Transfer Functions

ECM2105 rig engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ TRANSFER FUNCTIONS AND BLOCK DIAGRAMS 1. Introduction 2. channelise Function of Linear quantify-Invariant (LTI) remainss 3. Block Diagrams 4. doubled Inputs 5. tilt Functions with MATLAB 6. m Response summary with MATLAB 1. Introduction An important pace in the analysis and design of control schemes is the numeral modelling of the controlled process. at that place argon a number of mathematical representations to describe a controlled processDifferential equalitys You catch knowing this before. rapture obligation It is watchd as the proportionality of the Laplace transmogrify of the call forth covariant to the Laplace trans organise of the excitant variable, with all(prenominal) nix initial conditions. Block draw It is utilise to represent all types of placements. It aro employment be used, in concert with take out contribu tions, to describe the energise and effect relationships throughout the ashes. State-space-representation You pull up stakes theme this in an advanced maneuver Systems role course. 1. 1. Linear sequence-Variant and Linear Time-Invariant SystemsDefinition 1 A season-variable derivative instrument par is a derivative equation with maven or more(prenominal) of its coefficients be scats of prison term, t. For type fixateters case, the derivative instrument equation d 2 y( t ) t2 + y( t ) = u ( t ) dt 2 (where u and y atomic number 18 dependent variables) is time-variable since the term t2d2y/dt2 depends explicitly on t through the coefficient t2. An character of a time-varying trunk is a ballistic capsule governance which the mass of spacecraft changes during relief valve repayable to fuel consumption. Definition 2 A time-invariant derived operate equation is a differential equation in which no(prenominal) of its coefficients depend on the independent time var iable, t.For guinea pig, the differential equation d 2 y( t ) dy( t ) m +b + y( t ) = u ( t ) 2 dt dt where the coefficients m and b be constants, is time-invariant since the equation depends only implicitly on t through the dependent variables y and u and their derivatives. 1 ECM2105 encounter engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ Dynamic dusts that are described by linear, constant-coefficient, differential equations are called linear time-invariant (LTI) systems. 2. Transfer Function of Linear Time-Invariant (LTI) SystemsThe move work on of a linear, time-invariant system is defined as the ratio of the Laplace (driving snuff it) U(s) = transform of the issue ( solution wreak), Y(s) = y(t), to the Laplace transform of the remark u(t), under the assumption that all initial conditions are postal code. u(t) System differential equation y(t) Taking the Laplace transform with zero initial condition s, U(s) Transfer mapping System transportation system berth G (s) = Y(s) Y(s) U(s) A dynamic system bath be described by the followers time-invariant differential equation 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 Taking the Laplace transform and considering zero initial conditions we have (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 sell affaire surrounded by u(t) and y(t) is given 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 tape drive function of the system the g course of study of N(s) are called poles of the system and the roots of M(s) are called zeros of the system.By setting the denominator function to zero, we come what is referred to as the characteristic equation ansn + an- 1sn-1 + ??? + a1s + a0 = 0 We shall see later that the stability of linear, SISO systems is tout ensemble governed by the roots of the characteristic equation. 2 ECM2105 chasteness engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ A delegate function has the future(a)(a) properties The get rid of function is defined only for a linear time-invariant system. It is not defined for nonlinear systems. The direct function surrounded by a pair of gossip and turnout variables is the ratio of the Laplace transform of the output to the Laplace transform of the stimulant. every initial conditions of the system are set to zero. The enthrall function is independent of the comment of the system. To derive the expatriation function of a system, we use the quest processs 1. Develop the differential equation for the system by utilise the physical laws, e. g. Newtons laws and Kirchhoffs laws. 2. Take the Laplace tr ansform of the differential equation under the zero initial conditions. 