Quasi L2/L2 Hankel Norms and L2/L2 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 Theta at which past and future are to be 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, 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 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 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.