hw5 - a region of gain order to make a second-order...

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Unformatted text preview: a region of gain? order to make a second-order approximation. or a negative-feedback system? transient response. . . iPROBLEMS 1. For each of the root=loci shown in Figure P81, tell whether or not the 'sketch can be a root lpcus. If the sketch cannot be a root locus, explain why. Give all reasons. [Section: 8.4] 1'01 if!) srplane sAplane 5 3 ' (a) (b) ja) s—plane h ‘ 1 a (4) 1'91 1'60 s-plane s—plane 5 6 ' (9) (f) fa: if” s—plane . s—plane 6 , I 5 (g) (it) Figure P8.1 417 Problems 9. How can you tell from the root locus that the natural frequency does not change over It}. How would you determine whether or not a root locus plot crossed the real axis? 11. Describe the conditions that must exist for all closed-loop poles and zeros in 12. What rules for plotting the root locus are the same whether the system is a positive- 1_3. Briefly describe how the zeros of the open—loop system affect the root locus and the 2. Sketch the general shape of the root locus for each of the open-loop pole-zero plots shown in Figure P82. [Section: 8.4] if” jtu s-plane s—plane a a (a) (b) 1'60 jw s-plane s-plane U. 0' a (e) (d) 1'01 jw splane splane a i a (e) (D Figure P8.2 418' Chapter 8 Root Locus Techniques 3. Sketch the root locus for the unity feedback system shown in Figure PS3 for the following transfer func- tions: [Section: 8.4] Figure P83 Figure P8.4 K (s + 2)(S + 6) ' a. GS) = _—S2 + 83 + 25 7. Sketch the root locus of the unity feedback system K ( 2 + 4) shown .in Figure,P8.3, where S 2 " (s+5)(5+6) c. G 5 = M and find the break-in and breakaway points. () 2 3 [Section: 8.5] K d- Gm = ‘—3—7 8. The characteristic polynomial of a feedback control (3+1) (5+ ) system, which is the denominator of the closed- 4_ Let loop transfer function, is given by 33 + 252 + (20K + 7): + 100K. Sketch the root locus for this K(5 + 3) system. [Section: 8.8] G0") L" m 9. Figure P85 shows open~loop poles and zeros. There are two possibilities for the sketch of the root locus. . . . ‘ Sketch each of the two possibilities. Be aware that m Figure P83 [Section 8'5] only one can be the real locus for specific open-loop a. Plot the root locus. pole and zero values. [Section: 8.4] b. Write an expression for the closed-loop transfer function at the point where the three closed-loop poles meet. jt’D 5. Let HKo+1f 0“) = m with K > 0 in Figure P853). [Section: 8.5] a. Find the range of K for closed-loop stability. b. Sketch the system’s root locus. Figure P85 c. Find the position of the closed—loop poles when K = l and K = 2. 10. Plot the root locus for the unity feedback system shown in Figure P83, where 6. For the open-loop pole-zero plot shown in Figure 2 P8.4, sketch the root locus and find the break—in point. (KS) :- W [Section: 8.5] (5 "l" 3X3 — 3) For what range of K will the poles be in the right half- plane? [Section: 8.5] 11. For the unity feedback system shown in WW1“? Figure P83, where K(s2 — 9) G“) = W sketch the root locus and tell for what values of K the system is stable and unstable. [Section: 8.5] 12. Sketch -.the root locus for the unity feedback system shown in Figure P83, where Ids? + 2) G“) = (we???) Give the values for all critical points of interest. Is the system ever unstable? If so, for what range of K? [Section: 8.5] 13. For each system shown in Figure P86, make an accurate plot of the root locus and find the following: [Section: 8.5] a. The breakaway and break—in points b. The range of K to keep the system stable c. The value of K that yields a stable system with critically damped second-order poles d. The value of K that yields a stable system with a pair of second-order poles that have a damping ratio of 0.707 KG + 2)(s + l) (s — 2X5 — 1) System 1 K(s+2)(5+l) (2—25+2) S 03 System 2 Figure P8.6 14. Sketch. the root locus and find the range of K for stability for the unity feedback system shown in s; 419 Problems Figure P8.3 for the following conditions: [Section: 8.5] _ [{(s2 + 1) 3. 