Stabilization via Fully Actuated System Approach: A Case Study

In this note, a benchmark example system which is not stabilizable by a smooth state feedback controller is considered with the fully actuated system (FAS) approach. It is shown that a smooth controller exists which drives the trajectories starting from a large domain in the initial value space to the origin exponentially. Such a result brings about a generalization of Lyapunov asymptotical stability, which is termed as global exponential sub-stability. The region of attraction is allowed to be an unbounded open set of the initial values with closure containing the origin. This sub-stability result may be viewed to be superior to some local stability results in the Lyapunov sense because the region of attraction is much larger than any finite ball containing the origin and meanwhile the feasible trajectories are always driven to the origin exponentially. Based on this sub-stabilization result, globally asymptotically stabilizing controllers for the system can be provided in two general ways, one is through combination with existing globally stabilizing controllers, and the other is by using a pre-controller to first move an initial point which is not within the region of attraction into the region of attraction.