MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI

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1 MAHALASHMI ENGINEERING COLLEGE TIRUCHIRAPALLI-63. QUESTION BAN DEPARTMENT: ECE SUBJECT CODE: EC55 SEMESTER - III SUBJECT NAME: CONTROL SYSTEMS UNIT- SYSTEMS AND THEIR REPRESENTATION. Write Mao gai formula. May Mao Gai Formula The relatiohip betwee a iput variable ad a output variable of a igal flow graph i give by the et gai betwee iput ad output ode ad i kow a overall gai of the ytem. Mao gai formula i ued to obtai the overall gai trafer fuctio of igal flow graph. Gai P i give by P k P k k Where, P k i gai of k th forward path, i determiat of graph =-um of all idividual loop gai+um of gai product of all poible combiatio of two otouchig loop um of gai product of all poible combiatio of three otouchig loop + k i cofactor of k th forward path determiat of graph with loop touchig k th forward path. It i obtaied from by removig the loop touchig the path P k.. What i mathematical model of a ytem? May,06 The weight of mechaical tralatioal ytem i repreeted by the elemet MaM The elatic deformatio of the body ca be repreeted by a Sprig The frictioal exitig i rotatig mechaical ytem ca be repreeted by Dah pot B

2 3. What i cotrol ytem? May A cotrol ytem i a itercoectio of compoet or device o that the output of the overall ytem will follow a cloely a poible a deired igal. 4. What are the two major type of cotrol ytem? Nov 0 Cotrol ytem are baically claified a Ope-loop cotrol ytem Cloed-loop cotrol ytem 5. Why egative feedback i preferred i cotrol ytem? Nov 0 Negative Feedback ytem have the followig feature: - reduced effect of oliearitie ad ditortio - Icreaed accuracy - Icreaed badwidth - Le eitivity to variatio of ytem parameter - Tedecy toward ocillatio - Reduced effect of exteral diturbace 6. Ditiguih betwee ope loop ad cloed loop ytem Nov I ope-loop ytem the cotrol actio i idepedet of output. I cloed-loop ytem cotrol actio i omehow depedet o output. Each ytem ha at leat two thig i commo, a cotroller ad a actuator fial cotrol elemet. The iput to the cotroller i called referece iput. Thi igal repreet the deired ytem output. Cloed-loop ytem are alo called feedback cotrol ytem. Feedback i the property of the cloed-loop ytem which permit the output to be compared with the iput of the ytem o that appropriate cotrol actio may be formed a a fuctio of iput ad output. 7. What i liear ytem? Nov 8. Defie trafer fuctio. May 09 The trafer fuctio of a LTI ytem i the ratio of Laplace traform of the output variable to the Laplace traform of the iput variable aumig zero iitial coditio.

3 9. Write force balace equatio of ideal prig, ideal ma. Dec What i a aalogou ytem? Dec 09,05 Aalogou ytem may have etirely differet phyical appearace. For example, a give electrical circuit coitig of reitace, iductace ad capacitace may be aalogou to a mechaical ytem coitig of a uitable combiatio of ma, dahpot ad prig.

4 . Write the differetial equatio goverig the Mechaical ytem how i fig ad determie the trafer fuctio. 6 May, Nov, May 06 Solutio: Trafer fuctio = F Step : Free Body diagram for M The oppoig force actig o Ma M are f m, f b, f k, f b ad f k d w.k.t. f m m dt d B dt f b f k k d B dt f b f k k By Newto d Law: Applied force = Oppoig Force f f f f f m b b k k d d d M B B dt dt dt

5 Takig Laplace traform for equatio 0 BS B S M 0 B B B M ] [ ] [ B B B M B B M B Step : Free Body Diagram for M The oppoig force actig o Ma M are f m, f b, f k, f b w.k.t. dt d M f m dt d B f b f k dt d B f b By Newto d Law: t f f f f f b k b m t f dt dx B dt d B dt d M Takig Laplace traform: F B B M Sub value i the above equatio: ] [ F B B M B B B M

6 ][ [ F B B M B B B M B B M ] ][ [ k B B B M B B M B B M F Reult: The differetial equatio goverig the ytem are. 0 dt d B dt d B dt d M. t f dt dx B dt d B dt d M Trafer fuctio of the ytem i. ] ][ [ k B B B M B B M B B M F.Explai Sychro ad it type. 0 May Figure: Sychro error detector A ychro i a electromagetic traducer that i ued to covert agular haft poitio ito a electric igal. The baic elemet of a ychro i a ychro tramitter whoe cotructio i very imilar to that of the 3- alterator.

7 A ac voltage i applied to the rotor widig through lip-rig. The chematic diagram of ychro ychro tramitter- cotrol traformer pair i how above. vr t Vr i ct ; v t V i t co 0 r c v t V i t co r c v t V i t co 40 3 r c ; v t 3V i ti 40 r c v t 3V i ti 0 3 r c v t 3V i t i 3 r c Whe 0, v 0 3 t ad maximum voltage i iduced o S coil. Thi poitio of the rotor i defied a the electrical zero of the tramitter ad ued a referece poitio of the rotor. The output of the ychro tramitter i applied to the tator widig of a ychro cotrol traformer. Circulatig curret of the ame phae but of differet magitude flow through the two et of tator coil. The pair act a a error detector. The voltage iduced i the cotrol traformer rotor i proportioal to the coie of the agle betwee the two rotor ad i give by, e t V i t co ; where, 90. r c e t Vr i ct co90 Vr i ct i Vr i ct 0 The equatio above hold for mall agular diplacemet. Thu the ychro tramitter-cotrol traformer pair act a a error detector which give a voltage igal at the rotor termial of the cotrol traformer proportioal to the agular differece betwee the haft poitio. Equatio 0 i repreeted graphically i figure below for a arbitrary time variatio of.

8 It i ee that the output of the ychro error detector i a modulated igal, where the ac igal applied to the rotor of ychro tramitter act a carrier ad the modulatig igal i, em t ; V r 3.Write the rule for block diagram reductio techique. May, Nov 06 Block diagram reductio techique Block diagram: A block diagram i a hort had, pictorial repreetatio of caue ad effect relatiohip betwee the iput ad the output of a phyical ytem. It characterize the fuctioal relatiohip amogt the compoet of a cotrol ytem. Lower cae letter uually repreet fuctio of time. Upper cae letter deote Laplace traformed quatitie a a fuctio of complex variable or Fourier traformed quatity i.e. frequecy fuctio a fuctio of imagiary variable jω. Summig poit: It repreet a operatio of additio ad / or ubtractio. Negative feedback: Summig poit i a ubtractor. Poitive feedback: Summig poit i a adder. Stimulu: It i a exterally itroduced iput igal affectig the cotrolled output. Take off poit: I order to employ the ame igal or variable a a iput to more tha block or ummig poit, take off poit i ued. Thi permit the igal to proceed ualtered alog everal differet path to everal detiatio.