0(3) (s _ ”(s + 2)(s + 3) b. C(S) _ K(52 —23 + 2) 15‘. For the unity feedback system of Figure P83, where ' _ K(s+3) _ (s2 + 2)(s — 2)(s + 5) 0(5) sketch the root locus and find the range of K such that there' will be only two right-half—plane poles for the closed-loop system. [Section: 8.5] 16. For the unity feedback system of Figure P83, where K 0(5) = s(s+ 6)(s + 9) plot the root locus and calibrate your plot for gain. Find all the critical points, such as breakaways, asymptotes, jag-axis crossing, and so forth. [Section: 8.5] 17. Given the unity feedback system of Figure P83, make an accurate plot of the root locus for the following: _ K(32 — 2s + 2) 3. 6(3) _ m b. Gm K(s —1)(s —— 2) = (s+1)(s+2) Calibrate the gain for at least four points for each case. Also find the breakaway points, the jm-axis crossing, and the range of gain for stability for each case. Find the angles of arrival for Part a. [Section: 8.5] 18. Given the root locus shown in Figure P87, [Section: 8.5] a. Find the value of gain that will make the system marginally stable. b. Find the value of gain for which the closed-loop transfer function will have a pole on the real axis at 75. 420 Figure P8.7 19. Given the unity feedback system of Figure P83, where K(s+ l) s(s+2)(s+3)(s+4) do the following: [Section: 8.5] Gfs) = a. Sketch the root locus. b. Find the asymptotes. c. Find the value of gain that will make the system marginally stable. d. Find the value of gain for which the closed-loop transfer function will have a pole on the real axis at ——0.5. 20. For the unity feedback system of Figure \lllleypl-US P83, where - . "-- K (s + a!) s(s + 3)(s + 6) find the values of o: and K that will yield a second— order closed-loop pair of poles at —l :I: j 100. [Section: 8.5] . 5 0(5) = Q°"trolsiilut'°" 21. For the unity feedback system of Figure P83, where _ [{(s —1)(s — 2) C(S) s(s + 1) sketch the root locus and find the following: [Section: 8.5] a. The breakaway and break-in points b. The jcu-axis crossing c. The range of gain to keep the system stable Chapter 8 Root Locus Techniques d. The value of K to yield a stable system with second- order complex poles, with a damping ratio of 0.5 22. For the unity feedback system shown in Figure P83, where _ K(s +10)(s + 20) _ (s + 30)(s2 — 203 + 200) do the following: [Section: 8.7] (3(5) a. Sketch the root locus. b. Find the range of gain, K, that makes the system stable. c. Find the value of K that yields a damping ratio of 0.707 for the system’s closed-loop dominant poles. d. Find the value of K that yields closed-loop criti- cally damped dominant poles. 23. For the system of Figure P88 (a), sketch utteyPtUg the .root locus and find the following: ‘ [Section: 8.7] v w)— (s + l)(s + 2)(s + 3x; + 4) I (b) Figure P8.8 a. Asymptotes b. Breakaway points c. The range of K for stability d. The value of K to yield a 0.7 damping ratio for the dominant second-order pair To improve stability, we desire the root locus to cross the jay-axis at j5.5. To accomplish this, the open-loop func- tion is cascaded with a zero, as shown in Figure P8.8(b). e. Find the value of a and sketch the new root locus. f. Repeat Part c for the new locus. ’ 9. Compare the results of Part c and Part f. What improvement in transient response do you notice? .24. Sketch the root locus for the positiveqfeedback system shown in Figure P8.9. [Section: 8.9] Figure P8.9 25. Root loci are usually plotted for variations in the gain. Sometimes we are interested in the variation of the closed-loop poles as other parameters are changed. For the system shown in Figure P810, sketch the root locus as or is varied. [Section: 8.8] Figure 8.10 26. Given the unity feedback system shown in Figure P83, where K 0(3) = (5+ l)(s+2)(s+3) do the following problem parts by first making a second-order approximation. After you are finished with all of the parts, justify your second-order approx- imation. [Section: 8.7] a. Sketch the root locus. r’ b. “Find K for 20% overshoot. c. For K found in Part b, what is the settling time, and what is the peak time? d. Find the Ideations of higher-order poles for K "found in Pan [3. e. Find the range of K for stability. 