  This work studies the problem of controlling the mean-field density of
large-scale stochastic systems, which has applications in various fields such
as swarm robotics. Recently, there is a growing amount of literature that
employs mean-field partial differential equations (PDEs) to model the density
evolution and uses density feedback to design control laws which, by acting on
individual systems, stabilize their density towards a target profile. In spite
of its stability property and computational efficiency, the success of density
feedback relies on assuming the systems to be homogeneous first-order
integrators (plus white noise) and ignores higher-order dynamics, making it
less applicable in practice. In this work, we present a backstepping design
algorithm that extends density control to heterogeneous and higher-order
stochastic systems in strict-feedback forms. We show that the strict-feedback
form in the individual level corresponds to, in the collective level, a PDE (of
densities) distributedly driven by a collection of heterogeneous stochastic
systems. The presented backstepping design then starts with a density feedback
design for the PDE, followed by a sequence of stabilizing design for the
remaining stochastic systems. We present a candidate control law with stability
proof and apply it to nonholonomic mobile robots. A simulation is included to
verify the effectiveness of the algorithm.
