## The H Norm

The Hm norm of a natural mode can be expressed approximately in the closed-form as follows:

Property 5.2. Hx Norm of a Mode. Consider the ith mode (Ami,Bmi, Cmi) or (ai, ^, bmi, cmi). Its Hw norm is estimated as

Proof. In order to prove this, note that the largest amplitude of the mode is approximately at the ith natural frequency; thus,

OVnnx (CmiBmi ) _ llBmi II2 llCmi|I2

The modal Hm norms can be calculated using the Matlab function norm_Hinf.m given in Appendix A.10.

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Figure 5.3. H2 and H„ norms (a) of the second mode; and (b) of the system.

Example 5.2. In this example we illustrate the determination of the Hm norm of a simple structure, as in Example 5.1, and of its modes.

The Hm norm of the second mode is shown in Fig. 5.3(a) as the height of the second resonance peak. The Hm norm of the system is shown in Fig. 5.3(b) as the height of the highest (first in this case) resonance peak. The Hm norms of the modes, determined from the transfer function, are ||Gi|= 18.9229, ||G2|= 1.7454,

||G3||m= 1.2176, and the system norm is \\GL = fGj^ = 18.9619.

5.3.3 The Hankel Norm

This norm is approximately evaluated from the following closed-form formula:

Property 5.3. Hankel Norm of a Mode. Consider the ith mode in the statespace form (Ami, Bmi, Cmi), or the corresponding second-order form (mi, , bmi, cmi). Its Hankel norm is determined from

The modal Hankel norms can be calculated using the Matlab function norm_Hankel.m given in Appendix A.11.

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