Chinnam, Ratna Babu "Intelligent Quality Controllers for On-Line Parameter Design"
Computational Intelligence in Manufacturing Handbook
Edited by Jun Wang et al
Boca Raton: CRC Press LLC,2001

©2001 CRC Press LLC

17

Intelligent Quality
Controllers for On-Line

Parameter Design

17.1 Introduction

17.2 An Overview of Certain Emerging Technologies
Relevant to On-Line Parameter Design

17.3 Design of Quality Controllers
for On-Line Parameter Design

17.4 Case Study: Plasma Etching Process Modeling
and On-Line Parameter Design

17.5 Conclusion17.1 Introduction

Besides aggressively innovating and incorporating new materials and technologies into practical, effective,


Tolerance Design Phase —

Involves setting of tolerances on the nominal values of critical design
parameters. Tolerance design is considered to be an economic issue, and the loss function model
promoted by Taguchi can be used as a basis.

Ratna Babu Chinnam

Wayne State University

©2001 CRC Press LLC

Besides the basic parameter design method, Taguchi strongly emphasized the need to perform

robust

parameter design. Here “robustness” refers to the insensitive behavior of the product/process performance
to changes in environmental conditions and noise factors. Achieving this insensitivity at the design stage
through the use of designed experiments is a corner stone of the Taguchi methodology.
Over the years, many distinct approaches have been developed to implement Taguchi’s parameter
design concept; these can be broadly classified into the following three categories:
1. Purely analytical approaches
2. Simulation approaches
3. Physical experimentation approaches.
Due to the lack of precise mechanistic models (models derived from fundamental physics principles)
that explain product/process performance characteristics (in terms of the different controllable and
uncontrollable variables), the most predominant approach to implementing parameter design involves
physical experimentation. Two distinct approaches to physical experimentation for parameter design
include (i) orthogonal array approaches, and (ii) traditional factorial and fractional factorial design


controllable

parameters and

uncontrollable

parameters (note that the word

parameter

is equivalent to the word

factor

or

variable

normally used in parameter design literature).
1.

Controllable Parameters:

These are parameters that can be specified freely by the product/process
designer and or the user/operator of the product/process to express the intended value for the

©2001 CRC Press LLC

response. These parameters can be classified into two further groups:

its material/technology makeup fall under this category.
b.

Non-Fixed Controllable Parameters:

These are controllable parameters that can be freely
changed before or during the operation of the product/process (these factors are also referred
to as

signal

factors in the parameter design literature). For example, the cutting parameters
such as speed, feed, and depth of cut on a machining process can be labeled non-fixed
controllable parameters.
2.

Uncontrollable Parameters:

These are parameters that cannot be freely controlled by the pro-
cess/process designer. Parameters whose settings are difficult to control or whose levels are expensive
to control can also be categorized as uncontrollable parameters. These parameters are also referred
to as

noise factors

in the parameter design literature. These parameters can be classified into two
further groups:

constant


Ω

. However, the resistance of the
individual resistors will deviate from the nominal value affecting the performance of the
individual regulators. Please note that the parameter (i.e., resistance) is to some degree uncon-
trollable; however, the level/amplitude of the uncontrollable parameter for any given individual
regulator remains more or less constant for the life of that voltage regulator.
b.

Non-Constant Uncontrollable Parameters:

These parameters normally represent the environ-
ment in which the product/process operates, the loads to which they are subjected, and their
deterioration. For example, in machining processes, some examples of non-constant uncon-
trollable variables include room temperature, humidity, power supply voltage and current, and
amplitude of vibration of the shop floor.

