Robust Disturbance Rejection with Time Domain Specifications in Control Systems Design
|
|
- Chastity Hood
- 8 years ago
- Views:
Transcription
1 Robust Disturbace Rejectio with Time Domai Speciicatios i Cotrol Systems Desig Fabrizio Leoari Mauá Istitute o Techology, Electrical Egieerig Departmet Praça Mauá, o.1 CEP São Caetao o Sul SP Brazil abrizio@maua.br José Jaime Da Cruz Automatio a Cotrol Lab., Telecom a Cotrol Departmet Uiversity o Sao Paulo Av. Pro. Luciao Gualberto, Travessa 03, o. 158 CEP São Paulo SP Brazil jaime@lac.usp.br Abstract. The robust esig o compesators aimig isturbace rejectio with time omai speciicatios is iscusse i this paper rom a perspective o loop shapig. The costraits to provie robust isturbace rejectio are erive as uctios o the isturbace reerece moel, which cotais the associate time omai speciicatios. It is show that the larger is the istace betwee the omial plat moel a the reerece moel to be ollowe the more restrictive the esig costraits are. The problem ca be pose as a moel trackig compesator a the propose proceure may reuce the coservativeess ormally associate with its esig. The plat moel is assume subject to ustructure ucertaities a the esig speciicatios are writte i the usual orm o loop shape costraits. Hece techiques like H or LQG/LTR ca be applie as esig tools. I orer to illustrate the applicatio o the propose methoology we cosier a multivariable mixture tak as a example. Keywors. Disturbace Rejectio, Moel Matchig, 2D Cotrol, Robust Cotrol, Loop Shapig, Time Speciicatios 1. Itrouctio Nowaays most multivariable liear cotrol esig techiques are carrie out i the requecy omai. May textbooks cotai a exhaustive presetatio o the theme (Gree, 1995; Helto, 1998; Skogesta, 1996; Skogesta, Nevertheless i may practical situatios part o the speciicatios is give i the time omai. The use o H 2 a H theories requires that speciicatios i the time omai be expresse i the requecy omai beore they ca be applie. For servo problems, time omai costraits are ituitive. They are also quite atural i may regulatory problems. For SISO systems these speciicatios ca ote be traslate ito the requecy omai, but this is ot the case or MIMO systems i geeral Moel matchig a 2D cotrol are two approaches that ca be use to iirectly hale time omai speciicatios to solve servo problems a also may be aapte to treat the isturbace rejectio with time omai costraits. However, eve whe applie to servo problems, the loop gai is ormally mae too high i orer to make the iput/output traser matrix close eough to the ietity matrix so that the overall system behaves approximately as the reerece moel cotaiig the give time speciicatios. The objective o the esig approach presete i this paper is to aapt the Moel matchig a 2D cotrol schemes erivig coitios to guaratee robust isturbace rejectio with a prescribe time omai behavior. Attetio is restricte to the case where the isturbaces ca be measure. I particular, we propose a esig techique a argue that it may be less coservative tha usually. Coitios to guaratee setpoit trackig over give requecy rages withi prescribe accuracy are well kow a will be omitte here. Although the coitios or o measure isturbace rejectio a measuremet error rejectio are also well kow they are iclue i this paper just or the sake o completeess sice together with the measure isturbace rejectio they orm the so calle regulatory problem. The plat moel is assume subject to ustructure ucertaities a the esig speciicatios are writte i the usual orm o loop shape costraits. Hece techiques like H or LQG/LTR (Athas, 1986; Doyle, 1981 ca be applie as esig tools. I orer to illustrate the applicatio o the propose methoology we cosier a multivariable mixture tak as a example. The paper is structure as ollows. Sectio 2 cotais a prelimiary iscussio o how the isturbace rejectio problem with time omai speciicatios ca be pose as a moel trackig problem. I sectio 3 the esig coitios require or robust isturbace rejectio with time omai speciicatios are writte i both the orm o loop sesitivity costraits a i terms o costraits o the loop gai shape. A aalysis o the cotrol magitue associate to moel ollowig is perorme i sectio 4. A umerical example is presete i sectio 5 to illustrate the propose methoology. Sectio 6 cotais the coclusios o the paper.
