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 Variatin f fgain K d nt affect the lcatin f any ple H 4 Cled-lp TF KG T KG H K 4 6 K 8 4K K Lcatin f ple need t factr the denminatr difficult variatin f gain K d change the lcatin f ple Eay way t knw the CL ple lcatin w/ lving higher-rder characteritic equatin f CLTF fr variu gain K? Rt lcu graphical repreentatin f the cled-lp ple a a functin f ytem parameter Can be ued t deign ytem parameter f the high rder ytem t yield a deired ytem pecificatin Etimating cled-lp ple lcatin when gain K i varied uing pen-lp ple Repreent the ple f T a K varie
Vectr Repreentatin f Cmplex Number Vectr repreentatin f cmplex number Cmplex number σ jω Magnitude M and angle θ, a M θ Cmplex functin F a when σ j ω Cmplex functin F ha a zer at -a F a σ a jω a cmplex number a can be repreented by a vectr drawn frm the zer f the functin t the pint Ex F 7 5 j
Cmplicated functin m zi i : prduct F n where M: number f zer pi N: number f ple j Value f F at any pint Each cmplex factr vectr magnitude & angle Magnitude f F M where zer ple length length z i p i m zi i n p j i Magnitude f the vectr drawn frm t the pint p Magnitude f the vectr drawn frm t the pint z i p i Angle f F at any pint Meaured frm the pitive real axi θ angle t zer m i z i n j p angle t i ple
Ex Find F at the pint - j4 F Magnitude M zer length ple length j4 j4 j4 5 0 7 0.7 Angle r F angle t zer θ z θ p 0 θ p 4. 6.66 angle t 6.9 90 ple tan 04.00 6.6 4 F tan 4 4 4 tan tan 4.4
Prpertie f Rt Lcu Subject tracking camera ytem Knb 에해당 Equivalent cled lp TF Cled lp ple functin f gain K P, 0 ± 0 4K
Lcatin f C.L.ple vary with gain K Sytem perfrmance varie K ple K<5 Real : verdamped Ple plt K5 Repeated real: critically damped K>5 underdamped d d Cnnecting the C.L.ple crrepnding t frm K 0 t K C.L. ple lcu fr frm K 0 t K Rt lcu Rt lcu Path f cled-lp ple a gain i varied Uing rt lcu, we can eaily analyze the characteritic f the higher-rder ytem w/ calculating cled-lp ple
Prpertie f Rt Lcu Higher-rder ytem difficult t calculate the cled-lp ple lcatin fr variu gain K. Uing rt lcu, withut lving denminatr plynmial f cled lp TF, it i pible t have rapid ketch f cled-lp ple lcatin change fr variu gain K rt lcu C.L.T.F. KG T KG H Characteritic equatin i.e. C.L. ple the value that atify the characteritic equatin KG H 0 KG H k 80 KG H k 80 k 0, ±, ±, ±, L Any cmplex number that atifie thee cnditin cled-lp ple! KG H K G H Angle cnditin determine the rt lcu Magnitude cnditin Determine the pecific pitin crrepnding t pecific value f K n the rt lcu
Ex Open-lp TF KG H K 4 Cled-lp TF T K K 4 7K K If a ple a jb i a cled-lp ple fr me value f gain K a jb mut atify KG H k 80 and KG H Cnider a the firt tet pint KG H j 70.55 angle t zer 4 θ θ θ θ 56. angle t 7.57 80 i nt a cled-lp ple j ple 90 08.4 j What abut / j KG H 80 j K G H / i a cled-lp ple fr me value f gain K ple length zer length 4 j / j j / j / /... 0. j / i a cled-lp ple i.e. i a pint n the rt lcu when K 0.
