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141_1_Chapter4_problems_1

141_1_Chapter4_problems_1 - 208 Chapter 4 A First Analysis...

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Unformatted text preview: 208 Chapter 4 A First Analysis of Feedback + + R K H1< (a) Figure 4.24 w + K TF1 R0 2 > K d K I OFz H14 H2‘ Block diagrams for Problem 4.3 4.4 A unity feedback control system has the open—loop transfer function Gm = Sls + a)' (a) Compute the sensitivity of the closed-loop transfer function to changes in the parameter A. (b) Compute the sensitivity of the closed—loop transfer function to changes in the parameter a. (C) If the unity gain in the feedback changes to a value of ,8 7+~ l, compute the sensitivity of the closed—loop transfer function with respect to ,6. 4.5 Compute the equation for the system error for the filtered feedback system shown in Fig. 4.4. 4.6 If S is the sensitivity of the filtered feedback system to changes in the plant transfer function and T is the transfer function from reference to output, compute the sum of S+T.ShowthatS+T=lifF=H. (a) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the plant transfer function, G. (b) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the controller transfer function, Del. (c) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the filter transfer function, F. ((1) Compute the sensitivity of the filtered feedback system shown in Fig. 4.4 with respect to changes in the sensor transfer function, H. Problemsfor Section 4.2: Control QfSteady—State Error 4.7 Consider the DC—motor control system with rate (tachometer) feedback shown in Fig. 4.25(a)t (21) Find values for K’ and k; so that the system of Fig. 4.25(b) has the same transfer function as the system of Fig. 4.25(a). (b) Determine the system type with respect to tracking Br and compute the system Ky in terms of parameters K ’ and k;. (c) Does the addition of tachometer feedback with positive kt increase or decrease Ky? 4.8 Consider the system shown in Fig. 4.26, where (S+0t)2 $2 +wg' D(s) : K Problems ‘Uy + K , a K K 4. m g 1 O 9 + K r O ’_ 1’ s(1 + ms) k ‘9 ’ 5(1 + ms) 9 19s 4 l + k’ls (a) (b) figure 4.25 Control system for Problem 4.7 figure 4.26 Control system for Problem 4.8 figure 4.27 Control system for Problem 4.9 OY (a) Prove that if the system is stable, it is capable of tracking a sinusoidal reference input r = sin war with zero steady-state error. (Look at the transfer function from R to E and consider the gain at mo.) (b) Use Routh’s criterion to find the range of K such that the closed-loop system remains stable if (no 2 l and or = 0.25. 4.9 Consider the system shown in Fig. 4.27, which represents control of the angle of a pendulum that has no damping. (a) What condition must D(s) satisfy so that the system can track a ramp reference input with constant steady—state error? (b) For a transfer function D(s) that stabilizes the system and satisfies the condition in part (a), find the class of disturbances w(t) that the system can reject with zero steady-state error. 4.10 A unity feedback system has the overall transfer function Y (S) 0% R(s) (S) 52 + 2410,15 + mg, Give the system type and corresponding error constant for tracking polynomial reference inputs in terms of g“ and a)”. 210 Chapter 4 A First Analysis of Feedback Figure 4.28 Control system for Problem 4.12 4.11 Consider the second~order system We would like to add a transfer function of the form D(s) = $35? in series with C(s) in a unity feedback structure. (a) Ignoring stability for the moment, what are the constraints on K , a, and b so that the system is Type 1? (b) What are the constraints placed on K , a, and b so that the system is both stable and Type 1? (c) What are the constraints on a and b so that the system is both Type 1 and remains stable for every positive value for K? 4.12 Consider the system shown in Fig. 4.28(a). (a) What is the system type? Compute the steady—state tracking error due to a ramp input r(t) = r0t1(t). (b) For the modified system with a feed~forward path shown in Fig. 4.28(b), give the value of Hf so the system is Type 2 for reference inputs and compute the Ka in this case. (C) Is the resulting Type 2 property of this system robust with respect to changes in Hf? i.e., will the system remain Type 2 if Hf changes slightly? (a) (b) 4.13 A controller for a satellite attitude control with transfer function G : l/s2 has been designed with a unity feedback structure and has the transfer function D(s) = W. (a) Find the system type for reference tracking and the corresponding error constant for this system. (b) If a disturbance torque adds to the control so that the input to the process is u + w. what is the system type and corresponding error constant with respect to disturbance rejection? 