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10 4. i Derive the trafer fuctio for Armature cotrolled DC motor. May, Nov A. Trafer Fuctio of Armature Cotrolled DC Motor i ; T i i i f f M f f a T a d a eb b ; L di a Raia eb e dt dt d d J f T i dt dt 0 M T a I Laplace domai, E b b L R I E E ; a a a b J f0 TIa G T E R L J f a a 0 T b Neglectig L a, G / R / R ; / T a T a m J f0 T b Ra J f m where, f f0 T b / R a ad m T / Ra f ; m J / f. m ad m are called the motor gai ad time cotat repectively. Thee two parameter are uually upplied by the maufacturer.the block diagram model i,

11 B. Trafer Fuctio of a Field-cotrolled DC Motor Figure T i i i i ' M a f f a T f di f Lf R f i f dt e d d J f dt dt T i M T f ; We obtai, L R I E f f f J f TM T I f G T E L R J f f f

12 5. Explai the workig of AC ervomotor i cotrol ytem. A.C. Servomotor For low power applicatio a.c. motor are preferred, becaue of their light weight, ruggede ad o bruh cotact. Two phae iductio motor are motly ued i cotrol ytem. The motor ha two tator widig diplaced 90 electrical degree apart. The voltage applied to the widig are ot balaced. Uder ormal operatig coditio a fixed voltage from a cotat voltage ource i applied to oe phae. The other phae, called cotrol phae, i eergized by a voltage of variable magitude which i 90 out of phae w.r.t. the voltage of fixed phae. The torque peed characteritic of the motor i differet from covetioal motor. / R ratio i low ad the curve ha egative lope for tabilizatio.

13 The torque-peed curve i ot liear. But we aume it a liear for the derivatio of trafer fuctio. The troque i afuctio of both peed ad the r.m.. cotrol voltage, ie., T f, E. M Uig Tailor erie expaio about the ormal operatig poit TM 0, 0, E0 we get, TM TM TM TM0 E E0 0 E E E0 E E0 0 0 For poitio cotrol ytem, E 0 0, 0 0, T M 0 Thu, the above equatio may be implified a, T ke m J f where, M 0 ; k TM E E E0 0 ad T M m. E E0 0 Performig Laplace traform, ke m J f 0 k m Or, G ; where, E J f m 0 m m k f0 m ad m 0 J f m. Sice m i egative the traiet part i decayig a m i poitive. If m would poitive ad m f the traiet part will icreae with time ad the ytem would be utable. k ad m 0 are the lope of the torque-voltage ad torque-peed curve.

14 6. Determie the overall trafer fuctio CS/RS for the ytem how i fig. Nov 0,05 To implify the ier feedback loop to obtai the followig block diagram. To combie the two erie block ito oe. To obtai the trafer fuctio for the tadard feedback ytem.

15 7.Fid the trafer fuctio of the cloed-loop ytem below. To implify the ier feedback block. To get the followig block diagram.

16 To combie the two et of erie block. Calculate the overall trafer fuctio of the ytem 8. Fid the overall gai C / R for the igal flow graph how below May 09 G 6 G 7 R G G G 3 G 4 G 5 C H -H Figure Sigal flow graph of example

17 There are three forward path. The gai of the forward path are: P =G G G 3 G 4 G 5 P =G G 6 G 4 G 5 P 3 = G G G 7 There are four loop with loop gai: L =-G 4 H, L =-G G 7 H, L 3 = -G 6 G 4 G 5 H, L 4 =-G G 3 G 4 G 5 H There i oe combiatio of Loop L ad L which are o-touchig with loop gai product L L =G G 7 H G 4 H = +G 4 H +G G 7 H +G 6 G 4 G 5 H +G G 3 G 4 G 5 H + G G 7 H G 4 H Forward path ad touch all the four loop. Therefore =, =. Forward path 3 i ot i touch with loop. Hece, 3 = +G 4 H. The trafer fuctio T = C R P P P 3 3 G 4 H G G G G G G 7 3 H 4 G 5 G G 6 G G 4 G 4 5 G G H 5 6 G G G 3 G G G 4 7 G 5 H G 4 H G G 4 G 7 H H

18 9. Draw the aalogou electric circuit of the ytem below uig f-i aalogy. 0. Fid the overall gai C / R for the igal flow graph how below May 09 Fid the gai 6, 5, 3 for the igal flow graph how i Fig.4. b -h a c d 5 f 6 e 3 4 -g -i Figure 3 Sigal flow graph of MIMO ytem

19 Cae : 6 There are two forward path. The gai of the forward path are: P =acdef P =abef There are four loop with loop gai: L =-cg, L =-eh, L 3 = -cdei, L 4 =-bei There i oe combiatio of Loop L ad L which are otouchig with loop gai product L L =cgeh = +cg+eh+cdei+bei+cgeh Forward path ad touch all the four loop. Therefore =, =. The trafer fuctio T = 6 P P cg cdef abef eh cdei bei cgeh

20 Cae : 5 The modified igal flow graph for cae i how i Fig.5. b -h 5 d e c g -i Figure 4 Sigal flow graph of example 4 cae The trafer fuctio ca directly maipulated from cae a brache a ad f are removed which do ot form the loop. Hece, The trafer fuctio T= 5 P P cg cde be eh cdei bei cgeh Cae 3: 3 The igal flow graph i redraw to obtai the clarity of the fuctioal relatio a how i Fig.6.

21 -h c a b e 5 f i d Figure 5 Sigal flow graph of example 4 cae 3 -g There are two forward path. The gai of the forward path are: P =abcd P =ac There are five loop with loop gai: L =-eh, L =-cg, L 3 = -bei, L 4 =edf, L 5 =-befg There i oe combiatio of Loop L ad L which are otouchig with loop gai product L L =ehcg = +eh+cg+bei+efd+befg+ehcg Forward path touche all the five loop. Therefore =. Forward path doe ot touch loop L. Hece, = + eh The trafer fuctio T = 3 P P eh abef cg bei ac efd eh befg ehcg

22 MAHALASHMI ENGINEERING COLLEGE TIRUCHIRAPALLI-63. QUESTION BAN DEPARTMENT: EEE SUBJECT CODE: EE53 SEMESTER - III SUBJECT NAME: CONTROL SYSTEMS UNIT- TIME DOMAIN ANALYSIS PART A. What i traiet ad teady tate repoe? May. Traiet Repoe: It i the repoe of the ytem whe the iput chage from oe tate to aother. Steady State Repoe: It i the repoe a time t approache ifiity.. Name the tet igal ued i time repoe aalyi. May Name of the Sigal Time domai rt Laplace Traform Step A A / Uit Step / Ramp At A / Uit Ramp t / Parabolic At / A / 3 Uit Parabolic t / / 3 Impule δ t

