Control-System-Notes-by-HPK-Kumar.pdf

 This Notes not P

CONTROL SYSTEM

CONTROL SYSTEM

 NOTES

 NOTES

(For

(For BachelorBachelor of of Engineering)Engineering) Notes

Notes by:by:

PROF. SHESHADRI G. S

PROF. SHESHADRI G. S

Soft

Soft CopyCopy materialmaterial designeddesigned by:by:

KARTHIK KUMAR H P

KARTHIK KUMAR H P

Your

Your feedbacksfeedbacks cancan bebe mailedmailed to:to:

[email protected] [email protected]

INDEX INDEX

(1)

(1) Introduction

Introduction to

to Control

Control system

system

(2)

(2) Mathematical

Mathematical model

model of 

of linear

linear systems

systems

(3)

(3) Transfer

Transfer functions

functions

(4)

(4) Block

Block diagram

diagram

(5)

(5) Signal

Signal Flow

Flow Graphs

Graphs

(6)

(6) System

System Stability

Stability

(7)

(7) Root

Root Locus

Locus Plots

Plots

(8)

(8) Bode

Bode Plots

Plots

1

1

Control Systems 

Control Systems 

CONTROL SYSTEM

CONTROL SYSTEM

 NOTES

 NOTES

(For

(For BachelorBachelor of of Engineering)Engineering) Notes

Notes by:by:

PROF. SHESHADRI G. S

PROF. SHESHADRI G. S

Soft

Soft CopyCopy materialmaterial designeddesigned by:by:

KARTHIK KUMAR H P

KARTHIK KUMAR H P

Your

Your feedbacksfeedbacks cancan bebe mailedmailed to:to:

[email protected] [email protected]

INDEX INDEX

(1)

(1) Introduction

Introduction to

to Control

Control system

system

(2)

(2) Mathematical

Mathematical model

model of 

of linear

linear systems

systems

(3)

(3) Transfer

Transfer functions

functions

(4)

(4) Block

Block diagram

diagram

(5)

(5) Signal

Signal Flow

Flow Graphs

Graphs

(6)

(6) System

System Stability

Stability

(7)

(7) Root

Root Locus

Locus Plots

Plots

(8)

(8) Bode

Bode Plots

Plots

1

1

Control Systems 

Control Systems 

1 1

Control Systems 

Control Systems 

By: By: HPK Kumar  HPK Kumar 

Introduction to Control Systems

Introduction to Control Systems

By: By:

CIT, Gubbi. CIT, Gubbi.

C

Control ontrol SysSystetem m mmeeans any qans any quantiuantity of ity of intenterreest ist in a mn a machiachine ne or or mmeechanichanism sm iis ms maiaintaintainened or d or alalteterreed id inn accor

accordance dance wwiith th dedesisirreed d mmanneannerr. . OROR A

A sysystestem m wwhihich controlch controls the s the output quoutput quantiantity is calty is callleed a contrd a control ol sysystestemm..

Definitions:

Definitions:

1.1.  Co

 Cont

ntro

roll

lled

ed Var

Variab

iab le

le::

IIt it is ths the e ququantity or antity or condcondiitition thon that iat is ms meeasuasured & red & controlcontrollleed.d.

2.

2.  Co

 Cont

ntro

roll

ller

er::

Controller 

Controller  m meeans mans meeasuriasuring the ng the valvalue ue of of the the contrcontrolollleed varid variablable of the sye of the systestem m & & applapplyiying theng the m

manianipulpulateated varid variablable to the e to the syssystetem m to corto corrreect or ct or to lto liimmiit the det the deviviatiation of on of the mthe meeasured vasured valalue ue to theto the desired value.

desired value.

3.

3. Plant:

Plant:

A

A plant plant  i  is a pis a piecece of eqe of equiuipmpmeent, which int, which is a ses a set of t of mmachiachine ne parts fparts functiunctionioning togeng togethetherr. T. Thehe purpos

purpose e of of wwhihich ch iis s to to peperrforform m a a partiparticular cular opeoperratiation. on. EExxamamplplee: : FFurnaceurnace, , SpaSpace ce crafcraft t eetc.,tc.,

4.

4. System:

System:

A

Asystemsystem i is a coms a combibinatination of comon of componeponents that nts that wwororks togks togetetheher r & & peperrforformms ces cerrtaitain objn objeectictiveve..

5.

5.  Di

 Dist

stur

urba

banc

nce:

e:

A

Adisturbancedisturbance iis a sis a signal gnal that tethat tends tnds to affo affeect the ct the value of the outpvalue of the output of a sysut of a systetemm. . IIf f a dia disturbancesturbance is created inside the system, it is called

is created inside the system, it is called internalinternal. While an. While anexternalexternal di disturbance isturbance is ges genenerrateated outsid outside thde thee system.

system.

6.

6. Feedback Control:

Feedback Control:

IIt it is an s an operatioperation that, ion that, in the n the prpresesence ence of of didistursturbance bance tendtends to reduce s to reduce the the didifffference erence bebetwetweeen then the output of

output of a systea system m & & somsome refe reference ierence input.nput.

 7.

 7. Servo Mechanism:

Servo Mechanism:

A

A servo mechanismservo mechanism  i  is a fees a feedback contrdback controlollled ed systesystem m iin wn whihich the ch the output ioutput is soms some e mmeechanichanicalcal pos

posiition, tion, vveellocity or ocity or acceaccelleeratiration.on.

