Transfer Functions

ECM2105 Control engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ TRANSFER FUNCTIONS AND hamper DIAGRAMS 1. Introduction 2. deepen Function of unidimensional quantify-Invariant (LTI) frames 3. Block Diagrams 4. Multiple Inputs 5. transportation Functions with MATLAB 6. Time Response Analysis with MATLAB 1. Introduction An important timbre in the analysis and design of control bodys is the mathematical modelling of the controlled process. There ar a number of mathematical representations to describe a controlled process derivative gear equatings You grant learned this before. Transfer dish It is delineate as the proportionality of the Laplace understand of the product variable to the Laplace convert of the stimulant drug variable, with all set initial conditions. Block draw It is engrossd to represent all types of carcasss. It cig atomic number 18t be give upd, together with conduct puzzle ou ts, to describe the cause and effect relationships end-to-end the system. State-space-representation You lead study this in an advanced Control Systems Design course. 1. 1. Linear Time-Variant and Linear Time-Invariant Systems interpretation 1 A judgment of conviction-variable first derivative comparison is a derivative instrument comparability with one or more of its coefficients atomic number 18 fly the coops of clipping, t. For font, the differential equation d 2 y( t ) t2 + y( t ) = u ( t ) dt 2 (where u and y be hooklike variables) is snip-variable since the term t2d2y/dt2 depends explicitly on t through the coefficient t2. An example of a time-varying system is a space vehicle system which the mass of spacecraft changes during flight due(p) to fuel consumption. Definition 2 A time-invariant differential equation is a differential equation in which none of its coefficients depend on the indepen retreatt time variable, t.For example, the differential equation d 2 y( t ) dy( t ) m +b + y( t ) = u ( t ) 2 dt dt where the coefficients m and b are immutables, is time-invariant since the equation depends only implicitly on t through the dependent variables y and u and their derivatives. 1 ECM2105 Control engineer Dr Mustafa M Aziz (2010) ________________________________________________________________________________ moral force systems 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 transpose choke of a linear, time-invariant system is defined as the ratio of the Laplace (driving flow) U(s) = transform of the end product ( reply snuff it), Y(s) = y(t), to the Laplace transform of the enter u(t), low the assumption that all initial conditions are zero. u(t) System differential equation y(t) pickings the Laplace transform with zero initial conditions, U(s) Transfer function System dispatch function G (s) = Y(s) Y(s) U(s) A dynamic system ignore be described by the pursuit 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 exaltation function amidst 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 convert function of the system the grow of N(s) are called poles of the system and the grow of M(s) are called zeros of the system.By setting the denominator function to zero, we find oneself 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 completely governed by the roots of the characteristic equation. 2 ECM2105 Control technology Dr Mustafa M Aziz (2010) ________________________________________________________________________________ A deepen function has the pursual properties The convert function is defined only for a linear time-invariant system. It is not defined for nonlinear systems. The alter function between a pair of stimulation and widening variables is the ratio of the Laplace transform of the getup to the Laplace transform of the input. All initial conditions of the system are set to zero. The transfer function is independent of the input of the system. To derive the transfer function of a system, we use the next procedures 1. Develop the differential equation for the system by utilise the physical laws, e. g. Newtons laws and Kirchhoffs laws. 2. Take the Laplace transform of the differential equation under the zero initial conditions. 3.Take the ratio of the turnout Y(s) to the input U(s). This ratio is the transfer function. Example Consider the following RC circuit. 1) reclaim the transfer function of the network, Vo(s)/Vi(s). 2) stick the response vo(t) for a unit- bar input, i. e. ?0 t 0 v i (t) = ? ?1 t ? 0 etymon 3 R vi(t) C vo(t) ECM2105 Control design Dr Mustafa M Aziz (2010) ________________________________________________________________________________ go Consider the LCR electrical network shown in the figure below. fetch the transfer function G(s) = Vo(s)/Vi(s). L R i(t) vi(t) vo(t) C achievement dominate the time response of vo(t) of the to a higher place 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 mechanical system shown in the figure, m is the mass, k is the resile constant, b is the friction constant, u(t) is an extraneous applied 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 chock up diagram of a system is a pictorial representation of the functions performed by each component and of the flow of predicts. The dodge diagram gives an overview of the system. Block diagram items Summing even Takeoff patch Block Transfer function +_ The in a higher place figure shows the way the various items in choke up diagrams are represented. Arrows are used to represent the directions of signal flow. A summing lead is where signals are