Upper/lower bounds of generalized H₂ norms in sampled-data systems with convergence rate analysis and discretization viewpoint

This paper considers linear time-invariant (LTI) sampled-data systems and studies their generalized H₂ norms. They are defined as the induced norms from L₂ to L∞, in which two types of the L∞ norm of the output are considered as the temporal supremum magnitude under the spatial ∞-norm and -norm. The input/output relation of sampled-data systems is first formulated under their lifting-based treatment. We then develop a method for computing the generalized H₂ norms with operator-theoretic gridding approximation. This method leads to readily computable upper bounds as well as lower bounds of the generalized H₂ norms, whose gaps tend to 0 at the rate of 1/√N with the gridding approximation parameter N. An approximately equivalent discretization method of the generalized plant is further provided as a fundamental step to addressing the controller synthesis problem of minimizing the generalized H₂ norms of sampled-data systems. Finally, a numerical example is given to show the effectiveness of the computation method