hw7 355

hw7 355 - a. 502 N increased by at least a factor of- 5. A...

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Unformatted text preview: a. 502 N increased by at least a factor of- 5. A lag compensator of the form _ (s + 0.5) GclS) — is to be used. [Section: 9.4] a. Find the gain required for both the compensated and the uncompensated systems. b. Find the value of Kit) 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. MATLAE ;€'E§./\For the unify feedback system in Figure P91, with K _ a ._ .le G“) = (Wm/9W design a PIID controller that will yield a”peak time of 1.047 seconds and a damping ratio of 0.8, with zero error for a step input. [section: 9.4] 26. For the unity feedback system in Figure P9.l, with 27. Redo Problem 26 using MATLAB in the K \IllleYPLUS =p+4xy+mo+1m do the following: 6(5) Enntrol Solutions 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. “ATLAB b. Use MATLAB and verifyyour design. following way: fl. r- 1' Chapter 9 Design via Root Locus 29. 30. a. MATLAB will ask for the desired percent over-_ shoot, settling time, and PI compensator zero._.i b. MATLAB will design the PD controller’s zero. i; 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. 9. MATLAB will display the gain and transient response characteristics of the PID-compen- sated system. ‘ f. MATLAB will display the step response of the PID-compensated system. 9. MATLAB will display the ramp response of the; PID-compensated system. ‘ . If the system of Figure P93 operates with {damping A ratio of 0.517 for the dominant Second-order poles, find the location of all closed-loop poles and zeros. m (s + 2) Figure P93 For the unity feedback system in Figure,P9.1, with K G = —._.——. m so+mo+nw+m do the following: [Section: 9.5] 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. ' Him-As b. Use MATLAB and simulate your compensated system. / Given the system of Figure P9.4: [Section: 9.5] a. Design the value of K 1, as well as a in the feedback path of the minor loop, to yield a settling time of 1 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. ' c. Use MATLAB or any other computer -program to simulate the step --response to’tiié'entire closed—loop system. d. Adda PI compensator to reduce the major—loop steady—state error to zero‘and simulate the step response using MATLAB or any other com- ‘puter program. smug MATlAs Figure P9.4 31. Identify and' realize the following controllers with operational amplifiers. [Section: 9.6] s+0.01 s b. 5 +2 a. a. 32. Identify and realize the following com- pensators with passive networks. [Section: 9.6] s + 0.1 s + 0.01 “‘u-Io 5+2 [3' 5+5 c s+0.l 5+1 ' s+0.01 S-i—lO 33. Repeat Problem 32 using operational amplifiers. [Section: 9.6] a. 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, (2(5), to temperature, T(s), is (Thomas, 2005) _ T_(.Q _ {1 x10’6)52 +0314 x 1W9): +9.66 x1043} Gm Q(s) _ 53 + 0.0016352 + (5.272 x [OJ-Us + (3.538 x 10‘“) .m- 3; 503 Probiems .N .r The system is assumed to be in the closed-loop configuration shown in Figure P9.l. 'n" 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 PI controllehto obtain zero steady-state error for step inputs without appreciably changing the tran- sient response. Then explain why it is not possible to do so. c. Design a PI controller of the form 06(5) = K(s + z) s to zero while not changing significantly the tran- sient response. (Hint: Place the zero of the com- pensator in a position where the closed—loop poles of the uncompensated root locus will not be affected significantly.) d. UseSimulinktosimulatethesystems of Parts b and c and to verify the correctness of your design in Part c. that will reduce the step-response error samulink 35. Figure P9.5 shows a two-tank system. The liquid inflow to the upper tank can be controlled using a valve and is represented by F0. The upper tank’s outflow equals the lower tank‘s inflow and is repre- sented by F1. The outflow of the lower tank is F 2. The objective of the design is to control the liquid level, y(t), in the lower tank. The open-loop transmission Y(S) _ 021613 F0(s) w 52 +(a1—i— a4)s + (11614 (Romagnoli, 2006). The system will be controlled in a loop analogous to that of Figure P9.l, where the for this system is F0 Figure P9.5 Chapter 9 Design via Root Locus lower liquid level will be measured and compared to a corresponding controller, TC 22, that compares the, setpoint. The resultingen'orwill be fed to acontroller, actual temperature with a desired temperature set which in turn will open or close the valve feeding the 5! point, SP. The controller autornatically opens or upper tank. closes a valve to allow or prevent the flow of steam to change the temperature in the tank. The cone 3. Assuming a; = 0.04, a; = 0.0187, 4513 = l, and a4 spending block-diagram for this system is shown in 2 0.227, design a lag compensator to obtain a Figure,P9.6(b) (SmithagOOZ). Assume the following- step-response steady-state error of 10% without transfer functions: affecting the system’s transient response appreci~ ably- Gv(s) = 0'02 ; 01(5) = 70 ; yo) = _1——. . Verify your design through MATLAB “Am” 45 + 1 505 + 1 125 +1 simulatiOns. ‘ a. Assuming GAS) = K, find the value of K that will . Figure P9.6(a) shows a =treat-exchanger process whose “33”” i" a (lominam PO]e With E = 0-7- Obtain the purpose is to maintain the temperature of a liquid at a conespondmg Tr preseribed temperature. b. Design a PD controller to obtain the same damping The temperature is measured usingasensor anda factor as Part a but with a settling time 20% transmitter, TT 22, that sends the measurement to a smaller. Condensate return Figure P9.6 a. Heat-exchanger process; b. block diagram at .2; K Problems 505 c. Verifyyourresults through MATLAB “ATM simulation: 37. Repeat Problem 36, Parts b and c, using a lead compensator. 