Mechanical Engineers Handbook 2011 Part 11 doc

The pressure drop through ori®ce A causes a reduced pressure in chamber B and allows spool 1 to move upward andopen, the outlet or secondary port.. The continued ¯ow and pressure drop th

top and bottom of spool 1 The force of spring 2 and the difference in areabetween the top side and the bottom side of spool 1 will quickly move thespool 1 down and close the outlet port.

Figure 2.15 shows another version of the compound-type relief valve

This valve is referred to as a balanced-piston-type relief valve The operatingfunctions in Fig 2.15a are basically the same as those in Fig 2.14 The maindifference is in the method of ¯ow From the pilot valve 2, oil is returned tothe reservoir The pilot drain D is a hole or ori®ce passing through the mainspool 1 directly into the outlet port This feature requires fewer hydrauliclines, but it also allows outlet back pressure to adversely affect pilotoperations It is important that the outlet lines be unrestricted to ensure

minimum back pressure Ori®ce B in Fig 2.15a is placed differently, as is thepilot valve The port V in Fig 2.15 offers a new method of control for thecompound-type unit If port V is allowed to be open to the atmosphere or,

by a simple valve, to the reservoir, the pressure on the top side of the mainspool 1 will be relieved and the spool will immediately move upward andopen This practice is called venting and offers an auxiliary or additionalmethod of instantly relieving the system pressure without altering or affectingthe unvented operating setting of the valve A simple direct-operating reliefvalve, manually operated, will handle venting functions and further increasethe ¯exibility of this unit's operation Figure 2.15b shows a typical arrange-ment that allows manually controlled venting and=or automatic systempressure operation The vent valve is small since it is required to handleminor volumes

2.6.7 COMPOUND-TYPE SEQUENCE VALVES

Figures 2.16a and 2.16b show the revised compound relief valve Figure2.16a illustrates the unit designated as the Y type Since the outlet chamberbecomes the secondary port exposed to the system pressure when the valveopens, an external bleed line E (Fig 2.16a) must be used The pilot chamber

is no longer opened at the center ori®ce D of spool 1 The ori®ce D is nowused to ensure complete hydraulic balance of spool 1 when the unit is in itssequenced position The operation of the valve in Fig 2.16a starts when thesystem pressure at the inlet port passing through ori®ce A reaches the levelrequired to unseat pilot piston 3 The pressure drop through ori®ce A causes

a reduced pressure in chamber B and allows spool 1 to move upward andopen, the outlet or secondary port Oil passing through ori®ce D in the center

of spool 1 is now opened to system pressure, and the effective areas on both

sides of valve 1 are equal The continued ¯ow and pressure drop throughori®ce A maintains a lower pressure in chamber B, the valve remaining open.

In the event that the inlet pressure decreases, pilot piston 3 closes, thepressure in chamber B rises, and the valve closes

The valve presented in Fig 2.16a is dependent on the system pressure atthe point of operation Figure 2.16b shows another modi®cation of the basiccompound unit This model is identi®ed as the X type For this type an openpassage between the pilot chamber and the top of spool 1 is used The ori®ce

D through the center of spool 1 is eliminated

Operation of the X type is different from any type previously discussed

The purpose of this design is to ®ll the main circuit of a system with oil before

¯ow to the outlet or secondary circuit is allowed As the main circuit becomesfull, the pressure rises at the inlet of Fig.2.16b The ¯ow through ori®ce Acauses the opening of pilot 3, and, as a result, spool 1 opens As the valveopens, the full area of the bottom spool 1 is exposed to system pressure

Since the small guide area of the top side spool 1 is opened to the pilot drainchamber, the spool 1 is hydraulically unbalanced because of the differentialarea It is required that the system pressure be suf®cient to overcome the verylight force of spring 2 to remain open This X type does not close until thesystem pressure has decreased nearly to zero This valve's main purpose istherefore limited to controlling the sequence of ¯ow as a hydraulic system isput into operation

