Note 8 Root-Locus Techniques
|
|
|
- Silvester McKenzie
- 5 years ago
- Views:
Transcription
1 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 Nte 8 Rt-Lcu Techique Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd
2 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862. Vectr Repreetti f Cplex Nuer A cplex uer, deted y z, c e repreeted i the cplex pl y uig vectr, hw i the fllwig figure. The legth f the vectr i the gitude f the cplex uer d the gle eured cuterclckwie fr the rel xi i the gle f the cplex uer. I z θ z Re Suppe tht th p d re cplex uer. The utrcti f -p i l cplex uer, which c e repreeted y vectr drw fr the lcti f p t the lcti f, hw i the fllwig figure. If i vried, the vectr -p chge i th gitude d gle. p I -p Re Exple If -34j, repreet the vectr f,, d 2 i the cplex ple, d the evlute the gitude d gle f F. 2 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 2
3 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME Rt-Lcu Defiiti: The rt lcu i the plt f the rt f yte chrcteritic equti r the ple f cled-lp trfer fucti yte preter i vried. Cider the fllwig cled lp yte: R C _ 2 The cled lp trfer fucti i tied : G T G Fr G 2, the T 2 2 The chrcteritic equti i tied : The ple c e fud t e: 2 ± 4 4 ± 2 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 3
4 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 It c e ee fr the ve equti tht the ple chge the vlue f i vried. The chge i therefre ffect the tility d the perfrce f the yte. Fr the ve ecd rder yte, e h ω ζ 2 / 2 I ecd rder yte, the effect f the turl frequecy d dpig rti c e redily deteried. Fr higher rder yte, thi lyi i t viu. Thi i where the rt lcu deig ethd i ued. I the ve exple, the chge f ple with icree i c e tied d pltted : I Ple Ple j j j.4 -- j j j j j j j j j j j j j j j Rt-Lcu Re Clculti f the rt, give chge i gi, c e redily ccplihed y uig deig pckge uch MATLAB. Hwever, the rt lcu y l e ketched y hd y fllwig the prcedure tht i decried i the fllwig ecti. Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 4
5 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME Prpertie f the Rt-Lcu Cider the fllwig cled-lp yte R C G _ H The cled-lp trfer fucti i G T G H The chrcteritic equti: GH0 Or GH- Thi i tur require: Mgitude prperty: the gitude f the pe-lp trfer fucti i, i.e., G H r G H 2 Agle prperty: the gle f the pe-lp trfer fucti i dd ultiple f 80, i.e., G H 80 fr ±, ± 3, ± 5,... r G H 80 fr ±, ± 3, ± 5,... Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 5
6 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 Exple Suppe the pe-lp trfer fucti G H 2 Are j d 3 j the rt lcu? 2 If, wht i the crrepdig vlue f? Fiite d Ifiite the Ope-Lp Ple d Zer Cider the fllwig pe-lp trfer fucti: 2 G H 20 I thi exple, there re 2 fiite ple t - d -20 d e fiite zer t -2. Beide, fucti culd hve ifiite ple d zer. If the fucti pprche ifiity ge ifiity, the the fucti h ple t ifiity. If the fucti pprche zer pprche ifiity, it h zer t ifiity. Fr the ve exple, pprche ifiity: Thu, the fucti h ifiite zer. 2 li G H li 0 20 Plee t tht every fucti f h equl uer f ple d zer, icludig th ifiite d fiite ple d zer. I the ve exple, the fucti h tw fiite ple, e fiite zer d e ifiite zer, thu the uer f ple the uer f zer 2 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 6
