a r X i v :a s t r o -p h /0501477v 1 24 J a n 2005
astro-ph/0501477
Bayesian model selection and isocurvature perturbations
Mar′?a Beltr′a n,1,2Juan Garc′?a-Bellido,1Julien Lesgourgues,3Andrew R.Liddle,2and An?z e Slosar 4
1
Departamento de F′?sica Te′o rica C-XI,Universidad Aut′o noma de Madrid,Cantoblanco,28049Madrid,Spain
2
Astronomy Centre,University of Sussex,Brighton BN19QH,United Kingdom
3
LAPTH,F-74941Annecy-le-Vieux Cedex,France and INFN Padova,Via Marzolo 8,I-35131Padova,Italy
4
Faculty of Mathematics and Physics,University of Ljubljana,Slovenia
(Dated:February 2,2008)
Present cosmological data are well explained assuming purely adiabatic perturbations,but an admixture of isocurvature perturbations is also permitted.We use a Bayesian framework to compare the performance of cosmological models including isocurvature modes with the purely adiabatic case;this framework automatically and consistently penalizes models which use more parameters to ?t the data.We compute the Bayesian evidence for ?ts to a dataset comprised of WMAP and other microwave anisotropy data,the galaxy power spectrum from 2dFGRS and SDSS,and Type Ia supernovae luminosity distances.We ?nd that Bayesian model selection favours the purely adiabatic models,but so far only at low signi?cance.
PACS numbers:98.80.Cq
IFT-UAM/CSIC-05-02,LAPTH-1085/05,DFPD/05/A05,astro-ph/0501477
I.INTRODUCTION
Following recent developments in observational cos-mology,particularly observations by the Wilkinson Mi-crowave Anisotropy Probe (WMAP)[1],there exist com-pelling reasons to talk about a Standard Cosmological Model based on the ΛCDM paradigm seeded with purely adiabatic perturbations.In addition,there have been many attempts to analyze more general models featuring additional physics,either to constrain such processes or in the hope of discovering some trace e?ects in the data.A case of particular interest is the possible addition of an admixture of isocurvature perturbations to the adiabatic ones [2,3]which has been studied in the post-WMAP era by many authors [4,5,6,7,8,9,10,11].
The bulk of investigations so far have as a starting point chosen a particular set of parameters to de?ne the cosmological model under discussion,and then at-tempted to constrain those parameters using observa-tions,a process known as parameter ?tting .Based on such analyses,many parameters are determined to a high degree of accuracy.Much less attention has been directed at the higher-level inference problem of allowing the data to decide the set of parameters to be used,known as model comparison or model selection [12,13],although such techniques have been widely deployed outside of as-trophysics.Recently,one of us applied two model selec-tion statistics,known as the Akaike and Bayesian Infor-mation Criteria,to some simple cosmological models [14],and showed that the simplest model considered was the one favoured by the data.These criteria have recently been applied to models with isocurvature perturbations by Parkinson et al.[9],who concluded that the purely adiabatic model was favoured.
Those statistics are not however full implementations of Bayesian inference,which appears to be the most ap-propriate framework for interpretting cosmological data.The correct model selection tool to use in that context is the Bayesian evidence [12,13],which is the probability
of the model in light of the data (i.e the average likeli-hood over the prior distribution).It has been deployed in cosmological contexts by several authors [15],and the ratio of evidences between two models is also known as the Bayes Factor [16].1The Bayesian evidence can be combined with prior probabilities for di?erent models if desired,but even if the prior probabilities are assumed equal,the evidence still automatically encodes a prefer-ence for simpler models,implementing Occam’s razor in a quantitative manner.
Whenever one aims to decide whether or not a partic-ular parameter p should be ?xed (for example at p =0),one should use model selection techniques.If one car-ries out only a parameter-?tting exercise and then ex-amines the likelihood level at which p =0is excluded,such a comparison fails to account for the model dimen-sionality being reduced by one at the point p =0,and hence draws conclusions inconsistent with Bayesian in-ference.This typically overestimates the signi?cance at which the parameter p is needed.An example is spectral index running,which parameter ?tting favours at a mod-est (albeit unconvincing)con?dence level [1],but which is disfavoured by model selection statistics [14].In this paper we use the Bayesian evidence to com-pare isocurvature and adiabatic models in light of cur-rent data.We will closely follow the notation of Beltr′a n et al.[10],who recently carried out a parameter-?tting analysis of isocurvature models,and we use the same datasets.We follow the notation of that paper and pro-vide only a brief summary in this article.
