2.Mathematical Foundation

2.1 The transfer function concept

From the mathematical standpoint, algebraic and differential or difference equations can be used to describe the dynamic behavior of a system .In systems theory, the block diagram is often used to portray system of all types .For linear systems, transfer functions and signal flow graphs are valuable tools for analysis as well as for design

If the input-output relationship of the linear system of Fig.1-2-1 is known, the characteristics of the system itself are also known.

The transfer function of a system is the ratio of the transformed output to the transformed input.

systeminput output

a

TF(s)input output

b

)(

)(

)(

)()(

sr

sc

sinput

soutputsTF

Finger 1-2-1 input-output relationships (a) general (b) transfer function

(2-1)

Summarizing over the properties of a function we state:

1.A transfer function is defined only for a linear system, and strictly, only for time-invariant system.

2.A transfer function between an input variable and output variable of a system is defined as the ratio of the Lap lace transform of the output to the input.

3.All initial conditions of the system are assumed to zero.

4.A transfer function is independent of input excitation.

2.2 The block diagram. Figure 2-3-1 shows the block diagram of a linear feedback control

system. The following terminology often used in control systems is defined with preference to the block diagram.

R(s), r (t)=reference input.

C(s), c (t)=output signal (controlled variable).

B(s), b (t)=feedback signal.

E(s), e (t)=R(s)-C(s)=error signal.

G(s)=C(s)/c(s)=open-loop transfer function or forward-path transfer function.

M(s)=C(s)/R(s)=closed-loop transfer function

H(s)=feedback-path transfer function.

G(s)H(s)=loop transfer function.

G(s)

H(s)

Fig2-2-1

The closed –loop transfer function can be expressed as a function of G(s) and H(s). From Fig.2-2-1we write:

C(s)=G(s)c(s) (2-2)

B(s)=H(s)C(s) (2-3)

The actuating signal is written

C(s)=R(s)-B(s) (2-4)

Substituting Eq(2-4)into Eq(2-2)yields

C(s)=G(s)R(s)-G(s)B(s) (2-5)

Substituting Eq(2-3)into Eq(2-5)gives

C(s)=G(s)R(s)-G(s)H(s)C(s) (2-6)

Solving C(s) from the last equation ,the closed-loop transfer function of the system is given by M(s)=C(s)/R(s)=G(s)/(1+G(s)H(s)) (2-7)

2.3 Signal flow graphs

Fundamental of signal flow graphs

A simple signal flow graph can be used to represent an algebraic relation

It is the relationship between node i to node with the transmission function A, (it is also represented by a branch).

jiji XAX (2-8)

2.3.1 Definitions

Let us see the signal flow graphs

Definition 1: A path is a Continuous, Unidirectional

Succession of branches along which no node is passed

more than once. For example, to to to ,

and back to and to to are paths.

1X 2X3X

4X32 , XX

2X 1X 2X

4X

Definition 2: An Input Node Or Source is a node with only

outgoing branches. For example, is an input node. 1X

Definition 3: An Output Node Or Sink is a node with only

incoming branches. For example, is an output node.

4X

Definition 4: A Forward Path is a path from the input node to

the output node. For example, to to to and

to to are forward paths.

Definition 5: A Feedback Path or feedback loop is a path

which originates and terminates on the same node. For

example, to , and back to is a feedback path.

Definition 6: A Self-Loop is a feedback loop consisting of a

single branch. For example, is a self-loop.

1X 2X3X 4X

1X 2X4X

2X3X 2X

33A

Definition 7: The Gain of a branch is the transmission

function of that branch when the transmission function is

a multiplicative operator. For example, is the gain of

the self-loop if is a constant or transfer function.

Definition 8: The Path Gain is the product of the branch

gains encountered in traversing a

path. For example, the path gain of the forward path from,

to to to is  

Definition 9: The Loop Gain is the product of the branch

gains of the loop. For example, the loop gain of the

feedback loop from to and back is

33A

33A

1X2X 3X 4X

.2332 AA2X 2X3X

2.4 Construction of signal flow graphs

A signal flow graph is a graphical representation of a set of algebraic relationship, and it is a directed graph. The arrow represents the relationship between variables. In general, a variable can be represented by a node.

  Example: A typical feedback system. (In this case, a

dummy node and a branch are added because the output node C has all outgoing branch).

Example: Consider the following resistor network. There are five variables, .,,,, 21321 iiVVV

We can write 4 linear equations:

243

32

22

2

2332

21

11

1

11

11

iRv

vR

vR

i

iRiRv

vR

vR

i

i

Let as input node, the output node can be found as follows:

1v3v

 2.5 General input-output gain transferfunction Let denote the ration between the input and the

output. For the signal flow diagram representation, it becomes

  Definition 10: Non-touching two loops, paths, or

loop and path are said to be non-touching if they have no nodes in common.

R

CT

l

n

X

XT

iP

jkP

j

jkk

k P1)1(1

j

jj j

jj PPP 3211

= the ith forward path gain

= jth possible product of k non-touching loop gains

Definition 11: Signal Flow Graph Determinant (or characteristic function); is defined as follows:

1-(sum of all loop gains) + (sum of all gain-products of 2 non-touching loops) -(sum of all gain-products of 3 non-touching loops) +…

 i

iiiP

R

CT

The general formula for any signal flow graph is

= evaluated with all loops touching Pi eliminated.

Example:

032

1

PP

GP

,1,1,011

kjP

GHP

jk

There is only one forward path; hence

There is only one (feedback) loop. Hence

GHP 11 11

1011

GH

GP

R

CT

1

11

then

and

final

Example:

The signal flow graph of the resistance network,

determine the voltage gain

There is one forward path 1

3

v

vT

Hence the forward path gain is

21

431 RR

RRP

There are three feedback loops:

Hence the loop gains are

,1

311 R

RP ,

2

321 R

RP

2

431 R

RP

12P

21

433111 RR

RRPP

12312111 )(1 PPPP

21

43

2

4

2

3

1

31RR

RR

R

R

R

R

R

R

21

4332413121

RR

RRRRRRRRRR

There are no three loops that do not touch. Therefore

There are two non-touching loops, loops one and three. Hence

Gain-Product of the only two non-touching

loops =

 

Since all loops touch the forward path, Finally, 11

4332413121

4311

1

3

RRRRRRRRRR

RRP

v

vT

Problems

1.A control system has the block diagram of Fig1.

(a)Find the system transfer function c/r

(b) Redraw the block diagram to show the control force u as the output and find the transfer function u/r

(c) Redraw the diagram with the actuating signal εas the output and find the transfer function ε/r.

ccG pG

H

-

+r

Fig 1

u