2.6.8 PRESSURE-REDUCING VALVES

A low-pressure, low-volume ¯ow in addition to the main system pressure, high-volume ¯ow is required by some hydraulic systems Theextra pump can be eliminated by the use of a pressure-reducing valve tosupply the small ¯ow at reduced pressure

high-Figure 2.17 shows an X -model pressure-reducing valve The X -valvecombines the features of the direct-operated valve type with those of thecompound-type valve This valve incorporates a pilot 1 to control the action

of the main spool 3, thus being a compound valve The pressure-actuatedspool 3 seals because of its close ®t to the main body and because of a slidingaction that opens and closes the outlet or reduced-pressure port As system

¯ow begins, the inlet is supplied with oil at the main pressure-control-valvesetting The ¯ow to the outlet or reduced-pressure port is transmitted throughori®ce C , which is a narrow space between the reduced-diameter section ofspool 3 and the main body The ¯uid under pressure passes throughchamber D and exerts a force on the bottom area of spool 3 A very smallori®ce E carries the pressurized oil through the center of spool 3 intochamber A The areas of both ends of spool 3 are equal and under thesame pressure so that a state of hydraulic balance exists Spool 3 is thus helddown by the force of spring 4 Since a reduced pressure at the outlet isdesired, pilot 1 is adjusted to open at a pressure considerably lower than thepressure available at ori®ce C The ori®ce E has a smaller area than chamber Fluid

D, and the ¯ow of the oil from E to F causes a pressure drop The pressure inchamber F and on the top of spool 3 is lower than the pressure in chamber D

or the pressure on the bottom of spool 3 Hydraulic unbalance occurs now;spring 4 is overcome and spool 3 is moved upward As spool 3 movesupward, ori®ce C is reduced in size, opposing the ¯ow, and a pressure drop

is created between the inlet and outlet ports Port C will thus be consistentlychanged to increase or decrease the resistance to the ¯ow in order tomaintain a constant reduced pressure at the outlet As the ¯ow from theoutlet port increases in response to an increased low-pressure ¯ow demand,the spool will move downward and open ori®ce C As ¯ow is diminished,ori®ce C will be closed The maximum pressure available at the outlet is thesum of the forces of spring 2 and spring 4

This valve has three critical situations Ori®ce E is very small and it can

be very easily plugged by minute foreign bodies A constant ¯ow throughori®ce E to the drain port of the pilot valve is needed to maintain a constantdependable reduced pressure Ori®ce F must remain completely open Thepilot drain must have a free, unrestricted, unshared line to the reservoir The

®nal critical area of this valve is the close tolerance required between spool 3and the bore of the main body

allow the ¯ow from inlet to outlet The ori®ce C in poppet A serves as a drainfor chamber D It also exposes the top side of the poppet to the prevailingpressure of the outlet side when it is closed and of the inlet side when it isopen The system pressure need only overcome the force of spring B to holdthe valve open The pressure drop through this valve, from inlet to outlet, isthus equal to the force of spring B when the valve is properly sized withrespect to ¯ow volume Figure 2.18b shows the valve in the opened position.

There are situations in hydraulic circuit design when it is desirable tohave the automatic single-¯ow feature of the simple check valve for only aportion of the time and at any given time to be able to allow ¯ow in eitherdirection This situation occurs in working with load-lifting devices Thenormal single-¯ow characteristic allows the load to be lifted at any time andautomatically held It is also required that the ability to lower the load beincluded in the design A pilot-operated check valve will adequately performthis function

Figure 2.19 illustrates a pilot-operated check valve In Fig 2.19a, thecheck valve has a portion constructed in a manner similar to the simplecheck valve in Fig 2.18 A pilot piston D with a stem E and a pilot pressureport for external connection have been added

Figure 2.19b shows the valve when inlet pressure is high enough toovercome the force of spring B Pilot pressure is still 0 lb=in2, and thus theinlet pressure acts on the top side of piston D and holds the pilot stem Edownward The valve acts as the conventional check valve in Figs 2.19a and2.19b

Let us assume that we need to have the ¯ow from the outlet port to theinlet port A load has been lifted by allowing ¯ow from inlet to outlet, andthat it is now time to lower the load

