Maximizing regions of attraction via backstepping and CLFs with singularities

Abstract

When a nonlinear control law contains a singularity (that is, becomes unbounded on a set of points in the state space), the region of attraction is often only a subset of the region of feasibility of this control law. A method which is well known to suffer from this difficulty is feedback linearization. We show that the backstepping design (in its standard form) has an inherent ability to make the regions of feasibility and attraction coincide, thus maximizing the latter. The key observation that this paper provides is that a standard backstepping-style control Lyapunov function, which grows unbounded on the set where the control law becomes unbounded, has level sets that always remain in the feasibility region, which makes the feasibility region positively invariant. A simulation comparison with a feedback linearization design shows a dramatic improvement of the region of attraction with backstepping. Since our theorem imposes a strong assumption that the feasibility region for the first subsystem in the backstepping problem is positively invariant, we present examples (ranging from simple to fairly difficult) which demonstrate how this condition can be satisfied.