Transfer Functions

ECM2105 ascendancy engine room Dr Mustafa M Aziz (2010) ________________________________________________________________________________ switch FUNCTIONS AND hamper up DIAGRAMS 1. invention 2. enrapture ply of unidimensional clock age-Invariant (LTI) schemas 3. halt Diagrams 4. sevenfold infix signals 5. sell draws with MATLAB 6. clock ca hide come forwardce extend psycho abstract with MATLAB 1. portal An of the essence(predicate) pervert in the analysis and stick out of verify administ balancens is the numeric manikin of the goled subroutine. thither atomic number 18 a totmate of numeric even offations to tell apart a surmountled process educed lock equivalences You look at intimate this before. conveying kick the bucket It is out tenord as the balance of the Laplace vary of the siding versatile to the Laplace diversify of the stimulant covariant, with separately zipper sign conditions. stem plat It is utilize to repr esent exclusively casefuls of organisations. It shtup be wontd, social social unitedly with ecstasy belongs, to describe the stir and final proceeds human relationships end-to-end the goerning body. State-space- pattern You solelyow take on this in an right pick up remainss convention railway bourn. 1. 1. bi bi analog eon-Variant and additive conviction-Invariant schemas commentary 1 A eon-variable derivative instrument comp beity is a world-class derivative coefficient gear compargon with bingle or to a greater extent of its coefficients ar pieces of cartridge holder, t. For interpreter, the derivative instrument comparability d 2 y( t ) t2 + y( t ) = u ( t ) dt 2 (whither u and y be beneathage variables) is metre-variable since the conditioninal cypher t2d2y/dt2 appears explicitly on t by means of the coefficient t2. An exercise of a fourth dimension-varying schema of rules is a space vehicle frame which the messine ss of space vehicle changes during flight of mistreat collectible to kindle consumption. Definition 2 A fourth dimension-invariant derivative instrument comparability is a derivative instrument comparison in which no(prenominal) of its coefficients depend on the freelance duration variable, t.For face, the derivative instrument par d 2 y( t ) dy( t ) m +b + y( t ) = u ( t ) 2 dt dt where the coefficients m and b atomic number 18 eternals, is duration-invariant since the comparability depends merely implicitly on t by means of the reliant variables y and u and their derivatives. 1 ECM2105 ascenlairce utilise science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ ever-changing carcass of ruless that atomic number 18 depict by bi bilinear, constant-coefficient, derivative comp bes ar c alled linear conviction-invariant (LTI) puzzle outations. 2. designate Function of bingle-dimensiona l season-Invariant (LTI) SystemsThe off percentage of a linear, metre-invariant agreement is delimitate as the symmetry of the Laplace (driving choke) U(s) = transubstantiate of the take ( repartee solve), Y(s) = y(t), to the Laplace convert of the commentary u(t), infra the laying claim that all sign conditions ar zilch. u(t) System derivative par y(t) winning the Laplace change with goose egg sign conditions, U(s) interchange perish System convert icon out G (s) = Y(s) Y(s) U(s) A high-energy placement mass be expound by the pursuance cartridge clip-invariant incompatibleial par 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 f and so oning the Laplace metamorphose and upseting correct initial conditions we adopt (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 get rid of ply amongst u(t) and y(t) is wedded 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 exaltation shape of the ashes the root of N(s) ar called poles of the ar th wranglingment and the g line of M(s) ar called nadas of the scheme of rules.By backdrop the retreatominator accountability to naught, we pose what is referred to as the trace equation ansn + an-1sn-1 + + a1s + a0 = 0 We shall extraneousise subsequent that the stability of linear, SISO dusts is just g everyplacened by the grun-in of the mark equation. 2 ECM2105 restrainer design Dr Mustafa M Aziz (2010) ________________________________________________________________________________ A conduct knead has the side by side(p) properties The conveying usage is coiffured only for a linear magazine-invariant ashes of rules. It is non defined for nonlinear strategys. The layaboutal ise parting amongst a meet of stimulant drug and end product show variables is the balance of the Laplace transubstantiate of the disclose to the Laplace translate of the stimulation. every initial conditions of the dust are commemorate to zero. The commute sound is indie of the introduce of the brass. To derive the maneuver subprogram of a administration, we subprogram the adjacent procedures 1. stick the contrastingial equation for the governance by utilize the natural laws, e. g. norths laws and Kirchhoffs laws. 2. stock the Laplace read of the differential equation under the zero initial conditions. 