On well‐definability of the L∞/L2 Hankel operator and detection of all the critical instants in sampled‐data systems

Abstract

Abstract Because sampled‐data systems have h‐periodic nature with the sampling period h, an arbitrary Θ∈[0,h) is taken and the quasi L∞/L2 Hankel operator at Θ is defined as the mapping from L2(−∞,Θ) to L∞[Θ,∞). Its norm called the quasi L∞/L2 Hankel norm at Θ is used to define the L∞/L2 Hankel norm as the supremum of their values over Θ∈[0,h). If the supremum is actually attained as the maximum, then a maximum‐attaining Θ is called a critical instant and the L∞/L2 Hankel operator is said to be well‐definable. An earlier study establishes a computation method of the L∞/L2 Hankel norm, which is called a sophisticated method if our interest lies only in its computation. However, the feature of the method that it is free from considering the quasi L∞/L2 Hankel norm for any Θ∈[0,h) prevents the earlier study to give any arguments as to whether the obtained L∞/L2 Hankel norm is actually attained as the maximum, as well as detecting all the critical instants when the L∞/L2 Hankel operator is well‐definable. This paper establishes further arguments to tackle these relevant questions and provides numerical examples to validate the arguments in different aspects of authors' theoretical interests.

Keywords

• Combinatorial mathematics
• Discrete control systems