  In this paper, the multiple-source ellipsoidal localization problem based on
acoustic energy measurements is investigated via set-membership estimation
theory. When the probability density function of measurement noise is
unknown-but-bounded, multiple-source localization is a difficult problem since
not only the acoustic energy measurements are complicated nonlinear functions
of multiple sources, but also the multiple sources bring about a
high-dimensional state estimation problem. First, when the energy parameter and
the position of the source are bounded in an interval and a ball respectively,
the nonlinear remainder bound of the Taylor series expansion is obtained
analytically on-line. Next, based on the separability of the nonlinear
measurement function, an efficient estimation procedure is developed. It solves
the multiple-source localization problem by using an alternating optimization
iterative algorithm, in which the remainder bound needs to be known on-line.
For this reason, we first derive the remainder bound analytically. When the
energy decay factor is unknown but bounded, an efficient estimation procedure
is developed based on interval mathematics. Finally, numerical examples
demonstrate the effectiveness of the ellipsoidal localization algorithms for
multiple-source localization. In particular, our results show that when the
noise is non-Gaussian, the set-membership localization algorithm performs
better than the EM localization algorithm.