II.BAYESIAN EVIDENCE
A.Theoretical basis
The Bayesian evidence is the average likelihood of a model over its prior parameter space,namely
E= L(θ)pr(θ)dθ,(1)
whereθis the parameter vector de?ning the model, pr(θ)the normalized priors on those parameters(typi-cally taken to be top-hat distributions over some range), and L(θ)is the likelihood.In essence,it asks the ques-tion:‘If I consider the possible model parameters I was allowing before I knew about this data,on average how well did they?t the data?’.Generally speaking,models with fewer parameters tend to be more predictive,and provided that for some parameter choices they?t the data well,then the average likelihood can be expected to be higher.On the other hand,a simple model which cannot?t the data for any parameter choices will not gen-erate a good likelihood.The Bayesian evidence therefore sets up the desired tension between model simplicity and ability to explain the data.
Models are ranked in order of their Bayesian evidence, usually using its logarithm.The overall normalization is irrelevant.As the evidence is the(unnormalized) probability of the model,if two models are being com-pared,the odds of the one with the lower evidence is 1/(1+exp(?ln E)).What constitutes a signi?cant dif-ference is to some extent a matter of personal taste, but a useful guide is given by Je?reys[12]who rates ?ln E<1as‘not worth more than a bare mention’, 1
A signi?cant,but unavoidable,disadvantage of the use of the evidence is that it depends on the prior ranges chosen for the parameters.For instance,if one doubles the range of one parameter by allowing it to vary in a region where the likelihood is negligibly small,then the evidence will half.Indeed,one can make any model dis-favoured simply by extending its prior range inde?nitely in a direction where there is no hope of?tting the data. From a Bayesian point of view this is unsurprising;of course your belief in a model should be in?uenced by what you thought of it before the data came along,and the Bayesian analysis has the virtue of forcing you to make your assumptions explicit.
However,the prior width is not as crucial as one might na¨?vely expect.The main reason is that the likelihood is typically falling o?exponentially away from the best?t, while the parameter volume is growing only as a polyno-mial function.For example,consider a one-dimensional toy-model for which the likelihood is given by
L(x)=L0exp
?(x?μ)2
ied from zero to one,this interpolates between sampling from the prior and the posterior distributions.De?ning
E(λ)= Lλ(θ)pr(θ)dθ,(3) it can be shown that
ln E=ln E(1)
dλ
dλ= 10 ln L λdλ,(4)
where
ln L λ≡ ln L Lλpr(θ)dθ
q n
,(7)
(q is typically1.5–2and n an increasing inte-
ger).The thermodynamic integral is calculated
by the trapezoid rule after each additional point is
added.The points are added until the integral con-
verges to a user-speci?ed stopping accuracy?stop.
At each temperature the integral is calculated by
making a short burn-in at that temperature(typi-
cally400samples,since the chain must already be
roughly burned in from the previous step)and then
calculating ln L λfrom a further number(typi-
cally1000)of accepted samples.This approach has
the disadvantage that extra samples are needed for
burn-in at each temperature and that there might
be systematics associated with stepwise tempera-
ture change.However,it is less sensitive to the
quality of covariance matrix as a poorer covariance
matrix simply results in more samples being taken
to get enough accepted samples(note that we can-
not do the same for the continuous scheme without
biasing the result,unless one is willing to burn-in
at each‘continuous’temperature change step).
Additionally,we modify the proposal function so that
its width scales withλ?1(up to a certain width),which
ensures that at high temperatures the chain is sampling
randomly from the prior,rather than random-walking
with the step-size corresponding to theλ=1posterior.
These two methods have been extensively tested to
give results that are consistent and accurate to within a
unit of ln E for a single run.The?nal numbers for all
models were calculated using the continuous temperature
change method.Additionally we have performed a com-
parison with an analytic approximation to the posterior
and got results that are also consistent to better than one
unit of ln E in the adiabatic case,though slightly worse
in the isocurvature case.
