# Circuit Tolerance Design Using Belief Rule Base

Authors: Xiao-Bin Xu; Zheng Liu; Yu-Wang Chen; Dong-Ling Xu; Cheng-Lin Wen
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

A belief rule-based (BRB) system provides a generic nonlinear modeling and inference mechanism. It is capable of modeling complex causal relationships by utilizing both quantitative information and qualitative knowledge. In this paper, a BRB system is firstly developed to model the highly nonlinear relationship between circuit component parameters and the performance of the circuit by utilizing available knowledge from circuit simulations and circuit designers. By using rule inference in the BRB system and clustering analysis, the acceptability regions of the component parameters can be separated from the value domains of the component parameters. Using the established nonlinear relationship represented by the BRB system, an optimization method is then proposed to seek the optimal feasibility region in the acceptability regions so that the volume of the tolerance region of the component parameters can be maximized. The effectiveness of the proposed methodology is demonstrated through two typical numerical examples of the nonlinear performance functions with nonconvex and disconnected acceptability regions and high-dimensional input parameters and a real-world application in the parameter design of a track circuit for Chinese high-speed railway.


---

## Body

## 1. Introduction

Tolerance has become a crucial design consideration in integrated and discrete circuit designs due to the demand of improved product quality, longer product lifetimes, and shorter design cycle. Designers have to unceasingly seek a central point with the maximum tolerances in the space of circuit component parameters so as to maximize parametric yield and minimize costs while maintaining compliance with design specifications [1–3]. On the other hand, circuit reliability is closely linked to its yield, namely, only those products with high yield would have high reliability. So, tolerance design and yield optimization are also the effective ways to improve circuit reliability [4].In essence, there are mainly two kinds of methods for tolerance design and yield estimation, that is, the Monte Carlo based statistical methods and the deterministic methods [5–11]. Because the former requires numerous circuit simulations and computationally expensive analysis runs [5, 6], researchers have proposed alternative deterministic methods based on response surface modeling to approximate the performance function and the corresponding acceptability region (RA). Thus, the optimal center and tolerances (i.e., feasibility region RF) of component parameters can be found in the approximated region RA. The deterministic methods mainly include simplicial approximation [7], polyhedral approximation [8], quadratic approximation [9], ellipsoidal method [10], and neural network [11]. However, such approximation methods and low-order polynomial models may not be applicable to some complex cases in which ranges of parameter variables are wide, performance functions are highly nonlinear, and the feasibility regions are nonconvex and even disconnected [1–3]. Hence, there is a need to develop new methods that can be used to model and optimize the design in such a highly complex setting.This paper develops a novel method of the acceptability region approximation and tolerance optimization to obtain available feasibility region using belief rule-based (BRB) model. In the belief rule base, each possible consequent of a rule is associated with a belief degree. Such a rule base is capable of capturing highly nonlinear and continuous causal relationships between different factors [12, 13]. When applying a belief rule base, the input of an antecedent is transformed into a belief distribution over the referential values of an antecedent. The distribution is then used to calculate the activation weights of the rules in the rule base. Subsequently, inference in the belief rule base is through the combination of all the activated rules using the evidential reasoning (ER) approach [14, 15]. Compared with polynomial and neural network models, the model parameters in the BRB can be extracted not only from objective data, but also from experts’ subjective knowledge [16]. Moreover, the physical meanings of these parameters are easy to understand for experts and engineers, so they can intuitively participate in the whole course of system modeling [17]. The BRB modeling technique has been widely applied in nonlinear system modelling and decision support systems [16–21].In this paper, a BRB system is designed to model the complex nonlinear relationship between circuit component parameters (i.e., input variables to the BRB system) and a performance index of the circuit (i.e., output) by utilizing the limited knowledge from circuit simulations and its designers. Through rule inference in the BRB and clustering analysis, the acceptability regions can be separated from the value domain of component parameters. Then, an optimization method is presented to seek the optimal feasibility region in the acceptability regions to maximize the volume of tolerance region of the circuit parameters. The remainder of this paper is organized as follows. The research issue is expounded in Section2. Section 3 describes the use of the BRB modelling technique to approximate the acceptability regions. The tolerance optimization method is presented in Section 4. Section 5 shows some encouraging results obtained from two typical numerical examples of nonlinear performance functions with nonconvex and disconnected acceptability regions and high-dimensional input parameters and a real-world application in the parameter design of track circuit of Chinese high-speed railway.


## 2. Problem Formulation

Given a product performance or response specification [2–6](1)Lb⩽gu⩽Ub,here u=(u1,u2,…,uN) is a vector of design parameters. g is the performance index or function of an electrical circuit. Ub and Lb are constants, respectively, representing the upper and lower allowable limits of variation of the resonance performance. For discrete component circuits, these parameters may include, but are not limited to, resistances, capacitances, and inductances, whereas for integrated circuits these may be resistivities, linewidths, specific capacitances, and so forth. Commonly, the element ui of u is characterized by a nominal value ui_0 and a tolerance ti, i=1,2,…,N.The acceptability regionRA is defined as [3, 22](2)RA=u∣Lb⩽gu⩽Ub.If u∈RA, then the product is acceptable; otherwise it is unacceptable. The tolerance region RT is defined as [3, 22](3)RT=u∣ui-u0_i≤ti,i=1,2,…,N.When given a nominal valueu0=(u0_1,u0_2,…,u0_N) and a tolerance t=(t1,t2,…,tN), the corresponding parametric yield is defined as [3, 22](4)Yu=VRT∩RAVRT×100%=MN×100%.Here, N is the number of total products, M is the number of acceptable products (u∈RA), and  V(·) is the volume of a region. RF=RT∩RA is defined as the feasibility region which is a subset of RT at the intersection between RT and RA.When given the design constraintsRA, the actual goal of tolerance design is to maximize RF (i.e., to seek for the maximum t and corresponding u0) so that a 100% yield is achievable. In this case, the maximum RF is equal to the maximum RT [3, 22]. In tolerance design, the key step is to calculate performance g(u) and estimate the parametric yield. In most cases, the structures of the integrated circuit and analogous circuit are too complex to obtain analytical expressions of circuit performance functions. Hence, designers have to build the circuit simulator using some design software tools (e.g., HSPICE, Simulink) to evaluate the performance function g(u) and estimate yield by simulation runs [22, 23]. However, this kind of simulation-based tolerance design (e.g., Monte Carlo methods) requires numerous circuit simulations and computationally expensive analysis runs [2–6]. In the following section, instead of using a circuit simulator, we will build a BRB system to model the performance function by running as few simulations as possible and using experts’ knowledge. The proposed BRB system can be used to approximate the acceptability region and obtain the corresponding feasibility region.


## 3. Approximating Acceptability Region by Using a BRB System

As an extension of traditional IF-THEN rules, belief rules are the key parts of a BRB system. In a belief rule, each antecedent attribute takes a referential value, and each possible consequent is related to a belief degree [13]. To build a BRB system for circuit performance modelling and acceptability region approximating, we map the relationship between BRB system and circuit performance function in Table 1.Table 1
Belief rule-based system for circuit performance modelling.
BRB system
Circuit performance function
Antecedent attributes
Parameter inputsu1,u2,…,uN
The set of referential valuesAi = {Ai,j∣j=1,2,…,Ji}
Referential values of the design parameterui
Antecedent in thekth rule, i = 1,2,…,N, k = 1,2,…,K
Thekth referential value of the input vector u uk = (uk,1,uk,2,…,uk,N), uk,i∈Ai
Consequent in thekth rule {(D1,βk,1),(D2,βk,2),…,(DL,βk,L)}, ∑l=1Lβk,l≤1
D
l is the referential value of performance output, βk,l is the belief degree of Dl when u is taken as uk
Rule weightsθk∈[0,1]
Relative importance of thekth rule
Attribute weightsδi∈[0,1]
Relative importance ofui in the rule baseCorresponding to Table1, the kth (k=1,2,…,K) referential relationship between input and output of performance function, that is, the belief rule k in the BRB system, can be defined as (5)Rulek:IFu1isuk,1∧u2isuk,2∧⋯∧uNisuk,NTHEND1,βk,1,D2,βk,2,…,DL,βk,Lwith rule weightθk,k=1,2,…,Kand attribute weightδi,i=1,2,…,N;here, “∧” denotes “and.”To use the BRB system to approximate acceptability regionRA involves the following steps: (1) constructing referential values of parameter inputs and performance output about acceptability region RA; (2) generating new performance output by ER inference of belief rules; (3) training BRB by the selected samples near to RA.


