Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

values are chosen from their statistical distributions. The standard deviation(s) of
the constraint(s) and/or objective, are then found from the results of the Monte Carlo
analysis.
The formulation of this statistical approach for RPT is as follows:
�
JSR
�� �
min {αf (�x, �p0)+(1− α) σf (�x, �p)}
�x
(3.11)
s.t. �g(�x) ≤ 0
where �p0 denotes the nominal values of the uncertain parameters, α is a relative
weighting and σf is the standard deviation of the performance:
σ 2 f
= 1
N
N�
(f (�x, �pi) − µf) 2
i=1
(3.12)
where N is the number of uncertainty samples chosen to populate the output distri-
bution. The mean, µf, is simply the weighted average of the performance:
µf = 1
N
N�
f (�x, �pi) (3.13)
i=1
As with the other two methods discussed thus far, an uncertainty analysis is neces-
sary at each iteration of the tailoring parameters. If Monte Carlo analysis is used
then the performance at all values in the uncertainty sample space is computed and
Equations 3.13 and 3.12 are used to calculate the performance mean and standard
deviation, respectively.
The gradient of the objective is obtained by differentiating JSR directly and sub-
stituting the derivatives of Equations 3.12 and 3.13 appropriately:
∂JSR
∂x
=
(�x, �p0)
α∂f
∂x
+ (1−α) 1
Nσf
N�
�
∂f (�x, �pi)
(f (�x, �pi) − µf)
−
∂x
1
N
i=1
87
N�
j=i
(3.14)
�
∂f (�x, �pj)
∂x