Parametric Dependence of Large Disturbance Response for Vector Fields with Event-Selected Discontinuities

The ability of a nonlinear system to recover from a large disturbance to a desired stable equilibrium point depends on system parameter values, which are often uncertain and time varying. A particular disturbance acting for a finite time can be modeled as an implicit map that takes a parameter value to its corresponding post-disturbance initial condition in state space. The system recovers when the post-disturbance initial condition lies inside the region of attraction of the stable equilibrium point. Critical parameter values are defined to be parameter values whose corresponding post-disturbance initial condition lies on the boundary of the region of attraction. Computing such values is important in numerous applications because they represent the boundary between desirable and undesirable system behavior. Many realistic system models involve controller clipping limits and other forms of switching. Furthermore, these hybrid dynamics are closely linked to the ability of a system to recover from disturbances. The paper develops theory which underpins a novel algorithm for numerically computing critical parameter values for nonlinear systems with clipping limits and switching. For an almost generic class of vector fields with event-selected discontinuities, it is shown that the boundary of the region of attraction is equal to a union of the stable manifolds of the equilibria and periodic orbits it contains, and that this decomposition persists and the boundary varies continuously under small changes in parameter.