controlsystems9-04-13-130409135906-phpapp01.docx

8/18/2019 controlsystems9-04-13-130409135906-phpapp01.docx

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Dr.Y.NARASIMHA MURTHY Ph.D [email protected]

CONTROL SYSTEMS

INTRODUCTION :

A control system is one which can control any quantity of interest in a machine, mechanism or other equipment in order to achieve the desired performance or output.(or) A control system is an

interconnection of components connected or related in such a manner as to command, direct, or

regulate itself or another system. For example consider, the driving system of an automobile.

Speed of the automobile is a function of the position of its accelerator. he desired speed can be

maintained (or a desired change in speed can be achieved) by controlling pressure on the

accelerator pedal. his automobile driving system (accelerator, carburetor and engine!vehicle)

constitutes a control system.

"ontrol systems find numerous and widespread applications from everyday to extraordinary inscience, industry, and home. #ere are a few examples:

(a) #ome heating and air!conditioning systems controlled by a thermostat

(b) he cruise (speed) control of an automobile

(c) $anual control

(i) %pening or closing of a window for regulating air temperature or air quality

(ii) Activation of a light switch to regulate the illumination in a room (iii) #uman controlling the speed of an automobile by regulating the gas supply to the engine

(d) Automatic traffic control (signal) system at roadway intersections

(e) "ontrol system which automatically turns on a room lamp at night, and turns it off in

&ay light

he general bloc' diagram of a control system is shown below.

.

Fig.1 .Block diagram of a conrol !"!#m.

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he above method of representation of a control system is 'nown as bloc' diagram

representation where in each bloc' represents an element, a plant, mechanism, device etc.,

whose inner details are not indicated. ach bloc' has an input and output signal which are lin'ed

by a relationship characteriing the bloc'. *t may be noted that the signal flow through the bloc'

is unidirectional

Ba!ic Conrol S"!#m com$on#n! : he basic control system components are ob+ectives i.e

inputs or actuating signals to the system and the %utput signals or controlled variables etc.he

control system will control the outputs in accordance with the input signals.he relation beteen

these components is shown in the bloc' diagram.

he components of the control system changes as we move from openloop control system to

closed loop control systems. *n a closed loop control system ,the feedbac' control networ' playan important role in getting the correct output.

he general bloc' diagram of a control system with feed bac' is shown below. he error detector

compares a signal obtained through feedbac' elements, which is a function of the output

response, with the reference input. Any difference between these two signals gives an error or

actuating signal, which actuates the control elements. he control elements in turn alter the

conditions in the plant in such a manner as to reduce the original error.

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Fig.% .n#ral 'lock diagram of an a(omaic conrol !"!#m.

T"$#! of conrol !"!#m! :

here are basically two types of control systems (i) the open loop system and the (ii) closed loop

system. hey can both be represented by bloc' diagrams. A bloc' diagram uses bloc's to

represent processes, while arrows are used to connect different input, process and output parts.

O$#n loo$ Conrol S"!#m : Asystem which do not possess any feed bac' networ' ,and

contains only the input and output relationship is 'nown as a open loop control system

.

xamples of the open loop control systems are washing machines, light switches, gas ovens, burglar alarm system etc.

he drawbac' of an open loop control system is that it is incapable of ma'ing automatic

ad+ustments. ven when the magnitude of the output is too big or too small, the system cant

ma'e the necessary ad+ustments. For this reason, an open loop control system is not suitable for

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use as a complex control system.

Clo!#d loo$ conrol !"!#m : A closed loop system is one which uses a feed bac' control

between input and output. A closed loop control system compares the output with the expected

result or command status, then it ta'es appropriate control actions to ad+ust the input signal.herefore, a closed loop system is always equipped with a sensor, which is used to monitor the

output and compare it with the expected result.

he output signal is fed bac' to the input to produce a new output. A well!designed feedbac'system can often increase the accuracy of the output.

xamples for closed loop systems are air conditioners, refrigerators, automatic rice coo'ers,

automatic tic'eting machines, etc. For example An air conditioner, uses a thermostat to detect

the temperature and control the operation of its electrical parts to 'eep the room temperature at a preset constant.

%ne advantage of using the closed loop control system is that it is able to ad+ust its output

automatically by feeding the output signal bac' to the input. -hen the load changes, the error

signals generated by the system will ad+ust the output suitably. he limitation of a closed loop

control systems is they are generally more complicated and thus also more more expensive to

design.

Lin#ar )#r!(! Nonlin#ar Conrol S"!#m! : inear feedbac' control systems are idealied

models fabricated by the analyst purely for the simplicity of analysis and design

-hen the magnitudes of signals in a control system are limited to ranges in which system

components exhibit linear characteristics (i.e., the principle of superposition applies), the system

is essentially linear.

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/ut when the magnitudes of signals are extended beyond the range of the linear operation,

depending on the severity of the nonlinearity, the system should no longer be considered linear.

For instance, amplifiers used in control systems often exhibit a saturation effect when their input

signals become large0 the magnetic field of a motor usually has saturation properties. %ther common nonlinear effects found in control systems are the bac'lash or dead play between

coupled gear members, nonlinear spring characteristics, nonlinear friction force or torque

between moving members, and so on. 1uite often, nonlinear characteristics are intentionally

introduced in a control system to improve its performance or provide more effective control.

Also the analysis of linear systems is easy and lot of mathematical solutions are available for

their simplification.

2onlinear systems, on the other hand, are usually difficult to treat mathematically, and there areno general methods available for solving a wide class of nonlinear systems. *n practice , first a

linear!system is modeled by neglecting the nonlinearities of the system and the designed

controller is then applied to the nonlinear system model for evaluation or redesign by computer

simulation.

Di!ing(i!* '#+##n O$#nloo$ and Clo!#d loo$ conrol !"!#m!

Tim#,In)arian conrol S"!#m! :

-hen the parameters of a control system do not change with respect to time during the

operation of the system, the system is called a time!invariant system. *n practice, most physical

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systems contain elements that drift or vary with time. For example, the winding resistance of an

electric motor will vary when the motor is first being excited and its temperature is rising.

Another example of a time!varying system is a guided!missile control system in which the mass

of the missile decreases as the fuel on board is being consumed during flight. Although a time!varying system without nonlinearity is still a linear system, the analysis and design of this class

of systems are usually much more complex than that of the linear time!invariant systems.

Conin(o(! and Di!cr## Daa Conrol S"!#m! : A continuous!data system is one in which

the signals at various parts of the system are all functions of the continuous time variable t. he

signals in continuous!data systems may be further classified as ac or dc. *n control systems the

ac conrol !"!#m- means that the signals in the system are modulated by some form of modulation scheme. A dc conrol !"!#m- on the other hand, simply implies that the signals are

unmodulated , but they are still ac signals according to the conventional definition. he schematic

diagram of a closed loop dc control system is shown below. ypical waveforms of the signals in

responseto a step!function input are shown in the figure. ypical components of a dc control

system are potentiometers, dc amplifiers, dc motors, dc tachometers, and so on.

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*n ac control systems , the signals are modulated i.e the information is transmitted by an ac

carrier signal. #ere the output controlled variable behaves similarly to that of the dc system. *n

this case, the modulated signals are demodulated by the low!pass characteristics of the ac motor.

Ac control systems are used extensively in aircraft and missile control systems in which noiseand disturbance often create problems. /y using modulated ac control systems with carrier

frequencies of 344 # or higher, the system will be less susceptible to low!frequency noise.

ypical components of an ac control system are synchros, ac amplifiers, ac motors, gyroscopes,

accelerometers etc.

