Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

where the subscript indicates the i th performance metric. In order for the derivative of
Equation 2.15 to equal the derivative of the variance, σ2 z , its derivatives with respect
to both Σq and Λ must be zero. As a result, the Lagrange multiplier is computed by
solving the following equation:
A T Λi +ΛiA + C T C = 0 (2.16)
Equation 2.16 is similar in form to that of Equation 2.13 and is in fact simply another
Lyapunov equation in A T and C instead of A and B.
Then, taking the derivative of Equation 2.15 with respect to a design variable, x
and using well-known matrix identities gives:
∂σz
∂x
= tr
�
Σq
∂ � CT C ��
�
+ tr
∂x
Λi
�
∂A
∂x Σq
∂A
+Σq
T
∂x + ∂ � BB
∂x
T �
��
(2.17)
Equation 2.17 is referred to as the Governing Sensitivity Equation (GSE) and
provides an analytical method of obtaining cost function gradients for design opti-
mizations that use output variance as a performance metric. However, in order to
use this equation it is necessary to calculate the gradients of the state-space matrices.
Recall that these matrices are based on the modal representation of the structure and
therefore, in general, are not explicit functions of the design variables. The design
variables are related to these matrices through the modal quantities, ω and Φ as
follows:
∂A
∂x =
∂B
∂x =
∂C
∂x =
m�
j=1
∂A
∂ωj
m� ∂B
j=1
∂φj
m� ∂C
where the summations are performed over the modes included in the model.
j=1
∂φj
∂ωj
∂x
∂φj
∂x
∂φj
∂x
(2.18)
(2.19)
(2.20)
Since the natural frequencies and mode shapes of the model are obtained through
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