4.14 A compensated motor position control system is shown in Fig. 4.29. Assume that the sensor dynamics are H (s) = l. figure 4.29 Control system for Problem 4.14 Figure infill} Single input—single output unity feedback system with disturbance inputs Figure 4.3%. System using integral control Problems 211 (a) Can the system track a step reference input r with zero steady—state error? If yes, give the value of the velocity constant. (b) Can the system reject a step disturbance w with zero steady—state error? If yes, give the value of the velocity constant. (c) Compute the sensitivity of the closed-loop transfer function to changes in the plant pole at —-Z. (d) In some instances there are dynamics in the sensor. Repeat parts (a) to (c) for H (s) = (Egg—0) and compare the corresponding velocity constants. 4.15 The general unity feedback system shown in Fig. 4.30 has disturbance inputs wl, W2, and W3 and is asymptotically stable. Also, (a) Show that the system is of Type 0, Type I}, and Type ([1 -— 12) with respect to disturbance inputs W1, rug, and W3 respectively. W1 W2 W3 + l R G1(s) —> 02(3) —> Y 4.16 One possible representation of an automobile speed—control system with integral control is shown in Fig. 4.31. + kE k V60“ k1 1 3'2 >21 0V k3 k1 ¢ (a) With a zero reference velocity input (VC 2 0 ), find the transfer function relating the output speed 12 to the wind disturbance w. (b) What is the steady-state response of v if w is a unit ramp function? (c) What type is this system in relation to reference corresponding error constant? (d) What is the type and corres tracking the disturbance w? 4.17 For the feedback system shown in Fi Type 1 for K = 5. Give the corres not robust by using this value of or reference for K = 4 and K = 6. inputs? What is the value of the ponding error constant of this system in relation to g. 4.32, find the value of or that will make the system ponding velocity constant. Show that the system is and computing the tracking error e : r — y to a step Figure 4.32 R ow a + K Control system for C: S t 2 Y Problem 4.17 4.18 Suppose you are given the system de , , picted in Fig. 4.33(a), where the plant parameter a 18 subject to variations. + + R 1 x + 1 + E 4 U SHIT E Y RWW G(s):]r—OY _l 4 + .. 2 (a) (b) Figure 433 Control system for Problem 4.18 (3) Find G(s) so that the system shown in Fi from r to y as the system in Fig. 4.33(a). (b) Assume that a = l is the nominal value of th type and the error constant in this case? g. 4.33(b) has the same transfer function 4.19 Two feedback systems are shown in Fig. 4.34 (3) Determine values for K1, K2, and K3 so that (1) both systems exhibit zero stead y—state error to step inputs (that is, both are Type I), and (ii) their static velocity error constant KV = 1 when K0 2 l. Problems 115 OY (a) (b) figure 4.334 Two feedback systems for Problem 4.19 (b) Suppose K0 undergoes a small perturbation: K0 ~> K0 + 6K0. What effect does this have on the system type in each case? Which system has a type which is robust? Which system do you think would be preferred? 4.20 You are given the system shown in Fig. 4.35, where the feedback gain [3 is subject to variations. You are to design a controller for this system so that the output y(t) accurately tracks the reference input r(t). figure 43% R o + a ’l ADI-(Si ’ (S + 018+ 10) O Y Control system for Problem 4.20 (a) Let ,8 = 1. You are given the following three options for the controller Di(s): 2 kS-l—kI kps +kIS+k2 maze, 02m: ” S , “Gk—"T“ Choose the controller (including particular values for the controller constants) that will result in a Type 1 system with a steady—state error to a unit reference ramp of less than 1-15. (b) Next, suppose that there is some attenuation in the feedback path that is modeled by fl = 0.9. Find the steady—state error due to a ramp input for your chorce of D1- (s) in part (a). . (c) lffi = 0.9. what is the system type for part (b)? What are the values of the appropr1ate error constant? 