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Robust Parameter Design with Feed-Forward Control V. Roshan Joseph School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta, GA 30332-0205 ([email protected]) Accepted by Technometrics. Abstract When there exists strong noise factors in the process, robust parameter design alone may not be effective and a control strategy can be used to compensate for the effect of noise. In this article, a parameter design methodology in the presence of a feed-forward control is developed. In particular, performance measures for evaluating control factor settings in measurement systems, simple response systems, and multiple target systems are developed. Strategies for the design and analysis of experiments are discussed. The approach is illustrated using an example on gold plating. KEY WORDS: Experiments, Optimization, Process control, Quality engineering. 1

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Robust Parameter Designwith Feed-Forward Control

V. Roshan Joseph

School of Industrial and Systems Engineering

Georgia Institute of Technology

Atlanta, GA 30332-0205

([email protected])

Accepted by Technometrics.

Abstract

When there exists strong noise factors in the process, robust parameter design alone

may not be effective and a control strategy can be used to compensate for the effect of

noise. In this article, a parameter design methodology in the presence of a feed-forward

control is developed. In particular, performance measures for evaluating control factor

settings in measurement systems, simple response systems, and multiple target systems

are developed. Strategies for the design and analysis of experiments are discussed. The

approach is illustrated using an example on gold plating.

KEY WORDS: Experiments, Optimization, Process control, Quality engineering.

1

1. INTRODUCTION

Robust parameter design attempts to make the process insensitive to noise by appropri-

ately choosing levels for the control factors. Pioneered by Taguchi (1987) robust parameter

design has been recognized as an important tool for quality improvement. See Wu and

Hamada (2000), Wu and Wu (2000), and Vuchkov and Boyadjieva (2001) for the recent

developments. The robustness is achieved by exploiting the control-by-noise interactions.

Consider the following example, suppose the response y at time t follows the model

yt = 10 + 2x1 − qt − 0.5rt + x2rt + εt, (1)

where (x1, x2) are the control factors and (q, r) the noise factors, and ε is the random error

caused by the other noise factors in the process. For simplicity, let q, r, and ε be independent

with mean 0 and variance 1. Suppose initially the process was set at (x1, x2) = (0,−1) to

achieve the target 10 on average which results in a variance of 4.25. The robust parameter

design solution is to change the setting of x2 to 0.5. We see that by this change the effect of

r on y is removed. Thus the variance of y reduces to 2. Because q does not interact with x1

or x2, the approach cannot reduce/eliminate the variations caused by this noise factor. In

such cases tolerance design is often used as a remedy measure. In this example the tolerance

of q will be tightened so as to reduce the variation in y to a desired level.

An alternative approach is to compensate for the effect of noise. For example, we can

adjust x1 depending on q using the control law x1t = 0.5qt, so that the effect of q on

y is eliminated. Now the variation in y reduces to 1. This approach can be effectively

implemented if q is on-line measurable and x1 is on-line adjustable.

From the above discussions it is clear that the robust parameter design solution works

only if there exists control-by-noise interactions. The control solution does not require any

such conditions and therefore has a much wider applicability. But the control solution needs

on-line measurements of the noise and a controller to implement the control law. This

increases the cost of production. Moreover unlike robust parameter design this is not a

one-time activity and requires continuous monitoring and maintenance. Therefore it is not

2

prudent to jump into control systems without investigating the opportunities for robustness.

A cost-effective strategy would be to use robust parameter design to make the process as

robust as possible and then use a control system or tolerance design approach to further

improve the process.

A two-stage approach for quality improvement using first robust parameter design and

then control systems may not always work well. For example in (1) if V ar(ε) = (1 + x1x2)2,

then the robust parameter design solution will depend on the control law and vice versa. In

such cases we cannot decouple the methodology into two stages and arrive at the optimal

solution.

In many processes there exists strong noise which cannot be de-sensitized by parameter

design. The use of control systems is inevitable in such cases. When such noise factors are

known a priori, the experiments are designed and the optimization is carried out so that the

control law becomes robust to the other noise factors. Taguchi and his co-workers call such

experiments as double-signal experiments. See, for example, Fowlkess and Creveling (1995,

Section 6.5) and Wu and Wu (2000, Chapter 4). Many real case studies can be found in

Taguchi, Chowdhury, and Taguchi (2000, Chapters 4, 9, 14, and 18). The case studies are

quite successful, albeit using rudimentary modeling and optimization techniques.

In this article we develop a general methodology for robust parameter design of systems

with control. The article is organized as follows. In Section 2 we explain the application

of control in the optimization of measurement systems. The connection with the signal-

to-noise ratio analysis is established. In Section 3 we develop the optimization procedure

for feed-forward control in the simple response systems. The robust parameter design of

multiple target systems with feed-forward control is discussed in Section 4. The approach

is illustrated using an example on gold plating in Section 5. Some concluding remarks and

future research directions are given in Section 6.

