  Ensemble systems appear frequently in many engineering applications and, as a
result, they have become an important research topic in control theory. These
systems are best characterized by the evolution of their underlying state
distribution. Despite the work to date, few results exist dealing with the
problem of directly modifying (i.e., "steering") the distribution of an
ensemble system. In addition, in most of the existing results, the distribution
of the states of an ensemble of discrete-time systems is assumed to be
Gaussian. However, in case the system parameters are uncertain, it is not
always realistic to assume that the distribution of the system follows a
Gaussian distribution, thus complicating the solution of the overall problem.
In this paper, we address the general distribution steering problem for
first-order discrete-time ensemble systems, where the distributions of the
system parameters and the states are arbitrary with finite first few moments.
Both linear and nonlinear system dynamics are considered using the method of
power moments to transform the original infinite-dimensional problem into a
finite-dimensional one. We also propose a control law for the ensuing moment
system, which allows us to obtain the power moments of the desired control
inputs. Finally, we solve the inverse problem to obtain the feasible control
inputs from their corresponding power moments. We provide numerical results to
validate our theoretical developments. These include cases where the parameter
distribution is uniform, Gaussian, non-Gaussian, and multi-modal, respectively.
