Solution Manual for Feedback Control of Dynamic Systems, 8th Edition

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100Solutions Manual:Chapter 18th EditionFeedback Control of DynamicSystems..Gene F. Franklin.J. David Powell.Abbas Emami-Naeini....Assisted by:H. K. AghajanH. Al-RahmaniP. CoulotP. DankoskiS. EverettR. FullerT. IwataV. JonesF. SafaiL. KobayashiH-T. LeeE. ThuriyasenaM. Matsuoka

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Chapter 1An Overview and BriefHistory of Feedback Control1.1Problems and Solutions1. Draw a component block diagram for each of the following feedback controlsystems.(a) The manual steering system of an automobile(b) Drebbelís incubator(c) The water level controlled by a áoat and valve(d) Wattís steam engine with áy-ball governorIn each case, indicate the location of the elements listed below andgive the units associated with each signal.the processthe process desired output signalthe sensorthe actuatorthe actuator output signalthe controllerthe controller output signalthe reference signalthe error signalNotice that in a number of cases the same physical device may per-form more than one of these functions.Solution:(a) A manual steering system for an automobile:101

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102CHAPTER 1.AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL(b) Drebbelís incubator:(c) Water level regulator:(d) Fly-ball governor:

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1.1.PROBLEMS AND SOLUTIONS1032. Identify the physical principles and describe the operation of the thermo-stat in your home or o¢ ce.Solution:A thermostat is a device for maintaining a temperature constant at adesired value.It is equipped with a temperature sensor which detectsdeviation from the desired value, determines whether the temperaturesetting is exceeded or not, and transmits the information to a furnace orair conditioner so that the temperature in the room is brought back to thedesired setting.Examples: Tubes Ölled with liquid mercury are attachedto a bimetallic strip which tilt the tube and cause the mercury to slideover electrical contacts. A bimetallic strip consists of two strips of metalbonded together, each of a di§erent expansion coe¢ cient so that temper-ature changes bend the metal.In some cases, the bending of bimetallicstrips simply cause electrical contacts to open or close directly.In mostcases today, temperature is sensed electronically using,for example, a ther-mistor, a resistor whose resistance changes with temperature.Moderncomputer-based thermostats are programmable, sense the current fromthe thermistor and convert that to a digital signal.Fig 1.12A Paper Making Machine

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104CHAPTER 1.AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL3. A machine for making paper is diagrammed in Fig. 1.12. There are twomain parameters under feedback control: the density of Öbers as controlledby the consistency of the thick stock that áows from the headbox onto thewire, and the moisture content of the Önal product that comes out of thedryers. Stock from the machine chest is diluted by white water returningfrom under the wire as controlled by a control valve (CV). A meter suppliesa reading of the consistency. At the ìdry endî of the machine, there is amoisture sensor. Draw a signal graph and identify the seven componentslisted in Problem 1 for(a) control of consistency(b) control of moistureSolution:(a) Control of paper machine consistency:(b) Control of paper machine moisture:4. Many variables in the human body are under feedback control. For eachof the following controlled variables, draw a graph showing the processbeing controlled, the sensor that measures the variable, the actuator thatcauses it to increase and/or decrease, the information path that completesthe feedback path, and the disturbances that upset the variable. You mayneed to consult an encyclopedia or textbook on human physiology forinformation on this problem.

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1.1.PROBLEMS AND SOLUTIONS105(a) blood pressure(b) blood sugar concentration(c) heart rate(d) eye-pointing angle(e) eye-pupil diameterSolution:Feedback control in human body:VariableSensorActuatorInform ation pathDisturbancesa) Blood pressure-Arterial-Cardiac output-A§erent nerve-Bleedingbaroreceptors-Arteriolar/venousÖb ers-Drugsdilation-Stress,Painb) Blood sugar-Pancreas-Pancreas secreting-Blood áow to-Dietconcentrationinsulinpancreas-Exercise(Glucose)c) Heart rate-Diastolic volum e-Electrical stimulation-M echanical draw-Horm one releasesensorsof sino-atrial nodeof blood from heart-Exercise-Cardiac sym patheticand cardiac muscle-Circulatingnervesepinephrined) Eye p ointing-Optic nerve-Extraocular muscles-Cranial innervation-Head m ovem entangle-Im age detection-M uscle twitche) Pupil diam eter-Rods-Pupillary sphincter-Autonom ous-Ambient lightmusclessystem-Drugsf ) Blood calcium-Parathyroid gland-Ca from b ones to blood- Parathorm one-Ca need in b onesleveldetectors-Gastrointestinalhorm one a§ecting-Drugsabsorptione§ector sites5. Draw a graph of the components for an elevator-position control.Indi-cate how you would measure the position of the elevator car. Consider acombined coarse and Öne measurement system. What accuracies do yousuggest for each sensor?Your system should be able to correct for thefact that in elevators for tall buildings there is signiÖcant cable stretch asa function of cab load.Solution:A coarse measurement can be obtained by an electroswitch located beforethe desired áoor level.When touched, the controller reduces the motorspeed.A ìÖneî sensor can then be used to bring the elevator preciselyto the áoor level.With a sensor such as the one depicted in the Ögure,a linear control loop can be created (as opposed to the on-o§ type of thecoarse control).Accuracy required for the course switch is around 5 cm;for the Öne áoor alignment, an accuracy of about 2 mm is desirable toeliminate any noticeable step for those entering or exiting the elevator.