3

2. MEASUREMENT SYSTEMS

Taguchi, Chowdhury, and Taguchi (2000, Chapter 4) reports a case study on the accuracy

improvement of a disposable oxygen sensor used for open heart surgery. The sensor is used

for measuring the oxygen concentration in the blood. There are several control factors

for this measurement system such as sensor thickness, polymer molecular weight, dye-to-

polymer ratio, over coat size and type, etc. There are several noise factors such as time

at elevated temperature, exposure to ambient light, sensor thickness variation, etc. The

blood temperature is also a noise factor because it varies from operation to operation. It is

possible to continuously measure the blood temperature and give a feed-forward correction

using an algorithm in the monitor’s software. This experiment is different from the usual

robust parameter design experiments because no effort is made to de-sensitize the effect of

blood temperature on the measurements. The objective is to set the levels of the control

factors such that the oxygen concentration measurements become robust to the noise after

compensating for the effect of blood temperature.

We will first explain the robust parameter design without any control and then describe

it with control so that direct comparisons can be made. More details about measurement

systems without control can be found in Miller and Wu (1996) and Wu and Hamada (2000,

Chapter 11).

2.1 Without Control

Let M be the true value of the variable, Y the measured value, X the set of control

factors, and Z the set of noise factors. Assume that Y and M are nonnegative variables.

Let Y = f(X,Z,M). Suppose that there is no zero-correction required in the measurement

system, i.e., Y = 0 when M = 0. Then we can approximate the above relationship in the

operating range of M by

Y = β(X,Z)M. (2)

If the relationship is nonlinear, then adding an intercept term will improve the fit, but will

complicate the optimization. We will discuss the use of intercept term at the end of this

4

section.

Partition Z into {N,U} where N is the observable (or known) set of noise factors and U

the unobservable (or unknown) set of noise factors in the experiment. Let E[β(X,Z)|N] =

β(X,N) and V ar[β(X,Z)|N] = V (X,N). Because U is unobservable we may use the

following additive noise model

Y = β(X,N)M + ε, (3)

where E(ε|N) = 0 and V ar(ε|N) = V (X,N)M2. For the moment, we will assume that the

functions β and V are known to the experimenter and will derive the performance measures

for optimization. In real practice these functions are not known and are to be estimated from

the data by conducting experiments. We will consider the issue of estimation in Section 2.4.

From an observed value of Y , M can be estimated as

M =Y

E[β(X,N)]. (4)

Let E[β(X,N)] = β(X). Then the variation in the estimate is

V ar(M) =V ar(Y )

β2(X). (5)

The V ar(Y ) can be obtained as follows:

V ar(Y ) = E[V ar(Y |N)] + V ar[E(Y |N)]

= E[V (X,N)M2] + V ar[β(X,N)M ]

= σ2(X)M2,

where σ2(X) = E[V (X,N)] + V ar[β(X,N)]. Thus

V ar(M) =σ2(X)M2

β2(X).

Letting D be the feasible region for X, our objective is to find an X ∈ D to minimize the

variance of M . This can be done by maximizing the signal-to-noise (SN) ratio given by

SN(X) =β2(X)

σ2(X). (6)

5

Note that the V ar(M) is a function of M . In this particular case maximizing SN ratio

will minimize the variance for all the values of M . For some other models this need not be

the case and one may have to average the V ar(M) over the distribution of M to get the

performance measure. The approach can be summarized as follows

1. Find X∗ ∈ D to maximize SN(X) in (6).

2. Estimate M = Y/β(X∗).

It is pointed out that the SN ratio in (6) is different from Taguchi’s signal-to-noise

ratio, because the underlying modeling assumptions are different. Joseph and Wu (2002a,b)

has shown the validity of model (3) in multiple target systems and we adopt this for the

measurement systems.

Now consider a model with an intercept term. Let

Y = β0(X) + β1(X,N)M + ε,

where E(ε|N) = 0 and V ar(ε|N) = V (X,N)M2. We will continue to assume that the true

relationship between Y and M passes through the origin. The β0(X) is introduced to get a

better fit to the true model in the operating range of M . A reasonable estimator for M is

M = max

{0,

Y − β0(X)

β1(X)

},

where β1(X) = E[β1(X,N)]. Because of the nonlinearity in the estimator, obtaining an

explicit expression for the performance measure is difficult. But if P (Y ≤ β0(X)) is small

in the operating range of M , then we can take the estimator as M = (Y − β0(X))/β1(X).

Now the variance of M can be obtained as before. It is easy to show that the required

performance measure is the same as the SN ratio in (6) with β(X) replaced by β1(X).

2.2 With Control

We may not be able to compensate for all of the N in the usual operation of the system.

So divide N into {Q,R}, where we call Q the on-line noise factors and R the off-line

noise factors. The on-line noise factors can be easily measured and compensated during

6

the operation of the system. They have a large effect on the response, otherwise it is not

worthwhile to apply a compensation strategy on them. In the oxygen sensor example Q is the

blood temperature. The off-line noise factors are systematically varied in the experiment

but are unobserved during the operation of the system. The use of off-line noise factors

improves the efficiency of parameter design experiments. See the discussions in Berube and

Nair (1998) and Steinberg and Bursztyn (1998).

Consider the same model as in (3),

Y = β(X,Q,R)M + ε, (7)

where E(ε|Q,R) = 0 and V ar(ε|Q,R) = V (X,Q,R)M2. Because Q is on-line measurable,

a feed-forward correction will be applied based on the values of Q. A measurement system

with control is diagrammatically depicted in Figure 1. Feed-back control is not possible in

measurement systems, because the true value of the variable is not known during the usual

operation of the system.

