This paper investigates the input‐output gains, including ℓ₁‐ and ℓ_∞‐gains and L₁‐ and L_∞‐gains, of discrete‐time and continuous‐time positive periodic systems. For the discrete‐time case, the input–output gain can be characterized by linear inequalities. For the continuous‐time case, the input–output gain characterization problem turns into the existence problem of a positive periodic vector function. Based on some necessary and sufficient input–output gain conditions, we find the ℓ₁‐ (L₁‐) gain of discrete‐time (continuous‐time) positive periodic systems is equivalent to the of ℓ_∞‐ (L_∞‐) gain of the associated dual systems. Finally, two numerical examples are given to illustrate the results.

中文翻译：

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正周期系统的输入输出增益分析

本文研究的输入输出增益，包括ℓ ₁ -和ℓ _∞ -gains和大号₁ -和大号_∞ -gains，离散时间和连续时间周期正系统。对于离散时间，输入-输出增益可以用线性不等式表征。对于连续时间情况，输入输出增益表征问题变成了正周期矢量函数的存在问题。基于一些必要的和足够的输入输出增益的条件下，我们发现ℓ ₁ - （大号₁ - ）的离散时间（连续时间）正周期系统增益等同于的ℓ_∞ - （大号_∞ - ）相关联的双系统的增益。最后，给出了两个数值示例来说明结果。