ch1 - THE SOFTWARE In each chapter there are five problem...

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Unformatted text preview: THE SOFTWARE In each chapter, there are five problem types: I Exercises I Problems I Advanced Problems I Design Problemleontinuous Design Problem I MATLAB Problems In total, there are over 800 problems. The abundance of problems of increas— ing complexity gives students confidence in their problem- solving ability as they work their way from the exercises to the design and MATLAB problems. A set of M-files, the Modem Control Systems Toolbox, have been developed by the authors to supplement this Solution Manual. The M-files contain the scripts from each MATLAB problem in the Solution Manual. You may retrieve the M- files from the MCS website site at http:/Iwww.prenhall.comldorf. The Modern Control Systems Toolbox is for use with the program MATLAB, available from The MathWorks, Inc. located in Natick, MA, U.S.A. at (508) 653-1415. It is assumed that instructors (and students) have access to MATLAB and the Con- trol System Toolbox. All of the MATLAB solutions in this Solution Manual were developed and tested on a Wndow 98 platform using MATLAB 5.3 and the Con- trol System Toolbox Version 4.2. It is not possible to verify each solution on all the available computer platforms that are compatible with MATLAB. Please for- ward any incompatibities you encounter with the MA’I‘LAB scripts to one of the email addresses given below. 1. MATLAB is registered trademark of The MathWorks, Inc. w—v—A-J" "'Mm MW. ,7 _. Mm", -- . m W unn—_.‘ .. .M—_ We --"—.‘4—~-—- '— “ ~~ .._ mm M“. .,_..-——-... iv Preface ACKNOWLEDGEMENTS We wish to express our appreciation to the following reviewers: Dr. Hy Tran, University of New Mexico, and Ed Chan, University of California, Berkeley. We would also like to thank the following University of Texas at Austin graduate students for their assistance with the solutions of some of the problems: Eduardo Gildin and Rich Mrozinski. Finally, our gratitude goes to Lynda Ferrera Bishop for her assistance in typesetting the Solution Manual. OPEN LINES OF COMMUNICATION The authors and the staff at Prentice Hall would like to establish an open line of communication with the instructors using Modem Control Systems. We enc0ur- age you to contact Prentice Hall with comments and suggestions for this and fu— ture editions. Robert H. Bishop [email protected] Richard C. Dorf [email protected] Table of Contents TABLE OF CONTENTS 1. Introduction to Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Mathematical Models of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3. State Variable Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4. Feedback Control System Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5. The Performance of Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . 153 6. The Stability of Linear Feedback Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7. The Root Locus Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .232 8. Frequency Response Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .329 9. Stability in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 10. The Design of Feedback Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 444 11. The Design of State Variable Feedback Systems . . . . . . . . . . . . . . . . . . . . 521 12. Robust Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .566 13. Digital Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .620 Introduction to Control Systems EXERCISES -....____-___._:\-.__,_g There are, in general, no unique solutions to the following exercises and prob- lems. Other equally valid block diagrams may be submitted by the student. ELI A microprocessor controlled laser system: controller processor E1.2 A driver controlled cruise control system: Will! Actual auto Measurement Speedometer CHAPTER 1 Introduction to Control Systems E13 Although the principle of conservation of momentum explains much of the pro- cess of fly-casting, there does not exist a comprehensive scientific explanation of how a fly-fisher uses the small backward and forward motion of the fly rod to cast an almost weightless fly lure long distances (the current world—record is 236 ft). The fly lure is attached to a short invisible leader about lS-ft long, which is in turn attached to a longer and thicker Dacron line. The objective is cast the fly lure to a distant spot with deadeye accuracy so that the thicker part of the line touches the water first and then the fly gently settles on the water just as an insect might. Fl -flsher Wind dmurbanoa Mind and ' bod of the Adam I fly—I‘sher 5L3; - Vlslan of -_ the fly-fisher E1.4 An autofocus camera control system: Gunny trlp time for the bum convumbn tumor 1'— R 7 _ _ _ * _ k L fl — — — _ _ -‘ _ fl , . [Woflightar I E 80W) | l | J : Emma! .3 _;ET_—._ 1 Dhurluhlubhct . RMOIVH' ‘—*—_ I Sublet: I L l l WM__.