Extensive theoretical/numerical comparative studies on H 2 and generalised H 2 norms in sampled-data systems

This paper is concerned with linear time-invariant (LTI) sampled-data systems (by which we mean sampled-data systems with LTI generalised plants and LTI controllers) and studies their H 2 norms from the viewpoint of impulse responses and generalised H 2 norms from the viewpoint of the induced norms from L 2 to L ∞. A new definition of the H 2 norm of LTI sampled-data systems is first introduced through a sort of intermediate standpoint of those for the existing two definitions. We then establish unified treatment of the three definitions of the H 2 norm through a matrix function G(τ) defined on the sampling interval [0, h). This paper next considers the generalised H 2 norms, in which two types of the L ∞ norm of the output are considered as the temporal supremum magnitude under the spatial 2-norm and ∞-norm of a vector-valued function. We further give unified treatment of the generalised H 2 norms through another matrix function F(θ) which is also defined on [0, h). Through a close connection between G(τ) and F(θ), some theoretical relationships between the H 2 and generalised H 2 norms are provided. Furthermore, appropriate extensions associated with the treatment of G(τ) and F(θ) to the closed interval [0, h] are discussed to facilitate numerical computations and comparisons of the H 2 and generalised H 2 norms. Through theoretical and numerical studies, it is shown that the two generalised H 2 norms coincide with neither of the three H 2 norms of LTI sampled-data systems even though all the five definitions coincide with each other when single-output continuous-time LTI systems are considered as a special class of LTI sampled-data systems. To summarise, this paper clarifies that the five control performance measures are mutually related with each other but they are also intrinsically different from each other.