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Automatic Control Systems

Lecture-5Block Diagram Representation of Control Systems

Introduction• A Block Diagram is a shorthand pictorial representation of

the cause-and-effect relationship of a system.

• The interior of the rectangle representing the block usuallycontains a description of or the name of the element, gain,or the symbol for the mathematical operation to beperformed on the input to yield the output.

• The arrows represent the direction of information or signal flow.

dt

dx y

Introduction• The operations of addition and subtraction have a special

representation.

• The block becomes a small circle, called a summing point, withthe appropriate plus or minus sign associated with the arrowsentering the circle.

• The output is the algebraic sum of the inputs.

• Any number of inputs may enter a summing point.

• Some books put a cross in the circle.

Introduction

• In order to have the same signal or variable be an inputto more than one block or summing point, a takeoff (orpickoff) point is used.

• This permits the signal to proceed unaltered alongseveral different paths to several destinations.

Example-1• Consider the following equations in which 𝑥1 , 𝑥2 , 𝑥3 , are

variables, and 𝑎1, 𝑎2 are general coefficients or mathematicaloperators.

522113 xaxax

Example-1

522113 xaxax

Example-2• Draw the Block Diagrams of the following equations.

11

2

22

13

11

12

32

11

bxdt

dx

dt

xdax

dtxbdt

dxax

)(

)(

Canonical Form of A Feedback Control System

Characteristic Equation

• The control ratio is the closed loop transfer function of the system.

• The denominator of closed loop transfer function determines thecharacteristic equation of the system.

• Which is usually determined as:

)()(

)(

)(

)(

sHsG

sG

sR

sC

1

01 )()( sHsG

Example-31. Open loop transfer function

2. Feed Forward Transfer function

3. control ratio

4. feedback ratio

5. error ratio

6. closed loop transfer function

7. characteristic equation

8. Open loop poles and zeros if 9. closed loop poles and zeros if K=10.

)()()(

)(sHsG

sE

sB

)()(

)(sG

sE

sC

)()(

)(

)(

)(

sHsG

sG

sR

sC

1

)()(

)()(

)(

)(

sHsG

sHsG

sR

sB

1

)()()(

)(

sHsGsR

sE

1

1

)()(

)(

)(

)(

sHsG

sG

sR

sC

1

01 )()( sHsG

)(sG

)(sH

Reduction techniques

2G1G21GG

1. Combining blocks in cascade

1G

2G21 GG

2. Combining blocks in parallel

3. Eliminating a feedback loop

G

HGH

G

1

G

1H

G

G

1

Example-4: Reduce the Block Diagram to Canonical Form.

Example-4: Continue.

Example-5• For the system represented by the following block diagram

determine:1. Open loop transfer function


3. control ratio

4. feedback ratio

5. error ratio



8. closed loop poles and zeros if K=10.

Example-5– First we will reduce the given block diagram to canonical form

1s

K

Example-5

1s

K

ss

Ks

K

GH

G

11

1

1

Example-5 (see example-3)1. Open loop transfer function


3. control ratio

4. feedback ratio

5. error ratio




)()()(

)(sHsG

sE

sB

)()(

)(sG

sE

sC

)()(

)(

)(

)(

sHsG

sG

sR

sC

1

)()(

)()(

)(

)(

sHsG

sHsG

sR

sB

1

)()()(

)(

sHsGsR

sE

1

1

)()(

)(

)(

)(

sHsG

sG

sR

sC

1

01 )()( sHsG

)(sG

)(sH

Example-6• For the system represented by the following block diagram

determine:1. Open loop transfer function


3. control ratio

4. feedback ratio

5. error ratio




Reduction techniques

4. Moving a summing point behind a block

G G

G

G G

G

1

5. Moving a summing point ahead a block

7. Moving a pickoff point ahead of a block

G G

G G

G

1

G

6. Moving a pickoff point behind a block

8. Swap with two neighboring summing points

A B AB

Example-7

R

_+

_

+1G 2G 3G

1H

2H

++

C

• Reduce the following block diagram to canonical form.