### 3.1. Constructing Referential Values of Parameter Inputs and Performance Output about Acceptability RegionRA

The initial belief rules can be established in the following four ways [16]: (1) extracting rules from expert knowledge; (2) extracting rules by examining historical data; (3) using the precious rule bases if available; (4) random rules without any preknowledge. In our context, we use (1) and (2) to determine the parameters in BRB system as listed in Table 1.In the tolerance design, we first need to specify the upper and lower bounds of each design parameterui. Let lbi and ubi represent, respectively, the lower and upper bounds on the ith parameter ui. Thus, the space in which the vector u takes value can be constructed as (6)Su=u=u1,u2,…,uN∣lbi⩽ui⩽ubi,i=1,2,…,N.Generally speaking, we have to set Su large enough to relatively entirely include the acceptability region. The referential points are uniformly selected from Su, and then the circuit simulations are implemented to calculate the performance outputs of those points.As an illustrative example suppose there are two inputsu=(u1,u2) with the bounds 1⩽u1⩽14, 1⩽u2⩽9 and its performance function is g(u). We uniformly select 126 initial referential points as (1,1),(1,2),…,(14,9) shown in Figure 1 and calculate their performance outputs by circuit simulations. Assume we have known the acceptability region RA, which is nonconvex and shown in Figure 1. Then, we can find out 35 points in RA by constrains in (1). These points are divided into two parts, one is called as “inside point” (12 points), and the other is “inside boundary (IB) point” (23 points). From Figure 1, it can be seen that 126 initial referential points subdivide Su into 104 gridding cells. The boundary of RA passes through 26 cells, called boundary gridding (BG) cells and vertexes of each BG cell certainly contain one or two IB points. Hence, in each of BG cells, starting from its IB points, we can seek out three or two vertexes as external boundary (EB) points adjacent to the IB points, as demonstrated in the BG cell directly below RA. In this example, totally, 29 EB points can be found out.Figure 1
Selection of referential values of parameter inputs and performance output.As a result, we totally select 64 points (including 29 EB, 23 IB, and 12 inside points) as the referential values of parameter input. The sets of referential values can be listed as(7)A1=3,4,…,13,A2=2,3,…,8.Denote the referential values of parameter input as uk=(uk,1,uk,2), then uk∈A1×A2, k=1,2,…,K, K=77; here “×” denotes Cartesian product. Correspondingly, we can construct 77 belief rules.Next, we need to calculate the bounds of performance output as(8)Lb′≤guk≤Ub′,Lb′=minuk∈A1×A2guk,Ub′=maxuk∈A1×A2guk.Obviously, Lb′≤Lb, Ub′≥Ub. According to designers’ experiences, the L referential values of performance output can be selected from the interval [Lb′,Ub′] uniformly or not, denoted as {D1,D2,…,DL} and D1=Lb′<D2<⋯<DL=Ub′. For the kth rule, the input uk=(uk,1,uk,2) and its performance output is g(uk); then the consequent belief distribution can be given as (9)D1,βk,1,D2,βk,2,…,DL,βk,L.Here(10)βk,l=Dl+1-gukDl+1-Dl,Dl≤guk≤Dl+1,βk,l+1=guk-DlDl+1-Dl,Dl≤guk≤Dl+1,βk,c=0,c=1,2,…,l-1,l+2,…,L.Obviously, ∑l=1Lβk,l=1, g(uk)=βk,l+1Dl+1+βk,lDl.In Table1, the remaining rule weights θk and the attribute weights δi reflect the importance (or unimportance) of the kth rule and ith referential parameter input, respectively. Their values can be determined based on designers’ knowledge. When designers’ knowledge cannot be collected, θk and δi will be set equal to 1, respectively, which means equal importance.Note that, in practice, there may be more than one acceptability region in the parameter spaceSu; in this case, some available clustering analysis methods (e.g., k-means clustering [24]) can be used to recognize every disconnected acceptability region. Then, by using the proposed procedure as shown in Figure 1, the belief rules of all disconnected regions can be generated so as to compose a rule base for modelling the whole acceptability regions. In Section 5.1, an example of the disconnected acceptability regions will be given to show the procedure of BRB based modeling.


### 3.2. Generating New Performance Output by ER Inference of Belief Rules

Section3.1 gives the BRB system to describe the acceptability region RA. Actually, the BRB uses the grid-based mechanism to approximate RA and one vertex of gridding cell corresponds to one belief rule in rule base. Therefore, given a new input u=(u1,u2,…,uN), which certainly falls into a certain N-dimension cell. This input u can activate 2N rules of this cell. Thus, the activation weight of the kth rule, wk, is calculated as [16](11)wk=θk∏i=1Nαikδ-i∑k=1Kθk∏i=1Nαikδ-i;here, the relative attribute weight is defined as [16] (12)δ-i=δimaxi=1,2,…,Nδi.αik is the individual matching degree to which the input ui matches the ith antecedent referential value uk,i in the kth rule. Here, uk,i∈Ai, Ai={Ai,j∣j=1,2,…,Ji}, Ji is the number of referential values of ui, and Ai,1<Ai,2<⋯<Ai,Ji. αik is calculated as(13)αik=ui-Ai,qAi,q+1-Ai,q,Ai,q≤ui≤Ai,q+1,uk,i=Ai,q+1Ai,q+1-uiAi,q+1-Ai,q,Ai,q≤ui≤Ai,q+1,uk,i=Ai,q1,ui≤Ai,1=uk,iorui≥Ai,Ji=uk,i0,otherwise.Here, q=1,2,…,Ji-1.Having determined the activation weight of each rule in the rule base, the ER approach can be directly applied to combine the rules and generate final conclusions [14]. The output of the new input u by the combination is defined as(14)Ou=D1,β1,D2,β2,…,DL,βL.Since the rule consequent in the BRB system has the form of belief distribution, the result of reference in (14) is also a belief distribution, which expresses that if the input is given as u=(u1,u2,…,uN); then the consequent is D1 to a degree β1, D2 to a degree β,…, and DL to a degree βL. The analytical format of the ER algorithm can be used to calculate the combined belief degree βl in Dl as [17](15)βl=μ×∏k=1Kwkβk,l+1-wk∑l=1Lβk,lpppppppp-∏k=1K1-wk∑l=1Lβk,l·1-μ×∏k=1K1-wk-1.Here, (16)μ=∑l=1L∏k=1Kwkβk,l+1-wk∑l=1Lβk,lpppp-L-1∏k=1K1-wk∑l=1Lβk,l-1.Next, we can estimate performance output by the weighted average operator(17)gBRBu=∑l=1LDlβl.


### 3.3. The Parameter Optimization of the BRB System Using the Selected Training Samples

Although the initial belief rules can be constructed by the limited simulations and designers’ knowledge and the performance outputs can be estimated by ER inference of the initial belief rules, the estimation accuracy can be improved if the parameters in the belief rules are fine-tuned through learning from some selected training samples. The adjustable parameters in a rule base include belief degrees (βk,1,βk,2,…,βk,L), rule weights (θ1,θ2,…,θK), and attribute weight (δ1,δ2,…,δN).When we search for the available feasibility regionRF from the approximated acceptability region RA estimated by the BRB, the vertexes of the maximum RF certainly reach to the boundary points of RA, which are all included in the set of critical points {u∣g(u)=Lb,g(u)=Ub}. To a large extent, the estimation accuracy of the boundary points determines the accuracy of the optimal RF. Since the boundary points of RA fall into the boundary gridding (BG) cells as shown in Figure 1, we select the central points of the BG cells as training points and then obtain their performance outputs by circuit simulations as the training samples. Thus, given Q training points uq (q=1,2,…,Q), the error function d(P) between the simulated output g(uq) and the estimated output gBRB(uq) can be defined as [21](18)dP=1Q×∑q=1Qguq-gBRBuq2.Here, d(P) is function of the parameter set P(19)P=βk,l,θk,δi∣k=1,2,…,K,pppppppipppβk,ll=1,2,…,L,i=1,2,…,N.The objective of the training is to minimize the difference d(P) by adjusting the parameters P. Note that the optimal P can be obtained by using gradient-based search methods or nonlinear optimization software packages, such as thefmincon function in the Optimization Toolbox of MATLAB [18].For example, in the case of two-dimensional inputs as shown in Figure1, there are 26 BG cells. Correspondingly, 26 training points are selected, only the extra 26 circuit simulations are needed to get training samples. As a result, totally 152 circuit simulations (126 initial referential points and 26 training points) are required to construct the trained BRB with two-dimensional inputs.


