自动控制理论甲-第五周课件英文

Principle

of

Automatic

ControlSystem

Representation(Ch.5)浙江大学控制科学与工程学系IntroductionDetermination

of

the

overalltransferfunctionSimulation

diagramsSignal

flow

graphFrom

transfer

function

to

state

spacemodel21Simulation

diagramsSimulation

diagrams(State

variable

diagram)Phase

variablesSimulationdiagramsSimulation

diagrams

(see

P144.

5.6)Simulation

used

to

represent

the

dynamic

equations

of

asystem

may

show

the

actual

physical

variables

that

appear

inthe

system,

or

it

may

show

variables

that

are

used

purely

formathematical

convenience..The

simulation

diagram

is

similar

to

the

block

diagramused

to

represent

the

system

on

an

og

computer.The

basic

elements

used

in

simulation

diagram

are

idealintegrators,

ideal

amplifiers,

and

ideal

summers,

shown

inFig.5.16.Multipliers

and

dividers

may

be

used

for

nonlinearsystems425x2

x1dt1s2

1sX

(

s

)

1

X

(

s

)Integratorx1X1(s)Figure

5.16

Elements

used

in

simulation

diagram21x

x

dtIntegratorx1Amplifier

or

gainx1x2

Kx1Kx4

x1

x2

x3x2x3+-Summerx1

+LTSimulationdiagramsSimulation

diagrams

(seeP144.5.6)Basic

elements:

ideal

integrators,ideal

amplifiers,and

idealsummers.6LC

c

RC

c

vc(t)

edt

2

dt

dv

2

(t)

dv

(t)LCy

RCy

y

ueFig2.2yyyabub+Fig.5.17

b

R

L1L

C

Where:a

b

Let

y=vc(t),

u=ey

bu

ay

bySimulationdiagramsSimulation

diagrams

(see

P144.5.6)3SimulationdiagramsSimulation

diagrams

(seeP144.5.6)One

of

the

methods

used

to

obtain

a

simulationdiagramincludes

the

following

steps:Step

1:

Start

with

thedifferential

equationStep

2:

Rearrange

the

differential

equation

(by

order

ofderivative)Step

3:

Start

the

diagram

by

assuming

the

signal

placed

onthe

leftside

of

the

equation,is

available.

Note

the

number

ofintegratorsare

needed

to

determine

at

.Step

4:

Complete

the

diagram

by

feeding

back

the

appropriateoutputs

of

the

integrators

to

a

summer,

in

order

to

generate

theoriginal

signal

of

step

2.

Include

the

input

function

if

it

isrequired.78Step

1

&

2

:

When

y=vc

,

u=e

are

used,

rearrange

the

Eq.(2.12)

to

the

formy

bu

ay

byWhere:a

R

L1L

C

b

LCy

RCy

y

ueFig

2.2SimulationdiagramsSimulation

diagrams

(see

P144.5.6)Example.

Draw

the

simulation

diagram

for

the

RLC

circuit

ofFig.2.2.LC

c

RC

c

v

c

(

t

)

edt

2

dtdv

2

(

t

)

dv

(

t

)49Step

4

:

Completing

thesimulation

diagram

as

shown

in

Fig.

5.17byyyabub+Fig.5.17bWhere:a

R

L1L

C

b

SimulationdiagramsSimulation

diagrams

(see

P144.5.6)y

bu

ay

byStep

3

:

The

signal

y

is

integrated

twice,

as

shown

in

Fig.

5.17abelow.

bab+yu

yyy

x2y

x1y

x2

x15SimulationdiagramsSimulation

diagrams

(see

P144.

5.6)The

simulation

diagram

is

as

shown

in

Fig.5.17b.The

state

variables

are

selected

as

the

outputs

of

the

eachintegrator

in

the

simulation

diagram,

as

shown

in

the

Fig.So,

the

simulation

diagram

can

express

the

relations

of

statevariables,

which

is

called

state

variable

diagram.1011y

bu

ay

byubab+y

b

a

2

x

b

x1

0 1

x

0uy

x1y

x2y

x1y

x2

x1Simulation

diagramState

variable

diagramWhere:a

R

LL

C

1b

state

variable

diagramstate

space

model12y

x2y

x12

1y

x

xub-

a-

b+yub

ab+y

x22

1y

x

xy

x1

yFor

the

convenience,

two

forms

diagram

are

used.Where:a

R

L1L

C

b

L

LC

R

x

1

u1

0

0

2

LCx

1

x1y

x1Simulation

diagramSimulationdiagramsState

variable

diagram613State-space

Model∫∫x1=y-5-1x2u42SimulationdiagramsState

variable

diagram14Review:

state

space

modelL

R

L L

C

x

1

u

0

1

x

2

1

x

1

0y

x1andich

is

cUp

to

now,

tworepresentationsfor

expressingthe

systemstate

are

known:physical

and

phase.LCy

RCy

y

ueFig2.2

R

x

1

u1

0

0

2

LC L

LC

x

1

x1y

x1Note：same

RLC

circuit

of

Fig.2.5.(P31),

if

state

variables

are

selected

asx

1

v

c

,

wh

x

2

ialled

physical-variable

method

based

upon

theenergy-storage

elements

of

the

system,

the

state

space

model

isSo,

same

systemmay

berepresented

by

differentstatespace

model.The

difference

illustrates

thatstate

variables

are

not

unique.7Teral

block

diagram

representing

the

state

space

model

is

shownas

following.state

space

model)x(t)

Ax

(t)

Bu

(t

)y

(t

)

Cx

(t

DuState

equationOutput

equationFeedforwardmatrixSimulationdiagramsSimulation

diagrams

(see

P144.5.6)Different

sets

of

state

variables

may

be

selected

to

represent

asystem.The

selection

ofthe

state

variables

determines

the

A,

B,

C,

and

Dmatrices

of15SimulationdiagramsPhase

variables

(see

P146.

5.6)When

the

state

variables

are

the

dependent

variableand

the

derivatives

of

the

dependent

variable,

theyare

called

phase

variables.The

phase

variables

are

applicable

to

differentialequations

of

any

order

and

can

be

used

withoutdrawing

the

simulation

diagram.There

are

two

cases

required

to

consider

as

following.16817When

these

state

variables

are

usedTeral

differential

equation

that

contains

no

derivatives

of

input

isy

(

n

)

(

t

)

a

n

1

y

(

n

1)

(

t

)

a

0

y

(

t

)

uThe

state

variables

are

selected

as

the

phase

variables,

which

are.

Thus,defined

by

x1

y

(

t

)

,

x

2

x1

y(t)

,x3

x2

y(t)

,xn

xn1

y

(

n

1)

(

t)y

(

n

)

(

t

)

xn(

D

n

an

1

Dn

1

a

1

D

a

0

)

x

1

uSimulationdiagramsPhase

variables:

Case

1

(no

derivatives

in

the

input)xn

a

n

1

x

n

a

n

2

xn

1

a

1

x

2

a

0

x1

u(5.37)x

A

c

x

bcuy

cxStandard

forms:Note:subscripty

x1

1

0

0

0x1

x

0

x

0

2

x1

0

a

0

n

1

an

1

x

n

n

1

[

u

]

a1

an

2

xn

x

x2

x1

are:Companion

matrix友矩阵，能控标准形(Ac

,bc)xn

a

n

1

x

n

a

n

2

xn

1

a

1

x

2

a

0

x1

u(5.37)x

wx

w

101000010000118SimulationdiagramsPhase

variables:

Case

1

(no

derivatives

in

the

input)The

resulting

state

and

output

equations

using

phase

variables919

(c

w

Dw(

D

n

a0w

1n

11

c

w

1

D

c

D

c

)u1

0D

n

1

a

D

a

)

yY

(

s

)

cs

w

csw

1

c

s

c

w w

1

1

0U

(

s

)s

n

asn

1

a

s

an

1

1

0ux1ywc

s

w

c

s

w1

c

s

cw1

1

01s

n

a

s

n

1

a

s

an

1

1

0串联分解Phase

variables:

Case

2

(there

are

derivatives

in

the

input)P147(5.40-5.43)的解释20x

Ac

x

bc

uStandard

forms:Where

Ac

and

bc

arethe

same

as

Case

1.The

output

equation

is

different

from

Case

1.

it

depends

ondifferent

values

of

w:

w=n

or

w<n.xw

xw

1SimulationdiagramsPhase

variables:

Case

2

(there

are

derivatives

in

the

input)The

differential

equation

has

the

same

form

as

Case

1,that

is(

D

n

an

1

Dn

1

a

1

D

a

0

)

x

1

uux1ywc

s

w

c

s

w1

c

s

cw1

1

01s

n

a

s

n

1

a

s

an

1

1

0串联分解10xn

u

(

a

n

1

x

n

a

n

2

x

n

1

a

1

x

2

a

0

x

1

)y

c

n

xn

c

n

1

x

n

c

1

x

2

c

0

x1

x

n

a

n

1

x

n

a

n

2

x

n

1

a

1

x

2

a

0

x

1

u

(5.37)(1)

For

w=ny

c

w

xw

c

w

1

x

w

c

1

x

2

c

0

x11 1

ny

(c

a

c

)

(c

a

c

)

(c

a c

)

x

c

un

1

n

1

n

n

0 0

n

cx

du(5.44)21SimulationdiagramsPhase

variables:

Case

2

(there

are

derivatives

in

the

input)22) (c1

)In

this

case,

the

output

y

reducestoy

c0

c1

cw

0

0x

cx

xw

xw

1cw

i

0,

i

1

,

2

,

,

n

wandy

(c0

(

cn

1

an

1

cn

)x

[

c

n

]u1

0w

(c

w

D

cw

1w

1n

1D

c

D

c

)u(2)

For

w<n

(

D

n

aD

n

1

a

D

a

)

y1

0SimulationdiagramsPhase

variables:

Case

2

(there

are

derivatives

in

the

input)11

(c

D

w

cw w

1D

w

1

c

D

c

)u1

0(

D

n

a Dn

1

a

D

a

)

yn

1

1

0Differentialequation11

0

0

n

x

x

0

x1

0

n

0

1n

1

n

2

n

1

a

a2

[

u

]0

0

x

a

a

xState

spaceequations)

(c1

c1

cw

0)

(cn1

an1cn

)

uw=n

y

(c0

w<n

y

c0

0x

cxPhase

variablesxw

xw

1

x1

0100

x

2

0010

SimulationdiagramsSummary:

Phase

variablesThe

simulation

diagram

that

represents

the

system

above

is

shownin

P148Fig.5.18(w<n).23SimulationdiagramsSimulationdiagram2412Advantages:Only

two

summers

requiredDifferentiators

for

u(t)

are

avoided.(noise

accentuation)Fig.

5.18

Simulation

Diagram

for

(5.40)

with

phase

variablesSignal

flow

graphFlow-Graph

definitionsFlow-Graph

AlgebraGeneral

Flow-Graph

ysisThe

Mason

Gain

RuleState

transition

signal

flow

graph26??????Question?For

complexsystems,

the

block

diagram

method

maye

difficult

to

complete.

Ex.

as

Fig.

Below.G(s)

C(s)

G1G2G3G4G5G6

R(s) 1

G1G2G3G4G5G6H1

G2G3

H2

G4G5

H3

G3G4

H4

G2G3H2G4G5

H313Signal-Flow

Graphs

(SFG)

(see

P149

5.7)By

using

the

signal-flow

graph

model,

the

reduction(simplification)

procedure

(used

in

the

block

diagrammethod)

is

not

necessaryto

determine

the

relationshipbetween

system

variables.In

other

words,

with

signal-flow

graphmethod

,

itiseasier

to

deal

with

difficult

block

diagram.What

issignal-flow

graph?27AnSFG

isa

diagram

that

represents

a

set

of

simultaneous

equations.G(s)uySignal-Flow

Graph(SFG)

Definitions(see

P1495.7)信号流图是由节点和支路组成的信号传递网络An

SFG

consists

of

a

graph

in

which

nodes

are

connected

by

directed

branches.The

nodes

represent

each

of

systemvariables.A

branch

connected

between

two

nodes

acts

as

a

one-way

singlemultiplier:

the

direction

is

indicated

by

an

arrow

and

the

multiplicationfactor(transfer

function

or

gain)

is

placed

onthebranch.y

G

(

s

)

u28A：SFG只适用于线性系统，而方块图可以适用于非线性系统Q:Block

Diagram和SFG的适用性？1429xabcduvyww

au

bvx

cw

c(au

bv)y

dw

d(au

bv)Signal-Flow

Graph(SFG)

Definitions(see

P1495.7)A

node

performs

two

functions:Addition

of

the

signalson

all

ing

branchesTransmission

of

the

total

node

signal

(the

sum

of

allingsignals)

to

all

out-going

branchesxabcduvywEx.

