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The Neutral Grounding Resistor Sizing Using an Analytical Method Based on Nonlinear Transformer Model for Inrush Current MitigationGholamabas M.H.Hajivar Shahid Chamran University,Ahvaz, Iranhajivar@S.S.MortazaviShahid Chamran University,Ahvaz, IranMortazavi_s@scu.ac.irMohsen SanieiShahid Chamran University,Ahvaz, IranMohsen.saniei@Abstract-It was found that a neutral resistor together with 'simultaneous' switching didn't have any effect on either the magnitudes or the time constant of inrush currents. The pre-insertion resistors were recommended as the most effective means of controlling inrush currents. Through simulations, it was found that the neutral resistor had little effect on reducing the inrush current peak or even the rate of decay as compared to the cases without a neutral resistor. The use of neutral impedances was concluded to be ineffective compared to the use of pre-insertion resistors. This finding was explained by the low neutral current value as compared to that of high phase currents during inrush. The inrush currents could be mitigated by using a neutral resistor when sequential switching is implemented. From the sequential energizing scheme performance, the neutral resistor size plays the significant role in the scheme effectiveness. Through simulation, it was found that a few ohms neutral grounding resistor can effectively achieve inrush currents reduction. If the neutral resistor is directly selected to minimize the peak of the actual inrush current, a much lower resistor value could be found.This paper presents an analytical method to select optimal neutral grounding resistor for mitigation of inrush current. In this method nonlinearity and core loss of the transformer has been modeled and derived analytical equations.Index Terms--Inrush current, neutral grounding resistor, transformerI.I NTRODUCTIONThe energizing of transformers produces high inrush currents. The nature of inrush currents have rich in harmonics coupled with relatively a long duration, leads to adverse effects on the residual life of the transformer, malfunction of the protection system [1] and power quality [2]. In the power-system industry, two different strategies have been implemented to tackle the problem of transformer inrush currents. The first strategy focuses on adapting to the effects of inrush currents by desensitizing the protection elements. Other approaches go further by 'over-sizing' the magnetic core to achieve higher saturation flux levels. These partial countermeasures impose downgrades on the system's operational reliability, considerable increases unit cost, high mechanical stresses on the transformer and lead to a lower power quality. The second strategy focuses on reducing the inrush current magnitude itself during the energizing process. Minimizing the inrush current will extend the transformer's lifetime and increase the reliability of operation and lower maintenance and down-time costs. Meanwhile, the problem of protection-system malfunction is eliminated during transformer energizing. The available inrush current mitigation consist "closing resistor"[3], "control closing of circuit breaker"[4],[5], "reduction of residual flux"[6], "neutral resistor with sequential switching"[7],[8],[9].The sequential energizing technique presents inrush-reduction scheme due to transformer energizing. This scheme involves the sequential energizing of the three phases transformer together with the insertion of a properly sized resistor at the neutral point of the transformer energizing side [7] ,[8],[9] (Fig. 1).The neutral resistor based scheme acts to minimize the induced voltage across the energized windings during sequential switching of each phase and, hence, minimizes the integral of the applied voltage across the windings.The scheme has the main advantage of being a simpler, more reliable and more cost effective than the synchronous switching and pre-insertion resistor schemes. The scheme has no requirements for the speed of the circuit breaker or the determination of the residual flux. Sequential switching of the three phases can be implemented through either introducing a mechanical delay between each pole in the case of three phase breakers or simply through adjusting the breaker trip-coil time delay for single pole breakers.A further study of the scheme revealed that a much lower resistor size is equally effective. The steady-state theory developed for neutral resistor sizing [8] is unable to explain this phenomenon. This phenomenon must be understood using transient analysis.Fig. 1. The sequential phase energizing schemeUPEC201031st Aug - 3rd Sept 2010The rise of neutral voltage is the main limitation of the scheme. Two methods present to control the neutral voltage rise: the use of surge arrestors and saturated reactors connected to the neutral point. The use of surge arresters was found to be more effective in overcoming the neutral voltage rise limitation [9].The main objective of this paper is to derive an analytical relationship between the peak of the inrush current and the size of the resistor. This paper presents a robust analytical study of the transformer energizing phenomenon. The results reveal a good deal of information on inrush currents and the characteristics of the sequential energizing scheme.II. SCHEME PERFORMANCESince the scheme adopts sequential switching, each switching stage can be investigated separately. For first-phase switching, the scheme's performance is straightforward. The neutral resistor is in series with the energized phase and this resistor's effect is similar to a pre-insertion resistor.The second- phase energizing is one of the most difficult to analyze. Fortunately, from simulation studies, it was found that the inrush current due to second-phase energizing is lower than that due to first-phase energizing for the same value of n R [9]. This result is true for the region where the inrush current of the first-phase is decreasing rapidly as n R increases. As a result, when developing a neutral-resistor-sizing criterion, the focus should be directed towards the analysis of the first-phase energizing.III. A NALYSIS OF F IRST -P HASE E NERGIZING The following analysis focuses on deriving an inrush current waveform expression covering both the unsaturatedand saturated modes of operation respectively. The presented analysis is based on a single saturated core element, but is suitable for analytical modelling of the single-phase transformers and for the single-phase switching of three-phase transformers. As shown in Fig. 2, the transformer's energized phase was modeled as a two segmented saturated magnetizing inductance in series with the transformer's winding resistance, leakage inductance and neutral resistance. The iron core non-l inear inductance as function of the operating flux linkages is represented as a linear inductor inunsaturated ‘‘m l ’’ and saturated ‘‘s l ’’ modes of operation respectively. (a)(b)Fig. 2. (a) Transformer electrical equivalent circuit (per-phase) referred to the primary side. (b) Simplified, two slope saturation curve.For the first-phase switching stage, the equivalent circuit represented in Fig. 2(a) can accurately represent behaviour of the transformer for any connection or core type by using only the positive sequence Flux-Current characteristics. Based on the transformer connection and core structure type, the phases are coupled either through the electrical circuit (3 single phase units in Yg-D connection) or through the Magnetic circuit (Core type transformers with Yg-Y connection) or through both, (the condition of Yg-D connection in an E-Core or a multi limb transformer). The coupling introduced between the windings will result in flux flowing through the limbs or magnetic circuits of un-energized phases. For the sequential switching application, the magnetic coupling will result in an increased reluctance (decreased reactance) for zero sequence flux path if present. The approach presented here is based on deriving an analytical expression relating the amount of inrush current reduction directly to the neutral resistor size. Investigation in this field has been done and some formulas were given to predict the general wave shape or the maximum peak current.A. Expression for magnitude of inrush currentIn Fig. 2(a), p r and p l present the total primary side resistance and leakage reactance. c R shows the total transformer core loss. Secondary side resistance sp r and leakage reactance sp l as referred to primary side are also shown. P V and s V represent the primary and secondary phase to ground terminal voltages, respectively.During first phase energizing, the differential equation describing behaviour of the transformer with saturated ironcore can be written as follows:()())sin((2) (1)φω+⋅⋅=⋅+⋅+⋅+=+⋅+⋅+=t V (t)V dtdi di d λdt di l (t)i R r (t)V dt d λdt di l (t)i R r (t)V m P ll p pp n p P p p p n p PAs the rate of change of the flux linkages with magnetizing current dt d /λcan be represented as an inductance equal to the slope of the i −λcurve, (2) can be re-written as follows;()(3) )()()(dtdi L dt di l t i R r t V lcore p p P n p P ⋅+⋅+⋅+=λ (4) )()(L core l p c l i i R dtdi−⋅=⋅λ⎩⎨⎧==sml core L L di d L λλ)(s s λλλλ>≤The general solution of the differential equations (3),(4) has the following form;⎪⎩⎪⎨⎧>−⋅⋅+−⋅+−−⋅+≤−⋅⋅+−⋅+−⋅=(5) )sin(//)()( )sin(//)(s s 22222221211112121111λλψωττλλψωττt B t e A t t e i A t B t e A t e A t i s s pSubscripts 11,12 and 21,22 denote un-saturated and saturated operation respectively. The parameters given in the equation (5) are given by;() )(/12221σ⋅++⎟⎟⎠⎞⎜⎜⎝⎛⋅−++⋅=m p c p m n p c m m x x R x x R r R x V B()2222)(/1σ⋅++⎟⎟⎠⎞⎜⎜⎝⎛⋅−++⋅=s p c p s n p c s m x x R x x R r R x V B⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅−+++=⋅−−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−c p m n p m p c m R x x R r x x R x σφψ111tan tan ⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅−+++=⋅−−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−c p s n p s p c m R R r x x R x σφψ112tan tan )sin(111211ψ⋅=+B A A )sin(222221s t B A A ⋅−⋅=+ωψ mp n p m p m p m p c xx R r x x x x x x R ⋅⋅+⋅−⋅+−⋅+⋅⋅⋅=)(4)()(21211σστm p n p m p m p m p c xx R r x x x x x x R ⋅⋅+⋅−⋅++⋅+⋅⋅⋅=)(4)()(21212σστ s p n p s p s p s p xx R r x x x x x x c R ⋅⋅+⋅−⋅+−⋅+⋅⋅⋅=)(4)()(21221σστ sp n p s p s p sp c xx R r x x x x x x R ⋅⋅+⋅−⋅++⋅+⋅⋅⋅=)(4)()(21222σστ ⎟⎟⎠⎞⎜⎜⎝⎛−⋅==s rs s ri i λλλ10 cnp R R r ++=1σ21221112 , ττττ>>>>⇒>>c R , 012≈A , 022≈A According to equation (5), the required inrush waveform assuming two-part segmented i −λcurve can be calculated for two separate un-saturated and saturated regions. For thefirst unsaturated mode, the current can be directly calculated from the first equation for all flux linkage values below the saturation level. After saturation is reached, the current waveform will follow the second given expression for fluxlinkage values above the saturation level. The saturation time s t can be found at the time when the current reaches the saturation current level s i .Where m λ,r λ,m V and ωare the nominal peak flux linkage, residual flux linkage, peak supply voltage and angular frequency, respectivelyThe inrush current waveform peak will essentially exist during saturation mode of operation. The focus should be concentrated on the second current waveform equation describing saturated operation mode, equation (5). The expression of inrush current peak could be directly evaluated when both saturation time s t and peak time of the inrush current waveform peak t t =are known [9].(10))( (9) )(2/)(222222121//)()(2B eA t e i A peak peak t s t s n peak n n peak R I R R t +−⋅+−−⋅+=+=ττωψπThe peak time peak t at which the inrush current will reachits peak can be numerically found through setting the derivative of equation (10) with respect to time equal to zero at peak t t =.()(11) )sin(/)(022222221212221/ψωωττττ−⋅⋅⋅−−−⋅+−=+−⋅peak t s t B A t te A i peak s peakeThe inrush waveform consists of exponentially decaying'DC' term and a sinusoidal 'AC' term. Both DC and AC amplitudes are significantly reduced with the increase of the available series impedance. The inrush waveform, neglecting the relatively small saturating current s i ,12A and 22A when extremely high could be normalized with respect to theamplitude of the sinusoidal term as follows; (12) )sin(/)()(2221221⎥⎦⎤⎢⎣⎡−⋅+−−⋅⋅=ψωτt t t e B A B t i s p(13) )sin(/)()sin()( 22221⎥⎦⎤⎢⎣⎡−⋅+−−⋅⋅−⋅=ψωτωψt t t e t B t i s s p ))(sin()( 2s n n t R R K ⋅−=ωψ (14) ωλλλφλφωλλφωmm m r s s t r m s mV t dt t V dtd t V V s=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−+−⋅=+⋅+⋅⋅==+⋅⋅=−∫(8) 1cos 1(7))sin((6))sin(10The factor )(n R K depends on transformer saturation characteristics (s λand r λ) and other parameters during saturation.Typical saturation and residual flux magnitudes for power transformers are in the range[9]; .).(35.1.).(2.1u p u p s <<λ and .).(9.0.).(7.0u p r u p <<λIt can be easily shown that with increased damping 'resistance' in the circuit, where the circuit phase angle 2ψhas lower values than the saturation angle s t ⋅ω, the exponential term is negative resulting in an inrush magnitude that is lowerthan the sinusoidal term amplitude.B. Neutral Grounding Resistor SizingBased on (10), the inrush current peak expression, it is now possible to select a neutral resistor size that can achieve a specific inrush current reduction ratio )(n R α given by:(15) )0(/)()(==n peak n peak n R I R I R α For the maximum inrush current condition (0=n R ), the total energized phase system impedance ratio X/R is high and accordingly, the damping of the exponential term in equation (10) during the first cycle can be neglected; [][](16))0(1)0()0(2212=⋅++⎥⎦⎤⎢⎣⎡⋅−+===⎟⎟⎠⎞⎜⎜⎝⎛+⋅⋅n s p c p s pR x n m n peak R x x R x x r R K V R I c s σ High n R values leading to considerable inrush current reduction will result in low X / R ratios. It is clear from (14) that X / R ratios equal to or less than 1 ensure negative DC component factor ')(n R K ' and hence the exponential term shown in (10) can be conservatively neglected. Accordingly, (10) can be re-written as follows;()[](17) )()(22122n s p c p s n p R x m n n peak R x x R x x R r V R B R I c s σ⋅++⎥⎦⎤⎢⎣⎡⋅−+=≈⎟⎟⎠⎞⎜⎜⎝⎛+⋅Using (16) and (17) to evaluate (15), the neutral resistorsize which corresponds to a specific reduction ratio can be given by;[][][](18) )0()(1)0( 12222=⋅++⋅−⋅++⋅−+⋅+=⎥⎥⎦⎤⎢⎢⎣⎡⎥⎥⎦⎤⎢⎢⎣⎡=n s p c p s p n s p c p s n p n R x x R x x r R x x R x x R r R K σσα Very high c R values leading to low transformer core loss, it can be re-written equation (18) as follows [9]; [][][][](19) 1)0(12222s p p s p n p n x x r x x R r R K +++++⋅+==α Equations (18) and (19) reveal that transformers require higher neutral resistor value to achieve the desired inrush current reduction rate. IV. A NALYSIS OF SECOND-P HASE E NERGIZING It is obvious that the analysis of the electric and magnetic circuit behavior during second phase switching will be sufficiently more complex than that for first phase switching.Transformer behaviour during second phase switching was served to vary with respect to connection and core structure type. However, a general behaviour trend exists within lowneutral resistor values where the scheme can effectively limitinrush current magnitude. For cases with delta winding or multi-limb core structure, the second phase inrush current is lower than that during first phase switching. Single phase units connected in star/star have a different performance as both first and second stage inrush currents has almost the same magnitude until a maximum reduction rate of about80% is achieved. V. NEUTRAL VOLTAGE RISEThe peak neutral voltage will reach values up to peak phasevoltage where the neutral resistor value is increased. Typicalneutral voltage peak profile against neutral resistor size is shown in Fig. 6- Fig. 8, for the 225 KVA transformer during 1st and 2nd phase switching. A del ay of 40 (ms) between each switching stage has been considered. VI. S IMULATION A 225 KVA, 2400V/600V, 50 Hz three phase transformer connected in star-star are used for the simulation study. The number of turns per phase primary (2400V) winding is 128=P N and )(01.0pu R R s P ==, )(05.0pu X X s P ==,active power losses in iron core=4.5 KW, average length and section of core limbs (L1=1.3462(m), A1=0.01155192)(2m ), average length and section of yokes (L2=0.5334(m),A2=0.01155192)(2m ), average length and section of air pathfor zero sequence flux return (L0=0.0127(m),A0=0.01155192)(2m ), three phase voltage for fluxinitialization=1 (pu) and B-H characteristic of iron core is inaccordance with Fig.3. A MATLAB program was prepared for the simulation study. Simulation results are shown in Fig.4-Fig.8.Fig. 3.B-H characteristic iron coreFig.4. Inrush current )(0Ω=n RFig.5. Inrush current )(5Ω=n RFig.6. Inrush current )(50Ω=n RFig.7. Maximum neutral voltage )(50Ω=n RFig.8. Maximum neutral voltage ).(5Ω=n RFig.9. Maximum inrush current in (pu), Maximum neutral voltage in (pu), Duration of the inrush current in (s)VII. ConclusionsIn this paper, Based on the sequential switching, presents an analytical method to select optimal neutral grounding resistor for transformer inrush current mitigation. In this method, complete transformer model, including core loss and nonlinearity core specification, has been used. It was shown that high reduction in inrush currents among the three phases can be achieved by using a neutral resistor .Other work presented in this paper also addressed the scheme's main practical limitation: the permissible rise of neutral voltage.VIII.R EFERENCES[1] Hanli Weng, Xiangning Lin "Studies on the UnusualMaloperation of Transformer Differential Protection During the Nonlinear Load Switch-In",IEEE Transaction on Power Delivery, vol. 24, no.4, october 2009.[2] Westinghouse Electric Corporation, Electric Transmissionand Distribution Reference Book, 4th ed. East Pittsburgh, PA, 1964.[3] K.P.Basu, Stella Morris"Reduction of Magnetizing inrushcurrent in traction transformer", DRPT2008 6-9 April 2008 Nanjing China.[4] J.H.Brunke, K.J.Frohlich “Elimination of TransformerInrush Currents by Controlled Switching-Part I: Theoretical Considerations” IEEE Trans. On Power Delivery, Vol.16,No.2,2001. [5] R. Apolonio,J.C.de Oliveira,H.S.Bronzeado,A.B.deVasconcellos,"Transformer Controlled Switching:a strategy proposal and laboratory validation",IEEE 2004, 11th International Conference on Harmonics and Quality of Power.[6] E. Andersen, S. Bereneryd and S. Lindahl, "SynchronousEnergizing of Shunt Reactors and Shunt Capacitors," OGRE paper 13-12, pp 1-6, September 1988.[7] Y. Cui, S. G. Abdulsalam, S. Chen, and W. Xu, “Asequential phase energizing method for transformer inrush current reduction—part I: Simulation and experimental results,” IEEE Trans. Power Del., vol. 20, no. 2, pt. 1, pp. 943–949, Apr. 2005.[8] W. Xu, S. G. Abdulsalam, Y. Cui, S. Liu, and X. Liu, “Asequential phase energizing method for transformer inrush current reduction—part II: Theoretical analysis and design guide,” IEEE Trans. Power Del., vol. 20, no. 2, pt. 1, pp. 950–957, Apr. 2005.[9] S.G. Abdulsalam and W. Xu "A Sequential PhaseEnergization Method for Transformer Inrush current Reduction-Transient Performance and Practical considerations", IEEE Transactions on Power Delivery,vol. 22, No.1, pp. 208-216,Jan. 2007.。

基于经验模态分解和遗传支持向量机的多尺度大坝变形预测_张豪

1引言
为满足国家经济快速发展的需要，我国修建大量水库，成为世界兴建大坝最多的国家之一。然而，我国一半以上水库建于 20 世纪 50～70 年代，很多是“边勘测、边设计和边施工”，工程质量差，经过几十年的运行大多数处于病险状态，我国水库工程安全的状况并不乐观。据国家水利部的数据显示，全国共有病险水库 30 413 座，占水库总数的 36%，其中大型病险水库 145 座，中型 1 118 座，小型 29 150 座，分别占各类水库总数的 42%，42%和 36%。 1954～2003 年 50 a 中的溃坝事故共 3 481 起，其中大型水库 2 起，中型水库 123 起，小型水库 3 356 起，对人类生命、财产带来巨大损失[1-2]。为此，大坝的安全监测及稳定性评价成为国内外学者重要研究内容之一。大坝变形预测是大坝安全监测系统中关键组成部分，能否建立较为精确的大坝变形预测模型，对及时掌握大坝变形状态和确保大坝安全具有重要的意义。
良好的全局最优和预测能力，因此，支持向量用于大坝变形预测具有较高精度。大坝变形受温度、水压和时间效应等多因素共同影响，难以直接建立较为准确的变形量与影响因素之间的复杂非线性关系模型。因此，王新洲等[11-13]用小波算法把变形量分解成具有一定物理意义的一组不同尺度变形分量，然后分别针对每一个变形分量进行预测，最后再把各变形分量预测结果进行重构，得到最终变形预测结果，试验证明从不同尺度进行变形预测，有效提高变形预测精度；N. E. Huang 等[14-15]提出一种信号处理领域内解决非线性非平稳信号分析问题的新方法——经验模态分解(empirical mode decomposition， EMD)。相比傅立叶谱分析、小波分析等方法而言，它不仅可以很好地处理非平稳、非线性信号，而且具有计算更加简单、直观、基于经验和自适应的特点；李旋等[16-17]证实 EMD 能有效地把变形数据分解成具有不同特征尺度的平稳的窄带信号。基于上述研究，提出基于 EMD-SVM 的多尺度大坝变形预测模型。利用 EMD 算法将大坝变形数据分解成多个变形分量，针对不同的变形分量从大坝变形影响因素中选择对应的因素[18]，利用支持向量机构建各变形分量与对应影响因素之间复杂的非线性关系，建立变形分量预测模型，为提高支持向量机模型精度，用遗传算法优化其参数。将各变形分量预测模型进行相加，获得最终多尺度大坝变形预测模型，将各分量预测结果相加，得到最终变形预测结果。通过实例演示和精度分析，证实该方法的精度较高。

基于AMESim、MATLAB_与LabVIEW_的联合仿真虚拟平台技术

第 22卷第 10期2023年 10月Vol.22 No.10Oct.2023软件导刊Software Guide基于AMESim、MATLAB与LabVIEW的联合仿真虚拟平台技术董壮壮，王兆强，孙令涛，陆阳钧（上海工程技术大学机械与汽车工程学院，上海 201620）摘要：针对AMESim和MATLAB/Simulink的机电液系统联合仿真过程中参数设置较为繁琐、仿真结果可视化效果不够直观等问题，基于FMI标准化接口和ActiveX技术，利用LabVIEW进行人机交互界面设计与数据交互，研究了一种可定制化、参数设置集中化且仿真结果可视化的仿真虚拟平台技术。

初步应用实验结果表明，该虚拟平台可简便地对联合仿真模型进行参数设置与数据交互，结果准确、仿真效果直观，且仿真报告可自动化输出，有利于提高工作效率。

关键词：联合仿真；人机交互；多物理域；虚拟平台；数据交互DOI：10.11907/rjdk.231493开放科学（资源服务）标识码（OSID）：中图分类号：TP391.9 文献标识码：A文章编号：1672-7800（2023）010-0042-07Joint Simulation Virtual Platform Technology Based on AMESim，MATLAB and LabVIEWDONG Zhuangzhuang， WANG Zhaoqiang， SUN Lingtao， LU Yangjun（School of Mechanical and Automotive Engineering， Shanghai University of Engineering Science， Shanghai 201620， China）Abstract：In response to the problem of cumbersome parameter settings and insufficient visualization of simulation results in the joint simu⁃lation process of AMESim and MATLAB/Simulink electromechanical hydraulic systems，a customizable，centralized parameter settings，and visualized simulation results simulation virtual platform technology was studied using LabVIEW based on the standardized interface of FMI （Functional Mock up Interface） and ActiveX technology for human-machine interaction interface design and data exchange. The pre⁃liminary application experimental results showed that the virtual platform can easily set parameters and interact with data for joint simula⁃tion models， with accurate results and intuitive simulation effects. The simulation report can be automatically output， which is conducive to improving work efficiency.Key Words：joint simulation； human-computer interaction； multi-physical domain； virtual platform； data interaction0 引言目前，国内外仿真软件种类越来越多，仿真技术已经广泛地应用于汽车制造［1-4］、工程机械［5］、航空航天［6-7］等领域。

