Tài liệu Computational Intelligence In Manufacturing Handbook P17 doc

Due to the lack of precise mechanistic models models derived from fundamental physics principlesthat explain product/process performance characteristics in terms of the different control

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

Intelligent Quality Controllers for On-Line

Parameter Design17.1 Introduction

to recognize that it is through engineering design that we have the greatest opportunity to influence theultimate delivery of products and processes that far exceed the customer needs and expectations For the last two decades, classical experimental design techniques have been widely used for settingcritical product/process parameters or targets during design But recently their potential is being ques-tioned, for they tend to focus primarily on the mean response characteristics One particular designapproach that has gained a lot of attention in the last decade is the robust parameter design approach thatborrows heavily from the principles promoted by Genichi Taguchi [1986, 1987] Taguchi views the designprocess as evolving in three distinct phases:

1 System Design Phase — Involves application of specialized field knowledge to develop basic designalternatives

2 Parameter Design Phase — Involves selection of “best” nominal values for the important designparameters Here “best” values are defined as those that “minimize the transmitted variabilityresulting from the noise factors.”

3 Tolerance Design Phase — Involves setting of tolerances on the nominal values of critical designparameters Tolerance design is considered to be an economic issue, and the loss function modelpromoted by Taguchi can be used as a basis

Ratna Babu Chinnam

Wayne State University

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 stagethrough 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 parameterdesign 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 anduncontrollable variables), the most predominant approach to implementing parameter design involvesphysical experimentation Two distinct approaches to physical experimentation for parameter designinclude (i) orthogonal array approaches, and (ii) traditional factorial and fractional factorial designapproaches

The orthogonal array approaches are promoted extensively by Taguchi and his followers, and thetraditional factorial and fractional factorial design approaches are normally favored by the statisticalcommunity Over the years, numerous papers have been authored comparing the advantages and disad-vantages of these approaches Some of the criticisms for the orthogonal array approach include thefollowing [Box, 1985]: (i) the method does not exploit a sequential nature of investigation, (ii) the designsadvocated are rather limited and fail to deal adequately with interactions, and (iii) more efficient andsimpler methods of analysis are available

In addition to the different approaches to “generating” data on product/process performance, thereexist two distinct approaches to “measuring” performance:

1 Signal-to-Noise (S/N) Ratios — Tend to combine the location and dispersion characteristics ofperformance into a one-dimensional metric; the higher the S/N ratio, the better the performance

2 Separate Treatment — The location and dispersion characteristics of performance are evaluatedseparately

Once again, numerous papers have been authored questioning the universal use of the S/N ratiossuggested by Taguchi and many others The argument is that the Taguchi parameter design philosophyshould be blended with an analysis strategy in which the mean and variance of the product/processresponse characteristics are modeled to a considerably greater degree than practiced by Taguchi Numer-ous papers authored in recent years have established that one can achieve the primary goal of the Taguchiphilosophy, i.e., to obtain a target condition on the mean while minimizing the variance, within a responsesurface methodology framework Essentially, the framework views both the mean and the variances asresponses of interest In such a perspective, the dual response approach developed by Myers and Carter[1973] provides an alternate method for achieving a target for the mean while also achieving a target forthe variance For an in-depth discussion on response surface methodology and its variants, see Myersand Montgomery [1995] For a panel discussion on the topic of parameter design, see Nair [1992]

17.1.1 Classification of Parameters

A block diagram representation of a product/process is shown in Figure 17.1 A number of parameterscan influence the product/process response characteristics, and can be broadly classified as 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/processdesigner and or the user/operator of the product/process to express the intended value for the

response These parameters can be classified into two further groups: fixed controllable parametersand non-fixed controllable parameters.

a Fixed Controllable Parameters: These are parameters that are normally optimized by the uct/process designer at the design stage The parameters may take multiple values, called levels,and it is the responsibility of the designer to determine the best levels for these parameters.Changes in the levels of certain fixed controllable parameters may not have any bearing onmanufacturing or operation costs; however, when the levels of certain others are changed, themanufacturing and/or operation costs might change (these factors that influence the manu-facturing cost are also referred to as tolerance factors in the parameter design literature) Onceoptimized, these parameters remain fixed for the life of the product/process For example,parameters that influence the geometry of a machine tool used for a machining process, and/orits material/technology makeup fall under this category

prod-b Non-Fixed Controllable Parameters: These are controllable parameters that can be freelychanged 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 parameterssuch as speed, feed, and depth of cut on a machining process can be labeled non-fixedcontrollable parameters

