L∞/L2 Hankel norm analysis of linear periodically time-varying systems and detection of critical instants

This paper deals with the Hankel norm analysis of linear periodically time-varying (LPTV) systems, where $L_2$ is taken as the (past) input space and the (future) output is regarded as an element in $L_{\infty }$. This norm is known to be important because of its coincidence with the well-known $H_2$ norm for the special case of single-output linear time-invariant systems. For the period $h$ of LPTV systems, an arbitrary instant $\Theta \in [0,h)$ is first taken to separate past and future, and then the quasi $L_{\infty }/L_2$ Hankel norm at $\Theta$ is defined. A computation method for the quasi $L_{\infty }/L_2$ Hankel norm for each $\Theta$ is further derived, and for the $L_{\infty }/L_2$ Hankel norm defined as the supremum of the quasi $L_{\infty }/L_2$ Hankel norms over $\Theta \in [0,h)$, its alternative and direct computation method is also provided, which is actually completely free from dealing with any quasi $L_{\infty }/L_2$ Hankel norms. A relevant question of whether the supremum is attained as the maximum is also studied, in which case each maximum-attaining $\Theta$ is called a critical instant. In particular, it is discussed when and how the existence/absence of a critical instant can be determined and some or all critical instants can be detected without directly computing all (or any of) the quasi $L_{\infty }/L_2$ Hankel norms over $\Theta \in [0,h)$. Finally, numerical examples are provided to illustrate the arguments of this paper.