Modeling & Simulation of Dynamic Systems Dr. Imtiaz Hussain email: [email protected]@faculty.muet.edu.pk URL :

Modeling & Simulation of Dynamic Systems

Dr. Imtiaz Hussainemail: [email protected]

URL :http://imtiazhussainkalwar.weebly.com/

Lecture-2Block Diagram & Signal Flow Graph 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 usually contains a description of or the name of the element, or the symbol for the mathematical operation to be performed 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, with the appropriate plus or minus sign associated with the arrows entering 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 input

to more than one block or summing point, a takeoff point is used.

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

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

variables, and a1, a2,. . . , an , are general coefficients or mathematical operators.

112211 nnn xaxaxax

Exercise-1• Draw the Block Diagrams of the following equations.

11

22

2

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 the characteristic equation of the system.

• Which is usually determined as:

)()()(

)()(

sHsG

sG

sR

sC

1

01 )()( sHsG

Example-21. 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. closed loop poles and zeros if K=10.

)()()()(

sHsGsE

sB

)()()(

sGsE

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

2G1G 21GG

1. Combining blocks in cascade

1G

2G21 GG

2. Combining blocks in parallel

Reduction techniques

3. Moving a summing point behind a block

G G

G

5. Moving a pickoff point ahead of a block

G G

G G

G

1

G

3. Moving a summing point ahead of a block

G G

G

1

4. Moving a pickoff point behind a block

6. Eliminating a feedback loop

G

HGH

G

1

7. Swap with two neighboring summing points

A B AB

G

1H

G

G

1

Example-3• For the system represented by the following block diagram

determine:1. Open loop transfer function2. Feed Forward Transfer function3. control ratio4. feedback ratio5. error ratio6. closed loop transfer function7. characteristic equation 8. closed loop poles and zeros if K=10.

Example-3– First we will reduce the given block diagram to canonical form

1sK

Example-3

1sK

ss

Ks

K

GH

G

11

11

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. closed loop poles and zeros if K=10.

)()()()(

sHsGsE

sB

)()()(

sGsE

sC

)()()(

)()(

sHsG

sG

sR

sC

1

)()()()(

)()(

sHsG

sHsG

sR

sB

1

)()()()(

sHsGsR

sE

1

1

)()()(

)()(

sHsG

sG

sR

sC

1

01 )()( sHsG

)(sG

)(sH

Exercise-2• For the system represented by the following block diagram

determine:1. Open loop transfer function2. Feed Forward Transfer function3. control ratio4. feedback ratio5. error ratio6. closed loop transfer function7. characteristic equation 8. closed loop poles and zeros if K=100.

Example-4

R_+

_+

1G 2G 3G

1H

2H

+ +

C

Example-4

R_+

_+

1G 2G 3G

1H

1

2

G

H

+ +

C

Example-4

R_+

_+

21GG 3G

1H

1

2

G

H

+ +

C

Example-4

R_+

_+

21GG 3G

1H

1

2

G

H

+ +

C

Example-4

R_+

_+

121

21

1 HGG

GG

3G

1

2

G

H

C

Example-4

R_+

_+

121

321

1 HGG

GGG

1

2

G

H

C

Example-4

R_+

232121

321

1 HGGHGG

GGG

C

2G 4G1G

4H

2H

3H

)(sY)(sR

3G

1H

Example-5

Find the transfer function of the following block diagrams

Solution:

2G 4G1G

4H)(sY

3G

1H

2H

)(sRA B

3H4

1

G

4

1

G

I1. Moving pickoff point A behind block

4G

4

3

G

H

4

2

G

H

Example-5

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

Example-5

)(sR )(sY

4

142

G

HGH

332443

4321

1 HGGHGG

GGGG

3. Eliminate loop II & IIII

143212321443332

4321

1 HGGGGHGGGHGGHGG

GGGG

sR

sY

)()(

Example-5

2. Eliminate loop I & Simplify

3G

1H

2GB

3

1

G

2H

32GGB

23

1 HG

H

1G)(sR )(sY

4G

3

1

G

H

23212

32

1 HGGHG

GG

II

Example-5

)(sR )(sY

12123212

321

1 HGGHGGHG

GGG

3. Eliminate loop II

12123212

3214 1)(

)()(

HGGHGGHG

GGGG

sR

sYsT

4G

Example-5

Superposition of Multiple Inputs

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

Example-6

Example-6

Example-6

Exercise-3: Multi-Input Multi-Output System. Determine C1 and C2 due to R1 and R2.