2 Uncontrollable Parameters: These are parameters that cannot be freely controlled by the cess/process designer Parameters whose settings are difficult to control or whose levels are expensive

pro-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 twofurther groups: constant uncontrollable parameters and non-constant uncontrollable parameters

a Constant Uncontrollable Parameters: These are parameters that tend to remain constant duringthe life of the product or process but are not easily controllable by the product/process designer.Certainly, the parameters representing variation in components that make up the product/pro-cess fall under this category This variation is inevitable in almost all manufacturing processesthat produce any type of a component and is attributed to common causes (representing naturalvariation or true process capability) and assignable/special causes (representing problems withthe process rendering it out of control) For example, the nominal resistance of a resistor to

be used in a voltage regulator may be specified at 100 KΩ However, the resistance of theindividual resistors will deviate from the nominal value affecting the performance of theindividual 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 individualregulator remains more or less constant for the life of that voltage regulator

b Non-Constant Uncontrollable Parameters: These parameters normally represent the ment in which the product/process operates, the loads to which they are subjected, and theirdeterioration For example, in machining processes, some examples of non-constant uncon-trollable variables include room temperature, humidity, power supply voltage and current, andamplitude of vibration of the shop floor

environ-FIGURE 17.1 Block diagram of a product/process.

ControllableParameters

UncontrollableParameters

Fixed Non-Fixed

Constant Non-Constant

ProductorProcess

Response

17.1.2 Limitations of Existing Off-Line Parameter Design Techniques

Whatever the method of design, in general, parameter design methods do not take into account thecommon occurrence that some of the uncontrollable variables are observable during production [Pledger,1996] and part usage This extra information regarding the levels of non-constant uncontrollable factorsenhances our choice of values for the non-fixed controllable factors, and, in some cases, determines theviability of the production process and or the product This process is hypothetically illustrated for atime-invariant product/process in Figure 17.2 Here T0 and T1 denote two different time/usage instantsduring the life of the product/process Given the level of the uncontrollable variable at any instant, thethick line represents the response as a function of the level of the controllable variable Given the responsemodel, the task here is to optimize the controllable variable as a function of the level of the uncontrollablevariable In depicting the optimal levels for the controllable variable in Figure 17.2, the assumption made

is that it is best to maximize the product/process response (i.e., larger the response, the better it is) Thesame argument can be extended to cases where the product/process has multiple controllable/uncontrol-lable variables and multiple outputs In the same manner, it is possible to extend the argument to smaller-is-better and nominal-is-best response cases, and combinations thereof

Given the rapid decline in instrumentation costs over the last decade, the development of methodsthat 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 adesigned 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 secondthere 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.

Optimal Level at T1

Contr ollab

le Variab le

Contr ollab

le Variab le

Uncontr ollab

le Variab le

Uncontr ollab

le Variab le

factors and the observable uncontrollable factors while simultaneously minimizing the variance of theresponse and keeping the mean response on target Given the levels of the observable uncontrollablevariables, appropriate values for the controllable factors are generated on-line that meet the statedobjectives

As is also pointed out by Pledger [1996], if an observable factor changes value in wild swings, it wouldnot be sensible to make continuous invasive adjustments to the product or process (unless there is minimalcost associated with such adjustments) Rather, it would make sense to implement formal control oversuch factors Pledger derived a closed-form expression, using Lagrangian minimization, that facilitatesminimization of product or process variance while keeping the mean on target, when the model thatrelates the quality response variable to the controllable and uncontrollable variables is linear in parametersand involves no higher order terms However, as Pledger pointed out, if the model involves quadraticterms or other higher order interactions, there can be no closed-form solution

17.1.3 Overview of Proposed Framework for On-Line Parameter Design

Here, we develop some general ideas that facilitate on-line parameter design The specific objective is tonot impose any constraint on the nature of the relationship between the different controllable anduncontrollable variables and the quality response characteristics, and allow multiple quality responsecharacteristics In particular, we recommend feedforward neural networks (FFNs) for modeling thequality response characteristics Some of the reasons for making this recommendation are as follows:

1 Universal Approximation FFNs can approximate any continuous function f∈(RN, RM) over acompact subset of RN to arbitrary precision [Hornik et al., 1989] Previous research has also shownthat neural networks offer advantages in both accuracy and robustness over statistical methodsfor 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 abuilt-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 thatthe response characteristics change with time, this property will play an important role in achievingon-line parameter design of time-variant systems

Besides proposing nonparametric neural network models for “modeling” quality response istics of manufacturing processes, we recommend a gradient descent search technique and a stochasticsearch technique for “optimizing” the levels of the controllable variables on-line In particular, we consider

character-a neurcharacter-al network itercharacter-ative inversion scheme character-and character-a stochcharacter-astic secharacter-arch method thcharacter-at utilizes genetic character-algorithmsfor optimization of controllable variables The overall framework that facilitates these two on-line tasks,i.e., modeling and optimization, constitutes a quality controller Here, we focus on development of qualitycontrollers for manufacturing processes whose quality response characteristics are static and time-invari-ant Future research can concentrate on extending the proposed controllers to deal with dynamic andtime-variant systems In addition, future research can also concentrate on modeling the signatures of theuncontrollable variables to facilitate feedforward parameter design

17.1.4 Chapter Organization

The chapter is organized as follows: Section 17.2 provides an overview of feedforward neural networksand genetic algorithms utilized for process modeling and optimization; Section 17.3 describes anapproach to designing intelligent quality controllers and discusses the relevant issues; Section 17.4presents some results from the application of the proposed methods to a plasma etching semiconductormanufacturing process; and Section 17.5 provides a summary and gives directions for future work

17.2 An Overview of Certain Emerging Technologies

Relevant to On-Line Parameter Design

17.2.1 Feedforward Neural Networks

In general, feedforward artificial neural networks (ANNs) are composed of many nonlinear tional elements, called nodes, operating in parallel, and arranged in patterns reminiscent of biologicalneural nets [Lippman, 1987] These processing elements are connected by weight values, responsible formodifying signals propagating along connections and used for the training process The number of nodesplus the connectivity define the topology of the network, and range from totally connected to a topologywhere each node is just connected to its neighbors The following subsections discuss the characteristics

computa-of a class computa-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 twohidden layers is shown in Figure 17.3 (referred to as a three-layer network; normally, the input layer isnot counted) For convenience, the same network is denoted in block diagram form as shown in Figure17.4 with three weight matrices W(1), W(2), and W(3) and a diagonal nonlinear operator Γ with identicalsigmoidal elements γ following each of the weight matrices The most popular nonlinear nodal functionfor multilayer perceptron networks is the sigmoid [unipolar→γ(x) = 1/(1 + e –x) where 0 ≤γ(x) ≤ 1for –∞<x<∞ and bipolar→γ(x) = (1 – e –x )/(1 + e –x ) where –1 ≤γ(x) ≤ 1 for –∞<x<∞] It isnecessary to either scale the output data to fall within the range of the sigmoid function or use a linearnodal function in the outermost layer of the network It is also common practice to include an externally

FIGURE 17.3 A three-layer neural network.

FIGURE 17.4 A block diagram representation of a three-layer network.

HiddenLayer #2

OutputLayer

(1)

(1) (1)

(2)

(3) (2) (2) (2)

(2)

Σ Σ

γ

γ γ γ

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)

The weights of the network W(1), W(2), and W(3) are adjusted (as described in Section 17.2.1.2) to

minimize a suitable function of error e between the predicted output y of the network and a desired

output yd (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, the error-correction learning process involves the

determination of the vector θ*, which optimizes a performance function J based on the output error In

error-correction learning, the gradient of the performance function with respect to θ is computed and

adjusted along the negative gradient as follows:

Equation (17.3)

whereη is a positive constant that determines the rate of learning (step size) and s denotes the iteration step