## 3.1. Constructing Referential Values of Parameter Inputs and Performance Output about Acceptability RegionRA

The initial belief rules can be established in the following four ways [16]: (1) extracting rules from expert knowledge; (2) extracting rules by examining historical data; (3) using the precious rule bases if available; (4) random rules without any preknowledge. In our context, we use (1) and (2) to determine the parameters in BRB system as listed in Table 1.In the tolerance design, we first need to specify the upper and lower bounds of each design parameterui. Let lbi and ubi represent, respectively, the lower and upper bounds on the ith parameter ui. Thus, the space in which the vector u takes value can be constructed as (6)Su=u=u1,u2,…,uN∣lbi⩽ui⩽ubi,i=1,2,…,N.Generally speaking, we have to set Su large enough to relatively entirely include the acceptability region. The referential points are uniformly selected from Su, and then the circuit simulations are implemented to calculate the performance outputs of those points.As an illustrative example suppose there are two inputsu=(u1,u2) with the bounds 1⩽u1⩽14, 1⩽u2⩽9 and its performance function is g(u). We uniformly select 126 initial referential points as (1,1),(1,2),…,(14,9) shown in Figure 1 and calculate their performance outputs by circuit simulations. Assume we have known the acceptability region RA, which is nonconvex and shown in Figure 1. Then, we can find out 35 points in RA by constrains in (1). These points are divided into two parts, one is called as “inside point” (12 points), and the other is “inside boundary (IB) point” (23 points). From Figure 1, it can be seen that 126 initial referential points subdivide Su into 104 gridding cells. The boundary of RA passes through 26 cells, called boundary gridding (BG) cells and vertexes of each BG cell certainly contain one or two IB points. Hence, in each of BG cells, starting from its IB points, we can seek out three or two vertexes as external boundary (EB) points adjacent to the IB points, as demonstrated in the BG cell directly below RA. In this example, totally, 29 EB points can be found out.Figure 1
Selection of referential values of parameter inputs and performance output.As a result, we totally select 64 points (including 29 EB, 23 IB, and 12 inside points) as the referential values of parameter input. The sets of referential values can be listed as(7)A1=3,4,…,13,A2=2,3,…,8.Denote the referential values of parameter input as uk=(uk,1,uk,2), then uk∈A1×A2, k=1,2,…,K, K=77; here “×” denotes Cartesian product. Correspondingly, we can construct 77 belief rules.Next, we need to calculate the bounds of performance output as(8)Lb′≤guk≤Ub′,Lb′=minuk∈A1×A2guk,Ub′=maxuk∈A1×A2guk.Obviously, Lb′≤Lb, Ub′≥Ub. According to designers’ experiences, the L referential values of performance output can be selected from the interval [Lb′,Ub′] uniformly or not, denoted as {D1,D2,…,DL} and D1=Lb′<D2<⋯<DL=Ub′. For the kth rule, the input uk=(uk,1,uk,2) and its performance output is g(uk); then the consequent belief distribution can be given as (9)D1,βk,1,D2,βk,2,…,DL,βk,L.Here(10)βk,l=Dl+1-gukDl+1-Dl,Dl≤guk≤Dl+1,βk,l+1=guk-DlDl+1-Dl,Dl≤guk≤Dl+1,βk,c=0,c=1,2,…,l-1,l+2,…,L.Obviously, ∑l=1Lβk,l=1, g(uk)=βk,l+1Dl+1+βk,lDl.In Table1, the remaining rule weights θk and the attribute weights δi reflect the importance (or unimportance) of the kth rule and ith referential parameter input, respectively. Their values can be determined based on designers’ knowledge. When designers’ knowledge cannot be collected, θk and δi will be set equal to 1, respectively, which means equal importance.Note that, in practice, there may be more than one acceptability region in the parameter spaceSu; in this case, some available clustering analysis methods (e.g., k-means clustering [24]) can be used to recognize every disconnected acceptability region. Then, by using the proposed procedure as shown in Figure 1, the belief rules of all disconnected regions can be generated so as to compose a rule base for modelling the whole acceptability regions. In Section 5.1, an example of the disconnected acceptability regions will be given to show the procedure of BRB based modeling.


## 3.2. Generating New Performance Output by ER Inference of Belief Rules

Section3.1 gives the BRB system to describe the acceptability region RA. Actually, the BRB uses the grid-based mechanism to approximate RA and one vertex of gridding cell corresponds to one belief rule in rule base. Therefore, given a new input u=(u1,u2,…,uN), which certainly falls into a certain N-dimension cell. This input u can activate 2N rules of this cell. Thus, the activation weight of the kth rule, wk, is calculated as [16](11)wk=θk∏i=1Nαikδ-i∑k=1Kθk∏i=1Nαikδ-i;here, the relative attribute weight is defined as [16] (12)δ-i=δimaxi=1,2,…,Nδi.αik is the individual matching degree to which the input ui matches the ith antecedent referential value uk,i in the kth rule. Here, uk,i∈Ai, Ai={Ai,j∣j=1,2,…,Ji}, Ji is the number of referential values of ui, and Ai,1<Ai,2<⋯<Ai,Ji. αik is calculated as(13)αik=ui-Ai,qAi,q+1-Ai,q,Ai,q≤ui≤Ai,q+1,uk,i=Ai,q+1Ai,q+1-uiAi,q+1-Ai,q,Ai,q≤ui≤Ai,q+1,uk,i=Ai,q1,ui≤Ai,1=uk,iorui≥Ai,Ji=uk,i0,otherwise.Here, q=1,2,…,Ji-1.Having determined the activation weight of each rule in the rule base, the ER approach can be directly applied to combine the rules and generate final conclusions [14]. The output of the new input u by the combination is defined as(14)Ou=D1,β1,D2,β2,…,DL,βL.Since the rule consequent in the BRB system has the form of belief distribution, the result of reference in (14) is also a belief distribution, which expresses that if the input is given as u=(u1,u2,…,uN); then the consequent is D1 to a degree β1, D2 to a degree β,…, and DL to a degree βL. The analytical format of the ER algorithm can be used to calculate the combined belief degree βl in Dl as [17](15)βl=μ×∏k=1Kwkβk,l+1-wk∑l=1Lβk,lpppppppp-∏k=1K1-wk∑l=1Lβk,l·1-μ×∏k=1K1-wk-1.Here, (16)μ=∑l=1L∏k=1Kwkβk,l+1-wk∑l=1Lβk,lpppp-L-1∏k=1K1-wk∑l=1Lβk,l-1.Next, we can estimate performance output by the weighted average operator(17)gBRBu=∑l=1LDlβl.