The application of an independent external pressure to the pilot port willmove piston D upward, allowing ¯ow from outlet to inlet, thus lowering the

load To maintain the valve in an opened position, the following relation isrequired:

PP  DB> FsB‡ PC DTor

FP > FsB‡ Fibecause

P  A ˆ Fwhere Pp is the pilot pressure, DB is the bottom area D, FSB is the spring Bforce, Piis the inlet pressure, DT is the top area D, FP is the pilot force, and Fi

is the inlet force Also P is the pressure, A is the area, and F is the force

2.7.2 PARTIAL-FLOW-LIMITING CONTROLS

For a hydraulic cylinder, the speed in one direction can be controlled if asimple needle valve is located in the exhaust port of the cylinder This isreferred to as a meter-out application The exhaust pressure of a hydrauliccylinder is relatively stable and, thus, it maintains a reasonably accurate ¯owrate control with a simple needle valve Figure 2.20a shows a simple needle-valve meter-out control The unit depicted in Fig 2.20 has the additionalfeature of allowing the ¯ow to pass through a check valve B in one direction,thus being unaffected by the adjustment of the metering valve A Themetered ¯ow direction of these valves is usually indicated by an arrow onthe external surface of the unit A ®ne adjustment thread on the stem of valve

A provides a precise control of the ¯ow An adjustment of A is minor whilethe valve is subjected to system pressure Excessive looseness of the locknutfor valve A and excessive turning of valve A may damage the small valveseal If large adjustments are needed, they are best accomplished at 0 lb=in2

The ¯ow control illustrated in Fig 2.20d is far superior to those shown in Figs.

2.20a to 2.20c The unit shown in Fig 2.20d is adjustable at any time, evenwhile under maximum pressure

2.8 Hydraulic Pumps

Although many hydraulic pumps and motors appear to be interchangeable inthat they operate on the same principles and have similar parts, they oftenhave design differences that make their performances better as either motors

or pumps Moreover, some motors have no pump counterparts In thischapter only positive-displacement pumps are considered (those pumpsthat deliver a particular volume of ¯uid with each revolution of the inputdrive shaft) This terminology is used to distinguish them from centrifugalpumps and turbines

2.8.1 GEAR PUMPS

The simplest type of these pumps is the gear pump, shown in Fig 2.21, inwhich the ¯uid is captured in the spaces between the gear teeth and thehousing as the gears rotate Flow volume is controlled by controlling thespeed of the drive gear Although these pumps may be noisy unless welldesigned, they are simple and compact

2.8.2 GEROTOR PUMPS

Another version of the gear pump is the gerotor, whose cross section ispresented schematically in Fig 2.22 The internal gear has one fewer tooth

Figure 2.20 Single-needle valve ¯ow control (a) Metered ¯ow in both directions (b) Check valve for reverse

¯ow (c) Reverse ¯ow check valve shown open (d) Valve constructed to allow adjustment while under pressure

than the external gear, which causes its axis to rotate about the axis of theexternal gear The geometry is such that on one side of the internal gear thespace between the inner and outer gerotor increases for one-half of eachrotation and on the other side it decreases for the remaining half of therotation It consists of three basic parts: the ring, the outer gerotor, and theinner gerotor The number of the teeth varies, but the outer gerotor alwayshas one more tooth than the inner gerotor.

The ®gure shows the two kidney-shaped ports, namely, the suction portand discharge port The axis around which the inner element rotates is offset

by the amount e from the axis of the outer gerotor, which, driven by the innergerotor, rotates within the ring

2.8.3 VANE PUMPS

Figure 2.23a shows the sketch of a vane pump The drive shaft center line isdisplaced from the housing center line, having uniformly spaced vanesmounted in radial slots so that the vanes can move radially inward andoutward to always maintain contact with the housing Fluid enters throughport plates, shown in Fig 2.23b, at each end of the housing The advantages

of vane pumps over gear pumps are that they can provide higher pressuresand variable output without the need to control the speed of the primemover (electric motor, diesel engine, etc) The design modi®cation requiredfor a variable volume output from a pump having a circular interior crosssection is that of mounting the housing between end plates so that the axis ofthe cylinder in which the vane rotates may be shifted relative to the axis ofthe rotor, as shown in Fig 2.24 The maximum ¯ow is obtained when theyare displaced by the maximum distance (Fig 2.24a), and zero ¯ow isobtained when the axis of the rotor and the housing tend to coincide (Fig