In the three-layered network shown in Figure 17.3, x = (x1, …, x N)T denotes the input pattern vector

while y = (y1, …, y M)T is the output vector The vectors y(1) = (y1(1), …, y P(1)) and y(2) = (y1(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

as shown in Figure 17.3 Note that the first subscript in weight matrices denotes the neuron in the

next layer and the second subscript denotes the neuron in the current layer The vectors

where , , and are elements of , , and respectively If yd = (y d1,

…, y dM)T is the desired output vector, the output error of a given input pattern x is defined as e = y – yd

Typically, the performance function J is defined as

Equation (17.4)

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 thegradient 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 namebackpropagation) 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” throughthe network Linden and Kindermann [1989] have shown that the same mechanism of weight learningcan be used to iteratively invert a neural network model This approach is used here for on-line parameterdesign 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 neuralnetworks proceeds by a gradient descent search of the network input space, while error backpropagationtraining proceeds through a search in the synaptic weight space

Through iterative inversion of the network, one can generate the input vector, x, that gives an output

as close as possible to the desired output, yd By taking advantage of the duality between the synapticweights and the input activation values in minimizing the performance criterion, the iterative gradientdescent algorithm can again be applied to obtain the desired input vector:

Equation (17.6)

where η is a positive constant that determines the rate of iterative inversion and the superscript refers

to the iteration step For further information, see Linden and Kindermann [1989]

Due to their global convergence behavior, GAs are especially suited for the field of continuous eter optimization [Solomon, 1995] Traditional optimization methods such as steepest-descent, quadraticapproximation, Newton method, etc., fail if the objective function contains local optimal solutions Manypapers suggest (see, for example, Goldberg [1989] and Mühlenbein and Schlierkamp-Voosen [1994])that the presence of local optimal solutions does not cause any problems to a GA, because a GA is amultipoint search strategy, as opposed to point-to-point search performed in classical methods

param-17.3 Design of Quality Controllers for On-Line Parameter Design

The proposed framework for performing on-line parameter design is illustrated in Figure 17.6 In contrast

to the classical control theory approaches, this structure includes two distinct control loops The processcontrol loop “maintains” the controllable variables at the optimal levels, and will involve schemes such

as feedback control, feedforward control, and adaptive control It is the quality controller in the qualitycontrol loop that “determines” these optimal levels, i.e., performs parameter design The quality controllerincludes 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 invariant products and processes, and hence, the model building process can be carried out off-line Intime-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

time-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

words, the response is neither dependent on the history of the levels of the trollable variables nor dependent on past process response history.

controllable/uncon-2 The process time constant is relatively large in comparison with the rate of change of uncontrollable variables The assumption implies that there exists enough time to respond to changes in the levels

of the uncontrollable variables (i.e., perform parameter design) If the rate of change is too high,one ends up constantly chasing the uncontrollable variables and may not be able to enhance

FIGURE 17.5 The fundamental cycle and operations of a basic genetic algorithm.

FIGURE 17.6 Proposed framework for performing on-line parameter design.

Old Population

Evaluation

New Population

Recombination Selection

a) Basic genetic algorithm cycle.

b) Evaluation and contribution to the gene pool.

10 5 1 1 1

ResponseProcess

(or Product)

ProcessControlLoopControllableVariables

ParameterDesignLoopUncontrollableVariablesQuality

Controller

ProcessController

Set Points

product/process quality In fact, if the rate of change of uncontrollable variables is relatively high

in comparison with the process time constant, attempting to perform on-line parameter designmight even deteriorate product/process quality in the long term

3 Uncontrollable variables are observable during production and unit operation The need for this

assumption is rather obvious If the vast majority of significant uncontrollable variables areunobservable during product/process operation, one has to resort to off-line robust parameterdesign strategies

4 Uncontrollable variables are autocorrelated and change smoothly over time This assumption is

critical if one cannot or does not desire to significantly change the levels of the non-fixed lable variables in a more or less random fashion within different ranges For example, in apultrusion process that produces reinforced composite material, normally one cannot quicklychange the temperature of the pultrusion die (given the large mass and specific and thermalconductivity properties of most die materials) However, given “adequate” time, one can controlthe temperature of the die to follow a relatively smooth desired trajectory

control-5 Scales for the controllable variables and response variables are assumed to be continuous This