__ __._ _____._~,#_n-u._s,, ._ , 7 H Exercises ELS Tacking a sailboat as the wind shifts: IIII Id|umont Dulrod Saflboat Actual gallbolt ullboat direction dlroeilon E1.6 An automated highway control system merging two lanes of traffic: Bum, 9n or computer during Ipbllncl Dulrod 8‘ Actual FF Actuators can Haunt“ gap E13 Using the speedometer, the driver calculates the difference between the measured speed and the desired speed. The driver throotle knob or the brakes as necessary to adjust the speed. If the current speed is not too much over the desired speed, the driver may let friction and gravity slow the motorcycle down. llouummrrl Vllull Indlcltlon CHAPTER 1 introduction to Controi Systems E1.8 Human biofeedback control system: Doolred Body sensor Process Actual tGl'l’lp Problems PROBLEMS P1.l An automobile interior cabin temperature control system block diagram: controller “new Thermostat and Automobile Dashed a. air conditioning m um um turn + unit cabin :ahfn tuJigornrure oat ytho driver Minotaur-m Temperature Sensor temper-turn PL2 A human operator controlled valve system: ConuoIIIr Pm" Desired Fluld fluld output outplfl‘ VIIuII lndlntlon of lluld oulpm ‘ ' = opullor lunctlonl P13 A chemical composition control block diagram: Controller PM Dull-ad Chomloal chamlcal composfllon compoultlon Honoured chomlca! compoofllon CHAPTER 1 Introduction to Control Systems P1.4 A nuclear reactor control block diagram: Controller Freon: Motor and Amplifier Desired Output m “"91 power love! Ionization Chamber pawlanl PLS A light seeking control system to track the sun: Measurement controller Moan Motor, u m Dun! iihmuy Traleetory carriage, 9 “ aim. ' Photoeells Planner and gears caning! potflbn P1.6 If you assume that increasing worker’s wages results in increased prices, then by delaying or falsifying cost-of-living data you could reduce or eliminate the pres- sure to increase worker’s wages, thus stabilizing prices. This would work only if there were no other factors forcing the cost—of—living up. Government price and wage economic guidelines would take the place of additional “controllers” in the block diagram, as shown in the block diagram. Proms Onmllur Govt price guidelines Initial maul Govt wage guldeilnes P1.7 Assume that the cannon fires initially at exactly 5:00 p.m.. We have a positive feedback system. Denote by At the time lost per day, and the net time error by Probiems E T. Then the follwoing relationships hold: At = 4/3 min. + 3 min. = 13/3 min. and E7 = 12 days >< 13/3 min/day. Therefore, the net time error after 15 days is E1" = 52 min. P1.8 The student-teacher learning process: Convener m. Knowledge muscles Fantail CHAPTER 1 Introduction to Control Systems P110 An aircraft flight path control system using GPS: Fm" Actuators Controller c t Allarons. Dam Aoinpuue: elevators, Fug.“ “I M path 1- u o- o “I tigm alr wattle p erxdldneer. p. controller- 9 [—4 Honour-mam Globe: Posltlonlng I System mam pun P1.11 The accuracy of the clock is dependent upon a constant flow from the orifice; the flow is dependent upon the height of the water in the float tank. The height of the water is controlled by the float. The control system controls only the height of the water. Any errors due to enlargement of the orifice or evaporation of the water in the lower tank is not accounted for. The control system can be seen as: contra IIII' sum Float level How "am upper tank Actuul fififl “ to float tank new In flout tank P1.12 Assume that the turret and fantail are at 90°, if 9w =,é EFF-90°. The fantail Operates on the error signal 9,” - QT, and as the fantafl turns, it drives the turret to turn. SW = Wind angle 9,: = Fantaltnngte 8T = Turret angle Wind comran fantail - actuator - Sears 8: turret I Problems 9 P1.13 This scheme assumes the person adjusts the hot water for temperature control, and then adjusts the cold water for flow rate control. . | l Dellrod water temperature : 7 Actual l wular lemperfiure i and flow rule I Deliver! water flow me u‘ l Measured water flow vlsual and tOHC h Measured mar tempcrulure P1.14 If the rewards in a specific trade is greater than the average reward, there is a positive influx of workers, since 610) = f1(c(!) -r(t))- If an influx of workers occurs, then reward in specific trade decreases, since C(t) = —f2(61(t))- I'm P1.15 A computer controlled fuel injection system: Prom-I Controller High Pressure Fuel Electronlc Dellmd —~+ Suppl Pump and F Fun! m + c°mr°l Unit EIeetriinlc Fuel Pi'o'iaum injectors Fuel Pressure Hun-rad Sensor Full Pro-cure 10 CHAPTER 1 introduction to Control Systems 191.16 With the onset of a fever, the body thermostat is turned up. The body adjusts by shivering and less blood flows to the skin surface. Aspirin acts to lowers the ther- mal set—point in the brain. Controller Process Adjustments Desired temperature within the Body or net-polnt from body + tamporllura “tumult In the min P1.17 P118 ’8 ti Screw K a displacement + _ x(t) body Internal sensor Hitting