7 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME Sketchig the Rt Lcu Befre digitl cputer, ketchig f the rt lcu w perfred y tkig dvtge f the gle d gitude criteri. We will review thi ethd t gi further iight it rt lcu. Whe wrkig i idutry, yu re trgly dvied t ue deig tl uch Mtl. The fllwig rule fcilitte the ketchig f the rt lcu:. Nuer f Brche: The uer f rche f rt lcu equl t the uer f cled lp ple. There will e e rch fr ech cled lp ple. 2. Syetry: The rt lcu i yetricl with repect t the rel xi. 3. Rel-xi Seget: O the rel-xi the rt lcu exit ly i the eget where the uer f fiite pe-lp ple d zer tht re lcted the righthd ide f the eget i dd. Further t the gle criteri, rt the rel-xi ut tify the fllwig requireet: Σ gle fr fiite zer Σ gle fr fiite ple l80 fr, 3,. where Σ u. If we re ciderig pit the rel xi, t thi pit the fllwig c e erved The gulr ctriuti f pir f pe-lp cplex ple r zer i zer. 2 The gulr ctriuti f the pe-lp ple d zer the rel-xi, which re lcted the left-hd ide f the pit, i l zer. S the ctriuti t the gle ly ce fr the pe-lp ple d zer the rel-xi, which re lcted the right-hd ide f the pit cidered. The gulr ctriuti i 80 fr ech f uch pe-lp ple d zer. If the uer f thee pe-lp ple d zer i dd, the pit i therefre the rt lcu. Otherwie, it i t the rt lcu. 4. Strtig d edig pit: The rt lcu rigite t the ple f GH d terite t the zer f GH, icludig the zer t ifiity. The explti fr thi rule i fllw. The chrcteritic equti fr the yte y e expreed i the fr: Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 7
8 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 8 0 j j i i p z H G Thi equti y e rerrged d rewritte : 0 i i j j z p Thu fr 0 the rt f the chrcteritic equti re iply the ple f GH. A pprche ifiity ut rei fiite, the rche f the lcu pprch the zer f the pe-lp fucti. If the pe-lp fucti h zer t ifiity, tht i, if > i the uul ce with the del f phyicl yte the lcu will l pprch thee zer. Thi pect f the rt lcu will e prved i the dicui f e f the dditil rule hrtly. 5. Ayptte: The rt lcu pprche tright lie yptte the lcu pprche ifiity. Let the geerl fr f pe-lp fucti e rewritte i the fr: H G Fr t yte cidered, < d the ve equti y e writte : H G where -, i the uer f ifiite zer. H G li li li The rt lcu fr lrge vlue f the tifie the reltihip: 0 ] li[ ] [ li H G Thi equti h rt tht re give y
9 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 0 r r80 fr r ±, ± 3,... The gitude f thee rt pprche ifiity ecue f the upti tht i pprchig ifiity. The gle θ ctriuted y ech rt i therefre: r80 θ Thee gle re the gle f yptte f the rt lcu, ice geerlly the rt lci pprch thee gle d thu pprche ifiity. If we dete the vlue f t which the yptte iterect the rel xi σ, thi vlue i give y u f fiite ple - u f fiite zer σ uer f fiite ple - uer f fiite zer 6. Brekwy d Brek-i Pit: Thi rule ccer pit t which tw r re rche f the lcu ce tgether r rek wy. Pit f thi type re clled rekwy d rek-i pit. Ituitively, whe tw cled lp ple ve twrd ech ther lg the rel xi, they eet t the xiu vlue f the gi tht wuld led t rel ple. The rekwy pit therefre ccur t pit f xiu gi the rel xi. Siilrly, fr the rek i pit, cplex pir f ple retur t the rel-xi d, the gi icree t ifiity, the ple ve twrd the pe lp zer. Therefre, the gi t the rek-i pit i the iiu gi lg the rel-xi etwee tw zer. We wuld therefre e le t fid thee iiu d xiu pit y uig differetil clculu fllw. Fr pe lp trfer fucti GH, it c e hw tht Philip & Hrr, 2000 the rekwy/rek-i pit rt lcu will pper g the rt f the plyil: d[ G H ] 0 d r equivletly, N D N D 0 where N d D re the uertr d deitr plyil f GH. Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 9
