hw6 - Problems 497 4 What kind of compensation improves...

Info icon This preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problems 497 4. What kind of compensation improves transient response? 5. What kind of compensation improves both steady-state error and transient response? 6. Cascade compensation to improve the steady-state error is based upon what pole— zero placement of the compensator? Also, state the reasons for this placement. 7. Cascade compensation to improve the transient response is based upon what pole- zero placement of the compensator? Also, state the reasons for this placement. 8. What difference on the S—plane is noted between using a PD controller or using a lead network to improve the transient response? .9. In order to speed up a system without changing the percent overshoot, 'where must the compensated system’s poles on the s—plane be located in comparison to the uncompensated system’s poles? 10. Why is there more improvement in steady-state error if a PI controller is used instead of a lag network? 11: When compensating for steady-state error, what effect is sometimes noted in the transient response? . 12. A lag compensator with the zero 25 times as far from the imaginary axis as the compensator pole will yield approximately how much improvement in steady-state error? 13. If the zero of a feedback compensator is at —-3 and a closed-loop system pole is at -3.001, can you say there will be pole-zero cancellation? Why? 14. Name two advantages of feedback compensation. um PROBLEMS 1. Design a PI controller to drive the step \tiiieYPlUS response error to zero for the unity feed- back system shown in Figure P9.1“, where . 5 K (Emmi suutm" b. Use MATLAB to simulate your design for K = 1. Show both the input ramp and the output response on the same plot. G(s) = ————.— (S + ”(.5 + 3) (S + .10) . 3‘. The unity feedback system shown in Figure P9.1 with The- system operates With a damping ratio of 0.5. Compare the specifications of the uncompensated and GU) = K compensated systems. [Section: 9.2] (s + l)(s + 3)(s + 5) is operating with 10% overshoot. [Section: 9.2] a. What is thewvalue of the appropriate static error constant? Figure P9.1 b. Find the transfer function of a lag network so that the appropriate static error constant equals 4 with- 2. Consider the unity feedback system shown in out appreciably changing the dominant P0168 0f Figure P9.1, where the-untmmpensated system. K c. Use MATLAB or any other computer “ATM“ G(s) = ——m program to simulate the system to 5“ + 2)“ ‘l' 5) see the effect of your compensator. a. Design a :PI controller to drive the ramp response error to zero for any K that yields stability. 4. Repeat Problem 3 for G(s) = —-———. [Section: 9.2] [Section: 9.2] 5“ + 3X5 + 5) K 498 5. Consider the unity feedback system shown in Figure P9.1 with G(s) = K (S + 2)(s + 4)(s + 6) a. Design a compensator that will yield KP = 20 without appreciably changing the dominant pole location that yields a 10% overshoot for the uncompensated system. [Section: 9.2] b. Use MATLAB or any other computer program to simulate the uncom- pensated and compensated systems. c. Use MATLAB or any other computer program to determine how much time it takes the slow response of the lag compensator to bring the output to within 2% of its final compensated value. mATlAs “Ml—As 6. The unity feedback system shown in Figure P9.1 with K(s+6) Gfiy=o+2xa+no+s) is operating with a dominant—pole damping ratio of 0.707. Design a PD controller so that the settling time is reduced by a factor of 2. Compare the transient and” steady-state performance of the uncompensated: and “compgrflted systems. Describe any problems with year design. [Section: 9.3], I “ATLAB 7. Redo Problem 6 using MATLAB in the following way: a. MATLAB will generate the root locus for the uncompensated system along with the 0.707 damping ratio line. You will interactively select the operating pdint. MATLAB will then inform you of the coordinates of the operating point, the gain at the operating point, as well as the estimated 0/005, T5, T,,, 1;, can, and K,J repre- sented by a second-order approximation at the operating point. b. MATLAB will display the step response of the uncompensated system. c. Without further input, MATLAB will. calculate the compensated design point and wiil then ask you to input a value for the PD compensator zero from the keyboard. MATLAB will respond with a plot of the root locus showing the compensated design point. MATLAB will then allow you to Chapter 9 Design via Root Locus i keep changing the PD compensator value from the: keyboard 'until a root locus is plotted that goes through the design point. d. For the compensated system, MATLAB will inform you of the coordinates of the operating point, the gain at the operating point, as well as the estimated %05, T5, Tp, t, (on, and KD repre- sented by a second-order approximation at the operating point. e. MATLAB will then display the step response of the compensated system. 