Extensive theoretical/numerical comparative studies on <i>H</i><sub>2</sub> and generalised <i>H</i><sub>2</sub> norms in sampled-data systems
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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 <i>H</i><sub>2</sub> norms from the viewpoint of impulse responses and generalised <i>H</i><sub>2</sub> norms from the viewpoint of the induced norms from <i>L</i><sub>2</sub> to <i>L</i><sub>∞</sub>. A new definition of the <i>H</i><sub>2</sub> 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 <i>H</i><sub>2</sub> norm through a matrix function <i>G</i>(τ) defined on the sampling interval [0, <i>h</i>). This paper next considers the generalised <i>H</i><sub>2</sub> norms, in which two types of the <i>L</i><sub>∞</sub> 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 <i>H</i><sub>2</sub> norms through another matrix function <i>F</i>(θ) which is also defined on [0, <i>h</i>). Through a close connection between <i>G</i>(τ) and <i>F</i>(θ), some theoretical relationships between the <i>H</i><sub>2</sub> and generalised <i>H</i><sub>2</sub> norms are provided. Furthermore, appropriate extensions associated with the treatment of <i>G</i>(τ) and <i>F</i>(θ) to the closed interval [0, <i>h</i>] are discussed to facilitate numerical computations and comparisons of the <i>H</i><sub>2</sub> and generalised <i>H</i><sub>2</sub> norms. Through theoretical and numerical studies, it is shown that the two generalised <i>H</i><sub>2</sub> norms coincide with neither of the three <i>H</i><sub>2</sub> 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.
