《工程测试与信号处理》课程教学资源（文献资料）ACTUATORS2

534Chapter12Mechatronics:Sensors,Actuators,andControls12.3ACTUATORSLinear ActuatorsThetaskof a linearactuatoristoprovidemotioninastraight line.Wediscussthreewaystoachievelinear motion:1.Conversion of rotarymotion into linear motion.This can be accomplished using a linkage,asin the slider-crank mechanism, or using screw threads coupled to a rotary motion source.2. Use of a fluid pressure to move a piston in a cylinder. When air or another gas is used as theworkingfluid,thesystemiscalledapneumaticsystem.Whenafluidsuchasoilisusedastheworkingfluid, thesystemistermed hydraulic.3.ElectromagneticSlider-CrankMechanismA commonmeansofgenerating areciprocating linearmotion,or converting linear motion to rotarymotion,is the slider-crank mechanism,as illustrated inFigure12.30.Such a mechanismis thebasisof transforming the reciprocating motion of the piston in an internal combustion engine.This orsimilar linkages could also be applied in pick-and-place operations, or in a variety of automationapplications.Screw-drive linear motionAcommonmeansfortranslatingrotarymotionintolinearmotionisalead screw.Alead screwhashelical threads that are designed for minimum backlash to allow precise positioning.Numerousdesigns exist for such actuating threads. The basic principle is illustrated in Figure 12.31.The rotarymotion of thelead screw is translated into linear motion of thenut, with the torque required to drivethe lead screw directly related to the thrust the particular application requires.Figure 12.30 Slider-crank mechanismThrust2output区Nut区XThrustoutputFigure 12.31 Linear actuation using a leadTorqueinputscrew

E1C12 09/14/2010 13:54:12 Page 534 12.3 ACTUATORS Linear Actuators The task of a linear actuator is to provide motion in a straight line. We discuss three ways to achieve linear motion: 1. Conversion of rotary motion into linear motion. This can be accomplished using a linkage, as in the slider-crank mechanism, or using screw threads coupled to a rotary motion source. 2. Use of a fluid pressure to move a piston in a cylinder. When air or another gas is used as the working fluid, the system is called a pneumatic system. When a fluid such as oil is used as the working fluid, the system is termed hydraulic. 3. Electromagnetic Slider–Crank Mechanism A common means of generating a reciprocating linear motion, or converting linear motion to rotary motion, is the slider–crank mechanism, as illustrated in Figure 12.30. Such a mechanism is the basis of transforming the reciprocating motion of the piston in an internal combustion engine. This or similar linkages could also be applied in pick-and-place operations, or in a variety of automation applications. Screw-drive linear motion A common means for translating rotary motion into linear motion is a lead screw. A lead screw has helical threads that are designed for minimum backlash to allow precise positioning. Numerous designs exist for such actuating threads. The basic principle is illustrated in Figure 12.31. The rotary motion of the lead screw is translated into linear motion of the nut, with the torque required to drive the lead screw directly related to the thrust the particular application requires. Figure 12.30 Slider–crank mechanism. Thrust output Torque input Thrust output Nut Figure 12.31 Linear actuation using a lead screw. 534 Chapter 12 Mechatronics: Sensors, Actuators, and Controls

53512.3ActuatorsFigure 12.32 Precision translationtable. (UniSlidefrom Velmex,Inc).Common applications that employalead screw includetheworktableforamill,anda varietyofother precision positioning translation tables, such as the one shown in Figure 12.32.PneumaticandHydraulicActuatorsThe term“pneumatic"implies a component or system that uses compressed air as the energysource. On the other hand, a hydraulic system or component uses incompressible oil as the workingfluid. An example of a hydraulic system is the power steering on an automobile; such a system isillustrated in Figure 12.33. Hydraulic fluid is supplied at an elevated pressure from the powersteering pump.When a steering input is made from the driver, the rotaryvalve allows high-pressurefluid to enter the appropriate side of the piston, and aidin turning the wheels.By maintaining a directconnection between the steering column and the rack and pinion, the car can be steered even if thehydraulic system fails.SteeringcolumnRotaryvalveToFluid linesreservoir-FrompumpWFigure12.33SchematicdiagramofapowerRackPistonPinionsteering system

