lec-45 control system engineering

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Block Diagram fundamentals &

reduction techniquesLect# 4-5



Introduction

• Block diagram is a shorthand, graphicalrepresentation of a physical system, illustratingthe functional relationships among itscomponents.

OR

• Block !iagram is a shorthand pictorialrepresentation of the cause-and-effectrelationship of a system.



Introduction

• "he simplest form of the lock diagram is the single block,with one input and one output.

• "he interior of the rectangle representing the lock usually

contains a description of or the name of the element, or thesymol for the mathematical operation to e performed onthe input to yield the output.

• "he arro$s represent the direction of information or signal

flo$.

dt

d x y



Introduction• "he operations of addition and sutraction ha%e a special

representation.• "he lock ecomes a small circle, called a summing point,

$ith the appropriate plus or minus sign associated $ith thearro$s entering the circle.

• ny numer of inputs may enter a summing point.

• "he output is the algeraic sum of the inputs.

•&ome ooks put a cross in the circle.



Components of a Block Diagram for

a Linear Time Invariant System• &ystem components are alternati%ely called

elements of the system.

• Block diagram has four components'( Signals

( System/ block

( Summing junction

( Pick-off/ Take-off point



• )n order to ha%e the same signal or %ariale e an input to

more than one lock or summing point, a takeoff point isused.

• !istriutes the input signal, undiminished, to se%eraloutput points.

• "his permits the signal to proceed unaltered along se%eraldifferent paths to se%eral destinations.



Eample!"

• *onsider the follo$ing e+uations in $hich -, ., /, are

%ariales, and a1, a2 are general coefficients or

mathematical operators.

522113 −+= xa xa x



Eample!"

• *onsider the follo$ing e+uations in $hich -, ., /, are

%ariales, and a1, a2 are general coefficients or


522113

−+= xa xa x



Eample!#

• *onsider the follo$ing e+uations in $hich -, .,. . . , n, are %ariales, and a1, a2,. . . , an , are general coefficients or


112211 −−++=

nnn xa xa xa x



Eample!$

• !ra$ the Block !iagrams of the follo$ing e+uations.

1

1

2

2

2

13

1

1

12

32

11

bxdt

dx

dt

xd a x

dt x

bdt

dxa x

−+=

∫ +=

)(

)(



Topologies

• 0e $ill no$ eamine some common topologiesfor interconnecting susystems and deri%e thesingle transfer function representation for eachof them.

• "hese common topologies $ill form the asis for

reducing more complicated systems to a single lock.



CASCADE

• ny finite numer of locks in series may ealgeraically comined y multiplication oftransfer functions.

• "hat is, n components or blocks with transfer functions G 1 , G 2, . . . , G n, connected in cascade

are e+ui%alent to a single element 1 $ith atransfer function gi%en y



Eample

• 2ultiplication of transfer functions iscommutative; that is,

GiGj = GjGi for any i or j .



Cascade%

Figure:a) Cascaded Subsystems.

b) Equivalent Transfer Function.

The equivalent transfer function

is



arallel 'orm%

• 3arallel susystems ha%e a common input and anoutput formed y the algeraic sum of the outputsfrom all of the susystems.

Figure: Parallel Subsystems.



arallel 'orm%

Figure:

a) Parallel Subsystems.

b) Equivalent Transfer Function.

The equivalent transfer function is



'eed(ack 'orm%• "he third topology is the feedack form. Let us deri%e the

transfer function that represents the system from its inputto its output. "he typical feedack system, sho$n in figure'

Figure: Feedback Closed !oo") Control System.

The system is said to have negative feedback if the sign at the

summing junction is negative and positive feedback if the signis positive.



'eed(ack 'orm%

Figure:

a)Feedback Control System.

b)Sim"lified #odel or Canonical Form.

c) Equivalent Transfer Function.

The equivalent or closed$loo"

transfer function is



Characteristic Equation% The control ratio is the closed loo" transfer function of the

system.

% The denominator of closed loo" transfer function determines the

characteristic equation of the system.

% &hich is usually determined as:

)()(

)(

)(

)(

s H sG

sG

s R

sC

±=1

01 =± )()( s H sG



(. "en loo" transfer function

*. Feed For+ard Transfer function

,. control ratio

-. feedback ratio

. error ratio

/. closed loo" transfer function

0. characteristic equation

1. closed loo" "oles and 2eros if 34(5.

)()()(

)( s H sG

s E

s B=

)()(

)( sG

s E

sC =

)()(

)(

)(

)(

s H sG

sG

s R

sC

+=1

)()()()(

)()(

s H sG s H sG

s R s B

+=1

)()()(

)(

s H sG s R

s E

+=1

1

)()()(

)()(

s H sG sG

s R sC

+=1

01 =+ )()( s H sG

)( sG

)( s H



Characteristic Equation



)nity 'eed(ack System



*eduction techniques

2G1G 21GG

(. Combining blocks in cascade

1G

2G

21 GG +

*. Combining blocks in "arallel




,. #oving a summing "oint behind a block

G G

G



. #oving a "ickoff "oint ahead of a block

G G

G G

G

1

G

,. #oving a summing "oint ahead of a block

G G

G

1

-. #oving a "ickoff "oint behind a block




/. Eliminating a feedback loo"

G

H GH

G

1

0. S+a" +ith t+o neighboring summing "oints

A B A B

G

1= H

G

G

1




Block Diagram Transformation Theorems

The letter P is used to represent any transfer function, and W, X ,

Y, Z denote any transformed signals.



