  This paper introduces a new controllability notion, termed partial strong
structural controllability (PSSC), on a structured system whose entries of
system matrices are either fixed zero or indeterminate, which naturally extends
the conventional strong structural controllability (SSC) and bridges the gap
between structural controllability and SSC. Dividing the indeterminate entries
into two categories, generic entries and unspecified entries, a system is PSSC,
if for almost all values of the generic entries in the parameter space except
for a set of measure zero, and any nonzero (complex) values of the unspecified
entries, the corresponding system is controllable. We highlight that this
notion generalizes the generic property embedded in the conventional structural
controllability for single-input systems. We then give algebraic and
(bipartite) graph-theoretic necessary and sufficient conditions for
single-input systems to be PSSC. Conditions for multi-input systems are
subsequently given for a special case. It is shown the established results can
induce a new maximum matching based criterion for SSC over the system bipartite
graph representations.