8.

8. Open loop System:

Open loop System:

IIn an annOpen loop SystemOpen loop System, the , the control control actiaction on iis s iindendepependendent nt of the of the dedesisirreed oud output. tput. OROR Wh

When en the the output quantioutput quantity of the ty of the contrcontrol ol syssystem tem iis not fed back to s not fed back to the ithe input quantitnput quantity, the y, the contrcontrolol sys

system tem iis cals callled ed ananOpen loop System.Open loop System.

9.

9.  Cl

 Clos

osed

ed lo

loop

op Sy

Syst

stem

em::

IIn thn theeClosed loop Control SystemClosed loop Control System the control  the control actiaction ion is des depependendent on the dent on the desisirreed output, d output, wwheherree the

the output qoutput quantity iuantity is considerabls considerably controly controllleed by d by sesendinding a comng a commmand signal and signal to ito inpunput quant quantitity.ty.

Sheshadri G S

Sheshadri G S

Sheshadri G S Sheshadri G S

 2

 2

 2

Introduction to Control System

Introduction to Control System

By:

By:

HPK Kumar HPK Kumar

10

10.. Feed Back:

Feed Back:

Normally, the feed back signal has opposite polarity to the input signal. This is called negative Normally, the feed back signal has opposite polarity to the input signal. This is called negative ffeeeed back. Td back. The he advanadvantage tage iis the s the rreesultant sisultant signal gnal obtaiobtainened fd frrom om the the comcomparator parator bebeiing ding diffffeerreence nce of of thethe tw

two sio signalgnals is is of sms of smalalller mer magniagnitudetude. I. It can bt can be e handled ehandled easiasilly by by y the the contrcontrol ol syssystetemm. T. The he rresesulultiting sng siignalgnal is called

is calledActuating SignalActuating Signal ThiThis s sisignal gnal has has zezerro o valvalue ue wwhehen n the the dedesisirreed d outpuoutput t iis s obtaiobtainened. d. IIn n thatthat condition, control system will not operate.

condition, control system will not operate.

Effects of Feed Back: Effects of Feed Back:

L

Leet t the the sysystestem m has has opeopen n lloop oop gaigain n ffeeeed d back back lloop oop gaigain n Output Output sisignal gnal && IInpnput ut signasignal l ..

 Th

 Theen n tthhe e fefeeed d bbaacck k ssigignnaal l isis,,

With

With thithis s eeqn. qn. , w, we e can can wwrriite te the the eeffffeects cts of feof feeed d back back as as ffololllowows.s.

(a

(a))

Overall Gain:

Overall Gain:

E

Eqn. qn. showshows s that that the the gaigain n of of the the open open lloop oop syssystem tem iis s rrededuced uced by by a a ffactor actor iinn a f

a feeeed back sd back systeystemm. . HHeerre the the feee feed back sid back signal gnal iis nes negatigativeve. . IIf f the the feefeed back gain has posid back gain has posititive vve valalueue, , thethe ove

overralall l gaigain wn wiilll l be be rreeduceduced. Id. If the f the ffeeeed back d back gaigain n has nhas neegatigative ve valvalueue, the , the oveoverralall l gaigain mn may ay iincreasencrease..

(b

(b))

Stability:

Stability:

IIf a systef a system m iis abls able e to folto folllow ow the input comthe input commmand siand signalgnal, the sy, the systestem m iis sais said to bed to beStable.Stable.

A

A systesystem m iis sais said to bed to beUnstableUnstable, , iif f iits ts outpoutput ut iis s out out of of contcontrrolol. . IIn n eeqnqn. . , , iif f the the outpoutput ut of of thethe sy

systestem m iis is infinfininite te ffor or any fiany fininite ite inputnput. . TThihis shows shows that a stabls that a stable se systeystem m mmay beay becomcome ue unstabnstablle for e for cecerrtaitainn val

value of a fue of a feeeed back gaid back gain. Tn. Theherreefoforre if e if the the feefeed back id back is not prs not propeoperrlly usey used, d, the sthe sysystetem m can be harmcan be harmffulul..

(c

(c))

Sensitivity:

Sensitivity:

 Th

 This is ddeeppeenndds s oon n tthhe e ssyysstteem m ppaararammeetteersrs. . FoFor r a a ggooood d ccoonnttrorol sl syysstteemm, , it it is is ddeessirairabble le tthhaat t tthhe e ssyysstteemm should be i

should be insensensitinsitive ve to ito its parts paramameeteter r changechanges.s. Sensitivity,

Sensitivity,

S

S

_GG

=

=

 Th This is fufunnccttioion n oof f tthhe e ssyysstteem m ccaan n bbe e reredduucceed d bby y ininccrereaassining g tthhe e vvaalulu of

of . Thi. This s can can be be done done by by sesellecectiting ng proper proper feefeed d back.back.

G(S)

G(S)

H(S)

H(S)

R(S)

R(S)

B(S)

B(S)

E(S)

E(S)

C(S)

C(S)

--& & –– Hence, Hence,

=

=

₍₁₎

₍₁₎

Sheshadri G S Sheshadri G S 3 3

Control Systems 

Control Systems 

3 3

Control Systems 

Control Systems 

By: By: HPK Kumar  HPK Kumar 

(d

(d))

Noise:

Noise:

Examples are

Examples are brush &  brush & commutation commutation noisenoise in electrical machines, in electrical machines, VibrationsVibrations i in mn movioving systemng system e

etc.,tc.,. T. The he eeffffeect of ct of feefeed back on thd back on theese se noinoise se sisignalgnals wils will l be be greatlgreatly iy inflnflueuencenced by thd by the pe poioint at wnt at whihich thesch thesee si

signalgnals ars are intre introduceoduced id in the n the sysystestemm. . IIt it is posss possiiblble to redue to reduce tce the he eeffffeect of noict of noise se by pby prropeoper r dedesisign of fgn of feeeedd back

back systemsystem..