In all cases we?nd that the number of samples required
to accurately estimate the evidence and avoid systemat-
ics associated with covariance matrices,proposal widths
and similar is unexpectedly large;an order of magni-
tude larger than what is required for a simple param-
eter estimation.This makes the computation a chal-
lenging task as it is limited by the speed of the likeli-
hood evaluations which require generation of the model
4
ωb(0.018,0.032)AD-HZ,AD-n s,ISO
ωdm(0.04,0.16)AD-HZ,AD-n s,ISO
θ(0.98,1.10)AD-HZ,AD-n s,ISO
τ(0,0.5)AD-HZ,AD-n s,ISO ln[1010R rad](2.6,4.2)AD-HZ,AD-n s,ISO
n s(0.8,1.2)AD-n s,ISO
n iso(0,3)ISO
δcor(?0.14,0.4)ISO
√
TABLE I:The parameters used in the models.The sound horizonθwas used in place of the Hubble parameter.For the AD-HZ model n s was?xed to1and n iso,δcor,αandβwere?xed to0.In the AD-n s model,n s also varies.Every isocurvature model holds the same priors for the whole set of parameters.
power spectra.This also suggests that the uncertainties on evidence values already found in the literature may be underestimated,though we note that the high quality of the WMAP data makes this task considerably more di?cult than it was in the pre-WMAP era.Further in-vestigation into evidence estimation methods is clearly warranted and will be a focus of a forthcoming paper.
III.EVIDENCE FOR ISOCUR V ATURE MODELS
Our principal aim is to compare the evidence of isocur-vature models with purely adiabatic ones.We will follow the notation of Beltr′a n et al.[10].In general there are four types of isocurvature modes[3]—cold dark matter isocurvature(CDI),baryon isocurvature(BI),neutrino isocurvature density(NID),and neutrino isocurvature velocity(NIV)—but the?rst two are observationally indistinguishable[5]so we ignore the baryon isocurva-ture case.These modes can exist in any combination, and with correlations both amongst themselves and with the adiabatic modes.We will only allow a single type of isocurvature mode in any model,though we will allow a general spectral index both for the isocurvature modes and for their correlation with the adiabatic ones.
The?at prior ranges for all parameters are given in Table I.We consider two adiabatic models.AD-HZ is the simplest model giving a good?t to the data,with a Harrison–Zel’dovich spectrum and?ve variable param-eters.We also computed the evidence for an extended adiabatic model AD-n s in which we let n s vary.
For each isocurvature model there are four extra pa-rameters.As in Ref.[10]we parametrize the contribu-tion to the temperature and polarization angular power spectra from the adiabatic,isocurvature and correlation amplitudes at the pivot scale(k0=0.05Mpc?1)byαandβso that:
C l=(1?α)C ad l+αC iso l+2β
Model ln(Evidence)
TABLE II:Evidences for the four di?erent models studied, normalized to the AD-HZ evidence.The absolute value for that model was ln E=?854.1.
The parameterδcor is related to the spectral tilt of the correlation mode,n cor,and its boundaries are?xed by the pivot scale and the k min=4×10?5Mpc?1and k max=0.5Mpc?1scales used for the analysis.It is de?ned as
δcor≡n cor/ln|β|?1.(9) Thus the priors on the?rst seven parameters are theoret-ically motivated,whereas the priors on the last three are automatically set by the model.Throughout the analysis the equation of state parameter of the dark energy was set to?1.
We have used the following datasets:cosmic mi-crowave anisotropy data from the WMAP satellite in-cluding temperature–polarization cross-correlation[1], VSA[20],CBI[21]and ACBAR[22],matter power spec-trum data from the two-degree?eld galaxy redshift sur-vey(2dFGRS)power spectrum[23]and from the Sloan Digital Sky Survey[24],and the supernovae apparent magnitude–redshift relation[25].
We ran32independent computations of the evidence for each model.In all of them the stopping criterion was satis?ed after about2.5×105steps,so the total number of likelihood evaluations was approximately107per model. The results,given as the logarithm of the evidence,are described in Table II.We have expressed all the calcu-lated evidence values relative to the AD-HZ model,as the absolute value is just a particular of the likelihood code.We see from the table that the evidences are cal-culated to su?cient accuracy to draw conclusions,but that the comparison is rather inconclusive.Firstly,the two adiabatic models happen to produce the same evi-dence;as a further consistency check,we also looked at an adiabatic model with the prior range on n s doubled, and found that ln E fell by0.4,to be compared with the expected drop of ln2that would appear if the likelihood were insigni?cant throughout the extended range.Sec-ondly,by coincidence all three isocurvature models have the same evidence,with?ln E being1.0relative to AD-HZ in each case.According to the Je?reys’scale this is just at the edge of being worthy of attention.
