  This paper is concerned with the analysis of the $L_{2}$ induced norm of
continuous-time LTI systems where the input signals are restricted to be
nonnegative. This induced norm is referred to as the $L_{2+}$ induced norm in
this paper. It has been shown very recently that the $L_{2+}$ induced norm is
particularly useful for the stability analysis of nonlinear feedback systems
constructed from linear systems and static nonlinearities where the nonlinear
elements only provide nonnegative signals. For the upper bound computation of
the $L_{2+}$ induced norm, an approach with copositive programming has also
been proposed. It is nonetheless true that this approach becomes effective only
for multi-input systems, and for single-input systems this approach does not
bring any improvement over the trivial upper bound, the standard $L_2$ norm. To
overcome this difficulty, we newly introduce positive filters to increase the
number of positive signals. This enables us to enlarge the size of the
copositive multipliers so that we can obtain better (smaller) upper bounds with
copositive programming.