23 3. How i ytem claified depedig o the value of dampig? May Defiitio: It i defied a the ratio of the actual dampig to the critical dampig The repoe ct of ecod order ytem deped o the value of dampig ratio. The ecod order ytem ca be claified ito four type depedig upo the value of dampig ratio. Udamped ytem ζ = 0. Uder damped ytem 0 < ζ < 3. Critically damped ytem ζ = 4. Over damped ytem ζ > 4. Sketch the repoe of a ecod order uder damped ytem. May r t ct Iput t Output Repoe t 5. Lit the time domai pecificatio. Nov The performace characteritic of a cotrolled ytem are pecified i term of the traiet repoe to a uit tep i/p ice it i eay to geerate & i ufficietly dratic. The traiet repoe of a practical C.S ofte exhibit damped ocillatio before reachig teady tate. I pecifyig the traiet repoe characteritic of a C.S to uit tep i/p, it i commo to pecify the followig term. Delay time td Rie time tr 3 Peak time tp 4 Max over hoot Mp 5 Settlig time t

24 6. What are geeralized error & tatic error cotat? Nov Static Poitio Error Cotat The teady tate error of the ytem for a uit tep iput i e lim 0 G G 0 The tatic poitio error cotat p i defied by p lim G 0 G 0 by e Thu, the teady tate error i term of the tatic poitio error cotat p i give p For a type 0 ytem, p lim 0 T a T T b T T m T p ad e For a type or higher ytem p lim 0 N T a T T b T T m T p N ad e Repoe of a feedback cotrol ytem to a tep iput ivolve a teady tate error if there i o itegratio i the feed forward path. If a zero teady tate error for a tep iput i deired, the type of the ytem mut be oe or higher. 7. Defie poitio, velocity error cotat. Nov 0 Type of Error cotat Steady tate error e ytem p v a Uit tep iput /+ Uit ramp iput 0 0 / Uit parabolic iput 0 0 /

25 8. Defie dampig ratio May 0 It i defied a the ratio of the actual dampig to the critical dampig 9. Defie Peak overhoot & Peak time. May 0 Peak time :- t p It i the time required for the repoe to reach the t of peak of the overhoot. Maximum over hoot :- MP It i the maximum peak value of the repoe curve meaured from uity. The amout of max over hoot directly idicate the relative tability of the ytem. 0. Coider a firt order ytem give by GS=CS/RS = /S+a.Plot it uit tep repoe. May 09 Ct rt t=0 t 0 T T 3T t. Draw the repoe of ecod order ytem for uder damped cae ad whe iput i uit tep. May Repoe of Uder Damped Secod Order Sytem: For Uit Step Iput: The tadard form of cloed loop trafer fuctio of ecod order ytem i C R cojugate. For uder damped ytem, 0 < ζ < ad root of the deomiator are complex The root of the deomiator are

26 Sice ζ <, ζ i alo le tha Therefore, the damped frequecy of ocillatio, d d j The repoe i domai, R C For uit tep iput, rt = ad R = / C C B A C c B A. Put = 0 A A =. Equatig the co-efficiet of. C B A 0 = A + B 0 = + B B = - 3. Equatig co-efficiet of 0 = Aω ζ + C = ω ζ + C

27 C = - ω ζ C Let u add ad ubtract to deomiator of the ecod term C d d d Multiply ad divide by ω d d d d d C The repoe i time domai i give by } { d d d d L C L t c

28 e t co d t d e t i d t e t co d t d i d t e t co d t i d t e t co d t i d t c t e t co d t i d t Note: O cotructig o right agle triagle with ζ ad, we get ζ i θ = co θ = ζ ta -

29 c t e t i co d t co i d t c t e t i d t ta For cloed loop uder damped ecod order ytem: Uit tep repoe = e t i d t Step Repoe = A e t i d t r t ct Iput t Output Repoe t The repoe of the uder damped ecod order ytem for uit tep iput ketched ad oberved that the repoe ocillate before ettlig to a fial value. The ocillatio deped o the value of dampig ratio.

30 . Derive the expreio for Rie time, Peak time, Peak overhoot, delay time May The performace characteritic of a cotrolled ytem are pecified i term of the traiet repoe to a uit tep i/p ice it i eay to geerate & i ufficietly dratic. The traiet repoe of a practical C.S ofte exhibit damped ocillatio before reachig teady tate. I pecifyig the traiet repoe characteritic of a C.S to uit tep i/p, it i commo to pecify the followig term. Delay time td Rie time tr 3 Peak time tp 4 Max over hoot Mp 5 Settlig time t Delay time :- t d Repoe curve the t time. It i the time required for the repoe to reach 50% of it fial value for

31 Rie time :- t r It i the time required for the repoe to rie from 0% ad 90% or 0% to 00% of it fial value. For uder damped ytem, ecod order ytem the 0 to 00% rie time i commoly ued. For over damped ytem, the 0 to 90% rie time i commoly ued. 3 Peak time :- t p It i the time required for the repoe to reach the t of peak of the overhoot. 4 Maximum over hoot :- MP It i the maximum peak value of the repoe curve meaured from uity. The amout of max over hoot directly idicate the relative tability of the ytem. 5 Settlig time :- t It i the time required for the repoe curve to reach & tay with i a rage about

32 the fial value of ize pecified by abolute percetage of the fial value uually 5% to %. The ettlig time i related to the larget time cot., of C.S.

33 3. A uity feedback cotrol ytem ha a ope loop trafer fuctio GS= 0/SS+.Fid the rie time, percetage over hoot, peak time ad ettlig time. 4.For a uity feedback cotrol ytem the ope loop trafer fuctio GS = 0S+/ SS+.Fid a poitio, velocity ad acceleratio error cotat. Nov 5. Explai P, PI, PID, PD cotroller Nov 09,May 09 A automatic cotroller compare the actual value of the ytem output with the referece iput deired value, determie the deviatio, ad produce a cotrol igal that will reduce the deviatio to zero or a mall value. The maer i which the automatic cotroller

34 produce the cotrol igal i called the cotrol actio. The cotroller may be claified accordig to their cotrol actio a Two poitio or o-off cotroller Proportioal cotroller 3 Itegral cotroller 4 Proportioal-plu- itegral cotroller 5 Proportioal-plu-derivative cotroller 6 Proportioal-plu-itegral-plu-derivative cotroller A two poitio cotroller ha two fixed poitio uually o or off. A proportioal cotrol ytem i a feedback cotrol ytem i which the output forcig fuctio i directly proportioal to error. A itegral cotrol ytem i a feedback cotrol ytem i which the output forcig fuctio i directly proportioal to the firt time itegral of error. A proportioal-plu-itegral cotrol ytem i a feedback cotrol ytem i which the output forcig fuctio i a liear combiatio of the error ad it firt time itegral. A proportioal-plu-derivative cotrol ytem i a feedback cotrol ytem i which the output forcig fuctio i a liear combiatio of the error ad it firt time derivative. A proportioal-plu-derivative-plu-itegral cotrol ytem i a feedback cotrol ytem i which the output forcig fuctio i a liear combiatio of the error, it firt time derivative ad it firt time itegral. May idutrial cotroller are electric, hydraulic, peumatic, electroic or their combiatio. The choice of the cotroller i baed o the ature of plat ad operatig coditio. Cotroller may alo be claified accordig to the power employed i the operatio a Electric cotroller Hydraulic cotroller 3 Peumatic cotroller 4 Electroic cotroller. The block diagram of a typical cotroller i how i Fig.. It coit of a automatic cotroller, a actuator, a plat ad a eor. The cotroller detect the actuatig error igal ad amplifie it. The output of a cotroller i fed to the actuator that produce the iput to the

