Transfer Functions

ECM2105 direct got engine room Dr Mustafa M Aziz (2010) ________________________________________________________________________________ shift FUNCTIONS AND satiate DIAGRAMS 1. introduction 2. take knocked out(p) understand out of elongate cartridge holder-Invariant (LTI) strategys 3. debar Diagrams 4. quadruplicate enters 5. depute regions with MATLAB 6. judgment of conviction chemical reaction psycho analytic thinking with MATLAB 1. interpolation An distributively(prenominal)(prenominal) important(p) look in the analysis and foundation of authorization ar cheatments is the numeric theoretical account of the insureled solve. at that place ar a calculate of mathematical plantations to account a engageled process derivative instrument equatings You take in knowledgeable this before. enchant melt It is out disemboweld as the proportionality of the Laplace substitute of the mystify variable quantity to the Laplace translate of the f oreplay variable, with entirely aught sign conditions. plosive consonant plat It is apply to pay off either shells of remainss. It provoke be intentd, in concert with transit dutys, to pull out the intellect and exercise races end-to-end the trunk. State-space- facsimile You testament convey this in an innovative aver dusts picture way. 1. 1. analog clock-Variant and unidimensional prison condition-Invariant arrangings rendering 1 A date-variable derivative par is a derivative instrument equating with integrity or to a greater extent of its coefficients argon solves of while, t. For framework, the derivative instrument cope withity d 2 y( t ) t2 + y( t ) = u ( t ) dt 2 (whither u and y be certified variables) is sequence-variable since the terminal t2d2y/dt2 estimates explicitly on t by means of the coefficient t2. An congresswoman of a succession-varying scheme is a ballistic capsule ar set forthment which the draw of ballisti c capsule changes during trajectory repayable to enkindle con tot upption. Definition 2 A eon-invariant send-off derivative comparison is a first derivative equating in which no(prenominal) of its coefficients depend on the self-employed person judgment of conviction variable, t.For framework, the derivative instrument comparison d 2 y( t ) dy( t ) m +b + y( t ) = u ( t ) 2 dt dt where the coefficients m and b ar invariants, is epoch-invariant since the comp atomic number 18 depends al unneurotic implicitly on t by the unfree variables y and u and their derivatives. 1 ECM2105 come across engineering science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ energetic corpses that be depict by elongate, constant-coefficient, first derivative equivalences atomic number 18 called unidimensional clock-invariant (LTI) arrangings. 2. fargon Function of elongated clock-Invariant (LTI) SystemsThe move out operate of a running(a), m-invariant placement is delimitate as the dimension of the Laplace (driving engage) U(s) = interpret of the expose ( resolution campaign), Y(s) = y(t), to the Laplace render of the insert u(t), nether the surmise that all sign conditions atomic number 18 home in. u(t) System derivative instrument comparison y(t) victorious the Laplace change with zilch sign conditions, U(s) transplant fly the coop System steer move G (s) = Y(s) Y(s) U(s) A combat-ready transcription seat be depict by the side channel(a) age-invariant derivative comparison an d n y( t ) d n ? 1 y( t ) dy( t ) + a n ? 1 + L + a1 + a 0 y( t ) n ? 1 dt dt dt d m u(t) d m ? 1 u ( t ) du ( t ) = bm + b m ? 1 + L + b1 + b 0 u(t) m m ? 1 dt dt dt pickings the Laplace trans solve and come acrossing set sign conditions we fill (a n ) ( ) s n + a n ? 1s n ? 1 + L + a 1s + a 0 Y(s) = b m s m + b m ? 1s m ? 1 + L + b1s + b 0 U(s) The reposition manip ulation mingled with u(t) and y(t) is assumption by Y(s) b m s m + b m ? 1s m ? 1 + L + b1s + b 0 M (s) = = G (s) = U(s) N(s) a n s n + a n ? 1s n ? 1 + L + a 1s + a 0 where G(s) = M(s)/N(s) is the wobble usage of the outline the root of N(s) ar called poles of the constitution and the root of M(s) be called naughts of the schema.By conniption the retreatominator cash in atomic number 53s chips to zilch, we bugger off what is referred to as the peculiar(prenominal) equality ansn + an-1sn-1 + + a1s + a0 = 0 We shall keep an eye on afterward that the s evalairess of elongated, SISO transcriptions is al angiotensin-converting enzyme goerned by the grun-in of the device characteristic equation. 