As mentioned in Section II,these results are not reparametrization invariant,since changing the basis of parameters typically leads to a di?erent choice of priors. Various parameterizations have been used in the litera-ture.For instance,a change of pivot scale leads to an
5 AD-HZ0.0±0.1
CDI?1.0±0.2
NID?2.0±0.2
NIV?2.3±0.2
α∈[?1,1]in order to avoid
dealing with boundary e?ects and to have a posterior
distribution falling down to zero on the two ends of the
prior range.We could nevertheless instead have chosen
a?at prior forα.Similarly,the cross-correlation ampli-
tude can be parametrized either by the correlation angle
β∈[?1,1],as in Refs.[4,10],or by the amplitude of the
cross-correlation power spectrum2β
α,β)we vary(α,2β
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第四章 等效电路模型参数在线辨识 通过第三章函数拟合的方法可以确定钒电池等效电路模型中的参数，但是在实际运行过程中模型参数随着工作环境温度、充放电循环次数、SOC 等因素发生变化，根据离线试验数据计算得到的参数值估算电池SOC 可能会造成较大的估计误差。因此，在实际运行时，应对钒电池等效电路模型参数进行在线辨识，做出实时修正，提高基于模型估算SOC 的精度。 4.1 基于遗忘因子的最小二乘算法 参数辨识是根据被测系统的输入输出来，通过一定的算法，获得让模型输出值尽量接近系统实际输出值的模型参数估计值。根据能否实时辨识系统的模型参数，可以将常用的参数辨识方法分为离线和在线两类，离线辨识只能在数据采集完成后进行，不能对系统模型实时地在线调整参数，对于具有非线性特性的电池系统往往不能得到满意的辨识结果；在线辨识方法一般能够根据实时采集到的数据对系统模型进行辨识，在线调整系统模型参数。常用的辨识方法有最小二乘法、极大似然估计法和Kalman 滤波法等。因最小二乘法原理简明、收敛较快、容易理解和掌握、方便编程实现等特点，在进行电池模型参数辨识时采用了效果较好的含遗忘因子的递推最小二乘法。 4.1.1 批处理最小二乘法简介 假设被辨识的系统模型： 12121212()()()1n n n n b z b z b z y z G z u z a z a z a z ------+++==++++L L （4-1） 其相应的差分方程为： 1 1 ()()()n n i i i i y k a y k i b u k i ===--+-∑∑（4-2） 若考虑被辨识系统或观测信息中含有噪声，则被辨识模型式（4-2）可改写为： 1 1 ()()()()n n i i i i z k a y k i b u k i v k ===--+-+∑∑（4-3） 式中， ()z k 为系统输出量的第k 次观测值；()y k 为系统输出量的第k 次真值，()y k i -为系统输出量的第k i -次真值；()u k 为系统的第k 个输入值，()u k i -为 系统的第k i -个输入值；()v k 为均值为0的随机噪声。
