Domain of attraction in hybrid systems

Abstract

Domain of Attraction (DoA) is a set of initial conditions for which the system converges to the equilibrium point. In fact, it is a key problem in control engineering to guarantee stability within a workspace and avoid system failures. Classical applications include pendulum systems, tunnel diode circuits, mass-spring systems, negative-resistance oscillators and more recently, these have been found in other fields such as biology and ecology.

This thesis firstly addresses the estimation of the DoA for a class of hybrid nonlinear systems in both discrete and continuous-time. The state space is partitioned into several regions which are described by polynomial inequalities, and the union of all the regions is a complete cover of the state space. The system dynamics are defined on each region independently from the others by polynomial functions. The problem of computing the largest sublevel set of a Lyapunov function included in the DoA is considered. An approach is proposed for addressing this problem based on linear matrix inequalities (LMIs), which provides a lower bound of the sought estimate by establishing negativity of the Lyapunov function derivative on each region.

Secondly, a sufficient and necessary condition is firstly provided for establishing optimality of the found lower bound. This requires to solve linear algebra operations in typical cases. The problem of looking for variable Lyapunov functions that maximize the estimate of the DoA is also considered, describing several strategies where the proposed approach can be readily adopted.

Thirdly, the computation of static nonlinear output feedback controllers for increasing the DoA of an equilibrium point of continuous hybrid nonlinear polynomial systems is addressed. A dynamical system where the state space is partitioned into possibly overlapping regions, and where the vector field is defined independently among the regions by polynomial functions, will be considered. The computation of static nonlinear output feedback controller that increase the estimate of the DoA provided by a polynomial Lyapunov function is addressed. The controller can be common or vary among the regions that partition the state space. A strategy is proposed which provides guaranteed estimates of the increased DoA controllers and the controllers required to achieve them. Moreover, this strategy can be readily exploited with optimality test and variable Lyapunov functions through the use of approaches described.