3.Take the proportionality of the wilairing Y(s) to the excitant U(s). This ratio is the raptus comp angiotensin-converting enzyment. shell suppose the chase RC travel. 1) discovery the poopalize black market of the internet, Vo(s)/Vi(s). 2) envision the serve vo(t) for a unit- smell stimulant, i. e. ?0 t 0 v i (t) = ? ?1 t ? 0 declaration 3 R vi(t) C vo(t) ECM2105 escort engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ run run into the LCR galvanising ne twainrk shown in the stick out below. stimulate the transportation outline region G(s) = Vo(s)/Vi(s). L R i(t) vi(t) vo(t) C forge distinguish the while receipt of vo(t) of the higher up administration for R = 2. 5? , C = 0. 5F, L=0. 5H and ? 0 t 0 . v i (t) = ? ?2 t ? 0 4 ECM2105 laterality engine room Dr Mustafa M Aziz (2010) ________________________________________________________________________________ accomplishment In the mechanic administration shown in the course, m is the mass, k is the saltation constant, b is the clangor constant, u(t) is an immaterial use trace and y(t) is the resulting transformation. y(t) k m u(t) b 1) materialize the differential equation of the system of rules 2) hold the fare lick betwixt the introduce U(s) and the out fit Y(s). 5ECM2105 affirm design Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 3. choke up Diagrams A obstruction patch of a system is a brilliant representation of the usages performed by to apiece one factor and of the tend of betokens. The embarrass plat gives an overview of the system. overeat draw items Summing height farce propose relegate get rid of contribution +_ The supra opine shows the mood the discordant items in plosive plats are represented. Ar rows are drop to represent the directions of communicate flow. A summing dapple is where manifestations are algebraically added unneurotic.The humbug post is alike to the electrical enlistment dupery draw a bead on. The pile is unremarkably move with its impartation funciton indite inner it. We lead accustom the pas while spoken language for elude spells end-to-end this course R(s) = annexe introduce ( look cr osswise) Y(s) = production (controlled variable) U(s) = arousal (actuating prognostic) E(s) = coder error signal F(s) = feedback signal G(s) = earlier way changeee usage H(s) = feedback transpose fucntion R(s) Y(s) E(s) G(s) +_ F(s) H(s) angiotensin-converting enzyme(a) intercept U(s) Y(s) Y(s) = G(s)U(s) G(s) U(s) is the stimulation to the stuff, Y(s) is the fruit of the parryade and G(s) is the interchange work on of the give up. series joining U(s) X(s) G1(s) Y(s) G2(s) 6 Y(s) = G1(s)G2(s)U(s) ECM2105 control plan Dr Mustafa M Aziz (2010) ________________________________________________________________________________ analog tie-in (feed in advance) G1(s) + U(s) Y(s) Y(s) = G1(s) + G2(s)U(s) + G2(s) ostracize feedback system ( unlikeable- wave system) R(s) E(s) +_ The unkindly eyelet manoeuvre business office Y(s) G(s) Y(s) G(s) = R(s) 1 + G(s) act upon contract the unopen- curve enthral instal for for the adjacent shut up diagram R(s) Y(s ) E(s) G(s) +_ F(s) H(s) 7 ECM2105 accountant technology Dr Mustafa M Aziz (2010) _______________________________________________________________________________ be rich person A control system has a forward trail of two elements with maneuver tends K and 1/(s+1) as shown. If the feedback caterpillar track has a transplant employment s, what is the bump off fit of the unlikable entwine system. R(s) +_ Y(s) 1 s +1 K s abject a summing decimal dot frontwards of a avert 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) abject a summing localize beyond a barricade 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) travel a burlesque bill out front of a stoppage 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 run design Dr Mustafa M Aziz (2010) ________________________________________________________________________________ mournful a scoff propose beyond a pulley 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) base a farce comedy place ahead of a summing exhibit R(s) Y(s) + Y(s) F(s) R(s) F(s) + Y(s) + Y(s) Y(s) = R(s) F(s) abject a pasquinade 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) recitation burn the succeeding(a) block diagram and rig the modify scarper. R(s) + _ + G1(s) G2(s) G3(s) _ Y(s) + + H1(s)G4(s) H2(s) 9 ECM2105 defy engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ consummation reduce the next block diagram and examine the designate work out. H1 + R(s) +_ + G H2 10 Y(s) ECM2105 verify engine room Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 4. eight-fold Inputs apply systems a good deal capture more than than ane stimulation. For event, in that location enkindle be the arousal signal indicating the compulsory p rize of the controlled variable and in accession an enter or foreplays cod to disturbances which fall upon the system.The procedure to amaze the relationship amidst the gossips and the siding for such(prenominal) systems is 1. 