浅析电力系统模型参数辨识 （贵哥提供） 一、现状分析 随着我国电力事业的迅猛发展, 超高压输电线路和大容量机组的相继投入, 对电力系统稳定计算、以及其安全性、经济性和电能质量提出了更高的要求。现代控制理论、计算机技术、现代应用数学等新理论、新方法在电力系统的应用，正在促使电力工业这一传统产业迅速走向高科技化。 我国大区域电网的互联使网络结构更复杂，对电力系统安全稳定分析提出了更高的要求，在线、实时、精确的辨识电力系统模型参数变得更加紧迫。由于电力系统模型的基础性、重要性，国外早在上世纪三十年代就开始了这方面的分析研究，[1,2]国内外的电力工作者在模型参数辨识方面做了大量的研究工作。[3]随后IEEE相继公布了有关四大参数的数学模型。1990年全国电网会议上的调查确定了模型参数的地位，促进了模型参数辨识的进一步发展，并提出了研究发电机、励磁、调速系统、负荷等元件的动态特性和理论模型，以及元件在极端运行环境下的动态特性和参数辨识的要求。但传统的测量手段，限制了在线实时辨识方法的实现。 同步相量测量技术的出现和WAMS系统的研究与应用，使实现在线实时的电力系统模型参数辨识成为可能。同步相量是以标准时间信号GPS作为同步的基准，通过对采样数据计算而得的相量。相量测量装置是进行同步相量测量和输出以及动态记录的装置。PMU的核心特征包括基于标准时钟信号的同步相量测量、失去标准时钟信号的授时能力、PMU与主站之间能够实时通信并遵循有关通信协议。 自1988年Virginia Tech研制出首个PMU装置以来，[4]PMU技术取得了长足发展，并在国内外得到了广泛应用。截至2006年底，在我国范围内，已有300多台P MU装置投入运行，并且可预计，在不久的将来PMU装置会遍布电力系统的各个主要电厂和变电站。这为基于PMU的各种应用提供了良好的条件。 二、系统辨识的概念 系统模型是实际系统本质的简化描述。[5]模型可分为物理模型和数学模型两大类。物理模型是根据相似原理构成的一种物理模拟，通过模型试验来研究系统的
基于最小二乘模型的Bayes 参数辨识方法 王晓侃1，冯冬青2 1 郑州大学电气工程学院，郑州（450001） 2 郑州大学信息控制研究所，郑州（450001） E-mail ：wxkbbg@https://www.doczj.com/doc/fe18094848.html, 摘 要：从辨识定义出发,首先介绍了Bayes 基本原理及其两种常用的方法,接着重点介绍了基于最小二乘模型的Bayes 参数辨识,最后以实例用MATLAB 进行仿真,得出理想的辨识结果。 关键词:辨识定义；Bayes 基本原理；Bayes 参数辨识 中国图书分类号：TP273+.1 文献标识码：A 0 概述 系统辨识是建模的一种方法。不同的学科领域，对应着不同的数学模型，从某种意义上讲，不同学科的发展过程就是建立它的数学模型的过程。建立数学模型有两种方法:即解析法和系统辨识。L. A. Zadehll 于1962年曾对”辨识”给出定义[1]:系统辨识是在对输入和输出观测的基础上，在指定的一类系统中，确定一个与被识别的系统等价的系统。一般系统输出y(n)通常用系统过去输出y(n-m)和现在输入u(n)及过去输入u(n-m)的函数描述 y(n)=f(y(n-1),y(n-2)，...，y(n-m y ), u(n),u(n-1),... ,u(n-m u ))=f(x(n),n) x(n)=[y(n-1),y(n-2),...y(n-m y ), u(n),u(n-1),...,u(n-m u )]’ 这里f(,)为未知函数关系，一般情况为泛函数，可以是线性函数或非线性函数，分别对应于线性或非线性系统，通常这个函数未知，但是局部输入输出数据可以测出，系统辨识的任务就是根据这部分信息寻找确定函数或确定系统来逼近这个未知函数。但实际上我们不可能找到一个与实际系统完全等价的模型。从实用的角度来看，系统辨识就是从一组模型中选择一个模型，按照某种准则，使之能最好地拟合由系统的输入输出观测数据体现出的实际系统的动态或静态特性。接下来本文就以最小二乘法为基础的Bayes 辨识方法为例进行分析介绍并加以仿真[4]。 1 Bayes 基本原理 Bayes 辨识方法的基本思想是把所要估计的参数看做随机变量,然后设法通过观测与该参数有关联的其他变量,以此来推断这个参数。 设μ是描述某一动态系统的模型，θ是模型μ的参数，它会反映在该动态系统的输入输出观测值中。如果系统的输出变量z(k)在参数θ及其历史纪录(1) k D ?条件下的概率密度函 数是已知的，记作p(z(k)｜θ，(1) k D ?)，其中(1) k D ?表示（k-1）时刻以前的输入输出数据集 合，那么根据Bayes 的观点参数θ的估计问题可以看成是把参数θ当作具有某种先验概率密 度p （θ，(1) k D ?）的随机变量，如果输入u(k)是确定的变量，则利用Bayes 公式，把参数θ 的后验概率密度函数表示成[2] p (θ，k D ）= p (θ｜z (k ),u(k ), (1) k D ?)=p (θ｜z (k ),(1) k D ?) = (k-1) (k-1) p(z(k)/,D )p(/D ) (k-1)(k-1)p(z(k)/,D )p(/D )d θθθθθ∞∫?∞ (1) 在式(1)中,参数θ的先验概率密度函数p(θ｜(1) k D ?)及数据的条件概率密度函数p(z(k)｜θ，