35 plat accordig to the cotrol igal. The eor i a device that covert the output variable ito aother uitable variable to compare the output to referece iput igal. Seor i a feedback elemet of the cloed loop cotrol ytem. automatic cotroller referece iput rt error detector actuatig Cotroller error igal et output ut Amplifier Actuator Plat Output ct feed back igal bt Seor Figure 6 Block diagram of cotrol ytem Two Poitio Cotrol Actio I a two poitio cotrol actio ytem, the actuatig elemet ha oly two poitio which are geerally o ad off. Geerally thee are electric device. Thee are widely ued are they are imple ad iexpeive. The output of the cotroller i give by Eq.. u t U U e t e t Where, U ad U are cotat U= -U or zero The block diagram of o-off cotroller i how i Fig. Referece Error Detector Actuatig Error igal et Cotroller Output ut Iput rt Feed back igal bt

36 Proportioal Cotrol Actio The proportioal cotroller i eetially a amplifier with a adjutable gai. For a cotroller with proportioal cotrol actio the relatiohip betwee output of the cotroller ut ad the actuatig error igal et i u t e t p... Where, p i the proportioal gai. Or i Laplace traformed quatitie U E p.. 3 Whatever the actual mechaim may be the proportioal cotroller i eetially a amplifier with a adjutable gai. The block diagram of proportioal cotroller i how i Fig.3. Referece Iput rt Error Detector Actuatig Error igal et p Cotroller Output ut Feed back igal bt Figure 7 Block diagram of a proportioal cotroller Itegral Cotrol Actio The value of the cotroller output ut i chaged at a rate proportioal to the actuatig error igal et give by Eq.4 du t dt e t i or u t t i 0 e t dt..4 Where, i i a adjutable cotat. The trafer fuctio of itegral cotroller i U E i.5

37 If the value of et i doubled, the ut varie twice a fat. For zero actuatig error, the value of ut remai tatioary. The itegral cotrol actio i alo called reet cotrol. Fig.4 how the block diagram of the itegral cotroller. Referece Error Detector Actuatig Error igal et i Cotroller Output ut Iput rt Feed back igal bt Figure 8 Block diagram of a itegral cotroller Proportioal-plu-Itegral Cotrol Actio The cotrol actio of a proportioal-plu-itegral cotroller i defied by The trafer fuctio of the cotroller i t p u t e t e t dt..6 p T 0 i U E p T i...7 Where, p i the proportioal gai, T i i the itegral time which are adjutable. The itegral time adjut the itegral cotrol actio, while chage i proportioal gai affect both the proportioal ad itegral actio. The ivere of the itegral time i called reet rate. The reet rate i the umber of time per miute that a proportioal part of the cotrol actio i duplicated. Fig.5 how the block diagram of the proportioal-plu-itegral cotroller. For a actuatig error of uit tep iput, the cotroller output i how i Fig.6.

38 Referece Iput rt Error Detector Actuatig Cotroller Error igal et p T i output ut T i Feed back igal bt Figure 9 Block diagram of a proportioal-plu-itegral cotrol ytem ut P-I cotrol actio p p proportioal oly 0 T i t Figure 0 Repoe of PI cotroller to uit actuatig error igal Proportioal-plu-Derivative Cotrol Actio The cotrol actio of proportioal-plu-derivative cotroller i defied by u t e t p p T d de t dt..8 The trafer fuctio i U E p T d..9 Where, p i the proportioal gai ad T d i a derivative time cotat. Both, p ad T d are adjutable. The derivative cotrol actio i alo called rate cotrol. I rate cotroller the output i proportioal to the rate of chage of actuatig error igal. The derivative time T d i the time iterval by which the rate actio advace the effect of the proportioal cotrol actio. The derivative cotroller i aticipatory i ature ad

39 amplifie the oie effect. Fig.7 how the block diagram of the proportioal-plu-derivative. For a actuatig error of uit ramp iput, the cotroller output i how i Fig.8 referece iput rt error detector actuatig error igal et P Cotroller output ut T D S feed back igal bt Figure Block diagram of a proportioal-plu-derivative cotroller ut P-D cotrol actio proportioal oly T d 0 t Figure Repoe of PD cotroller to uit actuatig error igal Proportioal-plu-Itegral-plu-Derivative Cotrol Actio It i a combiatio of proportioal cotrol actio, itegral cotrol actio ad derivative cotrol actio. The equatio of the cotroller i u t e t p T p i t o e t dt p T d de t dt..0 or the trafer fuctio i U E p T i T d..

40 Where, p i the proportioal gai, T i i the itegral time, ad T d i the derivative time. The block diagram of PID cotroller i how i Fig.9. For a actuatig error of uit ramp iput, the cotroller output i how i Fig.0. referece iput rt error detector actuatig error igal et p T i T d T T i d Cotroller output ut feed back igal bt Figure 3 Block diagram of a proportioal-plu-itegral-plu-derivative cotroller ut PID cotrol actio PD cotrol actio proportioal oly 0 t Figure 4 Repoe of PID cotroller to uit actuatig error igal. Coider a ope loop trafer fuctio GS of ecod order ytem with uity feedback ytem. Expre dampig ratio of the cloed loop ytem i term of a, w, ad δ, where a i a cotat, w i the atural frequecy of ocillatio ad δ i the dampig ratio of ope loop ytem.ov

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42 UNIT III FREQUENCY DOMAIN ANALAYSIS PART A. What i frequecy repoe aalyi? May The Frequecy repoe i the teady tate repoe output of a ytem whe the iput to the ytem i a iuoidal igal. The frequecy repoe of a ytem i ormally obtaied by varyig the frequecy of the iput igal by keepig the magitude of the iput igal at a cotat value.. Defie gai cro over frequecy? Nov The phae cro over frequecy i the frequecy at which the magitude of the ope loop trafer fuctio i uity. 3. Defie gai cro over frequecy? Nov The phae cro over frequecy i the frequecy at which the phae of the ope loop trafer fuctio i Defie Phae Margi? Nov 0 The phae margi, γ i the amout of additioal phae lag at the gai cro over frequecy, W gc required to brig the ytem to the verge of itability. It i give by 80 gc 5. Defie Gai Margi? Nov 0 The gai margi, g i defied a the reciprocal of the magitude of ope loop trafer fuctio at phae cro over frequecy, W pc Gai Margi g = G j at pc 6. How do you calculate the gai margi from the polar plot? May 0 Gai Margi GM: o The gai margi i the reciprocal of magitude at the frequecy at which the o phae agle i -80. I term of db 0 GM i db 0log0 0log 0 G jwc 0log 0 x G jwc