Calibration procedures for a computational model of ductile fracture

$Calibration procedures for a computational model of ductile fracture$

Calibration procedures for a computational model of ductile fracture Z.Xue a,1,M.G.Pontin b,2,F.W.Zok b ,J.W.Hutchinson a,*aSchool of Engineering and Applied Sciences,Harvard University,Cambridge,MA,United States b Materials Department,University of California,Santa Barbara,CA,United Statesa r t i c l e i n f o Article history:Received 18August 2009Received in revised form 22October 2009Accepted 29October 2009Available online 1November 2009Keywords:Ductile fracture Computational fracture Shear fracture Damage parametersa b s t r a c tA recent extension of the Gurson constitutive model of damage and failure of ductile struc-tural alloys accounts for localization and crack formation under shearing as well as tension.When properly calibrated against a basic set of experiments,this model has the potential topredict the emergence and propagation of cracks over a wide range of stress states.Thispaper addresses procedures for calibrating the damage parameters of the extended consti-tutive model.The procedures are demonstrated for DH36steel using data from three tests:(i)tension of a round bar,(ii)mode I cracking in a compact tension specimen,and (iii)shearlocalization and mode II cracking in a shear-off specimen.The computational model is thenused to study the emergence of the cup-cone fracture mode in the neck of a round tensilebar.Ductility of a notched round bar provides additional validation.Ó2009Elsevier Ltd.All rights reserved.1.IntroductionProgress in computational fracture mechanics has paralleled advances in constitutive models that incorporate damage mechanisms.For many ductile structural alloys the mechanism governing failure is void nucleation,growth and coalescence.The grand challenge for these alloys is the development of a computational capability for predicting localization,crack for-mation and crack propagation under all states of stress.Capturing both tensile (mode I)and shear (mode II)fractures has been particularly challenging.When properly calibrated for a speciﬁc structural alloy,the Gurson model [1]and some of its close relatives,such as the Rousselier model [2],have shown considerable promise for characterizing mode I crack growth[3–8].In addition,the models have been used to simulate transitions from mode I crack growth to mixed mode shear crack-ing in the cup-cone fracture process of round tensile bars [9,10]and in three-dimensional through-cracks in thin plates [11].Such transition problems are generally more challenging because the constitutive models have not been developed to explic-itly address damage under shear dominated conditions.A recent extension of the Gurson model [12]speciﬁcally incorporates damage in shear,adding the ﬂexibility to address shear ruptures as well as tension dominated failures.This extension will be employed here in conjunction with a suite of three tests (round bar tension,mode I compact tension,and mode II shear-off)to calibrate the constitutive parameters for the structural steel,DH36.For veriﬁcation,the calibrated model is then used to study the failure details of several other problems.To put the overall objectives of this work into some perspective,it is noted that three parameters are required to calibrate the extended Gurson model:the initial void volume fraction,f 0,a shear damage coefﬁcient,k x (deﬁned below)and the ﬁnite element size,D .To accurately characterize localization and fracture,D must be on the order of the spacing between the voids 0013-7944/$-see front matter Ó2009Elsevier Ltd.All rights reserved.doi:10.1016/j.engfracmech.2009.10.007*Corresponding author.E-mail address:hutchinson@ (J.W.Hutchinson).1Present address:Schlumberger Reservoir Completions,Rosharon,TX,United States.2Present address:Ceradyne,Costa Mesa,CA,United States.Engineering Fracture Mechanics 77(2010)492–509Contents lists available at ScienceDirectEngineering Fracture Mechanicsj o u r n a l h o m e p a g e :w w w.elsevier.c om /loc ate/engfracmechthat dominate the fracture process,typically from tens to hundreds-of microns.With mesh requirements this ﬁne,it is only possible to predict the onset and propagation of cracks in relatively small components or in larger structures where the loca-tion of the failure can be anticipated in advance.In contrast,it would not be feasible to employ a fracture model of this type to analyze fractures in large structures where the failure locations cannot be anticipated.Under such circumstances,because the ﬁnite element size for a large structure is necessarily orders of magnitude greater than void spacing and often larger than plate thickness,coarser criteria based on a critical effective plastic strain or a through-thickness cohesive zone must be em-ployed.These criteria must also be calibrated for each material,but against tests that make no attempt to resolve the ﬁne scale fracture processes relevant for the present class of models.The two classes of fracture models complement each other.In principle,computations based on a ﬁne scale model could be used to calibrate a coarse scale model.2.The extended Gurson modelThe Gurson model is an isotropic formulation that employs the mean stress,r m =r kk /3,and the effective stress,r e ﬃﬃﬃﬃﬃﬃﬃ3J 2p ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3s ij s ij =2p ,where s ij ¼r ij À13r kk d ijis the stress deviator.The extended model [12]employs,in addition,the third stress invariantJ 3¼det ðs Þ¼13s ij s ik s jk ¼ðr I Àr m Þðr II Àr m Þðr III Àr m Þð1Þwhere the expression on the right is couched in terms of principal stresses,assumed to be ordered asr I P r II P r III .Thenon-dimensional metric x ðr Þ¼1À27J 3r 3e 2ð2Þlies in the range,06x 61,with x ¼0for all axisymmetric stress states,r I P r II ¼r III or r I ¼r II P r III ;ð3Þand x ¼1for all states comprised of a pure shear stress plus a hydrostatic contribution,r I ¼s þr m ;r II ¼r m ;r III ¼Às þr m ðs >0Þð4ÞThe original Gurson model was formulated and calibrated based on the mechanics of void growth under axisymmetric stress states.The extension [12]does not alter the model for these states.The extension modiﬁes the predictions for states with non-zero x ðr Þ.In particular,a contribution to damage growth under pure shear stress states is accounted for in the extension whereas the original Gurson model predicts no change in damage for states having r m ¼0.NomenclatureA 0;Across-sectional area of neck:initial,current Dcharacteristic element size D P ij plastic strain rate EYoung’s modulus f 0;f ;f c ;f fvoid volume fraction:initial,current,onset of coalescence,failure Hplate thickness J 3stress invariant k xshear damage coefﬁcient Nstrain hardening exponent q 1;q 2;q 3ﬁtting parameters in Gurson model Rpunch radius s ijstress deviator d punch displacemente fductility—true strain in neck at failure e P M ;r Mintrinsic true plastic strain and stress in tension (damage-free)e peak T ;r peak T true strain and stress at maximum nominal stress r ij ;r e ;r m true stress,effective stress,mean stress r I P r II P r III true principal stressesx measure of shearing relative to axisymmetric stressingZ.Xue et al./Engineering Fracture Mechanics 77(2010)492–509493The yield surface of the extended Gurson model is the same as the original.Including the ﬁtting parameters,q 1,q 2and q 3,introduced by Tvergaard [13],it is given in terms of the effective and mean stress measures byF ðr e ;r m ;f Þ¼r e r M 2þ2q 1f cos h 3q 2r m r M Àð1þq 3f 2Þð5ÞThe current state is characterized by f ,the ‘‘apparent”void volume fraction,and r M ,the current effective stress governing ﬂow of the damage-free matrix material.All quantities not labeled with the subscript M represent overall quantities asso-ciated with the bulk material.Normality implies that the plastic strain rate,D Pij ,is given byD Pij ¼1h P ij P kl _r kl ð6Þwhere P ij ¼@F r ij ¼3s ij r M þfq 1q 2r M sin h 3q 2r m r M d ij ð7ÞIn ﬁnite strain formulations,_rij is identiﬁed with the Jaumann rate of stress.The hardening modulus,h ,is identiﬁed in the Appendix A .If r m ¼0,P kk ¼0and the rate of plastic volume change vanishes,i.e.,D Pkk ¼0;this feature persists in the exten-sion.In the absence of nucleation,the extension of the Gurson model posits_f ¼ð1Àf ÞD p kk þk x f x ðr Þs ij D p ij r e ð8ÞThe ﬁrst contribution is that incorporated in the original model while the second is the crux of the extension.As previ-ously noted,the modiﬁcation leaves the constitutive relation unaltered for axisymmetric stress states.In a state of pureshear,however,(8)gives _f ¼k x f _cP =ﬃﬃﬃ3p ,where _c P is the plastic shear strain rate and k x is the shear damage coefﬁcient,the sole new parameter in the extended model.The inclusion of the second term in (8)rests on the notion that the volume of voids undergoing shear may not increase,but void deformation and reorientation contribute to softening and constitute an effective increase in damage [14–16].In addition,the second term can model damage generated by the nucleation in shear of tiny secondary voids in void sheets linking larger voids.Thus,in the extension,f is no longer directly tied to the plastic volume change.Instead,it must be regarded either as an effective void volume fraction or simply as a damage param-eter,as it is for example when the Gurson model is applied to materials with distinctly non-spherical voids.Further discus-sion and illustrations of the extension are given in [12],where the emphasis is on its role in shear localization.The remaining equations specifying the entire description of the model are listed in the Appendix A .Included is the speciﬁcation of the widely used technique [13]that accelerates damage from f ¼f c to f ¼f f ,at which point the material element is deleted.De-tails of the numerical algorithm used to implement the constitutive model in the ﬁnite element code ABAQUS Explicit [16]are also presented in the Appendix A .3.Outline of the calibration protocolThe elastic–plastic inputs into the extended Gurson Model are the Young’s modulus,E ,the Poisson’s ratio,m ,and the intrinsic stress–strain response of the damage-free material (f 0¼0).The two damage-related input parameters are theinitial Fig.1.Optical micrograph of polished and etched cross-section through DH36steel plate,showing a microstructure of ferrite (light)and pearlite (dark).494Z.Xue et al./Engineering Fracture Mechanics 77(2010)492–509effective void volume fraction,f0,and the shear damage coefﬁcient,k x.Additionally,because the constitutive model contains no material length scale,the size of theﬁnite element mesh,D,is calibrated through crack growth predictions,employing well-established procedures[4,7].This paper addresses the general task of calibrating the three fracture-related parameters:f0,k x and D.The procedures are demonstrated through experiments and analyses of DH36steel(Fig.1):a high strength alloy commonly used in ship con-struction.Following extensive prior work on calibration procedures for the standard Gurson model(e.g.,[4,7]),the present study employs data from a mode I fracture test and a round bar tensile test to identify intrinsic uniaxial stress–strain behav-ior,f0and D.Additionally,a shear-off test is added to the suite of tests to determine the shear damage coefﬁcient,k x.The paper is organized following closely the steps in the calibration protocol:Section4:Determination of the intrinsic stress–strain response of the undamaged material from round bar tensile tests and establishing that f0,k x and D have little inﬂuence on the plastic response until neck development is quite advanced.Section5:Determination of f0and D from compact tension mode I fracture tests and establishing that k x has little inﬂu-ence on crack growth prediction when the crack is planar.Section6:Determination of k x using data from shear-off tests and the previously determined f0and D.Section7:Discussion of the applicability of the calibrated constitutive model to the cup-cone failure mode as one illus-tration and the ductility of notched round bars as another.Possible variations in the identiﬁcation protocol for other materials are also discussed.The three calibration tests were conducted under quasi-static loading,while all simulations were carried out using the dynamic code ABAQUS Explicit.In order to minimize inertial effects and efﬁciently simulate the quasi-static tests in the ex-plicit code,a preliminary series of calculations with differentﬁxed applied loading rates was performed for each test con-ﬁguration.At some loading rate,as the rates decrease,the simulations converge to a quasi-static limit.That loading ratewas then employed in all subsequent calculations.Material strain rate dependence is ignored in the presentcomputations.Fig.2.Tensile specimen geometry andﬁnite element mesh.Z.Xue et al./Engineering Fracture Mechanics77(2010)492–5094954.Intrinsic plastic response of the undamaged materialThe plastic response of the undamaged material (f 0¼0)was obtained from quasi-static uniaxial tensile tests on round bars coupled with elastic–plastic ﬁnite element computations.The test geometry and ﬁnite element mesh are shown in Fig.2.The nominal axial strain e N was measured using a non-contacting laser extensometer over a central 12.7mm length within the gauge section.Prior to necking,the true (logarithmic)strain is given by e T ¼ln ð1þe N Þand the true stress by r T ¼r N ð1þe N Þ,where r N is the nominal stress (load/initial area).To ascertain the true response in the post-necking regime,computations were performed using an assumed form of the stress–strain relation (detailed below)and matching the pre-dicted nominal stress–strain curves with those obtained experimentally.To accurately capture strain localization,a ﬁnite strain formulation of elasto-plastic theory was employed in the ﬁnite element model.Four-node axisymmetric elements 0 0.1 0.2 0.3 0.4 0.5900800700600500400300200100N=0.200.1850.16Experimental True strain, εT0 0.1 0.2 0.3 0.47006005004003002001000N=0.200.1850.16Experimental Nominal strain, εNεσT peak T peak ,()of the true tensile stress–strain curve beyond the onset of necking and (b)the corresponding element analysis.Error bars represent the full range of experimental measurements from six tests.extensometer over a 12.7mm gauge length near the specimen center.The nominal strain,deﬁned as consistently employed in both the experiments and the ﬁnite element calculations.The 496Z.Xue et al./Engineering Fracture Mechanics 77(2010)492–509with reduced Gaussian integration (CAX4R in ABAQUS/Explicit [16])were used.The model was based on an axisymmetric mesh comprised of square section elements with size,D =50l m,providing more than 30elements across the gauge radius.The element size was selected to be consistent with the value emerging from the calibration of the mode I fracture data,pre-sented in the next section.Nevertheless,since the selected element size is already very much smaller than the macroscopic specimen dimensions and hence the strains are adequately resolved,further reductions in element size would have essen-tially no effect on the intrinsic (damage-free)stress–strain response.Additional computations were performed to demon-strate that f 0and k x do not affect the identiﬁcation of the true stress–strain curve even up to strains approaching that for rupture.The average true stress–strain curve from ﬁve tensile tests is plotted in Fig.3a.This curve was subsequently used to char-acterize the stress–strain response for stresses below that corresponding to the load maximum,denoted r peakT.To extrapolate beyond r peak T ,a true stress–strain curve of the form r T ¼r peak T ðe T =e peak TÞN was assumed.A preliminary estimate of the strain hardening exponent N was obtained by a least squares ﬁt of the small strain data.A series of ﬁnite element computations was then performed to ascertain the full nominal tensile stress–strain curve,using a range of values of N ,guided by the pre-ceding curve ﬁtting.As shown in Fig.3b,the results for N ¼0:185(and f 0¼0)accurately replicate the experimental mea-surements up to the onset of rupture (at a nominal strain of e N ¼0:32).In summary,the true stress–strain curve used tocharacterize the damage-free material (f 0¼0)is given by the experimental curve below r peakT and the power law extrapo-lation at stresses above r peak T .For e N <0:3,void growth has almost no effect on the tensile behavior of DH36.This result is demonstrated in Fig.4by comparing the experimental data with ﬁnite element computations based on a hardening exponent N ¼0:185and several representative initial void volume fractions (including the Mises limit,wherein f 0¼0).Other than f 0,k x and D ,the basic parameters characterizing the constitutive model that are used in all simulations in this paper are:E ¼210MPa ;m ¼0:3;N ¼0:185;q 1¼1:5;q 2¼1;q 3¼2:25;f c ¼0:15and f f ¼0:25ð9ÞThe comparisons show that the effects of void growth,manifested in a divergence in the stress–strain response from that of a Mises material,are important only very near the point of ﬁnal rupture for the DH36tensile specimen.Their effect is to accelerate the softening of the material such that the load drops more rapidly than that predicted for the damage-free mate-rial.Further details of the failure process in the neck,including formation of a cup-cone fracture surface,are presented in Section 7.5.Determination of f 0and D from compact tension testCompact tension tests were performed on specimens with the geometry shown in Fig.5a.Crack mouth opening displace-ment was measured using a non-contacting extensometer and a pair of ﬁducial tapes mounted on the specimen edge,sep-arated by a distance of 14mm.Optical images of the broad sample surface were periodically recorded.The experimental 0 0.1 0.2 0.3 0.47006005004003002001000f o = 0.0010.0020.003Experimental Nominal strain, εNf o =0(Mises)k ω = 0fraction f o on the computed nominal tensile stress–strain response.Over the pertinent range of experimental measurements up to the onset of fracture.Z.Xue et al./Engineering Fracture Mechanics 77(2010)492–509497measurements and observations are summarized in Figs.6and 7.Signiﬁcant nonlinearity due to plasticity is evident in both the load–displacement response and in the optical images at displacements above 0.5mm.Following an initial rising por-tion,the load–displacement curve reaches a maximum,at a displacement of about 3–4mm.This point corresponds to the emergence of a crack on the external surface of the sample (Fig.7d–f ).Further growth both at the surface and in the interior occurs under decreasing load.The corresponding ﬁnite element model is shown in Fig.5b.In the present analysis,deformations are restricted to be symmetric with respect to the mid-plane such that a symmetry boundary condition is applied to the mid-plane.Conse-quently,the region meshed is only one half of the full specimen.Eight-node brick elements with reduced Gaussian integra-tion (C3D8R in ABAQUS/Explicit [16])were used.Iterations on element size and meshing details were made prior to arriving at the mesh used to carry out the ﬁnal analysis.The smallest elements at the mid-plane in the vicinity of the crack tip have dimensions 30Â30Â50l m with 50l m in the through-thickness direction.Near the surface of the specimen and near the tip the element dimensions are 30Â30Â80l m.Approximately 100elements extend from the mid-plane to the surface in the vicinity of the crack tip.The 30l m in-plane mesh at the tip allows accurate resolution of the initial tip notch.Further away from the notch tip in the region of crack propagation,the in-plane dimensions of the mesh are approximately 50Â50l m.Relatively small differences in results were found from a series of computations with different meshes with ele-ment dimensions in the range from 30l m to 50l m.The mesh in Fig.5b is regarded as having a nominal (characteristic)size D =50l m.In order to improve computational efﬁciency,only the material in the region of crack propagation,whichstartsFig.5.(a)Compact tension test geometry employed in the experimental study and (b)corresponding ﬁnite element model.Specimen thickness is 12.5mm.Crack mouth opening displacements were measured using a non-contacting extensometer and a pair of ﬁducial tapes mounted on the specimen edge,separated by a distance of 14mm.The same deﬁnition was used in the subsequent ﬁnite element calculations.498Z.Xue et al./Engineering Fracture Mechanics 77(2010)492–509Z.Xue et al./Engineering Fracture Mechanics77(2010)492–509499from the notch tip to the left edge of the specimen and has width of7mm,was modeled using the extended Gurson model. Outside this region,the specimen was modeled using von Mises plasticity(i.e.,f0¼0and k x¼0).Load–displacement predictions for four values of f0(including f0¼0)and k x¼2are compared with the experimental results in Fig.6.Over the range plotted,the load of the damage-free specimen increases monotonically with displacementbecause there is no damage-induced softening or crack growth.In contrast,the prediction for f 0¼0:001follows the exper-imental curve closely for displacements as large as 5mm.Furthermore,it predicts that cracking initiates at the center of the notch front,at a displacement of about 1mm.Thereafter,the crack grows deeper into the specimen and spreads laterally from the center (Fig.7).Upon reaching the free surface,at a displacement of 3.6mm,the load reaches a maximum and a load fall-off ensues.These results agree well with the experimental measurements.The predictions for the two larger values of f 0clearly over-predict the effect of damage and cracking at displacements below 5mm.They are particularly deﬁcient in predicting the displacement at the load maximum.At displacements above 5mm,the experimental data fall below the numerical predictions for all three values of f 0.This discrepancy arises for two reasons.The symmetry imposed in the simulation precludes the transition to slant fractures that usually develop as the crack advances and the crack in the test is likely to have departed from the imposed symmetry.In addition,element deletion was used to mimic the crack propagation such that the element is deleted when f ¼f f .As the crack advances,it encounters larger elements in the mesh and these dissipate more energy prior to failure than the cali-brated elements with D =50l m.It is indeed observed from Fig.8for the case of the crack month opening displacement reaching 8mm that some of the deleted elements are much larger than D =50l m.It remains for the future to verify that predictions based on the present choices of f 0and D can replicate the present experimental results for larger displacements using a computational model with no symmetry restrictions,as well as a uniform mesh with the same calibrated element size throughout the region of crack propagation.Unfortunately,this would result in a signiﬁcant increase in computational size that would not be feasible for the calibration procedure.3Although the results in Fig.6b were computed with k x ¼2,the shear damage coefﬁcient has essentially no effect on these predictions.To illustrate this,results for f 0¼0:001computed with k x =2,2.5and 3are plotted in Fig.6a.The response under-goes only very slight softening with increasing k x but remains well within the range of the experimental data.The weak dependence on k x is consistent with the fact that mode I cracking occurs over the range of load–displacement data used for theﬁtting.Fig.7.Images of broad face of compact tension specimen with increasing crack mouth opening displacements.Arrows in the right column indicate the emerging near-surface crack.3More than ten days were required for each calculation based on the current mesh using a personal computer with memory requirements up to 1GB.The trade-off between efﬁciency and accuracy suggests that the present calibration strategy is a reasonable compromise.500Z.Xue et al./Engineering Fracture Mechanics 77(2010)492–509In summary,based on the agreement between prediction and experiment for displacements below5mm,the choicesf0¼0:001with D%50l m are made for DH36.6.Determination of k x from a shear-off testTheﬁxture in Fig.9was designed to create a controlled test in which shear localization gives way to mode II fracture[17].The corresponding load–displacement curve is used to infer the shear damage coefﬁcient,k x.In the test,a plate specimen(3mm thick)is clamped between two thick steel platens,each with a through-hole of diameter19.2mm.Cylindrical steelplungers,19.05mm in diameter,are inserted into each of the two holes,leaving a narrow(0.075mm)radial gap between theplunger surface and the hole.An additional pair of plungers with slightly reduced diameter(to accommodate Teﬂon bear-ings)is then inserted into the holes.The four plungers and the test specimen are then clamped together with a single boltpassing through open holes in each of three of the plungers and the test specimen and a threaded hole in the last plunger,asshown in Fig.9.With one side of the assembly placed on a stiff supporting base,the plunger on the opposite side is loadaxially in compression.The movement of the plungers induces shear deformation within a narrow cylindrical ring in thespecimen.Failure starts as shear localizations near the upper and lower surfaces of the plate which subsequently developinto mode II cracks as the deformation progresses into the plate.The experimental measurements are summarized in Fig.10.The coordinate axes are the nominal applied shear stress, s P=ð2p RHÞ(R being the plunger radius and H the plate thickness)and the normalized displacement,d=H.The resulting curves exhibit features reminiscent of those obtained in tension tests.That is,the initial linear region gives way to plasticityat a shear stress of r O=2%240MPa(r O being the tensile yield stress,obtained from Fig.3).Following a period of strain hard-ening,the load reaches a peak,at a displacement of d=H%0.3–0.4,and subsequently diminishes with increasing displace-ment.Scanning electron micrographs of a cross-section through a test specimen that had been interrupted followingloading to a displacement d=H%0.5are presented in Fig.11.They reveal a diffuse damage zone within the region of intenseshear as well as well-deﬁned shear cracks emanating from the specimen surface in the vicinity of the plunger periphery.A detail of theﬁnite element mesh is depicted in the inset of Fig.9a.Based on the prior calibrations,computations ofshear-off employ an initial void fraction f0¼0:001and element size D=50l m in the region of shear localizationandFig.8.Evolution of plastic strain and crack growth fromﬁnite element calculations of the compact tension test.Z.Xue et al./Engineering Fracture Mechanics77(2010)492–509501。

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2 Multiobjective Optimization Using Evolutionary Algorithms 3

3.1 The Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Reducing the Pareto Set by Clustering . . . . . . . . . . . . . . . . . 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1 Introduction
Many real-world problems involve simultaneous optimization of several incommensurable and often competing objectives. Usually, there is no single optimal solution, but rather a set of alternative solutions. These solutions are optimal in the wider sense that no other solutions in the search space are superior to them when all objectives are considered. They are known as Pareto-optimal solutions. Consider, for example, the design of a complex hardware/software system. An optimal design might be an architecture that minimizes cost and power consumption while maximizing the overall performance. However, these goals are generally con icting: one architecture may achieve high performance at high cost, another low-cost architecture might considerably increase power consumption|none of these solutions can be said to be superior if we do not include preference information (e.g., a ranking of the objectives). Thus, if no such information is available, it may be very helpful to get knowledge about those alternate architectures. A tool exploring the design space for Pareto-optimal solutions in reasonable time can essentially aid the decision maker to arrive at a nal design. Evolutionary algorithms (EAs) seem to be particularly suited for this task, because they process a set of solutions in parallel, eventually exploiting similarities of solutions by crossover. Some researcher suggest that multiobjective search and optimization might be a problem area where EAs do better than other blind search strategies Fonseca and Fleming, 1995] Valenzuela-Rendon and Uresti-Charre, 1997]. Since the mid-eighties several multiobjective EAs have been developed, capable of searching for multiple Pareto-optimal solutions concurrently in a single run. In spite of this variety, it is di cult to determine the appropriate algorithm for a given problem because it lacks extensive, quantitative comparative studies. The few comparisons available to date are mostly qualitative and restricted to two different methods quite often, the test problems considered are rather simple. As a consequence, it sometimes seems that every new application results in a new multiobjective EA. In this study we have chosen another way. Firstly, we carried out an extensive comparison of di erent multiobjective EAs that bases on two complementary quantitative measures|the test problem was a NP-hard 0/1 knapsack problem. The experience we gained from the experiments led to the development of a new ap1

一种改进的高斯频率域压缩感知稀疏反演方法(英文)

AbstractCompressive sensing and sparse inversion methods have gained a significant amount of attention in recent years due to their capability to accurately reconstruct signals from measurements with significantly less data than previously possible. In this paper, a modified Gaussian frequency domain compressive sensing and sparse inversion method is proposed, which leverages the proven strengths of the traditional method to enhance its accuracy and performance. Simulation results demonstrate that the proposed method can achieve a higher signal-to- noise ratio and a better reconstruction quality than its traditional counterpart, while also reducing the computational complexity of the inversion procedure.IntroductionCompressive sensing (CS) is an emerging field that has garnered significant interest in recent years because it leverages the sparsity of signals to reduce the number of measurements required to accurately reconstruct the signal. This has many advantages over traditional signal processing methods, including faster data acquisition times, reduced power consumption, and lower data storage requirements. CS has been successfully applied to a wide range of fields, including medical imaging, wireless communications, and surveillance.One of the most commonly used methods in compressive sensing is the Gaussian frequency domain compressive sensing and sparse inversion (GFD-CS) method. In this method, compressive measurements are acquired by multiplying the original signal with a randomly generated sensing matrix. The measurements are then transformed into the frequency domain using the Fourier transform, and the sparse signal is reconstructed using a sparsity promoting algorithm.In recent years, researchers have made numerous improvementsto the GFD-CS method, with the goal of improving its reconstruction accuracy, reducing its computational complexity, and enhancing its robustness to noise. In this paper, we propose a modified GFD-CS method that combines several techniques to achieve these objectives.Proposed MethodThe proposed method builds upon the well-established GFD-CS method, with several key modifications. The first modification is the use of a hierarchical sparsity-promoting algorithm, which promotes sparsity at both the signal level and the transform level. This is achieved by applying the hierarchical thresholding technique to the coefficients corresponding to the higher frequency components of the transformed signal.The second modification is the use of a novel error feedback mechanism, which reduces the impact of measurement noise on the reconstructed signal. Specifically, the proposed method utilizes an iterative algorithm that updates the measurement error based on the difference between the reconstructed signal and the measured signal. This feedback mechanism effectively increases the signal-to-noise ratio of the reconstructed signal, improving its accuracy and robustness to noise.The third modification is the use of a low-rank approximation method, which reduces the computational complexity of the inversion algorithm while maintaining reconstruction accuracy. This is achieved by decomposing the sensing matrix into a product of two lower dimensional matrices, which can be subsequently inverted using a more efficient algorithm.Simulation ResultsTo evaluate the effectiveness of the proposed method, we conducted simulations using synthetic data sets. Three different signal types were considered: a sinusoidal signal, a pulse signal, and an image signal. The results of the simulations were compared to those obtained using the traditional GFD-CS method.The simulation results demonstrate that the proposed method outperforms the traditional GFD-CS method in terms of signal-to-noise ratio and reconstruction quality. Specifically, the proposed method achieves a higher signal-to-noise ratio and lower mean squared error for all three types of signals considered. Furthermore, the proposed method achieves these results with a reduced computational complexity compared to the traditional method.ConclusionThe results of our simulations demonstrate the effectiveness of the proposed method in enhancing the accuracy and performance of the GFD-CS method. The combination of sparsity promotion, error feedback, and low-rank approximation techniques significantly improves the signal-to-noise ratio and reconstruction quality, while reducing thecomputational complexity of the inversion procedure. Our proposed method has potential applications in a wide range of fields, including medical imaging, wireless communications, and surveillance.。

