  When underlying probability density functions of nonlinear dynamic systems
are unknown, the filtering problem is known to be a challenging problem. This
paper attempts to make progress on this problem by proposing a new class of
filtering methods in bounded noise setting via set-membership theory and Monte
Carlo (boundary) sampling technique, called Monte Carlo set-membership filter.
The set-membership prediction and measurement update are derived by recent
convex optimization methods based on S-procedure and Schur complement. To
guarantee the on-line usage, the nonlinear dynamics are linearized about the
current estimate and the remainder terms are then bounded by an optimization
ellipsoid, which can be described as a semi-infinite optimization problem. In
general, it is an analytically intractable problem when dynamic systems are
nonlinear. However, for a typical nonlinear dynamic system in target tracking,
we can analytically derive some regular properties for the remainder. Moreover,
based on the remainder properties and the inverse function theorem, the
semi-infinite optimization problem can be efficiently solved by Monte Carlo
boundary sampling technique. Compared with the particle filter, numerical
examples show that when the probability density functions of noises are
unknown, the performance of the Monte Carlo set-membership filter is better
than that of the particle filter.
