Uniting control laws: On obstacle avoidance and global stabilization of underactuated linear systems

An important problem in safety-critical applications is that of simultaneous stabilization (of the origin) and obstacle avoidance. When obstacles are defined by bounded sets of the state space, state feedback controllers are necessarily discontinuous and can be naturally formulated in the framework of hybrid systems. In previous work, we developed such hybrid controllers for linear systems when an obstacle is prohibited from being centered around an induced equilibrium of a single-input system or for multi-input systems when the input matrix has full row rank. In this paper, we explore a design for simultaneous stabilization of the origin and avoidance of an obstacle for single-input systems when the obstacle is centered around an induced equilibrium. The presented Lyapunov-based feedback control design provably works for planar systems and may work for systems of even degree.