ISSN 1000-0054CN 11-2223/N 清华大学学报(自然科学版)J T singh ua Un iv (Sci &Tech ),1999年第39卷第3期 1999,V o l.39,N o.311/34 37～40 负荷建模和参数辨识的遗传进化算法* 朱守真, 沈善德, 郑宇辉, 李 力, 艾 芊, 曲祖义 清华大学电机工程与应用电子技术系,北京100084; 东北电力集团公司,沈阳110006 收稿日期:1998-06-23 第一作者:女,1950年生,副教授 *基金项目:国家攀登计划B(85-35) 文 摘 提出了一种用于电力系统负荷建模和参数辨识的遗传进化算法,该方法与传统的最小二乘法相比具有全局搜索优化特点,适用于非线性、不连续或微分不连续的各种负荷模型。该方法已成功用于工业负荷实测数据辨识及动态和静态负荷建模。在静态负荷建模上,辨识结果略优于传统的最小二乘法,且通用性更好,只需做极小的修改就可以用于各种形式的静态负荷模型。在动态负荷建模上算法不仅给出了更优秀的结果,而且表现出很好的稳健性。结果表明此方法在负荷建模中的优势。 关键词 遗传进化算法;负荷建模;参数辨识分类号 T M 761 电力负荷模型是电力系统分析、规划、运行和计算的基础,尤其在计算中对电力系统动态行为的模拟结果影响很大。不同的计算需要采用不同的负荷模型,常规采用以不同比例的恒定阻抗、恒定电流、恒定功率或考虑不同动静比例负荷模型的方式使计算结果相差很大,甚至会导致完全错误的结论[1,2]。研究表明建立符合实际的负荷模型是十分必要的。负荷特性具有时变、非线形、不确定等多种特点,且实际负荷的用电设备构成差别很大,尤其是当电压或电流变化时,负荷产生突变,这也增加了建模的难度和复杂性。参数辨识是负荷建模的核心,目前常用的有最小二乘法、辅助变量法、分段线性多项式等方法,其中传统的方法不能有效地克服负荷建模中的非线性和不连续性等问题,会产生多值性等误差。近年来ANN 方法在建模方面已取得成功,但该方法更侧重于模拟模型的动态过程,且形成的结果是非参数模型。 遗传进化算法是模拟自然界进化中优胜劣汰的 优化过程,原则上能以较大的概率找到全局的最优解,具有并行、通用、鲁棒性强,全局收敛性好等优 点。研究人员已在发电规划[3],发电调度[4],无功优化[5]中用算例证明了EP 方法比传统的梯度寻优技术更优越。 本文采用遗传进化算法对静态、动态负荷进行了实测建模。 1 电力负荷的数学模型 本文主要描述以负荷特性来分类的静态和动态模型的建模方法。1.1 静态负荷模型 静态负荷模型表示某一时刻负荷所吸收的有功功率和无功功率与同一时刻负荷母线电压和频率之间的函数关系。静态负荷模型一般以幂函数和多项式模型表示。 本文以幂函数模型为例进行计算,幂函数表示的静态负荷特性如下: P =P 0U a 1f a 2, Q =Q 0 U b 1 f b 2 . (1) 定义误差函数 E w = N i =1 [W m (i )-W c (i )] 2 N (2)式中:N 为测量点数,W m (i )分别表示第i 次有功或无功功率测量值,W c (i )表示利用第i 次采样U i ,f i 的值由式(1)得到的有功或无功计算值,X p 、X q 是待辨识参数的向量: X p =[P 0,a 1,a 2], X q =[Q 0,b 1,b 2]. (3) 辨识问题表述为极小值寻优问题,即搜索一组参数使误差E w 达到最小值。1.2 动态负荷的模型 动态负荷模型表示某一时刻负荷所吸收的有功
上海交通大学 硕士学位论文 Bouc-Wen滞回模型的参数辨识及其在电梯振动建模中的应用 姓名：周传勇 申请学位级别：硕士 专业：机械设计及理论 指导教师：李鸿光 20080201