Classification of Contr

Classification of Control Systems

ol Systems

 Th

 The e CoConnttrorol Sl Syysstteem m ccaan n bbe e cclalassssifieified d mmaaininly ly ddeeppeennddining g uuppoonn,, ((aa)) MMeethod othod of f anaanallysysiis & s & dedesign, sign, asasLinear & Non- Linear Systems.Linear & Non- Linear Systems.

((bb)) Th The e ttyyppe e oof f tthhe e ssigignnaal, l, aassTime Varying, Time Invariant, Continuous data, Discrete Time Varying, Time Invariant, Continuous data, Discrete data systemsdata systems etc., etc.,

((cc)) Th The e ttyyppe e oof f ssyysstteem m ccoommppoonneennttss, , aassElectro Mechanical, Hydraulic, Thermal, Pneumatic Control systemsElectro Mechanical, Hydraulic, Thermal, Pneumatic Control systems etc.,etc.,

((dd)) Th The e mmaain in ppuurprpoossee, , aassPosition control & Velocity control Systems.Position control & Velocity control Systems.

1.1. Linear & Non

Linear & Non

--

Linear Systems:

Linear Systems:

IIn a ln a liinenear ar sysystestemm, , the the priprinciplnciple of supe of supeerpositirposition can bon can be ae apppplliieed. Id. I n non- n non- lliinenear ar sysystestemm,, this principle can’t be applied. Therefore a linear system is that which obeys superposition this principle can’t be applied. Therefore a linear system is that which obeys superposition principle & homogeneity.

principle & homogeneity.

2.

2. Time Varying & Time Invariant Systems:

Time Varying & Time Invariant Systems:

Wh

Whiille e operatioperating a contrng a control ol syssystemtem, , iif f the the parparamameteeterrs ars are unaffe unaffeected bcted by the y the titimme, thee, then then the sys

system tem iis cals callleded Time Invariant Control System.Time Invariant Control System. MMost physiost physical cal systesystemms have pars have paramameteeterrss changi

changing wng wiith tith timme. Ie. If thif this vs varariiatiation ion is ms measueasurrablable e duriduring the ng the syssystem tem operatioperation theon then the n the syssystemtem is called

is called Time Varying System.Time Varying System.

IIf f thetherre e iis no non-s no non-lliinenearariity ity in the n the titimme ve vararyiying sysng systetemm, , thethen the n the systesystem m mmay be ay be calcallleed asd as

Linear Time varying System. Linear Time varying System.

3.

3.  Di

 Disc

scre

rete

te Da

Data

ta Sy

Syst

stem

em s:

s:

IIf the f the sisignal ignal is not s not conticontinuounuouslsly y varyivarying wng wiith timth time e but ibut it it is in the s in the forform m of pulof pulseses. Thens. Then the

the control control sysystestem m iis cals calllededDiscreDiscrete Data te Data Control System.Control System.

IIf f the the sisignal ignal is is in the n the forform m of pulof pulse se data, thedata, then the n the sysystestem m iis cals callleedd Sampled Data ControlSampled Data Control System.

System. Here the information supplied intermittently at specific instants of time. This has the Here the information supplied intermittently at specific instants of time. This has the advantage of Time sharing system. On the other hand, if the signal is in the form of digital advantage of Time sharing system. On the other hand, if the signal is in the form of digital code, the system is called

code, the system is called Digital Coded System.Digital Coded System.  Here use of Digital computers, µp, µc is  Here use of Digital computers, µp, µc is m

made usade use e of of such syssuch systemtems ars are e analanalyzeyzed by d by the Z- the Z- trtransfansfororm m theortheory.y.

4.

4.  Co

 Cont

ntin

inuo

uous

us Da

Data

ta Sy

Syst

stem

ems:

s:

IIf f the the sisignal gnal obtained at vobtained at variarious parts of ous parts of the the sysystestem m arare ve vararying contiying continuousnuouslly wy wiith tith timmee,, then the

then the systesystem m iis cals calllededContinuous Data Control Systems.Continuous Data Control Systems.

5.

5.  Ad

 Adap

apti

tive

ve Co

Cont

ntro

rol

l sy

syst

stem

ems:

s:

IIn somn some e contrcontrol ol syssystemtems, certais, certain paran parammeeters arters are e eeiither not constant or varther not constant or vary in any in an unk

unknownown mn manneannerr. . IIf f the the paramparameeteter r varivariatiations are lons are lararge ge or or rrapiapid, id, it mt may bay be de deesisirrablable te to deo desisigngn for the capability of continuously measuring them & changing the compensation, so that the for the capability of continuously measuring them & changing the compensation, so that the sy

systestem m peperrforformmance ance cricriteriteria can ala can alwways ays satisatisfisfieed. Td. Thihis is cals is callleedd Adaptive Control Systems.Adaptive Control Systems.

