Perturbation Theory Based
Robust Design Under Model
Uncertainty
XinJiang Lu
Han-Xiong Li
1
Department of Manufacturing Engineering and
Engineering Management,
City University of Hong Kong,
Hong Kong
In real-world applications, a nominal model is usually used to approximate the practical
system for design and control. This approximation may make the traditional robust design less effective because the model uncertainty still affects the system performance. In
this paper, a novel robust design approach is proposed to improve the system robustness
to the variations in design variables as well as the model uncertainty. The proposed
robust design consists of two separate optimizations. One is to minimize the variation
effects of the design variables to the performance based on the nominal model just as
what the traditional deterministic robust design methods do. The other is to minimize the
effect of the model uncertainty using the matrix perturbation theory. Through solving a
multi-objective optimization problem, the proposed design can improve the system robustness to the uncertainty. Simulation examples have demonstrated the effectiveness of the
proposed design method. 关DOI: 10.1115/1.3213529兴
Keywords: robust design, matrix perturbation theory, model uncertainty, multi-objective
optimization
1
Introduction
Robust performance is one of the most important concerns in
design of any system since uncontrollable variations exist in the
real industry, including manufacturing operations, variations in
material properties, and operating environment. If these variations
are not considered, they will degrade the performance and may
result in a failure in practice. Furthermore, the system robustness
will be important and very useful especially when the system
needs to calibrate 关1兴.
The concept of the robust design was introduced by Taguchi.
The fundamental principle in robust design is to improve the quality of a product by minimizing the performance sensitivity to
variations. By making a design more robust to variations, it is
possible to improve number of the eligible parts or use less experiment 关2兴. In past decades, much effort was dedicated to the
robust design. All the work can be classified into two categories:
the stochastic approaches and the deterministic approaches 关3兴.
The stochastic approaches use probabilistic information of the
design variables and the design parameters, usually their mean
and variance, to improve the system robustness. There are many
authors who contributed to the stochastic approaches. Parkinson
关4兴 discussed seven robust design methods using engineering
models. A robust design procedure, which integrated the response
surface methodology with the compromise decision support problem, was developed by Chen et al. 关5兴 to overcome the limitations
of Taguchi’s methods. Du and Chen 关6兴 checked several feasibility
modeling techniques for robust optimization. A formulation of
robust design based on the mathematical model, which considered
the stochastic nature of the parameters, was proposed by AlWidyan and Angeles 关7兴. Kalsi et al. 关8兴 incorporated robust design concepts into multidisciplinary design. Yu and Ishii 关9兴 defined the manufacturing variation pattern to represent the
characteristic patterns of the design variables and investigated its
effects on robust design and constraint activity. The main short1
Corresponding author.
Contributed by the Design for Manufacturing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 19, 2009;
final manuscript received July 28, 2009; published online October 7, 2009. Review
conducted by Panos Y. Papalambros.
Journal of Mechanical Design
coming of stochastic approaches is that the essential information
of probabilistic distributions may not be easy to obtain in practice
关3兴.
The deterministic approaches often use the gradient information
of the variations and employ the Euclidean norm method and the
condition number method to improve the system robustness. Ting
and Long 关10兴 used the condition number of the sensitivity matrix
to measure the robustness of the system. Zhu and Ting 关2兴 used
the theory of performance sensitivity distribution to study the system robustness. Caro et al. 关1兴 compared two robust indexes: the
Euclidean norm and the condition number of the sensitivity matrix
and provided a two-consecutive-step synthesis method for the tolerance design. A comprehensive survey paper about the robust
optimization was presented by Beyer and Sendhoff 关11兴. Li et al.
关3兴 and Gunawan and Azarm 关12,13兴 proposed the sensitivity region measures for the robust design, which did not need the gradient information of the variations.
Generally, all the above robust design methods only work for
an accurate model since they need to know the relationship between the performance and the design variables. However, in real
application, this accurate model is difficult to obtain due to complex boundary conditions, complex process or unknown dynamics. Thus, a realistic approximation from the system is often taken
as the nominal model developed from experiment or data modeling. This approximation will cause the model uncertainty. Thus,
the traditional robust design may not work well by using the
nominal model only because the model uncertainty still affects the
system performance.