Di!cr## Daa Conrol S"!#m

*f the signal is not continuously varying with time but it is in the form of pulses then the control

system is called &iscrete &ata "ontrol System.*f the signal is in the form of pulse data, then thesystem is called Sampled &ata "ontrol System. #ere the information supplied intermittently at

specific instants of time. his has the advantage of ime sharing system. %n the other hand, if

the signal is in the form of digital code, the system is called &igital "oded System. #ere use of

&igital computers, micro processors or microcontrollers are made use of such systems and are

analyed by the 5! transform theory.

Block Diagrammaic R#$r#!#naion :

*t is a representation of the control system giving the inter!relation between the transfer function

of various components. he bloc' diagram is obtained after obtaining the differential equation 6

ransfer function of all components of a control system. he arrow head pointing towards the

bloc' indicates the i7p 6 pointing away from the bloc' indicates the o7p.

Suppose 8(S) is the ransfer function then 8(S) 9 "(S) 7 :(S)

After obtaining the bloc' diagram for each 6 every component, all bloc's are combined to get a

complete representation. *t is then reduced to a simple form with the help of bloc' diagram

algebra.

Ba!ic #l#m#n! of a 'lock diagram

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/loc's ransfer functions of elements inside the bloc's Summing points a'e off points

Arrow

A control system may consist of a number of components. A bloc' diagram of a system is a

pictorial representation of the functions performed by each component and of the flow of signals.

he elements of a bloc' diagram are 'lock , 'ranc* $oin and !(mming $oin.

Block :

*n a bloc' diagram all system variables are lin'ed to each other through functional bloc's. he

functional bloc' or simply bloc' is a symbol for the mathematical operation on the input signalto the bloc' that produces the output.

S(mming $oin :

he bloc's are used to identify many types of mathematical operations, li'e addition andsubtraction and represented by a circle, called a summing point. As shown belowdiagram a

summing point may have one or several inputs. ach input has its own appropriate plus or minussign. A summing point has only one output and is equal to the algebraic sum of the inputs

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A ta'eoff point is used to allow a signal to be used by more than one bloc' or summing point

rro+ ; associated with each branch to indicate the direction of flow of signal

d)anag#! of Block Diagram R#$r#!#naion :

• *t is always easy to construct the bloc' diagram even for a complicated system• Function of individual element can be visualied• *ndividual 6 %verall performance can be studied• %ver all transfer function can be calculated easily

Limiaion! of a Block Diagram R#$r#!#naion

• 2o information can be obtained about the physical construction• Source of energy is not shown

Block diagram r#d(cion #c*ni/(#: /ecause of the simplicity and versatility, the bloc'

diagrams are often used by control engineers to describe all types of systems. A bloc' diagram

can be used simply to represent the composition and interconnection of a system. Also, it can be

used, together with transfer functions, to represent the cause!and!effect relationships throughout

the system. ransfer Function is defined as the relationship between an input signal and an outputsignal to a device.

0roc#d(r# o !ol)# Block Diagram R#d(cion ! :

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Step @= :epeat steps < to 3 till simple form is obtainedStep = %btain the ransfer Function of %verall System

Block diagram r(l#!

() "ombining bloc's in Darallel= -hen two bloc's are connected parallel as shown below ,the

resultant transfer function is equal to the algebraic sum (or difference) of the two transfer

functions.his is shown in the diagram below.

(?) liminating a feed bac' loop:he following diagram shows how to eliminate the feed bac'

loop in the resultant control system

(3) $oving a ta'e!off point beyond a bloc': he effect of moving the ta'eoff point beyond a bloc' is shown below.

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(@) $oving a a'e!off point ahead of a bloc': he effect of moving the ta'eoff point ahead of a bloc' is shown below.

$$licaion! of *# conrol !"!#m! : here are various applications of control systems which

include biological propulsion0 locomotion0 robotics0 material handling0 biomedical, surgical,

and endoscopic 0 aeronautics0 marine and the defense and space industries. here are also many

household and industrial application examples of the control systems, such as washing machine,

air conditioner, security alarm system and automatic tic'et selling machine, etc.

i2 3a!*ing mac*in# :he most commonly used house hold application is the washing machine.*t comes under

automatic control system ,where the machine automatically starts to pour water, add washing

powder, spin and wash clothes, discharge wastewater, etc. After the completion of all the

procedures, the washing machine will stop the operation.

#owever, this 'ind of machine only operates according to the preset time to complete the

whole washing process. *t ignores the cleanness of the clothes and does not generate feedbac'.

herefore, this 'ind of washing machine is of open loop control system.

ii2. ir condiion#r

he air conditioner is used to automatically control the temperature of the room.*n the air

conditioner the coolant circulated in the machine will absorb heat indoor, then it will be

transported from the vaporiation device to cooling device. he hot air is then blown to outdoor

by a fan. here is an ad+ustable temperature device equipped in the air conditioner for the users

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to ad+ust the extent of cooling. -hen the temperature of the cool air is lower than the preset one,

the controller of the air conditioner will stop the operation of the compressor to cease the

circulation of the coolant. he temperature sensor installed near the vaporiation device will

continuously measure the indoor temperature, and send the results to the controller for further processing.his operation will come under closed loop control system.hesimple bloc' diagram

of air conditioner system is shown below.

F##d 'ack Conrol S"!#m : he feed bac' control system is represented by the following

bloc' diagram .*n the diagram feed bac' signal is denoted by /(S) and the output is "(S).he

input function is denoted by :(S).

he open loop gain of the system is 8(S) and the feed bac' loop gain #(S). hen the feed bac' signal b(s) is given by

/(S) 9 #(S). "(S)

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Tran!f#r f(ncion : he input! output relationship in a linear time invariant system is given by

the transfer function.For a time invariant system it is defined as the ratio of aplace transform

of the out to the apalce transform of the input

he important features of the transfer functions are,

• he transfer function of a system is the mathematical model expressing the differentialequation that relates the output to input of the system.

• he transfer function is the property of a system independent of magnitude and the natureof the input .

• he transfer function includes the transfer functions of the individual elements. /ut at thesame time, it does not provide any information regarding physical structure of the system

• *f the transfer function of the system is 'nown, the output response can be studied for

various types of inputs to understand the nature of the system• *t is applicable to inear ime *nvariant system.• *t is assumed that initial conditions are ero.• *t is independent of i7p excitation.• *t is used to obtain systems o7p response.• *f the transfer function is un'nown, it may be found out experimentally by applying

'nown inputs to the device and studying the output of the system

From the above bloc' diagram 8(S) 9 "(S) 7 (S) 6 (S) 9 :(S) ; /(S)

So, "(S) 9 8(S) .(S) 9 8(S)B :(S)! /(S)

9 8(S) B :(S) ; #(S)."(S)

herefore

his is the transfer function of the closed loop control system

Properties of Systems : "or a#y co#$ro% &y&$'m $o (#)'r&$a#) *$&

+'r,orma#c' $h' ,o%%o-*# +ro+'r$*'& ar' v'ry *m+or$a#$.

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/*0.Linearity A &y&$'m *& &a*) $o ' %*#'ar *, *$ ,o%%o-& o$h $h' %a- o,

a))$*v*$y a#) %a- o, homo'#'*$y. Th' &y&$'m -h*ch )o #o$ ,o%%o- $h' %a- o,

homo'#'*$y a#) a))*$*v*$y *& ca%%') a #o#%*#'ar &y&$'m.