38. a. Find the transfer function of a motor whose torque- speed curve and load are given in Figure P9.7. Torque (b) 0.5 N—m Figure P9.8 5V you cannot use a different motor, an amplifier and I tachometer are inserted into the loop as shown in 0 .. , RPM Figure P9.8(b). Find the values of K1 and Kf to =0 60.000 . . 2” yield a percent overshoot of 25% and a settling time of 0.2 second. filo—r] 1 c. Evaluate the steady-state error specifications for both the uncompensated and the compensated systems. 4 4 “ 40. A position control is to be designed with a 20% 10 a overshoot andxa settling time of 2 seconds. You . r have on hand an amplifier and a power amplifier 1 N-m-s/rad _ whose caScaded transfer function is K1 /(5 + 20) F'gure P9'7 with which to drive the motor. Two lO-turn ‘pots h are available to convert shaft position into voltage. b. Design a tachometer compensator to yield a damp- A V0hage 0f i5” VORS is PIaCEd across the POW A dc ing ratio of 0.5 for a position control employing a 1110‘“ Whose transfer funCtiOH i5 0f the form power amplifier of gain 1 and a preamplifier of gain 5000. 90(5) K_ c... Compare the transient and steady-state character- Egg) 2 5(3 + a) istics of the uncompensated system and the com- ensated s stem. . . . . p y 1s also available. The transfer function of the motor 15 l 39_ You are given the motor whose transfer Weypws found. experimentally as follows. The motor and function is Shown in Figure P9_8(a)_ geared load are driven open-loop by applying a large. short, rectangular pulse to the armature. An oscillo- gram ofthe response shows that the motor reached 63% of its final output value at 1/2 second after the _ application of the pulse. Further, with a constant 10 Of a "Pity feedback SXStemfiCalcmate the patent volts dc applied to the armature, the constant output overshoot and settling time that #could be Speed was 100 rad/S_ expected. b. You want to improve the closed-loop response. a. Draw a complete block diagram of the system, Since the motor constants cannot be changed and specifying the transfer function of each compo- ,_ I 5 Qunlrol Sulutl"f1 a.4 If this motor were the forward transfer function ,. 506 Amplifier Chapter 9 Design via Root Locus Figure P9.9 nent when the system is Operating with 20% overshoot. b. What will the steady-state error be for a unit ramp input? c. Determine the transient response characteristics. d. If tachometer feedback is‘u sed around the motor, as shown in Figure P9.9, find the tachometer and the amplifier gain to meet the original specifications. Summarize the transient and steady-state charac- teristics. 41. A position control is to be designed with a 10% overshoot, a settling time of 1 second, and K, = 1000. You have on hand an amplifier and a power amplifier whose cascaded transfer function is K1 / (s + 40) with which to drive the motor. Two 10-turn pots are available to convert shaft position into voltage. A voltage of :i:203r volts is placed across the pots. A dc motor whose transfer function is of the form 60 (s) K Ea(s) _ 5(3 + a) is also available. The following data are observed from a dynamometer test at 50 V. At 25 N—m of torque, the motor turns at 1433 rpm. At 75 N-m of torque, the motor turns at 478 rpm. The speed measured at the load is 0.1 that of the motor. The equivalent inertia, includingthe-load, at the motor armature is 100:.kg-m2, and the equivalent viscous damping, including the load, at the motor armature is 50 N—m-s/rad. a. Draw a complete block diagram of the system, specifying the transfer function of each compo- nent ‘56 b. Design a passive compensator to meet the require— ments in the problem statement. c. Draw the schematic of the compensator showing all component values. Use an operational amplifier for isolation where necessary. d. Use MATLAB or any other computer program to simulate your system and Show that all requirements have been met. “int-As 42. Given the system shown in Figure P910, find the values of K and Kf so that‘rthe closed-loop dominant poles will have a damping ratio of 0.5 and the under-‘ damped poles of the minor loop will have a damping ratio of 0.8. ~ Power amplifier Tachometer Figure P9.10 43. Given the system in Figure P9.1 1, find the values of K and Kf so that the closed-loop system will have a 4.32% overshoot and the minor loop will‘have a. : damping ratio of 0.8. Compare the expected perfor- mance of the system without tachometer compensa- tion to the expected performance with tachometer compensation. Power amplifier Amplifier Plat“ Figure P9.11 44. In Problem 55 of Chapter 8, a head-position control 46. _. Clonsi‘derthe temperature control system \tt‘tleyPtt/s system for a floppy disk drive was designed to yield a settling time of 0.1 second through gain adjustment alone. Design a lead compensator to decrease the settling time to 0.05 second without changing the percent overshoot. Also, find the required loop gain. for a chemical process shown in Figure P9. 