3.Take the ratio of the output Y(s) to the infix U(s). This ratio is the budge function. mannikin Consider the hobby RC go. 1) name the manoeuver function of the profit, Vo(s)/Vi(s). 2) experience the rejoinder vo(t) for a unit- bill input, i. e. ?0 t < 0 v i (t) = ? ?1 t ? 0 upshot 3 R vi(t) C vo(t) ECM2105 Control applied science Dr Mustafa M Aziz (2010) ________________________________________________________________________________ consummation Consider the LCR electrical network shown in the figure below. Find the transfer function G(s) = Vo(s)/Vi(s). L R i(t) vi(t) vo(t) C bring Find the time reception of vo(t) of the above system for R = 2. 5? , C = 0. 5F, L=0. 5H and ? 0 t < 0 . v i (t) = ? ?2 t ? 0 4 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ Exercise In the mechanic system shown in the figur e, m is the mass, k is the contain constant, b is the encounter constant, u(t) is an external utilise force and y(t) is the resulting displacement. y(t) k m u(t) b 1) Find the differential equation of the system 2) Find the transfer function between the input U(s) and the output Y(s). 5ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 3. Block Diagrams A tote diagram of a system is a pictorial representation of the functions performed by individually component and of the flow of tapers. The satiate diagram gives an overview of the system. Block diagram items Summing lodge charade point Block Transfer function +_ The above figure shows the way the sundry(a) items in stuff diagrams are represented. Arrun-ins are used to represent the directions of signal flow. A summing point is where signals are algebraically added together.The hoax point is similar to the electrical circuit takeoff poi nt. The plosive speech sound is usually worn with its transfer funciton written inside it. We go away use the pursual terminology for close down diagrams throughout this course R(s) = reference input ( overleap) Y(s) = output (controlled variable) U(s) = input (actuating signal) E(s) = error signal F(s) = feedback signal G(s) = send path transfer function H(s) = feedback transfer fucntion R(s) Y(s) E(s) G(s) +_ F(s) H(s) Single block U(s) Y(s) Y(s) = G(s)U(s) G(s) U(s) is the input to the block, Y(s) is the output of the block and G(s) is the transfer function of the block.Series familiarity U(s) X(s) G1(s) Y(s) G2(s) 6 Y(s) = G1(s)G2(s)U(s) ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ Parallel club (feed forward) G1(s) + U(s) Y(s) Y(s) = G1(s) + G2(s)U(s) + G2(s) Negative feedback system ( unkindly- handbuild system) R(s) E(s) +_ The unkindly curl transfer function Y(s) G(s) Y(s) G(s) = R(s) 1 + G(s) Exercise Find the closed-loop transfer function for the chase block diagram R(s) Y(s) E(s) G(s) +_ F(s) H(s) 7 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) _______________________________________________________________________________ Exercise A control system has a forward path of two elements with transfer functions K and 1/(s+1) as shown. If the feedback path has a transfer function s, what is the transfer function of the closed loop system. R(s) +_ Y(s) 1 s +1 K s touching a summing point in advance of a block 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) base a summing point beyond a block 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) locomote a takeoff point ahead 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 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ Moving a takeoff point 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) Moving a takeoff point ahead of a summing point R(s) Y(s) + Y(s) F(s) R(s) F(s) + Y(s) + Y(s) Y(s) = R(s) F(s) Moving a takeoff 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) Exercise Reduce the quest block diagram and determine the transfer function. R(s) + _ + G1(s) G2(s) G3(s) _ Y(s) + + H1(s)G4(s) H2(s) 9 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ Exercise Reduce the pursuance block diagram and determine the transfer function. H1 + R(s) +_ + G H2 10 Y(s) ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 4. Multiple Inputs Control systems often have more than one input. For object lesson, there flock be the input signal indicating the postulate value of the controlled var iable and also an input or inputs due to disturbances which affect the system.The procedure to obtain the relationship between the inputs and the output for much(prenominal) systems is 1. 2. 3. 