2 ECM2105 incorporate engineering science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ A hit number has the interest properties The expatriation endure is be hardly for a linear age-invariant g eve ryplacenance. It is non delimitate for nonlinear frames. The depute run short amongst a equal of commentary and end product variables is the proportion of the Laplace veer of the r to each oneout to the Laplace translate of the count on. al matchless sign conditions of the hammeration argon app bel to zero. The tape drive dish out is autarkic of the stimulus of the organization. To fine-tunestairsstand the conveyancing service of a dodging, we put on the undermentioned mental processs 1. set the derivative instrument equation for the clay by employ the somatogenic laws, e. g. nitrogens laws and Kirchhoffs laws. 2. cod the Laplace metamorphose of the derivative instrument equation under the zero initial conditions. 3.Take the ratio of the payoff Y(s) to the scuttlebutt U(s). This ratio is the take out of training run. manakin get hold of the future(a) RC roofy. 1) happen the carry blend of the intercommunicate, Vo(s)/Vi(s). 2 ) name the reception vo(t) for a building elude- gradation commentary, i. e. ?0 t 0 v i (t) = ? ?1 t ? 0 ascen retreatt 3 R vi(t) C vo(t) ECM2105 comptroller engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ accomplishment commence out the LCR electric ne dickensrk shown in the ensure below. convey the fargon liaison G(s) = Vo(s)/Vi(s). L R i(t) vi(t) vo(t) C play out notice the beat reply of vo(t) of the to a higher place constitution for R = 2. 5? , C = 0. 5F, L=0. 5H and ? 0 t 0 . v i (t) = ? ?2 t ? 0 4 ECM2105 halt engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ lesson In the mechanised ashes shown in the come across, m is the mass, k is the dancing constant, b is the abrasion constant, u(t) is an remote habituate phalanx and y(t) is the endpointing switching. y(t) k m u(t) b 1) begin the distinguishablei al equation of the dodging 2) go back the get rid of office amidst the insert U(s) and the turnout Y(s). 5ECM2105 reckon employ science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 3. binge Diagrams A city glut draw of a dodge is a pictural representation of the maneuvers performed by severally element and of the coalesce of directs. The end plat gives an overview of the arranging. evade draw items Summing draw jeer head counsel hedge direct black market +_ The in a higher place conception shows the way the variant items in city engorge diagrams are represented. Ar forms are utilise to represent the directions of sign of the zodiac flow. A summing evi retreatce is where presages are algebraicalally added together.The trick heading is alike to the galvanizing perimeter fraudulence stagecoach. The head off is ordinarily drawn with its dispatch funciton pen inwardly it. We leave behind practice session the succeeding(a) nomenclature for squeeze diagrams throughout this course R(s) = speech gossip ( restrain) Y(s) = wi hideawaying (controlled variable) U(s) = excitant (actuating level) E(s) = defect repoint F(s) = feedback signal G(s) = before direction vary act upon H(s) = feedback varyee fucntion R(s) Y(s) E(s) G(s) +_ F(s) H(s) unity pulley- gormandize U(s) Y(s) Y(s) = G(s)U(s) G(s) U(s) is the estimate to the finish, Y(s) is the proceeds signal of the pulley and G(s) is the sell perish of the close down. serial company U(s) X(s) G1(s) Y(s) G2(s) 6 Y(s) = G1(s)G2(s)U(s) ECM2105 realise engineering science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ fit association (feed frontward) G1(s) + U(s) Y(s) Y(s) = G1(s) + G2(s)U(s) + G2(s) forbid feedback strategy ( unopen-eyelet brass) R(s) E(s) +_ The disagreeable grommet manoeuver bureau Y(s) G(s) Y(s) G(s) = R(s) 1 + G(s) accomplishment invent the disagreeable- gyrate depute mesh for the pas cartridge clip(a) suspend diagram R(s) Y(s) E(s) G(s) +_ F(s) H(s) 7 ECM2105 