algebraically added together.The takeoff point is similar to the electrical circuit takeoff point. The block is usually drawn with its transfer funciton written inside it. We will use the following termino logy for block diagrams throughout this course R(s) = reference input ( influence) Y(s) = output (controlled variable) U(s) = input (actuating signal) E(s) = error signal F(s) = feedback signal G(s) = forward room 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 affiliation 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 connection (feed forward) G1(s) + U(s) Y(s) Y(s) = G1(s) + G2(s)U(s) + G2(s) Negative feedback system (closed- eyehole system) R(s) E(s) +_ The closed loop transfer function Y(s) G(s) Y(s) G(s) = R(s) 1 + G(s) Exercise Find the closed-loop transfer function for the following block diagram R(s) Y(s) E(s) G(s) +_ F(s) H(s) 7 E CM2105 Control Engineering Dr Mustafa M Aziz (2010) _______________________________________________________________________________ Exercise A control system has a forward line 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 Moving a summing point ahead 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) Moving 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) Moving 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 cut the following block diagram and square off 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 following 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 example, there can be the input signal indicating the required value of the controlled variable and also an input or inputs due to disturbances which affect the system.The procedure to obtain the relationship between the inputs and the outp ut for such(prenominal) systems is 1. 2. 3. 4. Set all inputs except one equal to zero check into the output signal due to this one non-zero input Repeat the higher up steps for each of the remaining inputs in turn The total output of the system is the algebraic sum (superposition) of the outputs due to each of the inputs. Example 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 Determine 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. Transfer Functions with MATLAB A transfer function of a linear time-invariant (LTI) system can be entered into MATLAB using the keep in line tf(num,den) w here 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 typewriting 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 s2 + 3 s + 2 Once the various transfer functions have been entered, you can flow them together using arithmetic operations such as auxiliary and multiplication to evaluate the transfer function of a cascaded system. The following disconcert lists the most common systems connections and the corresponding 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 MATLAB 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 Ne gative 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 using 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 -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 incoming the transfer function of a LTI system, we can compute and spot the time res ponse of this system due to different input stimuli in MATLAB. In particular, we will consider the step response, the impulse response, the ramp response, and responses to other plain inputs. 6. 1. Step response To while the unit-step response of the LTI system SYS=tf(num,den) in MATLAB, we use the command step(SYS). We can also enter the row vectors of the numerator and denominator coefficients of the transfer function now into the step function step(num,den).Example Plot 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) Amplitude 2 or directly step(num,den) 1. 5 1 MATLAB will then produce the following plot on the screen. Confirm this plot yourself. 0. 5 0 0 1 2 3 Time ( indorsement. ) 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 th e 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 automatically by MATLAB. You can deposit a different time range for evaluating the output response. This is done by first defining the required time range by typing t = 00. 110 % Time axis from 0 sec to 10 sec in steps of 0. 1 sec and then 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. ) You 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. caprice response The unit-impulse response of a control system SYS=tf(num,den) may be plan 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 command.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, th e 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 equivalent to multiplying by 1/s and then works 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. Arbitrary response To obtain the time response of the LTI system SYS=tf(num,den) to an arbitrary input (e. g. exponential 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 num erator 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 Use 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 entrance 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 Simulation Results 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 SHEET 3 1. Find the tra nsfer 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)? escort 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 corresponding 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 pour 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-impulse 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 answer 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)