In this paper, a novel robust design approach is proposed to
improve the system robustness to both the model uncertainty and
variations in the design variables. In the proposed robust design,
only the norm bound of the perturbation sensitivity matrix is required to be known. This new design approach consists of two
separate optimizations.
1. Minimize the influence of the variations in the design variables to the performance just as what the traditional deterministic robust design methods do.
2. Minimize the influence of the model uncertainty using the
matrix perturbation theory.
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Finally, the robust design is obtained by solving a multiobjective optimization problem.
2
Problem Description
Consider the robust design problem with the model uncertainty
Y = f共d兲 + ⌬f共d兲
共1兲
where Y = 关y 1 ¯ y m兴T represent the performances, d = 关d1 ¯ dn兴T
are the design variables vector, f共d兲 = 关f 1共d兲 ¯ f m共d兲兴T are the
known nominal model, and ⌬f共d兲 = 关⌬f 1共d兲 ¯ ⌬f m共d兲兴T are the
model uncertainty, which includes the parameter uncertainty and
the structure uncertainty. The superscript T represents the transpose. For convenience, f共d兲 and ⌬f共d兲 are simply denoted as f
and ⌬f.
Taking Taylor series expansion of Y at the nominal values d0,
the performance variations ⌬Y can be approximated by the linear
series expansion
⌬Y = J • ⌬d
共2兲
Fig. 1 Influence of the model uncertainty to singular value ␴
where the nominal sensitivity matrix J0, the perturbation sensitivity matrix ⌬J, the sensitivity matrix J, and the variation ⌬d are
defined as
冏 冏
冏 冏
⳵f
J0 =
⳵d
共3a兲
共3b兲
d=d0
J = J0 + ⌬J
共3c兲
⌬d = d − d0
共3d兲
储⌬Y储22 = ⌬dT • B • ⌬d
m
兺 共⌬y 兲
i
2
and
B = J TJ
共4兲
i=1
Define
B0 = JT0 J0
According to the singular value decomposition 共SVD兲 theory, the
real symmetric matrix B and B0 may be decomposed as
B = ␨diag共␴1, . . . , ␴n兲␨T
共5兲
B0 = ␨0diag共␴01, . . . , ␴0n兲␨T0
共6兲
␴0i
are the singular values of J and J0, respectively,
where ␴i and
and the corresponding orthogonal eigenvectors are denoted as ␨i
and ␨0i , which are one element of ␨ = 关␨1 ¯ ␨n兴 and ␨0 = 关␨01 ¯ ␨0n兴.
The traditional deterministic robust design is to reduce the influence of the variations ⌬d based on the nominal model and
condition of J = J0 and B = B0. Thus, inserting Eq. 共6兲 into Eq. 共4兲,
the performance variations ⌬Y in the traditional robust method
may be expressed as follows:
n
储⌬Y储22 =
0
i
d
0
␴max
0
␴min
h共d兲 = 0
0 2
i
关x01, . . . ,x0n兴T = ␨T0 ⌬d
Case 1: There is no model uncertainty in the system, then J
= J 0.
Case 2: There is model uncertainty in the system, then J ⫽ J0.
Obviously, the traditional deterministic robust design methods, including the Euclidean norm method and the condition number
method, can work well in case 1. However, in case 2, since J is
not equal to J0 and B is not equal to B0, there will exist the
difference ⌬␴i between the singular values ␴0i and ␴i as shown in
Fig. 1. This difference will cause that the largest singular value
␴max or the condition number ␴max / ␴min of J is not minimal under
the traditional robust design. Thus, the traditional robust design
methods are less effective in this case. For example, design A in
Fig. 1 is obtained by the Euclidean norm method. Its singular
value will change significantly 共⌬␴A兲 due to the effect of ⌬J.
However, design B remains relatively the same for both J and J0.
Therefore, design B is less sensitive to the model uncertainty than
design A.
If the singular value variation ⌬␴ is very small, only the nominal variation ␴0 should be minimized so that the traditional deterministic robust design methods are still effective. Thus, the robust
design problem under the model uncertainty is decomposed into
two subproblems. One is to reduce the influence of the variations
⌬d of the design variables to the performance variations ⌬Y based
on the nominal model. The other is to reduce the influence of the
model uncertainty to the variations in the singular values, which
represent ⌬␴ in Fig. 1, so that the nominal singular value ␴0 is
close to the singular value ␴.
