| The topology of the configuration manifold for many physical systems is inherently troublesome from a control perspective. In fact, for systems having a compact configuration manifold, there exists no continuous control law that is globally asymptotically stabilizing for some desired equilibrium point in the manifold. Memoryless discontinuous state feedback control is sometimes employed to "solve" such topological issues; however, we show that global attractivity properties brought about by such controllers are not robust to arbitrarily small measurement noise.;In this work, we employ hybrid control to solve the robust global stabilization problem for such topologically constrained systems. In particular, we propose a control scheme that coordinates a family of Lyapunov-based control laws using state-based hysteretic switching. Using Lyapunov functions to generate the family of control laws instantly yields a condition on the family of Lyapunov functions that is equivalent to guaranteeing robust global asymptotic stability of the closed-loop system. Hence, with the proposed hybrid controller, the original robust global stabilization problem is transformed into one of finding a family of "synergistic" Lyapunov functions. While this problem is difficult in general, we provide some results for manifolds commonly encountered in engineering including the unit circle, unit 2-sphere, and the special orthogonal group of order three.;In many cases, the topological complexity of certain manifolds can be reduced by considering its representation in a "covering space." While this transformation makes certain aspects of a control design less difficult, it introduces some subtleties that are often ignored and can have very undesirable consequences for the closed-loop system. We carefully examine rigid body attitude control in the context of a quaternion representation and again provide a hybrid control scheme that provides robust global asymptotic tracking and synchronization of a network of rigid bodies. |