FIGURE 17.1

Block diagram of a product/process.
Controllable
Parameters
Uncontrollable
Parameters
Fixed
Non-Fixed
Constant
Non-Constant
Product
or

that utilize this additional information will facilitate optimal utilization of the capability of products/pro-
cesses. Pledger [1996] described an approach that explicitly introduces uncontrollable factors into a
designed experiment. The method involves splitting uncontrollable factors into two sets,

observable

and

unobservable

. In the first set there may be factors like temperature and humidity, while in the second
there may be factors such as chemical purity and material homogeneity that may be unmeasurable due
to time, physical, and economic constraints. The aim is to find a relationship between the controllable

FIGURE 17.2

On-line parameter design of a time-invariant product/process.
ResponseResponse
Optimal
Level at T1
Status
at T1
Status
at T0
Optimal
Level at T0
Controllable Variable
Controllable Variable
Uncontrollable Variable
Uncontrollable Variable

f

∈

(R

N

, R

M

) over a
compact subset of R

N

to arbitrary precision [Hornik et al., 1989]. Previous research has also shown
that neural networks offer advantages in both accuracy and robustness over statistical methods
for modeling processes (for example, Nadi et al. [1991]; Himmel and May [1993]; Kim and May
[1994]). However, there is some controversy surrounding this issue.

2. Adaptivity

. Most training algorithms for FFNs are incremental learning algorithms and exhibit a
built-in capability to adapt the network to changes in the operating environment [Haykin, 1999].
Given that most product and processes tend to be time-variant (nonstationary) in the sense that
the response characteristics change with time, this property will play an important role in achieving
on-line parameter design of time-variant systems.
Besides proposing nonparametric neural network models for “modeling” quality response character-

In general, feedforward artificial neural networks (ANNs) are composed of many nonlinear computa-
tional elements, called

nodes

, operating in parallel, and arranged in patterns reminiscent of biological
neural nets [Lippman, 1987]. These processing elements are connected by weight values, responsible for
modifying signals propagating along connections and used for the training process. The number of nodes
plus the connectivity define the topology of the network, and range from totally connected to a topology
where each node is just connected to its neighbors. The following subsections discuss the characteristics
of a class of feedforward neural networks.

17.2.1.1 Multilayer Perceptron Networks

A typical multilayer perceptron (MLP) neural network with an input layer, an output layer, and two
hidden layers is shown in Figure 17.3 (referred to as a three-layer network; normally, the input layer is
not counted). For convenience, the same network is denoted in block diagram form as shown in Figure
17.4 with three weight matrices

W

(1)

,

W

(2)

, and


x

) = 1/(1 +

e

–x

) where 0

≤
γ
(

x

)

≤

1
for –

(

x

) = (1 –

e

–x

)/(1 +

e

–x

) where –1

≤
γ
(


FIGURE 17.3

A three-layer neural network.

FIGURE 17.4

A block diagram representation of a three-layer network.
x
1
{
w
ij
}
{
w
ij
}
{
w
ij
}
x
2
x
N
Input
Layer
Hidden
Layer #1
Hidden

y
p
y
_
Q
y
Q
y
_
1
y
_
1
y
_
1
y
2
y
2
y
2
y
1
y
1
y
1
(1) (1)
(1)

γ
W
(1)
W
(2)
Γ Γ
W
(3)
Γ
xy
(1)
y
(2)
y

©2001 CRC Press LLC

applied

threshold

or

bias

that has the effect of lowering or increasing the net input to the nodal function.
Each layer of the network can then be represented by the operator
Equation (17.1)
and the input–output mapping of the MLP network can be represented by
Equation (17.2)

y

d

(error-correction learning), resulting in a mapping function

N

[

x

]. From a systems theoretic
point of view, multilayer perceptron networks can be considered as versatile nonlinear maps with the
elements of the weight matrices as parameters.
It has been shown in Hornik et al., [1989], using the Stone–Weierstrass theorem, that even an MLP
network with just one hidden layer and an arbitrarily large number of nodes can approximate any
continuous function over a compact subset of to arbitrary precision (universal
approximation). This provides the motivation to use MLP networks in modeling/identification of any
manufacturing process’ response characteristics.