The New Robust Design Methodology
The perturbation bound S is used to estimate ⌬␴ and defined as
共7兲
Furthermore, the robust design variables d can be figured out by
minimizing the largest singular value ␴0max of J0 as well as the
Euclidean norm method 关1,2兴 or minimizing the condition number
␴0max / ␴0min of J0 as well as the condition number method 关10兴
111006-2 / Vol. 131, NOVEMBER 2009
共8兲
where h共d兲 and l共d兲 are constraints from other design aspects.
There exist two cases.
3
兺 ␴ 共x 兲
i=1
with
subject to
冉 冊
min
l共d兲 ⱕ 0
From Eq. 共2兲, the performance variations ⌬Y may be easily expressed as
储⌬Y储22 =
or
d
d=d0
⳵ ⌬f
⌬J =
⳵d
with
C1共d兲:min max共␴i0兲
max共兩⌬␴i兩兲 ⱕ S
with
⌬␴i = ␴i − ␴i0
共9兲
If the perturbation bound S is very small by the selection of the
suitable design variables d, then the singular value ␴ may be close
to the nominal one ␴0, which means that the model uncertainty
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+ 储⌬J储2兲 • 储⌬J储2. Thus, the inequality 共15兲 becomes
max共␭i0 − K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2,0兲
ⱕ ␭i ⱕ ␭i0 + K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
where max共␭0i − K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2 , 0兲 means
mal value between ␭0i − K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
共16兲
that a maxiand zero is
chosen.
According to the definition of the singular value, we have
␴i = 冑␭i,
␴i0 = 冑␭i0
共17兲
and ␭i are very close, then their sinThus, if the eigenvalues
gular values ␴0i and ␴i are also very close. From the inequalities
共14兲 and 共17兲, if K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2 is very small, then
␭0i
1. ␴i is close to ␴0i
2. ⌬␴i is less sensitive to ⌬J
Fig. 2 The new robust design methodology
has small effect to the singular value ␴. Under this condition, the
traditional deterministic robust design methods are still effective.
Thus, these two subproblems can be accordingly transformed
into two minimizations as shown in Fig. 2. One is to minimize the
nominal sensitivity matrix J0 just as what the traditional methods
do. This minimization can be achieved by solving the optimization problem C1共d兲. The other is to minimize the perturbation
bound S. Then, based on these two minimizations, a multiobjective optimization problem is proposed to minimize the performance variations ⌬Y caused by both the model uncertainty ⌬f
and the variations ⌬d of the design variables, which is conceptually expressed as min 共⌬Y / ⌬d , ⌬f兲 in Fig. 2.
3.1 Minimization of the Perturbation Bound S. From the
equality 共4兲, the matrix B can be rewritten as
B = B0 + JT0 ⌬J + ⌬JTJ0 + ⌬JT⌬J
共10兲
The eigenvalues ␭0i 共i = 1 , . . . , n兲 of the matrix B0 are related
corresponding eigenvectors U0i 共i = 1 , . . . , n兲 by the equation
B 0U 0 = U 0⌳ 0
to the
共11兲
where U0 = 关U01 , . . . , U0n兴 and ⌳0 = diag共␭01 , . . . , ␭0n兲 are the right eigenvector set and the right eigenvalue set of B0, respectively.
According to Bauer–Fike theorem 共matrix perturbation theory兲
关14–17兴, if B0 has an additive perturbation ⌬B = JT0 ⌬J + ⌬JTJ0
+ ⌬JT⌬J, then a bound on the sensitivities of the eigenvalues is
given by
兩␭i −
␭i0兩
ⱕ K • 储⌬B储2
Moreover, if K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2 ⬇ 0, then ␴i will be approximately equal to ␴0i . Thus, minimizing the perturbation bound
S may be transformed into the minimization of K • 共2储J0储2
+ 储⌬J储2兲 • 储⌬J储2 as Eq. 共18兲
C2共d兲:min K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
d
subject to
h共d兲 = 0
l共d兲 ⱕ 0
3.2 Multi-Objective Optimization. The multi-objective optimization is constructed to have the trade-off between two minimizations C1 in Eq. 共8兲 and C2 in Eq. 共18兲. The robust design variables d can be figured out from the following multi-objective
optimization.
