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.

在本文中，我们关注“积极的” ${升}_2$输入信号被限制为非负的离散时间线性时不变系统的诱导范数。为了处理输入信号的非负性，我们采用协正规划作为分析的数学工具。然后，通过对同正锥应用内近似，我们推导出了数值上易于处理的半定规划问题，用于“正”的上限和下限计算${升}_2$诱导规范。该范数通常可用于对由 LTI 系统和非线性元素构建的反馈系统的稳定性分析，其中非线性元素仅提供非负信号。作为一个具体的例子，我们说明了“积极的”${升}_2$ 激活函数为修正线性单元的递归神经网络稳定性分析的诱导范数。