E1C12 09/14/2010 13:54:12 Page 535 Common applications that employ a lead screw include the worktable for a mill, and a variety of other precision positioning translation tables, such as the one shown in Figure 12.32. Pneumatic and Hydraulic Actuators The term ‘‘pneumatic’’ implies a component or system that uses compressed air as the energy source. On the other hand, a hydraulic system or component uses incompressible oil as the working fluid. An example of a hydraulic system is the power steering on an automobile; such a system is illustrated in Figure 12.33. Hydraulic fluid is supplied at an elevated pressure from the power steering pump. When a steering input is made from the driver, the rotary valve allows high-pressure fluid to enter the appropriate side of the piston, and aid in turning the wheels. By maintaining a direct connection between the steering column and the rack and pinion, the car can be steered even if the hydraulic system fails. Figure 12.32 Precision translation table. (UniSlide1 from Velmex, Inc). Pinion Steering column To reservoir From pump Rotary valve Fluid lines Rack Piston Figure 12.33 Schematic diagram of a power steering system. 12.3 Actuators 535

536Chapter12Mechatronics:Sensors,Actuators,andControlsCompressedairinletsPistonsealsPiston rodRodseals andbearingFigure12.34 Construction of a pneumatic cylinder.(Courtesy of ParkerHannifin,Inc.PneumaticActuatorsWhen compressed air is the energy source of choice, a pneumatic cylinder can create linear motion.Ingeneral, the purpose of a pneumaticcylinder is to provide linear motion between two fixedlocations.Figure12.34shows a pneumatic cylinder and acutawayof sucha cylinder.Byapplyinghigh-pressure compressed air to either side of the piston, linear actuation between two definedpositions can easilybeaccomplished.Soleniods"Solenoid"is a term used to describe an electromagnetic device that is employed to create linearmotion of a plunger,as shown inFigure 12.35.The initial force availablefrom a solenoid can bedeterminedfromSNIA(12.21)F82

E1C12 09/14/2010 13:54:12 Page 536 Pneumatic Actuators When compressed air is the energy source of choice, a pneumatic cylinder can create linear motion. In general, the purpose of a pneumatic cylinder is to provide linear motion between two fixed locations. Figure 12.34 shows a pneumatic cylinder and a cutaway of such a cylinder. By applying high-pressure compressed air to either side of the piston, linear actuation between two defined positions can easily be accomplished. Soleniods ‘‘Solenoid’’ is a term used to describe an electromagnetic device that is employed to create linear motion of a plunger, as shown in Figure 12.35. The initial force available from a solenoid can be determined from F ¼ 1 2 ð Þ NI 2 mA d2 ð12:21Þ Figure 12.34 Construction of a pneumatic cylinder. (Courtesy of Parker Hannifin, Inc.) 536 Chapter 12 Mechatronics: Sensors, Actuators, and Controls

53712.3ActuatorsPlungeC-frameFigure 12.35 Construction of a solenoid linear actuator.whereF=force on plungerN-number of turns of wirein theelectromagneticI=currentμ=magneticpermeability of air (4×10-7H/m)S= size of the air gapA=plungercross-sectionalareaWhen the electromagnetis actuated, theresulting magnetic force pulls the plunger into the C-frame.Because the air gapis largestwhen the electromagnet is actuated,the minimum force occurs atactuation and the force increases as the air gap decreases.Rotary ActuatorsStepper MotorsThere is a class of electric motors that has the primarypurpose of providing power to a process.Anexample wouldbethe electricmotorthatdrives an elevator,anescalator,or a centrifugal blower.Inthese applications the electric motor serves as a prime mover, with clear and specific requirementsfor rotational speed, torque, and power. However, some applications have stringent requirements forpositioning.Rotary positioning presents a significant engineering challenge, but one that is so ubiquitousthat it has been addressed through a variety of design strategies. One design strategy is to employ afree-rotatingDCmotorto supplythe motive power and imposeprecisecontrol on the resultingmotion through gearing and some control scheme.DC motors that are subject to feedback controlare generally described as servo-motors. While this may be appropriate and necessary for someapplications, the stepper motor has found wide-ranging applications in precision rotary motioncontrol, and is a better choice for many applications.Stepperor steppingmotors,astheirnameimplies,arecapableofmovingafraction ofarotationwith a great degree of precision. This is accomplished by the design of a rotor that aligns with themagnetic field generated by energized coils.The step size can range from 90 degrees to as little as0.5 degrees or less.Two common types of steppermotors are variable reluctance and unipolardesigns. The design of a variable reluctance stepping motor is illustrated in Figure 12.36. Let'sconsider the operation of this motor.There are three sets of windings, labeled 1, 2, and 3 in thefigure, and there are two sets of teeth on the rotor, labeled X and Y. With the windings labeled 1energized, the rotor snaps to a position where one set of the teeth are aligned with the windings.Thismotion is a result of the magnetic field generated by the windings. Suppose that winding 1 is