冦
0
储⌬B储2 ⱕ 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
兩␭i −
共14兲
␭i0 − K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
ⱕ ␭i ⱕ ␭i0 + K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
共15兲
are the eigenvalues of B = J J and
reSince ␭i and
spectively, ␭i and ␭0i are not smaller than zero. From the inequality 共15兲, if ␭0i is smaller than K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2, then the
lower bound of ␭i can be negative, which is contradictory with
0 ⱕ ␭0i . In order to avoid such a case, the lower bound should take
the maximal value between zero and ␭0i − K • 共2储J0储2
Journal of Mechanical Design
冧
B 0U 0 = U 0⌳ 0
l共d兲 ⱕ 0
ⱕ K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2
T
min C 共d兲
2
d
h共d兲 = 0
共13兲
The inequality 共14兲 may be rewritten as
␭0i
min C 共d兲
1
d
subject to
Then, from the inequalities 共12兲 and 共13兲, we obtain
␭i0兩
共18兲
The solution of Eq. 共18兲 guarantees that ⌬␴i is small and ␴0i is
close to ␴i.
In the proposed robust design, only the bound of 储⌬J储2 is required to be known. This bound is easy to be estimated by experiment or simulation data, such as, the local and dispersion modeling method 关18兴. This proposed method can work well only if the
variation in 储⌬J储2 is limited in the estimated bound.
共12兲
where ␭ and ␭ are the eigenvalue set of B and B0, respectively
and the condition number K is the ratio of the largest singular
value ␴U max to the smallest singular value ␴U min of U0.
According to the matrix norm theory 关14兴, we have
B 0U 0 = U 0⌳ 0
B0 = JT0 J0,
共19兲
The most common method to solve the multi-objective optimization is the weighted-sum 共WS兲 method, which optimizes the
weighted sums of several objectives 关15,19,20兴. The multiobjective optimization 共19兲 can be easily derived by the WS methods as below
min ␤
d
C1共d兲
C2共d兲
+ + 共1 − ␤兲
C1共d 兲
C2共d+兲
subject to
B 0U 0 = U 0⌳ 0
h共d兲 = 0
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close to the singular value ␴i. Then, these two subproblems are
accordingly achieved by the following two minimizations:
1. The first optimization can be solved by minimizing the
nominal sensitivity matrix J0 just as what the traditional
methods do. Using the singular value decomposition theory,
minimizing the nominal sensitivity matrix J0 can be transformed into minimizing the largest nominal singular value
␴0max. Here, the Euclidean norm is used as the robust index
because Caro et al. 关1兴 confirmed that the Euclidean norm is
more suitable as the robust index than the condition number.
2. The second optimization can be solved by minimizing the
perturbation bound S. According to Bauer–Fike theorem, the
minimization of the perturbation bound S may be transformed into the minimization of K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2.
Then, a WS method can balance these two minimizations to
make the performances less sensitive to both the model uncertainty and variations in the design variables. Since only the norm
bound of the perturbation sensitivity matrix is needed, this proposed method is easy to realize.
However, the proposed robust design depends on the estimation
accuracy of 储⌬J储2. If the estimation accuracy is too poor, the
proposed robust design may be conservative and become impractical. Moreover, since the performance variations ⌬Y are approximated by the linear series expansion in Eq. 共2兲, although the
higher order term neglected may be regarded as the model uncertainty, the proposed method will be also conservative when the
system model is highly nonlinear.
Fig. 3 Design details of the proposed approach
l共d兲 ⱕ 0
4
共20兲
where the objectives C1共d兲 and C2共d兲 are normalized by their
central values C1共d+兲 and C2共d+兲 with the central point d+ in the
design variables space and ␤ is a trade-off weight in the range 0
ⱕ ␤ ⱕ 1.
The design variables d can be solved from Eq. 共20兲 with a
properly choice of the weight factor ␤. Any value of ␤ corresponds to a Pareto optimal solution 关19兴. Since the suitable Pareto
solution is chosen by users based on their preferences, the weight
␤ is also decided by users in proportion to the objective’s relative
importance in the context of the problem.