4. Set all inputs save one equal to zero find the output signal due to this one non-zero input buy out the above stairs for each of the remaining inputs in figure out The total output of the system is the algebraic sum (superposition) of the outputs due to each of the inputs. lawsuit Find the output Y(s) of the block diagram in the figure below. D(s) R(s) +_ G1(s) + + H(s) Solution 11 Y(s) G2(s) ECM2105 Control Engineering Dr Mustafa M Aziz (2010) _______________________________________________________________________________ Exercise particularize the output Y(s) of the following system. D1(s) R(s) +_ G1(s) + + Y(s) G2(s) H1(s) + + D2(s) 12 H2(s) ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 5. T ransfer Functions with MATLAB A transfer function of a linear time-invariant (LTI) system can be entered into MATLAB utilise the involve tf(num,den) where num and den are row vectors containing, respectively, the coefficients of the numerator and denominator polynomials of the transfer function.For example, the transfer function G (s) = 3s + 1 s + 3s + 2 2 can be entered into MATLAB by typing the following on the command line num = 3 1 den = 1 3 2 G = tf(num,den) The output on the MATLAB command window would be Transfer function 3s+1 &8212&8212&8212&8212s2 + 3 s + 2 Once the various transfer functions have been entered, you can combine them together utilize arithmetic operations such as addition and multiplication to appreciate the transfer function of a cascaded system. The following table lists the most common systems connections and the same MATLAB commands to implement them.In the following, SYS refers to the transfer function of a system, i. e. SYS = Y(s)/R(s). System MA TLAB command Series connection R(s) Y(s) G1 G2 SYS = G1*G2 or SYS = series(G1,G2) Parallel connection G1 + R(s) SYS = G1 G2 or SYS = parallel(G1,G2) Y(s) G2 Negative feedback connection R(s) Y(s) +_ G(s) SYS = feedback(G,H) H(s) 13 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ R(s) Y(s) +_ G1 G2 H Example Evaluate the transfer function of the feedback system shown in the figure above development MATLAB where G1(s) = 4, G2(s) = 1/(s+2) and H(s) = 5s.Solution Type the following in the MATLAB command 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 produces the following output on the command window (check this result) Transfer function 4 &8212&8212-21 s + 2 Exercise Compute the closed-loop transfer function of the following system using MATLAB. R(s) +_ 1 s +1 14 s+2 s+3 Y(s) ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________ ________________________________________________ 6. Time Response Analysis with MATLABAfter entering the transfer function of a LTI system, we can count and spot the time reply of this system due to different input stimuli in MATLAB. In particular, we will consider the timbre resolution, the impulse resolution, the ramp chemical reaction, and chemical reactions to other simpleton inputs. 6. 1. Step result To darn the unit- mistreat response of the LTI system SYS=tf(num,den) in MATLAB, we use the command graduation(SYS). We can also enter the row vectors of the numerator and denominator coefficients of the transfer function directly into the pace function step(num,den).Example plat the unit-step response of the following system in MATLAB Y (s) 2s + 10 =2 R (s) s + 5s + 4 Solution Step Response 2. 5 num = 0 2 10 den = 1 5 4 SYS = tf(num,den) step(SYS) bounty 2 or directly step(num,den) 1. 5 1 MATLAB will whence produce the following spot on the screen. fend for this plot yourself. 0. 5 0 0 1 2 3 Time (sec. ) 4 5 For a step input of magnitude other than unity, for example K, simply multiply the transfer function SYS by the constant K by typing step(K*SYS). For example, to plot the response due to a step input of magnitude 5, we type step(5*SYS).Notice in the previous example that that time axis was scaled mechanically by MATLAB. You can stipulate a different time come out for evaluating the output response. This is done by first define the required time range by typing t = 00. 110 % Time axis from 0 sec to 10 sec in steps of 0. 1 sec and consequently introducing this time range in the step function as follows step(SYS,t) % Plot the step response for the given time range, t This produces the following plot for the same example above. 15 6 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) _______________________________________________________________________________ Step Response 2. 5 Amplitude 2 1. 5 1 0. 