1B2. The unc0mpensated system is - '_ operating with a rise time approxi- Qqniml Salutiof'il mately‘ the same as a second~order system with a peak time of 16 seconds and 5% overshoot. There is also considerable steady-state error. Design a PID controller so that the compensated system will have a rise time approximately equivalent to a second-order system with a peak time of 8 seconds and 5% over- shoot, and zero'steady-state error for a step input. s Steam-driven power generators rotate at a constant speed via a governor that maintains constant steam pressure in the turbine. In addition, automatic gen- eration control (AGC) or load frequency control (LFC) is added to ensure reliability and consistency despite load variations or other disturbances that can affect the distribution line frequency output. A _ PID DESll'ed Controller Amplifier temperature set point + 507 Problems ._; specific turbine-governor system can be described only using the block diagram of Figure P9.l in which 6(5) = Gc(s)Gg(s)G,(s)Gm(s), where (Kho- dabakksh ian, 2005) 63(3) 2 is the govemor’s transfer function 1 . . . G,(s) = 6m 1s the turbine transfer function 1 . Gm(s) = represents the machme and load transfer functions GAS) is the LFC compensation to be designed a. Assuming Gc(s) = K, find the value of K that will result in a dodiinant pole with l; = 0.7. Obtain the corresponding Ts. b. Design a PID controller to obtain the same damp- ing factor as in Part a, but with a settling time of 2 seconds and zero steady-state error to step input commands. c. Verify your results using a MATLAB simuation. “ATLAa 47. Repeat Problem 46 using a lag—lead compensator instead of a PID controller. Design for a steady-state error of 1% for a step "input command. 48. Digital versatile disk (DVD) players incorporate several control systems for their operations. The con— trol tasks include (I) keeping the laser beam focused on the disc surface, (2) fast track selection, (3) disc rOt'ation speed control, and (4) following a track accurately. In order to follow a track, tithe pickup- =head radial position is controlled via a voltage that operates a voice coil embedded in a magnet config- Actuator Chemical and heat Valve process Actual temperature 0.7 1 52 + 1.7s + 0.25 Temperature SCHSOI‘ Figure P9.12 Chemical process temperature control system 508 10(3) = uration. For a specific DVD player, the transfer func- .tion is given by E Vts) 0.63 v. = 0.36 32 0.04 52 l + s —l— l + s + 305.4 305.42 248.2 248.22) where x(t) = radial pickup position and v(r) = the coil input voltage (Birz‘anri, 2002). Assume that the system will be controlled in a closed-loop configuration, such as the one shown in Figure P91. Assuming that the plant, P(5), is cascaded with a proportional compensatoL Gc(s) =' K, plot the root locus of the system. b. Repeat Bart ausjng MATLAB if your root locus plot was created by any other tool. c. Find the range of ‘K for closed—loop stability, the resulting damping factor range, and the smallest settling time. a. “mm; d. Design a notch filter compensator so that the sys- tem’s dominant poles have a damping factor of E = 0.7 with a closed-loop settling time of 0.1 second. “Mum e. Simulate the system‘s step reSponse for Part c using MATLAB. f. Add a PI compensator to the system to achieve zero steady-state error for a step input without appre- ciably affecting the transient response achieved in Part D. g. Simulate the system’s step response for Part e using MATLAB. Compensator Chapter 9 Design via Root Locus 49 ""‘i i l l l i . A coordinate measuring machine (CMM) measures coordinates on three-dimensional objects. The accuracy of CMMs is affected by temperature changes as well as by mechanical resonances due to joint elasticity. These resonances are more pronounced when the machine has to go over abrupt changes of dimension, such as sharp comers at'high speed. Each of the machine links can be controlled in a closed-loop configuration, such as the one shown in Figure P9.13 for a specific machine with prismatic (sliding) links. In the figure, ch(s) is the commanded position and X(s) is the actual position. The minor loop uses a tachometer generator to obtain .. the joint speed, while the main loop controls the joint's 50. Cl.046§s(s2 + 1.155 + 0.33) position (02:81, 2003). a. Find the value of K that will result in a minor loop with i; z 0.5. b. Use a notch filter compensator, 643), for the external loop sothat it results in a closed-loop damping factor of = 0.7 with T, #4 seconds; “Mus c. Use MATLAB to simulate the compen- sated system’s closed-loop step response. Magnetic levitation systems are now used