基于机器学习正则化理论的永磁同步电机转矩跟踪型MTPA_控制方法

第27卷㊀第11期2023年11月㊀电㊀机㊀与㊀控㊀制㊀学㊀报Electri c ㊀Machines ㊀and ㊀Control㊀Vol.27No.11Nov.2023㊀㊀㊀㊀㊀㊀基于机器学习正则化理论的永磁同步电机转矩跟踪型MTPA 控制方法漆星,㊀郑常宝,㊀曹文平,㊀张倩(安徽大学电气学院,安徽合肥230601)摘㊀要:内置式永磁同步电机(IPMSM )中的最大转矩电流比控制(MTPA )是交流电机控制中的经典问题㊂电动汽车用IPMSM 要求其控制策略不仅能够满足MTPA ,还能够精确地跟踪转矩指令㊂为解决这一问题,引入机器学习中的正则化理论,将传统的MTPA 控制问题转化成机器学习中的L 1㊁L 2正则化问题进行求解㊂首先将MTPA 控制问题等效为机器学习中的L 2正则问题,再对上述L 2正则问题中的转矩约束条件进行L 1正则转矩建模,从而实现对IPMSM 的转矩跟踪;最后使用拉格朗日对偶方法,对正则化后的MTPA 问题进行最优化求解㊂理论分析和试验结果表明,将IPMSM 中的MTPA 控制问题转化为正则化问题求解后,可以得到兼顾最大转矩电流比和高转矩跟踪精度的最优电流分配方案㊂所提方法结构简单㊁易于解释,还可以避免由于模型误差和电感饱和特性而造成的性能降低,从而融合了模型驱动法和数据驱动法的优势㊂关键词:内置式永磁同步电机;最大转矩电流比;转矩跟踪;机器学习;正则化;拉格朗日对偶DOI :10.15938/j.emc.2023.11.014中图分类号:TM351文献标志码:A文章编号:1007-449X(2023)11-0138-11㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀收稿日期:2022-03-17基金项目:国家自然科学基金(51507001)作者简介:漆㊀星(1985 ),男,博士,讲师,研究方向为电机控制中的人工智能技术;郑常宝(1963 ),男,博士,教授,博士生导师,研究方向为交流电机及其控制㊁电力电子技术;曹文平(1969 ),男,博士,教授,博士生导师,研究方向为交流电机及其控制㊁电机故障诊断;张㊀倩(1984 ),女,博士,教授,博士生导师,研究方向为交流电机及其控制㊁电机优化设计㊂通信作者:漆㊀星Torque-tracking MTPA control strategy of permanent magnet synchronous motors based on machine learning regularization theoryQI Xing,㊀ZHENG Changbao,㊀CAO Wenping,㊀ZHANG Qian(College of Electrical Engineering,Anhui University,Hefei 230601,China)Abstract :Maximum torque per ampere (MTPA )in internal permanent magnet synchronous motor (IPMSM)is a classical problem in AC motor control.The control strategy of IPMSM for electric vehicle not only need to achieve the MTPA,but also need to accurately track the torque commands.In order tosolve above problem,a regularization concept from machine learning theory was introduced to transform the traditional MTPA problem into L 1and L 2regularization issues.Firstly,the MTPA control problem is equivalent to the L 2regularization issue,and then the L 1regularization torque modeling was carried out for the torque tracking.Finally,the Lagrange dual method was used to optimize the above regularization-based MTPA problem.Theoretical and experimental analysis show that the proposed method can achieve an optimal current distribution scheme which both the maximum torque current ratio and the high torquetracking accuracy can be considered.Moreover,the proposed method solves the problem in a simple andanalytical manner,and the solution is easy to be interpreted.Thus,it combines the advantages of model-driven and data-driven methods.Keywords:interior permanent magnet synchronous motor;maximum torque per ampere;torque tracking; machine learning;regularization;Lagrange duality0㊀引㊀言内置式永磁同步电机(interior permanent magnet synchronous motor,IPMSM)由于其高效㊁高功率密度㊁宽调速范围等优点,在工业控制领域中大量应用㊂IPMSM本身具有的凸极性可以产生磁阻转矩,相较于表贴式永磁电机具有更高的动力输出㊂然而,IPMSM的凸极性会使得电机内部的交㊁直轴电感不一致,进而引出IPMSM控制中的交㊁直轴电流分配问题㊂在实际工程中,为减小损耗㊁最大限度地利用磁阻转矩,一般采用最大转矩电流比(maximum torque per ampere,MTPA)方式对IPMSM中的交㊁直轴电流进行分配[1]㊂近年来,一些特定领域的高速发展对IPMSM中的MTPA控制策略提出新的需求㊂例如,在电动汽车㊁数控机床等应用领域,其IPMSM中的MTPA控制策略不仅要求能够找出满足最大转矩电流比的最优交-直轴电流,还要求能够精确地跟踪转矩指令,称为转矩跟踪型MTPA控制㊂在已知给定转矩指令的条件下,如何使得电机的实际输出转矩与指令转矩保持一致,也是转矩跟踪型MTPA控制策略研究中需要解决的问题㊂现如今主流的MTPA方法主要分为模型驱动法和数据驱动法两类㊂模型驱动方法是利用电机本身的电感㊁磁链等模型,或者是利用谐波注入㊁在线搜索等手段,通过公式解析的方法推导出IPMSM中交㊁直轴电流的最优设定值[2-4]㊂模型驱动法具有结构简单㊁容易实现和易于解释的特性,其缺点在于使用的是电机的近似模型而非精确模型,往往无法克服模型误差的问题,在处理电感中的交叉饱和效应时难度较大[5-6],从而导致实际转矩跟踪精度下降,往往不能满足转矩跟踪型MTPA方法的要求㊂另一类MTPA方法主要基于电机实测数据,称为数据驱动法,具体而言,是搜集电机的有限元分析数据[7]或者电机离线测试数据[8],再通过数据拟合或数据挖掘的方法建立MTPA问题的数据模型㊂数据驱动的方法不依赖电机的近似模型,并且在数据挖掘的过程中已经考虑了由于电感磁饱和或交叉磁饱和而引起的非线性,因此转矩跟踪精度优于模型驱动方法㊂不过与模型驱动法使用的解析表达不同,现有的数据驱动方法大多使用非解析的隐式表达,例如神经网络㊁随机森林㊁支持向量机等[9-11],或者以网格搜索的形式建立转速-转矩-电流形式查找表并存储在电机控制器的MCU中[12]㊂相较于模型驱动方法,数据驱动方法虽然具有较高的转矩精度,但是存在数据结构复杂㊁算法结果不易解释等缺陷㊂以上两种方法都具有各自的优缺点,而迄今为止还没有一种方法能够融合两种方法之间的优势,从而实现算法简洁㊁结果精确的转矩跟踪型MTPA控制㊂基于此,本文借鉴机器学习理论中的正则化思想,研究一种将MTPA问题转化成机器学习理论中的L1㊁L2正则化问题的方法㊂首先将MTPA控制问题等效为机器学习中的L2正则问题,再对上述L2正则问题中的转矩约束条件进行L1正则转矩建模,最后使用拉格朗日对偶方法,对正则化后的MT-PA问题进行优化求解㊂理论分析和实验结果表明,将IPMSM中的MTPA问题转化成机器学习中的正则化问题后,可以得到兼顾最大转矩电流比和高转矩跟踪精度的最优解,从而满足转矩跟踪型MTPA 的需求㊂本文方法结构简单㊁模型易于解释,又避免由于模型误差和电感饱和特性而造成的性能降低,从而融合模型驱动方法和数据驱动方法的优势㊂1㊀IPMSM的MTPA控制和机器学习正则化理论1.1㊀IPMSM的数学模型与MTPA控制假设IPMSM模型为线性,即交㊁直轴电感为恒值,并忽略温度变化引起的电阻变化,则IPMSM在d-q轴坐标系下的电压方程为:u sd=R s i sd+dψsdd t-ωeψsq;u sq=R s i sq+dψsqd t-ωeψsd㊂üþýïïïï(1)式中:u sd㊁u sq为d-q轴电压;i sd㊁i sq为d-q轴电流; R s为定子电阻;ωe为电角频率;ψsd和ψsq分别为d-q 轴磁链,其中:931第11期漆㊀星等:基于机器学习正则化理论的永磁同步电机转矩跟踪型MTPA控制方法ψsd =L d i sd +ψf ;ψsq =L q i sq ㊂}(2)其中:L d ㊁L q 分别为d㊁q 轴电感;ψf 为永磁体磁链㊂IPMSM 的转矩方程为T e =32n p [ψf i sq +(L d -L q )i sd i sq ]㊂(3)式中n p 为电机的极对数㊂MTPA 控制方法是IPMSM 控制中较为常用的方法,其目的是以最小铜损实现IPMSM 的最大转矩控制,以输出电流最小为优化目标,可将传统的模型驱动MTPA 控制方法用数学描述为:min(i 2sd +i 2sq );s.t.T e =32p [ψf i sq +(L d -L q )i sd i sq ];i2sd+i 2sqɤi2smax㊂}(4)对式(4)使用拉格朗日乘子法,可将其等效为L (i sd ,i sq ,λ)=i 2sd +i 2sq +λ{T e -1.5p [ψf i sq +(L d -L q )i sd i sq ]}㊂(5)式中:L (㊃)表示拉格朗日函数;λ为拉格朗日乘子㊂对式(5)求偏导,最终可得最优的d -q 轴电流的设定值为:i sd =-ψf +ψ2f+4(L d -L q )2i 2sq2(L d -L q );i sq =i 2smax-i sd ㊂üþýïïïï(6)可以看出,使用式(3)~式(6)的MTPA 方法需要预知电机的转矩模型,以及ψf ㊁L d 和L q 等模型参数,考虑到模型的非线性和电感的交叉饱和特性,实际电机运行过程中ψf ㊁L d 和L q 的精确值往往难以获得,因此模型驱动方法通常只能获得最优电流值的近似解而非精确解,从而影响到最终的转矩跟踪精度㊂同时,传统的MTPA 控制问题为转矩开环的电流分配问题㊂例如,从式(3)~式(6)可知,i sd ㊁i sq 的选取只与实际输出转矩T e 有关,与指令转矩无关,因此不能构成指令转矩闭环的转矩跟踪控制㊂1.2㊀机器学习中的正则化理论由于本文方法是建立在机器学习中的正则化理论上的,因此本节对正则化理论进行简要介绍㊂机器学习中的正则化理论是通过最小化系数矩阵来降低学习器训练过程中存在的泛化误差,防止学习器陷入过拟合㊂同时,正则化理论可将特征选择和学习器的训练过程融为一体,即在学习器训练过程中自动进行特征选择[13]㊂给定数据集D ={x ,y ɪR |(x 1,y 1), ,(x n ,y n )},以线性回归模型为例,未正则化时,学习器的最小化训练损失函数为arg min w {ðni =1(y i -w T x i )2}㊂(7)而正则化后,学习器的最小化训练损失函数为arg min w {ðni =1(y i -w T x i )2+λ w P }㊂(8)式中:w =[w 1,w 2, ,w n ]为线性回归模型的系数矩阵; ㊃ P 表示P 范数;λ为拉格朗日乘子㊂可以发现,加入正则项后,使得系数矩阵w 最小也是其优化训练损失函数的目标㊂称P =1时的式(8)求解问题为L 1正则问题,其中系数矩阵w 的L 1范数为w 1=|w 1|+|w 2|+ +|w n |㊂(9)称P =2时的式(8)求解问题为L 2正则问题,其中系数矩阵w 的L 2范数为w 2=w 21+w 22+ +w 2n ㊂(10)L 1正则和L 2正则在特征选择上的区别如图1所示,其中椭圆和菱形(圆形)区域的切点就是目标函数的最优解㊂可以发现L 1正则和L 2正则都有助于降低过拟合的风险,然而L 1正则中,均方误差(mean square error,MSE)等高线和L 1范数的交点大多在某项坐标轴上,这表明L 1正则更倾向于获得多项系数为0的稀疏模型,以使模型具有结构简洁㊁易于解释的特性;而在L 2正则中,MSE 等高线和L 2范数的交点大多不在坐标轴上,这表明L 2正则更倾向于获得各项系数尽可能小的精确模型,以使模型具有高精确性[14]㊂在本文的方法中,将会综合使用L 1和L 2正则技巧,从而使得本文方法兼具简洁性㊁易解释性和高精确性的优势㊂图1㊀L 1和L 2正则化示意图Fig.1㊀Demonstration of L 1and L 2regularizations41电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第27卷㊀2㊀基于正则化理论MTPA 控制方法2.1㊀转化为L 1和L 2正则问题的MTPA 控制方法本节将IPMSM 的MTPA 问题转化为L 1和L 2正则化问题,并给出方法的总体框架㊂若考虑转矩指令跟踪精度的要求,则可将转矩跟踪精度作为优化目标,即IPMSM 的转矩跟踪MTPA 问题可重写为以转矩跟踪精度为目标的电流最优分配问题,即:argmin i s (T ref e -T e )2;s.t.T e =f (i s );i s 2ɤi 2smax ㊂}(11)式中:T e 表示电机实际输出转矩;Trefe表示指令转矩;i s ={i sd ,i sq }为d -q 轴电流的向量表示㊂再使用拉格朗日松弛法,可得式(11)的等价形式为MTPA argmin i s {(Tref e-T e )2+λ( i s 2-i 2smax )} argmin i s [(T ref e -T e )üþýïïïMSE2+λ i s 2}L 2REG]㊂(12)至此得到了式(8)所示的MTPA 的L 2正则等价形式,根据1.2节正则项的定义,其中转矩误差项MSE 保证了转矩跟踪精度最优特性,而正则项L 2REG 使得输出电流最小,使其具有最大转矩电流比特性,从而兼具了转矩跟踪精度和MTPA 的要求㊂事实上,可将式(3)代入式(12)中的约束项T e =f (i s ),则可将式(12)中转化为模型驱动的转矩跟踪型MTPA 问题进行求解㊂然而,根据1.1节分析,由于ψf ㊁L d 和L q 的参数真实值往往未知,并且ψf ㊁L d 和L q 还可能存在着非线性和交叉饱和问题,使用模型驱动求解方法往往无法获得理想的转矩跟踪精度㊂因此,本文将采用数据驱动的L 1正则化方法进行转矩模型T e =f (i s )的求解㊂由此得到的最终模型为:MTPA argmin i s {(T refe-T e )2+λ i s 2},L 2正则;s.t.T e =f (i s ),L 1正则;i s 2ɤi 2smax ㊂}(13)或简写成MTPA arg min i s {(T ref e -f (i s )üþýïïïL 1正则)2+λ i s 2üþýïïïïïïïïïïL 2正则-λi 2smax }㊂(14)由此,便可以将转矩跟踪型MTPA 控制问题转化成机器学习中的L 1㊁L 2正则化问题㊂其中L 2正则可以保证算法结果的精确性,而L 1正则可以保证算法结构的简洁性和可解释性㊂在实践中,本文研究方法的实际操作可分为离线测试阶段和在线调节阶段,如图2所示㊂具体步骤为:1)在离线阶段采集电机的测试数据,包括不同转速下的转矩T e ㊁d 轴电流i sd 和q 轴电流i sq ;2)将步骤1中采集的数据存储至数据池中,并建立电机转矩的L 1正则模型,具体方法由2.2节给出;3)将步骤2中建立的L 1正则转矩模型代入式(14)中的L 2正则MTPA 问题,并使用拉格朗日对偶原理求解最优拉格朗日乘子λ,具体方法由2.3节给出;4)在求出步骤3中的最优λ后,使用优化理论完成最优i sd 和i sq 的求解,从而实现在线的转矩跟踪型MTPA 控制㊂图2㊀基于L 1、L 2正则化问题的MTPA 控制框图Fig.2㊀MTPA control block based on L 1and L 2regularization2.2㊀基于L 1正则问题的转矩建模本节分析式(13)㊁式(14)中,基于L 1正则化理论的转矩建模方法㊂首先借鉴传统转矩模型的结构建立字典库,再基于字典库,采用L 1正则化理论中的LASSO 回归方法建立结构最优的数据驱动转矩模型,使得建立的转矩模型兼具精确性㊁简洁性和可解释性㊂具体步骤为:1)采集电机的台架测试数据{T e ,i sd ,i sq };2)建立数据驱动的转矩模型结构为T e =ΞΘ(i s ),其中,Ξ=[ξ1, ,ξn ]为模型系数矩阵,Θ(i s )为在转矩模型中有可能出现的i sd ㊁i sq 的组合,称之为字典库[15]㊂借鉴式(3)中传统的转矩模型结构,认为Θ(i s )中的字典复杂度不会超过i s 的二次型结构,即Θ(i s )=[1i P1s i P2s ]=[1i sdi sq i 2sd i 2sqi sd i sq ]㊂(15)141第11期漆㊀星等:基于机器学习正则化理论的永磁同步电机转矩跟踪型MTPA 控制方法式中i P1s ={i sd ,i sq }和i P2s ={i 2sd ,i 2sq ,i sd i sq }分别表示i s 内部元素的一次型和二次型结构㊂则模型的展开式为T e =ΞΘ(i s )=ξ1+ξ2i sd +ξ3i sq +ξ4i 2sd +ξ5i 2sq +ξ6i sd i sq ㊂(16)3)使用L 1正则化理论中的LASSO 回归[16]进行最优模型系数ξ∗1~ξ∗6的求解,表示为LASSO Ξ=arg min ξ∗i{(ð6i =1ξi Θ(i s )-T e )2+λ i s 1}㊂(17)由1.2节分析可知,使用L 1正则可以获得稀疏的最优系数矩阵Ξ∗=[0, ,ξ∗k , ,0],即式(16)模型中大部分系数为0,如图3所示,由此获得的数据驱动转矩模型具有结构简洁㊁易于解释的特性㊂图3㊀基于L 1正则化的转矩建模Fig.3㊀Torque modelling based on L 1regularization4)求得最优稀疏系数矩阵Ξ∗后,代入式(14),最终可将IPMSM 的MTPA 问题转化为MTPA arg min i s {[T ref e -Ξ∗Θ(i s )]2+λ( i s 2-i 2smax )}㊂(18)式中:i s =(i sd ,i sq )为需要求解的d -q 轴电流值;λ为未知参数㊂2.3㊀使用拉格朗日对偶求解最优参数对式(18)进行分析可知,λ的选择会影响到最优i sd ㊁i sq 的求解:过低的λ值会导致过大的i s 绝对值,而过高的λ值会导致过大的转矩跟踪误差,如图4所示㊂因此在求解最优i sd ㊁i sq 前,须先确定最优λ值,记为λ∗㊂针对上述问题,本节使用拉格朗日对偶方法[17]求解λ∗,步骤如下:首先引入拉格朗日函数L (i s ,λ,i smax ),表达式为L (i s ,λ,i smax )=(T ref e -Ξ∗Θ(i s ))2+λ( i s 2-i 2smax )㊂(19)图4㊀λ的选择对最优i sd 、i sq 的影响Fig.4㊀Selection of λand its effect to i sd and i sq则可将式(18)重写为MTPA min i s min λL (i s ,λ,i smax )㊂(20)则与之对应的拉格朗日对偶问题为MTPA min i s min λL (i s ,λ,i smax )min λmin i s L (i s ,λ,i smax )=min λmin i s [(T ref e -Ξ∗Θ(i s ))2+λ( i s 2-i 2smax )]㊂(21)再令g (i s ,λ)=(T ref e -Ξ∗Θ(i s ))2+λ i s 2,对g (i s ,λ)以i sd ㊁i sq 为自变量求偏导,并令其为0,记为:g (i s ,λ)i sd[]i sd=-(T refe-Ξ∗Θ(i s )) Ξ∗Θ(i s ) i sd+λi sd =0; g (i s ,λ) i sq[]i sq=-(T ref e -Ξ∗Θ(i s)) Ξ∗Θ(i s ) i sq+λi sq =0㊂üþýïïïïïïïï(22)对式(22)求解可得i sd ㊁i sq 基于λ的表达式,记为i sd (λ)和i sq (λ)㊂再将i sd (λ)和i sq (λ)代入式(18),可将式(18)转化为最优λ∗求解问题,表示为min λ{[T ref e -Ξ∗Θ(i sd (λ),i sq (λ)]2+λ[i sd (λ)2+i sq (λ)2-i 2smax )]}㊂(23)对式(23)进行求极值,可求得最优λ∗,将λ∗代入式(18),便可求解式(18)中的L 2正则问题,进而可使用牛顿法或单纯形法[18]等经典优化算法得到最优值i ∗sd 和i ∗sq ㊂241电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第27卷㊀3㊀仿真算例分析为能直观地说明本文方法的操作步骤,并验证本文方法的有效性,在MATLAB/Simulink仿真环境下进行实际算例演示,仿真电机参数如表1所示㊂在本节中,分别考虑L d㊁L q为恒参数值和变参数值这两种情况进行算例分析㊂表1㊀电机参数Table1㊀Parameters of motor㊀㊀参数数值额定功率/kW60额定电流/A350额定转速/(r/min)3000额定转矩/(N㊃m)205控制器直流电压/V340极对数4定子电阻/Ω0.02永磁体磁链/Wb0.0773.1㊀恒L d、L q参数值的MTPA控制策略假定L d㊁L q数值不会随着i sd㊁i sq变化,在仿真模型中设L d=0.00033H㊁L q=0.00082H,根据式(3)得到传统的转矩模型为T e=0.462i sq-0.0029i sd i sq㊂(24)在本节的仿真中,不考虑电机的数学模型和转矩模型,而直接让电机运行于不同的转速和转矩,采集转矩T e㊁d轴电流i sd以及q轴电流i sq数据,并使用式(16)中的L1正则方法进行T e=ΞΘ(i s)求解,解得T e的模型为T e=Ξ∗[1i s i P2s]=ξ3ξ6éëêêêêêêêêùûúúúúúúúú1i sd i sq i2sd i2sq i sd i sq[]=0.4539i sq-0.0029i sd i sq㊂(25)将式(25)与式(24)中的传统转矩模型T e= 0.462i sq-0.0029i sd i sq进行比较,可以发现,L1正则方法虽然是数据驱动的模型,但是和传统的转矩模型相比结构相同㊁系数相似㊂因此,相较于以往的神经网络㊁支持向量机等数据驱动模型,本文方法具有结构简洁㊁易于解释的特性㊂同时,重复上述实验50次,得到的系数均相似,如图5中的箱线图所示,表明了算法的可重复性和可靠性㊂图5㊀L1正则转矩模型的系数箱型图Fig.5㊀Coefficient boxplot of L1regularization图6为分别使用式(25)的L1正则转矩模型和式(3)的传统转矩模型,在不同的转矩指令T∗e下,真实转矩与计算转矩之间的转矩误差比较,可以发现,两种方法差别不大,表明了L1正则转矩模型的精确性㊂图6㊀L1正则转矩模型和传统转矩模型的比较Fig.6㊀Torque comparison between L1regularization model and classical model根据式(18)和式(25),可得最终的L1㊁L2正则表达式为arg min i s{[T∗e-(0.4539i sq-0.0029i sd i sq)]2+λ( i s 2-i2smax)}㊂(26)再使用2.3节所述方法进行最优λ求解,将式(26)代入式(22),得:g(i s,λ)i sd=0.0029i sq(T∗e-0.4539i sq+0.0029i sd i sq)+λi sd=0;g(i s,λ)i sq=-0.4539(T∗e-0.4539i sq+0.0029i sd i sq)+λi sq=0㊂üþýïïïïïïïï(27)341第11期漆㊀星等:基于机器学习正则化理论的永磁同步电机转矩跟踪型MTPA控制方法再结合式(23),经过简化计算,解得最优λ∗近似值为λ∗=10.45392i smax+(0.0029T ∗e)2㊂(28)将式(28)代入式(18),在不同转矩指令T ∗e 下使用单纯形法求极值,最终可得不同T ∗e 下的最优电流指令i ∗sd 和i ∗sq ㊂至此,正则化的MTPA 方法计算结束㊂图7为使用正则化MTPA 方法和使用传统MT-PA 方法,在不同转矩指令Trefe={20,40,60,80, ,200N㊃m}下的效果比较㊂其中,传统的MTPA 方法使用式(3)~式(6)所示的方法㊂为清晰起见,在图7中分别标出了在转矩指令T ref e 为{80,100,120,140}时,两种比较方法的实测转矩T e 和实测电流绝对值|i s |数据㊂可以发现,在转矩跟踪精度以及i sd ㊁i sq 最优电流分配的性能指标上,使用正则化MTPA 方法与传统MTPA 方法相比优势并不显著,只是在电流较大时,正则化MTPA 方法在转矩跟踪精度上才显著优于传统MTPA 方法㊂图7㊀正则化MTPA 方法和传统MTPA 方法的比较Fig.7㊀Comparison between regularization method andclassical method3.2㊀变L d ㊁L q 参数值的MTPA 控制策略考虑磁饱和以及交叉磁饱和的影响,实际中L d ㊁L q 的值往往会跟随不同i sd ㊁i sq 的值发生变化,在实际中,测得某台电机的L d ㊁L q 变化如图8所示,根据图8中的i sd ㊁i sq 建立L d ㊁L q 的查找表㊂同样在弱磁区以下让电机运行于不同的转速和转矩,并采集转矩数据T e ㊁d 轴电流数据i sd 以及q 轴电流数据i sq ,最后使用式(16)中的L 1正则方法进行T e =ΞΘ(i s )求解,解得变L d ㊁L q 参数下的转矩T e 模型为T e =Ξ∗1i s i P2s []=00ξ30ξ5ξ6éëêêêêêêêêêùûúúúúúúúúú1i sd i sq i 2sd i 2sq i sd i sq[]=-0.0298i sd +0.3272i sq +0.0006i 2sq -0.0014i sd i sq ㊂(29)可以发现,虽然L d ㊁L q 的变化较为复杂,但基于L 1正则方法仍然给出了最简洁的转矩模型㊂图8㊀变L d ㊁L q 条件下的L d ㊁L q 值分布Fig.8㊀L d and L q distribution under varying L d ,L qconditions图9为分别使用式(29)的L 1正则转矩模型和式(5)的传统转矩模型,在不同的转矩指令T ∗e 下,真实转矩与计算转矩之间的误差比较㊂可以发现正则化MTPA 方法明显优于传统方法㊂441电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第27卷㊀图9㊀使用L 1正则模型和传统模型的转矩比较Fig.9㊀Torque comparison between L 1regularizationmodel and classical model根据式(18)和式(29),可得最终的L 1㊁L 2正则表达式为argmin i s {[T ∗e -(-0.0298i sd +0.3272i sq +0.0006i 2sq -0.0014i sd i sq )]2+λ( i s 2-i 2smax )}㊂(30)后续最优λ∗及i ∗sd 和i ∗sq 计算与3.1节类似㊂使用正则化MTPA 方法和使用传统MTPA 方法在不同转矩指令T ∗e ={40,60,80, ,180N㊃m}下的比较结果如图10所示㊂同样的,传统MTPA 方法由式(3)~式(6)所示,转矩模型采用式(24)中的数值㊂为清晰起见,在图10中分别标出了转矩指令T ref e 为{80,100,120,140}时的两种方法实测转矩T e 和实测电流绝对值|i s |数据㊂可以发现,使用正则化MTPA 方法在各个转矩指令下,转矩跟踪精度和电流分配上都较传统的MTPA 方法具有明显的提高㊂图10㊀正则化MTPA 方法和传统MTPA 方法比较Fig.10㊀Comparison between regularization method and classical method3.3㊀结果讨论从3.1节的结果中可以看出,在恒L d ㊁L q 的工况下,正则化MTPA 方法与传统MTPA 相比结果相差不大㊂这是因为在L d ㊁L q 的精确值已知㊁且为恒定的条件下,转矩模型较为简单,传统转矩模型和L 1正则转矩模型具有相同的结构和相近的系数,同时,在Simulink 仿真环境中的电机模型多为理想模型,不用考虑模型误差和外部环境干扰的问题㊂由此可以得出结论:在模型参数精确值已知且恒定,并且不存在模型误差和外部干扰的条件下,本文研究的正则化MTPA 方法相较传统MTPA 方法优势并不明显㊂不过,上述的条件过于理想,在实际工程中并不容易实现㊂而从3.2节的结果中可以看出,在L d ㊁L q 根据不同工况发生变化的条件下,正则化MTPA 方法较传统MTPA 具有明显优势㊂这是因为在变L d ㊁L q 的工况下,使用式(3)已经无法描述精确的转矩模型,从而造成了较大的转矩跟踪误差;而正则化MTPA 方法使用实际运行数据来对转矩进行数据建模,从而有效避免了由于电感饱和效应导致的模型误差以及实际运行环境带来的干扰,显著提升了转矩跟踪精度㊂同时,从式(25)和式(29)可以看出,虽然本文的方法也是数据驱动方法,但是使用L 1正则方法可以获得结构简洁㊁易于解释㊁利于工程实现的数据模型,这是其他数据驱动方法,例如神经网络㊁支持向量机等方法不具备的优势㊂综上所述,本文研究的正则化MTPA 方法适用于存在模型误差和实际环境干扰的运行场合,并且兼顾了精确性㊁简洁性和可解释性㊂4㊀实验验证4.1㊀实验环境与实验步骤所研究的方法在AVL 电机台架上进行试验验证,试验平台如图11所示,由被测电机㊁被测电机控制器以及测功机构成㊂实验时,被测电机运行于转矩模式,测功机运行于转速模式㊂被测内置式永磁同步电机与第3节仿真中的电机参数相同,被测电机控制器的主控芯片为TI 公司的TMS320280049型DSP㊂具体实验步骤如下:1)测功机以300r /min 转速步长分别运行于n ={300,600, ,3000r /min},被测电机在不同转速n 下,以不同的i sd ㊁i sq 电流运行,以便输出不同转矩㊂2)在步骤1所述的不同转速㊁转矩㊁电流条件541第11期漆㊀星等:基于机器学习正则化理论的永磁同步电机转矩跟踪型MTPA 控制方法下,分别采集被测电机的运行数据,包括转速n ㊁电机转矩T e ㊁d 轴电流i sd ㊁q 轴电流i sq ,记为{n ,T e ,i sd ,i sq },共计400组㊂3)使用步骤2采集的400组数据进行被测电机转矩的L 1正则化建模,L 1正则化建模使用式(17)所示的LASSO 回归实现,并用MATLAB 语言在PC 机中完成㊂所建立的L 1正则化转矩模型具有结构简单的特征,可以移植到电机控制器的DSP 中㊂4)将步骤3建立的转矩模型移植进被测电机控制器的DSP 中,并建立式(18)所示的L 2正则转矩跟踪型MTPA 模型,进而在不同的转矩指令T ref e 下实现兼顾转矩跟踪精度和最优转矩电流比的d -q 轴电流分配㊂图11㊀实验平台示意图Fig.11㊀Experimental test-bench4.2㊀实验结果与分析在电机转速为300r /min,转矩指令T refe={20,40,60,80, ,200N㊃m}的条件下,分别使用本文正则化方法和基于模型方法进行d -q 轴电流分配,并记录本文方法和基于模型方法的实际输出转矩T e ,如图12和图13所示,上述两种方法对转矩指令的跟踪误差比较如图14所示㊂可以发现,使用本文方法较传统方法转矩精确跟踪优化的目标函数,同时使用数据驱动的L 1正则进行转矩建模,从而规避了由于交叉饱和效应引起的模型误差㊂图12㊀使用本文方法的输出转矩示意图Fig.12㊀Torque output of the proposedmethod图13㊀使用基于模型方法的输出转矩示意图Fig.13㊀Torque output of the model-basedmethod图14㊀本文方法和基于模型方法转矩跟踪误差Fig.14㊀Torque tracking comparison between the pro-posed method and model-based method图15为本文方法和基于模型方法的i s 电流绝对值比较,可以发现,在转矩指令较小时,基于模型方法的i s 电流绝对值大于本文的正则化MTPA 方法,而在转矩指令较大时,基于模型方法的i s 电流绝对值小于本文的正则化MTPA 方法㊂这是由于图15所示的转矩跟踪误差所导致的㊂在转矩指令较低时,基于模型的方法会输出远大于指令的转矩,从而需要更高的电流值;而在转矩指令较高时,基于模型的方法会输出远小于指令的转矩,从而需要更低的电流值㊂而在转矩误差相近的区域,两种方法的电流值相差不大㊂这表明,基于模型的方法和本文的方法都是以最小电流作为优化指标,因此都可以输出较小的电流值㊂不过由于转矩误差的不同,两种方法在不同的转矩指令范围内输出的电流值也不尽相同㊂641电㊀机㊀与㊀控㊀制㊀学㊀报㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀第27卷㊀。

三轴稳定微小卫星主动磁阻尼姿态控制

摘要 : 针对三轴稳定微小卫星 ,用四元数法建立了速率阻尼阶段的动力学和运动学模型。根据地磁场强度矢量投影到轨道坐标系的简便旋转关系 ,设计了采用主动磁控的拟比例微分控制器 , 同时分析了磁力矩器的磁偶极矩。速率阻尼仿真结果表明 : 该控制法有效可行 ,具有一定的应用价值。关键词 : 三轴稳定微小卫星 ; 主动磁控 ; 四元数 ; 拟比例微分控制 ; 速率阻尼中图分类号 :V448 . 222 文献标识码 :A
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Abstract : The at titude dyna mic s and kinematic s model during r ate damping was built with quate rnion for a t hree2 a xis stabilized micro2sa tellite in t his paper . Acco rding to the simple a nd conve nient projection of the geomagnetic field vector in the or bit ref ere nce f rame , a qua si propor tio nal plus de riva tive controller was de signe d with active magnetic cont rol. And t he magnetic dipole moment wa s analyzed. The simulation results of the micro2satellite rate damping showe d that the cont rol method wa s valid and wa s very usef ul for application. Keywor ds: Thr ee2 a xis sta bilized micro2sa tellite ; Ac tive magnetic control ; Quate rnion ; Qua si proportional plus derivative control ; Ra te damping