I, *#+($ 1/$0 +ro)(c'& r'&+o#&' y1/$0 a#) *#+($ 2/$0 +ro)(c'& r'&+o#&'

y2/$0 $h'# $h' &ca%') a#) &(mm') *#+($ a11/$0 12/$0 +ro)(c'& $h' &ca%')

a#) &(mm') r'&+o#&' a1y1/$0 1y2/$0 -h'r' a1 a#) a2 ar' r'a% &ca%ar&. I$

,o%%o-& $ha$ $h*& ca# ' '$'#)') $o a# ar*$rary #(m'r o, $'rm& a#) &o ,or

r'a% #(m'r& .

/**0 Time Invariance A &y&$'m -*$h *#+($ /$0 a#) o($+($ y/$0 *& $*m'*#var*a#$ *, /$ $!0 *& cr'a$'& o($+($ y / $ $!0 ,or a%% *#+($& a#) &h*,$& $!.

/***). Causality A &y&$'m *& ca(&a% *, $h' o($+($ y/$0 a$ $*m' $ *& #o$ a

,(#c$*o# o, ,($(r' *#+($& a#) *$ )'+'#)& o#%y o# $h' +r'&'#$ a#) +a&$ *#+($& .

A%% a#a%o &y&$'m& ar' ca(&a% a#) a%% m'm'ory%'&& &y&$'m& ar' ca(&a% .

I, $h' &y&$'m *& ca(&a% $h'# $h*& *m+%*'& h/$0 ! $ 9 !. A%$'r#a$*v'%y h:#;!

# 9 !.

(iv).Stability : A &y&$'m *& &a*) $o ' a &$a%' *, ,or 'v'ry o(#)')*#+($

$h'r' '*&$& a o(#)') o($+($ .

Transfer Function: "or a o+'# %oo+ co#$ro% &y&$'m &ho-# '%o- $h'

$ra#&,'r ,(#c$*o# *& $h' ra$*o o,


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y )'#*$*o# $h' $ra#&,'r ,(#c$*o# *& >/S0

Laplace transform of theOutput

Laplace Transform of the Input

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Poles an !eros : I# $h' $ra#&,'r ,(#c$*o# o, a co#$ro% &y&$'m o$h

#(m'ra$or a#) )'#om*#a$or -*%% ' +o%y#om*a%&. I, $h'&' #(m'ra$or a#)

)'#om*#a$or ar' &o%v') $h' roo$& o, $h' #(m'ra$or ar' ca%%') C'ro& a#) $h'

roo$& o, $h' )'#om*#a$or ar' ca%%') $h' Poles. Th'&' +o%'& o, $h' $ra#&,'r

,(#c$*o# )'c*)'& $h' &$a*%*$y o, $h' co#$ro% &y&$'m.

Th' $ra#&,'r ,(#c$*o# +rov*)'& a a&*& ,or )'$'rm*#*# *m+or$a#$ &y&$'m

r'&+o#&' charac$'r*&$*c& -*$ho($ &o%v*# $h' com+%'$' )*'r'#$*a% 'E(a$*o#.

A& )'#') $h' $ra#&,'r ,(#c$*o# *& *v'# y $h' ,o%%o-*# '+r'&&*o# -*$h

var*a%' & F G

I$ *& a%-ay& 'a&y $o ,ac$or $h' +o%y#om*a%& *# $h' #(m'ra$or a#)

)'#om*#a$or a#) $o -r*$'

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$h' $ra#&,'r ,(#c$*o# *# $'rm& o, $ho&' ,ac$or& .$h'# $h' aov' $ra#&,'r

,(#c$*o# ca# ' -r*$$'# a&

-h'r' $h' #(m'ra$or a#) )'#om*#a$or +o%y#om*a%& N/&0 a#) D/&0 hav' r'a%

co'c*'#$& )'#') y $h' &y&$'mJ& )*'r'#$*a% 'E(a$*o# a#) K m?a#.

A& -r*$$'# *# $h' aov' 'E(a$*o# $h' C* & ar' $h' roo$& o, $h' 'E(a$*o#

N/&0 ! a#) ar' )'#') $o ' $h' &y&$'m Ceros" a#) $h' +* & ar' $h' roo$&

o, $h' 'E(a$*o# D/s0 !, a#) ar' )'#') $o ' $h' &y&$'m poles.

A%% o, $h' co'c*'#$& o, +o%y#om*a%& N/&0 a#) D/&0 ar' r'a% $h'r',or' $h'

+o%'& a#) K'ro& m(&$

' '*$h'r +(r'%y r'a% or a++'ar *# com+%' co#G(a$' +a*r&.

Th' &$a*%*$y o, a %*#'ar &y&$'m may ' )'$'rm*#') )*r'c$%y ,rom *$& $ra#&,'r

,(#c$*o#. A# #$h

or)'r %*#'ar &y&$'m *& a&ym+$o$*ca%%y &$a%' o#%y *, a%% o, $h'com+o#'#$& *# $h' homo'#'o(& r'&+o#&' ,rom a #*$' &'$ o, *#*$*a%

co#)*$*o#& )'cay $o K'ro a& $*m' *#cr'a&'& or

-h'r' $h' pi ar' $h' &y&$'m +o%'&. I# a &$a%' &y&$'m a%% com+o#'#$& o, $h'

homo'#'o(& r'&+o#&' m(&$ )'cay $o K'ro a& $*m' *#cr'a&'&. I, a#y +o%' ha&

a +o&*$*v' r'a% +ar$ $h'r' *& a com+o#'#$ *# $h' o($+($ $ha$ *#cr'a&'& -*$ho($

o(#) ca(&*# $h' &y&$'m $o ' (#&$a%'.

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Example : A %*#'ar &y&$'m *& )'&cr*') y $h' )*'r'#$*a% 'E(a$*o#

"*#) $h' &y&$'m +o%'& a#) K'ro&.

Solution: "rom $h' )*'r'#$*a% 'E(a$*o# $h' $ra#&,'r ,(#c$*o# *&

So$h' &y&$'m ha& a &*#%' r'a% K'ro a$ & L1?2 a#) a +a*r o, r'a% +o%'& a$ &

L3 a#) & L2.

SI#$L%TI&' I%*%#S: Th' &*m(%a$*o# )*aram *& &*m*%ar $o $h' )*aram

(&') $o r'+r'&'#$ $h' &y&$'m o# a# a#a%o com+($'r.Th' a&*c '%'m'#$&

(&') ar' *)'a% *#$'ra$or& *)'a% am+%*'r& a#) *)'a% &(mm'r& &ho-# *#

'%o- )*aram. A))*$*o#a% '%'m'#$& &(ch a& m(%$*+%*'r& a#) )*v*)'r& may '

(&') ,or #o#%*#'ar &y&$'m&.

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To o$a*# $h' &*m(%a$*o# )*aram $h' ,o%%o-*# &$'+& ar' $o ' ,o%%o-').

1. S$ar$ -*$h )*'r'#$*a% 'E(a$*o#.

2. # $h' %',$ &*)' o, $h' 'E(a$*o# +($ $h' h*h'&$or)'r )'r*va$*v' o, $h'

)'+'#)'#$ var*a%'. A r&$or)'r or h*h'ror)'r )'r*va$*v' o, $h' *#+($ may

a++'ar *# $h' 'E(a$*o#. I# $h*& ca&' $h' h*h'&$or)'r )'r*va$*v' o, $h' *#+($

*& a%&o +%ac') o# $h' %',$ &*)' o, $h' 'E(a$*o#. A%% o$h'r $'rm& ar' +($ o# $h'

r*h$ &*)'.