27. For/the unity feedback system shown in Figure P83, where Ids? 125 + 2) GQF=Q+2X&+®@+5M5+® do the following: [Section: 8.7] a. Sketch the root locus. b. Find-the asymptotes. c. Find the range of gain, K, that makes‘the system stable. 28. 29. 30. 31. 421 Problems d. Find the breakaway‘points. e. Find the value of K that yields a closed-loop step response with 25% overshoot. f. Find the location of higher-order closed-loop poles when the system. is operating with 25% overshoot. 9. Discuss the validity of your second-order approx- imation. h. Use MATLAB to obtain the closed- Ioop step response to validate or refute your second-order approximation. NthTLAR The unin feedback system shown in Figure 8.3, where Kn+mn+n G“) = W— is to be designed for minimum damping ratio. Find the following: [Section: 8.7] a. The value of K that will yield minimum damping ratio b. The estimated percent overshoot for that case c. The estimated settling time and peak time for that case d. The justification of a second-order approximation (discuss) e. The expected steady-state error for a unit ramp input for the case of minimum damping ratio For the unity feedback system shown in Figure P83, where [((s + 2) 5(5 + 6) (s +10) .find K to yield closed-loop complex poles with- a damping ratio of 0.55. Does your solution require a justification of a second-order approximation? Explain. ::[Section: 8.7] For the unity feedback system shown in WWW? Figure P8.3, where ‘ . G(s) G(s) = __ K(s + a) _ 5(5 +1)(s +10) tindt the value of or so that the system will have a settling tirrieofA seconds for large values of K. Sketch the resulting' root locus. [Section: 8.8] For the unity feedback system, shown in Figure 8.3, where K(s+6) Gts) — W 422 32. 33. 34.. 35. design K and a so that the dominant complex poles of the closed-loop function have a damping ratio of 0.45 and a natural frequency of 9/8 radfs. For‘the unity feedback system shown in Figure 8.3, where K 0(5) = so +5)“ + 4)(s + 8) do the following: [Section: 8.7] a. Sketch the root locus. b. Find the value of K that will yield a 10% overshoot. c. Locate all nondominant poles. What can you say about the second-order approximation that led to your answer in Part 13‘? d. Find the range of K that yields a stable system. mATLAa Repeat Problem 32 using MATLAB. Use one program to do the following: a. Display a root locus and pause. b. Draw a close—up ofthe root locus where the axes go from —2 to O on the real axis and —2 to 2 on the imaginary axrs. c. Overlay the 10% overshoot line on the close-up root locus. d. Select interactively the point where the root locus crosses the 10% overshoot line, and respond with the gain at that point as well as all of the.closed-loop poles at that gain. e. Generate the step response at the gain for 10% overshoot. For the unity feedback system shown in WWII! Figure 8.3., where " _ K(sz+4s+5) . 701+m+SXs+ao+4) do“the following: [Section: 8.7] 0(5) a. Find the gain, K, to yield a 1-second peak time if one assumes a second-order approximation. b. Check the accuracy of the second— onus order approximation-using MATLAB to simulate the system. For the unity feedback system shown in Figure P83, where _1__ K(s+2)(s+3) G“) "(s2 +2s+2J<s+—4' )(s+5)(s+6) Chapter 8 Root Locus Techniques 36. 37. 38. do the following: [Section: 8.7] a. Sketch the root locus. b. Find the jco-axis crossing and the gain, K, at the crossing. C. Find all breakaway and break-in points. (I. Find angles of departure from the complex poles. e. Find the gain, K, to yield a damping ratio of 0.3 . " for the closed-loop dominant poles. Repeat Parts. a through c and e of Froblem 35 for [Section'. 