一种带前馈的双闭环APFC控制方法

一种带前馈的双闭环APFC控制方法闫文华;王小鹏;鱼鹏飞【摘要】为了提高传统功率因数校正(PFC)数字控制的稳态性能、动态响应和功率因数,降低输入电流谐波畸变,提出了一种带前馈的双闭环有源功率因数校正(APFC)控制方法.首先,建立了Boost PFC控制系统的小信号模型,推导出了系统传递函数;然后,估算出了Boost拓扑结构主要器件的参数值.通过前馈的作用,系统可以快速响应调整输入电压的变化,利用电压环和电流环实现对输入电流和输出电压的调节.Simulink建模仿真表明:该方法能够在输入电压波动较大的情况下有效地稳定控制输出电压,提高系统的动态响应能力和输入功率因数,减小输入电流谐波畸变.%In order to improve the steady state performance,dynamic response and power factor of traditional power factor correction (PFC) digital control method and reduce the harmonic distortion of input current,a double closed loop active power factor correction (APFC) control method with feed-forward is proposed.Firstly,the small signal model of Boost PFC control system is built and the system transfer function is deduced,and then the parameters of the main device with Boost topology is estimated.By means of the feed-forward,the system can quickly respond to the change in input voltage.Furthermore,the use of voltage loop and current loop can achieve input current and output voltageregulation.Simulink modeling shows that this method can effectively control the output voltage in case of input voltage largelyfluctuating,improve the system dynamic response ability and input power factor,and reduce the input current harmonic distortion.【期刊名称】《测试科学与仪器》【年(卷),期】2017(008)003【总页数】7页(P264-270)【关键词】有源功率因数校正;前馈;双闭环控制;传递函数【作者】闫文华;王小鹏;鱼鹏飞【作者单位】兰州交通大学电子与信息工程学院,甘肃兰州730070;兰州交通大学电子与信息工程学院,甘肃兰州730070;北京交通大学机械与电子控制工程学院,北京100044【正文语种】中文【中图分类】TM464Abstract： In order to improve the steady state performance, dynamic response and power factor of traditional power factor correction (PFC) digital control method and reduce the harmonic distortion of input current, a double closed loop active power factor correction (APFC) control method with feed-forward is proposed. Firstly, the small signal model of Boost PFC control system is built and the system transfer function is deduced, and then the parameters of the main device with Boost topology is estimated. By means of the feed-forward, the system can quickly respond to the change in input voltage. Furthermore, the use of voltage loop and current loop can achieve input current and output voltage regulation. Simulink modeling shows that this method can effectively control the outputvoltage in case of input voltage largely fluctuating, improve the system dynamic response ability and input power factor, and reduce the input current harmonic distortion.Key words： active power factor correction; feed-forward; double closed loop control; transfer functionCLD number： TM464 Document code： AThe problems of power factor correction (PFC) caused by the large use of power electronic devices are becoming more and more serious. A large number of harmonics produced by nonlinear loads in the power network make the input current distorted[1], as a result, the power quality of the power network is seriously affected. Active power factor correction (APFC)[2-4] can effectively improve the quality of power network and enhance the power factor of the system. The three-phase high-power APFC digital control technology has become the research focus of PFC technology.Compared to Flyback[2], Buck[3] and other topologies, Boost[4-6] APFC converter has less high-frequency ripple component of input current and higher output voltage. Above all, its highlight is that the input power factor can be maintained over the entire input voltage range[4,7]. At present, the commonly used PFC digital control methods include peak current control method[8], average current control method[9], and so on. Peak current control method compares output voltage with reference voltage to generate error signal; and then generates input reference current by means of the compensator of digital signals, finally, inductor current andreference current are compared to produce the driving power of pulse-width modulation (PWM) signal.However, this method needs to add slope compensation in the input channel of the comparator, and the effects of improving the power factor (PF) and reducing the harmonic distortion rate (total harmonic distortion, THD) are not perfect[8]. The average current control method compares inductor current with reference current to produce a signal, which is averaged by current error amplifier; and then the signal comparator compares the amplified average current with the sawtooth wave to generate the PWM driver of the power transistor. When it comes to achieve projects through the instrumentality of digital signal processor (DSP) or other devices that contain both hardware and software, the problem of time delay increases the bandwidth and gain of current loop, resulting in heavier load on the current loop. Of course, the performances of steady-state and the dynamic response capability of the system are not perfect[9-11].The double closed loop control strategy of non-isolated Weinberg converter with input voltage feed-forward was proposed in Ref.[12]. The used level of power is 1-2 kW. But when it comes to high power applications, it is difficult to guarantee its own advantages. While the double closed loop control strategy of Boost converter with input voltage feed-forward can effectively guarantee the performance of power equipment in high power condition. Ref.[10] used analog control chipUC3854 to control Boost PFC, and its control circuit needed many components, therefore it was easily affected by temperature and aging ofcomponents. At the same time, the working point and system parameters were easy to drift. The digital control can overcome the shortcomings of analog control. Moreover, software control is beneficial to easier system upgrading, functional extension, and more complex control algorithms. Aiming at the defects of PFC digital control method mentioned above, a double closed loop average current APFC control method [12] with feed-forward is proposed, of which feed-forward component improves the systematical dynamic response of input voltage, current loop makes input current closer to sinusoidal wave, and voltage loop keeps output voltage stable. The proposed method not only enhances the power factor of input current and dynamic response capability of the system, but also reduces the harmonic distortion of input current.The core part of control method is mainly composed of feed-forward component and double closed-loop control part. The feed-forward component mainly can improve the systematic dynamatic response ability of jumping input voltage, and the double closed loop part can improve the input power factor of power electrical devices and reduce the harmonic distortion of the input current.1.1 Double closed loop average current control method with feed-forward As shown in Fig.1, AC input voltage is uin, input inductance current is iLa and DC output voltage is uoa, which are input into the A/D converter through the sampling circuit, with the analog quantity being converted into digital quantity. The error between the feedback output voltage uo filtered and reference voltage Uref is adjusted by voltage loop PI controllerto generate a DC value that is similar to constant. And then this DC value and the input voltage pass through the multiplier to generate the current reference value iref, After that, the error between the iref and input current iL is adjusted by the current loop PI controller to generate the uPI. At the same time, the error between Uref and uo to do division by Uref, as a result, the duty cycle component uFF of feed-forward part is generated. Finally, the sum of uFF and uPI is compared with the triangular carrier to generate the final PWM driver, thus the inductor current is controlled in real time to approximate the inductance average current.1.2 Control modelIn order to estimate the correlation coefficient of the proportional-integral (PI) controller, we can build a small signal model of Boost PFC control system and figure out its transfer function. Based on this, we can build a Simulink simulation model of the system. In PFC digital control strategy, the double closed loop PI control of voltage outer loop and current inner loop is adopted. Considering the effect of coupling between the double closed loop, relative to proportional-integral-differential (PID) or proportional-differential (PD) control, PI control has the advantages of simple design, fast rise time, in sensitiving to the initial value a reasonable initial value can converge to the optimal solution, which brings great convenience for parameter debugging, and its parameters is less than those the classical PID control. In addition, due to the obvious delay phenomenon in the numerical control of actual project and the limitation of switching frequency of power pipe, the PI control can better solve thelag effect and improve the stability of the system.The small signal model of Boost PFC control system is shown in Fig.2. Fm and He(s) are the PWM and sampling gain of the system, respectively; Hi(s) and HV(s) are the transfer functions of the current loop and voltage loop, respectively; , , gm=2kii/uin. The transfer functions Gud and Gig of the “control-output”, Gug and Gid of the “input-output” can be figured out as.In the average current control with feed-forward, the pulse width modulator gain Fm iswhere re is the slope of the saw-tooth wave, and rn is the slope of current error amplifier’s output signal.The current loop transfer function can be obtained from the current loop control loop, as shown in Fig.3, namely.According to the small signal model of Boost PFC control system, the open-loop transfer function of current loop Ti(s) can be obtained asThe transfer function of voltage loop is obtained from the voltage loop control loop (Fig.4).According to the small signal model and system function of the Boost PFC control system, the Simulink simulation model is set up, as shown in Fig.5. The parameters of the Boost inductor L and filter capacitor C are estimated asy.In order to test the effect of power factor correction and the dynamic response capability of the system in case of the jumping input voltage , it is verified on the basis of Simulink modeling, in which the three-phase input AC voltage is 176- 256 V, boost inductor is L=750 μH, filter capacitor is C=4 230 μF, power tube’s switching frequency is f=24 kHz, and the maximum load is about 15 kW. Fig.6 shows the steady-state waveform of the output DC voltage. It can be seen the steady-state output is DC 400V, and the ripple voltage peak-to-peak value is about 6 V.In order to compare the dynamic response capability of the system when the input AC voltage is abruptly changed, the input voltage is abruptly fallen from 220 to 176 V at 0.57 s. Fig.7(a) shows the dynamic response wave of the system’s output voltage without feed-forward. It can be seen that the output voltage Vo falls, rises, slightly falls and finally tends to be stable. However, the voltage overshoot is obvious and the whole adjustment process takes 0.35 s. Fig.7(b) shows the dynamic response wave of the system’s output voltage with feed-forward, which tells us that tending to be stale only take 0.09 s without the voltage overshoot. Fig.8(a) and Fig.9(a) are the system’s dynamic response waves of the input current and the inductance current without feed-forward, It can be seen that the regulation of input current Iin and inductance current IL are similar to those of Vo, the processes have concussive phenomenon, Iin and IL take 0.12 s and 0.16 s tend to be stable, respectively. Fig.8(b) and Fig.9(b) are the system’s dynamic response waves of the input current and theinductance current with feed-forward, respectively. It can be seen that Iin and IL take 0.06 s and 0.08 s tend to be stable, respectively, and the processes are nearly without concession. Apparently, the dynamic response speed of the system is obviously faster, and the dynamic response process is better than the system without feed-forward section. Table 1 presents the system’s PF and THD of the three-input power. It can be seen that with the increase of the load power, the PF of three input power covering A, B and C are greater than 0.99, and the trend of increase is almost close to 1; the THD of the three-input current is less than 5%, and the effect of PFC is ideal and thus reaches the desired goal.A double closed loop average current APFC control method with feed-forward is proposed, and a three-phase APFC control system based on Boost topology is designed. In view of the fast change of the input voltage, the dynamic response capability of the system is improved by feed-forward circuit. The current regulating loop limits the inductance current, at the same time the double closed loop structure reduces the phase lag. Simulink modeling and simulation show that the method can effectively control the output voltage stability, improve the system dynamic response ability and input Power factor of the system, and reduce the harmonic distortion of the input current, thus the ideal power factor correction effect is obtained.[1] LU Wei-guo, FANG Hui-min, YANG Yi-di, et al. Analysis and design of dynamic slope compensation for Boost PFC converter. Electric Power Automation Equipment, 2017, 37(5)： 1-6.[2] BEN Hong-qi, ZHANG Ji-hong. Active power factor correction technology in switching mode power supply. Beijing： Machinery Industry Press, 2010.[3] WANG Zhao-an, LIU Jin-jun. Power electronics technology. Beijing：China Machine Press, 2009.[4] CHEN Zhe. Boost APFC device design. Electrical Technology, 2010, (1)：45-50.[5] YAO Kai, RUAN Xin-bo, MAO Xiao-jing, et al. Reducing storage capacitor of a DCM Boost PFC converter. IEEE Transactions on Power Electronics, 2012, 27(1)： 151-160.[6] LIU Zhi-fei, DU Gui-ping, DU Fa-da. A novel high-efficiency bridgeless dual boost PFC converter. Power Electronics, 2017, 51(1)： 68-71.[7] Abramovitz A. Effect of the Ripple Current on Power Factor of CRM Boost APFC. IEEE International Power Electronics & Motion C, 2006, 17(3)：1-4.[8] YAN Chang-guo, GONG Ren-xi, LIU Xiao-yong. Modeling and design of staggered boost converter based on peak control. Science & Technology and Engineering, 2017, 17(2)： 44-48.[9] HAO Bing-jin, XU Yan. Active power factor orrection analysis based on average current control. Journal of Shandong Agricultural University (Natural science edition), 2012, 43(4)： 543-548.[10] LI Jing-ling. Research and simulation of control method for active power factor correction circuit. Electrotechnics Electric, 2010, 11(3)： 33-36.[11] HOU Ming-kai, CHEN Cheng-hu, CHENG Ming-yang. Design andanalysis of a single-phase low-frequency active power factor correction circuit： a symmetric trapezoidal current waveform approach. Electrical Engineering, 2016, 98(3)： 257-270.[12] CHENG Xin, LIU Zhongxin, SHEN An, et al. The double loop control strategy for dc/dc converter based on voltage feed-forward control. Power Electronics, 2016, 50(7)： 18-20.。

基于大数据的数学建模方法融入高职数学教学实践探究

基于大数据的数学建模方法融入高职数学教学实践探究王英(甘肃财贸职业学院甘肃兰州 730207)摘要：“数学建模”是指利用计算机将现实生活中遇到的实际问题用一定的数学方法表示出来，并在计算机上进行模拟运算。

通过对现实生活中问题的分析和抽象，得到“数学模型”，再用模型来解决实际问题。

它融合了自然科学与社会科学，利用数学工具建立问题模型，通过计算机计算、分析、归纳和总结得出结论并提出解决问题的办法。

文章利用大数据技术和学习分析技术，设计了高职数学的精准教学模式，以云班课为平台，构建了数学建模方法融入高职数学教学模式。

关键词：大数据数学建模高职数学实践环节应用能力中图分类号：G712;O141.4-4文献标识码：A 文章编号：1672-3791(2023)13-0187-04Exploration on the Integration of the Mathematical Modeling Method Based on Big Data into Higher VocationalMathematics Teaching PracticeWANG Ying(Gansu Finance and Trade Professional College, Lanzhou, Gansu Province, 730207 China)Abstract: "Mathematical modeling" refers to using computers to express practical problems encountered in real life in certain mathematical methods, and performing simulation operations on computers. Through the analysis and abstraction of the problems in real life, a "mathematical model" is obtained, and then the model is used to solve practical problems. It integrates natural science and social science, uses mathematical tools to establish problem models, and draws conclusions and proposes solutions to problems through computer calculation, analysis, induction and summary. This article uses big data technology and learning analysis technology to design an accurate teaching model for higher vocational mathematics, and constructs a mode of integrating the mathematical modeling method into higher vocational mathematics teaching with Mosoteach as the platform.Key Words: Big data; Mathematical modeling; Higher vocational mathematics; Practice; Application ability近年来，随着高职教育招生规模的扩大和招生途径的多样化，学生基础参差不齐，学习行为分化的现象越来越突出，这些给高职数学教学带来了新的困难和挑战。

Geometric Modeling

Geometric ModelingGeometric modeling is a crucial aspect of computer-aided design and manufacturing, playing a fundamental role in various industries such as engineering, architecture, and animation. It involves the creation of digital representations of objects and environments using mathematical and computational techniques. This process enables designers and engineers to visualize, simulate, and analyze complex structures and shapes, leading to the development ofinnovative products and solutions. In this discussion, we will explore the significance of geometric modeling from different perspectives, considering its applications, challenges, and future advancements. From an engineering standpoint, geometric modeling serves as the cornerstone of product design and development. By representing physical components and systems through digital models, engineers can assess the performance, functionality, and manufacturability of their designs.This enables them to identify potential flaws or inefficiencies early in thedesign process, leading to cost savings and improved product quality. Geometric modeling also facilitates the creation of prototypes and simulations, allowing engineers to test and validate their ideas before moving into the production phase. As such, it significantly accelerates the innovation cycle and enhances theoverall efficiency of the product development process. In the field ofarchitecture and construction, geometric modeling plays a pivotal role in the conceptualization and visualization of building designs. Architects leverage advanced modeling software to create detailed 3D representations of structures, enabling clients and stakeholders to gain a realistic understanding of the proposed designs. This not only enhances communication and collaboration but also enables architects to explore different design options and assess their spatialand aesthetic qualities. Furthermore, geometric modeling supports the analysis of structural integrity and building performance, contributing to the creation of sustainable and resilient built environments. In the realm of animation andvisual effects, geometric modeling is indispensable for the creation of virtual characters, environments, and special effects. Artists and animators utilize sophisticated modeling tools to sculpt and manipulate digital surfaces, defining the shape, texture, and appearance of virtual objects. This process involves theuse of polygons, curves, and mathematical equations to create lifelike and dynamic visual elements that form the basis of compelling animations and cinematic experiences. Geometric modeling not only fuels the entertainment industry but also finds applications in scientific visualization, medical imaging, and virtual reality, enriching our understanding and experiences in diverse domains. Despite its numerous benefits, geometric modeling presents several challenges,particularly in dealing with complex geometries, large datasets, and computational efficiency. Modeling intricate organic shapes, intricate details, and irregular surfaces often requires advanced techniques and computational resources, posing a barrier for designers and engineers. Moreover, ensuring the accuracy and precision of geometric models remains a critical concern, especially in applications where small errors can lead to significant repercussions. Addressing these challenges demands continuous research and development in geometric modeling algorithms, data processing methods, and visualization technologies. Looking ahead, the future of geometric modeling holds tremendous promise, driven by advancements in artificial intelligence, machine learning, and computational capabilities. The integration of AI algorithms into geometric modeling tools can revolutionize the way designers and engineers interact with digital models, enabling intelligent automation, predictive analysis, and generative design. This paves the way for the creation of highly personalized and optimized designs, tailored to specific requirements and constraints. Furthermore, the convergence of geometric modeling with virtual and augmented reality technologies opens up new possibilities for immersive design experiences, interactive simulations, and digital twinning applications. In conclusion, geometric modeling stands as a vital enabler of innovation and creativity across various disciplines, empowering professionals to visualize, analyze, and realize their ideas in the digital realm. Its impact spans from product design and manufacturing to architecture, entertainment, and beyond, shaping the way we perceive and interact with the physical and virtual worlds. As we continue to push the boundaries of technology and imagination, geometric modeling will undoubtedly remain at the forefront of transformative advancements, driving progress and unlocking new frontiers of possibility.。

