L₂/L₁ induced norm and Hankel norm analysis in sampled-data systems

Abstract

his paper is concerned with the L₂/L₁ induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the h-periodicity of the input-output relation of sampled-data systems is taken into account, where h denotes the sampling period; past and future are separated by the instant θ∈[O, h), and the norm of the operator describing the mapping from the past input in L₁ to the future output in L₂ is called the quasi L₂/L₁ Hankel norm at θ. The L₂/L₁ Hankel norm is defined as the supremum over θ∈[O, h) of this norm, and if it is actually attained as the maximum, then a maximum-attaining θ is called a critical instant. This paper gives characterization for the L₂/L₁ induced norm, the quasi L₂/L₁ Hankel norm at θ and the L₂/L₁ Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function H(φ) on [O, h) plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through H(φ), even though φ is a variable that is totally irrelevant to θ. The relevance of the induced/Hankel norm to the H₂ norm of sampled-data systems is also discussed.