Sheshadri G S

Sheshadri G S

4

Introduction to Control System

4

Introduction to Control System

By:

HPK Kumar

6. Optimal Control System:

Optimal Control System is obtained by minimizing and/or maximizing the performance index. This index depends upon the physical system & skill.

 7. Single Variable Control System:

In simple control system there will be One input & One output such systems are called

Single variable System (SISO – Single Input & Single Output).

8.  Multi Variable Control System:

In Multivariable control system  there will be more than one input & correspondingly more output’s (MIMO - Multiple Inputs & Multiple Outputs).

Comparison between Open loop & Closed loop Gain

Open Loop System Closed Loop System

1. An open loop system has the ability to perform accurately, if its calibration is good. If the calibration is not perfect its performance will go down.

2. It is easier to build.

3. In general it is more stable as the feed back is absent.

4. If non- linearity’s are present; the system operation is not good.

5. Feed back is absent. Example:

(i)  Traffic Control System.

(ii) Control of furnace for coal heating. (iii) An Electric Washing Machine.

1. A closed loop system has got the ability to perform accurately because of the feed back.

2. It is difficult to build.

3. Less Stable Comparatively.

4. Even under the presence of non-linearity’s the system operates better than open loop system.

5. Feed back is present. Example:

(i) Pressure Control System. (ii) Speed Control System. (iii) Robot Control System.

(iv)  Temperature Control System.

Note:

Any control system which operates on time basis is an Open Loop System.

Compensator

_System

H(S)

Identification & Parameter adjustment

R(s)

_E(s)

+

-B(s)

C(s)

Sheshadri G S 5

Control Systems 

5

Control Systems 

By:

HPK Kumar 

Block Diagram of Closed Loop System:

 Thermometer Block Diagram of Temperature Control System:

Temperature Control of Passenger Compartment Car:

Control Elements

Feed Back elements Ref. i/p

E(S)

Controlled o/p

E(S)

Plant

Controller Actuator Desired Temperature i/p Controller Sensor

Radiation Heat Sensor

Air Conditioner Passenger Car

Sensor O/p Sun Ambient Temperature * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

A/D Converter Interface

Relay Amplifier Interface

Programmed i/p Electric Furnace Heater Thermo meter Sheshadri G S 1

Control Systems 

1

Control Systems 

Designed By:

HPK Kumar

([email protected])

 Mathematical Models of Linear Systems

By:

CIT, Gubbi.

 A physical system is a collection of physical objects connected together to serve an objective. An idealized  physical system is called a Physical model. Once a physical model is obtained, the next step is to obtain Mathematical model. When a mathematical model is solved for various i/p conditions, the result represents the dynamic behavior of the system.

 Analogous System:

The concept of analogous system is very useful in practice. Since one type of system may be easier to handle experimentally than another. A given electrical system consisting of resistance, inductance & capacitances may be analogous to the mechanical system consisting of suitable combination of Dash pot,  Mass & Spring. The advantages of electrical systems are,

1.  Many circuit theorems, impedance conc epts can be applicable.

2.  An Electrical engineer familiar with electrical systems can easily analyze the system under study & can predict the behavior of the system.

3. The electrical analog system is easy to handle experimentally.

Translational System:

 It has 3 types of forces due to elements.

1.  Inertial Force:  Due to inertial mass,

2.  Damping Force [Viscous Damping]:  Due to viscous damping, it is proportional to velocity & is given by,

3. Spring Force: Spring force is proportional to displacement.





    .

 Damping force is denoted by either D or B or F

D

F

k

  .

M F(t)

F

m



t



 M.a  







2

_



2

Where,

   .

   .













  . .



 

Sheshadri G S

 2

Mathematical Models of Linear Systems 

Rotational  system:

([email protected])

 2

Mathematical Models of Linear Systems 

Designed By:

HPK Kumar 

Rotational  system:

1.  Inertial Torque:  2.  Dampi ng Torque :  3. Spring Torque



_



:

 Analogous quantities in translational & Rotational system:

The electrical analog of the mechanical system can be obtained by, (i) Force Voltage analogy: (F.V)

(ii) Force Current analogy: (F.I)

Sl. No. Mechanical Translational System Mechanical Rotational System F.V Analogy F.I Analogy

1. Force (F) Torque (T) Voltage (V) Current (I) 2. Mass (M) Moment of Inertia (M) Inductance (L) Capacitance (C) 3. Viscous friction (D or B or F) Viscous friction (D or B or F) Resistance (R) Conductance (G) 4. Spring stiffness (k) Torsional spring stiffness

(



_

)

 Reciprocal

of Capacitance (1/C)

 Reciprocal

of Inductance (1/L) 5.  Linear displacement (



) Angular displacement (



) Charge (q) Flux (



)

6.  Linear velocity (



)  Angular Velocity (w) Current (i) Voltage (v)

D’Alemberts Principle:

The static equilibrium of a dynamic system subjected to an external driving force obeys the following principle, “For any body, the algebraic sum of externally applied forces resisting motion in any given direction is zero”.

Example Problems:

(1) Obtain the electrical analog (FV & FI analog circuits) for the Machine system shown & also write the equations.

                  

   

_







   





    .





        

   

   

     

Where,

   







_



 





D

₂

D

₁





2

2









1

1





Ft

Free Body diagram

M

1





 

_







_





 





Ft

M

1

 



1

Control Systems 

1

Control Systems 

Designed By:

HPK Kumar 

([email protected])

G(S)

R(S)

C(S)

G(S) =





  1

 Transfer Functions

By: CIT, Gubbi.

