## The Hankel Norm

The Hankel norm of a discrete-time system is its largest Hankel singular value

where subscript d denotes a discrete-time system. In Chapter 4 we showed that the discrete-time Hankel singular values converge to the continuous-time Hankel singular values, see (4.23); therefore, the discrete-time Hankel norms converge to the continuous-time Hankel norms when the sampling time approaches zero,

For structures in the modal representation, each mode is independent, thus the norms of a single mode are independent as well (they depend on the mode properties, but not on other modes).

### 5.3.1 The H2 Norm

Define Aai as a half-power frequency at the ith resonance, Aai = , see [18], [33]. This variable is a frequency segment at the ith resonance for which the value of the power spectrum is one-half of its maximal value. The determination of the halfpower frequency is illustrated in Fig. 5.1. The half-power frequency is the width of the shaded area in this figure, obtained as a cross section of the resonance peak at the height of h /V2, where ht is the height of the resonance peak.

Consider the ith natural mode and its state-space representation (Ami, Bmi, Cmi), see (2.52). For this representation we obtain the following closed-form expression for the H2 norm:

Property 5.1. H2 Norm of a Mode. Let Gt (m) = Cmi (jmI - Ami )_1 Bmi be the transfer function of the ith mode. The H2 norm of the ith mode is

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