Extensive Theoretical/Numerical Comparative Studies on H2 and Generalised H2 Norms in Sampled-Data Systems

Extensive Theoretical/Numerical Comparative Studies on H2 and Generalised H2 Norms in Sampled-Data Systems
김정훈Tomomichi Hagiwara
sampled-data systems; H2 norm; impulse response
Issue Date
International journal of control
VOL 90, NO 11-2553
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 H2 norms from the viewpoint of impulse responses and generalised H2 norms from the viewpoint of the induced norms from L2 to L. A new definition of the H2 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 H2 norm through a matrix function G(τ) defined on the sampling interval [0, h). This paper next considers the generalised H2 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 H2 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 H2 and generalised H2 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 H2 and generalised H2 norms. Through theoretical and numerical studies, it is shown that the two generalised H2 norms coincide with neither of the three H2 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 relatedwith each other but they are also intrinsically different from each other.
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