The input- output relationship in a linear time invariant system is defined by the transfer function. The features of the transfer functions are,

(1)  It is applicable to Linear Time Invariant system.

(2)  It is the ratio between the Laplace Transform of the o/p variable to the Laplace Transform of the i/p variable. (3)  It is assumed that initial conditions are zero.

(4)  It is independent of i/p excitation.

(5)  It is used to obtain systems o/p response.

 An equation describing the physical system has integrals & differentials, the step involved in obtaining the transfer function are;

(1) Write the differential equation of the system. (2)  Replace the terms





by ‘S’ &



by 1/S.

(3)  Eliminate all the variables except the desired variables.

Impulse Response of the Linear System:

Taking L-1

 Here G(t) will be impulse response of the Linear System. This is called Weighing Function. Hence LT of the impulse response is the Transfer function of the system itself.

P ROBLEMS:

(1) Obtain the Transfer _‐Function(TF) of the circuit shown in circuit 1.0 

Solution:

i.e., the Laplace Transform of the system o/p will be simply the ‘Transfer function’ of the system.

   .  .1

 In a control system, when there is a single i/p of unit impulse function, then there will be some response of the Linear System.

The Laplace Transform of the i/p will be R(S) = 1

  



  

R 

C

 i

Circuit 1.0

R 

 i

_(S)

 Laplace Transformed network

1



Contd……

Sheshadri G S

([email protected])

 2

Transfer Functions 

Designed By:

HPK Kumar 

.., 





_

1

 

&





   

_

1



(2) Obtain the TF of the mechanical system shown in circuit 2.

  .



  

 .



 





  1

  

_



_

_{ }



_{1 }

1

_

_



Where,



= RC 3

Control Systems 

3

Control Systems 

Designed By:

HPK Kumar 

([email protected])

(3) Transfer Function of an Armature Controlled DC Motor in circuit 3.0:

The air gap flux ‘



’ is proportional to the field current. i.e.,

  

_

  



.



Where, ‘K  f ’ is a constant.

The torque developed by the motor ‘T m’ is proportional to the product of the arm current & the air gap flux.

 



  















Where, K a & K  f  are the constants.

Since the field current is constant,



_

  

_



_

Where, K T  is Motor – torque constant.

The motor back e.m.f is proportional to the speed & is given by,





 

 

_

Where, K bis back e.m.f constant.

The differential equation of the armature circuit is,



_

  

_



_

 

_







  



The torque equation is,





 



2





2







Taking LT for above equation, we get





 







  



 



  





--- (1)

















_



_



_{ }

 

_



_



‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ (A)





  



     

Taking LT for the torque equation & equating, we get









  



      

 R

a

 L

a

i

a

i

a

V 

_i

 _E 

_b

_F 

 I 

 f

=

Constant 

V 

 f

T 

m



  



 Let, ‘Ra’   Resistance of armature in

Ω

’s. ‘La’   Inductance of armature in H’s.

‘ia’ Armature current. & ‘i f ’  Field current.

‘V i’   Applied armature voltage.

‘E b’   Back e.m.f in volts.

‘T m’   Torque developed by the motor in N-m.



 Angular displacement of motor shaft in radians

 J  Equivalent moment of inertia of motor & load referred to the motor shaft.

F   Equivalent Viscous friction co-efficient of motor & load referred to the motor shaft. circuit 3.0

   



J,

F

([email protected])

4

Transfer Functions 

Designed By:

HPK Kumar 















  

2

  

_

_



 

--- (2)













 



_{ }

--- (B)

Taking LT for back e.m.f equation, we get





 



 

--- (C)

Substituting the values of I a(S)&E b(S) from equation (C) & (2) in equation (1), we get





 



 



   



_

  

_

   



  











 



  

_



   

_



 .





 

The block diagram representation of armature controlled DC Motor can be obtained as follows,

From equation (A),

From equation (B),

From equation (C),

The complete block diagram is as shown below,













 



   





   









 

1





 





V i(S)  E b(S)

- I a(S)





  



_{  }



(S)  I a(S)







 E b(S)



(S)





  

2

_{  }



(S)

1





 





V i(S)  E b(S)

- I a(S)







5

Control Systems 

5

Control Systems 

Designed By:

HPK Kumar 

([email protected])

(4) Transfer  function of Field Controlled DC Motor in circuit 4.0:

 In the field controlled DC motor, the armature current is fed from a constant current source.

 



  















Where, K a & K  f  are the constants.

The KVL equation for the field circuit is,





  







  





_



On Laplace Transform,





 







  



 









 



.



  



 

--- (1)



 









_

 _



_{ }

 



 _



_{ }

--- (A)

The torque equation is ,



_

  

_



_

Where, K T  is Motor – torque constant.





   

_





_

  

_

On Laplace Transform,





  



 



  



     





 



 



    . 



 













2

  

_

_



.







--- (2)







 









_{   }

 

---

(B) Substituting the value of



_



 from equation (2) in equation (1), we get

 R

 f

 L

 f

i

 f

V 

 f

T 

m



 I 

a = Constant   Let,

 R f   Field winding resistance.

 L f   Field winding inductance.

V  f   Field control voltage.

 I  f   Field current.

T m  Torque developed by motor.

