Scholar - SciOpen

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 $Θ \in [0, h)$ , 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 $Θ$ . The $L_{2} / L_{1}$ Hankel norm is defined as the supremum over $Θ \in [0, h)$ of this norm, and if it is actually attained as the maximum, then a maximum-attaining $Θ$ 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 $Θ$ 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 $H (φ)$ 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 $H (φ)$ , even though $φ$ is a variable that is totally irrelevant to $Θ$ . The relevance of the induced/Hankel norm to the $H_{2}$ norm of sampled-data systems is also discussed.