$L_2/L_1$ induced norm and Hankel norm analysis in sampled-data systems

This paper is concerned with the $L_2/L_1$ induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the $h$-periodicity of the input-output relation of sampled-data systems is taken into account, where $h$ denotes the sampling period; past and future are separated by the instant $\mathrm{\Theta }\in [0,h)$, and the norm of the operator describing the mapping from the past input in $L_1$ to the future output in $L_2$ is called the quasi $L_2/L_1$ Hankel norm at $\mathrm{\Theta }$. The $L_2/L_1$ Hankel norm is defined as the supremum over $\mathrm{\Theta }\in [0,h)$ of this norm, and if it is actually attained as the maximum, then a maximum-attaining $\mathrm{\Theta }$ is called a critical instant. This paper gives characterization for the $L_2/L_1$ induced norm, the quasi $L_2/L_1$ Hankel norm at $\mathrm{\Theta }$ and the $L_2/L_1$ Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function $H(\phi )$ on $[0,h)$ plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through $H(\phi )$, even though $\phi$ is a variable that is totally irrelevant to $\mathrm{\Theta }$. The relevance of the induced/Hankel norm to the $H_2$ norm of sampled-data systems is also discussed.