M - β

?

X

- Y - 1/E(β) - M

6Z

6 6N U

6 6R Q

6

Figure 1: Measurement System with Control

Let E[β(X,Q,R)|Q] = β(X,Q). Assume that β(X,Q) > 0 with probability 1. Then

from an observed value of Y , M can be estimated as

M =Y

β(X,Q). (8)

Notice that unlike in (4), the slope in (8) is allowed to change with the values of the on-line

noise. To obtain the performance measure for optimization we need to compute the variance

7

of M . To do this we proceed as follows. First note that M is an unbiased estimator of M ,

because

E(M |Q) =E(Y |Q)

β(X,Q)=

E[β(X,Q,R)|Q]M

β(X,Q)= M. (9)

For a given value of Q the variance is

V ar(M |Q) =V ar(Y |Q)

β2(X,Q)=

σ2(X,Q)M2

β2(X,Q)(10)

because

V ar(Y |Q) = E[V ar(Y |Q,R)|Q] + V ar[E(Y |Q,R)|Q]

= E[V (X,Q,R)M2|Q] + V ar[β(X,Q,R)M |Q]

= σ2(X,Q)M2,

where σ2(X,Q) = E[V (X,Q,R)|Q] + V ar[β(X,Q,R)|Q].

Using (9) and (10) we have

V ar(M) = E[V ar(M |Q)] + V ar[E(M |Q)]

= E

[σ2(X,Q)M2

β2(X,Q)

]+ V ar[M ]

= E

[σ2(X,Q)

β2(X,Q)

]M2.

Call SN(X,Q) = β2(X,Q)/σ2(X,Q), the signal-to-noise ratio for a given value of Q. Thus

the variance of M can be minimized by minimizing E{1/SN(X,Q)} for all values of M . An

equivalent performance measure to maximize is

PM(X) = 1/E{1/SN(X,Q)}= 1/E

{E[V (X,Q,R)|Q] + V ar[β(X,Q,R)|Q]

E2[β(X,Q,R)|Q]

}(11)

The approach can be summarized as

1. Find X∗ ∈ D to maximize PM(X) in (11).

2. Estimate M = Y/β(X∗,Q).

8

The signal-to-noise ratio in (6) can be written as

SN(X) =E2[β(X,Q,R)]

E[V (X,Q,R)] + V ar[β(X,Q,R)]. (12)

Comparing with (11) we see that the performance measure for optimization with control is

different.

2.3 Design of Experiments

We can think of the whole approach as to evaluate the performance of a control system at

different levels of control factor settings and to choose a setting that optimizes a performance

measure of the system. Thus a cross array design between X and the other factors is

intuitively appealing. Let D(a) be the design for the factors a. Let D(a)⊗D(b) denotes the

cross array design between two sets of factors a and b, which means that all the runs in D(b)

are repeated at each run of D(a). Then the cross array for the robust parameter design of

measurement systems with control is given by

D(X)⊗D(R)⊗D(Q)⊗D(M). (13)

If the model in (7) is true, then we do not need to vary M in the experiment and the functions

β and V can be estimated from the data by conducting the experiments at some fixed value

of M . However, using a design for M with at least two levels will allow us to verify this

modeling assumption and to possibly explore more elaborate models. The case studies in

Taguchi, Chowdhury, and Taguchi (2000) follow the design of experiments in (13). Because

of crossing of many factors, such experiments are much larger than the usual experiments

without a control system. One approach to reduce the number of runs is to use the noise

factor compounding technique (Taguchi, 1987). Because on-line noise has a special meaning

in experiments with control systems, the compounding should be applied only on R and not

on Q. Another approach to reduce the number of runs is to use single arrays instead of cross

arrays. Here apart from the main effects we are interested in the interaction between the

control factors and the off-line noise. Therefore we may reduce the design in (13) to

D(X,R,Q)⊗D(M). (14)

9

See Wu and Hamada (2000) and Wu and Zhu (2003) for the optimal selection of single arrays.

Their design criterion may require some modification for the use in control systems because

of the presence of two types of noise factors. Among the two-factor interactions, control ×off-line noise interactions are most important. The control × on-line noise interactions and

the control × control interactions will take the second and third places. The interactions

among the noise factors are least important. The three-factor interaction control × control

× off-line noise should get the same importance as that of control × control interactions.

The experiment should be designed in such a way that the important effects are at least

estimable. This task is more challenging in control systems because of the different kinds of

factors.

If the experimenter is not sure before conducting the experiment which noise factors

are to be compensated, then (13) should be used for the experiment. This will enable

the experimenter to identify the on-line noise from the set of noise factors considered in the

experiment. When the noise factors are of the inner noise type (variations around the nominal

value of the factor), they need not be included in the experiment. This can considerably

reduce the run size of the experiment. See Vuchkov and Boyadjieva (2001) for a detailed

treatment of robust parameter design with inner noise. The example to be presented in

Section 5 is of this kind.

2.4 Estimation

Let yijklp be the measured value of the characteristic at Xi,Qj,Rk,Ml, and replicate

p. Let Y ′ = Y/M . Consider the model in (7). Assume that the errors are indepen-

dent and follow normal distribution. Then y′ijklp = β(Xi,Qj,Rk) + ε′ijklp, where ε′ijklp ∼N(0, V (Xi,Qj,Rk)). Express the β(X,Q,R) and log V (X,Q,R) as linear models in X, Q,

and R. An iterative algorithm for obtaining maximum likelihood estimates (MLE) is given

below.