4.21 Consider the system shown in Fig. 4.36. (21) Find the transfer function from the reference input to the tracking error. (b) For this system to respond to inputs of the form r(t) = t" l (t) (where n < q)w1th zero steady-state error. what constraint is placed on the open—loop poles p] , p2. . . . , pq? Figure 4.3 e + E 1 Y Control system for R O 2 > (S + PrXS + F2) ' ' ‘ (S + 19.1) Problem 4.21 214 Chapter 4 A First Analysis of Feedback Problems 215 4.22 A linear ODE model of the DC motor with negligible armature inductance (La 2 0) figure $37 and with a disturbance torque w was given earlier in the chapter; it is restated here, in Automoblle R : Desired speed slightly different form, as speed—control system Y 2 Actual speed W = Road grade, % JR“ 9' + K a + R“ — = u ——w, Kl WI 6 m (l Kt where GM is measured in radians. Dividing through by the coefficient of 5m, we obtain R C 4 H, ’ k!’ Y 5m + aldm = vaa + COW, Hy where KtKe K1 1 “1: JR ’ 190:3?" COZY’ . . , . . a . a . . . (a) Find the transfer functions from W(s) and from R(s) to Y (S). Wléh fiOtatlng Phtehholmegerss h 1: FESSIblh/gm :15??er thehPOSthhlhg error betweenht) (b) Assume that the desired speed is a constant reference r, so thatR(S) = “/3. Assume an t e re erence angte V or e ” ref " m It a tac ometer we can meet“? t e that the road is level, so w(t) = 0. Compute values of the gains K, Hr, and Hf to motor speed 6m. Consider usmg feedback of the error e and the motor speed 6m in the guarantee that form lim y(t) = r0. 00 a, = K(e — Tpa‘m), t—> ‘ include both the open—loop (assuming Hy 2 0) and feedback cases (Hy 7+. 0) in where K and TD are controller gains to be determined. . . your discuss1on. (21) Draw a block diagram of the resulting feedback system showing both 9m and 6m as ‘ (c) Repeat part (b) assuming that a constant grade disturbance W(.r) z WO/s is present variables 1h the diagram representing the motor. . in addition to the reference input. In particular, find the variation in speed due to (1)) Suppose the numbers work 01“ SO that a1 = 65> 170 = 200» and CO = 10~ If there is _ the grade change for both the feed-forward and feedback cases. Use your results to no load torque (W = 0), What speed (in rpm) rCSUhS from Va = 100 V7 explain (1) why feedback control is necessary and (2) how the gain kp should be (c) Using the parameter values given in part (b), let the control be D = kp + sz and : chosen to reduce steady—state error. find kp and kD so that, using the results of Chapter 3, a step change in 9ref with zero g ((1) Assume that w(t) : 0 and that the gain A undergoes the perturbation A + 8A. load torque results in a transient that has an approximately 17% overshoot and that i Determine the error in speed due to the gain change for both the feed-forward and settles to within 5% of steady—state in less than 0.05 SEC. 3 feedback cases. How should the gains be chosen in this case to reduce the effects of 5/1? 4.25 Consider the multivariable system shown in Fig. 4.38. Assume that the system is stable. Find the transfer functions from each disturbance input to each output and determine the steady-state values of y1 and y2 for constant disturbances. We define a multivari- able system to be type k with respect to polynomial inputs at w if the steady-state value of every output is zero for any combination of inputs of degree less than k and at least one input is a non zero constant for an input of degree k. What is the system type with respect to disturbance rejection at W1? At wz‘? (d) Derive an expression for the steady—state error to a reference angle input, and compute its value for your design in part (c) assuming Qref = l rad. (e) Derive an expression for the steady—state error to a constant disturbance torque when emf : 0 and compute its value for your design in part (c) assuming w z 1.0. 4.23 We wish to design an automatic speed control for an automobile. Assume that (l) the car has a mass m of 1000 kg, (2) the accelerator is the control U and supplies a force on the automobile of 10 N per degree of accelerator motion, and (3) air drag provides a friction force proportional to velocity of 10 N ~ sec/m. (a) Obtain the transfer function from control input U to the velocity of the automobile. (b) Assume the velocity changes are given by figure @038 V“) z s + 0.02 Um + s + 002 Ww’ Multivariable system Where V is given in meters per second, U is in degrees, and W is the percent grade of R 1 £@_. y1 the road. Design a proportional control law U : —ka that will maintain a velocity w error of less than 1 m/sec in the presence of a constant 2% grade. + (c) Discuss what advantage (if any) integral control would have for this problem. R2 0““— Y 2 (d) Assuming that pure integral control (that is, no proportional term) is advantageous, select the feedback gain so that the roots have critical damping (r = l). 