Experimental design is an engineering method for improving productivity by minimizing the effect of design and process uncertainties but without eliminating the causes. The tools of experimental design include a measure of quality loss and a set of experiments referred to as orthogonal array. The objective is to find an optimal design in minimizing the quality loss by a few experiments. Experimental design has been found valuable in many engineering fields and business applications.
For adaptive noise and vibration control, the system parameters, including the sampling rate, the dimension of control filter, and the dimension of the identification model, must be determined in an optimum level. A quality loss index is first defined to characterize the system performance. Then, the matrix experiment that considers many design parameters simultaneously is analyzed in a systematic way to select an optimal design with minimal quality loss.
Experimental design method aims at how to reduce the variance in design process by the matrix experiment of studying several design parameters simultaneously. Design parameters are often referred as factors and each design parameter can have many levels. The objective is to select the design parameters and levels such that a combination of them with their levels yields the best possible performance. The additive model in matrix experiment (Phadke, 1989) states that if the system performance, also termed total quality loss, is written as a function of product of quality loss from different design parameters, then a reduced set of experiments can be formulated to cover all design parameters at all levels. A quality loss index QL is employed to characterize system performance. It is a continuous function in a measure of deviation from the target value and often expressed in decibels for the additivity of design parameters by
) ( 10log10 QL
−
η= , (3.1)
where η is called the signal-to-noise ratio, SNR. Since log is a monotonically increasing function, minimizing QL is equivalent to maximize η. The optimal level of a design parameter is the level that has the highest η, which has the minimum quality loss.
If the total quality loss can be formulated as a sum of quality loss from different design parameters, then a reduced set of experiments can be formulated to cover all design parameters at all levels. The order of this reduced set of experiments, i.e., the dimension of the orthogonal array, is significantly smaller than that of the “full factorial experiment.” After conducting all experiments in an orthogonal array, the data are analyzed to determine the main effects of design parameters and to obtain their optimal levels. Analysis of variance (ANOVA) is conducted to identify the relative influence of the design parameters in discrete terms.
At last, the optimum design predicted in the analysis should be validated to confirm its performance.
In the procedures of experimental design, one should first define factors and Table 3.1 Levels of Each Design Parameters.
A: Sampling rate (Hz)
B: Identification model dimension C: Control filter dimension D: Convergence rate Factor
Level
A B C D
1 4.069K 64 64 3.3E-12
2 4.883K 128 128 1.0E-12 3 6.104K 192 192 3.3E-13 4 8.138K 255 255 1.0E-13
levels, and select an orthogonal array. Selection of the design parameters and their associated levels, however, requires certain engineering experience. In adaptive noise control system, four design parameters are defined, including the sampling rate, identification model dimension, control filter dimension and convergence rate.
Table 3.2 L12 Orthogonal Array and the Experimental Results.
Data: attenuated amount of noise (dB) Exp. No. A B C D Pure tone
y1
10 multi-tone y2
50 multi-tone y3
SNR
η
1 1 1 1 1 9.10 3.00 1.50 47.23
2 1 1 2 4 9.00 1.40 *0.01 4.77
3 1 4 3 1 9.10 3.40 2.00 49.35
4 2 2 3 4 9.10 1.70 *0.01 4.77
5 2 3 4 2 9.10 2.90 1.10 44.96
6 2 4 2 3 8.90 2.00 0.20 30.75
7 3 4 1 4 7.30 *0.01 0.10 4.73
8 3 3 4 3 9.10 2.80 1.20 45.56
9 3 2 4 2 9.00 3.00 1.90 48.75
10 4 3 1 2 9.00 1.40 1.00 42.95
11 4 2 2 1 9.00 3.20 0.90 43.49
12 4 1 3 3 9.00 3.00 0.80 42.50
Open Loop -- -- -- -- *0.01 *0.01 *0.01 0.00 Optimal
Design
4 3 4 1 9.60 8.10 2.70 52.64
(*: No attenuation)
The levels of each design parameter in the noise control system are listed in Table 3.1. Selection of the orthogonal array is based on the quantities of design parameter, level, and the interaction between them. The orthogonal array of L12, L18, and L38 are recommended because the interactions between the design parameters are distributed to the matrix experiments. A standard L12(44) array as listed in Table 3.2 is preferred for studying the controller with parameters of four 4-level factors. The matrix experiment is to determine the best level for each design parameter such that SNR is maximized. Classical statistics method to find the optimum combination of parameter levels is to conduct a full factorial experiment, which would have required 44 or 256 experiments. However, if the total quality loss can be formulated as a product of quality loss from the different design parameters, then the optimal design can be determined by using only twelve experiments as listed in Table 3.2. The entries in the matrix represent the levels of the design parameters. Thus, experiment
#1 is to be conducted with each parameter at the first level. Note that the levels of each parameter are equally represented in the twelve experiments, and the columns are mutually orthogonal. Orthogonality is interpreted in a combined sense; that is, for any pair of columns, all combinations of parameter levels occur at an equal number of times.
