Properties of the Quasi L2/L2 Hankel Norms and the L2/L2 Hankel Operator of Sampled-Data Systems

Abstract

This paper aims at appropriately defining the $L_{2}/L_{2}$ Hankel operator and $L_{2}/L_{2}$ Hankel norm of sampled-data systems through setting a general time instant $\Theta$ at which the past and future is separated and introducing the associated quasi $L_{2}/L_{2}$ Hankel operator/norm at $\Theta$ . We first provide a method for computing the quasi $L_{2}/L_{2}$ Hankel norm for each $\Theta$ . We then discuss some important properties of the quasi $L_{2}/L_{2}$ Hankel norms over a sampling interval and show that the supremum of these norm is attained, and thus the $L_{2}/L_{2}$ Hankel operator always exists for stable sampled-data systems.