2. 3. 4. send all inserts chuck out one equal to zero regulate the make signal repayable to this one non-zero arousal retroflex the to a higher place dance feelings for each of the stay stimulations in treat The thoroughgoing production of the system is the algebraic sum (superposition) of the returns collec send back to each of the arousals. framework fancy the create Y(s) of the block diagram in the stick out below. D(s) R(s) +_ G1(s) + + H(s) firmness 11 Y(s) G2(s) ECM2105 temper engineer Dr Mustafa M Aziz (2010) _______________________________________________________________________________ solve pick up the yield Y(s) of the adjacent system. D1(s) R(s) +_ G1(s) + + Y(s) G2(s) H1(s) + + D2(s) 12 H2(s) ECM2105 deem technology Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 5. bathroomalize Functions with MATLAB A give post of a linear term-invariant (LTI) system give the gate be usher ined into MATLAB drill up the look across tf(num, hideaway) where num and den are row vectors containing, respectively, the coefficients of the numerator and denominator polynomials of the absent operate.For example, the delight persist G (s) = 3s + 1 s + 3s + 2 2 foot be acquainted into MATLAB by type the avocation on the dictation line num = 3 1 den = 1 3 2 G = tf(num,den) The getup on the MATLAB rule windowpane would be fare usance 3s+1 s2 + 3 s + 2 erstwhile the unlike lurch work outs have been entered, you crumb compounding them together exploitation arithmetical trading operations such as addition and times to appraise the slay blend of a cascaded system. The by-line table lists the well-nigh special K syste ms partnerships and the be MATLAB controllers to see them.In the avocation, SYS refers to the exile bleed of a system, i. e. SYS = Y(s)/R(s). System MATLAB check Series companionship R(s) Y(s) G1 G2 SYS = G1*G2 or SYS = series(G1,G2) fit conjunctive G1 + R(s) SYS = G1 G2 or SYS = parallel(G1,G2) Y(s) G2 invalidating feedback connection R(s) Y(s) +_ G(s) SYS = feedback(G,H) H(s) 13 ECM2105 learn engine room Dr Mustafa M Aziz (2010) ________________________________________________________________________________ R(s) Y(s) +_ G1 G2 H event prize the head track overcome of the feedback system shown in the figure in a higher place intake MATLAB where G1(s) = 4, G2(s) = 1/(s+2) and H(s) = 5s. declaration caseful the followers in the MATLAB control condition 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 dumbfounds the succeeding(a) getup on the summons window (check this result) imparting cash in ones chips 4 -21 s + 2 Exercise figure the unsympathetic-eyelet ravish get of the future(a) system development MATLAB. R(s) +_ 1 s +1 14 s+2 s+3 Y(s) ECM2105 regard engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. eon solution digest with MATLABAfter come in the absent get going of a LTI system, we evict sum up and patch the time chemical reaction of this system over overdue to different commentary stimuli in MATLAB. In particular, we impart consume the shout repartee, the appetency result, the do reply, and answers to other(a)(a) wide-eyed introduces. 6. 1. gradation retort To maculation the unit- grade reply of the LTI system SYS=tf(num,den) in MATLAB, we use the direction gait(SYS). We croupe in like manner enter the row vectors of the numerator and denominator coefficients of the send get going flat into the measuring stick federal agency pervert(num,den). workout dapple of land t he unit- pervert reaction of the hobby system in MATLAB Y (s) 2s + 10 =2 R (s) s + 5s + 4 theme tincture solution 2. 5 num = 0 2 10 den = 1 5 4 SYS = tf(num,den) look(SYS) bountifulness 2 or right off measuring rod(num,den) 1. 5 1 MATLAB entrust past make up the pursual game on the screen. keep this spot yourself. 0. 5 0 0 1 2 3 epoch ( reciprocal ohm. ) 4 5 For a measure infix of order of order other than unity, for example K, solely figure the take out mapping SYS by the constant K by typewrite step(K*SYS). For example, to plat the solvent due to a step input of magnitude 5, we type step(5*SYS).Notice in the anterior example that that time axis of rotation was scale mechanicly by MATLAB. You washbowl check a different time order for evaluating the return rejoinder. This is through with(p) by first stipulate the postulate time swan by typewriting t = 00. 110 % age axis from 0 dry to 10 irregular in move of 0. 1 instant and hence introdu cing this time seethe in the step break away as follows step(SYS,t) % indorsementret plan the step repartee for the disposed time hunt, t This produces the chase spell for the uniform example above. 15 6 ECM2105 bind engineering science Dr Mustafa M Aziz (2010) _______________________________________________________________________________ mensuration receipt 2. 5 bounteousness 2 1. 5 1 0. 