8.7] K(s+ 8) s(s +2) (.5 + 4)(s + 6) For the unity feedback system shown in Figure P83, where Gfs) '2 K (sr'wi- 3)(s2 + 45 + 5) do the following: [Section: 8.7] 0(3) Z a. Find the location of the closed loop dominant polesll the system is operating with 15% overshoot. b. Find the gain for Part a. c. Find‘all other closed-loop poles. d. Evaluate the accuracy of your second-order approximation. For the system shown in Figure P811, do the follow- ing: [Section‘ 8.7] Figure P8.11 a. Sketch the root locus. .b. Find the jag-axis crossing and the gain, K. at the crossing. C. Find the real-axis breakaway to two-decimal-place accuracy. d. Find angles of arrival to the complex zeros. 9. Find the closed-loop zeros. ;. f. Find the gain, K, for a closed-loop step response with 30% overshoot. ‘9. Discuss the validity of your second—order approx- imation. 39. Sketch the root locus for the system of Figure P8.12 and find the following: [Section: 8.7] _K—_. 5(5 + 3)(s + 7)(s + 8) (s + 30) (s2 + 20s + 200) Figure P8.12 _i a. The range ofi gain to yield stability ' b. The value of gain that will yield a damping ratio of 0.707 for the system’s dominant poles i c. The value of gain that will yield closed-loop poles 1 that are critically damped l “ATLAB 40; Repeat Problem 39 using MATLAB. The program will do the following in one program: a. Display a root locus and pause. b. Display a close-up of the root locus where the axes go from —2 to 2 on the real axis and —2 to 2 on the imaginary axis. c. Overlay the 0.707 damping ratio line- on th close—up root locus. ‘ d. Allow you to select interactively the point where l the root locus crosses the 0.707 damping ratio line, and respond by displaying the gain at that point as well as all of the closed-loop poles at that gain. The program will then allow you to select interactively the imaginary-axis crossing and respond with a display of the gain at that point as well as all of the closed—loop poles at that gain. Finally, the program will repeat the evaluation for critically damped dominant closed—loop poles. e. Generate the step response atthe gain for 0.707 damping ratio. {t_.u.:_:_i:m-uma .. .. _. ti; .\ Problems 6 423 41. Given the unity feedback system shown «Wilts in Figure P83, where 9 fl K(s + z) . ( 6(3) — s2(s +20) 501M do the following: [Section: 8.7]" a. If .2 = 6, find K so that the damped frequency of oscillation of the transient response is 10 rad/s. b. For the system of Part a, what static error constant (finite) can be specified? What is its value? c. The system is to be redesigned by changing the values of z and K. If the new specifications are %OS = 4.32% and Ts = 0.4 s,findthe new values of z and K. 42. Given the unity feedback system shown in Figure P83, where K (s +1)(s + 3)(s + 6)2 find the following: [Section: 8.7] C(s) : a. The value of gain, K, that will yield a settling time of 4 seconds 5b. The value of gain, K, that will yield a critically damped system 43. Let K(s — l) (s + 2)(s + 3) in Figure P83. [Section: 8.7]. G(s) = a. Find the range of K for closed-loop stability. b. Plot the root locus for K > 0. c. Plot the root locus for K < 0. d. Assuming a step input, what value of K will result in the smallest attainable settling time? e. Calculate the system’s ess for a unit step input assuming the value of K obtained in Part (I. f. Make an apprOXimate hand sketch of the unit step response of the system if K has the value obtained in Part (1. 4-4. Given the unity feedback system shown in Figure P83, where am K = s(s+ l)(s+5) 424 Chapter 8 Root Locus Techniques evaiuate the pole sensitivity or the closed-loop system 45. Figure P8. 