Example-7

R

_+

_

+1G 2G 3G

1H

1

2

G

H

++

C

Example-7

R

_+

_

+21GG 3G

1H

1

2

G

H

++

C

R

_+

_

+121

21

1 HGG

GG

3G

1

2

G

H

C

Example-7

R

_+

_

+121

321

1 HGG

GGG

1

2

G

H

C

Example-7

R

_+232121

321

1 HGGHGG

GGG

C

Example-7

Example 8

Find the transfer function of the following block diagram

2G 3G1G

4G

1H

2H

)(sY)(sR

1. Moving pickoff point A ahead of block2G

2. Eliminate loop I & simplify

324 GGG B

1G

2H

)(sY4G

2G

1H

AB3G

2G

)(sR

I

Solution:

3. Moving pickoff point B behind block324 GGG

1GB

)(sR

21GH 2H

)(sY

)/(1 324 GGG

II

1GB

)(sRC

324 GGG

2H

)(sY

21GH

4G

2G A3G 324 GGG

4. Eliminate loop III

)(sR

)(1

)(

3242121

3241

GGGHHGG

GGGG

)(sY

)()(

)(

)(

)(

32413242121

3241

1 GGGGGGGHHGG

GGGG

sR

sY

)(sR

1GC

324

12

GGG

HG

)(sY324 GGG

2H

C

)(1 3242

324

GGGH

GGG

2G4G1G

4H

2H

3H

)(sY)(sR

3G

1H

Example 9

Find the transfer function of the following block diagrams

Solution:

2G4G1G

4H

)(sY

3G

1H

2H

)(sR

A B

3H4

1

G

4

1

G

I1. Moving pickoff point A behind block

4G

4

3

G

H

4

2

G

H

2. Eliminate loop I and Simplify

II

III

443

432

1 HGG

GGG

1G

)(sY

1H

B

4

2

G

H

)(sR

4

3

G

H

II

332443

432

1 HGGHGG

GGG

III

4

142

G

HGH

Not feedbackfeedback

)(sR )(sY

4

142

G

HGH

332443

4321

1 HGGHGG

GGGG

3. Eliminate loop II & IIII

143212321443332

4321

1 HGGGGHGGGHGGHGG

GGGG

sR

sY

)(

)(

Example-10: Reduce the Block Diagram.

Example-11: Simplify the block diagram then obtain the close-loop transfer function C(S)/R(S). (from Ogata: Page-47)

Superposition of Multiple Inputs

Example-12: Multiple Input System. Determine the output C due to inputs R and U using the Superposition Method.

Example-13: Multiple-Input System. Determine the output C due to inputs R, U1 and U2 using the Superposition Method.

Example-14: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2.


When R1 = 0,

When R2 = 0,

Block Diagram of Armature Controlled D.C Motor

Va

iaT

Ra La

J

c

eb

(s)IK(s)cJs

(s)V(s)K(s)IRsL

am

abaaa


(s)V(s)K(s)IRsL abaaa


(s)IK(s)cJs ama

amplifier Motor Gearing Valve

Actuator

Water

container

Processcontroller

Float

measurement (Sensor)

Error

Feedback signal

resistance comparator

Desired

water level

Input

Actual

water level

Output

Fig. 1.8

For the Fig.1.1, The water level control system:

Figure 1.1

The DC-Motor control system

Desi red

rot at e speed nRegul at or Tr i gger Rect i f i er DC

mot or

Techomet er

Act uat or

Processcont rol l er

measurement ( Sensor )

comparat or

Act ual

rot at e speed n

Error

Feedback si gnal

Ref erence

i nput ur

Out put n

Fi g. 1. 9

auk

ua

uf

e

Fundamental structure of control systems

1) Open loop control systems

Cont rol l er Act uat or Process

Di st urbance

( Noi se)

I nput r ( t )

Ref erence

desi red out put

Out put c( t )

( act ual out put )

Cont rol

si gnal

Act uat i ng

si gnal

uk

uact

Fi g. 1. 10

Features: Only there is a forward action from the input to the output.

2) Closed loop (feedback) control systems

Cont rol l er Act uat or Process

Di st urbance

( Noi se)

I nput r ( t )

Ref erence

desi red out put

Out put c( t )

( act ual out put )

Cont rol

si gnal

Act uat i ng

si gnal

uk

uact

Fi g. 1. 11

measurementFeedback si gnal b( t )

+-

( +)

e( t ) =

r ( t ) - b( t )

Features:

1) measuring the output (controlled variable) . 2) Feedback.

not only there is a forward action , also a backward action between the output and the input (measuring the output and comparing it with the input).

Chapter 2 Mathematical models of systems2.1 Introduction

Controller Actuator Process

Disturbance

Input r(t)

desired outputtemperature

Output T(t)

actual

outputtemperature

Control

signal

Actuating

signal

uk uac

Fig. 2.1

temperaturemeasurement

Feedback signal b(t)

+

-(－)

e(t)=

r(t)-b(t)

1) Easy to discuss the full possible types of the control systems—in terms of the system’s “mathematical characteristics”.

2) The basis — analyzing or designing the control systems.

For example, we design a temperature Control system :

The key — designing the controller → how produce uk.

2.1.1 Why？

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