straint is necessary if one desires to work with gradient search techniques in the non-fixed trollable variable space in performing parameter design The assumption can be relaxed by usingtraditional mixed-integer programming methods and their variants to perform parameter design.Please note that it is certainly possible to perform product/process identification (model building)using neural networks even in the event that certain controllable variables and response variablesare noncontinuous Also, genetic algorithms discussed in Section 17.2.2 are extremely popular fortheir ability to solve mixed-integer programming type problems

con-Throughout the rest of this chapter, the focus will be on on-line parameter design of time-invariantstatic products and processes using artificial neural networks for product/process modeling Performingon-line parameter design of dynamic systems is not as challenging as it appears originally; however, thetask of dealing with time-variant products/processes is truly daunting Future research efforts will focus

on extending the proposed on-line parameter design methods to time-variant dynamic systems.Given the above discussion, the role of a quality controller can be broken into two distinct tasks:product/process identification and product/process parameter design

17.3.1 Identification Mode

Let x = (x1, …, x K , x K+1 , …, x N)T be a column vector of K controllable variables, x1 through x K , and

N-K uncontrollable variables, x K+1 through x N , where K ≤ N Let y = (y1, …, y M)T be a vector of M quality

response characteristics of interest The quality vector y is a function of x1 through x N, and hence can

be written (for time-invariant systems) as

Equation (17.7)

Here, f(x) = (f1(x), …, f M(x))T denotes a column vector of functions, where y i = f i (x) for i = 1, …, M.

In most cases, due to economic, time, and knowledge constraints, there exists no accurate mechanistic

model for f, and it has to be estimated in an empirical fashion We recommend MLPs for modeling f,

given their universal approximation properties [Hornik et al., 1989] and extreme success discussed inthe literature with regard to accurate approximation of complex nonlinear functions [Haykin, 1999]

In contrast to some pattern recognition problems and other function approximation problems, ingeneral, off-line planning, design, and execution of experiments for modeling product/process responsecharacteristics can be very time-consuming and expensive At the initial stage, it is not uncommon tosee fractional factorial designs being utilized for screening significant controllable and uncontrollablevariables Even second phase experiments tend to use some form of a central composite design, typicallyused for empirical modeling of response surfaces The point here is that the size of the data set normally

y=f x( )

available for product/process identification is very limited This makes division of the data set betweentraining and testing more difficult, but does not prevent it As the name implies, a training data set will

be used for training the MLP network to approximate f from Equation 17.7 as follows:

Equation (17.8)such that

Equation(17.9)

for some specified constant ε≥ 0 and a suitably defined norm (denoted by ) The testingdata set will facilitate evaluation of the generalization characteristics of the network, i.e., the ability of

the network to perform interpolations and make unbiased predictions We use an S-fold cross-validation

method [Weiss and Kulikowski, 1991] for designing (i.e., determining the architecture in terms of number

of hidden layers, nodes per different hidden layers, and connectivity) and building MLP models for

approximating quality response characteristics This involves dividing the data set into S mutually sive subsets, using S – 1 subsets for training the network (as discussed in Section 17.2.1.2) and the remaining subset for testing, repeating the training process S times, holding out a different subset for

exclu-each run, and totaling the resulting testing errors The performance of the different network

configura-tions under consideration will be compared using the S-fold cross-validation error, and the network with

the least error will be used for product/process identification Once the optimal configuration has beenidentified, the complete data set can be used for training the final network Several other guidelines arediscussed in the literature regarding selection of potential network configurations and their training, andare not repeated here [Haykin, 1999; Weigand et al., 1992; Solla, 1989; Baum and Haussler, 1989]

17.3.2 On-Line Parameter Design Mode

Once the product/process identification is completed, parameter design can be performed on-line usingthe MLP model, Let yd = (y d1 , …, y dM)T denote the vector of M desired/target quality response characteristics of interest The objective is to determine the optimal levels for the K controllable variables,

x1 through x K, to

minimize

Equation (17.10)for a suitably defined norm (denoted by ) on the output space In Equation 17.10, f(x) denotes the output of the product/process and hence f(x) – yd e d is the difference between the

product/process output and the desired output yd In the absence of any knowledge about f(x), the

objective is to minimize the performance criterion

Equation (17.11)The constraints would be those restricting the levels of the controllable variables to an acceptable domain