optimization based on parameter moments estimation for robust...
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OPTIMIZATION BASED ON PARAMETER MOMENTS ESTIMATION FOR ROBUST DESIGN
Laura Picheral1, Khaled Hadj-Hamou
1, Ghislain Remy
2, Member, IEEE and Jean Bigeon
1,
1G-SCOP – CNRS UMR 5272; Grenoble INP-UJF; 38000 Grenoble - France
2LGEP / SPEE-Labs, CNRS UMR 8507; SUPELEC; Univ. Pierre et Marie Curie P6; Univ. Paris-Sud 11;
91192 Gif sur Yvette CEDEX - France
ABSTRACT
This paper deals with robust design based on analytical
models. We consider that variability on design parameters are
represented by the parameter Mean and Standard Deviation.
We propose in this article a new robust design approach that
allows the designer either to find a robust solution in a
reasonable execution time or to be informed of the non-
existence of a robust solution for his design problem. The
process combines a reformulation of the model to integrate
parameter uncertainties and a deterministic optimization
algorithm with new specifications based on robust objectives.
The engineering application of an electrical actuator design is
presented and used to show the implementation and the
effectiveness of this new robust approach.
Index Terms—Robustness, Design optimization,
Statistical analysis, Analytical model.
1. INTRODUCTION
This paper attempts to consider robust design problems in
preliminary design steps. In real engineering, product design
problems may be subject to various unavoidable uncertainties.
Uncertainties mainly influence product performances and can
lead to wrong products. Uncertainties may affect every design
parameter, such as new environmental conditions, geometrical
parameters, material properties and so on.
Usually, design engineers use analytical models to get a first
design solution. The analytical model based methods aim to
use mathematical equations of technical and economic
characteristics of the device. This model consists of a set of
non-linear and non-convex equations corresponding to a set of
output parameters and involving many input parameters.
Theses equations come from FEM/RSM [1] or from
approximation of physical laws. The specifications are
established by the designer, defining the user requests in terms
of product performances and costs, including one or more
objective functions (device cost, power, volume) and the
parameter variation. These problems are more and more
complex because of the multi-physics constraints and the
increase of the number of constraints.
Robust design aims at optimizing product performance
parameters in making them less sensitive to design parameter
variability. Actually, robustness level is defined by the
designer through the specifications of the robust optimization.
Usually, the robust design process is achieved in two steps.
In the first step, the designers implement an optimization
method on the design model considering only nominal values
of the parameters. They mainly use stochastic methods such as
Genetic Algorithms [2] with execution times varying from a
few minutes to several hours. Then, the optimal solution
robustness is analyzed using a propagation of uncertainty
method. This method is mostly based on the Monte-Carlo
method requiring at least 106 model evaluations [3]. The
optimal solution could be a non-robust solution regarding the
robustness specifications. If this solution is non-robust, the
designers have to test the robustness of other solutions.
The approach we propose allows the designer either to find
the robust solution in a reasonable execution time or to be able
to proof the non-existence of such a robust solution. We show
in this paper how to fill in robust design objectives in a way
that fits with deterministic optimization for speeding up the
process. Actually, this approach aims to reformulate only once
the initial design model, taking the parameter uncertainties
into account. The resulting model we call “Robust design
model” is optimized. If there is no robust optimal solution, the
designers have to review the robust specifications.
In this paper, we present in detail this new approach for
implementing a robust optimization using deterministic
algorithms. First, we introduce the Propagation of Variance
method that allows transforming the model to integrate
parameter uncertainties. Then, we propose a robust
optimization formulation based on six-sigma approach. The
last section is dedicated to the implementation of the proposed
approach through a case study of an electromagnetic device.
2. APPROACH PROPOSED FOR ROBUST
OPTIMIZATION
The robust optimization approach proposed and
implemented in this paper is illustrated on Figure 1. In a
robust approach, the initial model is reformulated as a "Robust
design model" which integrates uncertainties on design
parameters. Then the “Robust design model" is solved using
an optimization algorithm under specific requirements for
models that integrate uncertainties. Finally, the optimal robust
solution is given, providing that it exists.
The process for finding this robust solution is well described
in [4]. It consists in a compromise between shifting (moving
the Mean) and/or shrinking (reducing the Standard Deviation)
the probability distribution of the performance parameters of
the product (Fig. 2). It is done by varying the Mean and the
Standard Deviation of the design parameters so that the
performance parameters can meet the requirements.
2012 25th IEEE Canadian Conference on Electrical and Computer Engineering (CCECE) 978-1-4673-1433-6/12/$31.00 ©2012 IEEE
Fig. 1 Proposed approach for robust optimization
Fig. 2 Process for finding robust optimal solution that meet requirements [4]
2.1. Model transformation
The aim of this transformation is to integrate variability on
design parameters in the initial model. Robust analysis
methods are introduced to compute the variability of the
product performances (output parameters), given the
variations of design parameters (model inputs). We assume
here that design parameters can be represented either by
distributions or by their nominal values. It is also assumed that
random variables are independent.