1. Initialize β(Xi,Qj,Rk) = y′ijk.. = 1LP

∑Ll=1

∑Pp=1 y′ijklp .

2. Compute s2ijk = 1

LP

∑Ll=1

∑Pp=1{y′ijklp − β(Xi,Qj,Rk)}2, where β(Xi,Qj,Rk) is the

10

predicted value of β at Xi,Qj, and Rk. Using s2ijk as the response, estimate the

parameters in log V (X,Q,R) using a gamma generalized linear model (GLM) with log

link.

3. Fit y′ijklp ∼ β(Xi,Qj,Rk) using weighted least squares with weights 1/V (Xi,Qj,Rk),

where V (Xi,Qj,Rk) is the predicted value of V at Xi,Qj, and Rk.

4. Repeat steps 2 and 3 until convergence.

This gives the MLE of the two functions β(X,Q,R) and V (X,Q,R). By plugging them

in (11) we get the MLE of the performance measure. If LP = 1, then the algorithm needs

some modification. For this case start with step 3 by initializing V (Xi,Qj,Rk) = 1.

If one is uncertain about the mean model, then it is better to omit the iterations between

steps 2 and 3. By doing so we can avoid the transmission of modeling errors in the mean

model to the variance model. The maximum likelihood approach is known to under esti-

mate the variance function because it does not adjust for the loss of degrees of freedom for

estimating the mean function. Suppose n is the number of parameters in β(X,Q,R). Then

we can inflate the s2ijk in the step 2 by a factor IJKLP/(IJKLP − n) to reduce the bias.

This method produces approximate restricted maximum likelihood estimate (See McCullagh

and Nelder, 1989, Section 10.5). If the distribution of ε is unknown, then one can use an

extended quasi-likelihood criterion for estimating the parameters by specifying appropriate

link and variance functions .

The modeling approach used so far is known as response modeling (Wu and Hamada,

2000, Chapters 10 and 11). There is another type of modeling that is commonly used in

parameter design literature, which is known as performance measure modeling. The estima-

tion procedure for performance measure modeling is different and is discussed below. By

absorbing R into U we can reduce the model in (3) to

Y = β(X,Q)M + ε, (15)

where E(ε|Q) = 0 and V ar(ε|Q) = σ2(X,Q)M2. In contrast to response modeling, here β

and log σ2 are not expressed as linear models in X and Q. Instead they are directly estimated

11

at each combination of Xi and Qj. To emphasize this difference we will denote β(Xi,Qj)

by βij and σ2(Xi,Qj) by σ2ij. We have

βij = y′ij... and σ2ij =

1

KLP

K∑

k=1

L∑

l=1

P∑

p=1

(y′ijklp − βij)2.

Then ˆSN ij = β2ij/σ

2ij. If the levels of Q in the experiment can be considered as a represen-

tative sample from its distribution, then the performance measure in (11) can be estimated

as

ˆPM i =

1

J

J∑

j=1

1ˆSN ij

−1

. (16)

This shows that the performance measure for optimization in a measurement system with

control is the harmonic mean of the SN ratios estimated at each of the on-line noise factor

levels. The log PM can be expressed as a linear model in X and estimated using ordinary

least squares. We note that if one intends to use the performance measure modeling, then

the experimental design in (14) should not be used. Performance measure modeling requires

crossing of the on-line noise factors with the other factors. Therefore the design in (13) or a

design of the type D(X,R)⊗D(Q)⊗D(M) should be used.

The model in (15) makes sense only if R has random levels in the experiment, which

is not the case in most robust parameter design experiments. Therefore the performance

measure modeling will not be as statistically efficient as the response modeling approach.

Also the performance measure modeling needs a cross array design for the experiment,

whereas response modeling can be used with all kinds of designs. Although the response

modeling approach looks superior, the performance measure modeling has some advantages

in control systems. The response modeling requires explicit modeling of Q. Therefore any

modeling errors will be transmitted to (11) and may affect the result. In performance measure

modeling ˆPM i in (16) can be estimated without explicit modeling of on-line noise and

therefore is insensitive to modeling inaccuracies. Once the optimal solution X∗ is obtained,

more data can be collected keeping X fixed at X∗ and the response can be modeled with

respect to Q to obtain the control law. This approach is particularly useful when β has a

nonlinear relationship with Q.

12

3. SIMPLE RESPONSE SYSTEMS

Suppose the quality characteristic of interest has a single target. In parameter design

literature such a variable is known as nominal-the-best characteristic. The robust parameter

design with such characteristics is refered to as static parameter design (Taguchi, 1987) or

parameter design in simple response systems (Miller and Wu, 1996). As an example, consider

the hypo chlorite bleaching process of rayon grade pulp. In this bleaching process the pulp is

treated with hypo chlorite solution to reduce the pulp viscosity to a desired level. There are

several control factors in the process, such as pH of the pulp, reaction temperature, reaction

time, etc. The noise factors include pulp consistency, pulp brightness, lignin content, etc.

