we directly reformulate the conditions of stability theorems for invariant sets to cast a Sum of Squares program. We therefore consider the cycle in its entirety, avoiding sampling the cycle and constructing transverse dynamics in regions of the cycle.
The proposed method is first defined for general n-dimensional nonlinear continuous dynamical systems. Then, for planar systems only, we propose a theoretical extension of the current stability theorems and we extend the method consequently. Finally, everything is validated simulating three planar systems.