2 2. Prelimiary Discussio Cosier the system represete i Fig. (1. P ( a K ( are respectively the plat a compesator traser matrices. The traser matrix Σ ( is a ilter a will be calle isturbace reerece moel. ( is the o measure isturbace relecte at the plat output, ( is the measure isturbace relecte at the plat iput, ( is the measuremet error, r ( is the system iput a y ( is the system output. All sigal a tras er matrices are assume to have compatible imesios. ( Σ ( ( ( r( K( u( P( y( e( ( Figure 1. Cotrol problem. Whe there is o measure isturbace ( ( = 0, the esig problem turs ito the classical oe a the coitios to guaratee setpoit trackig, o measure isturbace rejectio a measuremet error rejectio over give requecy rages withi prescribe accuracy are well kow. O the other ha, takig oly the measure isturbace, i.e., or r ( = ( = 0, the block iagram i Fig. (1 becomes as show i Fig. (2. This igure is equivalet to the oe cosiere i reerece (Jockheere, 1999 i orer to solve a aircrat propulsio cotrol problem. ( P( ( u( y( e( K( ( Figure 2. Moel trackig structure. Notice that the cotrol problem epicte i Fig. (2 is a twoegree o reeom problem. By simple block maipulatio it becomes the iagram i Fig. (3 the 2D cotrol with a preilter (Leoari, 2002a. ( K( v( e( P( ( y( K( Figure 3. The 2D cotrol structure. ( Desig aimig measure isturbace rejectio with time omai speciicatios ca thus be viewe as a moel matchig problem a i this case it ca be reuce to makig the closeloop traser matrix rom v ( to y ( close to K 1 ( s i the requecy rage where the matchig betwee Σ ( a the traser matrix rom r ( to y ( is sought (Kwakeraak, 1996.
3 I reerece (Maciejowski, 1989 (see page 14 he writes about the choice o the preilter: "we ca cosier irst the problem o esigig K ( to obtai esire S ( a T (, a subsequetly esig Σ ( to give a suitable" traser matrix rom ( to y (. A high loop gai is require i orer to get a closeloop traser matrix close to K 1 ( s a the choice o how high it is mae may give rise to a coservative esig i it is take higher tha ecessary. As propose i (Leoari, 2002b, the moel trackig problem is uerstoo i this paper i the ollowig sese. We look or a compesator K ( such that the orm o the traser matrix rom ( jω to e ( jω (see ig. 2 be below some prescribe value i a give requecy rage this requiremet will be calle moel ollowig. Aitioally we wish that the cotributios to the output y ( o both the o measure isturbace ( a the measuremet error ( be below give values i give requecy sets. I what ollows we aopt the perspective o a set membership or traser matrices o the plat moel cosierig, i particular, the multiplicative represetatio o the moelig error. We assume that a upper bou is give or the spectral orm o the multiplicative error matrix i the orm o a scalar uctio e ( ω (Doyle, Loop Shapig M I what ollows the symbol represets the Eucliea orm o complex vectors. i[, mi [ a [ ith, the miimum a the imum sigular values o [, respectively. eote the 3.1. Nomial Plat Moel Assume that the plat yamics are give by their omial moel. The, or the systems represete i Fig. (3, the ollowig set o equatios hol: ( I K( ( S( P( K ( (, y( = S( ( S( P( s (1 e ( = S ( ( ( S( (, (2 ( I K ( ( S( K ( (. u( = S( K( ( S( s (3 S ( = I P( K ( is the loop sesitivity matrix. (s P (s a K ( where ( 1 At this poit, sice we are obviously assumig that Σ is stable, it shoul be clear that system stability is etermie solely by the closeloop cotaiig. Because this is a classical situatio, stability is ot aresse i the paper Measure Disturbace Rejectio Assume that > 0 α (typically α << 1 measure isturbace i a give set o requecies { ω ω } is a give umber which expresses the esire accuracy o the rejectio o Ω i the sese that e jω ( jw α ( ω Ω Ω = R : ω or a give ω. Assumig that ( = ( = 0 ollowig coitio rom Eq. (2: (. Typically, to accomplish moel ollowig we get the α [ S( ( ω Ω (4 Hece, as it shoul be expecte, the sesitivity must ecrease as the istace 1 betwee the plat a the isturbace reerece moel icreases. The same occurs as α ecreases. Nevertheless, epeig o the speciic problem at ha, this coitio may be ot so restrictive a the sesitivity may be ot ecessarily low. 1 Distace is obviously uerstoo here as measure by the spectral orm o the ierece o the two traser matrices.