Also there is a large variation in the viscosity of the input pulp which is uncontrollable at the

hypo chlorite bleaching stage. If the input viscosity is large, the output viscosity will also

be large unless some corrective action is taken in the process. It is a futile exercise trying to

make the process robust against this particular noise factor and the use of control system is

inevitable. Usually the hypo chlorite solution dosage rate is adjusted to achieve the required

output viscosity. The dosage can be adjusted depending on the values of the input viscosity

or depending on the values of the output viscosity. This corresponds to the feed-forward and

feed-back control schemes. In this article we only consider the case of feed-forward control.

See Figure 2.

Let Y be the output quality characteristic and C the controlling variable. A controlling

variable is a control factor that is selected to continuously adjust to compensate for the effect

of noise. It is that control factor which is easiest to adjust on-line and has a large impact

on the output. For the other control factors (X), once their levels are chosen, they remain

fixed during the operation of the process.

Consider the model

Y = f(X,Q,R, C) + ε, (17)

where ε is the error caused by the unobserved noise factors (U), with E(ε|Q,R) = 0 and

V ar(ε|Q,R) = V (X,Q,R, C). The above relationship should be viewed as the steady state

relationship between the input and the output and therefore we will avoid indexing the

13

C - System

?

X

- Y

6Z

6 6N U

6 6Q R

6

Figure 2: Simple Response System with Feed-Forward Control

variables with time t. As done in Section 2, we will assume that the functions f and V are

known and derive the performance measure. In practice these functions will be estimated

from the data and will be plugged-in to obtain the estimate of the performance measure.

The design of experiments will be similar to those discussed in Section 2.3 with M replaced

by C.

Let T be the target value of the quality characteristic. Suppose we use a quadratic loss

function to measure the quality of the output, L = (Y − T )2. Then the expected loss for

given Q and R is

E(L|Q,R) = (f(X,Q,R, C)− T )2 + V (X,Q,R, C).

Based on the observed value of Q, we can adjust the value of C to minimize the expected

loss averaged over the distribution of R given Q. This gives us the control law:

C∗ = arg minC

E[E(L|Q,R)|Q] = arg minC

E(L|Q). (18)

Thus C∗ is a function of X and Q. Let C∗ = h(X,Q). Cost considerations or system

limitations may require that the value of C be between CL and CH . In such cases the above

optimization should be performed by restricting C in [CL, CH ]. Now Q is a set of noise

factors and their variations during the operation of the process are uncontrollable. So we

will select an X to minimize the expected loss at C = C∗ taken over the distribution of Q.

14

Thus our objective is to find an X ∈ D to minimize

PM(X) = E(L∗) = E{E[L∗|Q]} = E{E[E(L∗|Q,R)|Q]} (19)

= E{E[(f(X,Q,R, h(X,Q))− T )2|Q]}+ E{E[V (X,Q,R, h(X,Q))|Q]}.

We call PM(X) a performance measure independent of control (PerMIC). This is an exten-

sion of the concept of performance measure independent of adjustment (PerMIA) to control

systems. See Leon, Shoemaker, and Kacker (1987) and Leon and Wu (1992) for details

on PerMIA. The PerMIC can be compared to a PerMIA by treating C as an adjustment

parameter. They are different because of the presence of on-line noise factors. It is easy to

show that

E(L∗) = E{minC

E(L|Q)} ≤ minC

E{E(L|Q)} = E(L∗o),

where E(L∗o) is the minimum expected loss without control. Here E(L∗) is a PerMIC and

E(L∗o) is a PerMIA. The above inequality shows that instituting a control system will improve

the performance of a system. But it is beneficial to use a control system only if the reduction

in the quality loss E(L∗o)− E(L∗) is much larger than the cost of implementing the control

system. The approach can be summarized as follows:

1. Find X∗ ∈ D to minimize PM(X) in (19).

2. Adjust C using the control law C = h(X∗,Q).

For on-line implementation, the control law will be written as Ct = h(X∗,Qt). This con-

trol law is obtained by assuming the model in (17). Therefore the control law is meaningful

only if the process is in a state of statistical control. If the process goes out-of-control due

to some special causes, the above control law may no longer be valid. Therefore statistical

process control techniques should be employed to check the stability of the process and to

take corrective actions (see, Montgomery, 2001).

To simplify the computations in (18) we may decide to adjust C to achieve the mean

of Y at target. In engineering such an unbiased adjustment strategy makes a lot of sense.

Then C∗ = h(X,Q), if exists, is a solution of C from

E[f(X,Q,R, C)|Q] = T. (20)

15

For example, in the pulp bleaching process the hypo chlorite solution dosage rate is adjusted

depending on the input viscosity so as to get the output viscosity at target on an average.

Substituting (20) in (19) we obtain

PM(X) = E{V ar[f(X,Q,R, h(X,Q))|Q]}+ E{E[V (X,Q,R, h(X,Q))|Q]}. (21)

The performance measure can now be interpreted as the variation in Y after the control

action. In general, the performance measure with an unbiased adjustment is different from

the PerMIC in (19). An example is given below.

Suppose f(X,Q,R, C) = β(X,Q,R)C and V (X,Q,R, C) = V (X,Q,R)C2. Then from

(18)

C∗ = h(X,Q) =T

E[β(X,Q,R)|Q]

SN(X,Q)

1 + SN(X,Q),

where

SN(X,Q) =E2[β(X,Q,R)|Q]

E[V (X,Q,R)|Q] + V ar[β(X,Q,R)|Q],

and from (19) we obtain the PerMIC,

PM(X) ≡ E

{1

1 + SN(X,Q)

}.