to elevate and propel trains along tracks. A diagram of a demon- stration magnetic levitation system is shown in Figure P9. [4(a). Action between a permanent magnet attachedto the Ping-Pong ball, the object to be levi—_ tated, and an electromagnet provides the lift. The amount of elevation can be controlled through ‘V, applied to the electromagriet as shown in Figure, . P9.14(a). The elevation is controlled by using a photo- detector pair to detect the elevation of the Ping-Pong ball. Assume that the elevation. control system is represented by FigureiP9.14(b) and do the following (Cho, 1993): Plant 574.98 5(52 +14245 + 3447.91) Tachometer generator Figure P9.13 Problems 509 Amplifier circuit Electromagnet Control computer Permanent magnet Ping-Pong ball Datum line . (a) Compensator I : Plant Photocell (b) Figi] re P9.14 3. Magnetic levitation system (© 1993 lEEE); I). block diagram .3. Design a compensator, G45), to yield a settling Commanded Actual tlme of 0.1 second or less 1f the step response 15 to angle of angle of have no more than 1% overshoot. Specify the film" Controller Aircraft “Wk compensator’s poles, zeros, andigain. b. Cascade another compensator to minimize the “steady-state error and have the total settling time not exceed 0.5 second. This compensator should .pot appreciably affect the transient response designed in Part a. Specify the poles and zeros of this compensator. c. Use MATLAB or any other computer program to simulate the system to check your design. 51. The transfer function for an AFTI/F-l6 aircraft relat- ing angle of attack, or(t), to elevator deflection, 69 (I), is Figure P9.15 Simplified block diagram for angle of attack control MATME a. Find the range of K for stability. b. ‘Plot or sketch a root locus. c. Design a cascade compensator to yield 'zero steady- state error, a settling time of about 0.05 second, and a percent overshoot not greater than 20%. given by “mus (1(5) d. Use MATLAB or any other computer Gm = 5 (S) 2 program to simulate the system 9 to check your design. _ 0 072 (s + 23 )(s2 + 0.053 + 0.04) ' (S n- 0-7)(S + 1.?)(S2 + 9.083 + 0.04) 52. Figure P9.16 is a simplified block diagram of a self— Assume the blockdiagram shown infigure P9. 15 for guiding vehicle's bearing angle control. Design a lead controlling the angle of attack, or, and do the following compensator to yield a closed-100p step response with (Monahemr', 1992): 10% overshoot and a settling time of 1.5 seconds. wawmdflwméggg :._.:: ' ‘ - . .. 1 510 . ll . Deslred Contra er bearing angle Figure P9.16 Simplified block diagram of a self-guiding vehicle‘s bearihg angle control Progressive Analysis and Design Problems 53. High-speed rail pantograph. Problem 20 in Chapter 1 discusses the active control of a pantograph mechanism for high-speed rail systems. In Problem 74(b). Chapter 5 , you found the block diagram for the active pantograph control system. In Chapter 8, Pro- blem 64, you designed the gain to yield a closed—loop step response with 38% overshoot. A- plot of the step response should have shown a settling time greater than 0.5 second as well as a high-frequency oscilla- tion superimposed over the step response (O’Conner; 1997). We want to reduce the settling time to about 0.3 second, reduce the step response steady-state error to zero, and eliminate the high-frequency oscillation. A way of eliminating the high-frequency oscillation is to cascade a notch filter with the plant. Using the notch filter, 52 + 165 + 9200 GAS) = (s + 60)2 do the following: a. Design a PD centroller to yield asettling time of approximately 0.3 second with no more then 60% overshoot. b. Add a PI controller to yield zero steady-state error for step inputs. c. Use MATLAB to plot the PlD/notch-compensated closed— .Ioop step response. MATLAE — ACYBER, :EXyFllORATION “LABORATORY Experiment 9.1 Objectives To perfOrr'n a trade-off study for lead..compensatiori. To design a controller and see its effect upon steady-state error. 1 Minimum required software packages MATLAB, and the Control System Toolbar . J. 3‘. nus Chapter 9 Design via Root Locus s1+10s +50 54. Control of HIV/AIDS. It was shown in Chapter ti, Vehicle Stealing dynamics Actual bearing angle. Problem 66 that when the virus levels in an HIV/AIDS patient are controlled using RTIs the linearized plant model is Y(s) _ —5205— 10.3844 U1(s) _ .93 + 2.68m2 + 0.11s + 0.0126 13(5) = Assume that the system is embedded in a configura- - tion, such as the one shown in Figure P9.l, where, ' 6(5) = GE (5) P0). Here, 06(5) is a cascade compen— .. sator. For simplicity in this problem, choose the dc ‘ gain of Gc(s) less then zero to obtain a negative- feedback system (the negative signs of (Ms) and Pp) cancel out) (Craig, 2004). a. Consider the uncompensated system with GAS) : r —K. Find the value of K that Will place all closed- loop poles on the real axis. b. Use MATLAB to simulate the unit step response of the gain-compen- sated system. Note the %OS and the T; from the simulation. c. Designta PI compensator so that the steady-state ' error for step inputs is zero. Choose a gain value to make all poles real. C]. Use MATLABto