Quasi $L_{2}/L_{2}$ Hankel Norms and $L_{2}/L_{2}$ Hankel Norm/Operator of Sampled-Data Systems

This article is relevant to appropriately defining the
$L_{2}/L_{2}$
Hankel norm of sampled-data systems through setting a general time instant
${\mathit {\Theta }}$
at which past and future are to be separated and introducing the associated quasi
$L_{2}/L_{2}$
Hankel operator/norm at
${\mathit {\Theta }}$
. We first provide a method for computing the quasi
$L_{2}/L_{2}$
Hankel norm for each
${\mathit {\Theta }}$
, which is carried out by introducing a shifted variant of the standard lifting technique for sampled-data systems. In particular, it is shown that the quasi
$L_{2}/L_{2}$
Hankel norm can be represented as the
$l_{2}/l_{2}$
Hankel norm of a
${\mathit {\Theta }}$
-dependent discrete-time system. It is further shown that an equivalent discretization of the generalized plant exists, which means that the aforementioned discrete-time system can be represented as the feedback connection of the discretized plant and the same discrete-time controller as the one in the sampled-data system. It is also shown that the supremum of the quasi
$L_{2}/L_{2}$
Hankel norms at
${\mathit {\Theta }}$
belonging to a sampling interval is actually attained as the maximum, which means that what is called a critical instant always exists and
the

$L_{2}/L_{2}$
Hankel operator is always definable (as the quasi
$L_{2}/L_{2}$
Hankel operator at the critical instant). Finally, we illustrate those theoretical developments through a numerical example.