10 Lecture Nte f Ctrl Syte I - ME 43/Alyi d Sythei f Lier Ctrl Syte - ME862 The fllwig exple will e give i cl Exple Sketch the rt lcu fr the yte with pe-lp trfer fucti give y 3 5 G H. 2 Exple 2 Fid the rekwy d rek-i pit fr the yte give i Exple. Exple 3 Sketch the rt lcu fr the yte with pe-lp trfer fucti give y G H. 2 3 Exple 4 Sketch the rt lcu fr the yte with the fllwig pe-lp trfer fucti give y G H. 3 0 Ad the deterie where the rt lcu cre the igiry xi. Deprtet f Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Skt, S S7N 5A9, Cd 0
Note 7 Root-Locus Techniques
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Note 7 Root-Locu Techique Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd Lecture Note
Chapter 5. Root Locus Techniques
Chapter 5 Rt Lcu Techique Itrducti Sytem perfrmace ad tability dt determied dby cled-lp l ple Typical cled-lp feedback ctrl ytem G Ope-lp TF KG H Zer -, - Ple 0, -, - K Lcati f ple eaily fud Variati f
BC Calculus Review Sheet. converges. Use the integral: L 1
BC Clculus Review Sheet Whe yu see the wrds.. Fid the re f the uuded regi represeted y the itegrl (smetimes f ( ) clled hriztl imprper itegrl).. Fid the re f differet uuded regi uder f() frm (,], where
Chapter #5 EEE Control Systems
Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter / Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce,
Introduction to Modern Control Theory
Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4
Chapter #3 EEE Subsea Control and Communication Systems
EEE 87 Chter #3 EEE 87 Sube Cotrol d Commuictio Sytem Cloed loo ytem Stedy tte error PID cotrol Other cotroller Chter 3 /3 EEE 87 Itroductio The geerl form for CL ytem: C R ', where ' c ' H or Oe Loo (OL)
Chapter 8. Root Locus Techniques
Chapter 8 Rt Lcu Technique Intrductin Sytem perfrmance and tability dt determined dby cled-lp l ple Typical cled-lp feedback cntrl ytem G Open-lp TF KG H Zer -, - Ple 0, -, -4 K 4 Lcatin f ple eaily fund
Chap8 - Freq 1. Frequency Response
Chp8 - Freq Frequecy Repoe Chp8 - Freq Aged Prelimirie Firt order ytem Frequecy repoe Low-p filter Secod order ytem Clicl olutio Frequecy repoe Higher order ytem Chp8 - Freq 3 Frequecy repoe Stedy-tte
, we would have a series, designated as + j 1
Clculus sectio 9. Ifiite Series otes by Ti Pilchowski A sequece { } cosists of ordered set of ubers. If we were to begi ddig the ubers of sequece together s we would hve series desigted s. Ech iteredite
ELEC 372 LECTURE NOTES, WEEK 6 Dr. Amir G. Aghdam Concordia University
ELEC 37 LECTURE NOTES, WEE 6 Dr mir G ghdm Cocordi Uiverity Prt of thee ote re dpted from the mteril i the followig referece: Moder Cotrol Sytem by Richrd C Dorf d Robert H Bihop, Pretice Hll Feedbck Cotrol
Chapter #2 EEE Subsea Control and Communication Systems
EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio
Spherical refracting surface. Here, the outgoing rays are on the opposite side of the surface from the Incoming rays.