43 7. How do you calculate the gai margi from the polar plot? May 0 Phae margi i that amout of additioal phae lag at the gai croover frequecy required to brig the ytem to the verge of itability margially tabile Where if if 0 Φ =80 +Φ m Φ= Gjω g Φ >0 => +PM Stable Sytem m Φ <0 => -PM Utable Sytem m 8. How do you fid the tability of the ytem by uig polar plot? May 0 Stable: If critical poit -+j0 i withi the plot a how, Both GM & PM are +ve Utable: If critical poit -+j0 i outide the plot a how, Both GM & PM are -ve Margially Stable Sytem: If critical poit -+j0 i o the plot a how, Both GM & PM are ZERO

44 9. What i cut off rate? Nov 09 The lop of the log magitude curve ear the cut off frequecy i called cut off rate. 0. Coider the firt order ytem GS = /+ST.Draw it polar plot. Nov 09 PART B. Defie reoat peak ad reoat frequecy. May 09 Peak repoe M p : The peak repoe M p i defied a the maximum value of M that i give i Eq..4. I geeral, the magitude of M p give a idicatio of the relative tability of a feed back cotrol ytem. Normally, a large M p correpod to a large peak overhoot i the tep repoe. For mot deig problem it i geerally accepted that a optimum value M p of hould be omewhere betwee. &.5. Reoat frequecy p : The reoat frequecy M p p i defied a the frequecy at which the peak reoace

45 . Sketch bode plot for the followig trafer fuctio ad determie the ytem gai for the gai cro over frequecy to be 5rad/ec GS = / Solutio: The iuoidal trafer fuctio Gj ω i obtaied by replacig S by j ω i the give S- domai trafer fuctio Gj ω=j ω /+0. j ω+0.00 j ω Let =, Gj ω= j ω /+0. j ω+0.00 j ω MAGNITUDE PLOT The corer frequecie are ω c =/0.=5rad/ec ad ω c =/0.0=50rad/ec The variou value of Gj ω are below tabulatio, i the icreaig order of their corer frequecie. alo lope cotributed by each term ad the chage i lope at the corer frequecy TREM CORNER FREQUENCY SLOPE IN db/dec CHANGE IN SLOPE db/dec jω /+j0. ω c =/0.= =0 ω c =/0.0= Chooe a low frequecy ω l uch that ω l< ω c& Chooe a high frequecy ω h uch that ω h> ω c. Let ω l= 0.5 rad/ec ad ω h= 00 rad/ec Let A= Gjω i db Let u calculate A at ω l,, ω c, ω c, ω h At ω= ω l, A=0 log jω =0 logω =0 log0. 5 = -db

46 At ω= ω c, A=0 log jω =0log5 =8db At ω= ω c, A=[lope from ω c to ω c log ω c /ω c ]+A at ω= ω c =0log50/5+8=48db At ω= ω h, A=[lope from ω c to ω h log ω h /ω c ]+A at ω= ω c =0 log 00/50+48=48db Let poit a,b,c,d be the poit correpodig to frequecie ω l,, ω c, ω c, ω h repectively o the magitude plot. PHASE PLOT The phae agle Gjω a a fuctio of ω i give by Φ= Gjω= 80 -ta - 0.ω-ta ω The phae agle of Gjω are calculated for variou value of ω ω rad/ec Ta - 0. ω deg Ta ω deg Φ= Gjω CALCULATION OF Give that the gai croover frequecy i 5rad/ec At ω=5,gai= 8db

47 At ω=5,frequecy the db gai hould be zero 0log =-8db Log =-8/0 =0-8/0 = NOTE: The frequecy ω rad/ec i a corer frequecy.hece i the exact plot the db gai at ω=5rad/ec will be 3db le tha the approximate plot.therefore for exact plot the 0log=-5 Log=-5/0=>=0-5//0 = Cotruct the polar give trafer fuctio Example Problem : Cotruct the polar plot for the critically damped ytem defied by : Solutio : Limitig coditio : i ii iii 0: : : Tj Tj Tj i.e. Tj Tj T j j 0 0 e j j80 o Im Re w Tjw j0 -j0.44 -j0.64 -j0.6 -j0.5 -j0.8 -j0.6 -j0.06 -j0.0

48 4. Draw the bode plot give trafer fuctio.

49 5. For the followig T.F draw the Bode plot ad obtai Gai cro over frequecy wgc,phae cro over frequecy, Gai Margi ad Phae Margi. G = 0 / [ +3 +4] The iuoidal T.F of G i obtaied by replacig by jw i the give T.F Gjw = 0 / [jw +j3w +j4w] Corer frequecie: wc= /4 = 0.5 rad /ec ; wc = /3 = 0.33 rad /ec Chooe a lower corer frequecy ad a higher Corer frequecy wl= 0.05 rad/ec ; wh = 3.3 rad / ec Calculatio of Gai A MAGNITUDE PLOT wl ; A= 0 log [ 0 / 0.05 ] = db wc ; A = [Slope from wl to wc x log wc / wl ] + Gai A@wl = - 0 log [ 0.5 / 0.05 ] = db wc ; A = [Slope from wc to wc x log wc / wc ] + Gai A@ wc = - 40 log [ 0.33 / 0.5 ] + 38 = 33 db Awh ; A = [Slope from wc to wh x log wh / wc ] + Gai wc = - 60 log [ 3.3 / 0.33 ] + 33 = - 7 db

50 Calculatio of Gai cro over frequecy The frequecy at which the db magitude i Zero wgc =. rad / ec Calculatio of Phae cro over frequecy The frequecy at which the Phae of the ytem i - 80o wpc = 0.3 rad / ec Gai Margi The gai margi i db i give by the egative of db magitude of Gjw at phae cro over frequecy

51 UNIT IV STABILITY ANALAYSIS PART A. What i root locu? May, 09 The root of the cloed-loop characteritic equatio defie the ytem characteritic repoe Their locatio i the complex -plae lead to predictio of the characteritic of the time domai repoe i term of:dampig ratio, atural frequecy, w dampig cotat, firt-order mode Coider how thee root chage a the loop gai i varied from 0 to. State Nyquit tability Criterio. May If the ope loop ytem i utable, the for the cloed loop ytem to be table, the umber of pole of G H lyig i PHP ad umber of ecirclemet hould be i the couter clock wie directio i the G Plae. 3. What i the eceary coditio for tability? May A ytem i bouded iput bouded output table if all the root of characteritic equatio lie i the left half of the complex S plae. 4. What i characteritic equatio? May Coider a th-order ytem whoe the characteritic equatio which i alo the deomiator of the trafer fuctio i 5. Defie tability. Nov A liear time ivariat ytem i aid to be table if followig coditio are atified.. Whe ytem i excited by a bouded iput, output i alo bouded & cotrollable.. I the abece of iput, output mut ted to zero irrepective of the iitial coditio.