simulate the design in Part c for a unit step input. Compare the simulation to Part b. “ATLAS 590 Chapter 10 Frequency Response Techniques 31PR 0 B L E M S 1. Find analytical expressions for the mag- Wequ nitude and phase response for each 6(3) below. [Section: 10.1] i 5" G“) =m _ (3+5) '3' G“) ' (s+ 2)(s+4) c. 6(5) _ (s+3)(s+5) _ 5(5 + 2)(s +4) 2. For each function in Problem 1, make a plot the log- magnitude and the phase, using log-frequency in rad/s as the ordinate. Do not use asymptotic approxima- tions. [Section: 10.1] 3. For each function in Problem I, make a polar plot of the frequency response. [Section: 10.1] 4. For each function in Problem 1, sketch the Bode asymmwlmd asymptotic phase plots. Compare your resu ts w1t "your answers to Problem .1 . [Section: 10.2] 5. Sketch the Nyquist diagram for each of the systems in Figure P10.1. [Section: 10.4] 6. Draw the Figure P1031 polar plot 0 , - _= .. I-l—IIII J llllll-IIIIIII iiiiii ""ll IIIIIIII i! magnitude and phase curves shown in Figure P10.2. [Section: 10.1] _30 40 IIIIIIII 20 log M ['0 C Iiiiiifllllllll IIlIl-Ii'!!!!!!_!!!!!!! Elll-Illlllll-IIIIIIII Phase (degreesj | 5 G Illl-Illlllll-IIIIIIII 7200 g 0.1 1 10 100 Frequency (rad/s) 4" Figure P10.2 - Problems 591 7. Draw the separate magnitude and phase curves from the polar plot shown in Figure P103. [Section: 10.1] x 10‘3 ‘ System 1 Imaginary axis Figure No.4 Real axis x 10’3 Figure P103 11. For each system of Problem 10, find the Warm”: gain margin and phase margin if the L value of K in each part of Problem 10 ._ is [Section: 10.6] 00mmls°mtiafls 8. Write a program in MATLAB that will MATLAB do the following: a. K: 1000 a. Plot the Nyquist diagram of a system b' K2100 C. K:0.1' b. Display the real-axis crossing value and TLA frequency 12. Write a program in MATLAB that will “A a do the following: Apply your program to a unity feedback system with MS + 5) a. Allow a value of gain, K, to be entered from the G = ___——._.— (S) (s2 +65 +100)(s2 +4s+25) keyboard btiDIspIay the Bode plots of a system for the -9. Using the Nyquist criterion, find out whether each entered Value Of K system of Problem 5 is stable. [Section: 10.3] c. Calculate and display the gain and phase margin for the entered value of K 10. Using the N yqui st criterion, find the range wieyPLUs of K for stability for each of the systems Test your program on a unity feedback system in Figure PlO.4. [Section: 10.3] ' With 6(5) = K / [5(5 + 3)(5 + 12)]}' 592 13. 14. 15. 16. 17. 18. 19. Use MATLAB’S LTi Viewer to find the gain margin, phase margin; zero dB frequency, and 180° frequency for a unity feedback system with q 10,000 (5 + 5)(s+18)(s + 30) Use the following methods: GUI Too! 6(5) = a. The Nyquist diagram b. Bode plots Derive Eq. (10.54), the CIOSed-loop bandwidth in terms of C and a)" of a two-pole system. [Section: 10.8] For each closed~loop System with the following performance characteristics, find the closed-loop bandwidth: [Section: 10.8] \fll‘iBYPLUS e W . “"trur Solut'on a. I: = 0.2, T5 = 3 seconds b. i: = 0.2, Tp .2 3 seconds c. T, = 4 seconds, TP 2 2 seconds d. l: 2 0.3, T, = 4 seconds Consider the unity feedback system of stilevPLUS Figure 10.10. For each 6(3) that follows, .. use the M and N circles to make a plot of 1 ‘ the closed—loop frequency response: Clint,“ Samuel‘s [Section: 10.9] 10 a- 60 = 1000 b. 00) = W i 50(5 + 3) 6- 0(3) - Repeat Problem 16, using the Nichols chart in place of the M and N circles. [Section: 109] Using the results of Problem 16, estimate the percent overshoot that can be expected in the step response for each system shown. [Section: 10.10] Use the results of Problem 17 to estimate the percent overshoot if the gain term in the numerator of the forward path of each part of the problem is respec- tively changed as follows: [Section: 10.10] I- . 3.- . . c as Chapter 10 Frequency Response Techniques 20. 21. 22. .a. Make a Nichols plot of an open- a. From 10 to 30 b. From 1000 to 2500 c. From 50 to 75 Write a program in MATLAB that will do the following: a. Allow a value of gain, K, to be entered from the keyboard b. Display the closed-loop magnitude and phase frequency response plots of a unity feedback system with an open-loop transfer function, K 6(5) c. Calculate and display the peak magnitude, fre- quency of the peak magnitude, and bandwidth for the closed-loop frequency response and the ; entered value of K H Figure P10.5 ,r-v Test your program on the system of Figure P105 for K 2 40. Use MATLAB‘s LTI Viewer with the Nichols plot to find the gain margin, phase margin, zero dB frequency, and 180° frequency for a unity feed— back system with the forward—path transfer function GUI Too, _ _ 7(5—i—S) 6(5) _s(sl +4s+ 10) Write a «program in MATLAB that will do the following: loop transfer function b. Allow the user to read the Nichols plot display . and enter the value of Mp ‘ f c. Make closed-loop magnitude and phase plots 1 d. Display the expected values of percent over- shoot, settling time, and peak time e. Plot the closed-loop step response Test your program on a unity feedback system with the fonNard-path transfer function 6(5) = 23. Using