GUO Jian-bin (郭健彬), ZHAO Zi-tan (趙子覃), ZHAO Jian-yu (趙健宇), ZENG Sheng-kui (曾聲奎),2, HAO Zhi-peng (郝志鵬)

1 School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China 2 Science and Technology on Reliability and Environmental Engineering Laboratory, Beijing 100191, China

Integrated Robust Design Method Based on Maximum Tolerance Region

GUO Jian-bin (郭健彬)1,2*, ZHAO Zi-tan (趙子覃)1, ZHAO Jian-yu (趙健宇)1, ZENG Sheng-kui (曾聲奎)1,2, HAO Zhi-peng (郝志鵬)1

1SchoolofReliabilityandSystemsEngineering,BeihangUniversity,Beijing100191,China2ScienceandTechnologyonReliabilityandEnvironmentalEngineeringLaboratory,Beijing100191,China

Traditionally, parameter design is carried out prior to tolerance design. However, this two-step design strategy cannot guarantee optimal robustness for products’ quality. The proposed integrated robust design method determined the optimal parameter and tolerance simultaneously by calculating the maximum tolerance region, thereby improving the quality of products. In addition, the proposed method did not need uncertainty analysis to obtain the maximum tolerance region, so that the calculation cost could be decreased. And the method avoided the difficulty of gaining cost-tolerance function as maximum tolerance region represented both demand of cost and robust. Finally, an amplifier circuit case was conducted for verification purpose. Based on the results, the proposed approach could provide robust solution with optimal maximum tolerance region.

robustdesign;parameterdesign;tolerancedesign;maximumtoleranceregion;geneticalgorithm(GA)

Introduction

Robust design method is composed of two parts: parameter design and tolerance design[1-2]. The former is to find the best combination of parameters with minimizing parameters fluctuation influence on the system outputs, assuming the parameter tolerances are fixed[3-5]. The latter is to determine the parameter tolerances with the target of minimizing cost based on solution of parameter design[6-7]. Parameter design and tolerance design are sequential in traditional approach. The parameter tolerances are usually assumed fixed when determining the optimized solution in parameter design phase. However, this solution may not be optimal because the tolerances are changed in tolerance design phase.

Therefore, Bisgaard and Ankenman pointed out that these two phases should be implemented concurrently because the best solution of parameter design relied on the tolerance[8]. Jeang and Chang discussed a parameter and tolerance integrated design approach which aimed for quality loss[9-10]and could finally find an optimal solution comparing with the traditional. Wu proposed a mathematical formula corresponding to the model from Taguchi’s quadratic quality loss function to minimize the expected total cost for the parameter design of multiple dynamic quality characteristics[11]. Shenetal. proposed a simultaneous optimization method of robust parameter and tolerance design based on generalized linear models[12]. Mengetal. proposed an integrated robust design framework to help achieve robustness against both external and internal noises and improve loss function for robust tolerance design[13]. However, there are still some problems in these integrated approaches. (1) The optimization iterations may increase when the robust design variables, which include both parameters and tolerances, are increased. In this case, the substantial increase of uncertainty analysis times during optimization may face unacceptable calculation burden. (2) The tolerance optimization target of quality loss is built on the basis of cost-tolerance function. But it is difficult to get the accurate function during the product design in practice.

Referring to the theory of tolerance polyhedron[14], an integrated robust design method based on maximum tolerance region is proposed in this paper. This method conducts the integrated robust design by searching a maximum tolerance region in the feasible region. Once the maximum tolerance is determined, the optimal solution of both parameter design and tolerance design is found. The calculation cost can be decreased effectively since uncertainty analysis is unnecessary to obtain the maximum tolerance region. Besides, the maximum tolerance region avoids the difficulty of gaining cost-tolerance function as it represents both demands of cost and robust.

In this paper, for the fist the definition of maximum tolerance region is given and the reason chosen as the robust design index is explained. Then the algorithm of maximum tolerance region using genetic algorithm (GA) in order to avoid local optimal solution is demonstrated. At last, a circuit is applied as an example to verifing this method.

1 Basic Concept

Assuming systemy=f(x),x={x1,x2, …,xn},n>2, the boundary of feasible region is the design constraints, and the mathematical expression is as follows:

(1)

Considering that tolerances consist of different parameter deviations, there is a polyhedron regionT(φ) which is formed by the tolerances and centered on the demand design solutionφinside the feasible regionF,T?F.T(φ) is named tolerance region.

(2)

whereliis the tolerance of theith parameter andnis the parameter quantity.

The width ofxinT(φ) is

W(x)={2l1, 2l2, …, 2ln}.

(3)

Then the volume ofT(φ) is

(4)

The toleranceT(φ) is anN-dimension region formed by the allowed deviations of system parameters. If the variations of design factors and noise factors are in the range of tolerance region, the outputs of system are robust. There may be many tolerance regions for a design solution in a feasible regionF. As the 2-dimensional systemy=f(x1,x2) shown in Fig.1, there always exists one maximum tolerance regionTmax(φ)={2l1, 2l2, …, 2ln} with the maximum volumeVmax(T(φ)). That is the maximum tolerance region.