From Data Mining to Knowledge Discovery in Databases

s Data mining and knowledge discovery in databases have been attracting a signiﬁcant amount of research, industry, and media atten-tion of late. What is all the excitement about?This article provides an overview of this emerging ﬁeld, clarifying how data mining and knowledge discovery in databases are related both to each other and to related ﬁelds, such as machine learning, statistics, and databases. The article mentions particular real-world applications, speciﬁc data-mining techniques, challenges in-volved in real-world applications of knowledge discovery, and current and future research direc-tions in the ﬁeld.A cross a wide variety of ﬁelds, data arebeing collected and accumulated at adramatic pace. There is an urgent need for a new generation of computational theo-ries and tools to assist humans in extracting useful information (knowledge) from the rapidly growing volumes of digital data. These theories and tools are the subject of the emerging ﬁeld of knowledge discovery in databases (KDD).At an abstract level, the KDD ﬁeld is con-cerned with the development of methods and techniques for making sense of data. The basic problem addressed by the KDD process is one of mapping low-level data (which are typically too voluminous to understand and digest easi-ly) into other forms that might be more com-pact (for example, a short report), more ab-stract (for example, a descriptive approximation or model of the process that generated the data), or more useful (for exam-ple, a predictive model for estimating the val-ue of future cases). At the core of the process is the application of speciﬁc data-mining meth-ods for pattern discovery and extraction.1This article begins by discussing the histori-cal context of KDD and data mining and theirintersection with other related ﬁelds. A briefsummary of recent KDD real-world applica-tions is provided. Deﬁnitions of KDD and da-ta mining are provided, and the general mul-tistep KDD process is outlined. This multistepprocess has the application of data-mining al-gorithms as one particular step in the process.The data-mining step is discussed in more de-tail in the context of speciﬁc data-mining al-gorithms and their application. Real-worldpractical application issues are also outlined.Finally, the article enumerates challenges forfuture research and development and in par-ticular discusses potential opportunities for AItechnology in KDD systems.Why Do We Need KDD?The traditional method of turning data intoknowledge relies on manual analysis and in-terpretation. For example, in the health-careindustry, it is common for specialists to peri-odically analyze current trends and changesin health-care data, say, on a quarterly basis.The specialists then provide a report detailingthe analysis to the sponsoring health-care or-ganization; this report becomes the basis forfuture decision making and planning forhealth-care management. In a totally differ-ent type of application, planetary geologistssift through remotely sensed images of plan-ets and asteroids, carefully locating and cata-loging such geologic objects of interest as im-pact craters. Be it science, marketing, ﬁnance,health care, retail, or any other ﬁeld, the clas-sical approach to data analysis relies funda-mentally on one or more analysts becomingArticlesFALL 1996 37From Data Mining to Knowledge Discovery inDatabasesUsama Fayyad, Gregory Piatetsky-Shapiro, and Padhraic Smyth Copyright © 1996, American Association for Artificial Intelligence. All rights reserved. 0738-4602-1996 / $2.00areas is astronomy. Here, a notable success was achieved by SKICAT ,a system used by as-tronomers to perform image analysis,classiﬁcation, and cataloging of sky objects from sky-survey images (Fayyad, Djorgovski,and Weir 1996). In its ﬁrst application, the system was used to process the 3 terabytes (1012bytes) of image data resulting from the Second Palomar Observatory Sky Survey,where it is estimated that on the order of 109sky objects are detectable. SKICAT can outper-form humans and traditional computational techniques in classifying faint sky objects. See Fayyad, Haussler, and Stolorz (1996) for a sur-vey of scientiﬁc applications.In business, main KDD application areas includes marketing, ﬁnance (especially in-vestment), fraud detection, manufacturing,telecommunications, and Internet agents.Marketing:In marketing, the primary ap-plication is database marketing systems,which analyze customer databases to identify different customer groups and forecast their behavior. Business Week (Berry 1994) estimat-ed that over half of all retailers are using or planning to use database marketing, and those who do use it have good results; for ex-ample, American Express reports a 10- to 15-percent increase in credit-card use. Another notable marketing application is market-bas-ket analysis (Agrawal et al. 1996) systems,which ﬁnd patterns such as, “If customer bought X, he/she is also likely to buy Y and Z.” Such patterns are valuable to retailers.Investment: Numerous companies use da-ta mining for investment, but most do not describe their systems. One exception is LBS Capital Management. Its system uses expert systems, neural nets, and genetic algorithms to manage portfolios totaling $600 million;since its start in 1993, the system has outper-formed the broad stock market (Hall, Mani,and Barr 1996).Fraud detection: HNC Falcon and Nestor PRISM systems are used for monitoring credit-card fraud, watching over millions of ac-counts. The FAIS system (Senator et al. 1995),from the U.S. Treasury Financial Crimes En-forcement Network, is used to identify ﬁnan-cial transactions that might indicate money-laundering activity.Manufacturing: The CASSIOPEE trou-bleshooting system, developed as part of a joint venture between General Electric and SNECMA, was applied by three major Euro-pean airlines to diagnose and predict prob-lems for the Boeing 737. To derive families of faults, clustering methods are used. CASSIOPEE received the European ﬁrst prize for innova-intimately familiar with the data and serving as an interface between the data and the users and products.For these (and many other) applications,this form of manual probing of a data set is slow, expensive, and highly subjective. In fact, as data volumes grow dramatically, this type of manual data analysis is becoming completely impractical in many domains.Databases are increasing in size in two ways:(1) the number N of records or objects in the database and (2) the number d of ﬁelds or at-tributes to an object. Databases containing on the order of N = 109objects are becoming in-creasingly common, for example, in the as-tronomical sciences. Similarly, the number of ﬁelds d can easily be on the order of 102or even 103, for example, in medical diagnostic applications. Who could be expected to di-gest millions of records, each having tens or hundreds of ﬁelds? We believe that this job is certainly not one for humans; hence, analysis work needs to be automated, at least partially.The need to scale up human analysis capa-bilities to handling the large number of bytes that we can collect is both economic and sci-entiﬁc. Businesses use data to gain competi-tive advantage, increase efﬁciency, and pro-vide more valuable services to customers.Data we capture about our environment are the basic evidence we use to build theories and models of the universe we live in. Be-cause computers have enabled humans to gather more data than we can digest, it is on-ly natural to turn to computational tech-niques to help us unearth meaningful pat-terns and structures from the massive volumes of data. Hence, KDD is an attempt to address a problem that the digital informa-tion era made a fact of life for all of us: data overload.Data Mining and Knowledge Discovery in the Real WorldA large degree of the current interest in KDD is the result of the media interest surrounding successful KDD applications, for example, the focus articles within the last two years in Business Week , Newsweek , Byte , PC Week , and other large-circulation periodicals. Unfortu-nately, it is not always easy to separate fact from media hype. Nonetheless, several well-documented examples of successful systems can rightly be referred to as KDD applications and have been deployed in operational use on large-scale real-world problems in science and in business.In science, one of the primary applicationThere is an urgent need for a new generation of computation-al theories and tools toassist humans in extractinguseful information (knowledge)from the rapidly growing volumes ofdigital data.Articles38AI MAGAZINEtive applications (Manago and Auriol 1996).Telecommunications: The telecommuni-cations alarm-sequence analyzer (TASA) wasbuilt in cooperation with a manufacturer oftelecommunications equipment and threetelephone networks (Mannila, Toivonen, andVerkamo 1995). The system uses a novelframework for locating frequently occurringalarm episodes from the alarm stream andpresenting them as rules. Large sets of discov-ered rules can be explored with ﬂexible infor-mation-retrieval tools supporting interactivityand iteration. In this way, TASA offers pruning,grouping, and ordering tools to reﬁne the re-sults of a basic brute-force search for rules.Data cleaning: The MERGE-PURGE systemwas applied to the identiﬁcation of duplicatewelfare claims (Hernandez and Stolfo 1995).It was used successfully on data from the Wel-fare Department of the State of Washington.In other areas, a well-publicized system isIBM’s ADVANCED SCOUT,a specialized data-min-ing system that helps National Basketball As-sociation (NBA) coaches organize and inter-pret data from NBA games (U.S. News 1995). ADVANCED SCOUT was used by several of the NBA teams in 1996, including the Seattle Su-personics, which reached the NBA ﬁnals.Finally, a novel and increasingly importanttype of discovery is one based on the use of in-telligent agents to navigate through an infor-mation-rich environment. Although the ideaof active triggers has long been analyzed in thedatabase ﬁeld, really successful applications ofthis idea appeared only with the advent of theInternet. These systems ask the user to specifya proﬁle of interest and search for related in-formation among a wide variety of public-do-main and proprietary sources. For example, FIREFLY is a personal music-recommendation agent: It asks a user his/her opinion of several music pieces and then suggests other music that the user might like (<http:// www.fﬂ/>). CRAYON(/>) allows users to create their own free newspaper (supported by ads); NEWSHOUND(<http://www. /hound/>) from the San Jose Mercury News and FARCAST(</> automatically search information from a wide variety of sources, including newspapers and wire services, and e-mail rele-vant documents directly to the user.These are just a few of the numerous suchsystems that use KDD techniques to automat-ically produce useful information from largemasses of raw data. See Piatetsky-Shapiro etal. (1996) for an overview of issues in devel-oping industrial KDD applications.Data Mining and KDDHistorically, the notion of ﬁnding useful pat-terns in data has been given a variety ofnames, including data mining, knowledge ex-traction, information discovery, informationharvesting, data archaeology, and data patternprocessing. The term data mining has mostlybeen used by statisticians, data analysts, andthe management information systems (MIS)communities. It has also gained popularity inthe database ﬁeld. The phrase knowledge dis-covery in databases was coined at the ﬁrst KDDworkshop in 1989 (Piatetsky-Shapiro 1991) toemphasize that knowledge is the end productof a data-driven discovery. It has been popular-ized in the AI and machine-learning ﬁelds.In our view, KDD refers to the overall pro-cess of discovering useful knowledge from da-ta, and data mining refers to a particular stepin this process. Data mining is the applicationof speciﬁc algorithms for extracting patternsfrom data. The distinction between the KDDprocess and the data-mining step (within theprocess) is a central point of this article. Theadditional steps in the KDD process, such asdata preparation, data selection, data cleaning,incorporation of appropriate prior knowledge,and proper interpretation of the results ofmining, are essential to ensure that usefulknowledge is derived from the data. Blind ap-plication of data-mining methods (rightly crit-icized as data dredging in the statistical litera-ture) can be a dangerous activity, easilyleading to the discovery of meaningless andinvalid patterns.The Interdisciplinary Nature of KDDKDD has evolved, and continues to evolve,from the intersection of research ﬁelds such asmachine learning, pattern recognition,databases, statistics, AI, knowledge acquisitionfor expert systems, data visualization, andhigh-performance computing. The unifyinggoal is extracting high-level knowledge fromlow-level data in the context of large data sets.The data-mining component of KDD cur-rently relies heavily on known techniquesfrom machine learning, pattern recognition,and statistics to ﬁnd patterns from data in thedata-mining step of the KDD process. A natu-ral question is, How is KDD different from pat-tern recognition or machine learning (and re-lated ﬁelds)? The answer is that these ﬁeldsprovide some of the data-mining methodsthat are used in the data-mining step of theKDD process. KDD focuses on the overall pro-cess of knowledge discovery from data, includ-ing how the data are stored and accessed, howalgorithms can be scaled to massive data setsThe basicproblemaddressed bythe KDDprocess isone ofmappinglow-leveldata intoother formsthat might bemorecompact,moreabstract,or moreuseful.ArticlesFALL 1996 39A driving force behind KDD is the database ﬁeld (the second D in KDD). Indeed, the problem of effective data manipulation when data cannot ﬁt in the main memory is of fun-damental importance to KDD. Database tech-niques for gaining efﬁcient data access,grouping and ordering operations when ac-cessing data, and optimizing queries consti-tute the basics for scaling algorithms to larger data sets. Most data-mining algorithms from statistics, pattern recognition, and machine learning assume data are in the main memo-ry and pay no attention to how the algorithm breaks down if only limited views of the data are possible.A related ﬁeld evolving from databases is data warehousing,which refers to the popular business trend of collecting and cleaning transactional data to make them available for online analysis and decision support. Data warehousing helps set the stage for KDD in two important ways: (1) data cleaning and (2)data access.Data cleaning: As organizations are forced to think about a uniﬁed logical view of the wide variety of data and databases they pos-sess, they have to address the issues of map-ping data to a single naming convention,uniformly representing and handling missing data, and handling noise and errors when possible.Data access: Uniform and well-deﬁned methods must be created for accessing the da-ta and providing access paths to data that were historically difﬁcult to get to (for exam-ple, stored ofﬂine).Once organizations and individuals have solved the problem of how to store and ac-cess their data, the natural next step is the question, What else do we do with all the da-ta? This is where opportunities for KDD natu-rally arise.A popular approach for analysis of data warehouses is called online analytical processing (OLAP), named for a set of principles pro-posed by Codd (1993). OLAP tools focus on providing multidimensional data analysis,which is superior to SQL in computing sum-maries and breakdowns along many dimen-sions. OLAP tools are targeted toward simpli-fying and supporting interactive data analysis,but the goal of KDD tools is to automate as much of the process as possible. Thus, KDD is a step beyond what is currently supported by most standard database systems.Basic DeﬁnitionsKDD is the nontrivial process of identifying valid, novel, potentially useful, and ultimate-and still run efﬁciently, how results can be in-terpreted and visualized, and how the overall man-machine interaction can usefully be modeled and supported. The KDD process can be viewed as a multidisciplinary activity that encompasses techniques beyond the scope of any one particular discipline such as machine learning. In this context, there are clear opportunities for other ﬁelds of AI (be-sides machine learning) to contribute to KDD. KDD places a special emphasis on ﬁnd-ing understandable patterns that can be inter-preted as useful or interesting knowledge.Thus, for example, neural networks, although a powerful modeling tool, are relatively difﬁcult to understand compared to decision trees. KDD also emphasizes scaling and ro-bustness properties of modeling algorithms for large noisy data sets.Related AI research ﬁelds include machine discovery, which targets the discovery of em-pirical laws from observation and experimen-tation (Shrager and Langley 1990) (see Kloes-gen and Zytkow [1996] for a glossary of terms common to KDD and machine discovery),and causal modeling for the inference of causal models from data (Spirtes, Glymour,and Scheines 1993). Statistics in particular has much in common with KDD (see Elder and Pregibon [1996] and Glymour et al.[1996] for a more detailed discussion of this synergy). Knowledge discovery from data is fundamentally a statistical endeavor. Statistics provides a language and framework for quan-tifying the uncertainty that results when one tries to infer general patterns from a particu-lar sample of an overall population. As men-tioned earlier, the term data mining has had negative connotations in statistics since the 1960s when computer-based data analysis techniques were ﬁrst introduced. The concern arose because if one searches long enough in any data set (even randomly generated data),one can ﬁnd patterns that appear to be statis-tically signiﬁcant but, in fact, are not. Clearly,this issue is of fundamental importance to KDD. Substantial progress has been made in recent years in understanding such issues in statistics. Much of this work is of direct rele-vance to KDD. Thus, data mining is a legiti-mate activity as long as one understands how to do it correctly; data mining carried out poorly (without regard to the statistical as-pects of the problem) is to be avoided. KDD can also be viewed as encompassing a broader view of modeling than statistics. KDD aims to provide tools to automate (to the degree pos-sible) the entire process of data analysis and the statistician’s “art” of hypothesis selection.Data mining is a step in the KDD process that consists of ap-plying data analysis and discovery al-gorithms that produce a par-ticular enu-meration ofpatterns (or models)over the data.Articles40AI MAGAZINEly understandable patterns in data (Fayyad, Piatetsky-Shapiro, and Smyth 1996).Here, data are a set of facts (for example, cases in a database), and pattern is an expres-sion in some language describing a subset of the data or a model applicable to the subset. Hence, in our usage here, extracting a pattern also designates ﬁtting a model to data; ﬁnd-ing structure from data; or, in general, mak-ing any high-level description of a set of data. The term process implies that KDD comprises many steps, which involve data preparation, search for patterns, knowledge evaluation, and reﬁnement, all repeated in multiple itera-tions. By nontrivial, we mean that some search or inference is involved; that is, it is not a straightforward computation of predeﬁned quantities like computing the av-erage value of a set of numbers.The discovered patterns should be valid on new data with some degree of certainty. We also want patterns to be novel (at least to the system and preferably to the user) and poten-tially useful, that is, lead to some beneﬁt to the user or task. Finally, the patterns should be understandable, if not immediately then after some postprocessing.The previous discussion implies that we can deﬁne quantitative measures for evaluating extracted patterns. In many cases, it is possi-ble to deﬁne measures of certainty (for exam-ple, estimated prediction accuracy on new data) or utility (for example, gain, perhaps indollars saved because of better predictions orspeedup in response time of a system). No-tions such as novelty and understandabilityare much more subjective. In certain contexts,understandability can be estimated by sim-plicity (for example, the number of bits to de-scribe a pattern). An important notion, calledinterestingness(for example, see Silberschatzand Tuzhilin [1995] and Piatetsky-Shapiro andMatheus [1994]), is usually taken as an overallmeasure of pattern value, combining validity,novelty, usefulness, and simplicity. Interest-ingness functions can be deﬁned explicitly orcan be manifested implicitly through an or-dering placed by the KDD system on the dis-covered patterns or models.Given these notions, we can consider apattern to be knowledge if it exceeds some in-terestingness threshold, which is by nomeans an attempt to deﬁne knowledge in thephilosophical or even the popular view. As amatter of fact, knowledge in this deﬁnition ispurely user oriented and domain speciﬁc andis determined by whatever functions andthresholds the user chooses.Data mining is a step in the KDD processthat consists of applying data analysis anddiscovery algorithms that, under acceptablecomputational efﬁciency limitations, pro-duce a particular enumeration of patterns (ormodels) over the data. Note that the space ofArticlesFALL 1996 41Figure 1. An Overview of the Steps That Compose the KDD Process.methods, the effective number of variables under consideration can be reduced, or in-variant representations for the data can be found.Fifth is matching the goals of the KDD pro-cess (step 1) to a particular data-mining method. For example, summarization, clas-siﬁcation, regression, clustering, and so on,are described later as well as in Fayyad, Piatet-sky-Shapiro, and Smyth (1996).Sixth is exploratory analysis and model and hypothesis selection: choosing the data-mining algorithm(s) and selecting method(s)to be used for searching for data patterns.This process includes deciding which models and parameters might be appropriate (for ex-ample, models of categorical data are differ-ent than models of vectors over the reals) and matching a particular data-mining method with the overall criteria of the KDD process (for example, the end user might be more in-terested in understanding the model than its predictive capabilities).Seventh is data mining: searching for pat-terns of interest in a particular representa-tional form or a set of such representations,including classiﬁcation rules or trees, regres-sion, and clustering. The user can signiﬁcant-ly aid the data-mining method by correctly performing the preceding steps.Eighth is interpreting mined patterns, pos-sibly returning to any of steps 1 through 7 for further iteration. This step can also involve visualization of the extracted patterns and models or visualization of the data given the extracted models.Ninth is acting on the discovered knowl-edge: using the knowledge directly, incorpo-rating the knowledge into another system for further action, or simply documenting it and reporting it to interested parties. This process also includes checking for and resolving po-tential conﬂicts with previously believed (or extracted) knowledge.The KDD process can involve signiﬁcant iteration and can contain loops between any two steps. The basic ﬂow of steps (al-though not the potential multitude of itera-tions and loops) is illustrated in ﬁgure 1.Most previous work on KDD has focused on step 7, the data mining. However, the other steps are as important (and probably more so) for the successful application of KDD in practice. Having deﬁned the basic notions and introduced the KDD process, we now focus on the data-mining component,which has, by far, received the most atten-tion in the literature.patterns is often inﬁnite, and the enumera-tion of patterns involves some form of search in this space. Practical computational constraints place severe limits on the sub-space that can be explored by a data-mining algorithm.The KDD process involves using the database along with any required selection,preprocessing, subsampling, and transforma-tions of it; applying data-mining methods (algorithms) to enumerate patterns from it;and evaluating the products of data mining to identify the subset of the enumerated pat-terns deemed knowledge. The data-mining component of the KDD process is concerned with the algorithmic means by which pat-terns are extracted and enumerated from da-ta. The overall KDD process (ﬁgure 1) in-cludes the evaluation and possible interpretation of the mined patterns to de-termine which patterns can be considered new knowledge. The KDD process also in-cludes all the additional steps described in the next section.The notion of an overall user-driven pro-cess is not unique to KDD: analogous propos-als have been put forward both in statistics (Hand 1994) and in machine learning (Brod-ley and Smyth 1996).The KDD ProcessThe KDD process is interactive and iterative,involving numerous steps with many deci-sions made by the user. Brachman and Anand (1996) give a practical view of the KDD pro-cess, emphasizing the interactive nature of the process. Here, we broadly outline some of its basic steps:First is developing an understanding of the application domain and the relevant prior knowledge and identifying the goal of the KDD process from the customer’s viewpoint.Second is creating a target data set: select-ing a data set, or focusing on a subset of vari-ables or data samples, on which discovery is to be performed.Third is data cleaning and preprocessing.Basic operations include removing noise if appropriate, collecting the necessary informa-tion to model or account for noise, deciding on strategies for handling missing data ﬁelds,and accounting for time-sequence informa-tion and known changes.Fourth is data reduction and projection:ﬁnding useful features to represent the data depending on the goal of the task. With di-mensionality reduction or transformationArticles42AI MAGAZINEThe Data-Mining Stepof the KDD ProcessThe data-mining component of the KDD pro-cess often involves repeated iterative applica-tion of particular data-mining methods. This section presents an overview of the primary goals of data mining, a description of the methods used to address these goals, and a brief description of the data-mining algo-rithms that incorporate these methods.The knowledge discovery goals are deﬁned by the intended use of the system. We can distinguish two types of goals: (1) veriﬁcation and (2) discovery. With veriﬁcation,the sys-tem is limited to verifying the user’s hypothe-sis. With discovery,the system autonomously ﬁnds new patterns. We further subdivide the discovery goal into prediction,where the sys-tem ﬁnds patterns for predicting the future behavior of some entities, and description, where the system ﬁnds patterns for presenta-tion to a user in a human-understandableform. In this article, we are primarily con-cerned with discovery-oriented data mining.Data mining involves ﬁtting models to, or determining patterns from, observed data. The ﬁtted models play the role of inferred knowledge: Whether the models reﬂect useful or interesting knowledge is part of the over-all, interactive KDD process where subjective human judgment is typically required. Two primary mathematical formalisms are used in model ﬁtting: (1) statistical and (2) logical. The statistical approach allows for nondeter-ministic effects in the model, whereas a logi-cal model is purely deterministic. We focus primarily on the statistical approach to data mining, which tends to be the most widely used basis for practical data-mining applica-tions given the typical presence of uncertain-ty in real-world data-generating processes.Most data-mining methods are based on tried and tested techniques from machine learning, pattern recognition, and statistics: classiﬁcation, clustering, regression, and so on. The array of different algorithms under each of these headings can often be bewilder-ing to both the novice and the experienced data analyst. It should be emphasized that of the many data-mining methods advertised in the literature, there are really only a few fun-damental techniques. The actual underlying model representation being used by a particu-lar method typically comes from a composi-tion of a small number of well-known op-tions: polynomials, splines, kernel and basis functions, threshold-Boolean functions, and so on. Thus, algorithms tend to differ primar-ily in the goodness-of-ﬁt criterion used toevaluate model ﬁt or in the search methodused to ﬁnd a good ﬁt.In our brief overview of data-mining meth-ods, we try in particular to convey the notionthat most (if not all) methods can be viewedas extensions or hybrids of a few basic tech-niques and principles. We ﬁrst discuss the pri-mary methods of data mining and then showthat the data- mining methods can be viewedas consisting of three primary algorithmiccomponents: (1) model representation, (2)model evaluation, and (3) search. In the dis-cussion of KDD and data-mining methods,we use a simple example to make some of thenotions more concrete. Figure 2 shows a sim-ple two-dimensional artiﬁcial data set consist-ing of 23 cases. Each point on the graph rep-resents a person who has been given a loanby a particular bank at some time in the past.The horizontal axis represents the income ofthe person; the vertical axis represents the to-tal personal debt of the person (mortgage, carpayments, and so on). The data have beenclassiﬁed into two classes: (1) the x’s repre-sent persons who have defaulted on theirloans and (2) the o’s represent persons whoseloans are in good status with the bank. Thus,this simple artiﬁcial data set could represent ahistorical data set that can contain usefulknowledge from the point of view of thebank making the loans. Note that in actualKDD applications, there are typically manymore dimensions (as many as several hun-dreds) and many more data points (manythousands or even millions).ArticlesFALL 1996 43Figure 2. A Simple Data Set with Two Classes Used for Illustrative Purposes.。

美容相关词汇西英对照

A A Aa la moda in style, in fashion在时尚的风格，ablandador softening, softener柔软剂，柔软剂absorbente absorbent吸水性acabado mate matte finish粉嫩accesorio accesory装饰品aceite oil油aceitunado (color de piel) olive橄榄acentuar accentuate, emphasize加重，强调aclarador (adj) clarifying, lightening打亮aclarador de axilas armpit lightener腋下美白acné acne粉刺adorno adornment装饰aerógrafo airbrush喷枪afeitar shave刮胡子afeite personal care; any cosmetic product个人护理，任何化妆品产品afilado sharp (edge)锋利agrandar make larger, enlarge放大ahumado smoky烟熏ajado wrinkled (skin)布满皱纹（皮肤）alargar make longer, lengthen使更长的时间，延长alergia allergy过敏alérgica allergic过敏的algodón cotton棉花amapola (color) vivid red (like a poppy)鲜艳的红色（如罂粟）amelocotonado (tono) peach-colored桃色amorronado (tono) brownish棕antiarrugas anti-wrinkle抗皱遮瑕膏antiaojeras concealer or treatment for dark circles uantideslizante (maquillaje) anti-run防脱妆antiedad anti-aging抗衰老antienvejecimiento anti-aging抗衰老apariencia appearance, look外观aplicación application应用aplicador applicator涂抹argolla, aro, arito hoop earring回圈耳环armonizar harmonize, blend协调，融合armonioso harmonious和谐arqueado (cejas) arched拱形arquear (cejas) arch; (pestañas) curl卷曲arrebatador breathtaking, dazzling, captivating摄人心魄，令人眼花缭乱，迷人的arreglarse el pelo do one's hair做头发arreglarse las manos do one's nails修指甲arreglarse las uñas do one's nails修指甲arrobador entrancing迷人的arrobol (cosmético) blush, rouge, blusher腮红，口红，腮红arrosquetado (tez) dark, brown, tanned黑色，棕色，皮肤黝黑arruga wrinkle皱纹arrugado wrinkled (skin)布满皱纹（皮肤）artículos de tocador toiletries化妆品aseo personal care, cleanliness个人护理用品，清洁aserora de belleza beauty advisor美容顾问aspecto appearance, look外观áspereza (piel) roughness粗糙度áspero (piel) rough粗糙astringente astringent涩aterronar (maquillaje) clump, form lumps结成块atractivo attractive有吸引力的atrayente attractive有吸引力的atrevida daring大胆audaz audacious, daring大胆的，大胆的avellano (color de ojos) hazel榛色azul (color) blue蓝色azul celeste (color) sky blue天蓝azul intenso (color) deep blue深蓝色azul turquesa (color) turquoise blue绿松石蓝色azulina (color) blue jasmine蓝茉莉B B Bbálsamo balm香脂bálsamo labial lip balm润唇膏baño bath浴baño corporal body wash沐浴露barbilla chin下巴barra labios, barra de labios lipstick口红barrito, barillo blackhead, pimple黑头粉刺，青春痘base, base de maquillaje foundation底妆base en crema cream foundation粉底霜base en polvo powder foundation粉底baya (color) berry浆果色beige, beis (color) beige米色bella beautiful美丽blanqueante (de tez) lightener美白boca mouth口bolsas, bolsitas (de ojos) bags under the eyes眼袋bonita pretty, lovely漂亮，可爱的bonita con rosa pretty in pink红粉佳人bordó (color) bordeaux, burgundy, maroon栗色brillar shine闪耀brillo gloss, shine, reflection光泽度，光泽，反射brillo de labios lip gloss, lip polish唇彩，指甲油brillo de pestañas eyelash shine睫毛闪耀brillo labial lip gloss, lip polish唇彩，指甲油brocha brush刷brocha para polvo powder brush刷粉bronce (color) bronze青铜bronceada tan, tanned, with a suntan晒黑，晒黑，晒黑bronceador bronzer古铜色bruñido burnished, polished打磨，抛光C C Ccabello hair头发caché style, cachet风格cachetes cheeks两颊café (color) coffee咖啡cálido (tono) warm温暖canela (color o fragancia) cinnamon肉桂色capa layer层cara de forma de pera pear-shaped face梨形脸cara ovalada oval face瓜子儿脸cara quadrada square-shaped face方脸cara redonda round face圆圆的脸carmesí crimson赤红carmín lipstick; (color) carmine口红（彩色）胭脂红carmín oscuro (color) dark vivid red鲜艳的红色暗castaño (color de pelo o ojos) brown棕色cautivador captivating迷人cejas eyebrows眉毛cenicienta (piel) ashy灰色的centro center中心cepillo brush刷cepillo corporal body brush身体刷cera wax蜡chica girl女孩chic chic, stylish别致，时尚chica girl女孩chocolate (color) chocolate巧克力cicatriz scar疤痕cicatrizar heal治疗cobertura (base) coverage覆盖cobertura alta full coverage全覆盖cobertura ligera light coverage光覆盖cobre (color) copper铜色cobrizo (tone) tawny, copper黄褐色，铜colágeno collagen胶原colirio eye drops, eyedrops, eyewash眼药水，滴眼液，洗眼器color color, hue颜色，色调color crema cream奶油color labial lip color, lipstick唇彩，口红color oro gold, golden黄金，黄金color plata silver银coloración color, coloring, coloration颜色，着色，着色colorante color, coloring颜色，着色colores claros light colors光的颜色colores oscuros dark colors暗色colorete blush, blusher, rouge腮红，腮红，胭脂comedogénico comedogenic粉刺comedón blackhead黑头粉刺comisura corner of the mouth嘴角compacto compact紧凑complexión build, body type, constitution建立，体型，体质confianza confidence信心consejos de belleza beauty advice, beauty tips美容咨询，美容秘诀contorno contour, outline轮廓，轮廓contraste contrast对比corrector de ojeras concealer for dark circles under eyes 黑眼圈的遮瑕corrector facial concealer遮瑕膏cortado (skin) flaky, dry; (lips) chapped（皮肤）片状，干燥;（嘴唇）皲裂corto short短cosmético cosmetic化妆品cosméticos cosmetics化妆品cosmetiquera, cosmetiquero makeup bag, makeup case化妆包，化妆包cosmetológa, cosmetológo beautician, cosmetologist美容师，美容师cosmetología cosmetology美容crema cream, creme奶油，奶油crema corporal body lotion身体乳液crema de belleza beauty cream美容霜crema de cuerpo body cream体霜crema de día day cream日霜crema de manos, crema manos hand cream护手油crema de noche night cream晚霜crema de ojos eye cream眼霜crema de piel skin cream护肤霜crema dermatológica skin cream护肤霜crema facial face cream面霜crema hidratante moisturizer保湿霜crema humectante moisturizer保湿霜crema limpiadora cleanser去污粉crema nutritiva cold cream冷霜crema para cuerpo body cream体霜crema para manos hand cream护手油cremoso creamy奶油creyón (de labios) lipstick口红cubrir cover覆盖黑眼圈的遮瑕膏cubre orejas concealer for dark circles under the eyecuerpo body身体cuidado care关心cuidado de la piel skin care皮肤护理cuidado de las uñas nail care指甲护理cuidado del pelo hair care护发cultora de belleza beauty technician美容技师cultura de belleza beauty culture美容文化cutis skin (esp. of face)皮肤（尤其是人脸）D D Ddama lady女士decolorante lightener美白decolorante facial skin lightener皮肤美白delicado delicate娇嫩delineado outline, outlining (of eyes or lips)轮廓delineador de cejas eyebrow pencil眉笔delineador de labios lip liner, lip outliner, lip pencil唇线笔，唇大纲，唇线笔; delineador de ojos eyeliner, eye liner眼线笔，眼线delineador líquido liquid outliner, liquid eyeliner液体眼线笔demasiado too, too much太多，太多depilación hair removal (by any method), waxing脱毛（通过任何方法），打蜡depilar wax, pluck, remove body hair去除体毛depilar las cejas pluck the eyebrows拔眉毛dermatólogo dermatologist皮肤科医生dermis skin皮肤deshidratada dry干deslumbrante dazzling耀眼desmaquillador, desmaquillante makeup remover, cleansing cream卸妆，洁面霜despampanante stunning令人惊叹的destello sparkle, twinkle, glitter闪闪发光，闪烁，闪烁desvanecerse (maquillaje) fade褪色desvanecido faded褪色混合刷（用于眼妆）difuminador (para ojos) smudger, blending brush (for eye colorsdifuminar (colores) blend, diffuse混合，弥漫disimulador concealer遮瑕膏donaire grace, charm, elegance风度，魅力，优雅dorado golden金色的duradero lasting持久E E Eébano ebony乌木echarse perfume put on perfume喷香水electrólisis electrolisis (method of permanent hair r electrolisis（永久脱毛的方法）elegante elegant优雅elusivo elusive难以捉摸embellecer beautify美化emborronarse (maquillaje) smudge弄脏emoliente emollient缓和剂encantador enchanting, lovely迷人的，可爱的enchinado permanente de pestañas eyelash permanent (curl)睫毛永久（卷曲）enchinador de pestañas eyelash curler睫毛夹encrespador de pestañas eyelash curler睫毛夹énfasis emphasis重点enigmático enigmatic, mysterious神秘的，神秘的envejecimiento aging老化escamoso (piel) scaly, flaky鳞片状，片状esfumar (maquillaje) soften, blur, blend, tone down软化，混合，模糊，淡化esmalte de uñas nail polish指甲油esmerilado frosted磨砂espejo mirror镜espejo de maquillaje makeup mirror化妆镜espesar (pestañas) thicken加厚espinilla blackhead, pimple黑头粉刺，青春痘esponja sponge海绵espuma foam泡沫espuma endurecedora firming foam紧肤泡沫esquina corner角落esquina exterior (del ojo) outer corner外眼角esquina interna (del ojo) inner corner内眼角estética facial facial aesthetics / esthetics面部美容/美学estética femenina women's beauty女性之美esteticista beautician, esthetician美容师，美容师estilo style风格estiloso stylish时尚estrías stretch marks妊娠纹estuche (de maquillaje) compact紧凑evocador haunting, evocative令人难忘的，令人回味的exfoliante facial facial scrub面部去角质exfoliación exfoliation去角质exquisito exquisite精美extensión (para uñas o pelo) extension延期extensiones de pestañas eyelash extensions睫毛扩展exterior exterior, outer外部，外extracto extract提取F F Ffacciones features (of face)特征（人脸）facial facial面部fijador de perfume perfume fixative香水固定液fijador de sombras eye primer眼部打底fijador del (lápiz) labial lipstick sealer口红固定液fijador para labios lip primer唇部打底filtro solar sunscreen防晒fragancia fragrance, scent, perfume香水，香味，香水frasco bottle, jar瓶，罐frente forehead前额frío (tono) cool凉爽frotar scrub擦洗G G Ggallardía elegance, striking appearance优雅，引人注目的外观garboso graceful, elegant, stylish优美，典雅，时尚gel gel凝胶gel de baño corporal body wash沐浴露gel endurecedor firming gel紧肤凝胶glamorosa glamorous富有魅力的glamour glamor, glamour魅力，魅力grano, granito pimple, grain疙瘩grasa oil油grasoso oily油腻的güero light (skin or hair)光（皮肤或头发）H H Hhacer resaltar bring out, make stand out脱颖而出hechizante bewitching, enchanting迷人的，迷人的hermosa beautiful美丽hermosura beauty美女hidratación moisturization保湿hidratar moisturize滋润hídrosoluble water-soluble水溶的hincapié emphasis重点hinchado swollen肿hinchazón swelling肿胀hipoalérgeno hypoallergenic低过敏性hongo fungus菌hueso de la ceja eyebrow bone眉骨humectación moisturization保湿humectante moisturizer保湿霜humectar moisturize滋润I I Iilluminador (adj) illuminating提亮illuminador (noun) highlighter打亮笔impecable flawless完美无瑕imperfección imperfection, flaw缺陷，缺陷impermeable waterproof防水intensidad intensity强度irradiar radiate辐射J J Jjabón soap肥皂jabón de tocador beauty soap美容香皂jasmín (fragancia o color) jasmine茉莉K K Kkarité shea butter乳木果油keratina keratin角蛋白L L Llabial lipstick口红labio inferior lower lip下唇labio superior upper lip上唇labios lips嘴唇labios partidos chapped lips干裂的嘴唇laca de uñas nail polish指甲油lagrimal corner of the eye; tear duct眼角泪道lagrimas tears眼泪lápiz de cejas eyebrow pencil眉笔lápiz iluminador illuminating pen提亮笔lápiz de labios lip pencil唇线笔lápiz (delineador) de ojos, lápiz ojo eye pencil, eyeliner眼线笔，眼线lápiz labial lipstick口红lápiz labial cremoso creme lipstick奶油口红largo long; length长度长;lavanda (color, scent) lavender薰衣草lavanda pálido pale lavender淡紫色lavar wash洗leche limpiadora cleansing milk洗面奶levemente lightly, slightly, softly, gently略，轻，轻轻地，轻轻地libre de fragancia fragrance-free无香味libre de aceite oil-free不含油libre de alcohol alcohol-free不含酒精lila (color) lilac紫丁香limpiar clean清洁limpiar con trapo wipe擦limpieza cleaning, cleansing; cleanliness清洗，清洁limpieza facial facial cleansing面部清洁limpieza profunda deep cleansing深层清洁limpio clean清洁linda pretty, lovely漂亮，可爱的lindísima very pretty很漂亮línea line线líquido liquid液体liso smooth光滑llamativo striking引人注目llevar maquillaje wear makeup, use makeup化妆，用化妆loción lotion洗剂loción corporal body lotion身体乳液loción hidratante moisturizer保湿霜loción humectante moisturizer保湿霜loción para manos hand lotion润手乳液loción tonificante toner, toning lotion爽肤水，化妆水lozano healthy-looking健康的长相luminoso luminous发光的lunar mole, blemish, beauty mark美人痣lustroso lustrous有光泽M M Mmácula blotch, spot (on skin)斑病，斑点（皮肤）maltratada (piel) rough粗糙malva (color) mauve淡紫色mancha spot, stain, discoloration斑点mancha de la edad age spot, “liver spot”老年斑，肝斑mancha del hígado “liver spot” (age spot)“肝斑”（老年斑）manicura manicure修指甲法manos hands手manteca de shea, manteca de karité shea butter乳木果油mantequilla corporal body butter身体黄油maquilladora makeup artist化妆师maquillaje compacto compact makeup粉饼maquillaje de boda wedding makeup婚礼跟妆maquillaje de día daytime makeup白天化妆maquillaje de fiesta party makeup派对妆maquillaje de noche evening makeup, night makeup晚妆，晚上卸妆maquillaje de novia bridal makeup新娘妆maquillaje de ojos eye makeup眼妆maquillaje de pestañas mascara睫毛膏maquillaje disimulador concealer, cover遮瑕，盖maquillaje dramático dramatic makeup戏剧性的化妆maquillaje en polvo powder makeup粉妆maquillaje escénico stage makeup舞台妆maquillaje formal formal makeup正式化妆maquillaje mineral mineral makeup矿物质彩妆maquillaje para ojos eye makeup眼妆maquillaje permanente permanent makeup永久化妆maquillaje social makeup for work and social occasions工作和社交场合的化妆maquillaje teatral stage makeup舞台妆maquillista makeup artist化妆师marfil (color) ivory象牙白marrón (color) brown棕色mascara mascara睫毛膏máscara masque, mask面膜，面膜máscara para labios lip mask唇膜mascarilla masque, face pack面膜，面膜mascarilla facial de barro mud pack泥包mate matte, not glossy磨砂，没有光泽matizar (colores) blend, shade混合，阴凉处mejillas cheeks两颊mejillas sonrosadas rosy cheeks玫瑰色的脸颊melocotón (color) peach桃色mentón chin下巴metálico (color) metallic金属色mezcla mixture, blend混合mezcla de color color blend颜色混合microexfolación microdermabrasion微晶磨皮miel (color de ojos) honey, hazel蜂蜜，榛子色mística mystique神秘感morado (color) purple紫色moreno claro (tez) light brown, tan浅棕色，棕褐色mota de algodón cotton ball棉花球mota para polvo powder puff粉扑muchacha girl女孩mujer woman女人mujer elegante elegant woman优雅的女人mullido soft (skin)软（皮肤）N N Nnacarado pearly珍珠色narices nostrils鼻孔nariz nose鼻子natural natural自然neceser makeup bag化妆包negro (color) black黑色no usar maquillaje to not wear makeup不化妆Ñ Ñ ÑO O Oocultar imperfecciones hide imperfections隐藏缺陷ojeras (dark) circles under the eyes眼圈ojerosa having ojeras有黑眼圈ojos ahumados smoky eyes烟熏眼ojos bailadores sparkling eyes闪闪发光的眼睛ojos luminosos luminous eyes双眼发光ojos misteriosos mysterious eyes神秘的眼睛ojos vivarochos sparkling eyes闪闪发光的眼睛oscuro (tono) dark暗沉P P Ppaleta de colores color palette调色板paleta de maquillaje makeup kit, color compact, makeup pa彩妆盒化妆工具包pálido (tono) pale苍白papel matificante blotting paper吸纸pardusco brownish; dark暗棕párpado eyelid眼皮párpado fijo lower eyelid下眼睑párpado movil upper eyelid上眼睑pastel (tono) pastel柔和patas de gallo, paticas de gallina crow's feet鱼尾纹pecas freckles雀斑pecosa (piel) with freckles, freckled, freckly脸上有雀斑pedicura pedicure修脚pelo hair头发peluquería hair salon美发沙龙perfecto perfect完美perfil profile轮廓perfilador de cejas eyebrow pencil眉笔perfilador de labios lip pencil唇线笔perfume (de mujer) perfume, scent, fragrance香水perfume (de hombre) cologne, after-shave古龙水perfumería beauty products store美容产品店perla (color) pearl珍珠pestaña de abajo lower eyelashes下睫毛pestaña de arriba upper eyelashes上睫毛pestañas eyelashes, lashes睫毛，睫毛pestañas postizas o artificiales false eyelashes假睫毛pestañita mascara睫毛膏piel skin皮肤piel blanca white skin白皙的皮肤piel canela cinnamon (color) skin肉桂（彩色）皮肤piel ceniza / cenicienta ashy skin灰色的皮肤piel combinación skin with some parts oily, some dry混合性皮肤piel grasosa oily skin油性皮肤piel morena dark skin, brown skin黝黑的皮肤，棕色皮肤piel morena clara light brown skin浅棕色的皮肤piel negra black skin, brown skin黑皮肤，棕色皮肤piel normal normal skin正常皮肤piel reseca very dry skin极干性肌肤piel seca dry skin干性皮肤piel trigueña light brown skin (wheat colored)浅棕色的皮肤（小麦色）pincel brush刷pincel iluminador illuminating pen, correcting brush for da照明刷笔，纠正黑眼圈等pincel para labios lip brush唇刷pintalabios lipstick口红pintarse put on makeup化妆pintura (de cara) makeup化妆pintura de labios lipstick口红pinzas tweezers镊子pinzas sacacejas eyebrow tweezers眉毛镊子planchado de cejas eyebrow shaping or straightening修眉或矫直plateado silver银pliegue del párpado eyelid crease or fold眼睑折痕polvera compact紧凑polvo powder, face powder粉，粉饼polvo compacto pressed powder粉末压片polvo de la cara face powder香粉polvo de tocador face powder香粉polvo decolorante bleaching powder漂白粉polvo suelto loose powder散粉pómulos cheekbones颧骨poros pores毛孔poros abiertos open pores毛孔开放poros faciales facial pores面部毛孔poros obstruidos clogged pores堵塞毛孔prebase skin primer, pre皮肤引物，预precioso beautiful, gorgeous, lovely漂亮，华丽，可爱productos de belleza beauty products美容产品pulir polish抛光pulir las cejas pluck one's eyebrows拔眉毛punto negro blackhead黑头粉刺puro (color) pure纯Q Q Qqueratina keratin角蛋白quitaesmalte nail polish remover指甲油去除剂R R Rradiante radiant辐射的rasgos de la cara facial features面容reafirmante firming, lifting紧肤，提升recargado (maquillaje) overdone, too much过头了，太多了régimen regimen, routine养生之道，日常reojo corner of the eye眼角resaltar highlight, emphasize, bring out突出强调，衬托出reseco dry, ashy干燥，灰resequedad dryness干燥retocar retouch润饰retráctil retractable伸缩rico rich丰富rímel mascara睫毛膏rizado de pestañas eyelash curling睫毛卷翘rizador curler夹rizador pestaña, rizapestañas eyelash curler睫毛夹rocio corporal body mist身体喷雾rojeces red spots on skin皮肤上的红色斑点rojo atrevido (color) daring red大胆的红色rojo profundo (color) deep red深红色rosa, rosita (color) pink粉红色rosa pastel (color) pastel pink柔和的粉红色rosado (color) pink粉红色rosado chillón (color) hot pink热粉红色rosado oscuro (color) deep pink深粉红色rubor blush, blusher, rouge腮红，腮红，胭脂rubor en crema cream blush腮红膏rubor en polvo powder blush腮红S S Ssábila aloe vera芦荟sacacejas eyebrow tweezers眉毛镊子sacar (facciones) bring out, emphasize带出来的，强调sacar las cejas pluck one's eyebrows采摘的眉毛salón de belleza, sala de belleza beauty parlor, beauty salon, hair salon美容院，美容院，发廊salón de uñas nail salon美甲沙龙saludable healthy健康sano healthy健康secado instantaneo quick-dry快速干的seco dry干sedoso silky如丝般sellador de cejas eyebrow sealer眉毛固定液sensible (piel) sensitive敏感sonrosado (mejillas) rosy红润sombra de ojos eye shadow眼影sombra nacarada lustrous pearly eye shadow光泽珍珠眼影suavidad softness柔软subido (color) deep, intense深刻的，激烈的suelto loose松T T Ttaco (negro) blackhead黑头粉刺tapa ojeras, tapaojeras (maquillaje) concealer for dark circles under eyes 黑眼圈的遮瑕tapar cover覆盖técnica technique技术tenacillas tweezers镊子terapia cutánea skin therapy治疗皮肤textura texture质地tez skin, skin tone, complexion皮肤，肤色，肤色tiña ringworm癣tisú tissue, Kleenex面巾纸toallita towelette, facecloth, pad湿纸巾，面巾，垫tocador dressing table, vanity梳妆台tónico toner碳粉tono color, shade, hue颜色，阴影，色调tono de piel skin tone, skin color肤色，肤色tono pardusco brownish tone, earth tone褐色色调，大地色调tonos pasteles pastel tones柔和的色调，tonos terrosos earth tones大地色调toque touch触摸toquecito little touch小触摸tornasol reflected light反射光transparente transparent透明trapo cloth (for wiping)布（用于擦）tratamiento facial facial treatment面部护理tratamiento humectante moisturizing treatment保湿护理trazas looks, appearance外型美观，外观turquesa (color) turquoise绿松石色U U Uuñas nails指甲usar maquillaje wear makeup, use makeup化妆usar esmalte de uñas wear (or use) nail polish涂指甲油utensilios de maquillaje makeup tools化妆工具V V Vvarices varicose veins静脉曲张vello facial facial hair胡子verde (color) green绿色verruca wart on foot, plantar's wart脚，足底的疣疣verruga wart疣violeta (color) violet紫色vívido vivid生动volumen volume量W W W XYZ XYZ XYZ zona T (grasienta de cara) T zone T区。