## 3.3. The Parameter Optimization of the BRB System Using the Selected Training Samples

Although the initial belief rules can be constructed by the limited simulations and designers’ knowledge and the performance outputs can be estimated by ER inference of the initial belief rules, the estimation accuracy can be improved if the parameters in the belief rules are fine-tuned through learning from some selected training samples. The adjustable parameters in a rule base include belief degrees (βk,1,βk,2,…,βk,L), rule weights (θ1,θ2,…,θK), and attribute weight (δ1,δ2,…,δN).When we search for the available feasibility regionRF from the approximated acceptability region RA estimated by the BRB, the vertexes of the maximum RF certainly reach to the boundary points of RA, which are all included in the set of critical points {u∣g(u)=Lb,g(u)=Ub}. To a large extent, the estimation accuracy of the boundary points determines the accuracy of the optimal RF. Since the boundary points of RA fall into the boundary gridding (BG) cells as shown in Figure 1, we select the central points of the BG cells as training points and then obtain their performance outputs by circuit simulations as the training samples. Thus, given Q training points uq (q=1,2,…,Q), the error function d(P) between the simulated output g(uq) and the estimated output gBRB(uq) can be defined as [21](18)dP=1Q×∑q=1Qguq-gBRBuq2.Here, d(P) is function of the parameter set P(19)P=βk,l,θk,δi∣k=1,2,…,K,pppppppipppβk,ll=1,2,…,L,i=1,2,…,N.The objective of the training is to minimize the difference d(P) by adjusting the parameters P. Note that the optimal P can be obtained by using gradient-based search methods or nonlinear optimization software packages, such as thefmincon function in the Optimization Toolbox of MATLAB [18].For example, in the case of two-dimensional inputs as shown in Figure1, there are 26 BG cells. Correspondingly, 26 training points are selected, only the extra 26 circuit simulations are needed to get training samples. As a result, totally 152 circuit simulations (126 initial referential points and 26 training points) are required to construct the trained BRB with two-dimensional inputs.


## 4. Tolerance Optimization Methods

In this section, the proposed BRB can be used to identify the optimal feasibility region from the acceptability regions based on the design criterion of maximizing volume of tolerance region.


### 4.1. Choosing Initial Solution of Optimization

Firstly, we have to select an available point and its tolerance from the acceptability region as the initial solution of optimization, since an arbitrary initial point and its tolerance may cause the optimization process which is time-consuming and even becomes trapped in the local optima [3]. There are two ways of choosing the initial solution.The first way is to seek the initial solution directly from the inside points and inside boundary (IB) points that we have obtained by circuit simulations.ForN-dimensional space of design parameters, let SI={uzI∣z=1,2,…,Z} be the set including a total of Z inside points and IB points in which uzI=(uz,1I,uz,2I,…,uz,NI). Set uzI as the central point and search the other points on its both sides along every coordinate direction of the N-dimensional space. In the ith coordinate direction, we can find a certain pair of symmetrical points around uzI; their values in the ith coordination is denoted as xz,i and yz,i, respectively. Because of symmetrical characteristic, we have |uz,iI-xz,i|=|uz,iI-yz,i| and define tz,iI=|uz,iI-xz,i| as the tolerance of uzI in the ith coordinate. Thus, we need to search all pairs of the symmetrical points about uzI in all coordinate directions and find out such a tolerance vector tzI=(tz,1I,tz,2I,…,tz,NI) such that (2tz,1I×2tz,2I×⋯×2tz,NI) is the maximum volume. For z=1,2,…,Z, the initial solution of optimization can be found in SI which has the maximum volume of tolerance.Let us take the 2-dimensional space as an example. Figure2 shows the inside points and IB points in Figure 1. Searching every point, we can determine that the initial solution is uI=(8,5), tI=(3,1). Compared with the other points (e.g., uI=(5,6)), uI=(8,5) has the maximum volume of tolerance (2·3×2·1=12).Figure 2
Searching for the initial solution of optimization in the inside points.Instead of circuit simulations, we use the proposed BRB to estimate more inside points so as to increase the density of the cast points in the acceptability region. Hence, the second way is to search for the initial solution from these estimated points using the above procedure given in the first way. Obviously, the initial solution got by the second way is more accurate than that got by the first way, but it needs more computational loads.Note that, if there are multiple disconnected acceptability regions in the parameter spaceSu; then the above two ways can be implemented in every acceptability region to obtain the corresponding initial solution.


### 4.2. Specifying the Objective Function

According to the criterion of maximizing volume of tolerance region, the objective function can be defined as(20)maxVRFsubjecttoRF⊆RA.From (2), (3), and (17), it can be translated into (21)maxu0,t∏iN2ti,u0=u0_1,u0_2,…,u0_N,ppppppippppippppppppppt=t1,t2,…,tNs.t.Lb≤gBRBu0_1±t1j1,u0_2±t2j2,…,ppppippppppppppu0_N±tNjN≤Ub,s.pit.Lb≤gBRBu0_1±t1j1,u0_2,…,u0_N≤Ub,s.ipt.Lb≤gBRBu0_1,u0_2±t2j2,…,u0_N≤Ub,…,spi.t.Lb≤gBRBu0_1,u0_2,…,u0_N±tNjN≤Ub,pppppipppppppppippppppppipppppi0≤ti≤ubi-lbi2,pppppppppppppppfori=1,2,…,N;j1=1,2,…,T,ppppppppppppppj2=1,2,…,T,…,jN=1,2,…,T.RF is a cube (N=3) or a hypercube (N>3). Hence, if RA is convex, then T=1. We only need to ensure that 2N vertexes of RF fall into the acceptability region RA. If RA is nonconvex, then T⩾2 because it is necessary that the extra 2TN+2TN-2N points on the N sides of RF must fall into RA. T can be taken according to experts’ experiences about the shape of RA and the tradeoff between the computational burden and accuracy of optimization. The initial u0 and t is given by the methods in Section 4.1. Similar with the optimization of the BRB parameters in Section 3.3, the tolerance optimization also can be solved by thefmincon function in the MATLAB.


## 4.1. Choosing Initial Solution of Optimization

Firstly, we have to select an available point and its tolerance from the acceptability region as the initial solution of optimization, since an arbitrary initial point and its tolerance may cause the optimization process which is time-consuming and even becomes trapped in the local optima [3]. There are two ways of choosing the initial solution.The first way is to seek the initial solution directly from the inside points and inside boundary (IB) points that we have obtained by circuit simulations.ForN-dimensional space of design parameters, let SI={uzI∣z=1,2,…,Z} be the set including a total of Z inside points and IB points in which uzI=(uz,1I,uz,2I,…,uz,NI). Set uzI as the central point and search the other points on its both sides along every coordinate direction of the N-dimensional space. In the ith coordinate direction, we can find a certain pair of symmetrical points around uzI; their values in the ith coordination is denoted as xz,i and yz,i, respectively. Because of symmetrical characteristic, we have |uz,iI-xz,i|=|uz,iI-yz,i| and define tz,iI=|uz,iI-xz,i| as the tolerance of uzI in the ith coordinate. Thus, we need to search all pairs of the symmetrical points about uzI in all coordinate directions and find out such a tolerance vector tzI=(tz,1I,tz,2I,…,tz,NI) such that (2tz,1I×2tz,2I×⋯×2tz,NI) is the maximum volume. For z=1,2,…,Z, the initial solution of optimization can be found in SI which has the maximum volume of tolerance.Let us take the 2-dimensional space as an example. Figure2 shows the inside points and IB points in Figure 1. Searching every point, we can determine that the initial solution is uI=(8,5), tI=(3,1). Compared with the other points (e.g., uI=(5,6)), uI=(8,5) has the maximum volume of tolerance (2·3×2·1=12).Figure 2
Searching for the initial solution of optimization in the inside points.Instead of circuit simulations, we use the proposed BRB to estimate more inside points so as to increase the density of the cast points in the acceptability region. Hence, the second way is to search for the initial solution from these estimated points using the above procedure given in the first way. Obviously, the initial solution got by the second way is more accurate than that got by the first way, but it needs more computational loads.Note that, if there are multiple disconnected acceptability regions in the parameter spaceSu; then the above two ways can be implemented in every acceptability region to obtain the corresponding initial solution.