Introduction

37

• Alternative method to block diagram representation, developed by Samuel Jefferson Mason.

• Advantage: the availability of a flow graph gain formula, also called Mason’s gain formula.

• A signal-flow graph consists of a network in which nodes are connected by directed branches.

• It depicts the flow of signals from one point of a system to another and gives the relationships among the signals.

Fundamentals of Signal Flow Graphs

• Consider a simple equation below and draw its signal flow graph:

• The signal flow graph of the equation is shown below;

• Every variable in a signal flow graph is designed by a Node.• Every transmission function in a signal flow graph is designed by a

Branch. • Branches are always unidirectional.• The arrow in the branch denotes the direction of the signal flow.

axy

x ya

Signal-Flow Graph Models

Y1 s( ) G11 s( ) R1 s( ) G12 s( ) R2 s( )

Y2 s( ) G21 s( ) R1 s( ) G22 s( ) R2 s( )

Example-7: R1 and R2 are inputs and Y1 and Y2 are outputs

Signal-Flow Graph Models

a11 x1 a12 x2 r1 x1

a21 x1 a22 x2 r2 x2

Exercise-4: r1 and r2 are inputs and x1 and x2 are outputs

Signal-Flow Graph Models

34

203

312

2101

hxx

gxfxx

exdxx

cxbxaxx

b

x4x3x2x1

x0 h

f

g

e

d

c

a

xo is input and x4 is output

Example-8:

Construct the signal flow graph for the following set of simultaneous equations.

• There are four variables in the equations (i.e., x1,x2,x3,and x4) therefore four nodes are required to construct the signal flow graph.

• Arrange these four nodes from left to right and connect them with the associated branches.

• Another way to arrange this graph is shown in the figure.

Terminologies

• An input node or source contain only the outgoing branches. i.e., X1

• An output node or sink contain only the incoming branches. i.e., X4

• A path is a continuous, unidirectional succession of branches along which no

node is passed more than ones. i.e.,

• A forward path is a path from the input node to the output node. i.e.,

X1 to X2 to X3 to X4 , and X1 to X2 to X4 , are forward paths.

• A feedback path or feedback loop is a path which originates and terminates on

the same node. i.e.; X2 to X3 and back to X2 is a feedback path.

X1 to X2 to X3 to X4 X1 to X2 to X4 X2 to X3 to X4

Terminologies

• A self-loop is a feedback loop consisting of a single branch. i.e.; A33 is a self

loop.

• The gain of a branch is the transmission function of that branch.

• The path gain is the product of branch gains encountered in traversing a path.

i.e. the gain of forwards path X1 to X2 to X3 to X4 is A21A32A43

• The loop gain is the product of the branch gains of the loop. i.e., the loop gain

of the feedback loop from X2 to X3 and back to X2 is A32A23.

• Two loops, paths, or loop and a path are said to be non-touching if they have no nodes in common.

Consider the signal flow graph below and identify the following

a) Input node.b) Output node.c) Forward paths.d) Feedback paths (loops).e) Determine the loop gains of the feedback loops.f) Determine the path gains of the forward paths.g) Non-touching loops

Example-9:

Consider the signal flow graph below and identify the following

• There are two forward path gains;

Example-9:

Consider the signal flow graph below and identify the following

• There are four loops

Example-9:

Consider the signal flow graph below and identify the following

• Nontouching loop gains;

Example-9:

Consider the signal flow graph below and identify the following

a) Input node.b) Output node.c) Forward paths.d) Feedback paths.e) Self loop.f) Determine the loop gains of the feedback loops.g) Determine the path gains of the forward paths.