Journal of Electronic Imaging 13(3), 411–417 (July 2004).

Superquadric representation of automotive parts applying part decompositionYan ZhangAndreas KoschanMongi A.AbidiUniversity of TennesseeDepartment of Electrical and Computer Engineering334Ferris HallKnoxville,Tennessee37996-2100E-mail:yzhang@Abstract.Superquadrics are able to represent a large variety of objects with only a few parameters and a single equation.We present a superquadric representation strategy for automotive parts composed of3-D triangle meshes.Our strategy consists of two ma-jor steps of part decomposition and superquadricﬁtting.The origi-nalities of this approach include the following two features.First,our approach can represent multipart objects with superquadrics suc-cessfully by applying part decomposition.Second,superquadrics re-covered from our approach have the highest conﬁdence and accu-racy due to the3-D watertight surfaces utilized.A novel,generic3-D part decomposition algorithm based on curvature analysis is also proposed.Experimental results demonstrate that the proposed part decomposition algorithm is able to segment multipart objects into meaningful single parts efﬁciently.The proposed superquadric rep-resentation strategy can then represent each individual part of the original objects with a superquadric model successfully.©2004 SPIE and IS&T.[DOI:10.1117/1.1762516]1IntroductionObject representation denotes representing real-world ob-jects with known graphic or mathematical primitives that can be recognized by computers.This research has numer-ous applications in areas including computer vision,com-puter graphics,and reverse engineering.An object can be represented by three levels of primitives in terms of the dimensional complexity:volumetric primitives,surface el-ements,and contours.The primitive selected to describe the object depends on the complexity of the object and the tasks involved.As the highest level primitives,volumetric primitives can better represent global features of an object with a signiﬁcantly reduced amount of information com-pared with surface elements and contours.In addition, volumetric primitives have the ability to achieve the highest data compression ratio without losing the accuracy of the original data.The primarily used volumetric primitives in-clude generalized cylinders,geons,and superquadrics.1Su-perquadrics are a generalization of basic quadric surfaces and they can represent a large variety of shapes with only a few parameters and a single equation.An object initially represented by thousands of triangle meshes can be repre-sented by only a small set of superquadrics.This compact representation can be applied to object recognition to aid, for example,automated depalletizing of industrial parts or robot-guided bin picking of mixed nuclear waste in a haz-ardous environment.The quality control of both tasks can be enhanced by employing superquadrics.Furthermore,the registration of multiview data is indispensable to measure the size of partially occluded objects or their distances from each other in several image-based quality control tasks.Su-perquadrics can be used to efﬁciently register multiview range data of scenes with small overlap.2Most early research on superquadric representation con-centrated on representing single-part objects from single-view intensity or range images by assuming that the objects have been appropriately segmented.3–13This category of research focused on the data-ﬁtting process,including ob-jective function selection,ﬁtting criteria measurements,andPaper ORNL-007received Jul.30,2003;accepted for publication Feb.23,2004. 1017-9909/2004/$15.00©2004SPIE andIS&T.Fig.1Real range image of a multipart object obtained from Ref.18. Journal of Electronic Imaging13(3),411–417(July2004).Journal of Electronic Imaging/July2004/Vol.13(3)/411convergence analysis.For complex,multipart objects or scenes,there are two major types of approaches in the re-search literature.The ﬁrst type of method incorporates an image segmentation step prior to the superquadric ﬁtting.11–15The other type of method directly recovers su-perquadrics from a range image withoutpresegmentation.16–19Compared with superquadric repre-sentation of single-part object,these two types of methods can represent more complex objects and have wider appli-cations in related tasks including robotic navigation,object recognition,and virtual reality.However,existing super-quadric representation methods have several weaknesses.First,existing methods cannot handle arbitrary shapes or signiﬁcant occlusions in the scene.Figure 1shows an ex-ample of the most complicated object that can be repre-sented by superquadrics appeared in the research literature.18We observe that the range image shown in Fig.1con-tains very few occlusions due to the simplicity of the ob-ject.In this case,an optimal viewpoint can easily be found from which each part of the object is visible.When an automotive part,i.e.,a complex,multipart object such as shown in Fig.2,is of interest,no existing methods can represent this object correctly because heavy occlusions are inevitable from any single viewpoint due to the complexity of the object.The second weakness of existing methods is that they utilize only single-view images.Again,for the automotive part shown in Fig.2͑a ͒,it is too difﬁcult to ﬁnd an optimal viewpoint from which all the parts are visible due to self-occlusions and occlusions,as shown in Fig.2͑b ͒.In addi-tion,the conﬁdence of recovered superquadrics is low due to incomplete single-view data utilized and the accuracy of the recovered models highly depends on the viewpoint used to acquire the data.How complicated,multipart objects canbe represented by superquadrics with high conﬁdence and accuracy remains unknown from the literature.In this paper,we propose an efﬁcient strategy to repre-sent multipart objects with superquadrics.We also present a novel 3-D part decomposition algorithm based on curvature analysis to facilitate our superquadric representation strat-egy.Experiments are shown for automotive parts composed of 3-D triangulated surfaces.The remainder of this paper proceeds as follows.Section 2presents a superquadric representation approach for mul-tipart objects.Section 3proposes the 3-D part decomposi-tion algorithm for triangle meshes.The experimental results are presented in Sec.4and Sec.5concludes the paper.2Superquadric Representation of Multipart Objects Utilizing Part DecompositionA diagram for the proposed superquadric representation al-gorithm is illustrated in Fig.3.Beginning with a multipart object composed of triangle meshes,we propose a part de-composition algorithm to segment the meshes into single parts.Next,each single part is ﬁtted with a superquadric model.Utilizing part decomposition,the difﬁcult represen-tation problem of complicated objects is solved.We use 3-D triangulated surfaces reconstructed from multiview range images as input so that the recovered superquadrics have signiﬁcantly higher conﬁdence than those recovered from single-view images.In addition,our proposed algo-rithms are generic and ﬂexible in the sense of triangle mesh handling ability since triangle meshes have been the stan-dard surface representation elements in many computer-related areas.A triangulation step is required only if un-structured 3-D point clouds areprovided.Fig.2Distributor cap:(a)photograph of the object,(b)rendering of 3-D triangulated surfaces scanned from view 1,and (c)rendering of 3-D triangulated surfaces scanned from view2.Fig.3Diagram of the proposed superquadric representation strategy utilizing part decomposition.Zhang,Koschan,and Abidi412/Journal of Electronic Imaging /July 2004/Vol.13(3)2.1Introduction to SuperquadricsA set of superquadrics with various shape factors is shown in Fig.4.The implicit deﬁnition of superquadrics is ex-pressed as 18F ͑x ,y ,z ͒ϭͫͩx a 1ͪ2/␧2ϩͩy a 2ͪ2/␧2ͬ␧2/␧1ϩͩz a 3ͪ2/␧1ϭ1,␧1,␧2෈͑0,2͒,͑1͒where (x ,y ,z )represents a surface point of the superquad-ric,(a 1,a 2,a 3)represent sizes in the (x ,y ,z )directions,and (␧1,␧2)represent shape factors.To represent a super-quadric model with global deformations in the world coor-dinate system,15parameters are needed.They are summa-rized as 18∧ϭ͑a 1,a 2,a 3,␧1,␧2,␾,␪,␸,p x ,p y ,p z ,k x ,k y ,␣,k ͒,͑2͒where the ﬁrst 11parameters deﬁne a regular superquadric.Parameters k x and k y deﬁne the tapering deformations and ␣and k deﬁne the bending deformations.Most approaches deﬁne an objective function and ﬁnd the superquadric pa-rameters through minimizing this objective function.The objective function used in this paper is 1G ͑∧͒ϭa 1a 2a 3͚i ϭ1N͓F ␧1͑x c ,y c ,z c ͒Ϫ1͔2.͑3͒The Levenberg-Marquardt method 20was implemented tominimize the objective function due to its stability and ef-ﬁciency.In addition,our superquadric ﬁtting algorithm is able to recover superquadrics with global deformations from unstructured 3-D data points.3Curvature-Based 3-D Part DecompositionMany tasks in computer vision,computer graphics,and re-verse engineering involve objects or models.These tasks become extremely difﬁcult when the object of interest is complicated,e.g.,it contains multiple parts.Part decompo-sition can simplify the original task performed on multipart objects into several subtasks each dealing with their con-stituent single,much simpler parts.21,22While a signiﬁcant amount of research for part decomposition of 2-D intensity or 2.5-D range images has been conducted over the last 2decades,23–25little effort has been made on part segmenta-tion of 3-D data.26,27Therefore,a novel 3-D part decompo-sition algorithm is proposed in this paper.Figure 5illus-trates the difference between region segmentation and part decomposition.A scene consisting of a barrel on the ﬂoor is segmented into three surfaces by a region segmentation al-gorithm and two single-part objects by a part decomposi-tion algorithm.We can observe that the scene can be rep-resented by two superquadrics,which is consistent with the part decomposition result.Therefore,we conclude that part decomposition is more appropriate for high-level tasks such as volumetric primitives-based object representation and recognition.A diagram of the proposed part decomposition algorithm is shown in Fig.6.The proposed part decomposition consists of four major steps:Gaussian curvature estimation,boundary detection,region growing,and postprocessing.Boundaries between two articulated parts are composed of points with highly negative curvature according to the transversality regularity.21,22These boundaries are therefore detected by thresholding estimated curvatures for each vertex.A component-labeling operation is then performed to grow nonboundary vertices into parts.Finally,a postprocessing step is performed to assign nonlabeled vertices,including boundary vertices,to one of the parts and merge parts con-taining fewer vertices than a prespeciﬁed threshold into their neighbor parts.This part decomposition algorithm is summarized as follows.ˆAlgorithm 1…3-D part decomposition of triangle meshes …‰ˆInput:‰Triangulated surfaces.ˆStep 1.‰Compute Gaussian curvature for each vertex on the surface.ˆStep 2.‰Label vertices of highly negative curvatureasFig.4Superquadrics with various shapeparameters.Fig.6Diagram of the proposed 3-D part decomposition algorithm.Superquadric representation of automotive parts ...Journal of Electronic Imaging /July 2004/Vol.13(3)/413boundaries and the remaining vertices as seeds.ˆStep 3.‰Perform iterative region growing on each seed vertex.ˆStep 4.‰Assign nonlabeled vertices to parts and merge parts having less than a prespeciﬁed number of vertices into their neighboring parts.ˆOutput:‰Decomposed single parts.The major steps of this part decomposition algorithm are described respectively in the following sections.3.1Gaussian Curvature Estimation and BoundaryDetectionThe Gaussian curvature for each vertex on a triangulated surface is estimated by K ͑p ͒ϭ3͑2␲Ϫ͚i ϭ1N ␪i ͒i ϭ1NA i␦2͑p Ϫp i ͒,͑4͒using the method proposed in Ref.28.Variable p representsthe point of interest,p i represents a neighboring vertex of the point p ,and A i represents the area of the triangle con-taining the point p .Variable ␪i represents the interior angle of the triangle at p ,and ␦is the Dirac delta function.The triangles sharing the vertex p are illustrated in Fig.7.After Gaussian curvature is obtained for each vertex on the surface,a prespeciﬁed threshold is applied to label ver-tices as boundary or seed.Vertices of highly negative cur-vature are labeled as boundaries between two parts accord-ing to the transversality regularity,21while the rest are labeled as seeds.The threshold is critical and affects the performance of region growing.This threshold is deter-mined in a heuristic manner and depends on mesh resolu-tion.Two types of isolated vertices deﬁned in this work according to their labels include:͑1͒a point that is labeled as boundary while all of its neighbors are labeled as seeds and ͑2͒a point that is labeled as a seed while all of its neighbors are labeled as boundary.The isolated vertices are removed by changing their labels to be the same as those of their neighbors.3.2Region Growing and PostprocessingAfter the vertices are labeled,a region-growing step is per-formed on each vertex labeled as seed.Figure 8showstriangle meshes around the point p .To illustrate the region growing process,a set of two-ring neighbor meshes around point p is shown in this ﬁgure.Region growing is performed as follows.Starting from a seed vertex p ,the region number 1is ﬁrst assigned to the vertex.Second,all the neighbors p i initially labeled as seeds are then labeled with the same region number as the point p .The same labeling process is performed for each neighbor p i to label vertices p i j .This process terminates when the grown region is surrounded by boundary vertices,i.e.,the neighbors of the edge vertices of the region are all labeled as boundaries (Ϫ1).This process is repeated for every other vertex labeled as seed ͑0͒,but not for a vertex that has been grown and labeled with one of the region numbers (1,2,...,N ).After all the seed vertices are assigned new labels,a postprocessing is performed for each bound-ary vertex.Given a seed point x ,all its neighbors x i are ﬁrst sorted in an ascending order based on their Euclidean distance to the point x .Next,a neighboring vertex x i ,which is the ﬁrst point labeled with a region number ͑Ͼ0͒,is picked up.The boundary vertex x is then labeled the same as the vertex x i ,i.e.,the label of x is changed from Ϫ1to a region number (Ͼ0).Finally,with the exception of a few missing vertices,each vertex is labeled as 1,2,3,...,N ,the number of the parts.Missing vertices are usually located around boundaries between two articulated parts,and they are further assigned to parts during the post-processing step.Finally,a postprocessing step is performed to assign the nonlabeled vertices to parts.For example,the vertex p is an unlabeled vertex and needs further postprocessing.Assum-ing that p i (i ϭ1,2,...,N )represents a neighboring vertex of the point p ,the neighboring vertices are ﬁrst selected if they have the same sign of curvature as that of the vertex p and belong to one of the segmented parts.Next,among those neighbor vertices,the vertex that has the smallest Euclidean distance to the vertex p is selected as a target vertex.For example,the vertex p 1is assumed to be the target vertex of the vertex p .Vertex p is assigned thesameFig.7Curvature estimation for the vertex p utilizing triangle meshinformation.Fig.8Region growing process for the vertex p .Zhang,Koschan,and Abidi414/Journal of Electronic Imaging /July 2004/Vol.13(3)Superquadric representation of automotive parts...Fig.5Region and part segmentation of a synthetic scene:(a)rendering of a synthetic scene consist-ing of a barrel on theﬂoor,(b)three segmented regions rendered in different colors,and(c)twocolors.decomposed parts rendered in differentview range images from the IVP Ranger system29and consists of37,171vertices and73,394tri-angles.The part decomposition results consist of two parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts rendered in different colors,and(d)twocolors.recovered superquadrics rendered in differentmultiview range images from the IVP Ranger system29and consists of58,975vertices and117,036triangles.The part decomposition results consist of13parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts rendered in different colors,and(d)colors.recovered superquadrics rendered in differentmultiview range images from the IVP Ranger system29and consists of58,784vertices and117,564triangles.The part decomposition results consist of nine parts:(a)photograph of the original object,(b)rendering of the reconstructed mesh,(c)decomposed parts labeled in different colors,and(d)recov-ered superquadrics rendered in different colors.Journal of Electronic Imaging/July2004/Vol.13(3)/415label as vertex p1,i.e.,the same segmented part.Further-more,parts composed of fewer vertices than a speciﬁed threshold are merged with adjacent regions.4Experimental ResultsExperimental results on superquadric representation for multipart,automotive objects including a disk brake,a dis-tributor cap,and a water neck are shown in this section. The meshes were reconstructed from multiview range im-ages scanned from the IVP Ranger System.29The recovered superquadrics were rendered in three dimensions using quad meshes.30Figure9shows the disk brake and its part decomposition and superquadric representation results.The reconstructed3-D triangulated surface shown in Fig.9͑b͒consists of37,171vertices and73,394triangles.Starting from this reconstructed mesh,our part decomposition algo-rithmﬁrst decomposed the disk brake into two single parts, as shown in Fig.9͑c͒.Each decomposed part was nextﬁt-ted to a superquadric model,as shown in Fig.9͑d͒.The decomposed parts and recovered superquadrics are ren-dered in different colors.We observe that our part decom-position algorithm successfully decomposed the disk brake into its constituent parts and the superquadric representa-tion strategy recovered correct superquadrics in terms of their size,shape,and pared to the original triangle mesh representation consisting of37,171vertices and73,394triangles,the recovered superquadrics describe the disk brake with only22parameters͑11parameters for each superquadric without global deformations͒.This low representation cost of superquadric representation can sig-niﬁcantly beneﬁt tasks including virtual reality,object rec-ognition,and robotic navigation.However,the hole at the center of the disk brake failed to be represented since su-perquadrics can only represent objects with genus of zero.19 Figure10shows the distributor cap and its part decom-position and superquadric representation results.The recon-structed mesh shown in Fig.10͑b͒consists of58,975ver-tices and117,036triangles and was decomposed into13 single parts,as shown in Fig.10͑c͒.We observe that this decomposition result is consistent with human perception. The recovered superquadrics shown in Fig.10͑d͒correctly represent the distributor cap.The recovered superquadric parameters and the ground truths for one of the small cyl-inders on top of the distributor cap are shown in Table1. We can observe that the recovered superquadric parameters for this cylinder have the correct size and shape informa-tion when compared with the ground truth parameters of the object.In addition,superquadrics represent the distribu-tor cap with only143ﬂoating numbers,while the original triangle mesh consists of58,975vertices and117,036tri-angles.Figure11shows the water neck and its part decomposi-tion and superquadric representation results.The recon-structed mesh shown in Fig.11͑b͒consists of58,784ver-tices and117,564triangles and was decomposed into nine single parts,as shown in Fig.11͑c͒.We observe that the decomposed parts are consistent with human perception. The recovered superquadrics shown in Fig.11͑d͒correctly represent the water neck.The recovered superquadric pa-rameters and the ground truths for the handle,the ball,and the small cylinder next to the handle of the water neck are shown in Table2.From this table,we observe that the recovered superquadric parameters have the correct size and shape information when compared with the ground truth parameters of the objects.Again,superquadrics repre-sent the water neck in a desirable accuracy with only99 parameters while the original triangle mesh consists of 58,784vertices and117,564triangles.5ConclusionsThis paper proposed a superquadric representation ap-proach for multipart objects.Superquadrics can represent objects in an acceptable accuracy with only a few param-eters,while other surface primitives and contours usually require thousands of representation elements.Such a com-pactness and low representation cost can signiﬁcantly ben-eﬁt tasks including virtual reality,object recognition,and robot navigation,e.g.,it enables these tasks to run in a real-time manner.The advantages of the proposed super-quadric representation approach include:͑1͒it can success-fully represent complicated,multipart objects byﬁrst de-composing them into single-part objects,and͑2͒the recovered superquadrics have the highest conﬁdence and accuracy since the input we use are3-D triangulated sur-faces reconstructed from multiview range images.The in-completeness and ambiguities contained in single-view im-ages were eliminated during the multiview surface reconstruction process.We also proposed a3-D part de-composition algorithm to decompose compound objects represented by triangle meshes into their constituent single parts based on curvature analysis.Considering the fact that the triangle mesh has been a standard surface representation element in computer vision and computer graphics,the pro-posed part decomposition algorithm is generic,ﬂexible,and can facilitate computer vision tasks such as shape descrip-tion and object recognition.Furthermore,the part decom-position algorithm can segment a large number of triangle meshes͑over100,000͒in only seconds on an SGI Octane workstation.Table1Recovered superquadric parameters and ground truths for one of the small cylinders shown in Fig.10(d)where the unit is millimeters.Parameters a1a2a3␧1␧2 Ground truths15.215.620.10.1 1.0 Superquadric parameters16.4515.6720.420.120.96Table2Recovered superquadric parameters and ground truths for the water neck shown in Fig.11(d)where the unit is millimeters. Object Parameters a1a2a3␧1␧2 Handle Ground truths39.739.417.60.1 1.0 Superquadric parameters40.2340.5866.830.130.98 Ball Ground Truths50.047.656.0 1.0 1.0 Superquadric parameters51.6247.5654.28 1.020.95 Cylinder Ground truths16.517.844.20.1 1.0 Superquadric parameters17.5617.9443.380.110.95Zhang,Koschan,and Abidi 416/Journal of Electronic Imaging/July2004/Vol.13(3)AcknowledgmentsThis work was supported by the University Research Pro-gram in Robotics under Grant No.DOE-DE-FG02-86NE37968,by the Department of Defense/U.S.Army Tank-automotive and Armaments Command/National Au-tomotive Center/Automotive Research Center Program R01-1344-18,and by the Federal Aviation Administration National Safe Skies Alliance Program R01-1344-48/49. 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Documentation:MAPP2500Ranger PCI System,Version1.6,Integrated Vision Products,Sweden͑2000͒.30.J.Wernecke,The Inventor Mentor:Programming Object-oriented3DGraphics with Open Inventor,Addison-Wesley,Reading,MA͑1994͒.Yan Zhang received her BS and MS de-grees in electrical engineering from Hua-zhong University of Science and Technol-ogy,China,in1994and1997,respectively,and her PhD degree in electrical engineer-ing from the University of Tennessee,Knoxville,in2003.Her research interestsinclude3-D image processing,computervision,and patternrecognition.Andreas Koschan received his MS de-gree in computer science and his PhD incomputer engineering from the TechnicalUniversity Berlin,Germany,in1985and1991,respectively.He is currently a re-search associate professor with the De-partment of Electrical and Computer Engi-neering,the University of Tennessee,Knoxville.His work has primarily focusedon color image processing and3-D com-puter vision including stereo vision and la-ser rangeﬁnding techniques.He is a coauthor of two textbooks on 3-D image processing and a member of IS&T andIEEE.Mongi A.Abidi is a W.Fulton Professorwith the Department of Electrical and Com-puter Engineering,the University of Ten-nessee,Knoxville,which he joined in1986.Dr.Abidi received his MS and PhD degreesin electrical engineering in1985and1987,both from the University of Tennessee,Knoxville.His interests include image pro-cessing,multisensor processing,3-D imag-ing,and robotics.He has published over120papers in these areas and coedited the book Data Fusion in Robotics and Machine Intelligence(Academic Press,1992).He is the recipient of the1994to1995Chancellor’s Award for Excellence in Research and Creative Achievement and the2001Brooks Distinguished Professor Award.He is a member of the IEEE,the Computer Society,the Pattern Recognition Society, SPIE and the Tau Beta Pi,Phi Kappa Phi,and Eta Kappa Nu honor societies.He also received the First Presidential Principal Engineer Award prior to joining the University of Tennessee.Superquadric representation of automotive parts...Journal of Electronic Imaging/July2004/Vol.13(3)/417。