3. S$ar$ $h' )*aram y a&&(m*# $ha$ $h' &*#a% r'+r'&'#$') y $h' $'rm&

o# $h' %',$ &*)' o, $h' 'E(a$*o# *& ava*%a%'. Th'# *#$'ra$' *$ a& ma#y $*m'&

a& #'')') $o o$a*# a%% $h' %o-'ror)'r )'r*va$*v'&. I$ may ' #'c'&&ary $o

a)) a &(mm'r *# $h' &*m(%a$*o# )*aram $o o$a*# $h' )'+'#)'#$ var*a%'

'+%*c*$%y.

4. =om+%'$' $h' )*aram y ,'')*# acB $h' a++ro*ma$' o($+($& o, $h'

*#$'ra$or& $o a &(mm'r $o '#'ra$' $h' or**#a% &*#a% o, &$'+ 2.I#c%()' $h'

*#+($ ,(#c$*o# *, *$ *& r'E(*r').

Example: Dra- $h' &*m(%a$*o# )*aram ,or $h' &'r*'& R


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,'')acB %oo+& a#) -h'# a &y&$'m ha& mor' $ha# o#' *#+($ a#) o#' o($+($

$h' %ocB )*aram a++roach *& v'ry com+%'. #ence an alternate method is proposed

by S.E. $ason. his method is called signal flow graphs. *n these graphs each node represents a

system variable 6 each branch connected between two nodes acts as Signal $ultiplier. hedirection of signal flow is indicated by an arrow.

A signal flow graph is a diagram that represents a set of simultaneous equations. *t consists of a

graph in which nodes are connected by directed branches. he nodes represent each of the

system variables. A branch connected between two nodes acts as a one!way signal multiplier= the

direction of signal flow is indicated by an arrow placed on the branch, and the multiplication

factor (transmittance or transfer function) is indicated by a letter placed near the arrow.

So,in the figure above , the branch transmits the signal x


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Mi4#d nod#! g#n#ral nod#!2: hese have both incoming and outgoing branches. *n Fig. (a),

node w is a mixed node. A mixed node may be treated as a sin' node by adding an out going

branch of unity transmittance, as shown in Fig (b), for the equation x 9 au bv

a#) - c ca( cv

Fi-(a)

Fi- (b)

A +a$h *& a#y co##'c$') &'E('#c' o, ra#ch'& -ho&' arro-& ar' *# $h'

&am' )*r'c$*o# a#) A ,or-ar) +a$h '$-''# $-o #o)'& *& o#' $ha$ ,o%%o-&

$h' arro-& o, &(cc'&&*v' ra#ch'& a#) *# -h*ch a #o)' a++'ar& o#%y o#c'. I#

"*./a0 $h' +a$h (- *& a ,or-ar) +a$h '$-''# $h' #o)'& ( a#) .

Flo/rap0 %l-ebra : Th' ,o%%o-*# r(%'& ar' (&',(% ,or &*m+%*,y*# a

&*#a% o- ra+h

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Series pat0s (cascae noes). S'r*'& +a$h& ca# ' com*#') *#$o a &*#%'

+a$h y m(%$*+%y*# $h' $ra#&m*$$a#c'& a& &ho-# *# "* / A 0.

Pat0 -ain. Th' +ro)(c$ o, $h' $ra#&m*$$a#c'& *# a &'r*'& +a$h.

Parallel pat0s. Para%%'% +a$h& ca# ' com*#') y a))*# $h'

$ra#&m*$$a#c'& a& &ho-# *# "*/0.

'oe absorption. A #o)' r'+r'&'#$*# a var*a%' o$h'r $ha# a &o(rc' or

&*#B ca# ' '%*m*#a$') a& &ho-# *# "* /=0.

Feebac1 loop. Ac%o&') +a$h $ha$ &$ar$& a$ a #o)' a#) '#)& a$ $h' &am'

#o)'.

Loop -ain. Th' +ro)(c$ o, $h' $ra#&m*$$a#c'& o, a ,'')acB %oo+.

Th'&' r'&(%$& ar' &ho-# )*aramma$*ca%%y *# $h' ,o%%o-*# (r'& /A0 /0

a#) =0 -h'r' $h' or**#a% )*aram a#) 'E(*va%'#$ )*aram& ar' &ho-#.

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Ma!on! gain form(la: he relationship between an input variable and an output variable of asignal flow graphis given by the net gain between input and output nodes and is 'nown as overallgain ofthe system. $asons gain formula is used to obtain the over all gain (transfer function) of signal flow graphs. According to $asons gain formula 8ain is given by

-here, D' is gain of ' th forward path and G is determinant of graph. #ere the G is given by

G 9

two non touching loops ;sum of gain products of all possible combination of three non touching

loops)

G' is cofactor of 'th forward path determinant of graph with loops touching ' th forward path.*t is obtained from G by removing the loops touching the path D'.Finin- transfer function from t0e system 2o -rap0s is explaine

belo by example.

Example3 : $a*# $h' $ra#&,'r ,(#c$*o# o, $h' &y&$'m -ho&' &*#a% o-

ra+h *& &ho-# '%o-.

here are two forward paths= %ne is 8ain of path < = D=D>98>

here are four loops with loop gains =

83

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here are no non!touching loops.

G 9 touch all the loops. herefore, G8?#>88?

"or-ar) +a$h 1 $o(ch'& a%% $h' %oo+&. Th'r',or' O1 1.

Th' $ra#&,'r ,(#c$*o# T *& *v'# y

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S"!#m Sa'ili":

he study of stability of a control system is very important to understand the performance .

his means that the system must be stable at all times during operation. Stability may be used to

define the usefulness of the system. Stability studies include absolute 6 relative stability.

Absolute stability is the quality of stable or unstable performance. :elative Stability is the

quantitative study of stability.

he stability study is based on the properties of the ransfer Function. *n the analysis, thecharacteristic equation is very important ,which describe the transient response of the system.From the roots of the characteristic equation, following conclusions about the stability can bedrawn.

() -hen one or more roots lie on the imaginary axis 6 there are no roots on the :#S of S! plane, the response will be oscillatory without damping. Such a system will be termed ascritically stable.

(?) -hen one or more roots lie on the :#S of S!plane, the response will exponentially increase

in magnitude and there by the system will be Hnstable.

Stability 5 e6nitions :

A &y&$'m *& &$a%' *, *$& o?+ *& o(#)') ,or a#y o(#)') *?+ . or A &y&$'m

*& &$a%' *, *$& r'&+o#&' $o a o(#)') )*&$(r*# &*#a% va#*&h'&(%$*ma$'%y a& $*m' Q $ Q a++roach'& *##*$y.

A &y&$'m *& (# &$a%' *, *$J& r'&+o#&' $o a o(#)') )*&$(r*#

&*#a% r'&(%$& *# a# o?+ o,

*##*$' am+%*$()' or a# &c*%%a$ory &*#a%.

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I, $h' o?+ r'&+o#&' $o a o(#)') *?+ &*#a% r'&(%$& *# co#&$a#$

am+%*$()' or co#&$a#$ am+%*$()' o&c*%%a$*o#& $h'# $h' &y&$'m may '

&$a%' or (#&$a%' (#)'r &om' %*m*$') co#&$ra*#$&. S(ch a &y&$'m *&

ca%%') Limitely Stable system.