## 4.2. Specifying the Objective Function

According to the criterion of maximizing volume of tolerance region, the objective function can be defined as(20)maxVRFsubjecttoRF⊆RA.From (2), (3), and (17), it can be translated into (21)maxu0,t∏iN2ti,u0=u0_1,u0_2,…,u0_N,ppppppippppippppppppppt=t1,t2,…,tNs.t.Lb≤gBRBu0_1±t1j1,u0_2±t2j2,…,ppppippppppppppu0_N±tNjN≤Ub,s.pit.Lb≤gBRBu0_1±t1j1,u0_2,…,u0_N≤Ub,s.ipt.Lb≤gBRBu0_1,u0_2±t2j2,…,u0_N≤Ub,…,spi.t.Lb≤gBRBu0_1,u0_2,…,u0_N±tNjN≤Ub,pppppipppppppppippppppppipppppi0≤ti≤ubi-lbi2,pppppppppppppppfori=1,2,…,N;j1=1,2,…,T,ppppppppppppppj2=1,2,…,T,…,jN=1,2,…,T.RF is a cube (N=3) or a hypercube (N>3). Hence, if RA is convex, then T=1. We only need to ensure that 2N vertexes of RF fall into the acceptability region RA. If RA is nonconvex, then T⩾2 because it is necessary that the extra 2TN+2TN-2N points on the N sides of RF must fall into RA. T can be taken according to experts’ experiences about the shape of RA and the tradeoff between the computational burden and accuracy of optimization. The initial u0 and t is given by the methods in Section 4.1. Similar with the optimization of the BRB parameters in Section 3.3, the tolerance optimization also can be solved by thefmincon function in the MATLAB.


## 5. Numerical Studies

In this section, two numerical studies and an industrial case are given to illustrate the procedure of using the BRB system to solve tolerance design problem.


### 5.1. Two-Dimensional Rosenbrock Function Example

The Rosenbrock function is a well-known benchmark for assessing nonlinear numerical optimization algorithms [22]. In our context, the two-dimensional Rosenbrock function is considered as the performance function to construct the nonconvex and disconnected acceptability regions. The proposed method can be tested using these complex acceptability regions.Set the acceptability regions as(22)RA=u∣5≤gu≤10.Here, the performance function is (23)gu=100u2-u122+1-u12.The design parameters are -0.4⩽u1⩽+0.4, -0.4⩽u2⩽+0.4, u=(u1,u2). From Figure 3, it can be seen that the RA includes two disconnected and nonconvex subregions denoted as RA,1 and RA,2, respectively.Figure 3
The acceptability regionsRA,1, RA,2 and IB, EB inside and training points.According to the procedure of building BRB system given in Section3, suppose 17 referential values are uniformly selected for the two design parameters u1 and u2, respectively; we then have 172 initial referential points in Su={u=(u1,u2)∣-0.4⩽ui⩽0.4i,i=1,2}. The corresponding 289 circuit simulations are implemented to generate the initial referential performance outputs. In this numerical example, the circuit simulations are replaced by calculations of analytical formula (23). Based on the constraints given by (22), the 68 inside pints and 65 IB points are picked out from the 289 initial referential points as shown in Figure 3. Here, the k-means clustering algorithm is used to divide these 289 points into two parts so as to recognize RA,1 and RA,2. Thus, the 79 EB points can be found out from the 79 BG cells. As a result, we obtain the sets of antecedent referential values as(24)A1=-0.4,-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,ooo0,0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,A2=-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,0,0.15,ooo0.2,0.25,0.3,0.35,0.4.The referential values of parameter input uk=(uk,1,uk,2), uk∈A1×A2, k=1,2,…,K, K=128. By (8), we have Lb′=2, Ub′=14. Suppose L=7 referential values of performance output are uniformly taken from the interval [Lb′,Ub′]; then D1=2,D2=4,…,D7=14. Thus, we can construct the initial belief rules by (9) and (10) and list the partial parameters of the initial rule base in Table 2.Table 2
The partial parameters of the initial rule base.
k
u
k
,
1
u
k
,
2
β
k
,
1
β
k
,
2
β
k
,
3
β
k
,
4
β
k
,
5
β
k
,
6
β
k
,
7
1
−0.4
−0.15
0
0
0
0
0.215
0.785
0
2
−0.4
−0.1
0
0
0
0.64
0.36
0
0
3
−0.4
−0.05
0
0
0.815
0.185
0
0
0
4
−0.4
0
0
0.74
0.26
0
0
0
0
5
−0.4
0.3
0.04
0.96
0
0
0
0
0
6
−0.4
0.35
0
0.215
0.785
0
0
0
0
7
−0.4
0.4
0
0
0.14
0.86
0
0
0
8
−0.35
−0.2
0
0
0
0
0
0.8884
0.1116
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
121
0.35
−0.1
0
0.3134
0.6866
0
0
0
0
122
0.35
−0.05
0.3009
0.6991
0
0
0
0
0
123
0.35
0.3
0.2134
0.7866
0
0
0
0
0
124
0.35
0.35
0
0.2009
0.7991
0
0
0
0
125
0.35
0.4
0
0
0
0.9384
0.0616
0
0
126
0.4
−0.2
0
0
0
0
0
0.34
0.66
127
0.4
−0.15
0
0
0
0.015
0.985
0
0
128
0.4
−0.1
0
0
0.44
0.56
0
0
0In Table2, the other parameters θk and δi are set equal to 1, respectively.Next, set the 79 central points of the BG cells as training points and then obtain their performance outputs by circuit simulations as the training samples as shown in Figure3. By (18), we can optimize the parameter set of the initial rule base and get the trained rule base. Table 3 lists the partial parameters obtained by training; here, after training, δi is still equal to 1 for i=1,2.Table 3
The partial parameters in the trained rule base.
k
u
k
,
1
u
k
,
2
θ
k
β
k
,
1
β
k
,
2
β
k
,
3
β
k
,
4
β
k
,
5
β
k
,
6
β
k
,
7
1
−0.4
−0.15
1
0.0177
0.0005
0
0
0
0.9818
0
2
−0.4
−0.1
1
0
0
0
1
0
0
0
3
−0.4
−0.05
0.9789
0.0411
0.03
0.9289
0
0
0
0
4
−0.4
0
1
0.0982
0.9018
0
0
0
0
0
5
−0.4
0.3
1
0.0118
0.9882
0
0
0
0
0
6
−0.4
0.35
1
0.0185
0.11
0.8715
0
0
0
0
7
−0.4
0.4
0.9990
0.0185
0
0.9815
0
0
0
0
8
−0.35
−0.2
1
0.0283
0
0
0
0
0.9152
0.0565
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
121
0.35
−0.1
1
0.0308
0.0004
0
0.2358
0.7332
0
0
122
0.35
−0.05
0.9746
0.0625
0
0.9375
0
0
0
0
123
0.35
0.3
0.9990
0.5379
0.4620
0
0
0
0
0
124
0.35
0.35
1
0.0156
0.0587
0.9254
0
0
0
0
125
0.35
0.4
1
0.0158
0
0.1587
0.8255
0
0
0
126
0.4
−0.2
0.9677
0.0391
0.9609
0
0
0
0
0
127
0.4
−0.15
0.9878
0.0429
0.0039
0
0
0
0.2058
0.7474
128
0.4
−0.1
1
0.0308
0.0004
0
0.2355
0.7333
0
0We uniformly select 404 sample points fromRA,1 and RA,2 and calculate their performance outputs by the initial BRB and trained BRB. Figure 4 shows the relative errors of the trained BRB and the initial BRB, respectively. Obviously, the training process improves the estimation accuracy of the BRB system.Figure 4
The relative errors of the initial BRB and the trained BRB.After obtaining the trained BRB, the next step is to optimize tolerances by the proposed method in Section4. Firstly, according to the second way introduced in Section 4.1, we reuse the inside points in Figure 4 generated by the trained BRB to find out the initial solution RF,BRBI of optimization. Secondly, by using the objective function (T=10) in (21), we can get the optimal result RF,BRBo together with the optimal RFo by circuit simulations shown in Figure 5. Tables 4 and 5 list RF,BRBI, RF,BRBo, RFo, and the relative errors reBRBo and reBRBI between the estimated RF,BRBI, RF,BRBo, and RFo for RA,1 and RA,2, respectively. Here, reBRBo is defined as(25)reBRBo=100%×1N∑i=1Nuio-tio-uBRB,io-tBRBiooooooooooooooo+uio+tio-uBRB,io+tBRB,iooooooooooooooouio-tio-uBRB,io-tBRBio·2tio-1.reBRBI can be gotten using the same way, in this example, N=2. Obviously, RF,BRBo is very close to the optimal RFo.Table 4
Comparisons of the optimization results forRA,1.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
u
o
t
o
u
BRB
o
t
BRB
o
r
e
BRB
o
u
BRB
I
t
BRB
I
r
e
BRB
I
(−0.026, 0.266)
(0.186, 0.033)
(−0.0255, 0.265)
(0.191, 0.035)
4.4%
(−0.02, 0.26)
(0.15, 0.04)
20.3%Table 5
Comparisons of the optimization results forRA,2.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
u
o
t
o
u
BRB
o
t
BRB
o
r
e
BRB
o
u
BRB
I
t
BRB
I
r
e
BRB
I
(0.0164, −0.234)
(0.182, 0.033)
(0.0195, 0.234)
(0.185, 0.034)
2.4%
(0.02, −0.24)
(0.14, 0.04)
22.1%Figure 5
The approximation ofRA and optimization of RF by using the trained BRB.