E1C12 09/14/2010 13:54:13 Page 537 where F ¼ force on plunger N ¼ number of turns of wire in the electromagnetic I ¼ current m ¼ magnetic permeability of air (4p  107 H=m) d ¼ size of the air gap A ¼ plunger cross-sectional area When the electromagnet is actuated, the resulting magnetic force pulls the plunger into the C-frame. Because the air gap is largest when the electromagnet is actuated, the minimum force occurs at actuation and the force increases as the air gap decreases. Rotary Actuators Stepper Motors There is a class of electric motors that has the primary purpose of providing power to a process. An example would be the electric motor that drives an elevator, an escalator, or a centrifugal blower. In these applications the electric motor serves as a prime mover, with clear and specific requirements for rotational speed, torque, and power. However, some applications have stringent requirements for positioning. Rotary positioning presents a significant engineering challenge, but one that is so ubiquitous that it has been addressed through a variety of design strategies. One design strategy is to employ a free-rotating DC motor to supply the motive power and impose precise control on the resulting motion through gearing and some control scheme. DC motors that are subject to feedback control are generally described as servo-motors. While this may be appropriate and necessary for some applications, the stepper motor has found wide-ranging applications in precision rotary motion control, and is a better choice for many applications. Stepper or stepping motors, as their name implies, are capable of moving a fraction of a rotation with a great degree of precision. This is accomplished by the design of a rotor that aligns with the magnetic field generated by energized coils. The step size can range from 90 degrees to as little as 0.5 degrees or less. Two common types of stepper motors are variable reluctance and unipolar designs. The design of a variable reluctance stepping motor is illustrated in Figure 12.36. Let’s consider the operation of this motor. There are three sets of windings, labeled 1, 2, and 3 in the figure, and there are two sets of teeth on the rotor, labeled X and Y. With the windings labeled 1 energized, the rotor snaps to a position where one set of the teeth are aligned with the windings. This motion is a result of the magnetic field generated by the windings. Suppose that winding 1 is Coil C-frame Plunger Figure 12.35 Construction of a solenoid linear actuator. 12.3 Actuators 537

538Chapter12Mechatronics:Sensors,Actuators,andControlsNSFigure 12.37 Variable reluctance stepper motorFigure 12.36 Variable reluctance stepper motordesign having six poles and two windings.design.de-energized and winding 2 is energized. The rotor will turn until the teeth marked Y are alignedwith winding 2. This produces a 30-degree step.A useful characteristic of stepper motors is holding torque.As long as one of the windings isenergized, the rotor resists motion, until the torque produced by the winding to rotor interaction isovercome.The motor shown in Figure 12.37 is a variable reluctance design. Unipolar motors incor-porate permanent magnets as the rotor. Figure 12.37 shows a rotor having six magnetic poles andtwo sets of windings. The motor moves in 30-degree increments as the windings are alternatelyenergized.Flow-ControlValvesValves are mechanical devices intended to allow, restrict, throttle, or meter fluid flow through pipesorconduits.Flow-control valves are used to regulate eitherthe flow or thepressure of a fluid bytheirelectronic actuation. They generally function by allowing flow while in their open position, stoppingflow when closed, and metering flow or fluid pressure to a desired value with a position that issomewherebetweenthesesettings,whichiscalledproportional control.Thesevalvescontain avalvepositioning element that is driven by an actuator, such as a solenoid.Any valve type can becontrolled.Thecommon control valvedesignofferseithera singlechamberbody containingapoppet with valve seat or a multichamber body containing a sliding spool with multiple poppets.Flow-control valves are used to transfer gases, liquids, and hydraulic fluids.The application ratingsare as follows: general service, for working with common liquids and gases; cryogenic service, forfluids such as liquid oxygen; vacuum service, for low pressure applications; and oxygen service,forcontamination-freeflowofoxygen.The control valve can respond to signals from any type of process variable transducer. Thesignal determines the position of the actuating solenoid. A specific characteristic of any controlvalve refers to whether its nonenergized operating state is open or closed. This is referred to as its"fail position."The fail position of a control valve is determined by the nonenergized solenoidplunger position.This position is an important consideration for process safety.These valves come in various configurations reflecting their number of ports.A two-way valvehas twoports.Two-way position control takes on one of twovalues:open orclosed.Atwo-wayvalvehastwoconnections: supplyport (P)and serviceport(A).Most common household valves fall