17.2.1.2 Training MLP Networks Using Backpropagation Algorithm

If MLP networks are used to solve the identification problems treated here, the objective is to determine
an adaptive algorithm or rule that adjusts the weights of the network based on a given set of input–output
pairs. An error-correction learning algorithm will be discussed here, and readers can see Zurada [1992]
and Haykin [1999] for information regarding other training algorithms. If the weights of the networks
are considered as elements of a parameter vector

θ

denotes the iteration step.
In the three-layered network shown in Figure 17.3,

x

= (

x

1

, …,

x

N

)

T

denotes the input pattern vector
while

y

= (

y



(1)

) and

y

(2)

= (

y

1
(2)

, …,

y

Q

(2)

)

T

are the outputs at the first and the second hidden layers, respectively. The matrices
and are the weight matrices associated with the three layers








=
[]
() () ()
ΓΓΓΓΓΓΓΓ
321
321
.
fC
NM
∈ℜℜ(, )
ℜ
N
θθη
θ
ss
Js
s
+
()
=
()
∂
()

×






y
1
()
∈ℜ
P
,
y
2
()
∈ℜ
Q
,
y ∈ℜ
M
γ
yy
ii
11
() ()





where the summation is carried out over all patterns in a given training data set S. The factor 1/2 is used
in Equation 17.4 to simplify subsequent derivations resulting from minimization of J with respect to free
parameters of the network.
While strictly speaking, the adjustment of the parameters (i.e., weights) should be carried out by
determining the gradient of J in parameter space, the procedure commonly followed is to adjust it at
every instant based on the error at that instant. A single presentation of every pattern in the data set to
the network is referred to as an epoch. In the literature, a well-known method for determining this
gradient for MLP networks is the backpropagation method. The analytical method of deriving the
gradient is well known in the literature and will not be repeated here. It can be shown that the back-
propagation method leads to the following gradients for any MLP network with L layers:
Equation (17.5)
for neuron i in output layer L Equation (17.5a)
for neuron i in hidden layer l Equation(17.5b)
Here, denotes the local gradient defined for neuron i in layer l and the use of prime in sig-
nifies differentiation with respect to the argument. It can be shown that for a unipolar sigmoid function,
g
'
(x) = x(1 – x) and for a bipolar function, g
'
(x) = 2x(1 – x). One starts with local gradient calculations
for the outermost layer and proceeds backwards until one reaches the first hidden layer (hence the name
backpropagation). For more information on MLP networks, see Haykin [1999].
17.2.1.3 Iterative Inversion of Neural Networks
In error backpropagation training of neural networks, the output error is “propagated backward” through
the network. Linden and Kindermann [1989] have shown that the same mechanism of weight learning
can be used to iteratively invert a neural network model. This approach is used here for on-line parameter
design and hence the discussion. In this approach, errors in the network output are ascribed to errors
in the network input signal, rather than to errors in the weights. Thus, iterative inversion of neural
networks proceeds by a gradient descent search of the network input space, while error backpropagation
training proceeds through a search in the synaptic weight space.

=
∑
1
2
2
e ,
∂
()
∂
()
=
() ()
()
() ( )
Js
ws
sy s
ij
l
i
l
j
l
–
–
δ
1
δγ
i
L

+
()
()
=
()




() ()
∑
'
11
δ
i
l
s
()
()
γ
i
i
L
ys
'
(())
()
©2001 CRC Press LLC
Equation (17.6)
where

as feedback control, feedforward control, and adaptive control. It is the quality controller in the quality
control loop that “determines” these optimal levels, i.e., performs parameter design. The quality controller
includes both a model of the product/process quality response characteristics and an optimization routine
to find the optimal levels of the controllable variables. As was stated earlier, the focus here is on time-
invariant products and processes, and hence, the model building process can be carried out off-line. In
time-variant systems the quality response characteristics have to be identified and constantly tracked on-
line, and call for an experiment planner that facilitates constant and optimal investigation of the prod-
uct/process behavior.
In solving this on-line parameter design problem, the following assumptions are made:
1. Quality response characteristics of interest can be expressed as static nonlinear maps in the input space
(the vector space defined by controllable and uncontrollable variables). This assumption implies that
there exits no significant memory or inertia within the system, and that the process response state
is strictly a function of the “current” state of the controllable and uncontrollable variables. In other
xx
x
ss
Js
s
+
()
=
()
⋅
∂
()
∂
()
1 –
η