Bouc-Wen滞回模型的参数辨识 及其在电梯振动建模中的应用 摘 要 电梯导靴是连接轿箱系统与导轨的装置，它能起到导向和隔振减振的作用。同时，在电梯的运行过程中它又将导轨由于制造或安装所造成的表面不平顺度传递给轿箱系统，从而引起轿箱系统的水平振动。国内外学者在电梯水平振动的建模和分析中，往往把导靴视为线性弹簧-阻尼元件来建模而忽略了非线性因素。事实上导靴与导轨之间存在非线性的迟滞摩擦力，本文通过实验的方法，采用Bouc-Wen 滞回模型来建立导靴-导轨非线性摩擦力模型。 Bouc-Wen滞回模型因其微分形式的非线性表达式而使得其参数辨识存在较大的困难，本文利用模型中部分参数的不敏感性，通过数学变换将非线性参数辨识问题转化为线性参数辨识问题，从而使得问题大大简化，参数辨识的效果也能满足要求。 基于以上导靴-导轨间摩擦力模型，本文进而建立了轿箱-导轨耦合水平振动动力学模型，该模型将轿箱系统等效为2自由度的平面运动刚体，将导靴等效为质量-弹簧-阻尼单元，同时考虑了导靴-导轨间的非线性摩擦力，以及导靴靴衬与导轨间接触的不连续性等。 在建立了轿箱-导轨耦合水平振动动力学模型后，利用Matlab/Simulink，建立了相应的仿真模型，开展了几种典型导轨不
平顺度激励（弯曲、失调和台阶）下的仿真分析。研究结果表明，这些分析对于电梯结构优化设计和动力学建模与分析有理论指导意义。 关键词：迟滞，参数辨识，非线性，动力学建模，系统仿真
参数辨识 参数辨识的步骤 飞行器气动参数辨识是一个系统工程，包括四部分：①试验设计，使试验能为辨识提供含有足够信息量且信息分布均匀的试验数据；②气动模型结果确定，即从候选模型集中，根据一定的准则和经验，选出最优的气动模型构式；③气动参数辨识，根据辨识准则和数据求取模型中待定参数，这是气动辨识定量研究的核心阶段；④模型检验，确认所得气动模型是否确实反映了飞行器动力学系统中气动力的本质属性。这四个部分环环相扣，缺一不可，要反复进行，直到对所得气动模型满意为止。 参数辨识的方法 参数辨识方法主要有最小二乘算法、极大似然法、集员辨识法、贝叶斯法、岭估计法、超椭球法和鲁棒辨识法等多种辨识方法。虽然目前参数辨识的领域己经发展了多种算法，但是用于气动参数估计的算法主要有：极大似然法(ML)，广义Kalman滤波(EKF)法，模型估计法(EBM )、分割及多分割算法(PIA及MPIA)、最小二乘法，微分动态规划法等。 因为最小二乘法和极大似然法是两种经典的算法，目前己经发展得相当成熟。最小二乘法适于线性模型的参数辨识，可以用于飞行器系统辨识中很多的线性模型，如惯性仪表误差系数的辨识，线性时变离散系统初始状态的辨识及多项式曲线拟合等。目前最小二乘法已经广泛应用于工程实际中。而极大似然算法因其具有渐进一致性、估计的无偏性、良好的收敛特性等特点而被广泛应用于飞行器参数辨识领域。 最小二乘法大约是1975年高斯在其著名的星体运动轨道预报研究工作中提出来的。后来，最小二乘法就成了估计理论的奠基石。由于最小二乘法原理简单，编程容易，所以它颇受人们重视，应用相当广泛。 极大似然估计算法在实践中不断地被加以改进，这种改进主要表现在三个方
基于最小二乘法的系统参数辨识 研究生二队李英杰 082068 摘要：系统辨识是自动控制学科的一个重要分支，由于其特殊作用，已经广泛应用于各种领域，尤其是复杂系统或参数不容易确定的系统的建模。过去，系统辨识主要用于线性系统的建模，经过多年的研究，已经形成成熟的理论。但随着社会、科学的发展，非线性系统越来越受到人们的关注，其控制与模型之间的矛盾越来越明显，因而非线性系统的辨识问题也越来越受到重视，其辨识理论不断发展和完善本。文重点介绍了系统参数辨识中最小二乘法的基本原理，并通过热敏电阻阻值温度关系模型的辨识实例，具体说明了基于最小二乘法参数辨识在Matlab中的实现方法。结果表明基于最小二乘法具有算法简单、精度较高等优点。 1. 引言 所谓辨识就是通过测取研究对象在人为输入作用下的输出响应，或正常运行时的输入输出数据记录，加以必要的数据处理和数学计算，估计出对象的数学模型。这是因为对象的动态特性被认为必然表现在它的变化着的输入输出数据之中，辨识只不过是利用数学的方法从数据序列中提炼出对象的数学模型而已[1]。最小二乘法是系统参数辨识中最基本最常用的方法。最小二乘法因其算法简单、理论成熟和通用性强而广泛应用于系统参数辨识中。