  In this paper, we focus on the "positive" $l_2$ induced norm of discrete-time
linear time-invariant systems where the input signals are restricted to be
nonnegative. To cope with the nonnegativity of the input signals, we employ
copositive programming as the mathematical tool for the analysis. Then, by
applying an inner approximation to the copositive cone, we derive numerically
tractable semidefinite programming problems for the upper and lower bound
computation of the "positive" $l_2$ induced norm. This norm is typically useful
for the stability analysis of feedback systems constructed from an LTI system
and nonlinearities where the nonlinear elements provide only nonnegative
signals. As a concrete example, we illustrate the usefulness of the "positive"
$l_2$ induced norm for the stability analysis of recurrent neural networks with
activation functions being rectified linear units.