Bode plots, estimate the transient response of the systems in Figure P106. [Sectionz l0.10] 24. For the system of Figure Pl0.5, do the following: [Section: 10.10] Problems 593 ii" “i if c. Use MATLAB orany other program to wA‘l'LAB check your assumptions by simulat— ing the step response of the system. thiewas 7(5 + 5): 5(52 + 45 + 10) 25. The Bode plots for a plant, C(s), used in a unity feedback system are shown in Figure 1310.7. Do the following: nlrul Solutions 3. Find the gain margin, phase margin, zero dB frequency, 180° frequency, and the closed-loop bandwidth. b. Use your results in Pan a to estimate the damping ratio, percent overshoot, settling time, and peak time. 50(s + 3)(s + 5) 3(3 + 2)(s + 4)(s + 6) 26. Write a program in MATLAB that will minus use an open-loop transfer function, 6(5), to do the following: System 2 Figu re P10.6l a. Make a Bode plot b. Use frequency response methods to estimatethe percent overshoot, settling time, and peak time c. Plot the closed-loop step response a. Plot the Bode magnitude and phase plots. b. Assuming a second-order approximation, estimate the transient response of the system if K: 40. Phase (degrees) Test your program by comparing the results to those obtained for the systems of Problem 23. ESI!!!!!!--IIIIIII--IIIIIII -IIIInns-IIIIIII-IIIIIIII 0 _ 40 IIIIIIII-Iiflml-IIIIIIII -40 .IIIIIIIIIIIIIIIEEJIIIIII 40 .IIIIlll-Illlll-hslfllll ,80 .IIIIIII-IIIIIIII III”!!! 400 IIIIIIII-IIIIIIII-IIIIIIII 0.1 l 10 100 Frequency (rad/s) IIIIIIIIIIIIIIIl-IIIIIIII “00 iilliii-Illllll-IIIIIIII —150 " - “' a. so, IIIIIIIl-III-l..l-IIIIIII IIIIIIIl-llllllllh'lllllll —250 4 'h-e n“-.. IIIIIIIl-IIIIIIl-IIIIIIII 0.] 1 IO 100 Frequency (rad/s) Fig ure P1 0.7 594 27. The open—loop frequency response shown in Fig- ure P108 was experimentally obtained from a unity feedback system. Estimate the percent overshoot and steady—state error of the closed—loop system. [Sec- tions: 10.10, 10.11] 28. Consider the system in Figure P109. [Section: 10.12] 100 (3+ 5)(s +10) Figure P10.9 a. Find the phase margin if the system is stable for time delays of 0, 0.1, 0.2, 0.5, and 1 second. b. Find the gain margin if the system is stable for each of the time delays given in Part a. C. For what time delays mentioned inPart a is the system stable? (I. For each time delay that makes the system unstable, how much reduction in gain is required for the system to be stable? 60 40 EEIIIIIII-IIIEII l i l I l I 20 log M ['0 o _80 IIIIIII-IIIIIII -IIIIIIII-IIIIIII 0.1 29. 30. 31. - - a - - - - - Chapter 10 Frequency Response Techniques Given a unity feedback system with the forward-path transfer function K G s = ———-—-—-— U (s +1)(s+ 3)(s+,6) and a delay of 0.5 second, find the range of gain, K, to yield stability. Use Bode plots and frequency response techniques. [Section: 10.12]— Given a unity feedback system with the forward-path transfer function W) = K 3(3 + 3)(s+ 12) and a delay of 0.5 second, make a second-order approximation and estimate the percent overshoot if K = 40. Use Bode plots and frequency response techniques. [Section: 10.12] fl Use the MATIAB function pade(T,n) to modei the delay in Problem 30. Obtain the unit step response and evaluate your second-order approximation in Problem 30. “Mus IIIIIIII-IIIIIII :IIIIIII-IIIIIII II-E!!IIIII II-IIIEI Il-IIIII 10 Frequency (rad/s) 450 _--IIIIII 400 .—.—....mgqunII-llllllll-IIIIIIII 4,0 -IIIIIIIIIn.E!l!l-Illlllll-IIIIIIIII -IIIIllll-IIIIIIIESIIIIIIII-IIIIIIII -IIIIIIll-IIIIIIIIflIIIIIIIl-IIIIIIII —l40 Phase (degrees) .1; a 8 9-3 I I - - - i I I - - E 7‘ I - _ E I I - - - = -IIIIIIII-IIIIIIII-Ililllll-IIIIIIII "l -IIIIIIII-IIllllll-IIIIIIIl-IIIIIIII -IIIIIIII-IIIIIIII-IIIIIIIIEEIIIIIII IIIIIIII-IIIIIIII-IIIIIIII-IIIIIIII 0.1 1 10 100 Frequency (rad/s) ‘25- Figure P1D.8 Problems 595 - 80 60 g!!!“ III l I IIIIII 40 III-l! -II I IIIIII i 20 IIIIII I IIIIII g 0 IIIII --- III-I IIIIII ,20 IIIIII -II IIIL! !III|| 40 IIIIIIII-IIIIIIII-II IIIII-I I!!!“ _6o __ IIIIIIII-IIIIIIII-II IIIII-I IIIIII 0.0l 0.1 1 IO EGG Frequency (rad/s) —80 -"""" l|||||-IIIII|||-IIIII||| 400 --IIIIIII—!!-_ _. IIIIIIIIIIIIIIIIIILIIIIllll-IIIIIIII IIIIII|II-IIIlllll-Ii"l|||-IIIII||| 3 E —140 i... _‘ A .IIIIIIIl-IIIIIIll-IIIIIIIINIIIIIIII 0.01 O. IIIIIIIIIIIIIIIIIIn!!!!l. 1 1 10 100 Frequency (radjs) Figure P10.10 32. For the Bode plots shown in Figure 1510.10, determine 34. An overhead crane consists of a horizontally moving the transfer function by hand or via MATLAB. trolley of mass mrdragging a load of mass mL, which [Section: 10.13] dangles from its bottom surface at the end of a rope of fixed length, L. The position of the trolley is 33. Repeat Problem 32 for the Bode plots shown in Figure P10.ll. [Section: 10.13] m ' ---IIIII IIIII ll 0 II l lllll-lll v —100 f i—fi % IIIII-IIIIIII- IIIIII E II 100 1 l0 Frequency (rad/s) Figure P10.11 596 35. controlled in the feedback configuration shown in Figure 10.20. Here, 0(5) = KP(s), H "= 1, and Xfls) __F "1— 32 + (05 HS) = FMS) — m1" s2 (52 + awfi) The input is fT (t), the input force applied to the trolley. The output is x70), the trolley displacement. Also, we = fianda = (mL + mfl/m:n (Marrtt'nen, 1990). Make a qualitative Bode plot of the system assuming» a>1. A room’s temperature can be controlled by varying - the radiator power. Inia specific room, the transfer 36. 