4 Whe the rightha sie o (4 is much smaller tha 1, this coitio ca be rewritte approximately as mi K ( ( ω Ω. (5 α Similarly as above, this coitio shows that the loop gai shoul icrease with both the istace betwee P a Σ a the iverse oα No Measure Disturbace Rejectio Suppose that = { ω ω } Ω R : ω is a give requecy set where the o measure isturbace ( ( = a ( = 0. For a give α > 0 (typically α << 1 has its eergy. Assume also that 0 measure isturbace rejectio coitio as preomiatly we express the o e( ( α ( ω Ω. (6 From Eq. (2 we the get the ollowig wellkow suiciet coitio: [ S( α ( ω Ω, (7 which leas to the ollowig approximate coitio (Da Cruz, 1996: mi 1 α jω K ( ( ω Ω (8 whe α << Measuremet Error Rejectio Suppose that = { ω ω } Assume also that 0 rejectio coitio as Ω R : ω is a give requecy set where the measuremet error preomiatly has its eergy. ( = a ( = 0. For a give α > 0 (typically α << 1 we express the measuremet error y( ( α ( ω Ω. (9 where From Eq. (1 we the get the coitio o measuremet error rejectio: [ T( α ( ω Ω, (10 T ( = S( P( K (. (11 Alteratively, rom Eq. (1 it ollows the ollowig approximate orm (Da Cruz, 1996: K ( α ( ω Ω. (12
5 whe α << Moel Ucertaities First o all, recall that or the multiplicative moel error aopte the stability robustess coitio is give by (Doyle, 1981 [ T ( e ( ω ( ω < M Assumig that ( ω < 1 e M or Ω Ω. (13 ω, the coitios (4 a (7 above moiy respectively to (Gree, 1995: [ 1 em ( ω N ( α [ S( ( ω Ω (14 a [ S( α [ 1 ( ω ( ω Ω. (15 e M Assumig that α << 1, coitio (10 ca be rewritte i the ollowig approximate orm (Da Cruz, 1996: α [ T ( ( ω Ω. (16 1 e ( ω M As expecte the eect o moel ucertaity is to make the costraits o S a T more restrictive. From Eq. (1 we ca see that S( P( ( I K( P (, Σ ( a K ( exhibit low gais. Hece, the overall traser matrix is approximately equal to P ( value o ω is ot arbitrary but relate to both P ( a Σ ( has o arbitrary yamics at high requecies where, i geeral,. I view o this the. This meas that the moel ollowig coitio (14 oes ot ecessarily implies moel matchig. This is the reaso why we call the proceure moel trackig. Simulatios carrie out by ow iicate it is reasoable to expect ice matchig up to oe ecae above the reerece moel bawith. I geeral this is eough to esure goo moel ollowig. It shoul be emphasize that (14(16 together with (13 are the key coitios to esig robust compesators i orer to achieve moel trackig. To close this sectio recall that the coitios or robust esig ca also be expresse i terms o the loop gai i the usual way (Doyle, Cotrol Magitue Aalysis or Moel Followig I this sectio we restrict the aalysis to the omial plat. From Eq. (3 it is straightorwar to see that u( ( = K ( 1 [ I P( K ( [ P( ( (17 Assume that P, Σ a it ollows that [ P ( K ( 1 α are such that α 1 mi >> >> or ω Ω. I this case Eq. (18 leas to. Thus i coitio (5 hols, u( ( P( 1 [ P( ( (18 i we assume that both P a K are square a From Eq. (19 it the ollows immeiately that 1 P exists.