The most common robust analysis method is the Monte-
Carlo method [5][6]. It can easily be implemented but it
requires 104 to 10
6 samples to obtain accurate enough results
[3]. If we consider a one to a 10-minute execution time for an
analytical model, the time needed to achieve 104
Monte-Carlo
simulations is a week to two months. Even if the Latin
Hypercube Sampling and the Importance Sampling methods
reduce the number of samples the execution time still high.
Another way to characterize product performances
variability is to estimate its moments accurately and efficiently
and then rebuild the probabilistic distribution function. The
mathematical moment is expressed by (1):
(1)
where is the output, is the set of random input
parameters and is the joint probability density function
of the random inputs. Estimate (1) by numerical integration
such as a quadrature rule is practically unfeasible, mainly
when the number of input parameters is large. The Univariate
dimension-reduction method [7][8], the Performance moment
integration method [8] and the Percentile difference method
[8] have been proposed to handle this problem.
All these methods use numerical computations and this leads
to the loss of analytical links between input and output
moments. But those links are necessary when dealing with
deterministic optimization process. The Propagation of
Variance method [9] analytically gives the moments
expressions. They are obtained from a Taylor series expansion
around the parameter’s Mean and for the first and the second
order. We have limited this study to the evaluation of the first
two moment expressions which are the Mean ( ) and the
variance ( ) for a second order Taylor expansions:
(2)
where is the model output, are the model inputs, are
the input Means, represents the set of input Means and are the input Standard Deviations.
Equation (2) is obtained from a quadratic approximation of
the initial model. A second order Taylor approximation should
better fit the exact result of the Mean and variance than a first
order Taylor one, because engineering design problems are
rarely linear. Moreover, this method requires the evaluation of
the Gradient and the Hessian of the model. Practically, the
partial derivatives are often computed by finite differentiation.
To summarize the model reformulation, the Mean and the
Standard Deviation of each output of the initial model are
expressed using the Propagation of Variance method at the
second order (Fig. 3). This new model is polynomial.
Fig. 3. Robust design model
2.2. Optimization
Once the Robust design model is obtained by the
Propagation of Variance method, it can be implemented in the
optimization process. This new model has changed and this
leads to adapt the specifications (constraints and objective) in
a way that integrates variability on parameters. To do so, we
propose to base specifications on a quality approach such as
the six-sigma method [4]. This method aims at placing six
Standard Deviations ( ) of a parameter between its lower and
upper bounds (Fig. 4) so that 99.73% of the parameter's
values will be located between these bounds [4].
Fig. 4. Six-sigma strategy [4]
A constrained optimization problem involves an objective
function, analytic constraints and bound constraints:
(3)
where is the set of input parameters, is the set of output
parameters, is the objective function, is the set of analytic
constraints, and are the set of respectively the lower
ROBUST DESIGN MODEL
and the upper bounds of , and are the set of
respectively the lower and the upper bounds of .
The resulting robust design specifications are as follows:
(4)
where is the reformulated objective function, is the set
of reformulated analytic constraints, and are the set of
respectively the Means and the Standard Deviations of the
input parameters , and and are the set of respectively
the Means and the Standard Deviations of the output .
Notice that the designer establishes these new specifications
and that it requires a specific knowledge of the designed
product. Moreover, the Robust design model is a deterministic
one even if it takes into account stochastic uncertainties.
Hence, local and/or global deterministic optimization
algorithms can be applied on, so as stochastic algorithms.
3. APPLICATION
The aim of this part is to show the implementation of the
proposed robust optimization approach. We first introduce the
test problem dealing with the design of an electrical actuator
[10]. The PoV method has already been studied in [11] to
certify that it has all the features required to be usable in the
proposed approach for obtaining the “robust design model”.
The Pro@Design [12] software developed by the Design
Processing Technologies company was used both to test the
Propagation of Variance method and, if necessary, to perform
classical optimizations. The Pro@Design software contains
two main modules, “Robust” and “Optimization” modules.
The “Robust” module allows the user to propagate
uncertainties using either Monte-Carlo simulations on the
exact design model, Monte-Carlo simulations on an
approximate model achieved by Taylor decomposition or the
Propagation of Variance method. The “Optimization” module
is carried out using deterministic local optimization
algorithms. This module is used to generate the robust
problem specifications and to choose the more appropriate
algorithm for this problem.
3.1. Electromechanical actuator
The applications have been carried out on a design model
benchmark. The studied system is an electromechanical
actuator. The analytical model has been introduced in [10]. It
is characterized by 8 non-linear explicit equations, 20 design
continuous parameters, and 12 degrees of freedom (TABLE I).
3.2. Robust optimization approach
The procedure applied here is the one described on Figure 1.
Firstly, we have transformed the initial model (TABLE I) into
a robust design model which is only composed of Means and
Standard Deviations of input and output parameters (Fig. 3).
