Characterization of Quasi L∞/L2 Hankel Norms of Sampled-Data Systems

Abstract

This paper is concerned with the Hankel operator of sampled-data systems. The Hankel operator is usually defined as a mapping from the past input to the future output and its norm plays an important role in evaluating the performance of systems. Since the continuous-time mapping between the input and output is periodically time-varying (h -periodic, where h denotes the sampling period) in sampled-data systems, it matters when to take the time instant separating the past and the future when we define the Hankel operator for sampled-data systems. This paper takes an arbitrary Θ ϵ [0,h) as such an instant, and considers the quasi L∞/L2 Hankel operator defined as the mapping from the past input in L2(-∞, Θ) to the future output in L∞Θ, ∞). The norm of this operator, which we call the quasi L∞/L2 Hankel norm at Θ, is then characterized in such a way that its numerical computation becomes possible. Then, regarding the computation of the L∞L2 Hankel norm defined as the supremum of the quasi L∞L2 Hankel norms over Θ ϵ [0,h), some relationship is discussed between the arguments through such characterization and an alternative method developed in an earlier paper that is free from the computations of quasi L∞/L2 Hankel norms. A numerical example is studied to confirm the validity of the arguments in this paper.