In the proposed method, the weight ␤ is selected according to
the effect of the model uncertainty to the performances. If the
model uncertainty has smaller influence to the performances, the
objective function C1 actually plays a bigger role. Then a big ␤
should be chosen. When ␤ = 1, it becomes the traditional deterministic robust design. On the other hand, if the model uncertainty
has larger influence to the performances, the objective function C2
would be more important and a smaller ␤ should be chosen. When
␤ = 0, it only minimizes the influence from the model uncertainty.
3.3 Design Summary. In industrial application, assumptions
and idealization in a system often lead to model uncertainty. This
model uncertainty is usually neglected by using the nominal
model only for design and control. These designs derived from the
nominal model only will be less robust because the model uncertainty neglected still affects the system performances. The proposed robust design method is to minimize the effect of the model
uncertainty to the system performances.
The proposed robust design procedure is summarized in Fig. 3.
The robust design problem is decomposed into two subproblems:
One is to minimize the influence of the design variables to the
performance 共⌬Y / ⌬d兲 by using the nominal model only and the
other is to minimize influence of the model uncertainty to the
performance 共⌬␴ / ⌬f兲 by making the nominal singular value ␴0i
111006-4 / Vol. 131, NOVEMBER 2009
Case Study
4.1 Example 1: Structure Uncertainty Design. Consider the
structure uncertainty design problem
with
⌬f =
冤
f=
冋
Y = f + ⌬f
ln共兩csc共d1兲 − tan共d1兲兩兲 + 3d2
3d1 + d2
0.05 ln共兩csc共d1兲 − tan共d1兲兩兲 +
册
and
0.1 3
d + 0.01d2
3 1
0.01d1 − 0.05 cos共d2兲
冥
共21兲
From Eq. 共21兲, the performance variations ⌬Y can be expressed as
⌬Y = J0 • ⌬d + ⌬J • ⌬d
with
冤
冥 冤
1
3
,
J0 = sin共d1兲
3
1
0.05
0.01
+ 0.1d21
⌬J = sin共d1兲
0.01
0.05 sin共d1兲
⌬d =
冋 册
⌬d1
⌬d2
冥
共22兲
The matrix B0, the singular value ␴0i , and the condition number K
can be calculated using MATLAB program. The upper bound of
储⌬J储2 is estimated from simulation data.
The design objective is to select the design variable d1 from
d1 苸 关1 , 2.5兴 to have a robust performance against uncertainty.
Here, the nominal value d02 of the design variable d2 is equal to 5.
The maximal singular values of J0 and J are shown in Fig. 4.
From Fig. 4, it is clear that there exist the difference ⌬␴max between the maximal singular value ␴max and ␴0max due to the effect
of the model uncertainty.
From Fig. 5, all 兩⌬␭兩 are smaller than the bound S1, which is
equal to K • 共2储J0储2 + 储⌬J储2兲 • 储⌬J储2 in Eq. 共14兲. Moreover, the
bound S1 is minimal at d1 = 1, which means the minimal variation
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0
Fig. 4 Maximal singular values of J0 and J versus the design variable d1 „a… ␴max
and ␴max and „b… the difference ⌬␴max
兩⌬␴兩. From Fig. 6, it is clear that the lower and upper bounds of
␴max are very close at d1 = 1. This also means that ␴max is close to
its nominal singular ␴0max, which can be verified in Fig. 4.
Designs from different ␤ are shown in Table 1. The traditional
robust design, which is figured out by the Euclidean norm method
or the condition number method 共Eq. 共8兲; the results of these two
methods are the same in this case兲, are taken as ␤ = 1. From Eq.
共22兲, the performance variations may be rewritten as
储⌬Y储22 = 共r1⌬d1 + 3.01⌬d2兲2 + 共3.01⌬d1 + r2⌬d2兲2
with
r1 =
1.05
+ 0.1d21,
sin共d1兲
r2 = 1 + 0.05 sin共d1兲
共23兲
For the performance variations in Eq. 共23兲, smaller parameters r1
and r2 will have smaller performance variations, i.e., the better
robustness. From Table 1, it is clear that both r1 and r2 obtained
by the proposed robust design method with ␤ = 0.95, 0.9, and 0.85
are smaller than by the traditional robust design methods. Thus the
proposed robust design with ␤ = 0.95, 0.9, and 0.85 has the better
robustness than the traditional robust design. This is because the
proposed robust design method considers the model uncertainty,
while the traditional robust design methods do not.