 J, F

 J  Equivalent moment of inertia of motor & load referred to the motor shaft. F  Equivalent Viscous friction co-efficient of motor & load referred to the motor shaft.



  Angular displacement of motor shaft. circuit 4.0

([email protected])

6

Transfer Functions 

Designed By:

HPK Kumar 











2

_{  }

_

_.

_

_

_









  



 

The block diagram representation of  field controlled DC Motor can be obtained as follows,

From equation (A),

From equation (B),

The complete block diagram is as shown below,

(5) Obtain the TF 













 for the network shown in circuit 5.0:

Solution:

 Applying KVL to this circuit,





   

_



 





--- (1)

 



 





 



_{  }





.  



1  

1  















_

1

 



.



 

_

1



 

















_





_

_{1}

  

_

2

_{ }











 













2

   









 _

 

 _

 





  



_{  }



(S)  I  f(S)

1





 





V  f(S)  I  f(S)

1





 





V  f(S)  I  f(S)





  



_{  }



(S) circuit 5.0  R C  R C V i V o

 Laplace Transformed network  R  R

1



1



V i(S) V o (S)

 I(S) I 1(S) I 2(S)









_







1

τ

τ







2



τ



Let,

 

τ

 7

Control Systems 

100 k 

 Ω

1M

Ω

 7

Control Systems 

Designed By:

HPK Kumar 

([email protected])

(6) Find the TF 













 for the network shown in circuit 6.0:

Solution:

Writing KVL for loop (1), we get











 

1







10

5



10

_

5



 

2







10

_

5





₁

















. 

10

5

 1





1

2



--- (1)

Writing KVL for loop (2), we get









_



 







_



 10







_



  0





₂









_{10 11}



1







--- (2)











  

2







.

10

_

6

 

2







 







₁₀





6

. 

--- (3) Substituting for I 1(S) from equation (2) in (1), we get





₂







.



10 11













. 

10

5

 1







1

2





 







10

5

 1

 

2





 



10 11



1

 

1





 1



From equation (3) the above equation becomes,

 







10

5









10





6

. 



10 

2

  21  10



 Laplace Transformed network

Circuit 6.0 100 k 

 Ω

1M

Ω

10



 F 1



 F V i V i(S)













10



10



10





10





V 0(S)  Loop 1 Loop 2



















 

10 



  21   10

10

10

6



V 0(S)

1

10  11

 I 1(S) +  I 2(S)

1

 1



10



 _{  1}

V i(S)  8

Transfer Functions 

([email protected])

 8

Transfer Functions 

Designed By:

HPK Kumar 

 2

Block Diagrams 

PROBLEMS:

Reduce the Block Diagrams shown below:

(1)

Solution: By eliminating the feed-back paths, we get

Combining the blocks in series, we get

Eliminating the feed back path, we get + - - -+ -+ C(S)

-3

Control Systems 

(2)

Solution: Shifting the take-off ‘ ’ beyond the block‘ ’, we get

Combining and eliminating (feed back loop), we get

Eliminating the feed back path , we get

Combining all the three blocks, we get

R(S) - -C(S) R(S) - -C(S) R(S) - _- C(S) R(S) C(S) R(S) -C(S)

4

Block Diagrams 

(3)

Solution: Re-arranging the block diagram, we get

Eliminating loop & combining, we get

Eliminating feed back loop

Eliminating feed back loop , we get

C(S) R(S) - - -C(S) R(S) - -R(S) C(S) -C(S) R(S) C(S) R(S) --

-1

Control Systems 

 Signal Flow Graphs

By:

CIT, Gubbi.

For complicated systems, Block diagram reduction method becomes tedious & time consuming. An alternate method is that signal flow graphs developed by S.J . Mason. In these graphs, each node represents a system variable & each branch connected between two nodes acts as Signal Multiplier. The direction of signal flow is indicated by an arrow.

Definitions:

1. Node: A node is a point representing a variable.

2. Transmittance: A transmittance is a gain between two nodes.

3. _Branch: A branch is a line joining two nodes. The signal travels along a branch. 4. _{Input node [Source]:} It is a node which has only out going signals.

5. _{Output node [Sink]:} It is a node which is having only incoming signals.

6. _{Mixed node:} It is a node which has both incoming & outgoing branches (signals).

7. _Path: It is the traversal of connected branches in the direction of branch arrows. Such that no node

is traversed more than once.

8. _{Loop:It is a closed path.}

9. _{Loop Gain:} It is the product of the branch transmittances of a loop.

10._{Non-Touching Loops}: Loops are Non-Touching, if they do not possess any common node.

11._{Forward Path: It is a path from i/p node to the o/p node which doesn’t cross any node more than}

once.

12._{Forward Path Gain: It is the product of branch transmittances of a forward path.}

MASON’S GAIN FORMULA:

 The relation between the i/p variable & the o/p variable of a signal flow graphs is given by the net gain between the i/p & the o/p nodes and is known as Overall gain of the system.

Mason’s gain formula for the determination of overall system gain is given by,

Where, _– Path gain of forward path.

Determinant of the graph.

 The value of the for that part of the graph not touching the forward path.  T Overall gain of the system.

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Control Systems 

Signal flow graph:

No. of forward paths:

No. of individual loops:

(6) Reduce the Block Diagram shown.