### 5.2. Eight-Dimensional Quadratic Performance Function

In the yield estimation of integrated circuits, the high-dimensional quadratics or higher order polynomials are usually used to model the performances of interest. Here, we give an eight-dimensional quadratic function to test the proposed method. Without loss of generality, assume a circuit performance can be expressed as an 8-dimension quadratic function whose symmetric matrixes are diagonal or block diagonal [6, 25, 26]. The function for this example is taken as (26)gu=a0+Ju+12uTHu,where a0=6 and matrices J and H are as follows:(27)J=-11023124,H=120000002-100000000470000003-3000000001-5000000620000000011000000-13.Set the acceptability region as(28)RA=u∣346≤gu≤360.The design parameter vector u=(u1,u2,…,u8), Su={u∣4.5⩽ui⩽5.5i,i=1,2,…,8}.Suppose the numbers of the referential values uniformly selected forui are Ji=3 for i=1,4,6,8 and Ji=3 for i=2,3,5,7, respectively. The corresponding 1296 circuit simulations are implemented to generate the initial referential performance outputs. Based on the constraints given by (28), 165 IB points are picked out from 1296 initial referential points. Thus, 458 EB points can be found out. As a result, we obtain the initial rule base consisting of 623 belief rules, in which L=26 referential values of performance output are uniformly taken from the interval [Lb′,Ub′]; here, Lb′=302, Ub′=406, and then D1=302,D2=306,…,D26=406. The rule weights and attribute weights are all set equal to 1, respectively.By testing the selected 1373 sample points inRA, the maximum of the relative errors of the initial BRB is only 0.25%; it is accurate enough for tolerance design. Hence, the training process is not implemented any longer so as to reduce the computational burden. According to the second way introduced in Section 4.1, we reuse the 1373 sample points in RA generated by the initial BRB to find out the initial solution RF,BRBI of optimization. Secondly, by using the objective function in (21) (T=2), we can get the optimal result RF,BRBo together with the optimal RFo by circuit simulations. Table 6 lists RF,BRBI, RF,BRBo, RFo, and rBRBo and rBRBI between the estimated RF,BRBI, RF,BRBo, and RFo respectively. Obviously, RF,BRBo accurately approximates the optimal RFo.Table 6
Comparisons of the optimization results forRA.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
i
u
i
o
t
i
o
u
BRB
,
i
o
t
BRB
,
i
o
r
e
BRB
o
u
BRB
,
i
I
t
BRB
,
i
I
r
e
BRB
I
1
4.9929
0.0738
4.9891
0.0743
5.8%
4.99
0.065
43.2%
2
4.9913
0.0887
4.9896
0.0889
4.99
0.065
3
4.9994
0.0157
4.9969
0.0164
4.99
0.045
4
4.9925
0.0774
4.9891
0.0778
4.99
0.065
5
4.9922
0.0802
4.9893
0.0808
4.99
0.065
6
4.9928
0.0747
4.9891
0.0753
4.99
0.055
7
4.9915
0.0868
4.9895
0.0872
4.99
0.055
8
4.9940
0.0642
4.9888
0.0652
4.99
0.05


### 5.3. Application in the Tolerance Design of Railway Track Circuit

Railway track circuit is an essential component of information transmission system between track and vehicle and the automatic train control system [6, 27]. It uses a specific carrier frequency to transmit the coded control information to the train. The application considered in this paper concerns the parameters designs of the insulating section of a jointless track circuit named as ZPW 2000A widely used on Chinese high-speed railway lines. This device will first be described and the problem addressed will be exposed.The railway track is divided into different sections. Each one of them has a specific ZPW 2000A consisting of main track circuit and short track circuit [28] (see Figure 6). In main track circuit, a transmitter in sending end delivers an alternating current with the specific modulation frequency; a receiver in receiving end demodulates the currents signal transmitted along two rails and controls the track relay. The short track circuit is an electric insulating section composed of two tuning units (BA1 and BA2) and a track air-core inductor (SVA), which can achieve good signal insulation between adjacent track circuits through its resonance characteristic.Figure 6
System structure of ZPW 2000A.Figure7 shows the schematic diagram of the short track circuit of G2. We take the analysis of the resonance performance of the tuning unit 2 (BA2) as an example. L2  C2  C3 parallel resonance with the track inductance Lv happens at its frequency 2300 Hz. Lv is given as Lv=0.5L+0.5L//Ls. L(μH) and Ls(μH) are respective inductance values of rails and SVA in short track circuit of G2. “//” denotes parallel operator. The impedance value of L2  C2  C3  Lv circuit gets to maximum about 2 Ω. It is equivalent to “open circuit” and reduces the attenuation of G2 signal intensity. In our context, parallel resonance frequency g(L2,C2,C3) is considered as the performance function. In practical parameters designs, it is generally required that the nominal values of g is 2300 Hz [28]. The maximum frequency drifts are ±11 Hz. So we set the acceptability region as (29)RA=u∣2289Hz≤gu≤2311Hz.Here, the design parameters u1=L2, u2=C2, u3=C3, u=(u1,u2,u3), and Su={u∣96μH⩽u1⩽98μH,88μF⩽u2⩽92μF,261μF⩽u3⩽264μF}. The whole circuit model of ZPW 2000A has been built using the software Simulink in [29]. Here, we can use the Simulink-based circuit simulator to generate the simulation data of performance output g(u).Figure 7
The short track circuit of G2.We uniformly selectJ1=7, J2=9, and J3=8 referential values for u1, u2, and u3, respectively. The corresponding 504 circuit simulations are implemented to generate the initial referential performance outputs. Based on the constraints given by (29), 347 inside points and 41 IB points are picked out from the 504 initial referential points. Thus, the 41 EB points can be found out. As a result, we obtain the initial rule base consisting of 429 belief rules, in which L=10 referential values of performance output are uniformly taken from the interval [Lb′,Ub′]; here, Lb′=2287, Ub′=2314, and then D1=2287,D2=2290,…,D10=2314. The rule weights and attribute weights are all set equal to 1, respectively.Similar with example 2, the given initial BRB is accurate enough for tolerance design, so the training process is cancelled. Figure8 shows the approximated RA which is a complex polyhedron. The selected 2470 sample points are generated by the initial BRB, which are used to seek for the initial solution RF,BRBI. Using the objective function in (21) (T=2), we can get the optimal result RF,BRBo (see Figure 8) together with the optimal RFo by Simulink simulations. Table 7 lists RF,BRBI, RF,BRBo, RFo, and rBRBo and rBRBI between the estimated RF,BRBI, RF,BRBo, and RFo, respectively. Obviously, RF,BRBo accurately approximates the optimal RFo.Table 7
Comparisons of the optimization results forRA.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
i
u
i
o
t
i
o
u
BRB
,
i
o
t
BRB
,
i
o
r
e
BRB
o
u
BRB
,
i
I
t
BRB
,
i
I
r
e
BRB
I
1
97.0506
0.5335
97.0153
0.5402
6.5%
96.9
0.6
11.1%
2
91.0297
0.8900
91.1083
0.8917
91
0.9
3
262.2203
1.1877
262.2363
1.2363
262.2
1.2Figure 8
The approximatedRA and the optimal RF given by the initial BRB.