Sphericl refrctig urfce Here, the outgoig ry re o the oppoite ide of the urfce from the Icomig ry. The oject i t P. Icomig ry PB d PV form imge t P. All prxil ry from P which trike the phericl urfce will
The Reimann Integral is a formal limit definition of a definite integral
MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl
Motions of Infinite Mass-Spring Systems
Motio of Ifiite M-Sprig Syte KHIEM V. GO Deprtet of Aeroutic d Atroutic Stford Uiverity Stford, CA 9435-435 UITED STATES OF AMERICA Atrct: - I the tudy of phyicl, echicl, d electricl yte oe ofte ecouter
Chapter 3.1: Polynomial Functions
Ntes 3.1: Ply Fucs Chapter 3.1: Plymial Fuctis I Algebra I ad Algebra II, yu ecutered sme very famus plymial fuctis. I this secti, yu will meet may ther members f the plymial family, what sets them apart
Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
MA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
Measurement and Instrumentation Lecture Note: Strain Measurement
0-60 Meurement nd Intrumenttin Lecture Nte: Strin Meurement eview f Stre nd Strin Figure : Structure under tenin Frm Fig., xil tre σ, xil trin, trnvere trin t, Pin' rti ν, nd Yung mdulu E re σ F A, dl
ME2142/TM3142 Introduction to Feedback Control Systems First half: Professor POO Aun Neow Second half: Professor V Subramaniam
ME4/TM34 Itroductio to Feedbck Cotrol Syte Firt hlf: Profeor POO Au Neow Secod hlf: Profeor V Subri Textbook: Textbook: Moder Moder Cotrol Cotrol Syte Syte by by Richrd Richrd Doff Doff d d Robert Robert
EVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
Limit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
All the Laplace Transform you will encounter has the following form: Rational function X(s)
EE G Note: Chpter Itructor: Cheug Pge - - Iverio of Rtiol Fuctio All the Lplce Trform you will ecouter h the followig form: m m m m e τ 0...... Rtiol fuctio Dely Why? Rtiol fuctio come out turlly from
G x, x E x E x E x E x. a a a a. is some matrix element. For a general single photon state. ), applying the operators.
Topic i Qutu Optic d Qutu Ifortio TA: Yuli Mxieko Uiverity of Illioi t Urb-hpig lt updted Februry 6 Proble Set # Quetio With G x, x E x E x E x E x G pqr p q r where G pqr i oe trix eleet For geerl igle
BC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
2.Decision Theory of Dependence
.Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give
Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
Area, Volume, Rotations, Newton s Method
Are, Volume, Rottio, Newto Method Are etwee curve d the i A ( ) d Are etwee curve d the y i A ( y) yd yc Are etwee curve A ( ) g( ) d where ( ) i the "top" d g( ) i the "ottom" yd Are etwee curve A ( y)
Chapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
THE CONCEPT OF THE ROOT LOCUS. H(s) THE CONCEPT OF THE ROOT LOCUS
So far i the tudie of cotrol yte the role of the characteritic equatio polyoial i deteriig the behavior of the yte ha bee highlighted. The root of that polyoial are the pole of the cotrol yte, ad their
GUC (Dr. Hany Hammad) 9/19/2016
UC (Dr. Hny Hmmd) 9/9/6 ecture # ignl flw grph: Defitin. Rule f Reductin. Mn Rule. ignl-flw grph repreenttin f : ltge urce. ive gle-prt device. ignl Flw rph A ignl-flw grph i grphicl men f prtryg the reltinhip
ROUTH-HURWITZ CRITERION
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7 ROUTH-HURWITZ CRITERION The Routh-Hurwitz riterio i lytil proedure for determiig
MATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
Chapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING. SHEET 2 Bode Plot
CAIRO UNIVERITY FACULTY OF ENGINEERING ELECTRONIC & COMMUNICATION DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING HEET Bode Plot [] Drw the symptotic Bode plots for ech of the followig trsfer fuctios: ) ( 0.5
( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
ALIGNMENT CHART FOR CALCULATION OF REFRACTIVE INDEX FROM THE DEVIATION OF LIGHT BY A PRISM HonacB WrNcuntl, Yale Uniaersity, New Haaen, Conm.