E1C12 09/14/2010 13:54:13 Page 538 de-energized and winding 2 is energized. The rotor will turn until the teeth marked Y are aligned with winding 2. This produces a 30-degree step. A useful characteristic of stepper motors is holding torque. As long as one of the windings is energized, the rotor resists motion, until the torque produced by the winding to rotor interaction is overcome. The motor shown in Figure 12.37 is a variable reluctance design. Unipolar motors incor￾porate permanent magnets as the rotor. Figure 12.37 shows a rotor having six magnetic poles and two sets of windings. The motor moves in 30-degree increments as the windings are alternately energized. Flow-Control Valves Valves are mechanical devices intended to allow, restrict, throttle, or meter fluid flow through pipes or conduits. Flow-control valves are used to regulate either the flow or the pressure of a fluid by their electronic actuation. They generally function by allowing flow while in their open position, stopping flow when closed, and metering flow or fluid pressure to a desired value with a position that is somewhere between these settings, which is called proportional control. These valves contain a valve positioning element that is driven by an actuator, such as a solenoid. Any valve type can be controlled. The common control valve design offers either a single chamber body containing a poppet with valve seat or a multichamber body containing a sliding spool with multiple poppets. Flow-control valves are used to transfer gases, liquids, and hydraulic fluids. The application ratings are as follows: general service, for working with common liquids and gases; cryogenic service, for fluids such as liquid oxygen; vacuum service, for low pressure applications; and oxygen service, for contamination-free flow of oxygen. The control valve can respond to signals from any type of process variable transducer. The signal determines the position of the actuating solenoid. A specific characteristic of any control valve refers to whether its nonenergized operating state is open or closed. This is referred to as its ‘‘fail position.’’ The fail position of a control valve is determined by the nonenergized solenoid plunger position. This position is an important consideration for process safety. These valves come in various configurations reflecting their number of ports. A two-way valve has two ports. Two-way position control takes on one of two values: open or closed. A two-way valve has two connections: supply port (P) and service port (A). Most common household valves fall 1 1 X X Y Y 2 3 3 2 Figure 12.36 Variable reluctance stepper motor design. 2 2 1 S S S S N N N N 1 Figure 12.37 Variable reluctance stepper motor design having six poles and two windings. 538 Chapter 12 Mechatronics: Sensors, Actuators, and Controls

53912.3ActuatorsSliding spoolPoppetSolenoidactuator.800000Port1 (P)2 (T)3 (A)Figure 12.38 Three-way flow control valve (deactivated position shown).into this category.A three-wayvalve has three portconnections: supply (P),exhaust (T),and service(A).The service port may be switched between the supply and the exhaust.A four-way has fourconnections: supply (P), exhaust (T), and two service ports (A and B). The valve connects either P toA and B to T, or P to B and A to T. In general, an N-way valve has N-ports with N number of flowdirections available.An example ofa three-port sliding spool control valve is shown in Figure 12.38.The solenoid drives the spool, which contains two valve seats. In thefully activated position,portPis open to service port A. When the solenoid is deactivated, port Tis open to service port A (shown).For example,in one application this valve can beused to pressurize a system (open the system toport P)for a period of time and then adjust the systempressure to another value (open the system toport A) for a period of time.All valveports offer some level of flow resistance,and this is specified through a flowcoefficient, C Flow resistance can be adjusted in design by varying the internal dimensions of thevalve chamber and can be set operationallyby varying theelement position within the chamber.Theflow coefficient is found based on the formulation detailed in Chapter10 or simply asQ=CVAp(12.22)where Q is the steady flow rate through the valve and p is the corresponding pressure drop. Thisloss is also expressed in terms of a K-factor based on the average velocity through theportsAp=Kpu/2(12.23)Flow-control valves are classified in a number of ways: thetype ofcontrol, the number of portsin the valve housing, the specific function of the valve, and the type of valve element used in theconstruction of the valve.Directional control valves allow or prevent the flow of fluid throughdesignated ports. Flow can move in either direction. Check valves are a special class of directionalvalve that allowflow in only one direction.Proportional valves can be infinitely positioned tocontrolthe amount,pressure, and direction of fluid flow.In a proportional valve, the valve is opened by anamount proportional to the applied current.Thevalves are termed proportional because their outputflow is not exactly linear in relation to the input signal.These valves provide a way to controlpressure or flow rate with a high response rate.In the simplest application, a solenoid is used to turn a valve either on or off.In a moredemanding application, the solenoid is expected to cycle rapidly to open and close the valve.The