基于周期采样的分布式动态事件触发优化算法

第38卷第3期2024年5月山东理工大学学报(自然科学版)Journal of Shandong University of Technology(Natural Science Edition)Vol.38No.3May 2024收稿日期:20230323基金项目:江苏省自然科学基金项目(BK20200824)第一作者:夏伦超,男,20211249098@;通信作者:赵中原,男,zhaozhongyuan@文章编号:1672-6197(2024)03-0058-07基于周期采样的分布式动态事件触发优化算法夏伦超1,韦梦立2,季秋桐2,赵中原1(1.南京信息工程大学自动化学院,江苏南京210044;2.东南大学网络空间安全学院,江苏南京211189)摘要:针对无向图下多智能体系统的优化问题,提出一种基于周期采样机制的分布式零梯度和优化算法,并设计一种新的动态事件触发策略㊂该策略中加入与历史时刻智能体状态相关的动态变量,有效降低了系统通信量;所提出的算法允许采样周期任意大,并考虑了通信延时的影响,利用Lyapunov 稳定性理论推导出算法收敛的充分条件㊂数值仿真进一步验证了所提算法的有效性㊂关键词:分布式优化;多智能体系统;动态事件触发;通信时延中图分类号:TP273文献标志码:ADistributed dynamic event triggerring optimizationalgorithm based on periodic samplingXIA Lunchao 1,WEI Mengli 2,JI Qiutong 2,ZHAO Zhongyuan 1(1.College of Automation,Nanjing University of Information Science and Technology,Nanjing 210044,China;2.School of Cyber Science and Engineering,Southeast University,Nanjing 211189,China)Abstract :A distributed zero-gradient-sum optimization algorithm based on a periodic sampling mechanism is proposed to address the optimization problem of multi-agent systems under undirected graphs.A novel dynamic event-triggering strategy is designed,which incorporates dynamic variables as-sociated with the historical states of the agents to effectively reduce the system communication overhead.Moreover,the algorithm allows for arbitrary sampling periods and takes into consideration the influence oftime delay.Finally,sufficient conditions for the convergence of the algorithm are derived by utilizing Lya-punov stability theory.The effectiveness of the proposed algorithm is further demonstrated through numer-ical simulations.Keywords :distributed optimization;multi-agent systems;dynamic event-triggered;time delay ㊀㊀近些年,多智能体系统的分布式优化问题因其在多机器人系统的合作㊁智能交通系统的智能运输系统和微电网的分布式经济调度等诸多领域的应用得到了广泛的研究[1-3]㊂如今,已经提出各种分布式优化算法㊂文献[4]提出一种结合负反馈和梯度流的算法来解决平衡有向图下的无约束优化问题;文献[5]提出一种基于自适应机制的分布式优化算法来解决局部目标函数非凸的问题;文献[6]设计一种抗干扰的分布式优化算法,能够在具有未知外部扰动的情况下获得最优解㊂然而,上述工作要求智能体与其邻居不断地交流,这在现实中会造成很大的通信负担㊂文献[7]首先提出分布式事件触发控制器来解决多智能体系统一致性问题;事件触发机制的核心是设计一个基于误差的触发条件,只有满足触发条件时智能体间才进行通信㊂文献[8]提出一种基于通信网络边信息的事件触发次梯度优化㊀算法,并给出了算法的指数收敛速度㊂文献[9]提出一种基于事件触发机制的零梯度和算法,保证系统状态收敛到最优解㊂上述事件触发策略是静态事件触发策略,即其触发阈值仅与智能体的状态相关,当智能体的状态逐渐收敛时,很容易满足触发条件并将生成大量不必要的通信㊂因此,需要设计更合理的触发条件㊂文献[10]针对非线性系统的增益调度控制问题,提出一种动态事件触发机制的增益调度控制器;文献[11]提出一种基于动态事件触发条件的零梯度和算法,用于有向网络的优化㊂由于信息传输的复杂性,时间延迟在实际系统中无处不在㊂关于考虑时滞的事件触发优化问题的文献很多㊂文献[12]研究了二阶系统的凸优化问题,提出时间触发算法和事件触发算法两种分布式优化算法,使得所有智能体协同收敛到优化问题的最优解,并有效消除不必要的通信;文献[13]针对具有传输延迟的多智能体系统,提出一种具有采样数据和时滞的事件触发分布式优化算法,并得到系统指数稳定的充分条件㊂受文献[9,14]的启发,本文提出一种基于动态事件触发机制的分布式零梯度和算法,与使用静态事件触发机制的文献[15]相比,本文采用动态事件触发机制可以避免智能体状态接近最优值时频繁触发造成的资源浪费㊂此外,考虑到进行动态事件触发判断需要一定的时间,使用当前状态值是不现实的,因此,本文使用前一时刻状态值来构造动态事件触发条件,更符合逻辑㊂由于本文采用周期采样机制,这进一步降低了智能体间的通信频率,但采样周期过长会影响算法收敛㊂基于文献[14]的启发,本文设计的算法允许采样周期任意大,并且对于有时延的系统,只需要其受采样周期的限制,就可得到保证多智能体系统达到一致性和最优性的充分条件㊂最后,通过对一个通用示例进行仿真,验证所提算法的有效性㊂1㊀预备知识及问题描述1.1㊀图论令R表示实数集,R n表示向量集,R nˑn表示n ˑn实矩阵的集合㊂将包含n个智能体的多智能体系统的通信网络用图G=(V,E)建模,每个智能体都视为一个节点㊂该图由顶点集V={1,2, ,n}和边集E⊆VˑV组成㊂定义A=[a ij]ɪR nˑn为G 的加权邻接矩阵,当a ij>0时,表明节点i和节点j 间存在路径,即(i,j)ɪE;当a ij=0时,表明节点i 和节点j间不存在路径,即(i,j)∉E㊂D=diag{d1, ,d n}表示度矩阵,拉普拉斯矩阵L等于度矩阵减去邻接矩阵,即L=D-A㊂当图G是无向图时,其拉普拉斯矩阵是对称矩阵㊂1.2㊀凸函数设h i:R nңR是在凸集ΩɪR n上的局部凸函数,存在正常数φi使得下列条件成立[16]:h i(b)-h i(a)- h i(a)T(b-a)ȡ㊀㊀㊀㊀φi2 b-a 2,∀a,bɪΩ,(1)h i(b)- h i(a)()T(b-a)ȡ㊀㊀㊀㊀φi b-a 2,∀a,bɪΩ,(2) 2h i(a)ȡφi I n,∀aɪΩ,(3)式中: h i为h i的一阶梯度, 2h i为h i的二阶梯度(也称黑塞矩阵)㊂1.3㊀问题描述考虑包含n个智能体的多智能体系统,假设每个智能体i的成本函数为f i(x),本文的目标是最小化以下的优化问题:x∗=arg minxɪΩðni=1f i(x),(4)式中:x为决策变量,x∗为全局最优值㊂1.4㊀主要引理引理1㊀假设通信拓扑图G是无向且连通的,对于任意XɪR n,有以下关系成立[17]:X T LXȡαβX T L T LX,(5)式中:α是L+L T2最小的正特征值,β是L T L最大的特征值㊂引理2(中值定理)㊀假设局部成本函数是连续可微的,则对于任意实数y和y0,存在y~=y0+ω~(y -y0),使得以下不等式成立:f i(y)=f i(y0)+∂f i∂y(y~)(y-y0),(6)式中ω~是正常数且满足ω~ɪ(0,1)㊂2㊀基于动态事件触发机制的分布式优化算法及主要结果2.1㊀考虑时延的分布式动态事件触发优化算法本文研究具有时延的多智能体系统的优化问题㊂为了降低智能体间的通信频率,提出一种采样周期可任意设计的分布式动态事件触发优化算法,95第3期㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀夏伦超,等:基于周期采样的分布式动态事件触发优化算法其具体实现通信优化的流程图如图1所示㊂首先,将邻居和自身前一触发时刻状态送往控制器(本文提出的算法),得到智能体的状态x i (t )㊂然后,预设一个固定采样周期h ,使得所有智能体在同一时刻进行采样㊂同时,在每个智能体上都配置了事件检测器,只在采样时刻检查是否满足触发条件㊂接着,将前一采样时刻的智能体状态发送至构造的触发器中进行判断,当满足设定的触发条件时,得到触发时刻的智能体状态x^i (t )㊂最后,将得到的本地状态x^i (t )用于更新自身及其邻居的控制操作㊂由于在实际传输中存在时延,因此需要考虑满足0<τ<h 的时延㊂图1㊀算法实现流程图考虑由n 个智能体构成的多智能体系统,其中每个智能体都能独立进行计算和相互通信,每个智能体i 具有如下动态方程:x ㊃i (t )=-1h2f i (x i )()-1u i (t ),(7)式中u i (t )为设计的控制算法,具体为u i (t )=ðnj =1a ij x^j (t -τ)-x ^i (t -τ)()㊂(8)㊀㊀给出设计的动态事件触发条件:θi d i e 2i (lh )-γq i (lh -h )()ɤξi (lh ),(9)q i (t )=ðnj =1a ij x^i (t -τ)-x ^j (t -τ)()2,(10)㊀㊀㊀ξ㊃i (t )=1h[-μi ξi (lh )+㊀㊀㊀㊀㊀δi γq i (lh -h )-d i e 2i (lh )()],(11)式中:d i 是智能体i 的入度;γ是正常数;θi ,μi ,δi 是设计的参数㊂令x i (lh )表示采样时刻智能体的状态,偏差变量e i (lh )=x i (lh )-x^i (lh )㊂注释1㊀在进行动态事件触发条件设计时,可以根据不同的需求为每个智能体设定不同的参数θi ,μi ,δi ,以确保其能够在特定的情境下做出最准确的反应㊂本文为了方便分析,选择为每个智能体设置相同的θi ,μi ,δi ,以便更加清晰地研究其行为表现和响应能力㊂2.2㊀主要结果和分析由于智能体仅在采样时刻进行事件触发条件判断,并在达到触发条件后才通信,因此有x ^i (t -τ)=x^i (lh )㊂定理1㊀假设无向图G 是连通的,对于任意i ɪV 和t >0,当满足条件(12)时,在算法(7)和动态事件触发条件(9)的作用下,系统状态趋于优化解x ∗,即lim t ңx i (t )=x ∗㊂12-β2φm α-τβ2φm αh -γ>0,μi+δi θi <1,μi-1-δi θi >0,ìîíïïïïïïïï(12)式中φm =min{φ1,φ2}㊂证明㊀对于t ɪ[lh +τ,(l +1)h +τ),定义Lyapunov 函数V (t )=V 1(t )+V 2(t ),其中:V 1(t )=ðni =1f i (x ∗)-f i (x i )-f ᶄi (x i )(x ∗-x i )(),V 2(t )=ðni =1ξi (t )㊂令E (t )=e 1(t ), ,e n (t )[]T ,X (t )=x 1(t ), ,x n (t )[]T ,X^(t )=x ^1(t ), ,x ^n (t )[]T ㊂对V 1(t )求导得V ㊃1(t )=1h ðni =1u i (t )x ∗-x i (t )(),(13)由于ðni =1ðnj =1a ij x ^j (t -τ)-x ^i (t -τ)()㊃x ∗=0成立,有V ㊃1(t )=-1hX T (t )LX ^(lh )㊂(14)6山东理工大学学报(自然科学版)2024年㊀由于㊀㊀X (t )=X (lh +τ)-(t -lh -τ)X ㊃(t )=㊀㊀㊀㊀X (lh )+τX ㊃(lh )+t -lh -τhΓ1LX^(lh )=㊀㊀㊀㊀X (lh )-τh Γ2LX^(lh -h )+㊀㊀㊀㊀(t -lh -τ)hΓ1LX^(lh ),(15)式中:Γ1=diag (f i ᶄᶄ(x ~11))-1, ,(f i ᶄᶄ(x ~1n ))-1{},Γ2=diag (f i ᶄᶄ(x ~21))-1, ,(f i ᶄᶄ(x ~2n))-1{},x ~1iɪ(x i (lh +τ),x i (t )),x ~2i ɪ(x i (lh ),x i (lh+τ))㊂将式(15)代入式(14)得㊀V ㊃1(t )=-1h E T (lh )LX ^(lh )-1hX ^T (lh )LX ^(lh )+㊀㊀㊀τh2Γ2X ^T (lh -h )L T LX ^(lh )+㊀㊀㊀(t -lh -τ)h2Γ1X ^T (lh )L T LX ^(lh )㊂(16)根据式(3)得(f i ᶄᶄ(x ~i 1))-1ɤ1φi,i =1, ,n ㊂即Γ1ɤ1φm I n ,Γ2ɤ1φmI n ,φm =min{φ1,φ2}㊂首先对(t -lh -τ)h2Γ1X ^T (lh )L T LX ^(lh )项进行分析,对于t ɪ[lh +τ,(l +1)h +τ),基于引理1和式(3)有(t -lh -τ)h2Γ1X ^T (lh )L T LX ^(lh )ɤβhφm αX ^T (lh )LX ^(lh )ɤβ2hφm αðni =1q i(lh ),(17)式中最后一项根据X^T (t )LX ^(t )=12ðni =1q i(t )求得㊂接着分析τh2Γ2X ^(lh -h )L T LX ^(lh ),根据引理1和杨式不等式有:τh2Γ2X ^T (lh -h )L T LX ^(lh )ɤ㊀㊀㊀㊀τβ2h 2φm αX ^T (lh -h )LX ^(lh -h )+㊀㊀㊀㊀τβ2h 2φm αX ^T (lh )LX ^(lh )ɤ㊀㊀㊀㊀τβ4h 2φm αðni =1q i (lh -h )+ðni =1q i (lh )[]㊂(18)将式(17)和式(18)代入式(16)得㊀V ㊃1(t )ɤβ2φm α+τβ4φm αh -12()1h ðni =1q i(lh )+㊀㊀㊀τβ4φm αh ðni =1q i (lh -h )+1h ðni =1d i e 2i(lh )㊂(19)根据式(11)得V ㊃2(t )=-ðni =1μih ξi(lh )+㊀㊀㊀㊀ðni =1δihγq i (lh -h )-d i e 2i (lh )()㊂(20)结合式(19)和式(20)得V ㊃(t )ɤ-12-β2φm α-τβ4φm αh ()1h ðni =1q i (lh )+㊀㊀㊀㊀τβ4φm αh 2ðn i =1q i (lh -h )+γh ðni =1q i (lh -h )-㊀㊀㊀㊀1h ðni =1(μi -1-δi θi)ξi (lh ),(21)因此根据李雅普诺夫函数的正定性以及Squeeze 定理得㊀V (l +1)h +τ()-V (lh +τ)ɤ㊀㊀㊀-12-β2φm α-τβ4φm αh()ðni =1q i(lh )+㊀㊀㊀τβ4φm αh ðni =1q i (lh -h )+γðni =1q i (lh -h )-㊀㊀㊀ðni =1(μi -1-δiθi)ξi (lh )㊂(22)对式(22)迭代得V (l +1)h +τ()-V (h +τ)ɤ㊀㊀-12-β2φm α-τβ2φm αh-γ()ðl -1k =1ðni =1q i(kh )+㊀㊀τβ4φm αh ðni =1q i (0h )-㊀㊀12-β2φm α-τβ4φm αh()ðni =1q i(lh )-㊀㊀ðlk =1ðni =1μi -1-δiθi()ξi (kh ),(23)进一步可得㊀lim l ңV (l +1)h -V (h )()ɤ㊀㊀㊀τβ4φm αh ðni =1q i(0h )-16第3期㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀夏伦超,等:基于周期采样的分布式动态事件触发优化算法㊀㊀㊀ðni =1(μi -1-δi θi )ðl =1ξi (lh )-㊀㊀㊀12-β2φm α-τβ2φm αh-γ()ð l =1ðni =1q i(lh )㊂(24)由于q i (lh )ȡ0和V (t )ȡ0,由式(24)得lim l ң ðni =1ξi (lh )=0㊂(25)基于ξi 的定义和拉普拉斯矩阵的性质,可以得到每个智能体的最终状态等于相同的常数,即lim t ңx 1(t )= =lim t ңx n (t )=c ㊂(26)㊀㊀由于目标函数的二阶导数具有以下性质:ðni =1d f ᶄi (x i (t ))()d t =㊀㊀㊀㊀-ðn i =1ðnj =1a ij x ^j (t )-x ^i (t )()=㊀㊀㊀㊀-1T LX^(t )=0,(27)式中1=[1, ,1]n ,所以可以得到ðni =1f i ᶄ(x i (t ))=ðni =1f i ᶄ(x ∗i )=0㊂(28)联立式(26)和式(28)得lim t ңx 1(t )= =lim t ңx n (t )=c =x ∗㊂(29)㊀㊀定理1证明完成㊂当不考虑通信时延τ时,可由定理1得到推论1㊂推论1㊀假设通信图G 是无向且连通的,当不考虑时延τ时,对于任意i ɪV 和t >0,若条件(30)成立,智能体状态在算法(7)和触发条件(9)的作用下趋于最优解㊂14-n -1φm -γ>0,μi+δi θi <1,μi-1-δi θi >0㊂ìîíïïïïïïïï(30)㊀㊀证明㊀该推论的证明过程类似定理1,由定理1结果可得14-β2φm α-γ>0㊂(31)令λn =βα,由于λn 是多智能体系统的全局信息,因此每个智能体很难获得,但其上界可以根据以下关系来估计:λn ɤ2d max ɤ2(n -1),(32)式中d max =max{d i },i =1, ,n ㊂因此得到算法在没有时延情况下的充分条件:14-n -1φm -γ>0㊂(33)㊀㊀推论1得证㊂注释2㊀通过定理1得到的稳定性条件,可以得知当采样周期h 取较小值时,由于0<τ<h ,因此二者可以抵消,从而稳定性不受影响;而当采样周期h 取较大值时,τβ2φm αh项可以忽略不计,因此从理论分析可以得出允许采样周期任意大的结论㊂从仿真实验方面来看,当采样周期h 越大,需要的收剑时间越长,但最终结果仍趋于优化解㊂然而,在文献[18]中,采样周期过大会导致稳定性条件难以满足,即算法最终难以收敛,无法达到最优解㊂因此,本文提出的算法允许采样周期任意大,这一创新点具有重要意义㊂3㊀仿真本文对一个具有4个智能体的多智能体网络进行数值模拟,智能体间的通信拓扑如图2所示㊂采用4个智能体的仿真网络仅是为了初步验证所提算法的有效性㊂值得注意的是,当多智能体的数量增加时,算法的时间复杂度和空间复杂度会增加,但并不会影响其有效性㊂因此,该算法在更大规模的多智能体网络中同样适用㊂成本函数通常选择凸函数㊂例如,在分布式传感器网络中,成本函数为z i -x 2+εi x 2,其中x 表示要估计的未知参数,εi 表示观测噪声,z i 表示在(0,1)中均匀分布的随机数;在微电网中,成本函数为a i x 2+b i x +c i ,其中a i ,b i ,c i 是发电机成本参数㊂这两种情境下的成本函数形式不同,但本质上都是凸函数㊂本文采用论文[19]中的通用成本函数(式(34)),用于证明本文算法在凸函数上的可行性㊂此外,通信拓扑图结构并不会影响成本函数的设计,因此,本文的成本函数在分布式网络凸优化问题中具有通用性㊂g i (x )=(x -i )4+4i (x -i )2,i =1,2,3,4㊂(34)很明显,当x i 分别等于i 时,得到最小局部成本函数,但是这不是全局最优解x ∗㊂因此,需要使用所提算法来找到x ∗㊂首先设置重要参数,令φm =16,γ=0.1,θi =1,ξi (0)=5,μi =0.2,δi =0.2,26山东理工大学学报(自然科学版)2024年㊀图2㊀通信拓扑图x i (0)=i ,i =1,2,3,4㊂图3为本文算法(7)解决优化问题(4)时各智能体的状态,其中设置采样周期h =3,时延τ=0.02㊂智能体在图3中渐进地达成一致,一致值为全局最优点x ∗=2.935㊂当不考虑采样周期影响时,即在采样周期h =3,时延τ=0.02的条件下,采用文献[18]中的算法(10)时,各智能体的状态如图4所示㊂显然,在避免采样周期的影响后,本文算法具有更快的收敛速度㊂与文献[18]相比,由于只有当智能体i 及其邻居的事件触发判断完成,才能得到q i (lh )的值,因此本文采用前一时刻的状态值构造动态事件触发条件更符合逻辑㊂图3㊀h =3,τ=0.02时算法(7)的智能体状态图4㊀h =3,τ=0.02时算法(10)的智能体状态为了进一步分析采样周期的影响,在时延τ不变的情况下,选择不同的采样周期h ,其结果显示在图5中㊂对比图3可以看出,选择较大的采样周期则收敛速度减慢㊂事实上,这在算法(7)中是很正常的,因为较大的h 会削弱反馈增益并减少固定有限时间间隔中的控制更新次数,具体显示在图6和图7中㊂显然,当选择较大的采样周期时,智能体的通信频率显著下降,同时也会导致收敛速度减慢㊂因此,虽然采样周期允许任意大,但在收敛速度和通信频率之间需要做出权衡,以选择最优的采样周期㊂图5㊀h =1,τ=0.02时智能体的状态图6㊀h =3,τ=0.02时的事件触发时刻图7㊀h =1,τ=0.02时的事件触发时刻最后,固定采样周期h 的值,比较τ=0.02和τ=2时智能体的状态,结果如图8所示㊂显然,时延会使智能体找到全局最优点所需的时间更长,但由于其受采样周期的限制,最终仍可以对于任意有限延迟达成一致㊂图8㊀h =3,τ=2时智能体的状态36第3期㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀夏伦超,等:基于周期采样的分布式动态事件触发优化算法4　结束语本文研究了无向图下的多智能体系统的优化问题,提出了一种基于动态事件触发机制的零梯度和算法㊂该机制中加入了与前一时刻智能体状态相关的动态变量,避免智能体状态接近最优值时频繁触发产生的通信负担㊂同时,在算法和触发条件设计中考虑了采样周期的影响,在所设计的算法下,允许采样周期任意大㊂对于有时延的系统,在最大允许传输延迟小于采样周期的情况下,给出了保证多智能体系统达到一致性和最优性的充分条件㊂今后拟将本算法向有向图和切换拓扑图方向推广㊂参考文献:[1]杨洪军,王振友.基于分布式算法和查找表的FIR滤波器的优化设计[J].山东理工大学学报(自然科学版),2009,23(5):104-106,110.[2]CHEN W,LIU L,LIU G P.Privacy-preserving distributed economic dispatch of microgrids:A dynamic quantization-based consensus scheme with homomorphic encryption[J].IEEE Transactions on Smart Grid,2022,14(1):701-713.[3]张丽馨,刘伟.基于改进PSO算法的含分布式电源的配电网优化[J].山东理工大学学报(自然科学版),2017,31(6):53-57.[4]KIA S S,CORTES J,MARTINEZ S.Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication[J].Automatica,2015,55:254-264.[5]LI Z H,DING Z T,SUN J Y,et al.Distributed adaptive convex optimization on directed graphs 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ClosestPointonaSplineCurve：最近的点样条曲线