8. Design a PD controller for the system shown in Figure P92 to reduce the settling time by a factor of 4 while continuing to operate the system with 20% overshoot. Compare your performance to that ob tained in Example 9.7. L s(s + 5)(s + 15) rFigure F92 9. Considerthe unity feedback system shown in Figure . P9.l with__[Section: 9.3] ' K Q+W a. Find the location of the dominant poles to yield a 1.6 second settling time and an overshoot of 25%. G(s)'= b. If a compensator with a zero at —I is used to achieve the conditions of Part a, what must the angular contribution of the compensator pole be? c. Find the location of the compensator pole. d. Find the gain-required to meet the requirements stated in Part a. e. Find the location of other closed-loop poles for the compensated system. f. Discuss the validity of your second-order approx- imation. 9. Use MATLAB or any other computer program to simulate the compen- sated system to check your design. 10. The unity feedback system shown in Figure. P9.1 with amt; i1. 12. is to be designed for a settling time of 1.667 seconds and a 16.3% overshoot. If the compensator zero is placed at —.1, do the following: [Section: 9.3] a. Find the coordinates of the dominant poles. b. Find the compensator pole. c. Find the system gain. d. Find the location of all nondorninant poles. e. Estimate the accuracy of your second-order approximation. f. Evaluate the steady—state error characteristics. 9. Use MATLAB or any other computer mama program to simulate the system and evaluate the actual transient res- ponse characteristics for a step input. Given the unity feedback system of Figure P9. 1 , with K (5 -l- 6) 1 (s + 2)(s + 4)(s + 7)(s + 8) do the following: [Section: 9.3] G(s) = a. Sketch the root locus. b. Find the coordinates of the dominant poles for which i; = 0.8. c. Find the gain for whichc = 0.8. d? If the system is to be cascade-compensated so that T5 = 1 second and i; = 0.8. find the compensator poleif the compensator zero is at —4.5. e. Discuss the validity of "your second—order approx- iman'on. f. Use MATLAB or any other computer program to simulate the compen- sated and uricompensatedzsystems and compare the results to those expected. MATLAB “ATLAS Redo Problem 1 1 using MATLAB in the following way: a. MATLAB 'will ' generate the root locus for the uncompensated system along with the 0.8 damp- ing~ratio line. You will interactively select the oper- ating point. MATLAB will then inform you of the coordinates of the operating point, the gain at the operating point, as well as the estimated %05, T5, Tp, g, a)", ande represented by a second-order approximation at the operating point. 13. 499 Problems b. MATLAB will display the step response of the uncompensated system. c. Without further input, MATLAB will calculate the compensated design point and will then ask you to input a value for the lead compensator pole from the keyboard. MATLAB will respond with a plot of the root locus showing the com- pensated design point. MATLAB will then allow you to keep changing the lead compensator pole value from the keyboard until a root locus is plotted that goes through the design point. d. For the compensatedssystem, MATLAB will in— form you of the coordinates of the operating point, the gain at the operating point, as well as the estimated %05, T5, Tp, ;, can, and Kp repre- sented by a second—order approximation at the operating point. e. MATLAB will then-displayvthe step response of the compensated system. f. Change the compensator's zero location a few times and collect data on the compensated sys- tem to see if any other choices of compensator zero yield advantages over the original design. Consider the unity feedback System of Figure 139.1 with _ K — s(s + 20)(s + 40) The system is operating at 20% overshoot. Design a compensator to decrease the settling time by a factor of 2 without affecting the percent overshoot and do the following: [Section: 9.3] G(S) a. Evaluate the uncompensated system’s dominant poles, gain, and settling time. b. Evaluate the compensated system’s dominant poles and settling time. c. Evaluate the compensator’s pole and zero. Find the required gain. d. Use MATLAB or any other computer program to simulate the compen— sated and uncompensated systems' step response. Man-As 14. The unity feedback system shown in Figure P9.l with K G(s) : (s +15)(52 -l— 63 +13) is operating with 30% overshoot. [Section: 9.3] 500 a. Find the transfer function of a cascade compen- sator, the system gain, and the dominant pole location that will cut the settling time in half if the compensator zero is at —7. b. Find other poles and zeros and discuss your sec- ond-order approximation. c. Use MATLAB or any other computer program to simulate both the un— compensated and compensated sys- tems to see the effect of your compensator. “ATLAE 15. For the unity feedback system of Figure P9.1 with K 5(5 ~l—1)(s2 +10: +26) do the following: [Section: 9.3] G(s) = a. Find the settling time for the system if it is operat- ing with 15% overshoot. b. Find the zero of a compensator and the gain, K, so that the settling time is 7 seconds. Assume that the pole of the compensator is locatedat —15. c. Use MATLAB or any other computer program to simulate the system's step response to test the compensator. 