52 6. How the root of characteritic are related to tability? 7. What do you mea by domiat pole? Nov 0 8. What are break away poit? Nov 0, May 09 To obtai the breakaway poit of the locu, differetiate the characteritic dk polyomial with repect to S ad put 0. Solve foe S which give the breakaway poit. d There are more tha oe break away poit ad they are real ad complex cojugate alo.

53 9. How will you fid the root locu o real axi? O a give ectio of the real axi, root axi i foud i the ectio oly if the total umber of pole ad zero of GH to the right of the ectio i odd if the feedback i Ve ad i +Ve, root locu i foud o the real axi where eve umber of pole ad zero occurred to the right of the cocered poit. 0. What i abolute tability ad coditioal tability of a ytem with repect to a parameter? Nov 09. Uig Routh criterio determie the tability of the ytem whoe characteritic equatio i May Determie the locatio of root with repect to = - give that F = Sol: hift the origi with repect to = - = = 0 S = 0 S S S S S 0 5 Two ig chage, there are two root to the right of = - & remaiig are to the left of the lie = -. Hece the ytem i utable.

54 = 0 Fid the umber of root of thi equatio with poitive real part, zero real part & egative real part Sol: S S S S da A = 3S 4 48 = 0 = 3 d S S S S S S S 0-48 Lim Therefore Oe ig chage & ytem i utable. Thu there i oe root i R.H.S of the plae i.e. with poitive real part. Now olve A = 0 for the domiat root A = =0 Put S = Y 3Y = 48 Y =6, Y = 6 = 4

55 S = + 4 S = -4 S = S = j So S = j are the two part o imagiary axi i.e. with zero real part. Root i R.H.S. idicated by a ig chage i S = a obtaied by olvig A = 0. Total there are 6 root a = 6. Root with Poitive real part = Root with zero real part = Root with egative real part = 6 = 3 3. Explai the tep by tep procedure rule for cotructig root locu The root locu i ymmetrical about real axi. The root of the characteritic equatio are either real or complex cojugate or combiatio of both. Therefore their locu mut be ymmetrical about the real axi. A icreae from zero to ifiity, each brach of the root locu origiate from a ope loop pole o. with = 0 ad termiate either o a ope loop zero m o. with = alog the aymptote or o ifiity zero at. The umber of brache termiatig o ifiity i equal to m. 3 Determie the root locu o the real axi. Root loci o the real axi are determied by ope loop pole ad zero lyig o it. I cotructig the root loci o the real axi chooe a tet poit o it. If the total umber of real pole ad real zero to the right of thi poit i odd, the the poit lie o root locu. The complex cojugate pole ad zero of the ope loop trafer fuctio have o effect o the locatio of the root loci o the real axi.

56 4 Determie the aymptote of root loci. The root loci for very large value of mut be aymptotic to traight lie whoe agle are give by Agle of aymptote A 80 q m ; q 0,,, m - 5 All the aymptote iterect o the real axi. It i deoted by a, give by σ a um of pole um of zero m p p p z z m z m 3 6 Fid breakaway ad break-i poit. The breakaway ad break-i poit either lie o the real axi or occur i complex cojugate pair. O real axi, breakaway poit exit betwee two adjacet pole ad break-i i poit exit betwee two adjacet d zero. To calculate thi polyomial 0 mut be olved. The reultig root are d the breakaway / break-i poit. The characteritic equatio give by Eq.7, ca be rearraged a where, B A p p z B A 0 z p z ad m 4 The breakaway ad break-i poit are give by d d d d A B A d d B 0 5 Note that the breakaway poit ad break-i poit mut be the root of Eq.5, but ot all root of Eq.5 are breakaway or break-i poit. If the root i ot o

57 the root locu portio of the real axi, the thi root either correpod to breakaway or break-i poit. If the root of Eq.5 are complex cojugate pair, to acertai that they lie o root loci, check the correpodig value. If i poitive, the root i a breakaway or break-i poit. 7 Determie the agle of departure of the root locu from a complex pole Agle of departure from a complex p 80 um of agle of vector to a complex pole i quetio from other pole um of agle of vector to a complex pole i quetio from other zero 6 8 Determie the agle of arrival of the root locu at a complex zero Agle of arrival at complex zero 80 um of agle of vector to a complex zero i quetio from other zero um of agle of vector to a complex zero i quetio from other pole 7 9 Fid the poit where the root loci may cro the imagiary axi. The poit where the root loci iterect the j axi ca be foud by a ue of Routh tability criterio or b lettig = j i the characteritic equatio, equatig both the real part ad imagiary part to zero, ad olvig for ad. The value of thu foud give the frequecie at which root loci cro the imagiary axi. The correpodig value i the gai at each croig frequecy. 0 The value of correpodig to ay poit o a root locu ca be obtaied uig the magitude coditio, or product of legth betwee poit to pole product of legth betwee poit to zero 8

58 PHASE MARGIN AND GAIN MARGIN OF ROOT LOCUS Gai Margi It i a factor by which the deig value of the gai ca be multiplied before the cloed loop ytem become utable. Gai Margi Value of at imagiary Deig value of cro over 9 The Phae Margi Fid the poit j o the imagiary axi for which G j H j for the deig value of i.e. B j /A j. deig The phae margi i φ 80 argg jω Hjω 0 4. Sketch the root locu of a uity egative feedback ytem whoe forward path trafer fuctio i G. Solutio: Root locu i ymmetrical about real axi. There are o ope loop zerom = 0. Ope loop pole i at = 0 =. Oe brach of root locu tart from the ope loop pole whe = 0 ad goe to aymptotically whe. 3 Root locu lie o the etire egative real axi a there i oe pole toward right of ay poit o the egative real axi. 4 The aymptote agle i A = 80 q, m q m 0. Agle of aymptote i A = Cetroid of the aymptote i σ A um of pole um of m zero

59 6 The root locu doe ot brach. Hece, there i o eed to calculate the break poit. 7 The root locu depart at a agle of -80 from the ope loop pole at = 0. 8 The root locu doe ot cro the imagiary axi. Hece there i o imagiary axi cro over. The root locu plot i how i Fig. Figure 5 Root locu plot of / Commet o tability: The ytem i table for all the value of > 0. Th ytem i over damped. 5. The ope loop trafer fuctio i G. Sketch the root locu plot Solutio: Root locu i ymmetrical about real axi. There i oe ope loop zero at =-.0m=. There are two ope loop pole at =-, -=. Two brache of root loci tart from the ope loop pole whe = 0. Oe brach goe to ope loop zero at =-.0 whe ad other goe to ope loop zero aymptotically whe. 3 Root locu lie o egative real axi for -.0 a the umber of ope loop pole plu umber of ope loop zero to the right of =-0. are odd i umber. 4 The aymptote agle i A = 80 q, m q m 0. Agle of aymptote i A = 80.