with inlet port

2.8.4 AXIAL PISTON PUMP

An axial piston pump is shown in Fig 2.25 The major components are theswashplate, the axial pistons with shoes, the cylinder barrel, the shoeplate,the shoeplate bias spring, and the port plate The shoeplate and the shoe-plate bias spring hold the pistons against the swashplate, which is heldstationary while the cylinder barrel is rotated by the prime mover Thecylinder, the shoeplate, and the bias spring rotate with the input shaft, thusforcing the pistons to move back and forth in their respective cylinders in thecylinder barrel The input and output ¯ows are separated by the stationaryport plate with its kidney-shaped ports Output volume may be controlled bychanging the angle of the swashplate As angle a between the normal to theswashplate and the axis of the drive shaft in Fig 2.26b goes to zero, the

¯ow volume decreases If angle a increases (Fig 2.26a), the volume alsoincreases Axial piston pumps with this feature are known as overcenter axialpiston pumps

2.8.5 PRESSURE-COMPENSATED AXIAL PISTON PUMPS

For these pumps the angle a of the swashplate is controlled by a loaded piston that senses the pressure at a selected point in the hydraulicsystem As the pressure increases, the piston can decrease a in an effort todecrease the system pressure, as illustrated in Fig 2.26b Pressure compensa-tion is often used with overcenter axial piston pumps in hydrostatic transmis-sions to control the rotational speed and direction of hydraulic motors

spring-2.9 Hydraulic Motors

Hydraulic motors differ from pumps because they can be designed to rotate

in either direction, can have different seals to sustain high pressure at lowrpm, or can have different bearings to withstand large transverse loads so

they can drive sprockets, gears, or road wheels on vehicles A rotating valvethat distributes the pressure to the pistons in sequence causes the outputshaft to rotate in the desired direction The pistons are mounted in a blockthat holds the pistons perpendicular to the rotor Each piston slides laterally

on a ¯at surface inside the housing as it applies a force between the ¯atportion of the housing and the eccentric rotor Figure 2.27 shows a radialpiston pump with the pistons, 2, arranged radially around the rotor hub, 1

The rotor with the cylinders and the pistons are mounted with an eccentricity

in the pump house 3 The pistons, which can slide within the cylinders with aspecial seal system, pull and then push the ¯uid (the arrows on the ®gure)through a central valve 4

Orientation of the block relative to the housing is maintained by means

of an Oldham coupling The schematic principle of operation of an Oldhamcoupling is presented in Fig 2.28 The main parts are the end plate 1,coupling plate 2, and block 3, which contains the pistons and the eccentric

Figure 2.25 Axial piston pump (a) Overcenter axial pump without drive shaft shown (b) Basic parts for

axial piston pump

Figure 2.26 Simpli®ed schematic of the operation of the compensator piston in controlling the angle of theswashplate to control output ¯ow rate (a) Large displacement for full ¯ow (b) Zero displacement for no ¯ow.

portion of the shaft Slot a is cut into plate 1 and accepts track A, which is part

of plate 2 Track B is perpendicular to track A and is located on the oppositeside of plate 2 from track A Slot b is cut into the block and accepts track B

Thus, any displacement of the block relative to plate 1, which is attached tothe housing, can be decomposed into components parallel to tracks A and B

As the shaft turns the center of the eccentric and of the block, it will describe

a circle about the center of the housing, but the block itself will not rotate

Pistons, block, and housing, therefore, will always maintain their properorientation relative to one another

2.10 Accumulators

An accumulator is a tank that accumulates and holds ¯uid under pressure

Accumulators are used to maintain the pressure in the presence of ¯uctuating

¯ow volume, to absorb the shock when pistons are abruptly loaded orstopped, as in the case of planers, rock crushers, or pressure rollers, or tosupplement pump delivery in circuits where ¯uid can be stored during otherparts of the cycle The bladder-type accumulator is presented in Fig 2.29a