Fig.1 Relation between feasible region and tolerance region

The maximum tolerance regionTmax(φ) is exclusively determined by the location of design solutionφ. And oneφhas only oneVmax(T(φ)). It could be applied as an index in early design phase as it represents the robustness of the design solution. As shown in Fig.2, in the feasible regionF,Vmax(T(φ2))>Vmax(T(φ1)), so we believe solutionφ2is better thanφ1.

Fig.2 Comparison of design solutions

From the above analysis we can make the conclusions that: (1) the maximum tolerance region represents the optimal robust design because it may allow the maximum parameters variations without exceeding the feasible region; (2) in addition, the maximum parameters variation means designers may choose cheaper components, thereby achieving a robust and economic solution.

2 Genetic Algorithm (GA) Approach

GA is a probabilistic search method for solving optimization problems, especially for those problems without an analytical expression. The proposed GA developed for this problem is described in the following subsections.

As two types of variables, which are tolerance assignment as well as the design centering, are involved in the tolerance design, it is difficult to describe the complex interaction between them clearly. In this way, a feasible solution is to merge both variables and optimize them at the same time. The constrained optimization problem can be rewritten as

(5)

whereV(φi,li) is the defined objective function,φiis the design centering, which is usually taken from industrial standard Ω in practice, andliis the associated tolerance design.Dis the expected output, whileDΔis the maximum deviation of output.yhandylare the maximum and the minimum outputs respectively, which represent the worst case and can be seen as implicit functions of variables ofφiandli.Liis the maximum deviation of tolerance. Obviously, no analytical expressions of constraint is available, so GA plays a considerable role in solving it.

As mentioned above, the optimization variable should be expanded from (li) to (φi,li). Letxi=(φi,li), which represents a feasible solution to the optimization problem and could be called as an individual or chromosome in GA. Sincexiis continuous whileliis discreet, it is of great benefit to take float-encoding instead of binary-encoding. Besides, the fitness function is defined as the optimization objective, which is corresponding to the best solution. To maintain the uniformity over various problem domains, the fitness function is normalized to a convenient range of 0 to 1. As for constraints, they can be treated as a penalty term, which means that the fitness of an individual is set to zero if it violates the constraints. And the worst case of constraints can be obtained by using electronic design automation (EDA) tools.

When it comes to selection, a fitter individual receives a higher number of offspring and thus has a higher chance of surviving in the subsequent generation. Ranking selection as well is widely adopted, which tends to maintain an emphasis on the best solution so far.[10]

(6)

The detailed steps of the genetic approach are presented in Fig.3.

3 Case Study

In this paper, an amplifier circuit of control system, as shown in Fig.4, is adopted as an example to demonstrate the characteristics of the proposed algorithm. The aim is to enlarge the tolerance within power constraint. The power amplification is 3.0±0.2. By sensitivity analyzing, the impact of capacitances of the amplifier can be ignored since the biggest relative sensitivity coefficient is just 73×10-6. In this time, resistors are only considered in this case.

Fig.3 Flow chart of GA

Fig.4 Radio frequency amplifier circuit

The integrated robust design are based on the following steps.

(1) sensitivity analysis

Order the resistors by sensitivity analysis and choose ones which have great influences on the circuit. Since the relative sensitivity ofR1is much less than others, it will let out in the following analysis. Therefore, the specific model of this case can be expressed as follows:

(7)

(2) optimal parameter setting

The upper limit of tolerance is 10%, and the design centering is presented in Table 1 and it varies within 12%. The optimization accuracy is 0.1%. Besides, all the initial tolerances are set as 7%.

Table 1 Real design centering and tolerance of each component

(3) optimal design

GA is called to solve the problem and parameters used for the GA running are reported in Table 2. Both tolerance design and design centering are treated as variables, while the value of tolerance is taken from industry standard discreetly. The results are presented in Table 3.

Table 2 Parameters of GA

Table 3 Design centering and maximum tolerance of each component in the second scenario

(4) result verification

Select the resistors within the maximum tolerance region according to Table 3 and then substitute the real tolerance from Table 1 to the EDA model to verify the worst case. The analysis suggests that the worst cases are 3.1620 and 2.8031 respectively, which satisfy the requirement.

The analysis indicates that the tolerance of each component is larger after optimization, and the maximum tolerance region could be enlarged to 4.3761×10-10comparable to the real tolerance region is about 9.126×10-11. In addition, the design centering has an influence on tolerance region.

4 Conclusions

The proposed integrated robust design method based on maximum tolerance region determines the optimal parameter and tolerance simultaneously. A genetic algorism is addressed to search the maximum tolerance region. This method avoids the difficulty of uncertainty analysis and the difficulty of gaining cost-tolerance function. And it is suitable for product preliminary design, when the design and cost information is insufficient.

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Foundation item: National Natural Science Foundation of China (No. 61304218)

1672-5220(2014)06-0737-04

Received date: 2014-08-08

* Correspondence should be addressed to GUO Jian-bin, Email: guojianbin@buaa.edu.cn

CLC number: TB114.3 Document code: A