In

Fig.

above,

the

path

u-w-x

is

a

forward

path

between

the

nodes

uand

x.30A

path,

is

any

connectedsequence

of

branches

whosearrows

are

in

the

samedirection.A

forward

path

between

2

nodesis

one

that

follows

the

arrowsofsuccessive

branches

and

inwhicha

node

appears

only

once.Source

nodesSink

nodesMixed

nodes(independent

nodes)

:

have

only

outgoing

branches.(dependent

nodes)

:

have

only ingbranches.(general

nodes)Signal-Flow

Graph(SFG)

Definitions(see

P1495.7)There

are

3

types

of

nodes:15Signal-Flow

Graph(SFG)

Definitions输入节点(源点）

只有输出支路的节点称为输入节点。它一般表示系统的输入变量。输出节点（阱点）

只有输入支路的节点称为输出节点。它一般表示系统的输出变量。混合节点既有输入支路又有输出支路的节点称为混合节点。它一般表示相加点、分支点。通路

从某一节点开始沿支路箭头方向经过各相连支路到另一节点所构成的路径称为通路。通路中各支路增益的乘积叫做通路增益。前向通路是指从输入节点开始并终止于输出节点且与其它节点相交不多于一次的通路。该通路的各增益乘积称为前向通路增益。31Signal-Flow

Graph

Algebra

(seeP150)Original

graphxa

byzParallel

pathsxab

zEquivalent

graph

PathgainuyG1(s)G2

(s)uG1(s)+

G2

(s)ySeries

paths(cascadenodes)Original

graphxbyzaucEquivalent

graphu

aczbcxNode

absorption16Signal-Flow

Graph

Algebra

(see

P151)33feedback

loopC

GEB

HCE

R

BC

GR

GHCC

G1

GHRGeneral

Flow-Graph

ysisGenerally,

the

SFG

for

an

arbitrarily

complex

systemcan

be

represented

by

Fig.5.25a

in

P152.Note

that

all

the

source

nodes

are

brought

to

the

left,and

all

the

sink

nodes

are

brought

to

the

right.x1x2111y1y2y3y1y2y3Signal

flow

graphFig.5.25a3417x1x2Tay1y2y3Fig.5.25bTbTcTfTTdey1

Ta

x1

Td

x2y2

Tbx1

Tex2

y3

Tc

x1

Tf

x2General

Flow-Graph

ysisThe

effect

of

the

internal

nodes

can

be

factored

out

byordinary

algebraic

processes

to

yield

the

equivalent

graphrepresented

by

Fig.5.25b.The

T’s,

called

overall

graph

transmittances,

are

the

overalltransmittances

from

a

specified

source

node

to

a

specifieddependent

node.35General

Flow-Graph

ysisFor

linear

systems

the

principle

of

superposition

(线性系统迭加原则)can

be

used

to“solve”the

graph.

That

is,thesources

can

be

considered

one

at

a

time.

Then

the

outputsignal

is

equal

to

the

sum

of

the

contributions

produced

byeach

input.The

overall

transmittances(传输增益)can

be

found

by

theordinary

processes

of

linear

algebra.The

same

results

can

by

obtained

directly

from

the

SFG.The

fact

that

they

can

produce

answers

to

large

sets

oflinear

equations

by

inspection

gives

the

SFGs

their

powerand

usefulness.3618General

Flow-Graph

ysisNote

that

the

overall

transmittance

is

the

system

transfer

function.Is

it

a

general

method

that

can

be

availableto

get

system

transfer

function??Mason’s

Gain

Formula

(MGF):

1

n

n:

the

numberofthe

overalltransmittance

T

T

Ti

i

forward

pathsis

given

by

iWhereTi

is

gain

of

theith

forward

path

between

asource

and

a

sink

node

is

the

graph

determinant,

it

is

characteristic

polynomial,

too

1

L

1

L

2

L

3

n前向通路总数Δ

特征式Δi

式nTi

i

iT

1The

Mason

Gain

Rule

(see

P152-154)n:

the

number

offorward

paths

all

possiblecombinations

1

L

1

L

2

L

3

where:L1

is

the

gain

of

each

closed

path(single

loop)ΣL1

is

thesum

of

thegains

of

all

closed

paths

in

the

graph.L2

is

the

product

of

the

gain

of

2

nontouching

loops.ΣL2

is

the

sum

of

the

products

of

gains

in

all

possiblecombinations

of

nontouching

loops

taken

two

at

a

time.L3

is

the

product

of

the

gain

of

3

nontouching

loops

in

thegraph

.