Solution:

Shifting beyond , we get

C(S) R(S) C(S) R(S)

-R(S)

-

+

C(S)

-R(S)

-

+

C(S) R(S)

-

C(S) R(S) C(S)

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 Signal  Flow Graphs 

Eliminating feed back loop , we get

Eliminating feed back loop , we get

Eliminating the another feed back loop , we get

Signal flow graph:

-R(S)

-

+

C(S)

-R(S)

+

C(S)

-R(S)

+

C(S) R(S)

+

C(S) C(S) R(S) R(S) _C(S) Contd...

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 Signal  Flow Graphs 

Eliminating loop, we get

(9) Using Mason’s gain rule, obtain the overall TF of a control system represented by the signal flow graph shown below.

Solution:

No. of forward paths:

Individual loops:

 Two non-touching loops = 0

(10) Construct signal flow graph from the following equations & obtain the overall TF.

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Control Systems 

Substituting ‘x’ value in the block diagram. The block diagrambecomes,

Signal flow graph:

No. of forward paths:

No. of individual loops: Two non-touching loops = 0

(16) Obtain TF, using block diagram algebra & also by using Masons Gain Formula. Hence Verify the TF in both the methods.

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Control Systems 

1

Control Systems 

 System Stability

By:

CIT, Gubbi.

While considering the performance specification in the control system design, the essential & desirable requirement will be the system stability. This 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 & relative stability. Absolute stability is the quality of stable or unstable performance. Relative Stability is the quantitative study of stability.

 The stability study is based on the properties of the TF. In the analysis, the characteristic equation is of importance to describe the transient response of the system. From the roots of the characteristic equation, some of the conclusions drawn will be as follows,

(1) When all the roots of the characteristic equation lie in the left half of the S-plane, the system response due to initial condition will decrease to zero at time Thus the system will be termed as stable.

(2) When one or more roots lie on the imaginary axis & there are no roots on the RHS of S-plane, the response will be oscillatory without damping. Such a system will be termed as critically stable.

(3) When one or more roots lie on the RHS of S-plane, the response will exponentially increase in magnitude; there by the system will be Unstable.

Some of the Definitions of stability are,

(1) A system is stable, if its o/p is bounded for any bounded i/p.

(2) A system is stable, if it‟s response to a bounded disturbing signal vanishes ultimately as time „t‟ approaches infinity.

(3) A system is unstable, if it‟s response to a bounded disturbing signal results in an o/p of infinite amplitude or an Oscillatory signal.

(4) If the o/p response to a bounded i/p signal results in constant amplitude or constant amplitude oscillations, then the system may be stable or unstable under some limited constraints. Such a system is called Limitedly Stable system.

(5) If a system response is stable for a limited range of variation of its parameters, it is called Conditionally Stable System.

(6) If a system response is stable for all variation of its parameters, it is called Absolutely Stable system.

Routh-Hurwitz Criteria:

A designer has so often to design the system that satisfies certain specifications. In general, a system before being put in to use has to be tested for its stability. Routh-Hurwitz stability criteria may be used. This criterion is used to know about the absolute stability. i.e., no extra information can be obtained regarding improvement.

As per Routh-Hurwitz criteria, the necessary conditions for a system to be stable are, (1) None of the co-efficient‟ of the Characteristic equation should be missing or zero. (2) All the co-efficient‟ should be real & should have the same sign.

 2

 _{System Stability} 

A sufficient condition for a system to be stable is that each & every term of the column of the Routh array must be positive or should have the same sign. Routh array can be obtained as follows.

 The Characteristic equation is of the form,

………

0 0

0 0

0 0 0

Similarly we can evaluate rest of the elements,

 The following are the limitations of Routh-Hurwitz stability criteria, (1) It is valid only if the Characteristic equation is algebraic.

(2) If any co-efficient of the Characteristic equation is complex or contains power of „ ‟, this criterion cannot be applied.

(3) It gives information about how many roots are lying in the RHS of S-plane; values of the roots are not available. Also it cannot distinguish between real & complex roots.

Special cases in Routh-Hurwitz criteria:

(1) When the term in a row is zero, but all other terms are non-zeroes then substitute a small positive number for zero & proceed to evaluate the rest of the elements. When the column term is zero, it means that there is an imaginary root.

(2) All zero row: In the case, write auxiliary equation from preceding row, differentiate this equation & substitute all zero row by the co-efficient‟ obtained by differentiating the auxiliary equation. This case occurs when the roots are in pairs. The system is limitedly stable.

Problems:

C OMMENT ON THE STABILITY OF THE SYSTEM WHOSE CHARACTERISTIC EQUATION IS GIVEN BELOW : (1) 1 21 20 6 36 0 15 20 0 28 0 0 20 0 0 Where,

 The no. of sign changes in the column = zero. No roots are lying in the RHS of S-plane.

 The given System is Absolutely Stable.

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Control Systems 

(10) The open-loop TF of a unity feed back system is given by the above expression. Find the value of ‘K’ for which the system is just stable.

Solution:

 The characteristic equation is

1 23 2K

9 (15+K) 0

2K 0

0 0

2K 0 0

When the value of „K‟ is 61.68 the systemis just stable.

(11) Using Routh-Hurwitz criteria, find out the range of ‘K’ for which the system is stable. The characteristic equation is Solution: 1 (2K+3) 5K 10 0 10 0  The range of K is „ ‟

(12) A proposed control system has a system & a controller as shown. Access the stability of the system by a suitable method. What are the ranges of ‘K’ for the system to be stable?