ALGNMENT CHART FOR CALCULATON OF REFRACTVE NDEX FROM THE DEVATON OF LGHT BY A PRSM HcB WrNcutl, Yle Uierity, New He, Cm. Arcr A ligmet chrt i preeted by me f which the refrctive idex f prim i btied frm
Unit 1 Chapter-3 Partial Fractions, Algebraic Relationships, Surds, Indices, Logarithms
Uit Chpter- Prtil Frctios, Algeric Reltioships, Surds, Idices, Logriths. Prtil Frctios: A frctio of the for 7 where the degree of the uertor is less th the degree of the deoitor is referred to s proper
SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
Module Summary Sheets. C2, Concepts for Advanced Mathematics (Version B reference to new book)
MEI Mthemtics i Educti d Idustry MEI Structured Mthemtics Mdule Summry Sheets C, Ccepts fr Advced Mthemtics (Versi B referece t ew ) Tpic : Alger. Lgrithms. Sequeces d Series Tpic : Trigmetry.Trigmetricl
n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
Linford 1. Kyle Linford. Math 211. Honors Project. Theorems to Analyze: Theorem 2.4 The Limit of a Function Involving a Radical (A4)
Liford 1 Kyle Liford Mth 211 Hoors Project Theorems to Alyze: Theorem 2.4 The Limit of Fuctio Ivolvig Rdicl (A4) Theorem 2.8 The Squeeze Theorem (A5) Theorem 2.9 The Limit of Si(x)/x = 1 (p. 85) Theorem
Root locus ( )( ) The given TFs are: 1. Using Matlab: >> rlocus(g) >> Gp1=tf(1,poly([0-1 -2])) Transfer function: s^3 + 3 s^2 + 2 s
![Root locus ( )( ) The given TFs are: 1. Using Matlab: >> rlocus(g) >> Gp1=tf(1,poly([0-1 -2])) Transfer function: s^3 + 3 s^2 + 2 s](/thumbs/89/98203330.jpg "Root locus ( )( ) The given TFs are: 1. Using Matlab: >> rlocus(g) >> Gp1=tf(1,poly([0-1 -2])) Transfer function: s^3 + 3 s^2 + 2 s")
The given TFs are: 1 1() s = s s + 1 s + G p, () s ( )( ) >> Gp1=tf(1,ply([0-1 -])) Transfer functin: 1 ----------------- s^ + s^ + s Rt lcus G 1 = p ( s + 0.8 + j)( s + 0.8 j) >> Gp=tf(1,ply([-0.8-*i
8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
8.1 Arc Legth Wht is the legth of curve? How c we pproximte it? We could do it followig the ptter we ve used efore Use sequece of icresigly short segmets to pproximte the curve: As the segmets get smller
Algebra 2 Important Things to Know Chapters bx c can be factored into... y x 5x. 2 8x. x = a then the solutions to the equation are given by
Alger Iportt Thigs to Kow Chpters 8. Chpter - Qudrtic fuctios: The stdrd for of qudrtic fuctio is f ( ) c, where 0. c This c lso e writte s (if did equl zero, we would e left with The grph of qudrtic fuctio
Helping every little saver
Spt th diffc d cut hw u c fid I c spt thigs! Hlpig v littl sv Hw d u p i? I ch Just pp it f u chs. T fid u lcl ch just visit s.c.uk/ch If u pig i chqu, it c tk ud 4 wkig ds t cl Ov th ph Just cll Tlph
Mathematics. Trigonometrical Ratio, Functions & Identities
Mthemtics Tigmeticl Rti, Fuctis & Idetities Tble f tet Defiitis stems f Mesuemet f gles Relti betwee Thee stems f Mesuemet f gle Relti betwee c d gle 5 Tigmeticl Rtis Fuctis 6 Tigmeticl Rtis f llied gles
1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
[Q. Booklet Number]
![[Q. Booklet Number]](/thumbs/82/85468362.jpg "[Q. Booklet Number]")
6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,
Approximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why?
AB CALCULUS: 5.3 Positio vs Distce Velocity vs. Speed Accelertio All the questios which follow refer to the grph t the right.. Whe is the prticle movig t costt speed?. Whe is the prticle movig to the right?
, so the state may be taken to be l S ÅÅÅÅ
hw4-7.b: 4/6/04:::9:34 Hoework: 4-7 ü Skuri :, G,, 7, 8 ü. bel the eigefuctio of the qure well by S, with =,,. The correpodig wve fuctio re x = px i Iitilly, the prticle i kow to be t x =, o the tte y
Exponential and Logarithmic Functions (4.1, 4.2, 4.4, 4.6)
WQ017 MAT16B Lecture : Mrch 8, 017 Aoucemets W -4p Wellm 115-4p Wellm 115 Q4 ue F T 3/1 10:30-1:30 FINAL Expoetil Logrithmic Fuctios (4.1, 4., 4.4, 4.6) Properties of Expoets Let b be positive rel umbers.
ROBUST STABILITY OF CONVEX COMBINATION OF TWO FRACTIONAL DEGREE CHARACTERISTIC POLYNOMIALS
ct mechic et utomtic, vol.2 o.2 (28) ROBUST STABILITY OF CONVEX COMBINATION OF TWO FRACTIONAL DEGREE CHARACTERISTIC POLYNOMIALS Mikołj BUSŁOWICZ * * Chir of Automtic d Electroic, Fculty of Electricl Egieerig
Lesson 5: Does the Order Matter?
: Does the Order Mtter? Opeig Activity You will eed: Does the Order Mtter? sortig crds [dpted fro 5E Lesso Pl: Usig Order of Opertios to Evlute d Siplify Expressios, Pt Tyree] 1. Rerrge the crds so they
M098 Carson Elementary and Intermediate Algebra 3e Section 10.2
M09 Crso Eleetry d Iteredite Alger e Sectio 0. Ojectives. Evlute rtiol epoets.. Write rdicls s epressios rised to rtiol epoets.. Siplify epressios with rtiol uer epoets usig the rules of epoets.. Use rtiol
ALGEBRA 2/TRIGONMETRY TOPIC REVIEW QUARTER 3 LOGS
ALGEBRA /TRIGONMETRY TOPIC REVIEW QUARTER LOGS Cnverting frm Epnentil frm t Lgrithmic frm: E B N Lg BN E Americn Ben t French Lg Ben-n Lg Prperties: Lg Prperties lg (y) lg + lg y lg y lg lg y lg () lg
Pre-Calculus - Chapter 3 Sections Notes
Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied
MTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
STRUCTURES IN MIKE 21. Flow over sluice gates A-1
A-1 STRUCTURES IN MIKE 1 Fl ver luice gate Fr a give gemetry f the luice gate ad k ater level uptream ad dtream f the tructure, the fl rate, ca be determied thrugh the equati f eergy ad mmetum - ee B Pedere,
Intermediate Arithmetic
Git Lerig Guides Iteredite Arithetic Nuer Syste, Surds d Idices Author: Rghu M.D. NUMBER SYSTEM Nuer syste: Nuer systes re clssified s Nturl, Whole, Itegers, Rtiol d Irrtiol uers. The syste hs ee digrticlly
AP Calculus AB AP Review
AP Clulus AB Chpters. Re limit vlues from grphsleft-h Limits Right H Limits Uerst tht f() vlues eist ut tht the limit t oes ot hve to.. Be le to ietify lel isotiuities from grphs. Do t forget out the 3-step
A New Method for Finding an Optimal Solution. of Fully Interval Integer Transportation Problems
Applied Matheatical Scieces, Vl. 4, 200,. 37, 89-830 A New Methd fr Fidig a Optial Sluti f Fully Iterval Iteger Trasprtati Prbles P. Padia ad G. Nataraja Departet f Matheatics, Schl f Advaced Scieces,
UNIT 4 EXTENDING THE NUMBER SYSTEM Lesson 1: Working with the Number System Instruction
Lesso : Workig with the Nuber Syste Istructio Prerequisite Skills This lesso requires the use of the followig skills: evlutig expressios usig the order of opertios evlutig expoetil expressios ivolvig iteger
In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
A B= ( ) because from A to B is 3 right, 2 down.
8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.
Remarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
Multiplicative Versions of Infinitesimal Calculus
Multiplictive Versios o Iiitesiml Clculus Wht hppes whe you replce the summtio o stdrd itegrl clculus with multiplictio? Compre the revited deiitio o stdrd itegrl D å ( ) lim ( ) D i With ( ) lim ( ) D
Mathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Nme : Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits c e prite give to the istructor or emile to the istructor t jmes@richl.eu.
D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS STATISTICAL FOURIER ANALYSIS
STATISTICAL FOURIER ANALYSIS The Furier Represetati f a Sequece Accrdig t the basic result f Furier aalysis, it is always pssible t apprximate a arbitrary aalytic fucti defied ver a fiite iterval f the
Chapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Hamid R. Rabiee Arman Sepehr Fall 2010
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Hmid R. Riee Arm Sepehr Fll 00 Lecture 5 Chpter 0 Outlie Itroductio to the -Trsform Properties of the ROC of
10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
MATHEMATICIA GENERALLI
MATHEMATICIA GENERALLI (y Mhmmed Abbs) Lgithmi Reltis lgb ) lg lg ) b b) lg lg lg m lg m d) lg m. lg m lg m e) lg lg m lg g) lg lg h) f) lg lg f ( ) f ( ). Eetil Reltis ). lge. lge.... lge...!! b) e......
x = a To determine the volume of the solid, we use a definite integral to sum the volumes of the slices as we let!x " 0 :
Clculus II MAT 146 Integrtion Applictions: Volumes of 3D Solids Our gol is to determine volumes of vrious shpes. Some of the shpes re the result of rotting curve out n xis nd other shpes re simply given
Tranformations. Some slides adapted from Octavia Camps
Trformtio Some lide dpted from Octvi Cmp A m 3 3 m m 3m m Mtrice 5K C c m ij A ij m b ij B A d B mut hve the me dimeio m 3 Mtrice p m m p B A C H @K?J H @K?J m k kj ik ij b c A d B mut hve A d B mut hve
Fuzzy Neutrosophic Equivalence Relations
wwwijirdm Jur 6 Vl 5 ssue SS 78 Olie u eutrsphi Equivlee eltis J Mrti Je eserh Shlr Deprtmet f Mthemtis irml Cllege fr Wme Cimtre mil du di rkiri riipl irml Cllege fr Wme Cimtre mil du di strt: his pper
e) Dates of Violation(S) reference ( page of report /data sheet): March 25, 2010, page 2
e) tes f Vilti(S) referee ( pge f reprt /dt sheet): Mrh 25, 2, pge 2 ) Explti f use(s): gig trtrs wrkig rretig plt defiieies, d trubleshtig filter prbles. g) rretie ti(s): trtrs d P re wrkig t rret prbles
CHAPTER 7 SYMMETRICAL COMPONENTS AND REPRESENTATION OF FAULTED NETWORKS
HAPTER 7 SMMETRAL OMPOETS AD REPRESETATO OF FAULTED ETWORKS A uled three-phe yte e reolved ito three led yte i the iuoidl tedy tte. Thi ethod of reolvig uled yte ito three led phor yte h ee propoed y.
Fourier Series and Applications
9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o
42. (20 pts) Use Fermat s Principle to prove the law of reflection. 0 x c
4. (0 ts) Use Femt s Piile t ve the lw eleti. A i b 0 x While the light uld tke y th t get m A t B, Femt s Piile sys it will tke the th lest time. We theee lulte the time th s uti the eleti it, d the tke
Content: Essential Calculus, Early Transcendentals, James Stewart, 2007 Chapter 1: Functions and Limits., in a set B.
