  This paper addresses the problem of bearing formation control in $d$ $(d\ge
2)$-dimensional space by exploring persistence of excitation (PE) of the
desired bearing reference. A general concept of bearing persistently exciting
(BPE) formation defined in $d$-dimensional space is fully developed for the
first time. By providing a desired formation that is BPE, distributed control
laws for multi-agent systems under both single- and double-integrator dynamics
are proposed using bearing measurements (also velocity measurements for
double-integrator dynamics), which guarantee exponential stabilization of the
desired formation up to a translation vector. A key contribution of this work
is to show that the classical bearing rigidity condition on the graph topology,
required for achieving the stabilization of a formation up to a scaling factor,
is relaxed in a natural manner by exploring PE conditions imposed solely on a
specific set of desired bearing vectors. Simulation results are provided to
illustrate the performance of the proposed control method.