This design incorporates a one-piece cylindrical shell with semicircular ends

to better withstand system pressure A rubber bladder or separator bag isinstalled inside the outer shell It is this bladder that, when ®lled with a gasprecharge, supplies the energy to expel stored liquid at the desired time Apoppet valve is supplied in the lower end of the accumulator to prevent thebladder from being damaged by entering the ¯uid port assembly Thispoppet is held open by a spring but is closed once the accumulatorprecharge extends the bladder and causes it to contact the top of the

poppet The gas used for accumulator service is an inert gas such as nitrogen.Also, a small valve is used to ®ll the bladder with gas This valve is similar tothose used to ®ll auto tires A locknut is provided to anchor this valve and thebladder to the shell.

The piston-type accumulator is shown in Fig 2.29b Note that the pistontype resembles a conventional hydraulic cylinder minus the piston rod Thiscon®guration can be identi®ed as free or ¯oating piston operation The gasprecharge is on one side of the piston, and the system oil is on the other Twoseals are indicated on the piston head Therefore, the two rings keep thepiston head from cocking, but actual sealing is accomplished by the seal ring

on the gas side of the piston head

Figure 2.29 Internal construction of accumulators (a) Bladder-type accumulator (b) Piston-type lator

2.11 Accumulator Sizing

Accumulator size depends on the amount of ¯uid to be stored and the meansused to supply pressure to the ¯uid stored in the accumulator If a weightabove a piston is used, the accumulator must be large enough to hold the

¯uid and the volume of the weight and piston When gas pressure is used,either in a bladder or above a piston, the sizing of the accumulator requiresthat we consider the behavior of the gas as it is being compressed by theincoming ¯uid The gas equation is considered to be polytropic and includesisothermal and reversible adiabatic changes as special cases if the appro-priate value of the exponent is selected An isothermal process is one inwhich the compression is slow enough for the temperature of gas to remainconstant An adiabatic process is one that is so rapid that no heat is lost andthe temperature rises accordingly The polytropic gas equation is

b ˆ 1:24; for bladder-type accumulators

b ˆ 1:11; for piston-type accumulators

2.12 Fluid Power Transmitted

For calculating the power transmitted to a particular unit, it is necessary toknow the functional formula for power,

where F is the force, and v is the velocity

The force F can be written as

where p is the pressure and A is the cross-sectional area

If L is the distance traveled in time t by a point that moves with the ¯uid

¯owing through the hose or cylinder of cross-sectional area A, the powerrequired to move that ¯uid is

P ˆ pA@L@t ˆ pAv ˆ p@V@t : …2:20†The rate of change of time @V =@t is denoted by _Q in units of gallons perminute or liters per minute

1 gallon …1 gal† ˆ 231 in3; and 1 horsepower …hp† ˆ …ft lb=min†=33;000:

2.13 Piston Acceleration and Deceleration

To analyze piston behavior, we consider piston velocity and acceleration as afunction of the system parameters

The velocity is the volumetric ¯ow rate divided by the cross-sectionalarea Ac of the cylinder Thus,

During the motion, the rod and piston are accelerating, and theequilibrium equation is

dx

dt ˆ

pAcÿ …Fr‡ f †

when the piston starts from rest

When set equal to the piston maximum steady-state velocity vr, the timeneeded to accelerate to velocity vr is

ta ˆ mvr

The distance required for the piston to reach this velocity may be calculated

by integrating Eq (2.23) with respect to time and using the condition that themotion started from x ˆ 0 to obtain

The time needed by the piston to accelerate, move at constant velocity,and decelerate may be estimated by using the relationship

2.14 Standard Hydraulic Symbols

The time and effort to draw and modify design drawings for hydraulicsystems (Fig 2.30) can be greatly reduced by employing a set of standarddesign symbols to denote hydraulic components Two different conventionshave been accepted for joining and crossing hydraulic lines They are Fluid

presented in Fig 2.31a The main hydraulic lines are drawn as solid lines;pilot lines are drawn as long dashes; exhaust and drain line are drawn asshort dashes Check valves are drawn as in Fig 2.31b, where ¯ow is allowedfrom A to B, but not from B to A Figures 2.32 and 2.33 show other activationsymbol, and Fig 2.34 shows the symbolic circuit for the regenerativecylinder, initially shown in Fig 2.30.