Review Sheet: Chpter Cotet: Essetil Clculus, Erly Trscedetls, Jmes Stewrt, 007 Chpter : Fuctios d Limits Cocepts, Defiitios, Lws, Theorems: A fuctio, f, is rule tht ssigs to ech elemet i set A ectly oe
CE 59700: Digital Photogrammetric Systems. Lab2: Single Photo Resection
CE 9700: Digitl Phtgrmmetric Sstems L: Sigle Pht Resecti Ojectives Determie the Eterir Orietti Prmeters (EOPs f sigle pht usig lest squres justmet prceure. Am F. Hi Sme tset fr l # Y Give X Z Lscpe Me
CLOSED-LOOP SIGNAL SHAPING WITH INNER-LOOP MODEL MATCHING
Proceedig of the ASME 7 Itertiol Mechicl Egieerig Cogre d Expoitio IMECE7 Noveer 3-9 7 Tp Florid USA IMECE7-7344 CLOSED-LOOP SIGNAL SHAPING WITH INNER-LOOP MODEL MATCHING Withit Chtltgulchi Deprtet of
=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property
Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(
8.3 Sequences & Series: Convergence & Divergence
8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of
Chapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Dr. Hamid R. Rabiee Fall 2013
Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Dr. Hmid R. Rbiee Fll 03 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Outlie Itroductio to the -Trsform Properties
MATH Midterm Examination Victor Matveev October 26, 2016
MATH 33- Midterm Examiati Victr Matveev Octber 6, 6. (5pts, mi) Suppse f(x) equals si x the iterval < x < (=), ad is a eve peridic extesi f this fucti t the rest f the real lie. Fid the csie series fr
Logarithmic Scales: the most common example of these are ph, sound and earthquake intensity.
Numercy Itroductio to Logrithms Logrithms re commoly credited to Scottish mthemtici med Joh Npier who costructed tle of vlues tht llowed multiplictios to e performed y dditio of the vlues from the tle.
Intermediate Applications of Vectors and Matrices Ed Stanek
Iteredite Applictio of Vector d Mtrice Ed Stek Itroductio We decribe iteredite opertio d pplictio of vector d trice for ue i ttitic The itroductio i iteded for thoe who re filir with bic trix lgebr Followig
Mathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits shoul e emile to the istructor t jmes@richl.eu. Type your me t the top
Section IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
Linear predictive coding
Liner predictive coding Thi ethod cobine liner proceing with clr quntiztion. The in ide of the ethod i to predict the vlue of the current ple by liner cobintion of previou lredy recontructed ple nd then
Frequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
* power rule: * fraction raised to negative exponent: * expanded power rule:
Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures
Multi-objective Programming Approach for. Fuzzy Linear Programming Problems
Applied Mathematical Scieces Vl. 7 03. 37 8-87 HIKARI Ltd www.m-hikari.cm Multi-bective Prgrammig Apprach fr Fuzzy Liear Prgrammig Prblems P. Padia Departmet f Mathematics Schl f Advaced Scieces VIT Uiversity
OVERVIEW Using Similarity and Proving Triangle Theorems G.SRT.4
OVRVIW Using Similrity nd Prving Tringle Therems G.SRT.4 G.SRT.4 Prve therems ut tringles. Therems include: line prllel t ne side f tringle divides the ther tw prprtinlly, nd cnversely; the Pythgren Therem
x 2 x 3 x b 0, then a, b, c log x 1 log z log x log y 1 logb log a dy 4. dx As tangent is perpendicular to the x axis, slope
The agle betwee the tagets draw t the parabla y = frm the pit (-,) 5 9 6 Here give pit lies the directri, hece the agle betwee the tagets frm that pit right agle Ratig :EASY The umber f values f c such
B. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
11.2. Infinite Series
.2 Infinite Series 76.2 Infinite Series An infinite series is the sum f n infinite seuence f numbers + 2 + 3 + Á + n + Á The gl f this sectin is t understnd the mening f such n infinite sum nd t develp
[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
![[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.](/thumbs/78/77732161.jpg "[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.")
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
Identical Particles. We would like to move from the quantum theory of hydrogen to that for the rest of the periodic table
We wuld like t ve fr the quatu thery f hydrge t that fr the rest f the peridic table Oe electr at t ultielectr ats This is cplicated by the iteracti f the electrs with each ther ad by the fact that the