本文基于热敏电阻阻值与温度关系数据，介绍了最小二乘法的参数辨识在Matlab中的实现。 2. 系统辨识 一般而言，建立系统的数学模型有两种方法：激励分析法和系统辨识法。前者是按照系统所遵循的物化（或社会、经济等）规律分析推导出模型。后者则是从实际系统运行和实验数据处理获得模型。如图1 所示，系统辨识就是从系统的输入输出数据测算系统数学模型的理论和方法。更进一步的定义是L.A.Zadeh 曾经与1962 年给出的，即“系统辨识是在输入和输出的基础上，从系统的一类系统范围内，确立一个与所实验系统等价的系统”。另外，系统辨识还应该具有3 个基本要素，即模型类、数据和准则[5]。被辨识系统模型根据模型形式可分为参数模型和非参数模型两大类。所谓参数模型是指微分方程、差分方程、状态方程等形式的数学模型；而非参数模型是指频率响应、脉冲响应、传递函数等隐含参数的数学模型。在辨识工程中，模型的确定主要根据经验对实际对象的特性进行一定程度上的假设，如对象的模型是线性的还是非线性的、是参数模型还是非参数模型等。在模型确定之后，就可以根据对象的输入输出数据，按照一定的辨识算法确定模型的参数[4]。 图1 被研究的动态系统 3. 最小二乘法（LS）参数估计方法 对于参数模型辨识结构，系统辨识的任务是参数估计，即利用输入输出数据估计这些参数，建立系统的数学模型。在参数估计中最常用的是最小二乘法(LS)、
模态参数辨识方法综述 摘要：本文对模态分析和模态参数识别进行了综述，对当前识别方法的原理、识别精度及适用条件进行阐述和比较，提出环境激励下模态参数识别方法需解决的关键问题及模态分析在缺陷检测和结构优化中作用。 关键词：模态分析模态参数识别模态分析与缺陷检测结构工作模态 0引言 模态分析是将线性时不变系统振动微分方程组中的物理坐标变换为模态坐标，使方程组解耦，成为一组以模态坐标及模态参数描述的独立方程，坐标变换的变换矩阵为振型矩阵，其每列即为各阶振型。模态是机械结构的固有振动特性，每一个模态具有特定的固有频率、阻尼比和模态振型。这些模态参数可以由计算或试验分析取得，这样一个计算或试验分析过程称为模态分析。振动模态是弹性结构固有的、整体的特性。如果通过模态分析方法搞清楚了结构物在某一易受影响的频率范围内，各阶主要模态的特性，就可能预知结构在此频段内，在外部或内部各种振源作用下实际振动响应，而且一旦通过模态分析知道模态参数并给予验证，就可以把这些参数用于（重）设计过程，优化系统动态特性，或者研究把该结构连接到其他结构上时所产生的影响。模态分析的最终目标是识别出系统的模态参数，为结构系统的振动分析、振动故障诊断和预报、结构动力特性的优化设计提供依据。 解析模态分析可用有限元计算实现，而实验模态分析则是对结构进行可测可控的动力学激励，由激振力和响应的信号求得系统的频响函数矩阵，再在频域或转到时域采用多种识别方法求出模态参数，得到结构固有的动态特性，这些特性包括固有频率、振型和阻尼比等。有限元法是当前分析机械结构模态的主要方法，很多学者研究了单裂缝和多裂缝缺陷对不同结构动态特性的影响，但这些研究仅局限于出现缺陷结构的当前状态，考虑到缺陷在机械结构使用过程中的扩展，提出了模态分析与缺陷扩展理论相结合的方法分析缺陷的发展趋势，便于机械结构剩余寿命的评估，使已达到设计寿命的结构在失效前仍然发挥其功能，节约了经济成本。 一般模态识别方法是基于实验室条件下的频率响应函数进行的参数识别方法，它要求同时测得结构上的激励和响应信号。但是，在许多工程实际应用中，工作条件和实验室条件相差很大，对一些大型结构无法施加激励或施加激励费用很昂贵，因此要求识别结构在工作条件下的模态参数。工作模态参数识别方法与传统模态参数识别方法相比有如下特点：一、仅
Bouc-Wen模型因数字处理方便简单而得到较为广泛的应用，力可以表示为： 利用遗传算法工具箱（GA）对Bouc-Wen模型进行参数识别。 实验数据来源于对磁流变阻尼器（MR damper）进行性能测试，试验获得的数据包括力F，位移x，采用频率已知，速度和加速度可以由位移求导得出。 参数识别出现程序如下：（文件名：Copy_0_of_BoucWen） function j=myfung(x) y0=[0]; yy=y0; tspan=[]'; s=[]'; v=[]'; Ft=[]'; rr=max(size(s));%计算数据个数 i=1; while (i