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106CHAPTER 1.AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL6. Feedback control requires being able to sense the variable being controlled.Because electrical signals can be transmitted, ampliÖed, and processedeasily, often we want to have a sensor whose output is a voltage or currentproportional to the variable being measured. Describe a sensor that wouldgive an electrical output proportional to:(a) temperature(b) pressure(c) liquid level(d) áow of liquid along a pipe (or blood along an artery) force(e) linear position(f) rotational position(g) linear velocity(h) rotational speed(i) translational acceleration(j) torqueSolution:Sensors for feedback control systems with electrical output. Exam-ples(a) Temperature: Thermistor- temperature sensitive resistor with resis-tance change proportional to temperature; Thermocouple; Thyristor.Modern thermostats are computer controlled and programmable.(b) Pressure:Strain sensitive resistor mounted on a diaphragm whichbends due to changing pressure

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1.1.PROBLEMS AND SOLUTIONS107(c) Liquid level: Float connected to potentiometer. If liquid is conductivethe impedance change of a rod immersed in the liquid may indicatethe liquid level.(d) Flow of liquid along a pipe: A turbine actuated by the áow with amagnet to trigger an external counting circuit. Hall e§ect producesan electronic output in response to magnetic Öeld changes. Anotherway: Measure pressure di§erence from venturi into pressure sensoras in Ögure; Flowmeter.For blood áow, an ultrasound device like aSONAR can be used.(e) Position.When direct mechanical interaction is possible and for ìsmallî dis-placements, the same ideas may be used. For example a potentiome-ter may be used to measure position of a mass in an accelerator (h).However in many cases such as the position of an aircraft, the task ismuch more complicated and measurement cannot be made directly.Calculation must be carried out based on other measurements, forexample optical or electromagnetic direction measurements to severalknown references (stars,transmitting antennas ...); LVDT for linear,RVDT for rotational.(f) Rotational position.The most common traditional device is a poten-tiometer. Also common are magnetic machines in which a rotatingmagnet produces a variable output based on its angle.(g) Linear velocity. For a vehicle, a RADAR can measure linear velocity.In other cases, a rack-and-pinion can be used to translate linear torotational motion and an electric motor(tachometer) used to measurethe speed.(h) Speed: Any toothed wheel or gear on a rotating part may be used totrigger a magnetic Öeld change which can be used to trigger an elec-trical counting circuit by use of a Hall e§ect (magnetic to electrical)sensor.The pulses can then be counted over a set time interval toproduce angular velocity: Rate gyro; Tachometer

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108CHAPTER 1.AN OVERVIEW AND BRIEF HISTORY OF FEEDBACK CONTROL(i) Acceleration: A mass movement restrained by a spring measured bya potentiometer. A piezoelectric material may be used instead (a ma-terial that produces electrical current with intensity proportional toacceleration). In modern airbags, an integrated circuit chip containsa tiny lever and íproof massíwhose motion is measured generating avoltage proportional to acceleration.(j) Force, torque: A dynamometer based on spring or beam deáections,which may be measured by a potentiometer or a strain-gauge.7. Each of the variables listed in Problem 6can be brought under feedbackcontrol. Describe an actuator that could accept an electrical input and beused to control the variables listed. Give the units of the actuator outputsignal.Solution:(a) Resistor with voltage applied to it or mercury arc lamp to generateheat for small devices. a furnace for a building..(b) Pump:Pumping air in or out of a chamber to generate pressure.Else, a ítorque motoríproduces force..(c) Valve and pump: forcing liquid in or out of the container.(d) A valveis normally used to control áow.(e) Electric motor(f) Electric motor(g) Electric motor(h) Electric motor(i) Translational acceleration is usually controlled by a motor or engineto provide force on the vehicle or other object.(j) Torque motor.In this motor the torque is directly proportional tothe input (current).8.Feedback in Biology(a)Negative Feedback in Biology: When a person is under long term stress(say a couple of weeks before an exam!), hypothalamus (in the brain) se-cretes a hormone called CRF (Corticotrophin Releasing Factor) whichbinds to a receptor in the pituitary gland stimulating it to produce ACTH