E1C12 09/14/2010 13:54:13 Page 539 into this category. A three-way valve has three port connections: supply (P), exhaust (T), and service (A). The service port may be switched between the supply and the exhaust. A four-way has four connections: supply (P), exhaust (T), and two service ports (A and B). The valve connects either P to A and B to T, or P to B and A to T. In general, an N-way valve has N-ports with N number of flow directions available. An example of a three-port sliding spool control valve is shown in Figure 12.38. The solenoid drives the spool, which contains two valve seats. In the fully activated position, port P is open to service port A. When the solenoid is deactivated, port T is open to service port A (shown). For example, in one application this valve can be used to pressurize a system (open the system to port P) for a period of time and then adjust the system pressure to another value (open the system to port A) for a period of time. All valve ports offer some level of flow resistance, and this is specified through a flow coefficient, Cv. Flow resistance can be adjusted in design by varying the internal dimensions of the valve chamber and can be set operationally by varying the element position within the chamber. The flow coefficient is found based on the formulation detailed in Chapter 10 or simply as Q ¼ Cv ffiffiffiffiffiffi Dp p ð12:22Þ where Q is the steady flow rate through the valve and Dp is the corresponding pressure drop. This loss is also expressed in terms of a K-factor based on the average velocity through the ports, Dp ¼ KrU 2 =2 ð12:23Þ Flow-control valves are classified in a number of ways: the type of control, the number of ports in the valve housing, the specific function of the valve, and the type of valve element used in the construction of the valve. Directional control valves allow or prevent the flow of fluid through designated ports. Flow can move in either direction. Check valves are a special class of directional valve that allow flow in only one direction. Proportional valves can be infinitely positioned to control the amount, pressure, and direction of fluid flow. In a proportional valve, the valve is opened by an amount proportional to the applied current. The valves are termed proportional because their output flow is not exactly linear in relation to the input signal. These valves provide a way to control pressure or flow rate with a high response rate. In the simplest application, a solenoid is used to turn a valve either on or off. In a more demanding application, the solenoid is expected to cycle rapidly to open and close the valve. The Port 3 (A) Sliding spool Poppet Solenoid actuator Port 1 (P) Port 2 (T) Figure 12.38 Three-way flow control valve (deactivated position shown). 12.3 Actuators 539

540Chapter12Mechatronics:Sensors,Actuators,andControlstime between each signal cyclecoupled with the internalflowloss characterofthevalvedeterminesthe averageflow.Valveresponsetimecanbedefined in several waysbut all are consistent withthemethods used in Chapter 3. The 90% response time, tgo, is the time required to eitherfill or exhaust atarget device chamber through a valveport, in effect a step function response.There is a separateresponse time for filling or exhausting. Either way,(12.24)t9o=m+FVwhere m isthevalvelagtimebetween when the signal is applied and steadyflow is established at thedesignated port, F is the reciprocal of the average flow rate through the port, and V is the volume ofthetargetdevicechamber.For example,a valve having anFof 0.54ms/cc and a lagtime of 20msrequires tgo=155ms tofill a 250-cc chamber.Altermatively,the valve frequency response can befoundbycyclingthevalvewithasinewaveelectricalsignalandmeasuringtheflowratethroughthevalve.Thevalvefrequency bandwidth is thus established by its -3 dB point.12.4CONTROLSControl of a process or systemcan be exerted in a wide varietyof ways.Suppose ourgoal is tocreatea healthy lawn by appropriate watering.Each day we could monitor the weather forecast,take intoaccount theprobability of precipitation,and choose whethertowaterand forhowlong.We couldchoose to water all of the lawn or just those parts most subject to stress from heat and lack ofmoisture.If we choose to water,we could place the sprinklers and turn on the faucet (rememberingto shut off the flow at an appropriate later time)!All of thefunctions describedaboveforlawn care arecompletelyreasonablefor aperson toaccomplish,and they represent the functioning of an intelligent controller. Suppose we wish tointroduce some automation into the process.Atthesimplestlevel,a timer-basedcontrol systemcouldbe implemented,as shown inFigure12.39.The functioning of this system would be to open and close the faucet at predetermined times of the day.Atthesimplestlevel,thiscouldbeamechanicaltimerthatwateredthelawnonceeach24-hourperiodforapredetermined length oftime.Thistype of control is called an open loopcontrol.For this controlWater supplyTimeTo sprinklersFigure 12.39Open-loop control of a sprinkler system

