  Ensemble control deals with the problem of using a finite number of control
inputs to simultaneously steer a large population (in the limit, a continuum)
of control systems. Dual to the ensemble control problem, ensemble estimation
deals with the problem of using a finite number of measurement outputs to
estimate the initial condition of every individual system in the ensemble. We
introduce in the paper an innovative class of ensemble systems, termed
distinguished ensemble systems, and establish sufficient conditions for
controllability and observability of such systems. Every distinguished ensemble
system has two key components, namely distinguished control vector fields and
codistinguished observation functions. Roughly speaking, a set of vector fields
is distinguished if it is closed (up to scaling) under Lie bracket, and
moreover, every vector field in the set can be obtained by a Lie bracket of two
vector fields in the same set. Similarly, a set of functions is codistinguished
to a set of vector fields if the Lie derivatives of the functions along the
given vector fields yield (up to scaling) the same set of functions. We
demonstrate in the paper that the structure of a distinguished ensemble system
can significantly simplify the analysis of ensemble controllability and
observability, and can be used as a principle for ensemble system design. We
further address in the paper the existence of a distinguished ensemble system
for a given manifold. Specifically, we show that every connected, semi-simple
Lie group admits a set of distinguished vector fields, together with a set of
codistinguished functions. The proof is constructive, leveraging structure
theory of semi-simple real Lie algebras and representation theory. Examples
will be provided along the presentation illustrating key definitions and main
results.