From this comparison, it is clear that a larger ␤ should be
Fig. 5
円⌬␭円 and the bound S1
Journal of Mechanical Design
selected since the model uncertainty ⌬f is smaller than 5% of the
nominal model f. Thus, when the nominal model is dominant, a
large ␤ should be chosen.
To demonstrate the effectiveness of the proposed robust design
method, the verification is carried out by letting both ⌬d1 and ⌬d2
randomly vary in 关−0.05, 0.05兴. A total of 1000 samples are taken
to compare the performance variations ⌬Y with respect to ⌬d
under the model uncertainty. From Table 2, we can see that the
mean and variance of the performance variations ⌬Y gained by
the proposed robust design method with ␤ = 0.95, 0.9, and 0.85 are
smaller than the traditional design methods. It also shows that the
most robust design is achieved when ␤ = 0.95. The difference in
the performance variations is defined as
T = 储⌬Y p储22 − 储⌬Y C储22
共24兲
where ⌬Y p and ⌬Y C are the performance variations gained by the
proposed robust design method with ␤ = 0.95 and the traditional
robust design methods, respectively. The comparison in Fig. 7
shows that the proposed robust design can have more than 63%
共for T ⬍ 0兲 chance to have a better design than the traditional one.
In other words, for every 100 designs, the new approach can get
more than 63 better designs while the traditional one just can get
less than 37 better designs. Only if this percent is larger than 50%,
then the robustness is improved compared with the traditional
Fig. 6 Lower and upper bounds of ␴max
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Table 1 Robust design and performance under different ␤
Weight ␤
Design
r1
r2
1 共traditional
robust design兲
0.95
0.9
0.85
0.8
0.75
0.7
0.5
0.2
1.55
1.2905
1.05
1.35
1.2584
1.0488
1.25
1.2627
1.0474
1.15
1.2826
1.0456
1.05
1.3207
1.0434
1
1.3478
1.0421
1
1.3478
1.0421
1
1.3478
1.0421
1
1.3478
1.0421
dⴱ1
robust design methods and thus the proposed method is better than
the traditional methods.
4.2 Example 2: Robust Design of a Low-Pass Filter. The
RL circuit in Fig. 8 is used as the comparison study between the
traditional deterministic robust design methods and the proposed
robust design method. The design variables are the resistance R
and the inductance L. The current I is kept at a nominal value I0 of
10 A, while the amplitude V of the excitation voltage v共t兲
= V cos wt and its frequency w are uncontrollable. For this filter,
the steady-state current i共t兲 is harmonic of the form i共t兲
= I cos共wt + ␾兲, with I and ␾ as the amplitude and the phase of i共t兲
关7兴 as follows
I=
V
冑R 2 + w 2 L 2 ,
␾ = tan−1
冉 冊
wL
R
共25兲
The design variables d, the design parameters p, and the performance functions Y are
d=
冋册 冋册 冋册
R
L
,
p=
V
w
,
Y=
I
␾
The design model with the model uncertainty can be expressed as
共26兲
Only the nominal model f共d兲 is known to designers. The nominal
value of V0 and w0 are 110 V and 60 Hz, respectively, which have
the variations V = V0 ⫾ 5.5 and w = w0 ⫾ 6. The objective is to select the design variables R from R 苸 关0.05⍀ , 0.16⍀兴 and L from
L 苸 关7H , 11H兴 to have a robust performance against uncertainties.
Designs from different ␤ are shown in Table 3. To demonstrate
the effectiveness of the proposed robust design method, the verification is carried out by letting ⌬R, ⌬L, ⌬V, and ⌬w randomly
vary in 共−0.008, 0.008兲, 共−0.6, 0.6兲, 共−5.5, 5.5兲, and 共−6 , 6兲, respectively. A total of 1000 samples are taken to compare the performance variations ⌬Y with respect to ⌬d, ⌬p, and the model
uncertainty. From Table 3, we can see that the mean and variance
of performance variations ⌬Y gained by the proposed robust design method with ␤ = 0 and 0.2 are smaller than the other two
traditional design methods, evenly the Euclidean norm method
and the condition number method defined in Eq. 共8兲.