Solution: The characteristic equation is

16 (1+K) 8 K 0 K 0 (13) (i) K > 0 (ii) 192 – K > 0 K < 192 (iii) (192 – K)(15+K) – 162K > 0

(for the max. value of K)

From this evaluate for K,

Using,

Considering the positive value of „K ‟,

So, 0 < K < 61.68

(i) K > 0 (ii)

Considering the positive value of „K ‟,

(i) K > 0

(ii)

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Control Systems 

(17) 1 4 6 2 5 2 1.5 5 0 -1.666 2 0 6.8 0 0 2 0 0 (18) Solution: 1 11 6 6 10 0 6 0 (19) Solution:

No. of sign changes = 1  The system is Unstable.

(20) +ve 1 2 4 +ve 1 2 1 +ve _{3 0} -ve 1 0 +ve 0 0 +ve 1 0 0 (21) +ve 2 6 1 +ve 1 3 1 +ve -1 0 +ve 1 0 -ve 0 0 +ve 1 0 0 (22) 1 -5 2 -6 -2 0 -6 0

(i) No. of sign changes = 2. (ii)  Two roots lie on RHS of

S-plane.

(iii)  The system is Unstable.

No sign changes.

 The system is Absolutely Stable.

No. of sign changes = 2.  The system is Unstable.

No. of sign changes = 2.  The system is Unstable.

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Control Systems 

(36) The open-loop transfer function of a unity feed back control system is given by,

, using Routh-Hurwitz criteria. Discuss the stability of the closed loop-control system. Determine the value of ‘K’ which will cause sustained oscillations in the closed loop system. What are the corresponding oscillating frequencies?

Solution:

 The characteristic equation is

1 69 12 198 0 52.5 0 0 0 0 0 Hence,

(37) A feed back system has open-loop transfer function Determine the maximum value of ‘K’ for stability of the closed-loop system.

Solution:

Generally control systems have very low Band width which implies that it has very low frequency range of operations. Hence for low frequency ranges, the term can be replaced by . i.e.,

 The characteristic equation is ,

1 5 K 0 K 0 (i) (ii)

(iii)  The Auxiliary equation for the row is

When

(i)

(ii)

 The range of K is for the system to be stable.

1

Control Systems 

Designed By: HPK Kumar 

([email protected])

Root Locus Plots

By:

CIT, Gubbi.

 It gives complete dynamic response of the system. It provides a measure of sensitivity of roots to the variation in the parameter being considered. It is applied for single as well as multiple loop system. It can be defined as  follows,

 It is the plot of the loci of the root of the complementary equation when one or more parameters of the open-loop Transfer function are varied, mostly the only one variable available is the gain ‘K’ The negative gain has no physical significance hence varying ‘K’ from ‘0’ to ‘ ∞’ , the plot is obtained called the “Root Locus Point”.

Rules for the Construction of Root Locus

(1) The root locus is symmetrical about the real axis.

(2) The no. of branches terminating on ‘∞’ equals the no. of open-loop pole-zeroes.

(3)  Each branch of the root locus originates from an open-loop pole at ‘K = 0’ & terminates at open-loop zero corresponding to ‘K = ∞’.

(4)  A point on the real axis lies on the locus, if the no. of open-loop poles & zeroes on the real axis to the right of this  point is odd.

(5) The root locus branches that tend to ‘∞’, do so along the straight line.  Asymptotes making angle with the real axis is given by    180

0



 ,

Where, n=1,3,5,………

P = No. of poles & Z =No. of zeroes.

(6) The asymptotes cross the real axis at a point known as Centroid. i.e.,   ∑∑



(7) The break away or the break in points [Saddle points] of the root locus or determined from the roots of the equation_  0.

(8) The intersection of the root locus branches with the imaginary axis can be determined by the use of

Routh- Hurwitz criteria or by putting ‘  ’ in the characteristic equation & equating the real part and imaginary to  zero. To solve for ‘’ & ‘K’ i.e., the value of ‘’ is intersection point on the imaginary axis & ‘K’ is the value of

gain at the intersection point.

(9) The angle of departure from a complex open-loop pole(_) is given by, _  180 

              

Sheshadri G S

1

Control Systems 

1

Control Systems 

 Bode Plots

By:

CIT, Gubbi.

Sinusoidal transfer function is commonly represented by Bode Plot. It is a plot of magnitude against frequency. i.e., angle of transfer function against frequency.

 The following are the advantages of Bode Plot,

(1) Plotting of Bode Plot is relatively easier as compared to other methods.

(2) Low & High frequency characteristics can be represented on a single diagram.

(3) Study of relative stability is easier as parameters of analysis of relative stability are gain & phase margin which are visibly seen on sketch.

(4) If modification of an existing system is to be studied, it can be easily done on a Bode Plot.

Initial Magnitude: If

,

,

,

,

,

,

,

Phase Plot: Magnitude Plot:

GCF +ve PM GCF GCF GCF -ve PM line line PCF PCF PCF PCF +ve GM -ve GM 0 dB line 0 dB line Sheshadri G S

5

Control Systems 

Examination Problem (Mar/Apr’ 99):

(7) The sketch given shows the Bode Magnitude plot for a system. Obtain the Transfer function.

Solution: Since the initial slope is there must be zero at the origin. &

Examination Problem (Sep/Oct’ 99):

(8) Estimate the Transfer function for the Bode Magnitude plot shown in figure.

Solution: dB A B 40 (Z) C D E (P) (P) (DZ) dB (Z)