………..nontouching

loops:

loops

that

don’t

have

any

common

nodes19nTi

i

iT

1The

Mason

Gain

Rule

(seeP152-154)n:

the

number

offorward

paths

1

L

1

L

2

L

3

where:Δi

isthe

cofactor(

式,余因式）of

Ti

Δi

has

the

same

form

as

Δ.It

is

the

determinant

of

the

remaining

subgraph

when

the

forward

paththat

produces

Ti

is

removed直译：将第i条前向通路Ti移除，剩余图的特征式(特征式求法同上)另一个等价的说法：在Δ中，将与第i条前向通路相接触的回路除去后所余下的部分39Δi

is

equal

to

unity

when

the

forward

path

touches

all

the

loops

in

thegraph

or

when

the

graph

containsno

loops.40The

Mason

Gain

Rule

(see

P152-154)回路 通路的终点就是通路的起点，并且与任何其它节点相交不多于一次的通路称为回路。回路中各支路增益的乘积称为回路增益。不接触回路一个信号流图可能有多个回路，各回路之间没有任何公共节点，则称为不接触回路，反之称为接触回路。信号流图可以根据系统微分方程绘制，也可以由系统结构图按照对应关系得出。20

1

L

1

L

2

L

3

nTi

i

iT

1The

Mason

Gain

Rule

(seeP152-154)n

从输入节点到输出节点所有前向通路的条数Ti

从输入节点到输出节点第i条前向通路的增益；Δ

i

在Δ中，将与第k

条前向通路相接触的回路除去后所余下的部分，称为 式；∑L

1

所有各回路的回路增益之和；∑L

2

所有两两互不接触回路的回路增益乘积之和；∑L

3

所有三个互不接触回路的回路增益乘积之和；……….在回路增益中应包含代表反馈极性的正、负符号。n:

the

number

of

forward

paths41nTi

i

iT

1The

Mason

Gain

Rule

(see

P152-154)To

get

the

overall

transfer

function

of

a

system,

signal

flowgraphs

have

the

advantage

of

providing

a

systematic

rule

(noneed

to

simplify

graph)The

rule

is

based

onCramer’sRule

for

solving

simultaneousalgebraic

equations借助于 公式，不经任何结构变换，便可以直接求得系统的传递函数。n:

the

number

of

forward

paths

1

L

1

L

2

L

3

422143H

3H

4H1

H

2

H

5u

yH1

(s)H

3(s)H

2

(s)H

5

(s)uyH

6

H4The

flow

graph

form

is

as

followsSolution:The

Mason

Gain

Rule:

ExamplesExample-1:

find

the

overall

transferfunction.H

644H1

(s)H

2

(s)

H

5

(s)uyH

6Example-1Path

1:

H1

(s)H

2

(s)H5

(s)Path

2

:

H1

(s)H6

(s)H

3(s)

H4Step

1:

Identify

Loop

Gains

(--

magenta

in

the

graph)Loop

1:

H1

(s)H3

(s)Loop

2

:

H1

(s)H2(s)H4

(s)Step

2:

Identify

gains

of

forward

paths

from

u

to

y

(--

green

in

thegraph)The

Mason

Gain

Rule:

Examples22The

Mason

Gain

Rule:

ExamplesH1

(s)H

2

(s)

H

5

(s)uyH

6H

3(s)

H4Step

3:

Identify

Loops

not

touching

forward

path1:

NoneStep

4:

Identify

Loops

not

touching

forward

path2:

NoneStep

5:

Compute

determinants

of

path1

and

path2i

(s)

1

loop

gains

not

touching

path

i

gain

products

of

all

possible2

nontouching

loops

not

touching

path

i

gain

products

of

all

possible

3

nontouching

loops

not

touching

path

i45Example-1the

forward

paths.