60 5 Cetroid of the aymptote i σ A um of pole um of m 0.0 zero 6 The root locu ha break poit. d Break poit i give by d 0, 0; 3, 4 0 The root loci brake out at the ope loop pole at =-, whe =0 ad break i oto the real axi at =-3, whe =4. Oe brach goe to ope loop zero at =- ad other goe to alog the aymptotically. 7 The brache of the root locu at =-, - break at =0 ad are tagetial to a lie =-+j0 hece depart at The locu arrive at ope loop zero at The root locu doe ot cro the imagiary axi, hece there i o eed to fid the imagiary axi cro over. The root locu plot i how i Fig.. Figure 6 Root locu plot of +/+

61 Commet o tability: Sytem i table for all value of > 0. The ytem i over damped for > 4. It i critically damped at = 0, The ope loop trafer fuctio i G 4. Sketch the root locu. Solutio Root locu i ymmetrical about real axi. There are i oe ope loop zero at =-4m=. There are two ope loop pole at =0, -=. Two brache of root loci tart from the ope loop pole whe = 0. Oe brach goe to ope loop zero whe ad other goe to ifiity aymptotically whe. 3 Etire egative real axi except the egmet betwee =-4 to =- lie o the root locu. 80 q 4 The aymptote agle i A =, q 0,, m 0. m Agle of aymptote are A = Cetroid of the aymptote i σ A um of pole um of m 4.0 zero 6 The brake poit are give by d/d =0. d d 4.7, 6.88, ; Agle of departure from ope loop pole at =0 i 80. Agle of departure from pole at =-.0 i 0. 8 The agle of arrival at ope loop zero at =-4 i 80 9 The root locu doe ot cro the imagiary axi. Hece there i o imagiary cro over. The root locu plot i how i fig.3.

62 Figure 3 Root locu plot of +4/+ Commet o tability: Sytem i table for all value of. 0 > > : > over damped = : = critically damped > >.7 : < uder damped =.7 : = critically damped >.7 : > over damped. 7. The ope loop trafer fuctio i 0. G. Sketch the root locu. 3.6 Solutio: Root locu i ymmetrical about real axi. There i oe ope loop zero at = -0.m=. There are three ope loop pole at = 0, 0, -3.6=3. Three brache of root loci tart from the three ope loop pole whe = 0 ad oe brach goe to ope loop zero at = -0. whe ad other two go to aymptotically whe.

63 3 Root locu lie o egative real axi betwee -3.6 to -0. a the umber of ope loop pole plu ope zero to the right of ay poit o the real axi i thi rage i odd. 80 q 4 The aymptote agle i A =, q m 0, m Agle of aymptote are A = 90, Cetroid of the aymptote i σ A um of pole um of m.7 zero 6 The root locu doe brach out, which are give by d/d =0. d d , , ad 0,.55, 3.66 repectively. The root loci brakeout at the ope loop pole at = 0, whe =0 ad breaki oto the real axi at =-0.43, whe =.55 Oe brach goe to ope loop zero at =-0. ad other goe breakout with the aother locu tartig from ope loop ploe at = The break poit i at =-.67 with =3.66. The loci go to ifiity i the complex plae with cotat real part = The brache of the root locu at =0,0 break at =0 ad are tagetial to imagiary axi or depart at 90. The locu depart from ope loop pole at =-3.6 at 0. 8 The locu arrive at ope loop zero at =-0. at The root locu doe ot cro the imagiary axi, hece there i o imagiary axi cro over. The root locu plot i how i Fig.4.

64 Commet o tability: Sytem i table for all value of. Sytem i critically damped at =.55, It i uder damped for.55 > > 0 ad >3.66. It i over damped for 3.66 > > The ope loop trafer fuctio i G 6. Sketch the root locu. 5 Solutio: Root locu i ymmetrical about real axi. There are o ope loop zero m=0. There are three ope loop pole at =-0, -3 j4=3. Three brache of root loci tart from the ope loop pole whe = 0 ad all the three brache go aymptotically whe. 3 Etire egative real axi lie o the root locu a there i a igle pole at =0 o the real axi. 80 q 4 The aymptote agle i A =, q 0,, m 0,,. m Agle of aymptote are A = 60, 80, Cetroid of the aymptote i σ A um of pole um of m.0 zero 6 The brake poit are give by d/d =0. d d,, j.087ad j For a poit to be break poit, the correpodig value of i a real umber greater tha or equal to zero. Hece, S, are ot break poit. 7 Agle of departure from the ope loop pole at =0 i 80. Agle of departure from complex pole = -3+j4 i p 80 um of the agle of vector to a complex pole i quetio from other pole um of the agle of vector to a complex pole iquetio from zero

65 p ta Similarly, Agle of departure from complex pole = -3-j4 i φ p or The root locu doe cro the imagiary axi. The cro over poit ad the gai at the cro over ca be obtaied by Routh criterio 3 The characteritic equatio i The Routh array i For the ytem to be table < 50. At =50 the auxillary equatio i 6 +50=0. = ±j5. or ubtitute = j i the characteritic equatio. Equate real ad imagiary part to zero. Solve for ad. 3 jω 6ω 6 jω 5 jω ω jω ω 0, j5 0,50 The plot of root locu i how i Fig

66 Commet o tability: Sytem i table for all value of 50 > > 0. At =50, it ha utaied ocillatio of 5rad/ec. The ytem i utable for > Sketch the root locu of a uity egative feedback ytem whoe forward path trafer fuctio i GH. Commet o the 3 j 3 j tability of the ytem. Root locu i ymmetrical about real axi. There i oe ope loop zero at = - m =. There are three ope loop pole at = -, -3 ± j =3. All the three brache of root locu tart from the ope loop pole whe = 0. Oe locu tartig from = - goe to zero at = - whe, ad other two brache go to aymptotically zero at whe. 3 Root locu lie o the egative real axi i the rage =- to = - a there i oe pole to the right of ay poit o the real axi i thi rage. 4 The aymptote agle i A = 80 q, m q m 0,. Agle of aymptote i A = 90, Cetroid of the aymptote i σ A um of pole 3 3 um of zero m.5 6 The root locu doe ot brach. Hece, there i o eed to calculate break poit. 7 The agle of departure at real pole at =- i 80. The agle of departure at the complex pole at =-3+j i p 80 um of the agle of um of the agle of vector to a complex pole i quetio from other pole vector to a complex pole iquetio from zero

67 θ θ p ta - ata-, ta or 35 or 53.43, θ ta The agle of departure at the complex pole at =-3-j i p The root locu doe ot cro the imagiary axi. Hece there i o imagiary axi cro over. The root locu plot i how i Fig. Figure Root locu plot of +/++3+j+3-j Commet o tability: The ytem i table for all the value of > 0.