Ple and Zer at Infinity Infinite ple: if O.L.T.F. apprache, a apprache OLTF ha a ple at infinity Infinite zer: if O.L.T.F. apprache 0, a apprache OLTF ha a zer at infinity ex G G ha a ple at infinity ince ha a zer at infinity limg lim lim G lim 0 Ex KG H K Three finite ple 0, -, - N finite i zer K lim KG H 0 Fr every functin f number f finite i ple infinite i ple number f finite zer infinite zer Three zer at infinite finite ple 0 infinite ple 0 finite zer infinite zer
Shw hw lcatin f the CL ple mve a K varie frm 0 t Sketching the Rt Lcu K0~ 변할때 rt lcu가어떻게변하는지본다.. Mark pen lp ple with x and pen lp zer with. Draw rt lci n the real-axi t the left f an dd number f real ple plu zer K>0일때, 실수축상에서 Tet pint의오른쪽에있는 ple과 zer의수의합이홀수이면그점은rt luc상에있다 Satifying the angle cnditin KG H k 80. Rt lcu i.e. K 0 tart at finite and infinite pen-lp ple and end at finite and infinite pen-lp zer i.e. K 4. Draw the aymptte f rt lcu a. Real-axi intercept r center f aymptte finite ple finite zer σ a # finite ple# finite zer b. 양의실수축과이루는각도 k 80 θa k 0, ±, ±, ± # finite ple# finite zer Nte. - Number f branche f the rt lcu number f cled-lp lp ple ytem rder Branch: path that ne ple travere - The rt lcu i ymmetrical abut the real axi
Open-lp TF KG H K 4 Rt lcu tart at - and - meet between -and- - break ut int cmplex plane return t real axi mewhere between - and -4 end at -and-4
ex 8. End at zer at tinfinity it. Mark OL ple and zer. Draw lci n real axi t the left f dd number. Start at ple and end at zer 4. Draw aymptte σ a θ a 0 4 4 # π / π finite 4 k 80 ple# finite fr fr k 0 k 5 π / fr k π / zer End at zer at infinity 5π / π Check! # f aymptte # finite ple - #finite zer # f branche # f CL ple ytem rder 4 The rt lcu i ymmetrical abut the real axi # infinite zer #finite zer #finite ple 4 End at zer at infinity
Refining the Sketch breakaway frm real axi and mve int the cmplex plane A. Real-axi Breakaway and Break-in Pint σ σ : Breakaway pint : Break-in Pint 80 Break-in and Breakaway angle : n n: # f CL ple arriving at r departing frm the BA r BI n the real axi A K increae Tw ple 90 at BA Tw zer 90 at BI Determine the lcatin f BA and BI pint On the real axi between pen-lp p ple, gain K i maximum at BA pint On the real axi between pen-lp zer, gain K i minimum at BI pint dk d σ 0 where σ i an either BA r BI pint K Q KG H 0 G H Characteritic equatin
Ex 5 8 5 K K H KG 5 8 K 5 8 8 5 8 dk 6 6 5 8 8 5 8 σ σ σ σ d dk 8 45 0 5 8 6 6 σ σ σ σ A d i.8.45, σ BA and BI pint
B. j ω -Axi Cring A pint n the rt lcu that eparate the table peratin f the ytem frm the untable peratin A ple n j ω -axi at a certain gain K Hw t find the value f ω and K? Methd I Subtitute j ω directly int characteritic equatin 0 A ple n j ω -axi σ jω Methd II Ue Ruth table A rw f entire zer f Ruth table ple n j ω-axi pible
Ex find the frequency and gain K fr which the rt lcu cre the imaginary axi. C.L.T.F. T 4 7 K 4 8 K K Methd I Characteritic equatin: 4 7 4 8 K K 0 A j ω i a ple f cled-lp ytem, it mut atify characteritic equatin Subtitute t j ω int characteritic ti equatin jω 4 7 jω 4 jω 8 K jω K 0 4 ω j7ω 4ω 8 K jω K 0 Real part Imaginary part ω 4 4ω K 0 7ω 8 K ω 0 Slve fr ω and K ω ±.59 K 9.65 Rt lcu cre j ω -axi at ± j. 59 at a gain f K 9.65
Methd II Ue Ruth table K T 4 7 4 8 K K N ign change frm 4 t LHP ple remaining, but even plynmial f require ymmetric ple ple huld be n j jωω -axi Only rw can make all zer rw K 65K 70 0 Auming K>0, then K 9.65 Frm even plynmial uing rw 90 K K ± j.59 80.5 0.7 0 K 9.65 j.59 Rt lcu cre j ω-axi at ± j. 59 at a gain f K 9.65 K 0 K 0 Sytem i table fr 0 K < 9.65 K 9.65 j.59
C. Angle f Departure and Arrival Rt lcu tart at pen-lp ple ex. p and end at finite and infinite pen-lp zer ex. z Fr mre accurate rt lcu, need t knw the rt lcu departure angle frm the cmplex ple arrival angle t the cmplex zer If a cmplex number i n rt-lcu and cle ε t a cmplex ple p, i.e. i ne f the cledlp ple, it mut atify angle cnditin z p Tet pint KG H angle t zer k 80 6 4 θ θ θ θ θ θ angle t 5 ple if want t knw the departure angle frm a cmplex ple p θ θ θ6 θ θ4 θ5 k 80
C. Angle f Departure and Arrival Rt lcu tart at pen-lp ple ex. p and end at finite and infinite pen-lp zer ex. z Fr mre accurate rt lcu, need t knw the rt lcu departure angle frm the cmplex ple arrival angle t the cmplex zer If a cmplex number i n rt-lcu and cle ε t a cmplex ple p, i.e. i ne f the cledlp ple, it mut atify angle cnditin Tet pint KG H angle t zer k 80 6 4 θ θ θ θ θ θ angle t 5 ple if want t knw the departure angle frm a cmplex ple p θ θ θ6 θ θ4 θ5 k 80
z if want t knw the arrival angle frm a cmplex zer z Angle cnditin fr a cmplex number near z KG H angle t zer k 80 6 θ θ θ θ θ θ angle t θ θ θ θ θ θ k 80 6 4 5 4 5 ple Nte. and 대신각을찾고자하는 ple 이나 zer 까지의각도를직접구하면된다.