*eduction of Complicated Block Diagrams%



Eample!+% Continue,

6o+ever in this e7am"le ste"$- does not a""ly.

6o+ever in this e7am"le ste"$/ does not a""ly.



Eample!-% Simplify the Block Diagram,



Eample!-% Continue,



Eample!.% *educe the Block Diagram,



Eample!.% Continue,



Eample!/% *educe the Block Diagram, 0from 1ise% page!

#+#2



Eample!/% Continue,



Eample!3% 'or the system represented (y thefollo4ing (lock diagram determine%

. Open loop transfer function

. eed or$ard "ransfer function

/. control ratio

4. feedack ratio

5. error ratio

. closed loop transfer function

6. characteristic e+uation

7. closed loop poles and 8eros if 9:;.



Eample!3% Continue

( irst $e $ill reduce the gi%en lock diagram to canonical

form

1+ s

K



Eample!3% Continue

1+ s

K

s s

K

s

K

GH

G

1

1

1

1

+

+

+=

+

Eample 3% Continue



Eample!3% Continue

(. "en loo" transfer function

*. Feed For+ard Transfer function

,. control ratio

-. feedback ratio

. error ratio

/. closed loo" transfer function

0. characteristic equation

1. closed loo" "oles and 2eros if 34(5.

)()()(

)( s H sG

s E

s B=

)()(

)( sG s E

sC =

)()(

)(

)(

)(

s H sG

sG

s R

sC

+=1

)()(

)()(

)(

)(

s H sG

s H sG

s R

s B

+= 1

)()()(

)(

s H sG s R

s E

+=1

1

)()(

)(

)(

)(

s H sG

sG

s R

sC

+= 1

01 =+ )()( s H sG

)( sG

)( s H



• Eample!5% 'or the system represented (y the follo4ing(lock diagram determine%

. Open loop transfer function. eed or$ard "ransfer function

/. control ratio

4. feedack ratio

5. error ratio

. closed loop transfer function6. characteristic e+uation

7. closed loop poles and 8eros if 9:;;.



Eample!"6% *educe the system to a single transfer

function, 0from 1ise%page!#+$2,



Eample!"6% Continue,



Eample!""% Simplify the (lock diagram then o(tain the

close!loop transfer function C0S27*0S2, 0from 8gata%

age!+/2



Eample!"#% *educe the Block Diagram,

R

8 9

8

91G 2G 3G

1 H

2 H

99

C



Eample!"#%

R

8 9

8

9 1G 2G 3G

1 H

1

2

G H

99

C



Eample!"#%

R

8 9

8

9 21GG 3G

1 H

1

2

G H

99

C



Eample!"#%

R

8 9

8

9121

21

1 H GG

GG

− 3G

1

2

G H

C



Eample!"#%

R

8 9

8

9121

321

1 H GG

GGG

−

1

2

G H

C

l #



Eample!"#%

R

8 9

232121

321

1 H GG H GG

GGG

+−

C



Eample!"#%

R

321232121

321

1 GGG H GG H GG

GGG

++−

C



2G1G

1 H 2 H

)( s R )( sY

3 H

Eample "$% 'ind the transfer function of the follo4ing

(lock diagrams,



Solution:

(. Eliminate loo"

*. #oving "ickoff "oint ; behind block22

2

1 H G

G

+

1G

1 H

)( s R )( sY

3 H

B A

22

2

1 H G

G

+

2

221

G

H G+

1G

1 H

)( s R )( sY

3 H

2G

2 H

B A

II

I

22

2

1 H G

G

+

<ot a feedback loo"

)1(

2

2213

G

H G H H +

+



,. Eliminate loo"

)( s R )( sY

22

21

1 H G

GG

+

2

2213

)1(

G

H G H H

++

21211132122

21

1 H H GG H G H GG H G

GG

s R

sY

++++

=

)(

)(



Superposition of 9ultiple Inputs



Eample!"+% 9ultiple Input System, Determine the

output C due to inputs R and U using the Superposition9ethod,



Eample!"+% Continue,



Eample!"-% 9ultiple!Input System, Determine the

output C due to inputs R: U " and U # using the

Superposition 9ethod,



Eample!"-% Continue,



Eample!".% 9ulti!Input 9ulti!8utput System, Determine

C" and C# due to R" and R#,



Eample!".% Continue,



Skill ;ssessment Eercise%



;ns4er of Skill ;ssessment Eercise%

lec-45 control system engineering

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