Since the attenuated amount of sound pressure level between the open-loop and the closed-loop is the higher and the better, the quality loss is defined as
+ + +
= 2 2
2 2 1
1 1
1 1
n
L n y y y
Q L , (3.2)
where n is the number of data in each experiment and yi is the experimental data of the attenuated amount of the sound pressure level. The signal-to-noise ratio can then be calculated from Eq. (3.1) and (3.2). For example, experiment #1 in Table 3.2:
23 experiments are conducted in a similar way and the results are listed in Table 3.2.
Estimation of the design parameter effects is to construct a table by calculating SNR of each experiment as listed in Table 3.2. However, the optimal design does not necessary correspond to any row in the matrix experiment. Analysis of the effect from each design parameter is necessary. The effect of a design parameter at a specific level is defined by the average of SNR for each parameter and level. If the design parameter A at level 2, represented by A2, is in experiments 4, 5 and 6, and the average SNR for these experiments is given by
3
For example:
78
Table 3.3 SNR (dB) of Each Design Parameter and Level.
A B C D
Level 1 33.78 31.50 31.64 *46.69
Level 2 26.83 32.34 26.34 45.55
Level 3 33.01 *44.49 32.21 39.60
Level 4 *42.98 28.28 *46.42 4.76 Max-Min 16.15 16.21 20.08 41.93 (*: Maximum SNR in each column)
98 . 3 42
5 . 42 49 . 43 95 . 42
4 = + + =
ηA (3.8)
The overall mean of η for the experiments is given by 15 . 12 34
1 12
1
=
=
∑
= i
ηi
η (3.9)
A table of average of SNR for each design parameter and level is then constructed as listed in Table 3.3 with marking the highest average of SNR.
The predicted optimal level of a design parameter leading to minimum quality or maximum SNR is the level that has the highest SNR. Therefore, the matrix experiment shows that the optimum combination is A4B3C4D1, i.e., set A factor at level 4 (sampling rate, 8.138 KHz), B factor at level 3 (identification dimension, 192) and C factor at level 4 (control filter dimension, 255), and D factor at level 1 (convergence rate, 3.3E-12). Note that it does not necessarily correspond to any row in the orthogonal array. The experimental result of this predicted optimum design level is 9.6, 8.1 and 2.7 (dB) in pure tone, 10 multi-tone, 50 multi-tone noise attenuation, respectively. As was expected, the performance is actually better than the other experiments listed in Table 3.2.
Analysis of variance should be conducted to study the relative importance of design parameters. The sum of squares (SS) of a design parameter is defined by the square deviation from the overall mean. For example, the SS of A factor (sampling rate) is
0 . 133 )
( 4
4
1
2 =
−
=
∑
= i
A
A i
SS η η . (3.10)
The SSs of the other design parameters can be calculated in a similar way and they are listed in Table 3.4. Mean square is calculated by dividing the sum of square by the degree of freedom, where the degree of freedom is the number of levels subtracts 1.
Parameter sensitivity is defined by the ratio of mean square of a parameter to the total sum of mean squares. The higher the parameter sensitivity, the stronger its influence
on quality loss. Among all design parameters, the convergence rate (D factor) is shown to be most sensitive (69.98%), next by the control filter dimension (C factor, 13.14%).
In addition to the sensitivity analysis, the predicted optimal combination has to be verified to see if the error caused by the interaction among the design parameters is within an acceptable tolerance. An error variance of interaction (σe) is therefore defined by pooling half of the design parameters with lower sensitivity to estimate the interaction error. From Table 5.4, the error variance of interaction is calculated by
45 . of 47
DOF of
DOF
2 =
+
= +
B A
SS SSA B
σe (3.11)
where the design parameters with lower sum of square (A and B) are selected. By contrast, the error variance of dominant design parameters (σpre) has two components,
one caused by the error in estimating the overall mean and mean effects, and the other by the repetition error of the experiments. σpre in matrix experiment is then defined by
Table 3.4 Analysis of Variance.
Design Parameter
Degree of Freedom
Sum of Square (SS)
Mean Square (SS/DOF)
Parameter Sensitivity
A 3 133.0 *44.33 7.88%
B 3 151.7 *50.57 8.99%
C 3 221.8 73.93 13.14%
D 3 1181.0 393.67 69.98%
Interaction Error 6 284.7 47.45
(*: Smaller mean square in calculating the error variance of interaction.)
13 SNR of the dominant design parameters can be determined by the parameters with higher mean square
96
The predicted optimal combination can then be validated by the closeness between
ηdom and the SNR (η*) of the experiment with the predicted optimal combination,
pre
dom σ
η
η*− <2 , (3.14)
then the predicted optimal combination from the matrix experiment is close to the real optimum. Conversely, the interaction of design parameters may be significant and the predicted optimal combination is far from the real optimum. Another quality loss index or another set of design parameters and going through the experimental design procedure may be required until Eq. (5.14) is satisfied. In this study,
32 .
*−ηdom =6
η < 2σpre =17.34, the predict optimal combination from the matrix
experiments is validated to yield a near optimum performance. The error caused by the interaction among the design parameters is within an acceptable two-sigma tolerance.