I, a &y&$'m r'&+o#&' *& &$a%' ,or a %*m*$') ra#' o, var*a$*o# o, *$&

+aram'$'r& *$ *& ca%%')=o#)*$*o#a%%y S$a%' Sy&$'m.

I, a &y&$'m r'&+o#&' *& &$a%' ,or a%% var*a$*o# o, *$& +aram'$'r& *$ *&

ca%%') %bsolutely Stable system.

*out0/,urit7 Stability Criterion : his criterion is derived from the theory

of equations and is an algebraic method to determine the number of roots of a given equation

with positive real part.

he :outh!#urwit criterion is a method of finding whether a linear system is stable or not byexamining the locations of the roots of the characteristic equation of the system. *n fact, themethod determines only if there are roots that lie outside of the left half plane0 it does notactually compute the roots.

o determine whether this system is stable or not, chec' the following conditions

T+o n#c#!!ar" '( no !(ffici#n condiion! *a all *# roo! *a)# n#gai)# r#al $ar! ar#

a) All the polynomial coefficients must have the same sign.

b) All the polynomial coefficients must be nonero.

A sufficient condition for a system to be stable is that each 6 every term of the column of

the :outh array must be positive or should have the same sign. :outh array can be obtained asfollows

"onsider the "haracteristic equation of the form,

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Similarly rest of the elements , can be evaluated.

he limitations of the :outh!#urwit stability criteria are

(


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From the above table it is clear that the no. of sign changes in the <

st

column 9 ero. 2o roots arelying in the :#S of S!plane. So, the given System is Absolutely Stable.

E4am$l# %: Find the stability of a system whose characteristic equation is given below .

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"rom $h' aov' $a%' *$ *& c%'ar $ha$ $h' #o. o, &*# cha#'& *# $h' 1&$ co%(m#

2 a#) $-o roo$& ar' %y*# *# $h' RHS o, S+%a#'. So $h' *v'# Sy&$'m *&

(#&$a%'

*&&T L&C$S :

:oot locus is the plot of the loci of the root of the complementary equation when one or more

parameters of the open!loop ransfer function are varied, mostly the only one variable available

is the gain IK he negative gain has no physical significance hence varying IK from I4 to IL ,

the plot is obtained called the M:oot ocus DointN.

:oot locus gives the complete dynamic response of the system. *t provides a measure of

sensitivity of roots to the variation in the parameter being considered. *t is applied for single as

well as multiple loop system

*ules for t0e Construction of *oot Locus :

/1) he root locus is symmetrical about the real axis.

(>) he no. of branches terminating on IL equals the no. of open!loop pole!eroes.

(?) ach branch of the root locus originates from an open!loop pole at IK 9 4 6 terminates at

open!loop ero corresponding to IK 9 L.

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(3) A point on the real axis lies on the locus, if the no. of open!loop poles 6 eroes on the real

axis to the right of this point is odd.

(@) he root locus branches that tend to IL, do so along the straight line.

Asymptotes ma'ing angle with the real axis is given by

h'r' #135

P *& $h' #(m'r o, +o%'& a#) K *& $h' #(m'r o, K'ro&

/60 Th' a&ym+$o$'& cro&& $h' r'a% a*& a$ a +o*#$ B#o-# a& ='#$ro*)

/70 Th' r'aB a-ay or $h' r'aB *# +o*#$& :Sa))%' +o*#$&; o, $h' roo$ %oc(&

ar' )'$'rm*#') ,rom $h' roo$& o, $h' 'E(a$*o# )B ?)& !.

(O) he intersection of the root locus branches with the imaginary axis can be determined by the

use of :outh!#urwit criteria or by putting !8 9 in the characteristic equation 6 equating the

real part and imaginary to ero. o solve for P and K i.e., the value of IP is intersection point

on the imaginary axis 6 IK is the value of gain at the intersection point.

Q) he angle of departure from a complex open!loop pole Rd is given by Rd 9


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closed. he 2yquist criterion provides the information on absolute and relative stability of the

system.

et us suppose that the system transfer function is a complex function. /y applying "auchys

principle of argument to the open!loop system transfer function, we will get information about

stability of the closed!loop system transfer function and arrive at the 2yquist stability criterion

he importance of 2yquist stability lies in the fact that it can also be used to determine therelative degree of system stability by producing the so!called phase and gain stability margins.

hese stability margins are needed for frequency domain controller design techniques

he 2yquist plot is a polar plot of the function &(s) 9 < 9 8(s).#(s)

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he 2yquist criterion states that the number of unstable closed!loop poles is equal to the

number of unstable open!loop poles plus the number of encirclements of the origin of the

2yquist plot of the complex function &(s).

Continuous Time Feebac1 Control Systems : I, $h' &*#a%& *# a%% +ar$&

o, a co#$ro% &y&$'m ar' co#$*#(o(& ,(#c$*o#& o, $*m' $h' &y&$'m *& c%a&&*')

a& co#$*#(o(& $*m' ,'')acB co#$ro% &y&$'m. Ty+*ca%%y a%% co#$ro% &*#a%& ar'

o, %o- ,r'E('#cy a#) *, $h'&' &*#a%& ar' (# mo)(%a$') $h' &y&$'m *& B#o-#

a& a ).c. co#$ro% &y&$'m. Th'&' &y&$'m& (&' +o$'#$*om'$'r& a& 'rror

)'$'c$or& ).c am+%*'r& $o am+%*,y $h' 'rror &*#a% ).c. &'rvo mo$or a&

ac$(a$*# )'v*c' a#) ).c $achom'$'r& or +o$'#$*om'$'r& a& ,'')acB

'%'m'#$&. I, $h' co#$ro% &*#a% *& mo)(%a$') y a# a.c carr*'r -av' $h'

r'&(%$*# &y&$'m *& (&(a%%y r','rr') $o a& a# a.c co#$ro% &y&$'m. Th'&'

&y&$'m& ,r'E('#$%y (&' &y#chro& a& 'rror )'$'c$or& a#) mo)(%a$or& o, 'rror

&*#a% a.c am+%*'r& $o am+%*,y $h' 'rror &*#a% a#) a.c &'rvo mo$or& a&

ac$(a$or&. Th'&' mo$or& a%&o &'rv' a& )'mo)(%a$or& a#) +ro)(c' a# (#

mo)(%a$') o($+($ &*#a%.

iscrete ata Feebac1 Control Systems

D*&cr'$' )a$a co#$ro% &y&$'m& ar' $ho&' &y&$'m& *# -h*ch a$ o#' or mor'

+a#& o, $h' ,'')acB co#$ro% &y&$'m $h' &*#a% *& *# $h' ,orm o, +(%&'&.

U&(a%%y $h' 'rror *# &(ch &y&$'m *& &am+%') a$ (#*,orm ra$' a#) $h' r'&(%$*#

+(%&'& ar' ,') $o $h' co#$ro% &y&$'m. I# mo&$ &am+%') )a$a co#$ro% &y&$'m&

$h' &*#a% *& r'co#&$r(c$') a& a co#$*#(o(& &*#a% (&*# a )'v*c' ca%%') ho%)

)'v*c'. Ho%)& o, )*'r'#$ or)'r& ar' 'm+%oy') ($ $h' mo&$ commo# ho%)

)'v*c' *& a K'ro or)'r ho%). I$ ho%)& $h' &*#a% va%(' co#&$a#$ a$ a va%(''E(a% $o $h' am+%*$()' o, $h' *#+($ $*m' ,(#c$*o# a$ $ha$ &am+%*# *#&$a#$

(#$*% $h' #'$ &am+%*# *#&$a#$ .Th'&' &y&$'m& ar' a%&o B#o-# a& &am+%')

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)a$a co#$ro% &y&$'m&.