Now suppose we use an unbiased adjustment strategy. Then from (20)

C∗ = h(X,Q) =T

E[β(X,Q,R)|Q].

We see that the optimal adjustment is smaller from the unbiased adjustment by a factor

SN(X,Q)/(1 + SN(X,Q)). A similar shrinkage property was noted by Leon, Shoemaker,

and Kacker (1987) in the case of static parameter design without control. From (21) we

obtain the performance measure

PM(X) ≡ E

{1

SN(X,Q)

}, (22)

which is similar to (11) in a measurement system. We see that this performance measure is

different from the PerMIC.

The robust parameter design of simple response systems with control was also studied

by Pledger (1996) and Mevik et.al. (2001). In their approach, the loss is not averaged over

16

the on-line noise distribution to find X∗ and therefore X∗ is a function of Q. This approach

is difficult to implement in practice particularly when there are many control factors in the

system as all of them will have to be varied with the on-line noise. To overcome this situation

Berget and Naes (2002a,b) have proposed obtaining robust parameter design solution for

some specified categories and an optimal sorting for the raw materials (the on-line noise)

for selecting the categories. Our approach is different because only one control factor (C) is

selected from the set of control factors and varied with the on-line noise, which is the most

commonly used form of control in practice. The other control factors remain fixed at the

robust parameter design X∗. This is probably a better approach in continuous processes or

when there is too much of fluctuations in the on-line noise as only one factor need to be

adjusted frequently. However, in batch processes, one may consider using more than one

controlling variable.

4. MULTIPLE TARGET SYSTEMS

The optimization of multiple target systems (also known as dynamic parameter design)

comprises an important class of problems in robust parameter design. A detailed description

of it can be found in Miller and Wu (1996), Tsui (1999), and Joseph and Wu (2002a,b). As

an example, consider the injection molding process (Wu and Wu, 2000, page 102). Because

of the diversity of products, the products will have different dimensions. Depending on the

customer requirements, the different part dimensions can be achieved by changing the mold

dimensions. Fine adjustments to compensate for the various noise factors in the process

can be done through the injection pressure. Thus, here the mold dimension is the signal

factor (M) and the injection pressure is the controlling variable (C). In this article we only

consider the case of feed-forward control, see Figure 3.

Let

Y = f(X,Q,R,M,C) + ε, (23)

where E(ε|Q,R) = 0 and V ar(ε|Q,R) = V (X,Q,R,M, C). Suppose for a given customer

intent (T ) the loss is measured using the quadratic loss function, L = (Y − T )2. The

17

T©©*HHj

M -

C -System

?

X

- Y

6Z

6 6N U

6 6Q R

6

Figure 3: Multiple Target System with Feed-Forward Control

performance measure for multiple target systems without control is discussed in Joseph and

Wu (2002a,b). Using the concept of PerMIC this can be extended for multiple target systems

with control as

PM(X) =∫

E(L∗) dF (T ) =∫

minM

E{minC

E(L|Q)} dF (T )

=∫

E{(f(X,Q,R,M∗, C∗)− T )2}+ E{V (X,Q,R,M∗, C∗)} dF (T ), (24)

where F (T ) is the distribution of the customer intent. Thus we have the optimal control law

C∗ = arg minC

E(L|Q) (25)

and the optimal signal adjustment

M∗ = arg minM

E{minC

E(L|Q)}. (26)

Let C∗ = h(X,Q,M, T ) and M∗ = g(X, T ). The approach can be summarized as

1. Find X∗ ∈ D to minimize PM(X) in (24).

2. Adjust M depending on T as M = g(X∗, T ).

3. Adjust C depending on Q and T as C = h(X∗,Q, g(X∗, T ), T ).

18

In many systems there may not exist a unique solution to (25) and (26). In such cases

M∗ can be determined by fixing C at some value. Also the optimal adjustments in (25) and

(26) can be replaced by unbiased adjustments if they exist. We illustrate these situations

with two examples.

Example 1: Let f(X,Q,R,M, C) = β(X,Q,R)MC and V (X,Q,R,M, C) = V (X,Q,R)M2C2.

Clearly there is no unique solution to (25) and (26). Suppose [CL, CH ] is the preferred in-

terval for adjusting C. Then we will find the signal adjustment by fixing C at some value in

the interval, say, C0 = (CL + CH)/2. For unbiased adjustment

M∗ =T

C0E[β(X,Q,R)].

Again using unbiased adjustment for C, we obtain C∗ = T/E[β(X,Q,R)M |Q]. At M = M∗,

C∗ = C0E[β(X,Q,R)]

E[β(X,Q,R)|Q].

Note that here C∗ does not depend on T . The performance measure in (24) evaluated at

the above M∗ and C∗ becomes

PM(X) ≡ E

{E[V (X,Q,R)|Q] + V ar[β(X,Q,R)|Q]

E2[β(X,Q,R)|Q]

}= E

{1

SN(X,Q)

}, (27)

which is the as the same performance measure in (22). It is important to note that in this par-

ticular case we do not need to know the distribution of T to obtain the performance measure.