a baseball is arguably one of the most difficult feats in all of sports. Given that pitchers may throw the ball at speeds of 90 mph (or higherl), batters have only about 0.1 second to make the decision to swing—with bat speeds aproach— ing 90 mph. The key to hitting a baseball along distance is to make contact with the ball with a high bat velocity. This is more important than the bat’s weight, which is usually around 33 ounces (compared to '13! Cobb’s bat which was 41 ounces!). Since the pitcher can throw a variety of pitches (fast ball, curve ball, slider, etc), a batter must decide if the ball is going to enter the strike zone and if possible, decide the type of pitch. The batter uses his/her vision as the sensor in the feedback loop. A high degree of eye-hand coordination is key to success-— that is, an accurate feedback control system. Define the following variables: p = output pressure, )2, = spring force = K 1:, fd = diaphragm force = A p, and f” = valve force = f,r - far. The motion of the valve is described by j} = fu/m where m is the valve mass. The output pressure is proportional to the valve displacement, thus p 2 cy , where c is the constant of proportionality. Constant of proportionality Valve position Output pressure 9(1) Problems P1.19 A control system to keep a car at a given relative position offset from a lead car: Foalllon ol Follower follower car W Fuel Lead car P9! throttle (MD Actuator Throttle Vlcleo camera & procosslng al - orlthms Controller Danlrod retell“ poaltlon P120 A control system for a high-performance car with an adjustable wing: no", Prue-II Controller condfllom Computer 1mm . &ad|ustable am adheslon - wlng adhesion Tlre Internal straln gauges P1.21 A control system for a twin-lift helicopter system: Measurement Illa-m6 lap-ration dlltmoo Helloopter Honoured lltlludo 11 12 CHAPTER 1 Introduction to Control Systems P122 The desired building deflection would not necessarily be zero. Rather it would be prescribed so that the building is allowed moderate movement up to a point, and then active control is applied if the movement is larger than some predetermined amount. Prom” 00on let Hydraulic . actuator stiffeners Dulrod Deflection dotlectlon Building Hummus-1t Strain gauges on truss structure 6.1mm P123 The human-like face of the robot might have micro-actuators placed at strate- gic points on the interior of the malleable facial structure. Cooperative control of the micro-actuators would then enable the robot to achieve various facial ex- pressions. Controller Prue.“ Electro- m m Danna Amplifier mechanical " m actuator Wm“ Pl.24 We might envision a sensor embedded in a “gutter” at the base of the windshield which measures water levels—higher water levels corresponds to higher inten— sity rain. This information would be used to modulate the wiper blade speed. controller Prone“ Wiper blade and motor par blade spud Electronic Control Unit “Illul'll'l'llnl Water depth Measured sensor Probiems P125 A feedback control system for t Actunnr Reaction control Cowman” Controi lawr orhl‘l poanlon + lets Hal-mod orbit pool‘uun Radar or GPS he space traffic control: Applied Satellite :1 § r. Actual arbll ponhlon 13 1 4 CHAPTER 1 Introduction to Control Systems DESIGN PROBLEMS CDP1.1 The machine tool with the movable table in a feedback control configuration: Controller Precast Machine "' Amplifier and a m I 333.33. . positioning motor gallewnh position x ' X I Mancunian“ [ Posltlon Measured poaltlon sensor DP1.1 Use the stereo system and amplifiers to cancel out the noise by emitting signals 180° out of phase with the noise. ’ Process Controller ' 2 Stereo ‘ systemfi ‘ speakers Nola In nhln Dallrod not” = 0 cum—r 11K O IUIO 33.5! Honour-d urn-f: apood speed sensor Design Problems DP1.3 An automoted cow milking system: Mauuumm new location Controller Mumor Motor and Robot arm and COW 3"“ mm Dulr-d cup gears cup grlpper mm‘e" locfllnn Minimum-m Monuer sup lac-um DP1.4 A feedback control system for a robot welder: Cantrell-r Proan Computer Duh-ad Wold palm“ and amplifier gum Honour-mun! Honoured posltlon DPLS A control system for one wheel of a traction control system: Antisle ‘ Em" controller 1 mun Wheel dynamlcs sens” + "nomad Illa: Mull Illp Antiskid control Ier Vehicle dynamlcs Vohlcla Ipood 15 16 Dulrld “fill a 0 CHAPTER 1 Introduction to Control Systems DP1.6 A vibration damping system for the Hubble Space Telescope: Actuator Slgnll to prom“ canmm annual the Gyro and utter Spacecraft + . computer reaction dynamlcs wheels Honour-mam Hate gyro sensor Measurement of 0.05 Hz liner Jlttor oi vlbrn: Ion ...
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This note was uploaded on 04/28/2008 for the course EE 328 taught by Professor Thomas during the Fall '07 term at S. Alabama.

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ch1 - THE SOFTWARE In each chapter there are five problem...

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