6 sup 0 u( ( ( 1 [ P ( [ P( (19 This equatio shows that the worstcase relative icremet o the cotrol magitue is approximately the same as the relative istace betwee both the plat a the isturbace reerece moels. Hece isturbace reerece moels that are istat rom the plat moel require a large cotrol magitue to be ollowe. This is i accorace with coitio (14 which shows that the larger is the is tace betwee the plat a the isturbace reerece moels the more restrictive the coitio o moel ollowig is. 5. Numerical Example I orer to illustrate the applicatio o the propose methoology we cosier the stirre tak o reerece (Kwakeraak, Its omial liearize state moel is give by 0, x& ( t = ( 0 0,02 x t 0,25 0,01 0 y = x( t u( t 0,75 (20 Sice the mai cotributio o this work ocus o the isturbace rejectio with time omai speciicatios, other esig speciicatios are omitte i this examp le. Besies, as moel ucertaities turs the esig costraits just more restrictive, they are omitte i this istace a may be cosiere iclue i the esig speciicatios. Cosier as cotrol system speciicatios that isturbace at both cotrol chaels shall have its iluece o the output rejecte as i iltere by a seco orer system 0.25 s Σ ( = ( i = 1,2 2 s 0.7s 0.25 ii, (21 withi a tolerace o 10% (i.e., α = 0. 1 i the requecy rage that extes up to oe ecae above the reerece moel bawith. From the time omai speciicatios we thus have Σ11( = 0 0 Σ ( 22 s s, (22 The H mixesesitivity ramework will be use to obtai K ( (Gree, 1995; Helto, 1998; Skogesta, 1996; Skogesta, To simpliy we assume that moel ucertaities have alreay bee take ito accout i the eiitio o the weightig matrix W1 ( 10 s =. (23 Oce the plat P ( is i the rightha sie o (4 its iput variables may be scale i such a way as to get it close to the ietity i low requecies. I this case coitio (4 ca be rewritte as α [ S( ( ω Ω, (24 where S ( u S u is a square osigular matrix with compatible imesios 2. 2 Obviously, S u is a part o the compesator.
7 For simplicity S u is cosiere here as a costat matrix a equal to the plat iverse at low requecies, amely, S u = lim P 1 (. (24 s 0 Notice that scalig the plat iput variables oes ot chage the multiplicative moelig error. Hece both perormace a stability robustess coitios are ot aecte. Figure (4 shows the sigular value Boe plots o matrices relevat or the moel ollowig esig. W ( is the weightig matrix or the mixesesitivity proceure. 1 s 40 b 20 0 S ( i u 20 [ W ( j i 1 ω 40 [ ( i Σ ra/sec Figure 4. Boe plots or Moel Followig. Notice that i [ W1 ( is situate 20B above. With the sake o illustratio, the time respose o the closeloop system a the cotrol variables have bee evaluate or the moel ollowig esig Σ y( ( s ( sec Figure 5. Closeloop time respose.
8 Two uit steps isturbaces applie at istats 10 sec. a 30 sec. with positive a egative sigs, respectively, have bee cosiere. Simulatio results have bee plotte i Fig. (5. As it ca be see, the process outputs ollow closely the correspoig oes o the isturbace reerece moel. The cotrol time history is plotte i Fig. ( u 1 ( t 5 u 2 ( t 0 sec Figure 6. Cotrol time history. 6. Coclusios This work iscusse the robust cotrol esig or the rejectio o measure isturbace with time omai speciicatios. The moel trackig problem has bee pose as a atural way o ealig with time omai speciicatios i a requecy esig cotext. It has bee show that the moel ollowig coitio epes irectly o the istace betwee the reerece moel a the plat omial moel the larger the istace betwee both moels the more restrictive the coitio is. It has bee show that the relative icrease i the cotrol magitue to attai moel ollowig is approximately the same as the relative istace betwee the plat a the isturbace reerece moels. Hece, as expecte, the cotrol magitue icreases with the istace betwee the omial a isturbace reerece moels. Sice the isturbace reerece moel Σ ( is a explicit part o the compesator, small ajustmets may geerally be oe ater the esig has bee complete. This possibility ca be useul i practical applicatios where ie tuig is require urig cotrol systems startup. The mixe sesitivity ormulatio o the H cotrol theory has bee use i a umerical example to illustrate the applicatio o the methoology. Nevertheless, sice the esig coitios are ultimately expresse as costraits o Boe Diagrams o the system, ay loopshapig techique coul be equally use. 7. Ackowlegmets F. Leoari is grateul to Mauá Istitute o Techology or its partial iacial support. 8. Reereces Athas, M., 1986, "A Tutorial O The LQG/LTR Metho". Proceeigs o America Cotrol Coerece, Seattle, WA. Da Cruz, J. J., 1996 "Multivariable Robust Cotrol", Eusp, São Paulo, SP, (i Portuguese. Doyle, J. C.; Stei, G., 1981 "Mutivariable Feeback Desig: Cocepts or a Classical/Moer Sythesis". IEEE Trasactios o Automatic Cotrol, Vol. 26, pp Leoari, F.; Da Cruz, J. J., 2002 "Robust Moel Trackig a 2D Cotrol Desig". Proceeigs o the 10 th Meiterraea Coerece o Cotrol a Automatio, Lisbo. Leoari, F., 2002 "Robust Multivariable Cotrol System Desig with Time Domai Speciicatios". PhD. Thesis Epusp. São Paulo, SP (i Portugues e.
