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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