ABSTRACT
The relationships between product quality and process variation sources such as fixture errors and machine errors in a multistage manufacturing process are modeled in a linear state space form. They have the same linear model structure because the process faults are assumed to be of smaller magnitude as compared to nominal product dimensions. The chain like state space model can be further transformed into an input-output model and can be generally expressed as:
(10.1)
where
y
is an vector of product quality measurements,
Γ
is an
n
×
p
constant system matrix determined by product/process designs,
u
is a
p
×
1 random vector representing process faults,
ε
is an random vector representing measurement noises, unmodeled faults, and high-order nonlinear terms. Equation 10.1 is equivalent to Equation 9.4. The term
u
in Equation 10.1 is actually in Equation 9.4, and
ε
is actually in Equation 9.4. Without loss of generality, we can further assume that the columns of
Γ
matrix are of unit length. This can be achieved through simple scaling.