16. A unity feedback control system has the following forward transfer function: [Section: 9.3] G(s) = K 52(5 + 4)(s +12) a. Design a lead compensator to yield a closed-loop step response with 20.5% overshoot and a settling time of 3 seconds. Be sure to specify the value of K. b. Is your second—order approximation valid? c. Use MATLAB or any other computer “Am” program to simulate and compare the transientresponseofthecompensated system to the predicted transient response. 17. For the unity feedback,system of Figure P9.1, with _ K Gts) — mm the damping ratio for the dominant poles is to be 0.4, and the settling time is to be 0.5 second. [Section: 9.3] a. Find the coordinates of.the dominant poles. b. Find the location of the compensator zero if the compensator pole is at —15. Chapter 9 Design via Root Locus c. Find the required system gain. d. Compare the performance of the uncompensated and compensated systems. e. Use MATLAB or any other computer program to simulate the system to check your design. Redesign if necessary. “Ml-As 18. Consider the unity feedback system of Figure P9.1, with K G“) = m a. Show that the system cannot operate with a settling . time of 2/3 second and a percent overshoot of 1.5% with a simple gain adjustment. b. Design a lead compensator so that the system ' meets the transient response characteristics-oi Part a. Specify the compensator's pole, zero, and the required gain. 19. Given the unity feedback system of Figure P9.1 with K 0(3) = W Find the transfer function of a lag-lead compensator i . that will yield a settling time 0.5 second shorter than . that of the uncompensated system, with a damping ratio of 0.5, and improve the steady-state error bya factor of 30. The compensator zero is at ;5. Also, find the compensated system’s gain. Justify any second- order approximations or verify the design through simulation. [Section_: 9.4] 20. Redo Problem 19 using a lag—leradrcom— “TL“ pensator and MATLAB in the following way: a. MATLAB will generate the root locus for the” uncompensated system along with the 0.5, damping-ratio line. You will interactively select the operating point. MATLAB will then proceed to inform you of the coordinates of the operat- ing point, the gain at the operating point, aswell as the estimated %05, T5, Tp, E, can, and KR represented by a second-order approximation, at the operating point. b. MATLAB will display the step response of the uncompensated system. 21. c. Without further input, MATLAB will calculate the compensated design point and will then ask you to.input a.value for the lead compensator pole from the keyboard. MATLAB will respond with a plot of the root locus showing the com- pensated design point. MATLAB will then allow you to keep changing the lead compensator pole value from the keyboarduntil a root locus is plotted that goes through the design point. d. For the compensated system, MATLAB will "inform you of the coordinates of the operating point, the gain at the operating point, as well as the estimated %05, T5, TD, (1, run, and'Kp repre- sented by a second-order approximation at the operating point. e. MATLAB will then display the step response of the compensated _System. f. Change the compensator‘s zero location a few times and collect data on the compensated system to see if any other choices of the compensator zero yield advantages over the original design. 9. Using the steady-state error of the uncompen- sated system, add a lag compensator to yield an improvement of 30 times over the uncOmpen- sated system’s steady-state error, with minimal effect on the designed transient response. Have MATLAB plot the step response. Try several values for the lag compensator's pole and see the effect on the step response. Given the-uncompensated unity feedback system of Figure P9.1, with _K__ s(s +1)(s —l— 3) do the following: [Section: 9.4] Gfs) = a. Design a compensator to yield the following spe- cifications: settling time = 2.86 seconds; percent overshoot = 4.32%; the steady-state error is to be improved by a factor of 2 over the uncompensated system. b. Compare the transient and steady—state error spe- cifications of the uncompensated and compen- sated systems. c. Compare the gains of the uncompensated and compensated systems. d. Discuss the validity of your second-order approx- imation. 501E Problems e. Use MATL‘AB or any other computer program to simulate the uncompen— sated and compensated systems and verify the specifications. 22. For the unity feedback system given in WWW-Us Figure P9.1 with ' - K 6(3) : s(s+ 5)(s—l— 11) do the following: [Sectiom 9.4] a. Find the gain, K, for the uncompensated system to operate with 30% overshoot. b. Find the peak time and K, for the uncompensated system. c. Design a lag-lead compensator to decrease the peak time by a factor of 2, decrease the percent overshoot by a factor of 2, and improve the steady- state error by a factor of 30. Specify all poles, zeros, and gains. 