68 0. The ope loop trafer fuctio i GH Sketch the root locu plot. Commet o the tability of the ytem. Solutio: Root locu i ymmetrical about real axi. There are o ope loop zero m=0. There are four ope loop pole =4 at =0, -0.5, -0.3 ± j Four brache of root loci tart from the four ope loop pole whe = 0 ad go to ope loop zero at ifiity aymptotically whe. 3 Root locu lie o egative real axi betwee = 0 to = -0.5 a there i oe pole to the right of ay poit o the real axi i thi rage. 80 q 4 The aymptote agle i A =, q m 0,,,3. m Agle of aymptote i A = 45, 35, 5, ±35. 5 Cetroid of the aymptote i σ A um of pole um of m 0.3 zero 0.75 The value of at =-0.75 i The root locu ha break poit. = = Break poit are give by d/d = 0 d d = , j.89

69 There i oly oe break poit at Value of at = i The agle of departure at real pole at =0 i 80 ad at =-0.5 i 0. The agle of departure at the complex pole at = j3.48 i -9.8 p 80 um of the agle of vector to a complex pole i quetio from other pole um of the agle of vector to a complex pole iquetio from zero 3.48 θ ta 84.6 or ta 86.4, θ ta p The agle of departure at the complex pole at = j3.48 i 9.8 p The root locu doe cro the imagiary axi, The cro over frequecy ad gai i obtaied from Routh criterio. The characteritic equatio i =0 or =0 The Routh array i

70 The ytem i table if 0 < < 6.3 The auxiliary equatio at 6.3 i = 0 which give = ± j.3 at imagiary axi croover. The root locu plot i how i Fig.. Figure 7 Root locu plot of / Commet o tability: Sytem i table for all value of 6.3 > > 0. The ytem ha utaied ocillatio at =.3 rad/ec at =6.3. The ytem i utable for > The ope loop trafer fuctio i locu. G. Sketch the root Solutio: Root locu i ymmetrical about real axi. There are o ope loop zero m=0. There are three ope loop pole =3 at = -0, -4, - j4. Three brache of root loci tart from the three ope loop pole whe = 0 ad to ifiity aymptotically whe. 3 Root locu lie o egative real axi betwee = 0 to = -4.0 a there i oe pole to the right of ay poit o the real axi i thi rage.

71 80 q 4 The aymptote agle i A =, q m 0,,, 3 m Agle of aymptote are A = 45, 35, 5, Cetroid of the aymptote i σ A um of.0 pole um of zero m The root locu doe brach out, which are give by d/d =0. d Break poit i give by d , ; j.45, The root loci brakeout at the ope loop pole at = -.0, whe = 64 ad breaki ad breakout at =-+j.45, whe =00 7 The agle of departure at real pole at =0 i 80 ad at =-4 i 0. The agle of departure at the complex pole at = - + j4 i -90. p 80 um of the agle of vector to a complex pole i quetio from other pole um of the agle of vector to a complex pole iquetio from zero θ θ θ p 4 ta 63.4 or ata4, ta 63.4, θ The agle of departure at the complex pole at = - j4 i 90 8 ta p

72 8 The root locu doe cro the imagiary axi, The cro over poit ad gai at cro over i obtaied by either Routh array or ubtitute = j i the characteritic equatio ad olve for ad gai by equatig the real ad imagiary part to zero. Routh array 4 3 The characteritic equatio i The Routh array i For the ytem to be table > 0 ad > 0. The imagiary croover i give by 080-8=0 or = 60. At = 60, the auxiliary equatio i = 0. The imagiary cro over occur at = j 0. or The root locu plot i how i Fig.3. 4 put ω jω ω Equate real ad imagiary 8ω ω 36ω jω 80ω 36 jω j jω ω 8ω 3 0, ω part to zero j 60 jω 0; 0 j 0 0

73 Figure 8 Root locu plot of / Commet o tability: For 60 > > 0 ytem i table = 60 ytem ha taied ocillatio of 0 rad/ec. > 60 ytem i utable.. Give that GS HS = /SS+ S+0 = /SxS/+x0S/0+ = 0.05 /S+0.5S+0.S The ope loop trafer fuctio ha a pole at origi. Hece chooe the Nyquit cotour o S-plae ecloig the etire right half plae except the origi The Nyquit cotour ha four ectio C, C, C3, C4 the mappig of each ectio i performed eparately ad the overall Nyquit plot i obtaied by combiig the idividual ectio Mappig of ectio C I ectio c,ω varie from 0 to. The mappig of ectio C i give by the locu GjωHjω a ω i varied from 0 to. Thi polar plot of GjωHjω GSHS= 0.05/S+0.5S+0.S Let S=jω GjωHjω S= 0.05/ jωs+0.5 jω +0. jω = 0.05/-0.6ω +jω-0.05ω whe the locu of Gjω Hjω croe real axi the imagiary term will be zero ad the correpodig frequecy i the phae croover frequecy

74 ω PC. ω=+ ω=0 C C4 R σ C3 ω=- At ω=ω pc ω pc ω pc = ω pc =0 At ω=ω pc = 4.47 rad/ec GjωHjω = 0.05/-0.6ω = The ope loop ytem i type - ad third order ytem. Alo it i a miimum phae ytem with all pole. Hece the polar plot of GjωHjω tart at -90 axi at ifiity croe real axi at ad ed at origi i ecod quadrat. The ectio C ad it mappig how below diagram A ad B ω= jv ω=0 ω= σ Sectio c i -plae Mappig of ectio c i GSHS

75 Mappig of ectio C The mappig of ectio C from S-plae to GSHS Plae i obtaied by lettig Lt S=R Re jθ i GSHS ad varyig θ from + / to - /. Sice S Re jθ ad R, the GSHS ca be approximately a how below [ ie +ST ST ] GSHS = 0.05/S+0.5S+0.S 0.05/S x 0.5S x 0.S = /S 3 Lt Let S=R Re jθ GSHS S=R Re jθ /S 3 = Lt = 0e -j3θ whe θ=+ /, GSHS = 0e -j3 / a whe θ=- /, GSHS = 0e +j3 / b jω C S=Plae jv GSHS plae R u Sectio c i S-Plae From the equatio a ad b we ca tudy that ectio c i S-plae figa i mapped a circular are of zero radiu aroud origi i GSHS plae with argumet phae varyig from -3 / ad +3 / a how i fig b Mappig of ectio C3