"* D*&cr'$' Da$a "'')acB =o#$ro% Sy&$'m&

D*&cr''$ )a$a co#$ro% &y&$'m& *# -h*ch a )**$a% com+($'r *& (&') a& o#' o,

$h' '%'m'#$& ar' B#o-# a& )**$a% co#$ro% &y&$'m&. Th' *#+($ a#) o($+($ $o

$h' )**$a% com+($'r m(&$ ' *#ary #(m'r& a#) h'#c' $h'&' &y&$'m&

r'E(*r' $h' (&' o, )**$a% $o a#a%o a#) a#a%o $o )**$a% co#v'r$'r&

Tim# r#!$on!# anal"!i! : *t is an equation or a plot that describes the behavior of a system and

gives information about it with respect to time response specification as overshooting, settlingtime, pea' time, rise time and steady state error. ime response is formed by the transient

response and the steady state response.

ime response 9 ransient response Steady state response.

ransient time response or 2atural response describes the behavior of the system in its first short

time until arrives the steady state value. *f the input is step function then the output or the

response is called step time response and if the input is ramp, the response is called ramp timeresponse .. etc.

Tran!i#n R#!$on!#: he transient response is defined as the part of the time response thatgoes to ero as time becomes very large. hus yt(t) has the property

Lim "2 8 ;

,,


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he time required to achieve the final value is called transient period. he transient response may

be exponential or oscillatory in nature. %utput response consists of the sum of forced response

(form the input) and natural response (from the nature of the system).he transient response is

the change in output response from the beginning of the response to the final state of theresponse and the steady state response is the output response as time is approaching infinity (or

no more changes at the output). he behavior of a system in transient state is shown below.

S#ad" Sa# R#!$on!#: he steady state response is the part of the total response that remains

after the transient has died out. For a position control system, the steady state response when

compared to with the desired reference position gives an indication of the final accuracy of the

system. *f the steady state response of the output does not agree with the desired reference

exactly, the system is said to have steady state error.

R#!$on!# o a Uni S#$ In$( =Fir! Ord#r

"onsider a feedbac' system with 8)s) 9


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he closed loop transfer function of the system is given by

For a unit step input : (s) 9 < 7 s and the output is given by

*nverse aplace transformation yields

he plot of c(t) Us t is shown below

he response is an exponentially increasing function and it approaches a value of unity as t !!! V L

At t 9 T the response reaches a value,

which is ?.> percent of the steady value. his time, T is 'nown as the time constant of the

system. %ne of the important characteristics about the system is its speed of response or how

fast the response is approaching the final value. he time constant T is indicative of this measure

and the speed of response is inversely proportional to the time constant of the system.Another

important characteristic of the system is the error between the desired value and the actual value

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under steady state conditions. his quantity is 'nown as the steady state error of the ! system and

is denoted by e ss.

he error (s) for a unity feedbac' system is given by

For the system under consideration 8(s) 9 < 7 Ts and :(s) 9 < 7s herefore

As t W L e (t) W 4 . hus the output of the first order system approaches the reference input,

which is the desired output, without any error. *n other words, we say a first order system trac'sthe step input without any steady state error.

R#!$on!# o a Uni Ram$ In$( :

o study the response of a unit ramp let us consider a feedbac' system with 8)s) 9 is given by

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he time response is obtained by ta'ing inverse aplace transform of above equation

&ifferentiating the above equation we get

his is the response of the system to a step input. hus no additional information about thespeed of response is obtained by considering a ramp input.So, no additional information aboutthe speed of response is obtained by considering a ramp input.

/ut the effect on the steady state error is given by

hus the steady state error is equal to the time constant of the system. he first order system,therefore, can not trac' the ramp input without a finite steady state error. *f the time constant is

reduced not only the speed of response increases but also the steady state error for ramp input

decreases. #ence the ramp input is important to the extent that it produces a finite steady state

error.

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he response of a first order system for unit ramp input is shown below.

R#!$on!# o a Uni S#$ In$( , S#cond Ord#r S"!#m

et us consider a type S K

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is 'nown as the characteristic polynomial of the system and &(s) 9 4 is 'nown as thecharacteristic equation of the system. he above qn. is normally put in standard from, given by,

he poles of ( s), or, the roots of the characteristic equation

S> >X Pn s Pn > 9 4

are given by,

-here is 'nown as the damped natural frequency of the system. *f X V


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possible. *f it deviates from this desired output, the performance of the system is not satisfactory

under steady state conditions. he steady state error reflects the accuracy of the system. Among

many reasons for these errors, the most important ones are the type of input, the type of the

system and the nonlinearities present in the system. Since the actual input in a physical system isoften a random signal, the steady state errors are obtained for the standard test signals, namely,

step, ramp and parabolic signals.

Error Con!an! : et us consider a feedbac' control system as shown below.

he error signal (s) is given by (s) 9 : (s) ! # (s) " (s)

/ut " (s) 9 8 (s) (s)

From the above equations we have

Applying final value theorem, we can get the steady state error ess as,

he above equation shows that the steady state error is a function of the input :(s) and the open

loop transfer function 8(s). et us consider various standard test signals and obtain the steady

state error for these inputs.

0ro$orional- In#gral and D#ri)ai)# Conroll#r 0ID Conrol2 :

4!


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A Droportional;*ntegral;&erivative (D*&) controller is a three!term controller which is

considered as a standard controller in industrial settings. . *t can be found in virtually all 'inds of

control equipments, either as a stand!alone (single!station)controller or as a functional bloc' in

Drogrammable ogic "ontrollers (D"s)and &istributed "ontrol Systems (&"Ss).An integral control eliminates steady state error due to a velocity input, but its effect on dynamic

response is Zdifficult to predict as the system order increases to three. *t is 'nown that a

derivative term in the forward path improves the damping in the system. %ne of the best!'nown

controllers used in practice is the D*& controller, where the letters stand for $ro$orional-

in#gral-and d#ri)ai)#. he integral and derivative components of the D*& controller have

individual performance implications, and their applications require an understanding ofthe basics

of these elements. #ence a suitable combination of integral and derivative controls results in a proportional, integral and derivate control, usually called D*& control. he transfer function of

the D*& controller is given by,

he diagram below gives the ideal D*& controller.

he overall forward path transfer function is given by,

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and the overall transfer function is given by,

Droper choice of Kp, K& and Kr results in satisfactory transient and steadystate responses. he

process of choosing proper Kp, K&, at Kr for a given system is 'nown as tuning of a D *&

controller.Bod# $lo! = /ode plots are graphs of the steady state response of stable continuous time inear

time invariant systems for sinusoidal inputs,plotted as change in magnitude and phase versus

frequency on logarithmic scale./ode plots are a visual description of the system.So,these /ode

plots are used for the representation of sinusoidal transfer function . *n this representation the

magnitude of 8(+P) in db, i.e, >4 log 8(+P) is plotted against Zlog PZ.Similarly phase angle of

8(+P) is plotted against Zlog PZ .#ence the abscissa is logarithm of the frequency and hence the

plots are 'nown as logarithmic plots. he plots are named after the famous mathematician #. -./ode.

he transfer function 8(+P) can be written as 8(+P) 9 8(+P)[( P) -here [( P) is the angle

8(+P) .

