  An important open problem in the synthesis of passive controllers is to
obtain a passive network that realizes an arbitrary given impedance function
and contains the least possible number of elements. This problem has its
origins in electric circuit theory, and is directly applicable to the
cost-effective design of mechanical systems containing the inerter. Despite a
rich history, the problem can only be considered solved for networks that
contain at most two energy storage elements, and in a small number of other
special cases. In this paper, we solve the minimal network realization problem
for the class of impedances realized by series-parallel networks containing at
most three energy storage elements. To accomplish this, we develop a novel
continuity-based approach to eliminate redundant elements from a network.
