Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

FRF Magnitude: T z to OPD [µm/Nm]
10 4
10 2
10 0
10 −2
10 −2
Unconstrained
with ACS Model
10 −1
10 0
Frequency [Hz]
Figure 2-3: Frequency response function of SCI: disturbance input Tz to performance,
’-.’: without ACS model, ’–’: with ACS.
values are listed in Table 2.2. This method of modeling the ACS is simplistic, but will
suffice for a model of this fidelity. The Fx and Fy transfer functions are not shown in
Figure 2-3 since they are not affected by the rigid body modes.
2.3.2 Performance Gradients
A large number of optimization techniques, such as steepest descent, conjugate gradi-
ent and Newtons method (see Appendix A), require the gradient of the performance
with respect to the design variable, x. In the following discussion the gradients of the
output variance are derived. For a more thorough presentation of these equations the
reader is referred to the thesis by H. Gutierrez [50].
To begin, notice that the performance metric is defined by Equation 2.14 with the
constraint that Σq satisfies Equation 2.13. Therefore, the Lagrangian of the variance
is written by augmenting the expression for the variance with a symmetric Lagrange
multiplier matrix, Λi:
� σ 2 zi
�
L = CiΣqC T i +Λi
48
10 1
�
AΣq +ΣqA T �
T
+ BB
10 2
(2.15)