37. function from indoor radiator power, Q, to room temperature, Tin °C is (Thomas, 2005) __ T(s) Qts) (1 x 10-6);2 + (1.314 x 10‘9)s+ (2.66 x 10-13) : s3 +0.00163s2 + (5.272‘Y10-7)s + (3.538 x 10-“) The system is controlled in the closed-loop config- uration shown in Figure 10.20 with C(s) = KP(s), H = l. a. Draw the corresponding Nyquist diagram for K = l. b. Obtain the gain and phase margins. c. Find the range of K for the closed-loop stability. Compare your result with that of Problem 61, Chapter 6. The open-loop dynamics from dc voltage armature to angular position of a robotic manipulator joint is 48500 given by P(s) z m (Low, 2005). a. Draw by hand a Bode plot using asymptotic approximations for magnitude and phase. “mus b. Use MATLAB to plot the exact Bode plot and compare with your sketch from Part a. Problem 49, Chapter 8 discusses a magnetic levitation system with a plant transfer function P(s) = 1300 m (Galvfto, 2003). Assume that the plant is in cascade with an M(s) and that the system will be controlled by the loop shown in Figure 10.20, where G(s) = M (§)P(s) and H: 1. For each M(s) shown Chapter 10 Frequency Response Techniques 38. 39. P0) = 40. below, draw the Nyquist diagram when K = 1, and find the range of closed-loop stability for K > 0. a: M5) = .-K K(s + 200) s + 1000 c. Compare your results with those obtained in Problem 49, Chapter 8. b. Mo) = — The simplified and linearized model for the transfer function of a certain bicycle from steer angle (8) to roll angle (cp) is given by (Astrom, 2005) _ 99(3) _ 10(5 + 25) PmTW— 521-25 Assume the rider can be represented by a gain K, and that the closed-100p system is shown in Figure 10.20 with G(s) = KP(s) and H = 1. Use the Nyquist sta- bility criterion to find the range of K for closed-loop stability. The control of the radial pickup position of a digital versatile disk (DVD) was discussed in Problem 48, Chapter 9. There, the open-loop transfer function from coil input voltage ~to radial pickup position was given as (Bittann‘, 2002) " 0.63 t e1+0.36 + s2 1+0.04 + 5‘2 305.45 305.42 248525 243.32 Assume the plant is in cascade with a controller, _ 0.5(s + 1.63) Mm s(s + 0.27) and in the closed-loop configuration shown in Figure 10.20, where 0(5) = M(s)P(s) and H = 1. Do the following: a. Draw the open-loop frequency response in a Nichols chart. b. Predict the system’s response to a unit step input. Calculate the %0S, cfinal, and Ts. c. Verify the results of Part b using MATLAB simulations. “ATLAE The soft Arm, used to feed people with disabilities, was discussed in Problem 57 in Chapter 6. Assuming the system block diagram shown in Figure P1012, use frequency response techniques to determine the fol- lowing (Kara, I992): 41. 42. Desired spoon position Controller 1000§s + 0.01 its +6! 3(5' + 20)(s + 100) Desired position Controller X0014- ." 1001 000 I (s + SOON: + 800) I s(s + 100) I I Problems 597 Actual spoon position C(r) Robot arm Figure P10.13 Floppy disk drive block diagram a. Gain margin, phase margin, zero dB frequency, and 180° frequency ' b. Is the system stable? Why? A floppy disk dIiVe was discussed in Problem 55 in Chapter 8. Assuming the system block diagram shown in Figure P10. 13, use frequency response techniques to determine the following: a. Gain margin, phase margin, zero dB frequency, 180° frequency, and closed—loop bandwidth b. Percent overshoot, settling time, and peak time c. Use MATLAB to simulate the closed- tam-As mlgop step response and compare the results to those obtained in Part b. Industrial robots, such as that shown in Figure P10. 14, require accurate models for design of high perfor- mance. Many transfer function models for industrial robots assume interconnected rigid bodies with the drive-torque source modeled as a pure gain, or first- order system. Since the-motith associated with the robot are connected to the drives through flexible linkages rather than rigid linkages, past modeling does not explain the resonances observed. An accu- rate, small-motion, linearized model has been devel- oped that takes into consideration the flexible drive. The transfer function (s2 + 8.945 + 44.72) 6(5) -.= 999-13Wm relates the angular velocity of the robot base to electrical current commands (Good, 1985). Make a Bode plot of the frequency response and identify the resonant frequencies. . The charge-coupled device (CCD) that is used in video movie cameras to convert images into electrical signals can be used as part of an automatic focusing system in cameras. Automatic focusing can be imple- mented by focusing the center of the image on a charge-coupled device array through two lenses. The separation of the two images on the CCD is related to the focus. The camera senses the separation, and a computer drives the lens and focuses the image. FIGURE P10.14 AdeptOne, a four- or five-axis industrial robot, is used for assembly, packaging, and other manufacturing tasks. r'iril‘ .i "‘M n l l ‘ ! fil "I ll n: .9—.