Ex find the angle f departure frm the cmplex ple and ketch the rt lcu Open lp ple at -, -±j Departure angle frm -j θ θ θ θ4 k 80 θ θ θ θ k 80 tan 08.4 4 5.6 4 90 tan 80
D. Pltting and Calibrating the Rt Lcu Hw t find exact pint at which rt lcu atifie certain cnditin and the gain at that pint? Methd I Ex Find exact pint at which rt lcu cre the 0.45 damping line and the gain at that pint K KG H 4 ζ cθ θ θ c 0.45 6.5 On the ζ 0.45 line, nly a pint with radiu r 0.747 atifie angle cnditin KG H k 80 θ θ θ θ4 θ5 zi i G H n p m j i zer length G H ple length ple length A C D E At that pint, the crrepnding gain K. 7 G H zer length B
Methd II Ex Find exact pint at which rt lcu cre the 0.58 damping line and the gain at that pint G 4 6 C R G G α 0 4 ζω ω 0.76ω α n n n α 0.76ω ω 0.76ω α ω αω n n n n n θ c 0.58 ζ 0.58 69.0 0.76ω n α 0 0.76ω nα ωn 4 αω n ω n.8 ζ 0.58 d.007 ± j.67 CL ple pair n ζ 0.58 damping line C B A θ 80-69.0 0 ζω n ± jωn ζ.007 ± j.67-6 θ θ -4-0 K ple length G H zer length A B C θ θ θ 80
Cmment T be table All C.L. ple mut be in LHP If #O.L. ple - #finite zer There i a vlue f the gain K beynd which rt rci enter the RHPi..e ytem becme untable
Nnminimum-Phae Sytem Skip! Minimum phae ytem: all the ple and zer lie in LHP Nnminimum phae ytem: at leat ne ple r zer lie in RHP G K T a T H & T > 0 a G K T a T T 80 T ±80 k Phae hift thi i why it i called nnminimum phae K T ± 80 T K a a k 0 Lcu Nt n here!
Generalized Rt Lcu S far, rt lcu a a functin f the frward-path gain K Hw t draw rt lcu fr variatin f ther parameter? Ex rt lcu fr variatin f the value f pen lp ple p? 0 KG H gain K wa a multiplying factr f the functin but P i nt. p Denminatr f C.L.T.F. fr a rt lcu fr gain K variatin KG H Fr a rt lcu fr p variatin, we need a CLTF denminatr pg H KG T KG H KG p 0 p 0 0 Ilating p 0 p 0 0 p 0 p H 0 P i nw a multiplying factr f the pen lp tranfer functin G H 0 Path f cled lp ple a p i increaed
Summary f the Step fr Cntructing Rt Lci. Cnvert int a frm in which ytem parameter i a multiplying factr. Lcate the pen-lp ple and zer n the -plane. The branche tart frm pen-lp pple and end at finite r infinite zer 4. Draw the rt lcu n the real axi 5. Determine the aymptte f rt lcu 6. Find the breakaway and break-in pint 7. Dt Determine the angle f fdeparture/arrival lf frm/t a cmplex ple/zer l/ 8. Find the pint where the rt lci may cr the imaginary axi 9. Lcate the cled-lp ple n the rt lcu and determine the crrepnding gain K by ue f the magnitude cnditin
Rt Lcu fr Pitive-Feedback Sytem KG T KG H Pitive feedback ytem A cled-lp ple exit when KG H 0 KG H 60 k k 0, ±, ±, ± Angle cnditin fr a rt lcu f pitive feedback ytem KG H 60 k Rule fr ketching rt lcu Read by yurelf