130:) shows a robot equipped to perform if the second—order, underdamped closed-loop poles arc welding. A similar device can be configuredasa are set for [Section: 8.10] six-degrees-of-freedo‘m industrial robot that can I transfer objects according to a desired program a' C: 0591 Assume the block diagram of the swing motionf b- E = 0456 system shown in Figure P8.13(b). If K: 64,510, ; . c. Which of the two previous cases has more desir— make a second-order approximation and estimate 3 able sensitivity? the following (Hardy, 1967): -‘ Load ' actuator Input ' Rem position + __K__ pOSIuon _ 3‘2 + 75 + 1220 Network Pressure Position feedback (I?) Figure P8.13 a. Robot equipped to perform arc welding; I). block diagram for swing motion system a. Damping ratio b. Percent overshoot c. Natural frequency d. Settling time a. Peak time What can you say about your original second-order approximation? i 46. During ascent the automatic steering program aboard the space shuttle provides the interface between the low-rate processing of guidance (commands) and the high-rate processing of flight control (steering in response to the commands). The function performed is basically that of smoothing. A simplified represen- tation of a maneuver smoother linearized for coplanar maneuvers is shown in Figure P8.14. Here 6CB(S) is the commanded body angle as calculated by gui- dance, and 633(5) is the desired body angle sent to flight control after smoothing.3 Using the methods of Section 8.8, do the following: ‘ a. Sketch a root locus where the roots vary as a function of K3. b. Locate the closed-loop zeros. c. Repeat Parts 3 and b for a root locus sketched as a function of K2. Figure P8.14 Block diagram of smoother 47-. Repeat Problem 3 but sketch your root loci for negative values of K. [Section: 8.9] 48. Large structures in space, such as the space station, have to be stabilized against unwanted vibration. One niethod is to use an active vibration absorber to control‘ the structure, as shown in Figure P8.15(a) (Brunei; 1992). Assuming that all values except the JSource: Rockwell International. vibration absorber Output structure acceleration C (5) Input force / Active vibration absorber (b) Figure P8.15 3. Active vibration absorber (©1992 AIAA); b. control system block diagram. mass ofthe active vibration absorben areknown and are equalto unity, do the following: a. Obtain 6(5) and Ho) = H1(s)H2(5) in the block diagram representation of the system of Figure 8.115(1)), which shows that the active vibration absorber acts as a feedback element to control the structure. (Hint: Think of K5 and Dc as pro- ducing inputs to the structure.) b. Find the steady-state position of the structure for a force disturbance input. c. Sketch the root locus for the system as a function of active vibration absorber mass, MC. 49. Figure P8.16 shows the block diagram of the closed- loop control of the linearized magnetic levitation Figure P8.16 Linearized magnetic levitation system block diagram 426 50. 51. system described in Chapter 2, Problem 58. (Galvao, 2003). Assuming/1 = 1300 and n = 860, draw the root locus and find the range of K for closed—loop stability when: 3. 0(3) = K; F) K(s + 200) 5+ 1000 The simplified transfer function model from steering angle 6(5) to tilt angle rp(s) in a bicycle is given by b. 0(5) '= () +V gas aVS Z G's):m=ESZ-fl.g. In this model it represents the vertical distance from the center of mass to the floor, so it can be ureadily verified that the model is open-loop unstable. (Astrb‘m, 2005). Assume that for a specific bicycle, a = 0.6 m, b = 1.5 m, h = 0.8 m, andg = 9.8 m/sec. In orderto stabilize the bicycle, it is assumed that the bicycle is placed in the closed-loop configuration shown in Figure P83 and that the only available control vari- able is V, the rear wheel velocity. a. Find the range of V for closed—loop stability. b. Explain why the methods presented in this chapter cannot be used to obtain the root locus. . MATLM c. Use MATLAB to obtain the system's rootlocps. A technique to control the steering of a vehicle that follows a line located in the middle of a lane is to define a look-ahead point and measure vehicle devia- tions with respect to the point. A linearized model for such a vehicle is b1K V all 012 —biK T V . i; _ sz r 11:1 — 6121 (122 52K d “if Yg 0 1 0 0 Y3. 1 0 U 0 where V = vehicle’s lateral velocity, r = vehicle‘s yaw velocity, 1,0 = vehicle’s yaw position, and Ya 2 the y-axis coordinate of the vehicle’s center of gravity. K is a parame...
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