-——‘ .. _. 598 Desired lens 1mm“ + Computer and CCD _ system .a....;.:‘.:s,.c FIGURE P10.15 a. A cutaway view of a Nikon 35-min camera showing parts of the CCD automatic focusing system; 1}. functional block diagram; c. block diagram the three references to Popular Photography in this chapter’s Bibliography.) The automatic focus system is a position control, where the desired position of the lens is an input selected by pointing the camera at the subject. The output is the actual position of the lens. The Nikon camera in Figure P10.15(a) uses a CCD automatic focusing system. Figure P10.15(b) shows the automatic focusing feature represented as a posi- tion control system. Assuming the simplified model Desired Roll roll angle anglefld“) + Transducer FigUre P10.‘|6 Block diagram of a ship’s roll-stabilizing system Chapter 10 Frequency Response Techniques Actual lens position shown in Figure P10.15(c), draw the Bode plots and estimate thepercent overshoot for a step input. 44. A ship’s roll can be stabilized with a control system. A voltage applied to the fins’ actuators creates a roll torque that is applied to the ship. The ship, in response to the roll torque, yields a roll angle. Assuming the block dial gram for the roll control system shown in Figure P1016, determine the gain and phase margins for the system. Ship F11] roll ' Actual actuatOr dynamIcs mur angle, 60(5) 4 52 + O.9s+9 45. The linearized model of a particular netWork link working under TCP/IP and controlled using a random early detection (RED) algorithm can be described by Figure 1020 where G(s) = M (s)P(s), H = l, and (Holler, 2001) 00051 14062534115 Mm _ s + 0.005 ’ P“) ” (s + 2.67)(s +10) a. Plot the Nichols chart for L = 1. Is the system closed-100p stable? b. Find the range‘ of L for closed-loop stability. c. Use the Nichols chart to predict %OS and T5 for L = 0.95. Make a hand sketch of the expected unit step response. d. Verify Part c with a Simulink unit step response simulation. gunu link 46. In the TCP/IP network link of Problem 45, let L = 0.8, but assume that the amount of delay is an unknown variable. a. Plot the Nyquist diagram of the system for z‘ero delay, and obtain‘ the phase margin. b. Find the maximum delay allowed for closed-loop stability. 47. Thermal flutter of the Hubble Space Telescope (HST) produces errors for the pointing control system. Ther- mal flutter of the solar arrays occurs when the space- em: passes from sunlight to darkness and when the spacecraft is in daylight. In passing from daylight to darkness, an end-to-end bending'oscillation of fre— quency f1 rad/s is experienced. Such oscillations interfere with the pointing control’ system of the HST. A filter with the transfer function G (S) _ 196(52 +5 + 0.25)(s2 +1.26s+ 9.87) f _ (s2 + 0.0155 + 0.57)(s% + 0.0835 +172) is proposed to be placed in cascade with the controller to reduce the bending (Wie, 1992). a. Obtain the frequency response of the filter and estimate the bending frequencies that will be reduced. b. Explain why this filter will reduce the bending oscillations if these oscillations are thought to Problems 599 'ts be disturbances at the output of the Control system. Progressive Analysis and Design Problems 48. High-speed rail pantograph. Problem 20 in Chapter 1 discusses active control of a pantographu mechanism for high-speed rail systems. In Problem 74(a), Chapter 5, you found the block diagram for the active pantograph control system. In Chapter 8, Pro— blem 64, you designed the gain to yield a closed-loop step response with 30% overshoot. A plot of the step response should have shown a settling time greater than 0.5 second as well as a high-frequency oscillation superimposed over the step response. In Chapter 9, Problem 53, we reduced the settling time to about 0.3 second, reduced the step response steady—state error to icro, and eliminated the high-frequency oscillations by using a notch filter (O’Connor, 1997). Using the equivalent forward transfer function found in Chapter 5 cascaded with the notch filter specified in Chapter 9, do the following using frequency response techniques: a. Plot the Bode plots for a total equivalent gain of l and find the gain margin, phase margin, and 180° frequency. b. Find the range of K for stability. c. Compare your answer to Part b with your answer to Problem 65, Chapter 6. Explain any differences. 9. Control of HIVIAIDS. The linearized model for an HIV/AIDS patient treated with RTIs was obtained in Chapter 6 as (Craig, 2004); Yo) _ —520s — 10.3844 U1(s) _ s3 4; 2.681752 + 0.11s + 0.0126 P0) 3 a. Consider this plant in the feedback configuration in Figure 10.20 with G(s) = P(s) and H(s) = 1. Obtain the Nyquist diagram. Evaluate the system for closed-loop stability. Vb. Consider this plant in the feedback configuration in Figure 10.20 with 0(5) = —P(s) and H(s) : 1. Obtain the;Nyquist diagram. Evaluate the system for closed-loop stability. Obtain the gain and phase margins. ...
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hw7 355 - a. 502 N increased by at least a factor of- 5. A...

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