Example 2: In many systems, the signal factor and the controlling variable are the same.

For example, in an automobile, for the steering system the rotation of the steering wheel can

be used for both signal and control; for the braking system the force applied on the pedal can

be used for both signal and control. Such a factor can be represented as M + C, where M

denotes the initial adjustment to achieve the target and C represents the fine adjustments

of the factor around its initial value. Consider the following case: f(X,Q,R,M,C) =

β(X,Q,R)(M + C) and V (X,Q,R,M, C) = V (X,Q,R)(M + C)2. Here M should be

determined by fixing C = 0. Thus, for unbiased adjustments M∗ = T/E[β(X,Q,R)] and

19

C∗ = T/E[β(X,Q,R)|Q]− T/E[β(X,Q,R)]. The performance measure is the same as the

one in (27). It is pointed out that for automobiles, in practice, the driver will use feed-back

control for adjustments, a topic that is not developed in this article.

5. AN EXAMPLE

We illustrate the approach using a case study in Taguchi and Wu (1989) on the opti-

mization of gold plating process to achieve uniform thickness. The customer is interested

only in a single target for the thickness and therefore this problem belongs to the class of

simple response systems. The factors and levels of the experiment are given in Table 1.

The original experiment does not deal with any control system. But the factors such as

gold concentration (X1), temperature(X2) , pH (X3), and nickel concentration (X9) cannot

be maintained at a constant value. In this analysis we will explicitly consider their inner

noise, i.e. the variation around their nominal values. We can represent such a variable by

X = x + e, where x denotes the nominal value which can be controlled and e denotes the

error in the variable around its nominal value. The e will either belong to Q or R depending

on whether we can compensate for its variations during the usual operation of the process.

As mentioned in Taguchi and Wu (1989, page 233) the plating time, which is not considered

in this experiment, can be used to vary the plating thickness. So we will take the plating time

as the controlling variable. Let C0 be the existing plating time. We assume the following

model

Y = β(X,Q,R)C/C0 + ε, (28)

where ε follows a normal distribution with mean 0 and variance V (X,Q,R)(C/C0)2. The

model can be justified using Faraday’s law on electro-plating. For the variance term we

have neglected the variations introduced during the plating time (0, C). A more elaborate

model for the variance in plating is studied in Joseph and Wu (2002a). The data for a 29−5III

experiment is given in Table 2.

The model (28) can be estimated as follows. Because the whole experiment is conducted

at C = C0, the data is from the model Y = β(X,Q,R)+ε, where ε|Q,R ∼ N(0, V (X,Q,R)).

20

Table 1: Factors and levels

Factors Notation Levels

− +

Gold concentration X1 0.725 oz/gal 1.125 oz/gal

Temperature X2 100 oF 110 oF

pH X3 4.25 4.40

Current density x4 1.0 amps/ft2 2.0 amps/ft2

Barrel speed x5 18 rev/min 9 rev/min

Load size x6 1/3 full 1/2 full

Anode size x7 1/4′′ × 3′′ 1′′ × 3′′

Anode placement x8 max. distance 1/2 way to barrel

Nickel concentration X9 620 ppm 700 ppm

Now compute the mean Y and variance s2 for each run. Then using s2 as the response and

by using a gamma GLM with log-link we get the following model for log-variance

log V (X,Q,R) = 4.1538 + 0.5741x4 + 0.5001x6.

The two levels of the variables are coded as -1 and 1. Using Y as the response and by using

weighted least squares with weights equal to 1/V (X,Q,R) we get the model for mean as

β(X,Q,R) = 63.6791 + 2.4657X2 + 5.4289X3 + 1.6364x4 − 1.5906x7.

As discussed in Section 2.4, by iterating the above two steps we will get the MLEs, but we

stop at this stage to avoid the carry over of the lack-of-fit term in the mean model to the

variance model. We see that, of the four random factors, only X2 and X3 appear in the

models. Suppose we decide to compensate for the variations in X3. Following our notations,