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1.1.PROBLEMS AND SOLUTIONS109(Adrenocorticotropic hormone), which in turn stimulates the adrenal cor-tex (outer part of the adrenal glands) to release the stress hormone Glu-cocorticoids (GC). This in turn shuts down (turns o§ the stress response)for both CRF and ACTH production by negative feedback via the blood-stream until GC returns to its normal level. Draw a block diagram of thisclosed-loop system.(b)Positive Feedback in Biology: This happens in some unique circum-stances.Consider the birth process of a baby.Pressure from the headof the baby going through the birth canal causes contractions via secre-tion of a hormone called Oxytocin which causes more pressure which inturn intensiÖes contractions. Once the baby is born, the system goes backto normal (negative feedback). Draw a block diagram of this closed-loopsystem.Solution:(a) Negative Feedback in Biology - StressStress induced negative feedback(b) Positive Feedback in Biology - Child birthChild birth induced positive feedback

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2000Solutions Manual:Chapter 28th EditionFeedback Controlof Dynamic Systems..Gene F. Franklin.J. David Powell.Abbas Emami-Naeini....Assisted by:H. K. AghajanH. Al-RahmaniP. CoulotP. DankoskiS. EverettR. FullerT. IwataV. JonesF. SafaiL. KobayashiH-T. LeeE. ThuriyasenaM. MatsuokaCopyright (c) 2019 Pearson EducationNo part of this publication may be reproduced, storedin a retrieval system, or transmitted, in any form or by any means, electronic,mechanical, photocopying, recording, or otherwise, without the prior permissionof the publisher.

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Chapter 2Dynamic ModelsProblems and Solutions for Section 2.11. Write the di§erential equations for the mechanical systems shown in Fig. 2.43.For (a) and (b), state whether you think the system will eventually decayso that it has no motion at all, given that there are non-zero initial condi-tions for both masses, and give a reason for your answer.Also, for part(c), answer the question for F=0.2001

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2002CHAPTER 2.DYNAMIC MODELSFig. 2.43Mechanical systemsSolution:The key is to draw the Free Body Diagram (FBD) in order to keep thesigns right.For (a), to identify the direction of the spring forces on theobject, letx2= 0and Öxed and increasex1from0. Then thek1springwill be stretched producing its spring force to the left and thek2springwill be compressed producing its spring force to the left also. You can usethe same technique on the damper forces and the other mass.(a)m1m2x1x2k (x -x )212k (x -x )212k (x32- y)k x11b x11.Free body diagram for Problem 2.1(a)m1x1=k1x1b1_x1k2(x1x2)m2x2=k2(x2x1)k3(x2y)There is friction a§ecting the motion of mass 1 which will continueto take energy out of the system as long as there is any movement ofx1:Mass 2 is undamped; therefore it will tend to continue oscillating.However, its motion will drive mass 1 through the spring; therefore,the entire system will continue to lose energy and will eventuallydecay to zero motion for both masses.

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2003m1m2x1x2x2k (x -x )212k (x -x )212k x11b x12.x2x2k3Free body diagram for Problem 2.1(b)m1x1=k1x1k2(x1x2)m2x2=k2(x2x1)b1_x2k3x2Again, there is friction on mass 2 so there will continue to be a lossof energy as long as there is any motion; hence the motion of bothmasses will eventually decay to zero.m1m2x1x2k (x - x )212k (x - x )212k x11b (x - x )112b (x - x )112....FFree body diagram for Problem 2.1 (c)m1x1=k1x1k2(x1x2)b1( _x1_x2)m2x2=Fk2(x2x1)b1( _x2_x1)The situation here is similar to part (a).It is clear that the relativemotion between mass 1 and 2 would decay eventually, but as longas mass 1 is oscillating, it will drive some relative motion of the twomasses and that will cause energy loss in the damper.So the entiresystem will eventually decay to zero.2. Write the di§erential equations for the mechanical systems shown in Fig. 2.44.State whether you think the system will eventually decay so that it hasno motion at all, given that there are non-zero initial conditions for bothmasses, and give a reason for your answer.

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2004CHAPTER 2.DYNAMIC MODELSFig. 2.44 Mechanical system for Problem 2.2Solution:The key is to draw the Free Body Diagram (FBD) in order to keep thesigns right. To identify the direction of the spring forces on the left sideobject, letx2= 0and increasex1from0. Then thek1spring on the leftwill be stretched producing its spring force to the left and thek2springwill be compressed producing its spring force to the left also. You can usethe same technique on the damper forces and the other mass.m1m2x1x2x2k (x -x )212k x11k1b (x - x )212b (x - x )212....Free body diagram for Probelm 2.2Then the forces are summed on each mass, resulting inm1x1=k1x1k2(x1x2)b1( _x1_x2)m2x2=k2(x1x2)b1( _x1_x2)k1x2Therelativemotion betweenx1andx2will decay to zero due to thedamper.However, the two masses will continue oscillating togetherwithout decay since there is no friction opposing that motion and áexureof the end springs is all that is required to maintain the oscillation of thetwo masses.However, note that the two end springs have the same springconstant and the two masses are equalIf this had not been true, the twomasses would oscillate with di§erent frequencies and the damper wouldbe excited thus taking energy out of the system.3. Write the equations of motion for the double-pendulum system shown inFig. 2.45.Assume the displacement angles of the pendulums are smallenough to ensure that the spring is always horizontal.The pendulum

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