9 Gree, M.; Limebeer, D. J. N., 1995 "Liear Robust Cotrol". Pretice Hall, New Jersey. Helto, J.W.; Merio, O., 1998 "Classical Cotrol Usig H Methos", SIAM, Philaelphia. Jockheere, E. A.; Yu, G. R., 1999 "Propulsio Cotrol o Cripple Aircrat by H Moel Matchig". IEEE Trasactios o Cotrol Systems Techology, Vol. 7(2, pp Kwakeraak, H., 1996 "How Robust are H Optimal Cotrol Systems?", Proceeigs o the XI Brazilia Coerece o Automatics, São Paulo, pp Kwakeraak, H.; Siva, R., 1970 "Liear Optimal Cotrol Systems", Wiley, New York. Maciejowski, J. M., 1989 "Multivariable Feeback Desig". Aiso Wesley, Wokigham. Skogesta,, S.; Postlethwaite, I., 1996 Multivariable Feeback Cotrol: Aalysis a Desig, Joh Wiley & Sos. Zhou, K.; Doyle, J. C., 1998 "Essetials o Robust Cotrol". Pretice Hall, New Jersey.
Systems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised
More informationChapter XIV: Fundamentals of Probability and Statistics *
Objectives Chapter XIV: Fudametals o Probability ad Statistics * Preset udametal cocepts o probability ad statistics Review measures o cetral tedecy ad dispersio Aalyze methods ad applicatios o descriptive
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationStudy on the application of the software phase-locked loop in tracking and filtering of pulse signal
Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College
More informationBENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationThe evaluation mode of hotel housekeeping management
Arica Joural o Busiess Maagemet Vol. 5(34), pp. 1349-1353, 8 December, 11 Available olie at http://www.acaemicjourals.org/ajbm DOI: 1.5897/AJBM11.189 ISSN 1993-833 11 Acaemic Jourals Full Legth Research
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationU.C. Berkeley CS270: Algorithms Lecture 9 Professor Vazirani and Professor Rao Last revised. Lecture 9
U.C. Berkeley CS270: Algorithms Lecture 9 Professor Vazirai a Professor Rao Scribe: Aupam Last revise Lecture 9 1 Sparse cuts a Cheeger s iequality Cosier the problem of partitioig a give graph G(V, E
More informationEntropy of bi-capacities
Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics
More informationOverview on S-Box Design Principles
Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea
More informationINVESTMENT PERFORMANCE COUNCIL (IPC)
INVESTMENT PEFOMANCE COUNCIL (IPC) INVITATION TO COMMENT: Global Ivestmet Performace Stadards (GIPS ) Guidace Statemet o Calculatio Methodology The Associatio for Ivestmet Maagemet ad esearch (AIM) seeks
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationA Combined Continuous/Binary Genetic Algorithm for Microstrip Antenna Design
A Combied Cotiuous/Biary Geetic Algorithm for Microstrip Atea Desig Rady L. Haupt The Pesylvaia State Uiversity Applied Research Laboratory P. O. Box 30 State College, PA 16804-0030 haupt@ieee.org Abstract:
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationCOMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S 2 CONTROL CHART FOR THE CHANGES IN A PROCESS
COMPARISON OF THE EFFICIENCY OF S-CONTROL CHART AND EWMA-S CONTROL CHART FOR THE CHANGES IN A PROCESS Supraee Lisawadi Departmet of Mathematics ad Statistics, Faculty of Sciece ad Techoology, Thammasat
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationINVESTMENT PERFORMANCE COUNCIL (IPC) Guidance Statement on Calculation Methodology
Adoptio Date: 4 March 2004 Effective Date: 1 Jue 2004 Retroactive Applicatio: No Public Commet Period: Aug Nov 2002 INVESTMENT PERFORMANCE COUNCIL (IPC) Preface Guidace Statemet o Calculatio Methodology
More informationBasic Measurement Issues. Sampling Theory and Analog-to-Digital Conversion
Theory ad Aalog-to-Digital Coversio Itroductio/Defiitios Aalog-to-digital coversio Rate Frequecy Aalysis Basic Measuremet Issues Reliability the extet to which a measuremet procedure yields the same results