23. The unity feedback system shown in Figure P9.] with . K G“) = (52 +4s+ 8)(s+ 10) is to be designed to meet the following specifications: Overshoot'. Less than 25% Settling time: Less than 1 second Kp = 10 Do the following: [Section: 9.4] 3. Evaluate the performance of the uncompensated system operating at 10% overshoot. b. Design a passive compensator to meet the desired specifications. 1:. Use MATLAB to simulate the comp- ensated system. Compare the response with the desired specifica- tions. “ATLAB 24. Consider the unity feedback system in Figure P9.1, with _K__ (s + 2) (s + 4) The system is operated with 4.32% overshoot. In order to improve the steady-state error, KP is to be C(s) = 502 Chapter 9 Design via. Root Locus increased by at least a factor of S.~A lag‘ compensator of the form (5 +0.5) (3 + 0.1) is mine used. [Section: 9.4] GAS) = a. Find the gain required for both the compensated and the uncompensated systems. b. Find the value of K p for both the compensated and the uncompensated systems. c. Estimate the percent overshoot and settling time for both the compensated and the uncompensated systems. d. Discuss the validity of the second-order approx- imation used for your results in Part c. e. Use MATLAB or any other computer program to simulate the step response for the uncompensated and compensated systems. What do you notice about the compensated system's response? f. Design a lead compensator that will correct the objection you notice in Part e. “ATLAs ’2\5.For the unity feedback system in Figure P91, with \/ 02(3) (5+1) @mQA design a P’ID controller that will yield a peak time of 1.047 seconds and a damping ratio of 0.8, with zero 29 error for a step input. [Section: 9.4] 26. For the unity feedback system in Figure P9.l, with K \{t‘ileyPtUs (s + 4)(3 + 6)(s +10) do the following: 0(5) = f: ‘ . 5 ontrial smutla" a. Design a controller that will yield no more than 25% overshoot and no more than a 2 second settling time for a step input and zero steady-state error for step and ramp inputs. “ATLAS, b. Use MATLAB and verify your design. 27. Redo Problem 26 using MATLAB in the following way: 28. . For the unity feedback system in Figure P9,] with} 30: a. MATLAB will ask .tor the desired percent over- shoot,.settling time, and PI compensator zero. b. MATLAB will design‘the PD controller's zero. c. MATLAB will display the root locus of the PID- . compensated system with the desired percent : overshoot line. d. The user will interactively select the intersection of the root locus and the desired percent over- shoot line. e. MATLAB will display the gain and transient response characteristics of the PID- -compen sated system. - 1 f. MATLAB will display the step response of the . FIB-compensated system. 1 g. “MATIAB will display the ramp reSponse of the f FIB-compensated system. If- the system of Figure P93 operates with a damping _ ratio of 0.517 forthe dominant second-order poles. ‘ find the location of all closedrloop poles and zeros. l (s + 2) Figure P93 K s(s+ 2)(s+4)(S+ 6) do the following: [Section: 9.5] GlSJ= a. Design rate feedback to yield a step response with no more than 15% overshoot and no more than3 seconds settling time. Use Approach 1. b. Use MATLAB and simulate your compensated system. MATLAs Given the system of Figure P94: :[Section: 9.5] a. Design the value of K1, as well as a in the feedbaclr . : path of the minor loop, to yield a settling time all : second with 5% overshoot for the step] response - b.. Design the value of K to yield a major-loop . response with 10% overshoot for a step input. “AW-AR c. Use MATLAB many other computer program tolrr‘simulate the step response toTthe'entire closed-loop system. d. Add a PI 'cornpensator to reduce the major-bop steady-state error to zerogand simulate the step response using MATLAB or any other com- puter program. “ATLAB Figure P9.4 31. Identify and realize the following controllers with operational amplifiers. [Section: 9.6] s-l—0.01 s b.s+2 a. 32. Identify and realize the following com— pensators with passive networks. [Section: 9.6] a s+0.1 5+0.01 19 5+2 V O b. Lt/ v 5+5 t”l c s + 0.1 s + 1 ' 5+0.01 s+'10 E33. Repeat Problem 32 using operational amplifiers. [Section: 9.6] Design Problems .34. The room temperature of an 11 m2 room is to be controlled by varying the power of an indoor radiator. For this specific room the open-loop transfer function from radiator power, Q(s), to temperature, T(s), is (Thomas, 2005) , To) , (1 x 10—6)s2 + (1.314 x10’9)s+{2.66 x 10-“) G _ . _—___m (r) 9(5) s3 + 0.0016332 + (5.272 x 10-7); + (3.533 x 10~11) 503 Problems The system is assumed to be in. the closed-loop configuration shown in Figure P9.1. a. For a unit step input, calculate the steady-state error of the system. b. Try using the procedure of Section 9.2 to design a PT controller to obtain z...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern