## Norms of a Structure with a Filter

In structural testing or in controller design a structure is often equipped with a filter. The filter models disturbances or shapes the system performance. In the following we will analyze how the filter addition impacts the structural and modal norms. Consider a filter with a diagonal transfer function F(a). The diagonal F(a) of order i represents the input filter without cross-coupling between the inputs. Similarly, the diagonal F(a) of order r represents the output filter without cross-

coupling between the outputs. Denote by at the magnitude of the filter response at the ith natural frequency a,. = \F (at )| = 4 F >i ) F (at ). (5.28)

¿?crmax(F ) |
¿?crmax(G) | |

dco |
dco |

The filter is smooth if the slope of its transfer function is small when compared to the slope of the structure near the resonance, that is, at the half-power frequency for a = [at -0.5Aai, mi + 0.5Acoi], (5.29)

for i = 1,...,n. Above, <rmax(X) denotes the maximal singular value of X and Aai denotes the half-power frequency at the ith resonance. The smoothness property is illustrated in Fig. 5.8.

5.5.1 The H2 Norm

With the above assumptions the following property is valid:

Property 5.7. H2 Norm of a Structure with a Filter. The norm of a structure with a smooth filter is approximately an rms sum of scaled modal norms n l|2

and the norm of the ith mode with a smooth filter is a scaled norm

where the scaling factor ai is given by (5.28).

Proof. Note that for the smooth filter the transfer function GF preserves the properties of a flexible structure given by Property 2.1; thus,

In the above approximation we used (5.25), the trace commutative property tr(AB) = tr(BA), and the following inequality:

T3 0

3 10

T3 0

3 10

structure | ||

j^1— |
-----J \ h | |

, , 1 |

frequency, rad/s

frequency, rad/s

Figure 5.8. Comparing magnitudes of a transfer function of a structure (solid line) and a smooth filter (dashed line).

Property 5.7 says that a norm of a smooth filter in series with a flexible structure is approximately equal to the norm of a structure scaled by the filter gains at natural frequencies.

For a single-input-single-output system we obtain the following:

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