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationOn the Capacity of Hybrid Wireless Networks
O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA 0002 + IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights,
More informationNUMERICAL METHOD FOR UNCERTAINTY ASSESSMENT OF THE TOPOGRAPHIC PARAMETERS DERIVED FROM DEMS
NUMERICAL METHOD FOR UNCERTAINTY ASSESSMENT OF THE TOPOGRAPHIC PARAMETERS DERIVED FROM DEMS B. Bajat a, S.Rath b, D. Blagojevic a a Uiversity of Belgrae, Faculty of Civil Egieerig, Departmet of Geoesy,
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationAnalyzing Longitudinal Data from Complex Surveys Using SUDAAN
Aalyzig Logitudial Data from Complex Surveys Usig SUDAAN Darryl Creel Statistics ad Epidemiology, RTI Iteratioal, 312 Trotter Farm Drive, Rockville, MD, 20850 Abstract SUDAAN: Software for the Statistical
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationChair for Network Architectures and Services Institute of Informatics TU München Prof. Carle. Network Security. Chapter 2 Basics
Chair for Network Architectures ad Services Istitute of Iformatics TU Müche Prof. Carle Network Security Chapter 2 Basics 2.4 Radom Number Geeratio for Cryptographic Protocols Motivatio It is crucial to
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More informationMeasuring Magneto Energy Output and Inductance Revision 1
Measurig Mageto Eergy Output ad Iductace evisio Itroductio A mageto is fudametally a iductor that is mechaically charged with a iitial curret value. That iitial curret is produced by movemet of the rotor
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationNow here is the important step
LINEST i Excel The Excel spreadsheet fuctio "liest" is a complete liear least squares curve fittig routie that produces ucertaity estimates for the fit values. There are two ways to access the "liest"
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationUniversal coding for classes of sources
Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric
More information3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationAutomatic Tuning for FOREX Trading System Using Fuzzy Time Series
utomatic Tuig for FOREX Tradig System Usig Fuzzy Time Series Kraimo Maeesilp ad Pitihate Soorasa bstract Efficiecy of the automatic currecy tradig system is time depedet due to usig fixed parameters which
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationA Multifractal Wavelet Model of Network Traffic
A Multifractal Wavelet Model of Network Traffic Proect Report for ENSC80 Jiyu Re School of Egieerig Sciece, Simo Fraser Uiversity Email: re@cs.sfu.ca Abstract I this paper, a ew Multifractal Wavelet Model
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationStatistical inference: example 1. Inferential Statistics
Statistical iferece: example 1 Iferetial Statistics POPULATION SAMPLE A clothig store chai regularly buys from a supplier large quatities of a certai piece of clothig. Each item ca be classified either
More informationEstimating Probability Distributions by Observing Betting Practices
5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationTime Value of Money, NPV and IRR equation solving with the TI-86
Time Value of Moey NPV ad IRR Equatio Solvig with the TI-86 (may work with TI-85) (similar process works with TI-83, TI-83 Plus ad may work with TI-82) Time Value of Moey, NPV ad IRR equatio solvig with
More informationAn Agent-based Online Shopping System in E-commerce
Vol. 2, No. 4 Computer ad Iormatio Sciece A Aget-based Olie Shoppig System i E-commerce Zimig Zeg Ceter or Studies o Iormatio Resources, Wuha Uiversity, Wuha 430072, Chia E-mail: zmzeg977@63.com The paper
More informationGREY LEVEL CO-OCCURRENCE MATRICES: GENERALISATION AND SOME NEW FEATURES