E1C12 09/14/2010 13:54:13 Page 540 time between each signal cycle coupled with the internal flow loss character of the valve determines the average flow. Valve response time can be defined in several ways but all are consistent with the methods used in Chapter 3. The 90% response time, t90, is the time required to either fill or exhaust a target device chamber through a valve port, in effect a step function response. There is a separate response time for filling or exhausting. Either way, t90 ¼ m þ F8 ð12:24Þ where m is the valve lag time between when the signal is applied and steady flow is established at the designated port, F is the reciprocal of the average flow rate through the port, and 8 is the volume of the target device chamber. For example, a valve having an F of 0.54 ms/cc and a lag time of 20 ms requires t90 ¼ 155 ms to fill a 250-cc chamber. Alternatively, the valve frequency response can be found by cycling the valve with a sine wave electrical signal and measuring the flow rate through the valve. The valve frequency bandwidth is thus established by its 3 dB point. 12.4 CONTROLS Control of a process or system can be exerted in a wide variety of ways. Suppose our goal is to create a healthy lawn by appropriate watering. Each day we could monitor the weather forecast, take into account the probability of precipitation, and choose whether to water and for how long. We could choose to water all of the lawn or just those parts most subject to stress from heat and lack of moisture. If we choose to water, we could place the sprinklers and turn on the faucet (remembering to shut off the flow at an appropriate later time)! All of the functions described above for lawn care are completely reasonable for a person to accomplish, and they represent the functioning of an intelligent controller. Suppose we wish to introduce some automation into the process. At the simplest level, a timer-based control system could beimplemented, as shown in Figure 12.39. The functioning of this system would be to open and close the faucet at predetermined times of the day. Atthe simplestlevel, this could be a mechanicaltimer that watered the lawn once each 24-hour period for a predetermined length of time. This type of control is called an open loop control. For this control Water supply Timer To sprinklers Figure 12.39 Open-loop control of a sprinkler system. 540 Chapter 12 Mechatronics: Sensors, Actuators, and Controls

54112.4ControlsWater supplyClocktimerCCTotalizingflowmeterTo sprinklersFigure12.40Feedbackcontrolofa sprinklersystemsystem there areno sensorsto monitorthe amount ofwaterappliedtothelawn; infact,all that thecontrolsystem is accomplishing is to open and, later, to close the faucet.More advanced automatic control systems implement a closed-loop control.For the presentexample, it might be desired to apply 100 gallons (379 liters) of water to the lawn. A flow meter thatsensedthetotal waterflowthat hadoccurredcould be usedtoprovidefeedback tothecontrol systemtoallowthefaucettobeclosed when theflowtotaled 100gallons.Suchflowmeters arecommon and serveas water meters. The term closed-loop or feedback control simply means that the variable that is to becontrolledis being measured,and that thecontrol system in some way uses this measurementto exert thecontrol.Figure 12.40 llustrates a control system designed to apply 100 gallons of water to the lawn.Thereare two inputs to the controller: the time of day and the output of the totalizing flow meter. At theappropriate time of day,the controller opens the valve.The totalizing flow meter output is used by thecontrollertoclosethevalveafterthetotal flowreaches100gallons.Thistypeofbinarycontrol schemeis termed on-off control.Thevalve controlling the water flow is eitherfully open or fully closed.Probably the most familiar form of a binary on-off control system is the thermostatfor a homefurnace or air conditioner.Figure 12.41 shows the status of a home furnace and a time trace of theinside temperature during a winter day. A schematic representation of this control system is shownFurnaceOnOnoffoffsuroanceInsideDeadbandtemperatureFigure 12.41 Operation of on-offcontroller with a dead band