Since 8(+P) consists of many multiplicative factors in the numerator and denominator it is

convenient to ta'e logarithm of 8(+P) to convert these factors into additions and subtractions,

which can be carried out easily.

o plot the magnitude plot , magnitude is plotted against input frequency on a logarithmicscale.*t can be approximated by two lines and it forms the asymptotic (approximate) magnitude/ode plot of the transfer function=

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(i) for angular frequencies below Pc it is a horiontal line at 4 d/ since at low frequencies the

term P 7 Pc is small and can be neglected, ma'ing the decibel gain equation above equal toero,

ii2 for angular frequencies above Pc it is a line with a slope of \>4 d/ per decade since at highfrequencies the term P 7 Pc dominates and the decibel gain expression above simplifies to !>4log (P 7 Pc ) which is a straight line with a slope of \>4 d/ per decade.

hese two lines meet at the corner frequency. From the /ode plot, it can be seen that for frequencies well below the corner frequency, the circuit has an attenuation of 4 d/,corresponding to a unity pass band gain, i.e. the amplitude of the filter output equals theamplitude of the input. Frequencies above the corner frequency are attenuated ; the higher thefrequency, the higher the attenuation.

0*a!# $lo

he phase /ode plot is obtained by plotting the phase angle of the transfer function given by

] 9 ! tan ; (P7Pc ) Uersus P , where P and Pc are the input and cutoff angular frequencies

respectively.

For input frequencies much lower than corner, the ratio P7Pc is small and therefore the phaseangle is close to ero. As the ratio increases the absolute value of the phase increases and

becomes ;3@ degrees when P9 Pc As the ratio increases for input frequencies much greater than the corner frequency, the phase angle asymptotically approaches \Q4 degrees. he

frequency scale for the phase plot is logarithmic.

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he main advantage of using the /ode plots is that multiplication of magnitudes can be

converted into addition. Also a simple method of plotting an approximate log!magnitude curve is

obtained. *t is based on asymptotic approximations. Such approximation by straight line

asymptotes is sufficient if only rough information on the frequency response characteristics is isneeded.he phase angle curves can be drawn easily if a template for the phase angle curve of


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• -rite the frequency corresponding to each of the point of the plot.

&ain Margin: 8ain $argin is defined as Mthe factor by which the system gain can be increasedto drive the system to the verge of instabilityN.

For !a'l# !"!#m!, Pgc Y P pc_8(+)#(+)_ at P9 P pc Y <8$ 9 in positive d/ .

$ore positive the 8ain $argin , more stable is the system

For marginall" !a'l# !"!#m!- Pgc 9 P pc

_8(+)#(+)_ at P9P pc 9


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05SE MR&IN: Dhase $argin is defined as M the additional phase lag that can be introduced

before the system becomes unstableN.

• et IA be the point of intersection of _8(+)#(+)_ plot and a unit circle centered at theorigin.

• &raw a line connecting the points I% 6 IA and measure the phase angle between the line%A and ve real axis. his angle is the phase angle of the system at the gain cross over frequency. 8(+Pgc)#(+gc) 9 [gc

• *f an additional phase lag of D$ is introduced at this frequency, then the phase angle8(+Pgc)#(+Pgc) will become


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R#lai)# and a'!ol(# !a'ili"= A stable system is a dynamic system with a bounded responseto a bounded input. A necessary and sufficient condition for a feedbac' system to be stable is

that all the poles of the system transfer function have negative real partsA system is considered marginally stable if only certain bounded inputs will result in a bounded

output.

*n practical systems, it is not sufficient to 'now that the system is stable but a stable system must

meet the specifications of relative stability which is a quantitative measure of how fast the

transients die out in the system.

:elative stability of a system is usually defined in terms of two design parameters!phase marginand gain margin.

he relative stability of a system can be defined as the property that is measured by the relative

real part of each root or pair of roots. he relative stability of a system can also be defined in

terms of the relative damping coefficients of each complex root pair and, therefore, in terms of

the speed of response and overshoot instead of settling time.

Tim# Domain nal"!i! >! Fr#/(#nc" Domain nal"!i!

• Uariable frequency, sinusoidal signal generators are readily available and precisionmeasuring instruments are available for measurement of magnitude and phase angle. hetime response for a step input is more difficult to measure with accuracy.

• *t is easier to obtain the transfer function of a system by a simple frequency domaintest.%btaining transfer function from the step response is more tedious.

• *f the system has large time constants, it ma'es more time to reach steady state at eachfrequency of the sinusoidal input. #ence time domain method is preferred over frequencydomain method in such systems.

• *n order to do a frequency response test on a system, the system has to be isolated and thesinusoidal signal has to be applied to the system. his may not be possible in systemswhich can not be interrupted. *n such cases, a step signal or an impulse signal may begiven to the system to find its transfer function. #ence for systems which cannot beinterrupted, time domain method is more suitable.

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• he design of a controller is easily done in the frequency domain method than in timedomain method. For a given set of performance measures in frequency domain , the parameters of the open loop transfer function can be ad+usted easily.

• he effect of noise signals can be assessed easily in frequency domain rather than time

domain.• he most important advantage of frequency domain analysis is the ability to obtain the

relative stability of feedbac' control systems. he :outh #urwit criterion is essentiallya time do main method which determines the absolute stability of a system .

• Since the time response and frequency response of a system are related through Fourier transform , the time response can be easily obtained from the frequency response. hecorrelation between time and frequency response can be easily established so that thetime domain performance measures can be obtained from the frequency domainspecifications and vice versa.

Fr#/(#nc" R#!$on!# of a Conrol S"!#m : o study the frequency response of a controlsystem let us consider second order system with the transfer function,

he steady state sinusoidal response is obtained by substituting s 9 + P in the above equation

2ormalising the frequency P, with respect to the natural frequency Pn by defining a variable

-e have,

From the above equation the magnitude and angle of the frequency response is obtained as,

and

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he time response for a unit sinusoidal input with frequency %E is given by,

he magnitude and phase of steadystate sinusoidal response for variable frequency can be plotted and are shown in the Fig. (a) and (b).

An important performance measure, in frequency domain, is the bandwidth of the system.

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From Fig. , we observe that for X Y 0.707 and H V Hr the magnitude decreases monotonically.

he frequency Hb where the magnitude becomes 4.4 is 'nown as the cut off frequency. At

this frequency, the magnitude will be

Conrol !"!#m Com$#n!aor!:

A control system is usually required to meet three time response specifications, namely, steady

state accuracy, damping factor and settling time. o get the desired design with minimum errors

and to ad+ust the parameters of the overall system to satisfy the design criterion an additional

subsystem called Z"ompensator I must be used.his compensator may be used in series with the plant in the forward path or in the feedbac'

path shown in Fig. below. he compensation in the forward path is 'nown as series or cascade

compensation and the later is 'nown as feedbac' compensation. he compensator may be a

passive networ' or an active networ'.

S#ri#! com$#n!aion F##d'ack com$#n!aion

*n general the series compensation is much simpler than the feed bac' compensation./ut the

series compensation frequently require additional amplifiers to increase the gain and to provide

isolation.i.e to avoid power dissipation the series compensator is inserted in the lowest energy

point in the feed forward path.*n general the number of components required in feed bac'

compensation will be less than the number of components in series compensation.

here are three important types of compensators.(i) ead "ompensator (ii) ag compensator and

(iii) ag!ead compensator.