Example-10:

Input and output Nodes

a) Input node

b) Output node

Example-10:

(c) Forward PathsExample-10:

(d) Feedback Paths or Loops

Example-10:

(d) Feedback Paths or Loops

Example-10:

(d) Feedback Paths or LoopsExample-10:

(d) Feedback Paths or Loops

Example-10:

(e) Self Loop(s)Example-10:

(f) Loop Gains of the Feedback LoopsExample-10:

(g) Path Gains of the Forward PathsExample-10:

Mason’s Rule (Mason, 1953)

• The block diagram reduction technique requires successive

application of fundamental relationships in order to arrive at the

system transfer function.

• On the other hand, Mason’s rule for reducing a signal-flow graph

to a single transfer function requires the application of one

formula.

• The formula was derived by S. J. Mason when he related the

signal-flow graph to the simultaneous equations that can be

written from the graph.

Mason’s Rule:• The transfer function, C(s)/R(s), of a system represented by a signal-flow graph

is;

Where

n = number of forward paths.Pi = the i th forward-path gain.∆ = Determinant of the system∆i = Determinant of the ith forward path

• ∆ is called the signal flow graph determinant or characteristic function. Since ∆=0 is the system characteristic equation.

n

iiiP

sR

sC 1

)()(

Mason’s Rule:

∆ = 1- (sum of all individual loop gains) + (sum of the products of the gains of all possible two loops that do not touch each other) – (sum of the products of the gains of all possible three loops that do not touch each other) + … and so forth with sums of higher number of non-touching loop gains

∆i = value of Δ for the part of the block diagram that does not touch the i-th forward path (Δi = 1 if there are no non-touching loops to the i-th path.)

n

iiiP

sR

sC 1

)()(

Systematic approach

1. Calculate forward path gain Pi for each forward path i.

2. Calculate all loop transfer functions3. Consider non-touching loops 2 at a time4. Consider non-touching loops 3 at a time5. etc6. Calculate Δ from steps 2,3,4 and 57. Calculate Δi as portion of Δ not touching forward

path i

62

Example-11: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph

2211 PP

R

CTherefore,

24313242121411 HGGGLHGGGLHGGL ,,

There are three feedback loops

Example-11: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph

∆ = 1- (sum of all individual loop gains)

There are no non-touching loops, therefore

3211 LLL

243124211411 HGGGHGGGHGG

Example-11: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph

∆1 = 1- (sum of all individual loop gains)+...

Eliminate forward path-1

∆1 = 1

∆2 = 1- (sum of all individual loop gains)+...

Eliminate forward path-2

∆2 = 1

Example-11: Continue

67

Exercise-5: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph

Exercise-6

• Find the transfer function, C(s)/R(s), for the signal-flow graph in figure below.

Example-12: Apply Mason’s Rule to calculate the transfer function of the system represented by following Signal Flow Graph

332211

3

1 PPPP

sR

sC iii

)()(

There are three forward paths, therefore n=3.

Example-12: Forward Paths

722 AP 76655443321 AAAAAP

766554423 AAAAP

Example-12: Loop Gains of the Feedback Loops

23321 AAL

34432 AAL

45543 AAL

56654 AAL

67765 AAL

776 AL

2334427 AAAL

6776658 AAAL

23344557729 AAAAAL

23344556677210 AAAAAAL

Example-12: two non-touching loops

31LL

41LL

51LL

61LL

81LL

42LL

52LL

62LL

82LL

53LL

63LL64LL

74LL

75LL 87LL

Example-12: Three non-touching loops

31LL

41LL

51LL

61LL

81LL

42LL

52LL

62LL

82LL

53LL

63LL64LL

74LL

75LL 87LL

G1 G4G3

From Block Diagram to Signal-Flow Graph Models

－－

－

C(s)R(s)G1 G2

H2

H1

G4G3

H3

E(s) X1

X2

X3

R(s) C(s)

－ H2

－ H1

－ H3

X1 X2 X3E(s)1 G2

Example-13:

1;

)(1

143211

14323234321

GGGGP

HGGHGGHGGGG

14323234321

4321

1)(

)(

HGGHGGHGGGG

GGGG

sR

sCG

R(s)

－ H2

1G4G3G2G11 C(s)

－ H1

－ H3

X1 X2 X3E(s)

From Block Diagram to Signal-Flow Graph ModelsExample-13:

G1

G2

+－

+

－

－

－

+ C(s)R(s) E(s)

Y2

Y1X1

X2

－

Exercise-7

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