E1C12 09/14/2010 13:54:13 Page 541 system there are no sensors to monitor the amount of water applied to the lawn; in fact, all that the control system is accomplishing is to open and, later, to close the faucet. More advanced automatic control systems implement a closed-loop control. For the present example, it might be desired to apply 100 gallons (379 liters) of water to the lawn. A flow meter that sensed the total water flow that had occurred could be used to provide feedback to the control system to allow the faucet to be closed when the flow totaled 100 gallons. Such flow meters are common and serve as water meters. The term closed-loop or feedback control simply means that the variable that is to be controlled is being measured, and that the control system in some way uses this measurement to exert the control. Figure 12.40 illustrates a control system designed to apply 100 gallons of water to the lawn. There are two inputs to the controller: the time of day and the output of the totalizing flow meter. At the appropriate time of day, the controller opens the valve. The totalizing flow meter output is used by the controller to close the valve after the total flow reaches 100 gallons. This type of binary control scheme is termed on–off control. The valve controlling the water flow is either fully open or fully closed. Probably the most familiar form of a binary on–off control system is the thermostat for a home furnace or air conditioner. Figure 12.41 shows the status of a home furnace and a time trace of the inside temperature during a winter day. A schematic representation of this control system is shown Water supply To sprinklers Clock timer Totalizing flow meter Figure 12.40 Feedback control of a sprinkler system. Inside temperature Furnace On Off Off Disturbance Dead band On Figure 12.41 Operation of on–off controller with a dead band. 12.4 Controls 541

542Chapter12Mechatronics:Sensors,Actuators,andControls68°F(20℃C)RelayThermostatFurnaceFigure 12.42 Components of a thermostatic control for a home furnace.in Figure 12.42. A key element here is that there is the possibility of a disturbance that wouldinfluence the rate of change of the inside temperature. Suppose a delivery arrives and the doorremains open for a period of time.The thermostat must then respond to this disturbance and attemptto maintain the inside temperature at the set point.Essentially all practical implementations of on-off control systems require a dead band thatcreates a hysteresis loop in the control action. This is illustrated in Figure 12.43. The dead band iscentered around zero error, and the action of the controller depends on the magnitude of the error.Here the error is defined ase = T setpoint -- T roomRecall that we are considering a furnace thermostat under winter conditions.As the roomtemperaturefalls relative to the set point, the error becomes a larger positive number.When the errorreaches the value corresponding to“Furnace ON"in Figure 12.43,the furnace begins to add heat tothe conditioned space.Room temperature begins to rise and the error decreases towards zero.Thefurnaceremainsonuntiltheroomtemperaturereachesapredeterminedvaluethatisgreaterthantheset point. Here the error is negative. At this temperature, the furnace is turned off and roomtemperature begins once again to decrease.Because of the dead band in the controller, no furthercontrol action occurs until the error reaches the “"Furnace ON"error magnitude.ControlactionONZeroFurnaceFurnaceOFF/ONerrorOFFFigure12.43ControlactionandErrorhysteresisloopofabinaryon-off一一Deadbandcontroller with a dead band

E1C12 09/14/2010 13:54:13 Page 542 in Figure 12.42. A key element here is that there is the possibility of a disturbance that would influence the rate of change of the inside temperature. Suppose a delivery arrives and the door remains open for a period of time. The thermostat must then respond to this disturbance and attempt to maintain the inside temperature at the set point. Essentially all practical implementations of on–off control systems require a dead band that creates a hysteresis loop in the control action. This is illustrated in Figure 12.43. The dead band is centered around zero error, and the action of the controller depends on the magnitude of the error. Here the error is defined as e ¼ Tsetpoint Troom Recall that we are considering a furnace thermostat under winter conditions. As the room temperature falls relative to the set point, the error becomes a larger positive number. When the error reaches the value corresponding to ‘‘Furnace ON’’ in Figure 12.43, the furnace begins to add heat to the conditioned space. Room temperature begins to rise and the error decreases towards zero. The furnace remains on until the room temperature reaches a predetermined value that is greater than the set point. Here the error is negative. At this temperature, the furnace is turned off and room temperature begins once again to decrease. Because of the dead band in the controller, no further control action occurs until the error reaches the ‘‘Furnace ON’’ error magnitude. Thermostat Relay 68 ºF (20 ºC) Furnace Figure 12.42 Components of a thermostatic control for a home furnace. Deadband Furnace ON Furnace OFF Control action Zero error ON OFF Error Figure 12.43 Control action and hysteresis loop of a binary on–off controller with a dead band. 542 Chapter 12 Mechatronics: Sensors, Actuators, and Controls