of a b value The symbol b is immediately followed by a number that denotesthe diameter of the particles involved according to the relation

bdˆNumbers of particles of diameter d downstream from the filterNumber of particles of diameter d upstream from filter :Most ¯uid ®lters are not rated for particles less than 3 mm; b may be taken as1.0 to d < 3

2.16 Representative Hydraulic System

A simple hydraulic circuit to provide bidirectional control of a hydrauliccylinder is shown in Fig 2.35 It includes a motor with a clutch between themotor and pump, a ®lter in the motor intake lane from the reservoir, and amanually operated directional valve

Figure 2.32 Activation symbols

Figure 2.33 Other activation symbols.

McGraw-3 R M White, Fluid Mechanics McGraw-Hill, New York, 1999.

4 W C Orthwein, Machine Component Design West Publishing Company, St.Paul, MN, 1990

5 J J Pippenger and T G Hicks, Industrial Hydraulics, 2nd ed McGraw-Hill,New York, 1970

6 E F Brater and H W King, Handbook of Hydraulics McGraw-Hill, NewYork, 1980

7 M E Walter Ernst, Oil Hydraulic Power and Its Industrial Applications.McGraw-Hill, New York, 1949

8 R P Lambeck, Hydraulic Pumps and Motors: Selection and Application forHydraulic Power Control Systems Dekker, New York, 1983

9 R P Benedict, Fundamentals of Pipe Flow Wiley & Sons, New York, 1977

10 W G Holzbock, Hydraulic Power and Equipment Industrial Press, NewYork, 1968

11 L S McNickle, Jr., Simpli®ed Hydraulics McGraw-Hill, New York, 1966

3.1 Transfer Functions for Standard Elements 616

3.2 Transfer Functions for Classical Systems 617

4 Connection of Elements 618

5 Poles and Zeros 620

6 Steady-State Error 623

6.1 Input Variation Steady-State Error 623

6.2 Disturbance Signal Steady-State Error 624

7 Time-Domain Performance 628

8 Frequency-Domain Performances 631

8.1 The Polar Plot Representation 632

8.2 The Logarithmic Plot Representation 633

8.3 Bandwidth 637

9 Stability of Linear Feedback Systems 639

9.1 The Routh±Hurwitz Criterion 640

9.2 The Nyquist Criterion 641

9.3 Stability by Bode Diagrams 648

10 Design of Closed-Loop Control Systems by Pole-Zero Methods 649

10.1 Standard Controllers 650

10.2 P-Controller Performance 651

10.3 Effects of the Supplementary Zero 656

10.4 Effects of the Supplementary Pole 660

10.5 Effects of Supplementary Poles and Zeros 661

10.6 Design Example: Closed-Loop Control of a Robotic Arm 664

11 Design of Closed-Loop Control Systems by Frequential Methods 669

12 State Variable Models 672

611

T his chapter, ``Control,'' is an introduction to automation for technical

students and engineers who will install, repair, or develop automaticsystems in an industrial environment It is intended for use inengineering technology programs at the postsecondary level, but it is alsosuitable for use in industrial technology curricula, as well as for in-serviceindustrial training programs Industrial managers, application engineers, andproduction personnel who want to become familiar with control systems or

to use them in production facilities will ®nd this chapter useful

The text requires an understanding of the principles of mechanics and

¯uid power, as well as a familiarity with the basics of mathematics Althoughnot essential, a good knowledge of the principles of physics is also helpful

1 Introduction

Control engineering is concerned with the automation of processes in order

to provide useful economic products These processes can be conventionalsystems, such as chemical, mechanical, or electrical systems, or moderncomplex systems such as traf®c-control and robotic systems Control engi-neering is based on the foundation of feedback theory and linear systemanalysis The aim of the control system is to provide a desired systemresponse