From this comparison, it is clear that a small ␤ should be selected since the nominal model is not always dominant compared
with the model uncertainty, for example, the nominal model ␾0
= tan−1共w0L / R兲 and the model uncertainty ⌬␾ = tan−1共wL / R兲
− tan−1共w0L / R兲 are taken as −2.11 and 11.2, respectively when w
changes from w0 = 60 to w = 59, and L = 10 and R = 0.1. Thus, when
the model uncertainty has the larger effect to the system performances, then a smaller ␤ should be chosen.
Moreover, the comparison between the proposed robust design
with ␤ = 0.2 and the traditional robust designs, are carried out in
Fig. 9. The differences T1 and T2 of the performance variations are
defined as
T1 = 储⌬Y p储22 − 储⌬Y E储22
共27a兲
T2 = 储⌬Y p储22 − 储⌬Y C储22
共27b兲
where ⌬Y p, ⌬Y E, and ⌬Y C are the performance variations gained
by the proposed robust design method, the Euclidean norm
method, and the condition number method, respectively.
It is clear in Fig. 9 that the new approach has about 70.7% 共for
T1 ⬍ 0兲 and 73.3% 共for T2 ⬍ 0兲 chances to get the better design
than the traditional one. Only if this percent is larger than 50%,
then the robustness is improved compared with the traditional
methods. So the proposed robust design method is more robust
than the other two design methods, because the proposed robust
design method considers the model uncertainty.
4.3 Example 3: Robust Design of a Damper. The damper
design example in Fig. 10 is taken from the Ref. 关1兴. The design
variables are mass M and damping coefficient Cd to be determined with the aim of keeping the magnitude of displacement X0
at a nominal value of 3 m while the magnitude F0 of the excitation
force F共t兲 = F cos共␻ · t兲 and its pulsation ␻ undergo considerable
variations. The displacement is equal to X共t兲 = X cos共␻ · t + ␾兲
where ␾ is the phase. Moreover, the following relations exist:
X=
␻冑
C2d
F
+␻ M
2
2
␾ = tan−1
,
冉 冊
␻M
Cd
共28兲
The design variables d, the design parameters p, and the performance functions Y are
d=
冋 册 冋册 冋册
M
Cd
,
p=
F
w
,
Y=
X
␾
The design model with the model uncertainty can be expressed as
共29兲
Table 2 Performance comparison under ⌬d and the model uncertainty
Weight ␤
1
0.95
0.9
0.85
0.8
0.75
1.55
1.35
1.25
1.15
1.05
1
Design dⴱ1
0.0182
0.0181
0.0181
0.0181
0.0182
0.0183
储⌬Y储22 Mean
Variance 2.7082⫻ 10−4 2.6607⫻ 10−4 2.6669⫻ 10−4 2.6951⫻ 10−4 2.6955⫻ 10−4 2.7884⫻ 10−4
111006-6 / Vol. 131, NOVEMBER 2009
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Fig. 10 Damper
Fig. 7 Comparison of the proposed robust design „␤ = 0.95…
with the traditional robust design „␤ = 1…
Fig. 8 A low-pass filter
Only the nominal model f共d兲 is known to designers. The nominal
value of F0 and w0 are 200 N and 20 rad/s, respectively, and the
design parameters p have the variations F = F0 ⫾ 10 and w
= w0 ⫾ 2. The objective is to select the design variables M from
M 苸 关2 kg, 3 kg兴 and Cd from Cd 苸 关25 N s / m , 55 N s / m兴 to
have a robust performance against both the model uncertainty and
the parameter variations.
Designs from different ␤ as shown in Table 4 are compared
with the Euclidean norm method 共␤ = 1兲 and the condition number
method in Table 4. To demonstrate the effectiveness of the proposed robust design method, the verification is carried out by
letting ⌬M, ⌬Cd, ⌬F, and ⌬w randomly vary in 共−0.1, 0.1兲,
共−2 , 2兲, 共−10, 10兲, and 共−2 , 2兲, respectively. A total of 1000
samples are taken to compare the performance variations ⌬Y with
respect to ⌬d, ⌬p, and the model uncertainty. From Table 4, we
can see that the mean and variance of the performance variations
⌬Y gained by the proposed robust design method with ␤ = 0, 0.2,
and 0.4 are smaller than the other two traditional design methods.