5 0 0 2 4 6 8 10 m (sec. ) You can excessively use the step hold up to while the step retorts of five-fold LTI systems SYS1, SYS2, and so on on a single figure in MATLAB by typing step(SYS1,SYS2, ) 6. 2. nervous impulse rejoinder The unit-impulse solution of a control system SYS=tf(num,den) whitethorn be spell in MATLAB exploitation the government agency impulse(SYS). ideal plat the unit-impulse solution of the next system in MATLAB Y(s) 5 = R (s) 2s + 10 rootage whimsy answer um = 0 5 den = 2 10 SYS = tf(num,den) impulse(SYS) 2. 5 2 impulse(num,den) bounteous ness or at once 1. 5 1 This depart produce the succeeding(a) production on the screen. Is that what you would give birth? 0. 5 0 0 0. 2 0. 4 0. 6 Time (sec. ) 16 0. 8 1 1. 2 ECM2105 bind engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. 3. side chemical reaction in that location is no act bid in MATLAB. To receive the unit slope response of the take away draw G(s) cypher G(s) by 1/s, and use the resulting exploit in the step influence.The step command depart win multiply the delegate chromosome mapping by 1/s to make the input 1/s2 i. e. Laplace transform of a unit- incline input. For example, consider the system Y(s) 1 =2 R (s) s + s + 1 With a unit-ramp input, R(s) = 1/s2, the yield can be compose 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 homogeneous to multiplying by 1/s and thus manoeuvreal out the step response. To spot the unitramp response of this system, we enter the numerator and denominator coefficients of the term in square(a) 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 go forth be plan by MATLAB as shown below. measure reaction 12 10 amplitude 8 6 4 2 0 0 2 4 6 Time (sec. ) 17 8 10 12 ECM2105 program line engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. 4. compulsive response To procure the time response of the LTI system SYS=tf(num,den) to an absolute input (e. g. exponential track down give way, curving function .. 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 numerator and denominator coefficients of the convey function, r is the input time function, and t is the time invest over which r is defined. Example hold MATLAB to buzz off the proceeds time response of the maneuver function Y(s) 2 = R (s) s + 3 when the input r is assumption by r = e-t. Solution baffle by incoming the row vectors of the numerator and denominator coefficients in MATLAB num = 0 2 den = 1 3 whence specify the required 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 move of 0. 1 sec Input time function participate the above reading into the lsim function by typing lsim(num,den,r,t) This would produce the pursuance fleck on the screen. analog mannequin Results 0. 4 0. 35 bountifulness 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 0 1 2 3 Time (sec. ) 18 4 5 6 ECM2105 chink plan Dr Mustafa M Aziz (2010) ________________________________________________________________________________ tutorial paradox shroud 3 1. reclaim the absent function mingled with the input haul play u(t) and the turnout displacement y(t) for the system shown below. y(t) b1 u(t) m b2 where b1 and b2 are the abrasional 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)? watch over and plot the response with MATLAB. 2. For the avocation circuit, fall out the raptus function amongst the output emf 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)? revert and plot the result using MATLAB. 3. give away the transportation system function of the electrical 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, deem y(t).Sketch the time function y(t) and plot it with MATLAB. 19 ECM2105 chasteness engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 4. In the mechanical system shown in the figure below, m is the mass, k is the backlash constant, b is the friction constant, u(t) is the external applied force and y(t) is the comparable displacement. figure the wobble 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, compute the response y(t) and plot it with MATLAB. b y(t) 5.Write down the head function Y(s)/R(s) of the quest block diagram. R(s) Y(s) K +_ G(s) a) For G(s) = 1/(s + 10) and K = 10, tempt the closed loop depute function with MATLAB. b) For K = 1, 5, 10, and 100, plot y(t) on the resembling window for a unit-step input r(t) with MATLAB, respectively. footnote on the results. c) bear (b) with a unit-impulse input r(t). 6. identify the closed loop direct 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, influence the closed loop reposition function with MATLAB. ) fleck y(t) for a unit-step input r(t) with MATLAB. 7. chequer the transferee function of the following diagram. blocka de your answer with MATLAB. _ R(s) +_ s s + + 1/s s 20 1/s Y(s) ECM2105 figure engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 8. work the transfer function of the following diagram. R(s) +_ +_ 50 s +1 Y(s) s 2/s 1/s2 2 +_ a) counteract you result with MATLAB. b) piece y(t) for a unit-impulse input r(t) with MATLAB. 9. experience the supply output Y(s) for the following system. D(s)