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:

0 0

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