The commercial software Pro@Design has been used in this
part of the application only to achieve classical optimizations
of this new robust model. Actually, Pro@Design software is
able to compute the Mean and the Standard Deviation of the
output parameters when the Mean and the Standard Deviation
of the input parameters are given. It only gives the numerical
results. Unfortunately, it doesn’t give the analytical equations
that link the Mean and Standard Deviation of the output
parameters and the Mean and Standard Deviation of the input
parameters. This transformation is hand-realized for this
example because of the small number of model equations.
The robust design model is implemented using the
"Optimization" module of Pro@Design software and robust
specifications are established from initial specifications in
order to integrate parameter variability.
TABLE I
THE ELECTROMECHANICAL ACTUATOR MODEL
Active parts
volume
Form factor
Global heating up
of the winding
Number of pole
pairs
Leakage
coefficient
No-load magnetic
radial
Thickness of the
magnetic field
Electromagnetic torque
The design parameters are classified into three types:
- the “fixed input parameters” , , and for which
only nominal value are considered;
- the “varying input parameters” , , , , , and
considered using their mean and standard deviation: ,
, , , , , ;
- the “output parameters” , , , , , , and
also characterized by their mean and standard deviation:
,
,
,
,
, .
The robust specifications based on (4) are:
Objective function: as it is the upper
bound of the probability distribution of .
Constraints on input parameters: each varying input parameter
is constrained as follow:
(5)
Moreover, the Standard Deviation value is linked to a
manufacturing process. Hence, we know the lower value that a
parameter’s Standard Deviation can take. We constrain the
Standard deviation of each optimizable parameter to be higher
than a minimum value:
(6)
Constraints on output parameters: all output parameters are
constrained as follows except the electromagnetic torque ,
the global heating up of the winding and the number of
pole pairs :
(7)
As the problem consists in dimensioning a machine capable of
developing a at least equal to , the lower bound of
the probability distribution of is:
(8)
The probability distribution of the global heating up of the
winding is bounded as follow:
(9)
The number of pole pairs is a discrete parameter. Thus,
several integer values are specified.
3.3. Results and discussion
The robust optimization is performed using both a SQP-
based algorithm and a discrete optimization with some
heuristics dedicated to engineering design. The optimization
process reaches the optimal solution in 11 SQP-iterations.
Initially, based on the first robust specifications, the
optimization process didn’t reach any robust solution. This
result is verified using a deterministic algorithm based on
interval branch and bound [10]. Thus, to obtain an optimal
robust solution, the robust specifications (lower and upper
bounds of parameters) are relaxed. The two last columns of
TABLE II show the resulting Mean and Standard Deviation of
both input and output parameters. Besides, the optimization
model described in (3) is implemented and used as a reference.
The optimal solution is obtained in 17 SQP-iterations and
illustrated in the second column of TABLE II.
Results indicate that the robust optimization approach
converges to a solution similar to the one of the initial model
(comparison between the column of optimal values and the
column of the Mean). Moreover, this robust solution is
reached by fewer iterations than for an optimization of the
initial model. This is certainly due to the fact that the robust
model is polynomial while the initial model (TABLE I) is not.
The evolution of the uncertain performance (torque)
during the optimization is based on the shift and shrink
process (Fig. 2) until it meets the robust requirements (8).
Indeed, the optimized values are and
at the beginning and and
at the end of the process.
TABLE II
RESULTS OF THE ROBUST OPTIMIZATION
Parameter
Optimal
nominal values
from the initial model
Mean Standard
Deviation
Varying input parameters
0,1549 0,1591 0,9883 E-4
0,8 0,8030 0,9924 E-3
0,06196 0,0637 0,9917 E-4
0,001 0,0014 0,9999 E-6
0,0033 0,0032 0,9934 E-5
0,003 0,0030 0,9954 E-5
0,6619E7 6,637 E6 99,9998
Output parameters
0,004931 0,0047 0,1302 E-4
0,3698 0,3546 0,8107 E-3
5 5 0,0031
0,1606 0,1739 0,4607 E-3
1,0E11 0,9908 E11 0,3063 E9
2,5 2,5 0,0042
10 10,1257 0,0419
0,5138E-3 0,5407 E-3 0,1467 E-5
The objective function we implement here is based on the
technical parameter (the product active part volume). If we
were able to link the volume of each part of the electrical
actuator with the raw material cost, and the Standard
Deviation of the design parameters with the manufacturing
machine cost, we could have a relevant objective function
regarding the industrial context.
4. CONCLUSIONS
The purpose of this approach is to allow the designer either
to find a robust solution in a reasonable execution time or to
be informed of the non-existence of a robust solution of his
design problem. We chose the Propagation of Variance
method to reformulate the initial model in a new model
constituted by the analytical links between input moments and
output moments. We also proposed a reformulation for
optimization specifications to integrate robustness issues.
The approach has been implemented to an analytical model
of a engeeneering problem that can be considered as
simplified since it doesn’t integrate any functionals
(differential equations), implicit equations, or discrete
parameters. The reformulation of the initial model into a
robust design model was handmade and should be automated
for further and extensive studies. Finally, it would be useful to
introduce costs into the formulation of the objective function
and link them with parameter Means and Standard Deviations.
But that requires a deeper knowledge of the industry context,
which we do not have at this time of the project.
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