## 5.1. Two-Dimensional Rosenbrock Function Example

The Rosenbrock function is a well-known benchmark for assessing nonlinear numerical optimization algorithms [22]. In our context, the two-dimensional Rosenbrock function is considered as the performance function to construct the nonconvex and disconnected acceptability regions. The proposed method can be tested using these complex acceptability regions.Set the acceptability regions as(22)RA=u∣5≤gu≤10.Here, the performance function is (23)gu=100u2-u122+1-u12.The design parameters are -0.4⩽u1⩽+0.4, -0.4⩽u2⩽+0.4, u=(u1,u2). From Figure 3, it can be seen that the RA includes two disconnected and nonconvex subregions denoted as RA,1 and RA,2, respectively.Figure 3
The acceptability regionsRA,1, RA,2 and IB, EB inside and training points.According to the procedure of building BRB system given in Section3, suppose 17 referential values are uniformly selected for the two design parameters u1 and u2, respectively; we then have 172 initial referential points in Su={u=(u1,u2)∣-0.4⩽ui⩽0.4i,i=1,2}. The corresponding 289 circuit simulations are implemented to generate the initial referential performance outputs. In this numerical example, the circuit simulations are replaced by calculations of analytical formula (23). Based on the constraints given by (22), the 68 inside pints and 65 IB points are picked out from the 289 initial referential points as shown in Figure 3. Here, the k-means clustering algorithm is used to divide these 289 points into two parts so as to recognize RA,1 and RA,2. Thus, the 79 EB points can be found out from the 79 BG cells. As a result, we obtain the sets of antecedent referential values as(24)A1=-0.4,-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,ooo0,0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,A2=-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,0,0.15,ooo0.2,0.25,0.3,0.35,0.4.The referential values of parameter input uk=(uk,1,uk,2), uk∈A1×A2, k=1,2,…,K, K=128. By (8), we have Lb′=2, Ub′=14. Suppose L=7 referential values of performance output are uniformly taken from the interval [Lb′,Ub′]; then D1=2,D2=4,…,D7=14. Thus, we can construct the initial belief rules by (9) and (10) and list the partial parameters of the initial rule base in Table 2.Table 2
The partial parameters of the initial rule base.
k
u
k
,
1
u
k
,
2
β
k
,
1
β
k
,
2
β
k
,
3
β
k
,
4
β
k
,
5
β
k
,
6
β
k
,
7
1
−0.4
−0.15
0
0
0
0
0.215
0.785
0
2
−0.4
−0.1
0
0
0
0.64
0.36
0
0
3
−0.4
−0.05
0
0
0.815
0.185
0
0
0
4
−0.4
0
0
0.74
0.26
0
0
0
0
5
−0.4
0.3
0.04
0.96
0
0
0
0
0
6
−0.4
0.35
0
0.215
0.785
0
0
0
0
7
−0.4
0.4
0
0
0.14
0.86
0
0
0
8
−0.35
−0.2
0
0
0
0
0
0.8884
0.1116
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
121
0.35
−0.1
0
0.3134
0.6866
0
0
0
0
122
0.35
−0.05
0.3009
0.6991
0
0
0
0
0
123
0.35
0.3
0.2134
0.7866
0
0
0
0
0
124
0.35
0.35
0
0.2009
0.7991
0
0
0
0
125
0.35
0.4
0
0
0
0.9384
0.0616
0
0
126
0.4
−0.2
0
0
0
0
0
0.34
0.66
127
0.4
−0.15
0
0
0
0.015
0.985
0
0
128
0.4
−0.1
0
0
0.44
0.56
0
0
0In Table2, the other parameters θk and δi are set equal to 1, respectively.Next, set the 79 central points of the BG cells as training points and then obtain their performance outputs by circuit simulations as the training samples as shown in Figure3. By (18), we can optimize the parameter set of the initial rule base and get the trained rule base. Table 3 lists the partial parameters obtained by training; here, after training, δi is still equal to 1 for i=1,2.Table 3
The partial parameters in the trained rule base.
k
u
k
,
1
u
k
,
2
θ
k
β
k
,
1
β
k
,
2
β
k
,
3
β
k
,
4
β
k
,
5
β
k
,
6
β
k
,
7
1
−0.4
−0.15
1
0.0177
0.0005
0
0
0
0.9818
0
2
−0.4
−0.1
1
0
0
0
1
0
0
0
3
−0.4
−0.05
0.9789
0.0411
0.03
0.9289
0
0
0
0
4
−0.4
0
1
0.0982
0.9018
0
0
0
0
0
5
−0.4
0.3
1
0.0118
0.9882
0
0
0
0
0
6
−0.4
0.35
1
0.0185
0.11
0.8715
0
0
0
0
7
−0.4
0.4
0.9990
0.0185
0
0.9815
0
0
0
0
8
−0.35
−0.2
1
0.0283
0
0
0
0
0.9152
0.0565
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
⋮
121
0.35
−0.1
1
0.0308
0.0004
0
0.2358
0.7332
0
0
122
0.35
−0.05
0.9746
0.0625
0
0.9375
0
0
0
0
123
0.35
0.3
0.9990
0.5379
0.4620
0
0
0
0
0
124
0.35
0.35
1
0.0156
0.0587
0.9254
0
0
0
0
125
0.35
0.4
1
0.0158
0
0.1587
0.8255
0
0
0
126
0.4
−0.2
0.9677
0.0391
0.9609
0
0
0
0
0
127
0.4
−0.15
0.9878
0.0429
0.0039
0
0
0
0.2058
0.7474
128
0.4
−0.1
1
0.0308
0.0004
0
0.2355
0.7333
0
0We uniformly select 404 sample points fromRA,1 and RA,2 and calculate their performance outputs by the initial BRB and trained BRB. Figure 4 shows the relative errors of the trained BRB and the initial BRB, respectively. Obviously, the training process improves the estimation accuracy of the BRB system.Figure 4
The relative errors of the initial BRB and the trained BRB.After obtaining the trained BRB, the next step is to optimize tolerances by the proposed method in Section4. Firstly, according to the second way introduced in Section 4.1, we reuse the inside points in Figure 4 generated by the trained BRB to find out the initial solution RF,BRBI of optimization. Secondly, by using the objective function (T=10) in (21), we can get the optimal result RF,BRBo together with the optimal RFo by circuit simulations shown in Figure 5. Tables 4 and 5 list RF,BRBI, RF,BRBo, RFo, and the relative errors reBRBo and reBRBI between the estimated RF,BRBI, RF,BRBo, and RFo for RA,1 and RA,2, respectively. Here, reBRBo is defined as(25)reBRBo=100%×1N∑i=1Nuio-tio-uBRB,io-tBRBiooooooooooooooo+uio+tio-uBRB,io+tBRB,iooooooooooooooouio-tio-uBRB,io-tBRBio·2tio-1.reBRBI can be gotten using the same way, in this example, N=2. Obviously, RF,BRBo is very close to the optimal RFo.Table 4
Comparisons of the optimization results forRA,1.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
u
o
t
o
u
BRB
o
t
BRB
o
r
e
BRB
o
u
BRB
I
t
BRB
I
r
e
BRB
I
(−0.026, 0.266)
(0.186, 0.033)
(−0.0255, 0.265)
(0.191, 0.035)
4.4%
(−0.02, 0.26)
(0.15, 0.04)
20.3%Table 5
Comparisons of the optimization results forRA,2.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
u
o
t
o
u
BRB
o
t
BRB
o
r
e
BRB
o
u
BRB
I
t
BRB
I
r
e
BRB
I
(0.0164, −0.234)
(0.182, 0.033)
(0.0195, 0.234)
(0.185, 0.034)
2.4%
(0.02, −0.24)
(0.14, 0.04)
22.1%Figure 5
The approximation ofRA and optimization of RF by using the trained BRB.


