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

Hagiwara, Tomomichi, Hara, Hiroki

This article is relevant to appropriately defining the L₂/L₂ Hankel norm of sampled-data systems through setting a general time instant Θ at which past and future are to be separated and introducing the associated quasi L₂/L₂ Hankel operator/norm at Θ . We first provide a method for computing the quasi L₂/L₂ Hankel norm for each Θ , 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 L₂/L₂ Hankel norm can be represented as the l₂/l₂ Hankel norm of a Θ -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 L₂/L₂ Hankel norms at Θ belonging to a sampling interval is actually attained as the maximum, which means that what is called a critical instant always exists and the L₂/L₂ Hankel operator is always definable (as the quasi L₂/L₂ Hankel operator at the critical instant). Finally, we illustrate those theoretical developments through a numerical example.

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

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