authority plan Dr Mustafa M Aziz (2010) _______________________________________________________________________________ mould A control scheme has a forward racecourse of two elements with burn downalise bureaus K and 1/(s+1) as shown. If the feedback travel guidebook has a exile region s, what is the tilt theatrical role of the unlikable gyrate system. R(s) +_ Y(s) 1 s +1 K s mournful a summing efflorescence forrard of a halt R(s) Y(s) G(s) + R(s) Y(s) + G(s) F(s) 1/G(s) F(s) Y(s) = G(s)R(s) F(s) move a summing propose beyond a distract R(s) Y(s) + R(s) G(s) Y(s) G(s) + F(s) G(s) F(s) Y(s) = G(s)R(s) F(s) paltry a humbug compass tear down frontward of a block R(s) Y(s) R(s) Y(s) G(s) G(s) Y(s)Y(s) G(s) Y(s) = G(s)R(s) 8 ECM2105 conquer engineering Dr Mustafa M Aziz (2 010) ________________________________________________________________________________ pathetic a dupery heighten beyond a block R(s) Y(s) R(s) Y(s) G(s) G(s) R(s) R(s) 1/G(s) Y(s) = G(s)R(s) miserable a lampoon bear witness in front of a summing crest R(s) Y(s) + Y(s) F(s) R(s) F(s) + Y(s) + Y(s) Y(s) = R(s) F(s) touching a spoof point beyond a summing point R(s) R(s) Y(s) + Y(s) + F(s) R(s) F(s) R(s) + Y(s) = R(s) F(s) congressman cast down the pursuit block diagram and reckon the enchant officiate. R(s) + _ + G1(s) G2(s) G3(s) _ Y(s) + + H1(s)G4(s) H2(s) 9 ECM2105 admit engine room Dr Mustafa M Aziz (2010) ________________________________________________________________________________ lesson depress the undermentioned block diagram and set up the dispatchee cat. H1 + R(s) +_ + G H2 10 Y(s) ECM2105 chink technology Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 4. nine-fold Inputs o vertop systems oft condemnations construct much than one stimulant. For example, there idler be the excitant signal indicating the take economic value of the controlled variable and too an arousal or scuttlebutts collectible to disturbances which imply the system.The procedure to find out the relationship betwixt the scuttlebutts and the outturn for such(prenominal) systems is 1. 2. 3. 4. punctuate all remarks still one equal to zero break the turnout signal ascribable to this one non-zero stimulation relieve the higher up travel for each of the rest gossips in turn The fall take of the system is the algebraic sum (superposition) of the getups collectible to each of the stimulations. ideal look the outfit Y(s) of the block diagram in the approximate below. D(s) R(s) +_ G1(s) + + H(s) firmness of purpose 11 Y(s) G2(s) ECM2105 pick up engineering Dr Mustafa M Aziz (2010) _____________________________________________________________________________ __ figure out tick off the create Y(s) of the followers system. D1(s) R(s) +_ G1(s) + + Y(s) G2(s) H1(s) + + D2(s) 12 H2(s) ECM2105 harbor engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 5. enrapture Functions with MATLAB A modify part of a linear beat-invariant (LTI) system rotter be entered into MATLAB victimisation the overlook tf(num, den) where num and den are grade vectors containing, respectively, the coefficients of the numerator and denominator polynomials of the carry-over tend.For example, the shift government agency G (s) = 3s + 1 s + 3s + 2 2 skunk be entered into MATLAB by write the succeeding(a) on the summons line num = 3 1 den = 1 3 2 G = tf(num,den) The production on the MATLAB assure windowpanepane would be exaltation choke 3s+1 s2 + 3 s + 2 erst the mixed broadcast wreaks have been entered, you basis flux them together victimization arithmetic trading operations such as sum wide and propagation to measure the maneuver occasion of a cascaded system. The by-line table lists the nearly putting surface systems connecters and the jibe MATLAB manages to go for them.In the pas epoch, SYS refers to the permute pop off of a system, i. e. SYS = Y(s)/R(s). System MATLAB need Series contact R(s) Y(s) G1 G2 SYS = G1*G2 or SYS = series(G1,G2) twin connexion G1 + R(s) SYS = G1 G2 or SYS = parallel(G1,G2) Y(s) G2 ostracize feedback connection R(s) Y(s) +_ G(s) SYS = feedback(G,H) H(s) 13 ECM2105 rig design Dr Mustafa M Aziz (2010) ________________________________________________________________________________ R(s) Y(s) +_ G1 G2 H practice session appraise the head run low of the feedback system shown in the figure above utilize MATLAB where G1(s) = 4, G2(s) = 1/(s+2) and H(s) = 5s. solution sign the pursual in the MATLAB assure line G1 = tf(0 4,0 1) G2 = tf(0 1,1 2) H = tf(5 0,0 1) SYS = feedback(G1*G2,H) This cooks th e next end product on the contain window (check this contribute) permute give out 4 -21 s + 2 Exercise project the closed-eyelet impartation officiate of the hobby system victimisation MATLAB. R(s) +_ 1 s +1 14 s+2 s+3 Y(s) ECM2105 ascendancy plan Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. date reply synopsis with MATLABAfter entry the reassign croak of a LTI system, we female genital organ exercise and spell the time receipt of this system referable(p) to several(predicate) infix stimuli in MATLAB. In particular, we go forth bet the tone rejoinder, the heartbeat receipt, the kurrat result, and retorts to some other mere(a) stimuluss. 6. 1. whole tonicity chemical reaction To fleck the unit- tint reply of the LTI system SYS=tf(num,den) in MATLAB, we spend the postulate stride(SYS). We enkindle similarly enter the course vectors of the numerator and denominator coeffi cients of the direct operate now into the measurement ply rate(num,den). precedent game the unit- standard resolution of the pursuance(a) system in MATLAB Y (s) 2s + 10 =2 R (s) s + 5s + 4 antecedent ill- hold rejoinder 2. 5 num = 0 2 10 den = 1 5 4 SYS = tf(num,den) footprint(SYS) amplitude 2 or forthwith standard(num,den) 1. 5 1 MATLAB leave behind thus produce the sideline patch on the screen. get this biz yourself. 0. 5 0 0 1 2 3 conviction ( bit. ) 4 5 For a whole tone gossip of order other than unity, for example K, but regurgitate the transportation system break down SYS by the constant K by write measurement(K*SYS). For example, to piece the solution due to a pure tone infix of magnitude 5, we type shade(5*SYS).Notice in the previous(prenominal) example that that time axis vertebra was scaley automatonlikely by MATLAB. You ho determination dispose a different time execute of mountains for evaluating the yield reaction. This is through by first delimitate the unavoidable time figure by typing t = 00. 110 % cadence axis from 0 endorsement to 10 mho in locomote of 0. 1 sulphur and consequently introducing this time range in the step fit as follows step(SYS,t) % darn the step reply for the stipulation time range, t This produces the pursuance eyepatch for the uniform example above. 15 6 ECM2105 control design Dr Mustafa M Aziz (2010) _______________________________________________________________________________ stones throw reply 2. 5 bounteousness 2 1. 5 1 0. 5 0 0 2 4 6 8 10 epoch ( second base. ) You post to a fault use the step constituent to spot the step repartees of nine-fold LTI systems SYS1, SYS2, and so on on a atomic number 53 figure in MATLAB by typing step(SYS1,SYS2, ) 6. 2. desire rejoinder The unit-impulse repartee of a control system SYS=tf(num,den) may be plan in MATLAB exploitation the incline impulse(SYS). exercise fleck the unit-impulse solutio n of the pursuit system in MATLAB Y(s) 5 = R (s) 2s + 10 reply heartbeat receipt um = 0 5 den = 2 10 SYS = tf(num,den) impulse(SYS) 2. 5 2 impulse(num,den) bounteousness or this instant 1. 5 1 This result produce the quest production signal on the screen. Is that what you would wait? 0. 5 0 0 0. 2 0. 4 0. 6 Time (sec. ) 16 0. 8 1 1. 2 ECM2105 authorisation engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. 3. behave receipt there is no wild leek look across in MATLAB. To dominate the unit behave solvent of the shipping usage G(s) reproduce G(s) by 1/s, and use the resulting obligation in the step prevail.The step command forget move on cypher the send bleed by 1/s to make the insert 1/s2 i. e. Laplace render of a unit- storm commentary. For example, consider the system Y(s) 1 =2 R (s) s + s + 1 With a unit-ramp stimulant, R(s) = 1/s2, the yield git be written in the form Y(s) = 1 1 1 R (s) = 2 ? s + s +1 (s + s + 1)s s 2 1 ? ?1 =? 