## 5.2. Eight-Dimensional Quadratic Performance Function

In the yield estimation of integrated circuits, the high-dimensional quadratics or higher order polynomials are usually used to model the performances of interest. Here, we give an eight-dimensional quadratic function to test the proposed method. Without loss of generality, assume a circuit performance can be expressed as an 8-dimension quadratic function whose symmetric matrixes are diagonal or block diagonal [6, 25, 26]. The function for this example is taken as (26)gu=a0+Ju+12uTHu,where a0=6 and matrices J and H are as follows:(27)J=-11023124,H=120000002-100000000470000003-3000000001-5000000620000000011000000-13.Set the acceptability region as(28)RA=u∣346≤gu≤360.The design parameter vector u=(u1,u2,…,u8), Su={u∣4.5⩽ui⩽5.5i,i=1,2,…,8}.Suppose the numbers of the referential values uniformly selected forui are Ji=3 for i=1,4,6,8 and Ji=3 for i=2,3,5,7, respectively. The corresponding 1296 circuit simulations are implemented to generate the initial referential performance outputs. Based on the constraints given by (28), 165 IB points are picked out from 1296 initial referential points. Thus, 458 EB points can be found out. As a result, we obtain the initial rule base consisting of 623 belief rules, in which L=26 referential values of performance output are uniformly taken from the interval [Lb′,Ub′]; here, Lb′=302, Ub′=406, and then D1=302,D2=306,…,D26=406. The rule weights and attribute weights are all set equal to 1, respectively.By testing the selected 1373 sample points inRA, the maximum of the relative errors of the initial BRB is only 0.25%; it is accurate enough for tolerance design. Hence, the training process is not implemented any longer so as to reduce the computational burden. According to the second way introduced in Section 4.1, we reuse the 1373 sample points in RA generated by the initial BRB to find out the initial solution RF,BRBI of optimization. Secondly, by using the objective function in (21) (T=2), we can get the optimal result RF,BRBo together with the optimal RFo by circuit simulations. Table 6 lists RF,BRBI, RF,BRBo, RFo, and rBRBo and rBRBI between the estimated RF,BRBI, RF,BRBo, and RFo respectively. Obviously, RF,BRBo accurately approximates the optimal RFo.Table 6
Comparisons of the optimization results forRA.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
i
u
i
o
t
i
o
u
BRB
,
i
o
t
BRB
,
i
o
r
e
BRB
o
u
BRB
,
i
I
t
BRB
,
i
I
r
e
BRB
I
1
4.9929
0.0738
4.9891
0.0743
5.8%
4.99
0.065
43.2%
2
4.9913
0.0887
4.9896
0.0889
4.99
0.065
3
4.9994
0.0157
4.9969
0.0164
4.99
0.045
4
4.9925
0.0774
4.9891
0.0778
4.99
0.065
5
4.9922
0.0802
4.9893
0.0808
4.99
0.065
6
4.9928
0.0747
4.9891
0.0753
4.99
0.055
7
4.9915
0.0868
4.9895
0.0872
4.99
0.055
8
4.9940
0.0642
4.9888
0.0652
4.99
0.05


## 5.3. Application in the Tolerance Design of Railway Track Circuit

Railway track circuit is an essential component of information transmission system between track and vehicle and the automatic train control system [6, 27]. It uses a specific carrier frequency to transmit the coded control information to the train. The application considered in this paper concerns the parameters designs of the insulating section of a jointless track circuit named as ZPW 2000A widely used on Chinese high-speed railway lines. This device will first be described and the problem addressed will be exposed.The railway track is divided into different sections. Each one of them has a specific ZPW 2000A consisting of main track circuit and short track circuit [28] (see Figure 6). In main track circuit, a transmitter in sending end delivers an alternating current with the specific modulation frequency; a receiver in receiving end demodulates the currents signal transmitted along two rails and controls the track relay. The short track circuit is an electric insulating section composed of two tuning units (BA1 and BA2) and a track air-core inductor (SVA), which can achieve good signal insulation between adjacent track circuits through its resonance characteristic.Figure 6
System structure of ZPW 2000A.Figure7 shows the schematic diagram of the short track circuit of G2. We take the analysis of the resonance performance of the tuning unit 2 (BA2) as an example. L2  C2  C3 parallel resonance with the track inductance Lv happens at its frequency 2300 Hz. Lv is given as Lv=0.5L+0.5L//Ls. L(μH) and Ls(μH) are respective inductance values of rails and SVA in short track circuit of G2. “//” denotes parallel operator. The impedance value of L2  C2  C3  Lv circuit gets to maximum about 2 Ω. It is equivalent to “open circuit” and reduces the attenuation of G2 signal intensity. In our context, parallel resonance frequency g(L2,C2,C3) is considered as the performance function. In practical parameters designs, it is generally required that the nominal values of g is 2300 Hz [28]. The maximum frequency drifts are ±11 Hz. So we set the acceptability region as (29)RA=u∣2289Hz≤gu≤2311Hz.Here, the design parameters u1=L2, u2=C2, u3=C3, u=(u1,u2,u3), and Su={u∣96μH⩽u1⩽98μH,88μF⩽u2⩽92μF,261μF⩽u3⩽264μF}. The whole circuit model of ZPW 2000A has been built using the software Simulink in [29]. Here, we can use the Simulink-based circuit simulator to generate the simulation data of performance output g(u).Figure 7
The short track circuit of G2.We uniformly selectJ1=7, J2=9, and J3=8 referential values for u1, u2, and u3, respectively. The corresponding 504 circuit simulations are implemented to generate the initial referential performance outputs. Based on the constraints given by (29), 347 inside points and 41 IB points are picked out from the 504 initial referential points. Thus, the 41 EB points can be found out. As a result, we obtain the initial rule base consisting of 429 belief rules, in which L=10 referential values of performance output are uniformly taken from the interval [Lb′,Ub′]; here, Lb′=2287, Ub′=2314, and then D1=2287,D2=2290,…,D10=2314. The rule weights and attribute weights are all set equal to 1, respectively.Similar with example 2, the given initial BRB is accurate enough for tolerance design, so the training process is cancelled. Figure8 shows the approximated RA which is a complex polyhedron. The selected 2470 sample points are generated by the initial BRB, which are used to seek for the initial solution RF,BRBI. Using the objective function in (21) (T=2), we can get the optimal result RF,BRBo (see Figure 8) together with the optimal RFo by Simulink simulations. Table 7 lists RF,BRBI, RF,BRBo, RFo, and rBRBo and rBRBI between the estimated RF,BRBI, RF,BRBo, and RFo, respectively. Obviously, RF,BRBo accurately approximates the optimal RFo.Table 7
Comparisons of the optimization results forRA.
R
F
o
R
F
,
BRB
o
R
F
,
BRB
I
i
u
i
o
t
i
o
u
BRB
,
i
o
t
BRB
,
i
o
r
e
BRB
o
u
BRB
,
i
I
t
BRB
,
i
I
r
e
BRB
I
1
97.0506
0.5335
97.0153
0.5402
6.5%
96.9
0.6
11.1%
2
91.0297
0.8900
91.1083
0.8917
91
0.9
3
262.2203
1.1877
262.2363
1.2363
262.2
1.2Figure 8
The approximatedRA and the optimal RF given by the initial BRB.


## 6. Conclusions

In this paper, a BRB system is designed to model the acceptability region and optimize the feasibility region of circuit parameters so that the volume of the tolerance region of the circuit can be maximized. Its virtues can be demonstrated by several examples in this paper.The main advantages of this new method are as follows.(i)
The physical meanings of the parameters and structures of the BRB system are transparent and intuitively easy to understand by experts and engineers, so they can participate in the main steps of system modeling (e.g., determining the number of rules by considering the inside points and IB points, choosing the training samples from the BG cells, and determining the attribute weights and rule weights).(ii)
The proposed BRB system is applicable to complex cases, such as highly nonlinear performance function and nonconvex and disconnected feasibility regions.(iii)
The proposed optimization algorithm provides alternative ways to obtain the initial solutions of the optimization problem so as to avoid local optima with a higher chance and improve the efficiency of the algorithm.Like all other deterministic methods, the proposed method also suffers from exponential explosion of computational costs when the dimension of a design parameter space increases. However, when there are strong correlations between the design parameters, the correlation analysis methods can be used to find out less independent variables or principal components so as to reduce the dimension of design parameter space as analyzed in [30].



---

*Source: 908027-2015-06-08.xml*