Compound Noise - MIT Department of Mechanical Engineering

390 J. SINGH ET AL.
improvement to maximum possible improvement is called the improvement ratio. The average of this measure
for all replicates is shown in the eighth column of Table I.
From Table I we can see that compound noise strategy works reasonably well for the Op Amp, PNM, journal
bearing and slider crank, but not nearly as well for the CSTR and the temperature control circuit.
CONDITIONS FOR COMPOUND NOISE TO BE COMPLETELY EFFECTIVE
Hou 5 gave conditions under which a compound noise strategy will predict the robust setting for a system.
One of the conditions for compound noise to work is the existence of extreme settings. The extreme settings
of compound noise are those that maximize and minimize the system’s response to capture noise variations.
However, we found that in some systems (e.g., PNM; Tawfik and Durand 13 ) the compound noise strategy would
work even if extreme settings do not exist. So the conditions outlined in Theorem 4 of Hou 5 are sufficient but
not necessary conditions for compound noise to work.
For strong hierarchy systems the response y can be expressed as
m∑ m∑
y = f(x 1 ,x 2 ,...,x l ) + β j z j +
j=1
j=1 i=1
l∑
γ ij x i z j (1)
where f(x 1 ,x 2 ,...,x l ) is a general function in the control factors x i ,z j (j = 1,...,m)denotes the noise
factors affecting the system, β j denotes the effect intensity of noise factor j and γ ij denotes the effect intensities
of control-by-noise interactions present in the system. Control factors can have two settings, either −1 or1.
(Control factor variability can be represented as separate noise factors.) Noise factors are in the range −1 to1
and are independent, with zero mean and a variance of 1. Noise factors are symmetric about 0. The variance of
y with respect to z j termsisgivenby
( l∑ ( m∑ )
Var z (y) = C + 2 β j γ ij x i +
i=1 j=1
l∑
( m∑ )
γ ij γ kj
)x i x k
j=1
where C is a constant. The robust setting of the system is such that the variance of the response is minimized
with respect to noise factors. It can be represented as
[ ( l∑ ( m∑ ) l∑
( m∑ )]
X ≡ arg min β j γ ij x i + γ ij γ kj
)x i x k (3)
x i
i=1 j=1 j=1
where X is the setting of the x i terms that minimizes the variance.
If we follow Taguchi’s formulation of compound noise, based on the directionality of noise factors, then
responses at two levels of compound noise will be given by
m∑
( l∑
y + = f(x 1 ,x 2 ,...,x l ) + β j + γ ij x i
)sign(β j ) (4)
y − = f(x 1 ,x 2 ,...,x l ) −
j=1
k=1
i