5 0 0 2 4 6 8 10 Time (sec. ) Y ou can also use the step function to plot the step responses of multiple LTI systems SYS1, SYS2, etc. on a single figure in MATLAB by typing step(SYS1,SYS2, ) 6. 2. longing response The unit-impulse response of a control system SYS=tf(num,den) whitethorn be plotted in MATLAB using the function impulse(SYS). Example Plot the unit-impulse response of the following system in MATLAB Y(s) 5 = R (s) 2s + 10 Solution Impulse Response um = 0 5 den = 2 10 SYS = tf(num,den) impulse(SYS) 2. 5 2 impulse(num,den) Amplitude or directly 1. 5 1 This will produce the following output on the screen. Is that what you would expect? 0. 5 0 0 0. 2 0. 4 0. 6 Time (sec. ) 16 0. 8 1 1. 2 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. 3. Ramp response There is no ramp command in MATLAB. To obtain the unit ramp response of the transfer function G(s) multiply G(s) by 1/s, and use the resulting function in the step comm and.The step command will further multiply the transfer function by 1/s to make the input 1/s2 i. e. Laplace transform of a unit-ramp 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 output can 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 combining weight to multiplying by 1/s and then working out the step response. To plot the unitramp response of this system, we enter the numerator and denominator coefficients of the term in square 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 plotted by MATLAB as shown below. Step Response 12 10 Amplitude 8 6 4 2 0 0 2 4 6 Time (sec. ) 17 8 10 12 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 6. 4. discretional response To obtain the time response of the LTI sys tem SYS=tf(num,den) to an arbitrary input (e. g. exponential function function, sinusoidal 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 transfer function, r is the input time function, and t is the time range over which r is defined. Example utilise MATLAB to obtain the output time response of the transfer function Y(s) 2 = R (s) s + 3 when the input r is given by r = e-t. Solution Start by entering the row vectors of the numerator and denominator coefficients in MATLAB num = 0 2 den = 1 3 Then 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 steps of 0. 1 sec Input time function Enter the above information into the lsim function by typing lsim(num,den,r,t) This would produce the following plot on the screen. Linear pretence Res ults 0. 4 0. 35 Amplitude 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 0 1 2 3 Time (sec. ) 18 4 5 6 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ TUTORIAL PROBLEM airplane 3 1. Find the transfer function between the input force 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 frictional 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)? pinch and plot the response with MATLAB. 2. For the following circuit, find the transfer function between the output voltage across the inductor 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 and plot the result using MATLAB. 3. Find the transfer 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, compute y(t).Sketch the time function y(t) and plot it with MATLAB. 19 ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 4. In the mechanical system shown in the figure below, m is the mass, k is the spring constant, b is the friction constant, u(t) is the external applied force and y(t) is the agree displacement. Find the transfer 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 transfer function Y(s)/R(s) of the following block diagram. R(s) Y(s) K +_ G(s) a) For G(s) = 1/(s + 10) and K = 10, determine the closed loop transfer function with MATLAB. b) For K = 1, 5, 10, and 100, plot y(t) on the same window for a unit-step input r(t) with MATLAB, respectively. Comment on the results. c) Repeat (b) with a unit-im pulse input r(t). 6. Find 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, determine the closed loop transfer function with MATLAB. ) Plot y(t) for a unit-step input r(t) with MATLAB. 7. Determine the transfer function of the following diagram. Check your say with MATLAB. _ R(s) +_ s s + + 1/s s 20 1/s Y(s) ECM2105 Control Engineering Dr Mustafa M Aziz (2010) ________________________________________________________________________________ 8. Determine the transfer function of the following diagram. R(s) +_ +_ 50 s +1 Y(s) s 2/s 1/s2 2 +_ a) Check you result with MATLAB. b) Plot y(t) for a unit-impulse input r(t) with MATLAB. 9. Determine the total output Y(s) for the following system. D(s)