In order to be controlled, a process can be represented by a block, asshown in Fig 1.1 The input±output relationship represents the cause andeffect relationship of the process The simplest system of automation is theopen-loop control system, which consists of a controller that provides theinput size for the process (Fig 1.2)

13 Nonlinear Systems 678

13.1 Nonlinear Models: Examples 678

13.2 Phase Plane Analysis 681

13.3 Stability of Nonlinear Systems 685

13.4 Liapunov's First Method 688

13.5 Lipaunov's Second Method 689

14 Nonlinear Controllers by Feedback Linearization 691

15 Sliding Control 695

15.1 Fundamentals of Sliding Control 695

15.2 Variable Structure Systems 700

A Appendix 703

A.1 Differential Equations of Mechanical Systems 703

A.2 The Laplace Transform 707

A.3 Mapping Contours in the s-Plane 707

A.4 The Signal Flow Diagram 712

References 714

Figure 1.1

Plant

A closed-loop control system (Fig 1.3) uses a feedback signal consisting

of the actual value of the output This value is compared with the prescribed

or desired input, and the result of the comparison de®nes the system error

This size is ampli®ed and used to control the process by a controller Thecontroller acts in order to reduce the error between the desired input and theactual output Moreover, several quality criteria are imposed in order toobtain a good evolution of the global system

water level measured by a ¯oat that controls, by a mechanical system, thevalve that, in turn, controls the water ¯ow out of the tank The plant isrepresented by the water tank, and the water ¯owing into the tank represents

a disturbance of the system If the water level increases, the ¯oat moves upand, by a mechanical system representing the controller, initiates the opening

of the valve If the water level decreases, the ¯oat moves down and initiatesthe closing of the valve The size of the closing or opening can be adjustedfrom a reference panel that provides the prescribed values The systemoperates as a negative feedback control system because a difference isobtained between the prescribed value and the output; the water level andthe variations of the water level are eliminated by compensation through thevalve functions

This system represents one of the simplest control systems Effort isnecessary in order to eliminate transient oscillations and to increase theaccuracy of the control system

2 Signals

The differential equations associated with the components of control systems(Appendix A) indicate that the time evolution of the output variable x0…t† is afunction of the input variable xi…t† In order to obtain the main characteristics

of these elements it is necessary to use standard input signals

(a) The impulse function d…t†: The unit impulse is based on a rectangularfunction f …t† such as

fe…t† ˆ 1=e; 0  t  e0; t > e;



…2:1†where e > 0 As e approaches zero, the function fe…t† approaches theimpulse function d…t†, where

…1

…1

0 d…t ÿ a†f …t†dt ˆ f …a†: …2:3†The impulse input is useful when one considers the convolutionintegral for an output x0…t† in terms of an input xi…t†,

x0…t† ˆ

…t

0h…t ÿ t†xi…t†dt ˆ lÿ1fh…„†xi…„†g; …2:4†where h…s†; xi…s† are the Laplace transforms of h…t†; xi…t†, respec-tively (Appendix B) If the input is the impulse function d…t†,

This signal is shown in Fig 2.1.

(c) The ramp input: The standard test signal has the form (Fig 2.2)

where a is the amplitude, o is the frequency of the signal (Fig 2.3),and T is the period This signal is used when we analyze theresponse of the system when the frequency of the sinusoid isvaried So, several performance measures for the frequencyresponse of a system The Laplace transform is

3.1 Transfer Functions for Standard Elements

Transfer functions for standard elements are the following:

(a) Proportional element: For this element, the output is proportional tothe input,

(b) Integrating element: The output is de®ned by the integral of theinput,

3.2 Transfer Functions for Classic Systems

We consider the linear spring±mass±damper system described in AppendixA.1, Eq (A1.2), with zero initial conditions

Ms2Xo…s† ‡ kfsXo…s† ‡ kXo…s† ˆ AXi…s†: …3:15†

Figure 3.1

Transfer

The transfer function will be

In order to represent a complex system, an interconnecting of blocks isused This representation offers a better understanding of the contribution ofeach variable than is possible to obtain directly from differential equations.(a) Cascade connection: In this case, the output of the ®rst element isalso the input in the second element (Fig 4.2),