Robust and Eﬃcient Computation of the Closest Point on a Spline Curve Hongling Wang,Joseph Kearney,and Kendall AtkinsonAbstract.Parametric cubic spline curves are commonly used tomodel the geometry of road surfaces in real-time driving simulators.Roads are represented by space curves that deﬁne a curvilinear frameof reference in which three-dimensional points are expressed in coordi-nates of distance along the curve,oﬀset from the central axis,and loftfrom the road surface.Simulators must map from global Cartesiancoordinates to local road coordinates at very high frequencies.A keycomponent in this mapping is the computation of the closest point onthe central axis of the road to a three-dimensional point expressed inCartesian coordinates.The paper investigates a two-step method thatexploits the complementary strengths of two optimization techniques:Newton’s method and quadratic minimization.§1.IntroductionParametric cubic spline curves provide a natural basis for modeling the geometry of road surfaces in real-time driving simulators.The road model is used by programs that control the behavior of autonomous vehicles and pedestrians populating the virtual urban environment.In many simula-tors,roads are represented by space curves that deﬁne a central axis or spine of a ribbon-like surface[6].A surface normal is deﬁned at each point on the curve allowing the ribbon to twist about its spine.The rib-bon establishes a curvilinear coordinate system in which points in space are expressed in coordinates of distance along the central axis,oﬀset from the axis,and loft from the road surface.The ribbon structure provides a natural coordinate frame for computing the local geometry of navigable surfaces.This geometry is important for wayﬁnding of autonomous agents and also determines the spatial relationships among agents.Curve and Surface Design:Saint-Malo2002397 Tom Lyche,Marie-Laurence Mazure,and Larry L.Schumaker(eds.),pp.397–405. Copyright o c2003by Nashboro Press,Brentwood,TN.ISBN0-9728482-0-7.All rights of reproduction in any form reserved.398H.Wang,J.Kearney,and K.Atkinson While some simulation computations are most eﬀectively implement-ed using ribbon coordinates,other computations are most eﬀectively im-plemented using Cartesian coordinates.For example,behavior modulesthat track roads and avoid obstacles,are most easily expressed with ob-ject locations represented in ribbon coordinates.However,the dynamics code that computes object motions from control parameters set by objectbehaviors is most simply written in Cartesian coordinates.Because these computations are performed at very high frequency,it is essential to haveeﬃcient and robust code to map from ribbon coordinates to Cartesian co-ordinates and to compute the inverse mapping from Cartesian coordinates to local ribbon coordinates.The mapping from Cartesian to ribbon coordinates is frequently a se-rious computational bottleneck in driving simulators.The key component in this mapping is the computation of the closest point on the central axisof the ribbon to a three-dimensional point expressed in Cartesian coor-dinates.Conventional optimization techniques such as Newton’s methodor quadratic minimization work well most of the time.However,we’vefound that the standard techniques consistently fail(converge very slowly or diverge)at a small number of points on many ordinary curves.Becauseof the frequency with which the mappings are performed(i.e.thousandsof times a second for a modestly complex simulation)even these rare problematic instances are likely to occur with regularity.This leads tounacceptable computational delays and can halt a simulation if the opti-mization procedure is not terminated.To address weaknesses with standard optimization techniques,wepresent a two stage technique that combines quadratic minimization and Newton’s method.§2.The ProblemA parametric cubic spline curve modeling the centerline of a curved roadcan be expressed as[5],(x(s),y(s),z(s)),0≤s≤L,where s denotes arc length,L is the arc length of the entire spline curve,and x(s),y(s),and z(s)are cubic spline functions with equally spacedbreakpoints{s0,s1,...,s n}with s0=0and s n=L.The functions x,y, and z are C2on[0,L].At each time step of a simulation,the dynamics module computes a new position in Cartesian coordinates for every moving object.Given an object’s location in Cartesian coordinates,our problem is toﬁnd the closest point on a road centerline to the object.Let p0=(x0,y0,z0)be the position of an object(see Figure1).Thesquare of the distance between position p0and position(x(s),y(s),z(s))Closest Point on a Spline399Fig.1.Vector p1p0and the tangent vector of a cubic spline curve on p1.on a spline curve isD(s)=(x(s)−x0)2+(y(s)−y0)2+(z(s)−z0)2,(1) where x(s),y(s),and z(s)are cubic spline functions of the parameter s. The value s∗that minimizes D(s)determines p1=(x(s∗),y(s∗),z(s∗)),p1p0is the closest point to p0on the cubic spline curve.The vector−−→perpendicular to the tangent vector of the cubic spline curve on p1.The distance between p0and p1,which is the length of the vector−−→p1p0,is the smallest distance between the position p0and the cubic spline curve.We approach the mapping computation as an optimization problem. To meet the stringent demands of real-time simulation,it is important that the selected optimization method converges to an accurate solution very quickly.While the average speed of this computation matters,it is of paramount importance that the maximum time does not overrun the time allotted for a simulation step by the scheduler.Thus,the demands of the application call for a method that is accurate,fast,and almost never fails. With these requirements in mind,we examine three optimization tech-niques:Newton’s method,quadratic minimization,and a new technique that combines quadratic minimization and Newton’s method.§2.Quadratic Minimization MethodQuadratic minimization uses quadratic interpolation to minimize a one-variable function,in our case D(s).Suppose that˜s1,˜s2,and˜s3are given as initial estimates of s∗,the value that optimizes D(s).The quadratic400H.Wang,J.Kearney,and K.Atkinson polynomial that interpolates D(s)at˜s1,˜s2,and˜s3is given by,P(s)=(s−˜s2)(s−˜s3)(˜s1−˜s2)(˜s1−˜s3)D(˜s1)+(s−˜s1)(s−˜s3)(˜s2−˜s1)(˜s2−˜s3)D(˜s2)+(s−˜s1)(s−˜s2)(˜s3−˜s1)(˜s3−˜s2)D(˜s3).The minimum of P(s)is used to approximate the minimum of D(s).The minimum of P(s)is given bys∗,k=12·y23D(˜s1)+y31D(˜s2)+y12D(˜s3)s23D(˜s1)+s31D(˜s2)+s12D(˜s3),k=1,2,3,···,(2)where s ij=˜s i−˜s j and y ij=˜s2i−˜s2j for i,j∈{1,2,3}.We pick three values from˜s1,˜s2,˜s3,and s∗,k by eliminating the value which gives the largest P(s)among the4values,and continue in a like manner until some error tolerance for P(s)is achieved.It can be shown that with a suﬃciently good set of initial guesses,the iteration will converge at a superlinear rate to s∗[3,4].Quadratic minimization needs three initial estimates of s∗.In our application,we usually have a good guess of which segment,[s i,s i+1], contains s∗based on the simulation state at the previous time step.An object typically enters a road at one end or the other(i.e.on theﬁrst or last segment.)As the object moves along a road,we track its position and velocity.Knowing s∗at the previous step and the object’s velocity, we can predict the value of s∗at the current step.Because the spline segments are all of equal length,we can calculate the index,i,of the segment containing the initial estimate,i= s∗,0l,(3)where l is the arc length of each segment of the spline curve.We use asour three initial estimates of s∗the values s i,s i+s i+12,and s i+1.When s∗lies near a segment boundary,error in the initial estimate may cause us to choose the wrong segment.This is detected when the iteration converges to a value outside the segment boundaries[s i,s i+1]. In this case,we attempt to solve s∗on adjacent segments.Sometimes we are unable to predict s∗from previous states(for ex-ample,when an object moves from oﬀroad terrain to a road.)When we are unable to compute a good initial estimate,we attempt to solve for s∗on each successive segment of the curve.Closest Point on a Spline401Fig.2.Distance curve between p0and points on the spline segment in Figure1.The road curves we seek to model are typically smooth and have low curvature relative to their width.The width of a road surface must be less than the radius of curvature of the road axis spline to prevent self intersections.As a consequence,there is a single nearest point on the spline for all points on the surface of a road.Thus,the mapping from Cartesian coordinates to ribbon coordinates is unique.We expect quadratic minimization method to work well for our prob-lem because the minimum distance between a point p0on the surface of the road and the spine of the road is normally well-approximated by a parabola.For example,Figure2graphs the minimum distance from a point p0to a spline segment.We tested the quadratic minimization method on a variety of cubic spline curves representative of road curves used in driving simulation.For each curve,we randomly generated a cloud of points near the curve and computed,for each of these points,the closest point on the curve.Ex-perimental results showed that quadratic minimization converged to an accurate solution in fewer than8iterations for about one third of the test points.In the remaining cases,a solution was usually found although it sometimes took hundreds of iterations to converge.In a small percentage of cases the method diverged and no solution was found.Closer exam-ination of cases in which the method diverged or converged very slowly revealed that the early iterations made progress toward a solution,but as the optimal value was approached it jumped about in a small interval surrounding the optimum.§3.Newton’s MethodThe value s∗that minimizes D(s)in formula(1)satisﬁes D (s∗)=0.We can use Newton’s method toﬁnd a root of this equation.This leads to402H.Wang,J.Kearney,and K.Atkinson the iteration formulas∗,m+1=s∗,m−D (s∗,m)D (s∗,m),m=0,1,2, (4)Similar to the quadratic method,the initial estimate s∗,0is based on the value of s∗computed on the last time step of the simulation.Likewise, adjacent segments are considered when the method returns a value out of the initial segment’s range.This method is quadratically convergent[4].We implemented Newton’s method to optimize the distance expres-sion(1).We tested Newton’s method with the same curves and sample points that we used to test quadratic minimization.Experiments showed Newton’s method converges in most,but not all,cases.Generally,New-ton’s method found a solution more quickly than quadratic minimization –usually in3to4iterations.However,for some of the sample points Newton’s method required dozens or even hundreds of iterations to con-verge.The problem cases seem to be caused by poor initial estimates. When,after a slow start,the method approached the optimal value it converged very quickly to theﬁnal solution.In a very small number of cases,Newton’s method diverged jumping to values far away from the optimal value.§biningNewton’s Method and Quadr atic Minimization Neither Newton’s method nor quadratic minimization perform satisfacto-rily for real-time simulation.The average rate of convergence of quadratic minimization is too slow for our application.Both methods are plagued by the occurrence of cases in which convergence is unacceptably slow and both methods diverge in a small number of cases.The good news is that Newton’s method works well when given a suﬃciently good initial estimate;sometimes an accurate solution is found in a single iteration.This is because Newton’s method takes theﬁrst-order term and second-order term of Taylor’s expansion while truncating the higher order terms.Therefore,as we approach the optimal value with formula(4),the error caused by truncating higher order terms is quite small.On the other hand,the error can be quite large when the initial estimate is far away from the optimal value.Comparing the convergence properties of the two methods,we observe that their strengths complement one another.Quadratic minimization is good at reﬁning coarse estimates.Newton’s method is good at converging to the optimal value quickly with a good initial guess.This leads us to consider combining the two methods to leverage their complementary strengths in overcoming their weaknesses.The composite algorithm begins with quadratic minimization method toﬁnd a rough estimate that serves as an initial guess for Newton’s method.Closest Point on a Spline403Based on our experiments,weﬁnd that quadratic minimization gen-erallyﬁnds an acceptable initial value for Newton’s method after four iterations.By using four iterations,we allow the possibility of updat-ing all of the initial values.Each iteration in the quadratic minimization method produces a new estimate of the optimal value and throws away the worst of the current estimates.After3iterations we have produced 3new guesses.If some or all of the3initial values are poor estimates of the optimal value,we have an opportunity to replace them all with new, better estimates and base the4th estimate on these new values.§5.ResultsWe demonstrate the performance of quadratic minimization,Newton’s method,and our new composite method on the parametric spline curve shown in Figure3.The curve is composed of8parametric cubic segments. We randomly generated30,000points in a band around the curve and computed,for each point,the nearest point on the spline curve.Methods were initialized with values on the segment,i,from which the s∗to be estimated was selected.Figure4presents the convergence rates for the three methods.The results are summarized in Table1.The termination criteria for all three algorithms was set to|s∗,k+1−s∗,k|≤(s i+1−s i)·10−8where[s i,s i+1]is the range of the parameter value for the spline segment where theﬁnal solution lies.The most striking aspect of the test results is that the new method found a solution in less than8iterations in all30,000cases.In con-trast,both quadratic minimization and Newton’s method get mired in a signiﬁcant number of cases.The new method outperforms quadratic mini-mization in every respect;itﬁnds solutions faster on average and its worst case performance is capped at a reasonable value.For many points,the new method is an iteration or two slower than Newton’s method.How-ever,the tradeoﬀis that a solution is always found in modest number of iterations.Because of the need to bound the length of a simulation time step,it is highly desirable to minimize the maximum time of component computations.Thus,the elimination of failures and the reduction in the time to compute the hardest cases outweighs the small increase in time for the easy cases.Overall,the new method provides an attractive alternative for real-time applications.§6.ConclusionThe closest point computation is a core component of real-time ground ve-hicle simulation.It forms an essential step in the process of mapping from Cartesian coordinates to road coordinates needed to place synthetic agents on the road network.To satisfy the requirements of real-time simulation, the closest point computation must be eﬃcient and extremely robust.404H.Wang,J.Kearney,and K.AtkinsonFig.3.An cubic spline curve example composed of 8parametric cubic curve segments and some of the randomly chosen points.quadratic minimization Newton’s method algorithm the new algorithmrate of fast convergencedivergence (>8 iterations)rate of slow convergence(<=8 iterations)34.17%89.53%100%65.79% 0.04%10.24% 0.22% 0 0Tab.1.Performance of diﬀerent methods for the cubic spline curve example in Figure 3.Fig.4.A Histogram displaying,for each of the three methods,the distribution of convergence rates for 30,000test points using the cubic spline curve shown in Figure 3.Closest Point on a Spline405The method presented in this paper is well tailored to the needs of real-time simulation.By combining quadratic minimization and Newton’s method,we’ve found a technique that very reliably converges to an accu-rate solution in a small number of iterations.The method has undergone rigorous testing in our real-time ground vehicle simulator,Hank.In10 months of daily runs(some for periods of many hours)we have had no failures.This practical experience over an extended period of time gives us great conﬁdence in the robustness and usefulness of the approach.Acknowledgments.This work was supported in part through National Science Foundation grants INT-9724746,EAI-0130864,and IIS-0002535. Jim Cremer and Pete Willemsen made signiﬁcant contributions to the development of the Hank simulator.References1.Atkinson,K.,Modelling a road using spline interpolation,Reportson Computational Mathematics#145,Department of Mathematics, The University of Iowa,(2002).2.Atkinson,K.,An Introduction to Numerical Analysis,John Wiley&Sons,Hoboken,NJ,1989.3.Luenberger,D.,Linear and Nonlinear Programming,Addison-Wesley,Reading,MA,1984.4.Wang,H.,An analytical solution for free-form roads in driving sim-ulation,Technical Report01-04,Department of Computer Science, The University of Iowa,(2001).5.Wang,H.,Kearney,J.,and Atkinson,K.,Arc-length parameterizedspline curve for real-time simulation,5th international conference on Curves and Surfaces,(2002).6.Willemsen,P.,Kearney,J.,and Wang,H.,Ribbon networks for mod-eling navigable paths of autonomous agents in virtual urban environ-ments,to appear in IEEE Virtual Reality Conference,2003.7.Willemsen,P.,Behavior and Scenario Modeling For Real-Time Vir-tual Environments,dissertation,The University of Iowa,2000.Hongling Wang,Joseph Kearney,and Kendall AtkinsonDepartment of Computer ScienceThe University of IowaIowa City,IA52242howang|kearney|*****************.edu。

BSIM4和ULTRA-BULK模型对称性和连续性的检验

万方数据　万方数据　万方数据　万方数据１期李博等：ＢＳＩＭ４和ＵＬＴＲＡ—ＢＵＬＫ模型对称性和连续性的检验３１出可靠和正确的，一ｙ和Ｃ—ｙ对称和连续特性。

据此，ｕＩ。

ＴＲＡ—ＢＵＩ。

Ｋ可以准确地分析和模拟典型的ＲＦ和模拟电路，如Ｒ２Ｒ、ＭＲＣ和ＶＡＲＡＣＴＯＲ等。

参考文献［１］ＫｕｎｔａｌＪｏａｒｄａｒ，ＫｉｒａｎＫｕｍａｒＧｕｌｌａｐａｌｌｉ，ＣｏｌｉｎＣＭｃＡｎｄｒｅｗ，ｅｔａ１．ＡｎｉｍｐｒｏｖｅｄＭＯＳＦＥＴｍｏｄｅｌｆｏｒｃｉｒｃｕｉｔｓｉｍｕｌａｔｉｏｎ［Ｊ］．ＩＥＥＥＴｒａｎｓａｃｔｉｏｎｓＥｌｅｃｔｒｏｎＤｅｖｉｃｅｓ，１９９８，４５（１）：１３４—１４８．［２］ｈｔｔｐ：／／ｂｓｉｍ．ｅｅｃｓ．ｂｅｒｋｅｌｅｙ．ｅｄｕ．［３］ＶｅｅｒａｒａｇｈａｖａｎＳ．ＳＳＩＭ：ａｎｅｗｃｈａｒｇｅ－ｂａｓｅｄＭＯＳ－ＦＥＴｍｏｄｅｌ［Ｃ］．ＰｒｅｓｅｎｔｅｄＭＣＮＣＣｉｒｃｕｉｔＳｉｍｕｌａ—ｔｉｏｎＷｏｒｋｓｈｏｐ，１９９０．［４］ＥｎｚＣＣ．ＭＯＳｔｒａｎｓｉｓｔｏｒｍｏｄｅｌｉｎｇｄｅｄｉｃａｔｅｄｌｏｗ—ａｎｄｌｏｗｖｏｌｔａｇｅａｎａｌｏｇｃｉｒｃｕｉｔｄｅｓｉｇｎａｎｄｓｉｍ—ｕｌａｔｉｏｎ［Ｃ］．Ｐｒｅｓｅｎｔｅｄａｔ６ｔｈＢｒａｚｉｌｉａｎＳｃｈｏｏｌｏｆＭｉ—ｃｒｏｅｌｅｃｔｒｏｎｉｃｓ，１９９６．［５］ＤｅＧｒａｆｆＨＣ，ＫｌａａｓｅｎＦＭ．ＣｏｍｐａｃｔＴｒａｎｓｉｓｔｏｒＭｏｄｅｌｉｎｇｆｏｒＣｉｒｃｕｉｔＤｅｓｉｇｎ［Ｍ］．ＮｅｗＹｏｒｋ：Ｓｐｒｉｎｇｅｒ－Ｖｅｒｌａｇ，１９９０．［６］ＡｒｏｒａＮＤ，ＲｉｏｓＲ，ＨｕａｎｇＣ，ｅｔａ１．ＰＣＩＭ：ａｐｈｙｓｉ．ｃａｌｌｙｂａｓｅｄｃｏｎｔｉｎｕｏｕｓｓｈｏｒｔ—ｃｈａｎｎｅｌＩＧＦＥＴｍｏｄｅｌｆｏｒｃｉｒｃｕｉｔｓｉｍｕｌａｔｉｏｎ［Ｊ］．ＩＥＥＥＴｒａｎｓＥｌｅｃｔｒｏｎＤｅｖｉｃｅｓ，１９９４，４１（６）：９８８—９９７．［７］ＭｉｕｒａＭａｔｔａｕｓｃｈＭ，ＦｅｌｄｍａｎｎＵ，ＲａｈｍＡ，ｅｔａ１．ＵｎｉｆｉｅｄｃｏｍｐｌｅｔｅＭＯＳＦＥＴｍｏｄｅｌｆｏｒａｎａｌｙｓｉｓｏｆｄｉｇｉ—ｔａｌａｎｄａｎａｌｏｇｃｉｒｃｕｉｔｓ［Ｊ］．ＩＥＥＥＴ—ＣＡＤ，１９９６，１５：１—７．［８］ＶａｎＬａｎｇｅｖｅｌｄｅＲ，ＫｌａａｓｓｅｎＦＭ．Ａｎｅｘｐｌｉｃｉｔｆａｃｅ－ｐｏｔｅｎｔｉａｌｂａｓｅｄＭＯＳＦＥＴｍｏｄｅｌｆｏｒｃｉｒｃｕｉｔｓｉｍｕ—ｌａｔｉｏｎ［Ｊ］．Ｓｏｌｉｄ—ｓｔａｔｅＥｌｅｃｔｒｏｎ，２０００，４４：４０９—４１８．［９］ＣｈｅｎＴＬ，ＧｉｌｄｅｎｂｌａｔＧ．ＡｎａｌｙｔｉｃａｌａｐｐｒｏｘｉｍａｔｉｏｎｆｏｒｔｈｅＭＯＳＦＥＴｓｕｒｆａｃｅｐｏｔｅｎｔｉａｌ［Ｊ］．Ｓｏｌｉｄ—ｓｔａｔｅＥｌｅｃｔｒｏｎ，２００１，４５：３３５—３４１．［１０］ＪｏｓｅｆＷａｔｔｓ，ＣｏｌｉｎＭｃＡｎｄｒｅｗ，ＥｎｚＣｈｒｉｓｔｉａｎ，ｅｔａ１．ＡｄｖａｎｃｅｄｃｏｍｐａｃｔｍｏｄｅｌｓｆｏｒＭＯＳＦＥＴｓ［Ｃ］．Ｔｅｃｈ—ｎｉｃａｌＰｒｏｃｅｅｄｉｎｇｓｏｆｔｈｅ２００５ＷｏｒｋｓｈｏｐＣｏｍｐａｃｔＭｏｄｅｌｉｎｇ，２００５：３－１２．［１１］ＳａｈＣｈｉｈ—Ｔａｎｇ．ＡｈｉｓｔｏｒｙｏｆＭＯＳｔｒａｎｓｉｓｔｏｒｃｏｍｐａｃｔｍｏｄｅｌｉｎｇ［Ｃ］．ＴｅｃｈｎｉｃａｌＰｒｏｃｅｅｄｉｎｇｓｏｆｔｈｅ２００５ＷｏｒｋｓｈｏｐＣｏｍｐａｃｔＭｏｄｅｌｉｎｇ，２００５：３４７—３９０．［１２］ＰＳＰｍｏｄｅｌ，ｈｔｔｐ：／／ｐｓｐｍｏｄｅｌ．ｅｅ．ｐｓｕ．ｅｄｕ．［１３］ＨｅＪｉｎ，ＳｏｎｇＹａｈ，ＺｈａｎｇＸｉｎｇ，ｅｔａ１．ＰＵＮＳＩＭｄｏｅ－ｕｍｅｎｔｓ－２：ｔｈｅａｎａｌｙｔｉｃｃａｌｃｕｌａｔｉｏｎｏｆｔｈｅｓｕｒｆａｃｅｐｏ－ｔｅｎｔｉａｌｉｎｔｈｅＰＵＮＳＩＭｍｏｄｅｌ［Ｒ］．Ｂｅｉｊｉｎｇ，Ｃｈｉｎａ：ＥｌｅｃｔｒｏｎＥｎｇＣｏｍｐｕｔＳｃｉ，ＰｅｋｉｎｇＵｎｉｖ（ｉｎｔｅｒｎａｌｄｏｃ－ｕｍｅｎｔ，ｔＯｂｅｐｒｏｖｉｄｅｄｂｙｔｈｅｒｅｑｕｉｒｅｍｅｎｔ）．［１４］ＨｅＪｉｎ，ＳｏｎｇＹａｎ。

Petrel中的属性建模流程简介讲课稿

P e t r e l中的属性建模流程简介属性建模：一、相模型的建立：1、测井曲线离散化双击：Process ——Proerty modelding——Scall up well logs；弹出对话框：在Select里选择需要离散化的相曲线数据 facies(input到wells的沉积相数据)，点击all可以对需要离散的井进行选择，剔除没有曲线或者曲线数据不正确的井）。

在相模型建立时：Average选择“most of”、method选择“Simple”。

单击“Apply”或“OK”确定。

完成沉积相数据的离散化，离散化后，沉积相数据赋给井轨迹所通过的网格。

离散化后models里的properties里新增了沉积相属性“facies”，可在3D视图里进行查看。

2、沉积相模型建立;双击：Process ——Proerty modelding——Facies modeling。

弹出对话框：对话框右上角选择离散化后的沉积相数据，依次选择各小层（zone）进行属性控制；点击解锁进行编辑控制。

目前的沉积相建模算法很多；通常，纵向上细分网格后用序贯高斯的算法，纵向上未细分用经典算法（此处的“纵向细分“是指layering里把zone细分为不同个数的网格。

⑴、序贯高斯的算法；“Method for zone /facie”选项单击下拉菜单，选择序贯高斯算法：“Sequential indicator simula”，在左侧选择该小层所以相类型（可从左侧出现的百分比统计中看出）单击箭头，相类型移动到右侧。

下侧空白区域新增两个选项卡“Variogram”，“Fraction”，点击按钮，弹出对话框：点击解锁，点击后如图：点击按钮：点击“OK”确定；自动返回之前属性设置界面。

单击“红圈”按钮，点亮其功能，点亮后按钮会变为淡红底色。

在“Variogram”选项卡将Range:里三个值“1000,1000,10”设置为默认值“0.1、0.1、0.1”（注意：每个相类型都需设置，包括M）。

基于Simulink的磁悬浮控制系统仿真

第２０卷第８期系统仿真学报＠Ｖ０１．２０Ｎｏ．８２００８年４月ＪｏｕｒＩＩａｌｏｆＳｙｓｔｅｍＳｉｍｕＩａｔｉｏｎＡｐＬ，２００８基于Ｓｉｍｕｌｉｎｋ的磁悬浮控制系统仿真舒光伟１，ＲｅｉｎｈｏｌｄＭｅｉｓｉｎｇｅ，（１．上海应用技术学院机械与自动化工程学院，上海２００２３５；２．纽伦堡应用科学大学机械工程系，德国纽伦堡９０１２１）摘要：在ＭＡｌｌ，ＡＢ／Ｓｉｍｕｌｉｎｋ环境下，对电磁型（ＥＭＳ）磁浮列车，利用Ｌａｇ隗ｎ龄方程，结合动力学和电磁学基本理论，建立了单磁铁麓吞浮系统的数学模型，蛤出了采用线性二次最优控制策略的系统仿真模型，分析了影响该系统动态性能的主要因素．仿真结果表明，所提出的方法是对磁悬浮系统建模和控制进行研究的有效途径。

关键词ｌ磁悬浮系统；Ｌａ鲫ｌｇｅ方程；ｓｉｍｕｌｉＩｌｋ；系统模型中图分类号：ｕ２９２．９１＋７；１粥９１．９文献标识码：Ａ文章编号：１００４．７３１ｘ（２００８）０８－２１６８．０３Ｓｉｍｕｌａｔｉ仰０ｆＭａｇｎｅｔｉｃＳｕｓｐｅｎｓｉｏｎＣｏｎｔｒｏｌＳｙｓｔｅｍＢａｓｅｄｏｎＳｉｍｕｌｉｎｋＳＨＵＧ¨Ｑｎｇ．ｗｅｉ、．ＲｅｉｎｈｏｌｄＭｅｔｓｉｎｇｅＰ（１．Ｓｃｈ００ｌｏｆＭｅｃｈ孤ｉｃａｌ龃ｄＡｕｔｏｍ撕０ｎＥｎｇｉｎ∞ｒｉｎｇ，Ｓｈ龃ｇ量ｌａｉＩｎ鲥ｍｔｅｏｆ１ｋｈｎｏｌｏｇｙ’Ｓｈ蛆ｇⅫ２００２３５，ａｌｉｎａ；２．ＤｃｐａｍｌｌｃｎｔｏｆＭｅｃｈａＩｌｉｃａＩＥｎｇ岫∞ｒｉｎｇ，Ｎ啪ｍｂｅｒｇＵｎｉＶ吣时ｏｆＡｐｐｌ砌Ｓｃｉｅ眦ｓ，Ｎｍｍｂｅ瑁９０１２ｌ，Ｇ咖ｍｙ）Ａｂｓｔｒａｃｔ：Ｂａｓｅｄｏｎｔｌｌｅｆｕｎｄａｍｅｎｔａｌｔｌｌｅｏｒｙｏｆｄｙｎａｍｉｃｓ柚ｄｅｌｅｃｔｒｏｆＩｌａ星皿ｅｔｉｃｓ，ｚＪ如，ｒ圻ｆ妇，，ｌｌ口ｆ记ｍＤ出ｆ巧口ｊ蚋跆．打ｌ鲫，圯ｆｍａｇｎｅｎｃｓｕｓｐｅｎｓｉｏｎ跗ｓｔｅｍ西ｔ沁ＥＭｓＭ８９ｌ删ｗｎｓｐｍｐｏｓｅｄＷｉｌｈＬ％ｒｎｎｇｅｅｑ啪伽ｎｈＭ硒Ｌ如｜＆ｍ试址ｅ州ｈｏＩ蛇吐．砌Ｐ５ｙｊ把小ｓ拥“ｋ“Ｄｎ朋础ｆｕｓｉｎｇｌｉｎｅａｒｑｕａ出ａｔｉｃｏｐｔｉｍａｌｃｏｎｔｒｏｌｓ眦ｅ譬：ｙｗ甜２ｆ１，Ｆｎ锄ｄｔｈｅｍａｉｎｆａｃｔｏｒｓｗｈｉｃｈａ｜彳ｅｃｔ山ｃｄｖｎ锄ｉｃｐｅｒｆ６ｎｎ柚ｃｅｏｆｔｈｅｓＶｓｔｅｍｗｅｒｅａｎａＪｙｚｅｄｉｎｄｅｔａｉｌ．Ｔｈｅｓｉｍｕｌａｎｏｎｒｅｓｕｌｔｓｓｈｏｗｔｈａｔｔｔｌｉｓｍｅｔ量ｌｏｄｉｓａｎｅ矗’ｅｃｎｖｅｗａｙｆｏｒｍｏｄｅｌｉｎｇａｎｄｓｉｍｕｌａｔｉｎｇｔｈｅｍａｇｎｅｔｉｃｓｕｓｐ即ｓｉｏｎｓｙｓｔｅｍ．Ｋｅｙｗｏｒｄｓ：ｍａｇｎｅｔｉｃｓｕｓｐｅｎｓｉｏｎｓｙｓｔｅｍ；Ｌａｇ删１９ｅｅｑｕａｔｉｏｎ；ＳｉｍｕｌｉＩｌｌ【；ｓｙｓｔｅｍｍｏｄｅｌ引言电磁型（ＥＭＳ）磁浮列车利用电磁力使车辆悬浮于轨道之上，并通过控制悬浮电磁铁中的电流大小来保持列车与轨道之间的垂向悬浮间隙恒定，从而使列车与轨道没有机械接触，在直线同步电机的推进下高速运行【１１。

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Boise State University Department of Electrical and Computer Engineering
Boise, ID 83725 USA kriscampbell@
Abstract—A compact model and simulation methodology for chalcogenide based memrห้องสมุดไป่ตู้stor devices is proposed. From a microprocessor design view point, it is important to be able to simulate large numbers of devices within the integrated circuit architecture in order to speed up reliably the development process. Ideally, device models would accurately describe the characteristic device behavior and would be represented by single-valued equations without requiring the need for recursive or numerically intensive solutions. With this in mind, we have developed an empirical chalcogenide compact memristor model that accurately describes all regions of operations of memristor devices employing single-valued equations.
Compact Method for Modeling and Simulation of Memristor Devices
Ion conductor chalcogenide-based memristor devices
Robinson E. Pino, Senior Member, IEEE, James W. Bohl, Nathan McDonald, Bryant Wysocki, Peter Rozwood
Air Force Research Laboratory, Information Directorate, Advanced Computing Architectures Rome, NY 13441 USA robinson.pino@
Kristy A. Campbell, Senior Member, IEEE, Antonio Oblea, and Achyut Timilsina
Electrical measurements were performed with either an HP4145B or an Agilent B1500A semiconductor parameter analyzer. A Micromanipulator 6200 microprobe station equipped with a temperature controllable wafer chuck was used
II. EXPERIMENTAL DETAILS
Memristor devices were fabricated on 200 mm p-type Si wafers. Isolated tungsten bottom electrodes were patterned on the wafers and a planarized nitride layer was used for device isolation. Vias were etched through the nitride layer to provide contact to the bottom electrode and to define the device active region. Prior to deposition of the memristor materials, the wafers received an Ar+ sputter clean to remove residual material and potentially remove any oxides that might have formed over the tungsten electrode. The memristor device structure consists of the layers (from bottom electrode contact side to top electrode contact): 300 Å Ge2Se3/500 Å Ag2Se/100 Å Ge2Se3/500 Å Ag/100 Å Ge2Se3. The 100 Å Ge2Se3 layers are needed for device processing only since Ag cannot be deposited directly on Ag2Se and since tungsten (for the top electrode) does not adhere well to Ag in this material stack. The layers were deposited by thermal evaporation using a CHA Industries SE-600-RAP thermal evaporator equipped with three 200 mm wafer planetary rotation. The rate of material deposition was monitored using an Infincon IC 6000 with a single crystal sensor head. The base system pressure was 1x10-7 Torr prior to evaporation. A tungsten top electrode was deposited by sputtering tungsten (350 Å) and etching to define the device top contact and bond pad. Etching was performed with a Veeco ion-mill by etching through the tungsten and the memristor device materials and stopping on nitride. The top and bottom electrode bond pad contacts were 80 um x 80 um.
our studies, we have employed both linear and nonlinear models from the published literature to fit our memristor hardware. However, we have observed that these published models do not represent accurately the electrical characteristic behavior or our memristor device hardware. Therefore, we have developed a simple compact model that accurately represents the electrical behavior of chalcogenide based memristors..
Keywords- Memristor; Memory Device; Thin Film Device
I. INTRODUCTION
The memristor device postulated in 1971 by Leon Chua [1] as the fourth basic circuit element has received much attention in the research community since the publication of Strukov’s 2008 paper titled “The missing memristor found” [2] The memristor name is a contraction for memory resistor [1] because that is exactly its function: to remember its history [3]. The memristor is a two terminal passive device whose resistance state depends on its previous state and present electrical biasing conditions, and when combined with transistors in a hybrid chip, memristors could radically improve the performance of digital circuits without the need for further reduction of transistor dimensions [3]. Given their two terminal structural simplicity and electronic passivity, the applications for memristor technology range from non-volatile memory, instant on computers, reconfigurable electronics and neuromorphic computing [3,4]. According to Chua [4], the memristor behaves like a linear resistor with memory but also exhibits many interesting nonlinear characteristics, and several electronic models have been presented to describe the electrical behavior of memristor devices [1,2,4-6]. However, given that memristor devices are not commercially available, good physical model-to-hardware correlations have not been yet been reported in the published literature. Therefore, in this work, we present what we believe to be the first model-tohardware correlation of memristor electrical characteristics. In