Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 908027, 12 pages

http://dx.doi.org/10.1155/2015/908027

## Circuit Tolerance Design Using Belief Rule Base

^{1}Institute of System Science and Control Engineering, School of Automation, Hangzhou Dianzi University, Hangzhou 310018, China^{2}Manchester Business School, The University of Manchester, Manchester M15 6PB, UK

Received 30 October 2014; Accepted 31 December 2014

Academic Editor: Gang Li

Copyright © 2015 Xiao-Bin Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 (). Thus, the optimal center and tolerances (i.e., feasibility region ) of component parameters can be found in the approximated region . 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 Section 2. 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]here is a vector of design parameters. is the performance index or function of an electrical circuit. and 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 of is characterized by a nominal value and a tolerance , .

The acceptability region is defined as [3, 22]If , then the product is acceptable; otherwise it is unacceptable. The tolerance region is defined as [3, 22]

When given a nominal value and a tolerance , the corresponding parametric yield is defined as [3, 22]Here, is the number of total products, is the number of acceptable products (), and () is the volume of a region. is defined as the feasibility region which is a subset of at the intersection between and .

When given the design constraints , the actual goal of tolerance design is to maximize (i.e., to seek for the maximum and corresponding ) so that a 100% yield is achievable. In this case, the maximum is equal to the maximum [3, 22]. In tolerance design, the key step is to calculate performance 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 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.