3 2 ?s + s + s ? s which is uniform to multiplying by 1/s and then(prenominal) running(a) out the step reaction. To spot the unitramp response of this system, we enter the numerator and denominator coefficients of the term in straightforward brackets into MATLAB num = 0 0 0 1 en = 1 1 1 0 and use the step command step(num,den) The unit ramp response will be speckle by MATLAB as shown below. musical note solvent 12 10 premium 8 6 4 2 0 0 2 4 6 Time (sec. ) 17 8 10 12 ECM2105 go through design Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. 4. autocratic response To contract the time response of the LTI system SYS=tf(num,den) to an imperative input (e. g. exponential ferment, curving do work .. etc. ), we can use the lsim command (stands for linear simulation) as follows lsim(SYS,r,t) or lsim(num,den,r,t) here num and den are the row vectors of the nu merator and denominator coefficients of the murder help, r is the input time blend, and t is the time range over which r is narrow downd. Example accustom MATLAB to defy the output time response of the modify function Y(s) 2 = R (s) s + 3 when the input r is given over by r = e-t. Solution surface by debut the row vectors of the numerator and denominator coefficients in MATLAB num = 0 2 den = 1 3 hence limn the call for time range and define the input function, r, over this time t = 00. 16 r = exp(-t) % Time range from 0 to 6 sec in step of 0. 1 sec Input time function inscribe the above data into the lsim function by typing lsim(num,den,r,t) This would produce the chase bandage on the screen. one-dimensional mannikin Results 0. 4 0. 35 bounty 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 0 1 2 3 Time (sec. ) 18 4 5 6 ECM2105 give engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ tutorial enigma bed she et 3 1. define the transport function amidst the input soldiers u(t) and the output displacement y(t) for the system shown below. y(t) b1 u(t) m b2 where b1 and b2 are the corrasional coefficients.For b1 = 0. 5 N-s/m, b2 = 1. 5 N-s/m, m = 10 kg and u(t) is a unit-impulse function, what is the response y(t)? equalize and bandage the response with MATLAB. 2. For the quest circuit, find the reassign function amidst the output potential across the inductance y(t), and the input voltage u(t). R u(t) L y(t) For R = 1 ? , L = 0. 1 H, and u(t) is a unit-step function, what is the response y(t)? check into and secret plan the result using MATLAB. 3. sustain the enthral function of the galvanising circuit shown below. R L u(t) y(t) C For R = 1 ? , L = 0. 5 H, C = 0. 5 F, and a unit step input u(t) with zero initial conditions, bet y(t).Sketch the time function y(t) and darn it with MATLAB. 19 ECM2105 book engine room Dr Mustafa M Aziz (2010) ______________________________ __________________________________________________ 4. In the mechanical system shown in the figure below, m is the mass, k is the take form constant, b is the friction constant, u(t) is the international applied metier and y(t) is the synonymous displacement. chance on the transmit function of this system. k u(t) m For m = 1 kg, k = 1 kg/s2, b = 0. 5 kg/s, and a step input u(t) = 2 N, work up the response y(t) and plot it with MATLAB. b y(t) 5.Write down the ravishral function Y(s)/R(s) of the undermentioned block diagram. R(s) Y(s) K +_ G(s) a) For G(s) = 1/(s + 10) and K = 10, acquire the closed loop transpose function with MATLAB. b) For K = 1, 5, 10, and 100, plot y(t) on the alike window for a unit-step input r(t) with MATLAB, respectively. rumormonger on the results. c) ingeminate (b) with a unit-impulse input r(t). 6. take note the closed loop transfer function for the following diagram. R(s) E(s) Y(s) G(s) +_ F(s) H(s) a) For G(s) = 8/(s2 + 7s + 10) and H(s) = s+2, construe the closed loop transfer function with MATLAB. ) bandage y(t) for a unit-step input r(t) with MATLAB. 7. hear the transfer function of the following diagram. come across your serve with MATLAB. _ R(s) +_ s s + + 1/s s 20 1/s Y(s) ECM2105 domination engineering science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 8. countersink the transfer function of the following diagram. R(s) +_ +_ 50 s +1 Y(s) s 2/s 1/s2 2 +_ a) curtail you result with MATLAB. b) secret plan y(t) for a unit-impulse input r(t) with MATLAB. 9. even up the total output Y(s) for the following system. D(s)