let X2 = x2 + r2 and X3 = x3 + q3. Thus, we get

β(X,Q,R) = 63.6791 + 2.4657(x2 + r2) + 5.4289(x3 + q3) + 1.6364x4 − 1.5906x7

21

Table 2: Data of gold plating experiment

Run X1X9x4X2x7x8x5X3x6 Gold plating thickness in micro inches

1 −−−−−−−−− 63 52 57 60 51 64 57 56 61 57 58 51 54 60 54 64 50 54 55 49

2 −−−−+ + + + + 59 60 63 61 58 66 53 63 57 61 61 60 73 81 59 62 68 53 60 72

3 −−+ +−−−+ + 60 66 78 83 64 76 89 106 68 78 52 86 91 69 71 57 73 96 83 60

4 −−+ + + + +−− 67 77 66 51 53 55 59 58 62 62 81 76 60 58 53 59 55 54 70 60

5 −+−+−−+−+ 71 74 78 63 62 67 47 69 49 58 54 80 71 67 62 47 51 66 57 49

6 −+−+ + +−+− 70 58 69 65 65 74 71 75 75 65 70 64 65 66 55 70 71 74 65 75

7 −+ +−−−+ +− 61 66 65 74 66 73 73 65 83 81 60 75 77 62 60 69 65 60 89 76

8 −+ +−+ +−−+ 68 62 54 51 43 59 57 64 53 52 48 58 46 46 47 42 51 48 50 44

9 +−+−−+ +−+ 66 47 67 56 55 56 49 53 39 54 42 46 66 89 42 68 61 46 92 58

10 +−+−+−−+− 48 63 69 60 89 81 63 53 68 76 53 67 66 68 69 65 89 67 74 70

11 +−−+−+ + +− 75 76 75 70 71 70 84 68 75 73 80 76 69 71 70 75 67 68 76 66

12 +−−+ +−−−+ 58 55 47 49 59 45 53 56 41 53 61 52 55 55 54 50 53 52 56 55

13 + + + +−+−−− 64 65 57 76 54 54 65 60 64 67 62 62 67 57 67 58 55 61 64 56

14 + + + + +−+ + + 45 79 77 72 71 99 50 74 77 74 72 96 75 89 98 77 41 77 96 75

15 + +−−−+−+ + 84 53 56 64 61 74 57 56 69 65 72 57 48 64 64 67 55 68 56 55

16 + +−−+−+−− 61 60 55 50 54 51 50 56 57 55 53 52 57 52 53 52 54 55 55 49

22

and

V (X,Q,R) = exp(4.1538 + 0.5741x4 + 0.5001x6).

Assume that q3 and r2 follow normal distributions with means 0 and variances σ23 = (.05/.075)2 =

0.4444 and σ22 = (2/5)2 = 0.16 respectively. From (22) we have

ˆPM(X) = E

{exp(4.1538 + .5741x4 + .5001x6) + 2.46572σ2

2

(63.6791 + 2.4657x2 + 5.4289(x3 + q3) + 1.6364x4 − 1.5906x7)2

}.

We want to minimize ˆPM(X) with each variable restricted in the interval [−1, 1]. Using

Lemma 1 given in Appendix, we immediately get x∗2 = 1, x∗3 = 1, x∗6 = −1, and x∗7 = −1.

The lemma is not applicable to x4. So we evaluate the ˆPM(X) by Monte Carlo simulation

over a grid of values of x4, with all other variables at their optimal settings, to find x∗4 = −1.

At the optimal setting, the control law is given by

C = TC0/(71.5279 + 5.4289q3),

where T is the target gold thickness.

Without control, the performance measure will be

ˆPM o(X) =E[V (X,Q,R)] + V ar[β(X,Q,R)]

E2[β(X,Q,R)]

=exp(4.1538 + .5741x4 + .5001x6) + 2.46572σ2

2 + 5.42892σ23

(63.6791 + 2.4657x2 + 5.4289x3 + 1.6364x4 − 1.5906x7)2,

the minimization of which leads to the same setting as with control. The percentage reduction

in the variance of gold thickness with control compared to without control can be calculated

as 36.15%.

6. CONCLUSIONS

In this article we have formulated and developed a methodology for robust parameter design

of systems with feed-forward control. The approach can be used to obtain the optimal

control law and robust parameter design solution in a single stage. A new concept called

performance measure independent of control (PerMIC) is introduced for the optimization.

Performance measures are derived for the commonly encountered systems. We have shown

23

that, in general, the optimization procedures for systems with control and without control

are different. With control, the focus is on reducing the variation in the response after

compensating for the effect of noise. The usefulness of incorporating a control strategy with

robust parameter design was demonstrated using an example on gold plating.

The methodology is primarily developed for improving the steady state performance of

the system and therefore the models used in this article do not entertain the dynamics in the

system. Modeling the dynamics will help one to devise better on-line control strategies. This

is particularly true in the case of feed-back control. Robust parameter design with feed-back

control is an important area that needs investigation and we leave this as a topic for future

research.

ACKNOWLEDGMENTS

I am thankful to an anonymous referee for the very valuable comments and suggestions.

I am also thankful to Professor Jan Shi and Professor C. F. Jeff Wu for the help and support

throughout this research. The research is a part of author’s Ph.D. thesis at the University of

Michigan under the supervision of Professor C. F. Jeff Wu and was supported by a National

Science Foundation grant DMI-0217395.

APPENDIX : A LEMMA ON OPTIMIZATION

Suppose we want to optimize a function E[g(X,Q)] with respect to X, such as (11), (21),

(22), etc. Even if g(X,Q) has a simple functional form, E[g(X,Q)] can be a highly nonlinear

and complicated function of X. Often it is not even possible to get a closed form expression

of E[g(X,Q)] in terms of X, and we may have to resort to some Monte Carlo simulation

or numerical integration techniques to evaluate the expectation at different levels of X and

search for the optimal solution. See Spall (2003) for some algorithms. The following lemma

is sometimes useful for optimization.

Lemma 1 If X∗ = arg minX∈D g(X,Q0) is independent of Q0, then arg minX∈D E[g(X,Q)] =

X∗, where Q0 is any value in the support of Q.

24

Proof: Let X∗ = arg minX∈D g(X,Q0). Because X∗ is independent of the choice of Q0,

g(X,Q) ≥ g(X∗,Q) for all X ∈ D with probability 1, which implies E[g(X,Q)] ≥ E[g(X∗,Q)]

for all X ∈ D and therefore X∗ = arg minX∈D E[g(X,Q)]. ♦

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