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*n a networ' ,when a sinusoidal input signal ei is applied at its input and if the steady state output

eo has a phase lead then the networ' is called a lead networ'. Similarly *n the networ' for a

sinusoidal input ei ,if the steady state output has a phase lag ,then the networ' is called a lag

networ'.Similarly in a networ' for a sinusoidal input .if the steady state out put has both phase lag and

lead ,but different frequency regions ,then the networ' is called lag!lead networ'.8enerally the

phase lag occurs at low frequency regions and the phase lead occurs at higher frequency

region.A compensator having the characteristics of a lead networ' ,lag networ' or lead!lag

networ' is called a lead compensator ,lag compensator or lag!lead compensator.

he lead compensators, lag compensators and lag!lead compensators are be designed either

using electronic components li'e operational amplifiers or using :oot ocus methods. he firsttype are called electronic lag compensators .he second type are :oot locus compensators. he

:oot locus compensators are have many advantages over electronic type.

ead compensation basically speeds up the response of the control system and increase the

stability of the system. ag compensation improves the steady ;state accuracy of the system but

reduces the speed of the response. *f improvements in both transient response and steady state

response is required ,both lead compensator and lag compensator are used simultaneously .*n

general using a single lag!lead compensator is always economical.

ag!lead compensation combines the advantage of lag and lead compensation. Since, the lag ;

lead compensator possesses two poles and two eros ,such a compensation increases the order of

the system by > ,unless the cancellation of the poles and eros occurs in the compensated system.

L#ad Com$#n!aor : *t has a ero and a pole with ero closer to the origin. he general form of

the transfer function of the load compensator is

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Subsisting

ransfer function s

Lag Com$#n!aor :

*t has a ero and a pole with the ero situated on the left of the pole on the negative real axis. he

general form of the transfer function of the lag compensator is

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herefore, the frequency response of the above transfer function will be

2ow comparing with

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herefore,the transfer function is

Lag,L#ad Com$#n!aor

he lag!lead compensator is the combination of a lag compensator and a lead compensator. he

lag!section is provided with one real pole and one real ero, the pole being to the right of ero,

whereas the lead section has one real pole and one real ero with the ero being to the right of

the pole.

he transfer function of the lag!lead compensator will be

he figure below shows the lag lead compensator

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he above transfer functions are comparing with

hen

herefore, the transfer function is given by

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!a# )aria'l# anal"!i! :

Analysis and the design of feedbac' control systems by the classical design methods (root locus

and frequency domain methods) based on the transfer function approach are inad#/(a# and not

convenient. So, the development of the !a#,)aria'l# a$$roac*, too' place .his methos has

the following advantages over the classical approach.. *t is a very powerful technique for the design and analysis of linear or nonlinear, time!variant

or time!invariant, and SISO or MIMO systems.

?. *n this technique, the nth order differential equations can be expressed as In equations of first

order. hus, ma'ing the solutions easier.

3. Hsing this approach, the system can be designed for optimal conditions with respect to given

performance indices.

D#finiion!:

Sa# : he !a# of a dynamical system is a minimal set of variables x(t)x?(t) xn(t)

such that the 'nowledge of these variables at t = t 4 (initial condition), together with the

Knowledge of inputs u(t), u?(t) um (t) for t 4, completely determines the behavior of

the system for t Y t4.

Sa#,>aria'l#! :he variables x(t),x?(t) xm(t) such that the 'nowledge of these variables at t 9 t4 (initial

condition), together with the 'nowledge of inputs u(t), u?(t) um(t) for t t4, completely

determines the behavior of the system for t Y t 4 0 are called !a#,)aria'l#!. *n other words, the

variables that determine the state of a dynamical system, are called !a#,)aria'l#!.

Sa#,>#cor :

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*f n state variables x(t), x?(t) xn(t) are necessary to determine the behavior of a

dynamical system, then these n state!variables can be considered as n components of a vector

4(t), called !a#,)#cor.

Sa#,S$ac# : he n dimensional space, whose elements are the n state!variables, is called !a#,

!$ac#. Any state can be represented by a $oin in the state!space.

Sa# #/(aion of a lin#ar im#,in)arian !"!#m :

For a general system of the Figure shown below the state representation can be arranged in the

form of n first!order differential equations as

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(>)

quation > is the !a# #/(aion for time!invariant systems. #owever, for time!varying !"!#m!,

the function vector f (.) is dependent on time as well, and the vector equation may be given as

.(?)quation (?) is the !a# #/(aion for im#,)ar"ing !"!#m!.

he output "(t) can, in general, be expressed in terms of the state vector 4(t) and input vector ((t)

as

For im#,in)arian !"!#m! :

,,,,,,,,,,,,,,,,,,,,,,,, ,,,,,3)

For im#,)ar"ing !"!#m!: ,,,,,,,,,,,,,,,,?2

STTE MODEL OF LINER SIN&LE,IN0UT,SIN&LE,OUT0UT SYSTEM:

he state model of a linear single!input!single!output system can be written as

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()

-here

d 9 ransmission "onstant , u(t) 9 *nput or "ontrol Uariable (scalar)and, y(t) 9 %utput Uariable (scalar).

he bloc'!diagram representation of the state model of linear single!input!single!output system

is shown below.

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A very important conclusion is that the derivatives of all the state!variables are ero at the

equilibrium point. hus, the system continues to lie at the equilibrium point unless otherwise

disturbed.

SOLUTION OF STTE E@UTIONS FOR LINER TIME,IN>RINT SYSTEMS :

he state equation of a linear time!invariant system is given by

For a homogeneous (unforced) system

So , we have ..(:

et us consider the following state space equation ,

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2ow, ta'e the aplace ransform (with ero initial conditions)

-e want to solve for the ratio of (s) to H(s), which gives the transfer function.So, we need to

remove 1(s) from the output equation. -e solve the state equation for 1(s)

he matrix ](s) is called the state transition matrix. 2ow , put this into the output equation

2ow solving for transfer function ,we get

his is the method of determining the transfer function from state variable analysis (SUA).

ckno+l#dgm#n D'ar R'a)'r I )o#J$ c%a*m a#y o-#'r&h*+ ,or $h*& ma$'r*a% a&

*$ *& a co%%'c$*o# ,rom var*o(& ooB& a#) -'&*$'& a#) o$h'r ar$*c%'&. I hav'

&*m+%y a))') my '+'r*'#c' a#) $ry $o +rov*)' $h*& ma$'r*a% ,or $ho&' +'o+%'

-ho ar' +r'+ar*# ,or NT?U>= a#) o$h'r #*#''r*# 'am&.P%'a&'

)o-#%oa) $h*& ma$'r*a% ,or yo(r +r'+ara$*o# .($ #'v'r maB' *$ comm'rc*a%. I$

*& +(r'%y m'a#$ ,or &$()'#$& -ho ar' *# #'') o, $h*& ma$'r*a%.

R','r'#c'& 1A($oma$*c co#$ro% &y&$'m& .=. V(o Pr'#$*c'Ha%% o, I#)*a.

2. Mo)'r# =o#$ro% #*#''r*# V. a$a Pr'#$*c)Ha%% o, I#)*a.

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3. =o#$ro% Sy&$'m& #*#''r*# I.>. Nara$h M. >o+a%W *%'y

a&$'r#

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