54312.4ControlsControlactionControllerWater sourFlowmeterFigure 12.44 Flow rateGate valvcontrol systemMany control systems are designed to maintain a specified set point without a dead band;clearly this is not possible using on-off control.Consider again watering a lawn. Suppose that due tovarying water pressure thelawn was being watered nonuniformly.Wemightchooseto control boththewaterflow rate and the total waterflow applied to the lawn.A schemefor accomplishing this isshown in Figure 12.44.Thekey components of the control system are a flow meter, the controller,and an actuator that can control the position of a gate valve. Here a desired flow rate is set, say,to 2gallons per minute. The task of the controller is to vary the position of the gate valve in order tomaintain the flow rate at the set point.DynamicResponseAs another example of closed-loop control, consider the cruise-control system on an automobile.Oncea desired speed is set,thesystemvaries thethrottleposition toensurethatthesetpointismaintained.Anotherissueintheanalysisanddesignofcontrol systemsliesinthedynamicresponseof the physical process,the controller,and the actuators.An automobile does not respondinstantaneously to changes in the throttle position and time is required for a stepper motor tochange the position of a valve. The complexities of the combined responses of the physical andcontrol systems, especially in the presence of disturbances, are the subject of the remainder of thissection.LaplaceTransformsInChapter3,weusedLaplacetransforms tomodeltheresponseofsimplesystems.Nowletus applyLaplace transforms to understand how control systems function.Thefundamental basis of theapplication of Laplace transforms to control systems is the solution of a mathematically well-posedinitial value problemConsider an initial-value problem (here time is the independent variable) that is described by anordinary differential equation.If we apply the Laplace transform to a differential equation, weconvert the differential equation to an algebraic equation. For partial differential equations in timeand one spatial variable,the Laplace transform converts thepartial differential equation into anordinarydifferentialequationinthespacevariable.AppendixCreviewstheapplicationof Laplacetransforms and provides a table of Laplace transform pairs.We illustrate the application of the Laplacetransform through solution of first-order andsecond-order differential equations, which are important for control systems

E1C12 09/14/2010 13:54:13 Page 543 Many control systems are designed to maintain a specified set point without a dead band; clearly this is not possible using on–off control. Consider again watering a lawn. Suppose that due to varying water pressure the lawn was being watered nonuniformly. We might choose to control both the water flow rate and the total water flow applied to the lawn. A scheme for accomplishing this is shown in Figure 12.44. The key components of the control system are a flow meter, the controller, and an actuator that can control the position of a gate valve. Here a desired flow rate is set, say, to 2 gallons per minute. The task of the controller is to vary the position of the gate valve in order to maintain the flow rate at the set point. Dynamic Response As another example of closed-loop control, consider the cruise-control system on an automobile. Once a desired speed is set, the system varies the throttle position to ensure that the set point is maintained. Another issue in the analysis and design of control systems lies in the dynamic response of the physical process, the controller, and the actuators. An automobile does not respond instantaneously to changes in the throttle position and time is required for a stepper motor to change the position of a valve. The complexities of the combined responses of the physical and control systems, especially in the presence of disturbances, are the subject of the remainder of this section. Laplace Transforms In Chapter 3, we used Laplace transforms to model the response of simple systems. Now let us apply Laplace transforms to understand how control systems function. The fundamental basis of the application of Laplace transforms to control systems is the solution of a mathematically well-posed initial value problem. Consider an initial-value problem (here time is the independent variable) that is described by an ordinary differential equation. If we apply the Laplace transform to a differential equation, we convert the differential equation to an algebraic equation. For partial differential equations in time and one spatial variable, the Laplace transform converts the partial differential equation into an ordinary differential equation in the space variable. Appendix C reviews the application of Laplace transforms and provides a table of Laplace transform pairs. We illustrate the application of the Laplace transform through solution of first-order and second-order differential equations, which are important for control systems. Flow meter Controller Gate valve Control action Water source Figure 12.44 Flow rate control system. 12.4 Controls 543