4.24 Consider the automobile speed control system depicted in Fig. 4.37. 215 Chapter 4 A First Analysis of Feedback Figure 4.39 Speed-control system for a magnetic tape—drive Figure 4.40 Control system for Problem 4.27 Problems for Section 4.3: The Three—Term Controller. PID Control 4.26 The transfer functions of speed control for a magnetic tape—drive system are shown in Fig. 4.39. The speed sensor is fast enough that its dynamics can be neglected and the diagram shows the equivalent unity feedback system. (a) Assuming the reference is zero, what is the steady—state error due to a step distur~ bance torque of 1 N ~ m? What must the amplifier gain K be in order to make the steady-state error em 5 0.01 rad/sec? (b) Plot the roots of the closed~loop system in the complex plane, and accurately sketch the time response of the output for a step reference input using the gain K computed in part (a). (c) Plot the region in the complex plane of acceptable closed—loop poles corresponding to the specifications of a 1% settling time of ts f 0.1 sec and an overshoot MP 3 5%. ((1) Give values for kp and k0 for a PD controller, which will meet the specifications. (e) How would the disturbance—induced steady—state error change with the new control scheme in part (d)? How could the steady—state error to a disturbance torque be eliminated entirely? Disturbance torque Torque Tape Amplifier motor ‘ dynamics Reference + 10 Torque + i 1 Speed, 0, O 2 ’ i 0.55 + 1 2 > Js + b J = 0.10 kg-mz b = 1.00 N~m'sec 4.27 Consider the system shown in Fig. 4.40 with Pl control. (3) Determine the transfer function from R to Y. (b) Determine the transfer function from W to Y. (c) What is the system type and error constant with respect to reference tracking? ((1) What is the system type and error constant with respect to disturbance rejection? kps + k, U 10 S2+s+20 T. 4.28 Consider the second—order plant with transfer function (3(5) and in a unity feedback structure 1 :(s+l)(5s+1) figure are DC Motor speed-control block diagram for Problems 4.29 and 4.30 4.29 R0 4.30 4.31 Problems 217 (3) Determine the system type and error constant with respect to tracking polynomial reference inputs of the system for P [D = kp], PD [D = kp + sz], and PID [D = kp + [(1/5 + sz] controllers. Let kp = 19, k1 = 0.5, and kD = 4/19. (b) Determine the system type and error constant of the system with respect to distur— bance inputs for each of the three regulators in part (a) with respect to rejecting polynomial disturbances w(t) at the input to the plant. (c) Is this system better at tracking references or rejecting disturbances? Explain your response briefly. ((1) Verify your results for parts (a) and (b) using MATLAB by plotting unit step and ramp responses for both tracking and disturbance rejection. The DC—motor speed control shown in Fig. 4.41 is described by the differential equation )3 + 60y : 600w; ._ 1500w, where y is the motor speed, va is the armature voltage, and w is the load torque. Assume the armature voltage is computed using the Pl control law I va : —— (kpe + k[/ 6dr), 0 (a) Compute the transfer function from W to Y as a function of kp and k1. Wheree = r ~y. (b) Compute values for kp and k1 so that the characteristic equation of the closed-loop system will have roots at ~60 :: 60j. TW 1500 + s+60 2 >D >600 V [CY For the system in Problem 4.29, compute the following steady-state errors: (a) to a unit—step reference input; (1)) to a unit—ramp reference input; (c) to a unit—step disturbance input; ((1) for a unit—ramp disturbance input. (e) Verify your answers to (a) and (d) using MAT LAB. Note that a ramp response can be generated as a step response of a system modified by an added integrator at the reference input Consider the satellite-attitude control problem shown in Fig. 4.42 where the normalized parameters are J = 10 spacecraft inertia, N-m-secz/rad 9r 2 reference satellite attitude, rad. ...
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