It is clear that a small ␤ should be selected since the nominal
model is not always dominant compared with the model uncertainty. For example, the nominal model ␾0 = tan−1共w0M / Cd兲 and
the model uncertainty ⌬␾ = tan−1共wM / Cd兲 − tan−1共w0M / Cd兲 are
taken as 0.3323 and 0.4779, respectively when w changes from
the nominal value w0 = 20 to w = 18 and M = 2.5 and Cd = 40. Thus,
Table 3 Performance comparison under ⌬d, ⌬p, and the model uncertainty
Weight ␤
1
共Euclidean norm method兲
Design Rⴱ
Design Lⴱ
Mean
储⌬Y储22
Variance
0. 076
10.0103
0.1169
0.0086
0.8
0.6
0.4
0.2
0
0.075
0.074
0.073
0.148 0.149
10.0374 10.0641 10.0904 6.4920 6.4091
0.1183 0.1199 0.1215 0.0841 0.0842
0.0086 0.0087 0.0087 0.0080 0.0079
Condition
number method
0.068
10.21
0.1315
0.0091
Fig. 9 Performance comparison under ⌬d, ⌬p, and the model uncertainty: „a… comparison with the Euclidean norm method
and „b… comparison with the condition number method
Journal of Mechanical Design
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Table 4 Performance comparison under ⌬d, ⌬p, and the model uncertainty
Weight ␤
Design Rⴱ
Design Lⴱ
储⌬Y储22 Mean
Variance
1
共Euclidean norm method兲
2.668
39.9644
8.1287
0.0076
0.8
0.6
0.4
0.2
0
2.6560 2.6390 2.6060 2.5360 2.5220
40.2829 40.7273 41.5686 43.2658 43.5919
8.1286 8.1286 8.1284 8.1284 8.1285
0.0075 0.0074 0.0073 0.0071 0.0069
Condition
number method
3
25.059
8.1594
0.0154
Fig. 11 Comparison under ⌬d, ⌬p, and the model uncertainty: „a… the Euclidean norm method versus proposed method
and „b… the condition number method versus proposed method
when the model uncertainty has the larger effect to the system
performances, a smaller ␤ should be chosen similarly as in example 2.
In Fig. 11, the proposed robust design with ␤ = 0.2 is compared
with the traditional robust design methods, including the Euclidean norm method and the condition number method. It is clear
that it has about 56.3% 共for T1 ⬍ 0兲 and 80.5% 共for T2 ⬍ 0兲
chances to have a better design than the traditional one. Only if
this percent is larger than 50%, then the robustness is improved
compared with the traditional methods. So the proposed robust
design method is more robust than the other two traditional design
methods because the proposed robust design method considers the
model uncertainty.
5
Conclusion
In this paper, the novel robust design method is proposed to
design the system robustness. This new design approach considers
not only the variations in design variables but also the model
uncertainty. The proposed robust design approach consists of two
separate optimizations. One is to minimize the influence of the
model uncertainty using the matrix perturbation theory and the
other is to minimize the variation influence of the design variables
just as what the traditional deterministic robust design methods
do. Through solving the multi-objective optimization, the robust
design can be obtained to have the good robustness to both the
model uncertainty and the variations in the design variables d.
Simulation examples are used to compare the proposed method
with two traditional methods: the Euclidean norm method and the
condition number method. The comparisons show that the proposed robust design method is more robust than the traditional
methods when both the model uncertainty and the variations in the
design variables exist. This is because the proposed robust design
method considers the model uncertainty while the traditional
methods do not.
111006-8 / Vol. 131, NOVEMBER 2009
Acknowledgment
The project is partially supported by a project from RGC of
Hong Kong under Grant No. CityU: 117208, a project from City
University of Hong Kong under Grant No. 9360131, and a UGC
Special Equipment Project Grant: SEG_CityU 01.
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