Iteratioal Joural of Computer Sciece, Egieerig a Iformatio Techology (IJCSEIT), Vol.2, No.2, April 2012 GREY LEVEL CO-OCCURRENCE MATRICES: GENERALISATION AND SOME NEW FEATURES Bio Sebastia V 1, A. Uikrisha
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More informationAdaptive H Tracking Control Design via Neural Networks of a Constrained Robot System
Proceedigs of the 44th IEEE Coferece o Decisio ad Cotrol, ad the Europea Cotrol Coferece 25 Seville, Spai, December 12-15, 25 WeIB196 Adaptive H Trackig Cotrol Desig via Neural Networks of a Costraied
More informationExploratory Data Analysis
1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationPontifícia Universidade Católica do Rio Grande do Sul FACULDADE DE ADMINISTRAÇÃO, CONTABILIDADE E ECONOMIA PÓS-GRADUAÇÃO
Potifícia Uiversiae Católica o Rio Grae o Sul FACULDADE DE ADMINISTRAÇÃO, CONTABILIDADE E ECONOMIA PÓS-GRADUAÇÃO TEXTO PARA DISCUSSÃO Nº02/2008 LOCATION, COMMUNICATION, AND CONTROL WITHIN A VERTICALLY
More informationNon-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationSubject CT5 Contingencies Core Technical Syllabus
Subject CT5 Cotigecies Core Techical Syllabus for the 2015 exams 1 Jue 2014 Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which ca be used to model ad value
More informationA BitTorrent Proxy for Green Internet File Sharing: Design and Experimental Evaluation
A BitTorret roxy for Gree Iteret File Sharig: Desig a Experimetal Evaluatio GIUSEE ANASTASI, IARIA GIANNETTI, ANDREA ASSAREA 2 Dept. of Iformatio Egieerig, Uiversity of isa, via Diotisalvi 2, 5622 isa,
More informationHow To Solve The Homewor Problem Beautifully
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationDAME - Microsoft Excel add-in for solving multicriteria decision problems with scenarios Radomir Perzina 1, Jaroslav Ramik 2
Itroductio DAME - Microsoft Excel add-i for solvig multicriteria decisio problems with scearios Radomir Perzia, Jaroslav Ramik 2 Abstract. The mai goal of every ecoomic aget is to make a good decisio,
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationHow to read A Mutual Fund shareholder report
Ivestor BulletI How to read A Mutual Fud shareholder report The SEC s Office of Ivestor Educatio ad Advocacy is issuig this Ivestor Bulleti to educate idividual ivestors about mutual fud shareholder reports.
More informationTradigms of Astundithi and Toyota
Tradig the radomess - Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore
More informationConvention Paper 6764
Audio Egieerig Society Covetio Paper 6764 Preseted at the 10th Covetio 006 May 0 3 Paris, Frace This covetio paper has bee reproduced from the author's advace mauscript, without editig, correctios, or
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationIntelligent Sensor Placement for Hot Server Detection in Data Centers - Supplementary File
Itelliget Sesor Placemet for Hot Server Detectio i Data Ceters - Supplemetary File Xiaodog Wag, Xiaorui Wag, Guoliag Xig, Jizhu Che, Cheg-Xia Li ad Yixi Che The Ohio State Uiversity, USA Michiga State
More informationLECTURE 13: Cross-validation
LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M
More informationLearning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value
More informationModule 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur
Module The Sciece of Surface ad Groud Water Versio CE IIT, Kharagpur Lesso 8 Flow Dyamics i Ope Chaels ad Rivers Versio CE IIT, Kharagpur Istructioal Objectives O completio of this lesso, the studet shall
More informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationWhich Codes Have Cycle-Free Tanner Graphs?
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 6, SEPTEMBER 1999 173 Which Coes Have Cycle-Free Taer Graphs? Tuvi Etzio, Seior Member, IEEE, Ari Trachteberg, Stuet Member, IEEE, a Alexaer Vary,
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More information