Control of a Driftless Bilinear Vector Field on $n$-Sphere.

In this paper, we consider a multi-input driftless bilinear system evolving on the $n$ -dimensional sphere $S^{n}$. We first provide examples drawn from rigid body mechanics that provide the motivation for the control of bilinear systems on $S^{n}$. For the general framework, we establish the global controllability on $S^{n}$ and propose two linear control laws on $S^{n}$ that achieve asymptotic stabilization of an equilibrium point with an almost global domain-of-attraction. Further, the asymptotically stable closed-loop system trajectories are shown to be arcs on the geodesics of $S^{n}$ for a particular choice of the equilibrium point. Next, we propose two linear time-varying control laws to achieve trajectory tracking on $S^{n}$ and show the asymptotic stability of the tracking error. A distributed control is designed for the consensus of multiagent bilinear systems on $S^{n}$ with an undirected tree as the communication graph. The consensus manifold is shown to have an almost global domain of attraction. [ABSTRACT FROM AUTHOR]