Therefore1

2

1Step

6:

Compute

the

determinant

of

thesystem(s)

1

all

loop

gains

gain

products

of

all

2

loops

that

do

not

touch

gain

products

of

all

3

loops

that

do

not

touch

(s

)

1

H1

H

3

H1

H

2

H4Step

7:

Use

Mason’s

Rule

to

get

the

overall

transfer

functionG(s)

Y

(s)

H1

H2

H5

H1

H6

U

(s)

1

H1

H3

H1

H2

H446The

Mason

Gain

Rule:

ExamplesIn

this

case,

there

are

no

loops

that

do

nottouch23Example-2:

Find

the

overall

transferfunction

of

the

system

as

Fig.(a)The

Mason

Gain

Rule:

ExamplesSolution:47L1L2L3L4Step

1:

Identify

Loop

GainsThere

are

4

loops：L1=

-G1G2H1，L2=

-G2G3H2L3=

-G1G2G3，L4=

-G1G4Where

only

L2

and

L4

nontouching，L2L4=(-G2G3H2)*(-G1G4)Step

2:

The

system

determinantΔ-

L1

-

L2

-

L3

-

L4

L2L4G1G2H1G2G3H2G1G2G3G1G4

G1G2G3G4H2Example-2:

Find

the

overall

transfer

function

of

the

system

as

Fig.(a)The

Mason

Gain

Rule:

Examples4824L1L3P1The

Mason

Gain

Rule:

ExamplesExample-2:

Find

the

overall

transfer

function

of

the

system

asFig.(a)Step

3:

Identify

gains

of

forwardpaths

from

r

to

cThere

are

two

forward

paths,

n=2.P1=G1G2G3，it

touches

each

loop,

therefore

Δ1=1.P2=G1G4

，it

does

not

touch

with

loop

2,

L2=

-G2G3H2，thenΔ2=（1+

G2G3H2）P2L

L4249L1L3P1R(s)

1

12

2G1G2G3

G1G4(1

G2G3H2

)1

G1GH1

G2G3H2

G1G2G3

G1G4

G1G2G3G4

H2P2L4L2The

Mason

Gain

Rule:

ExamplesExample-2:

Find

the

overall

transfer

function

of

the

system

as

Fig.(a)Step

4:

Get

overall

transfer

function

by

Mason’s

ruleC(s)

1(P

P

)5025511

2

5L

L

L

1i

L

RCs

RCssoThere

are

six

groups

oftwo-loops

nottouching

each

other，theyareⅠ-Ⅱ、Ⅰ-Ⅲ、Ⅰ-Ⅴ、Ⅱ-Ⅲ、Ⅲ-Ⅳand

Ⅳ-Ⅴ.ThereforeExample-3:

Find

Uc/Ur

of

the

system

as

Fig.

Below.Solution:

Step

1:

Identify

LoopGainsThere

are

5

feedback

loops,

and

the

loop

transmittances

arethe

same5

L

L

i

j6R2C

2

s2The

Mason

Gain

Rule:

ExamplesThere

is

one

set

ofthree

loops

not

touching

，that

is

Ⅰ-Ⅱ-Ⅲ，theni

j

kR3C3s3

LL

L

1

Step

2:

The

system

determinant

isR2C

2

s2

R3

C3

s3

1

1

Li

Li

Lj

Li

Lj

Lk5

6

1RCsThere

is

only

one

forward

path,

n=1.1P1

R3C3s3Example-3:

Find

Uc/Ur

of

the

system

as

Fig.Step

3:

Identify

gains

offorward

paths

from

Ur

to

UcIt

touches

each

loop,

therefore

Δ1=1.The

Mason

Gain

Rule:

Examples5226Example-3:

Find

Uc/Ur

of

the

system

asFig.1

1R3C3s3Uc

P1

Ur

1

1RCsR2C

2

s2

R3C3s3

1

R3C

3s3

R2C

2

s2

RCs

1Step

4:

Get

overall

transfer

function

by

Mason’s

ruleThe

Mason

Gain

Rule:

Examples53There

are

4

feedback

loops:L1,

L2,

L3

and

L4.

And4

setsof

two-loops

are

not

touchingeach

other:

L1L3,

L1L4,

L2L3

and

L2L4.

There

is

noneset

ofthree

loops

nottouching.27(s

)

1

L1

L2

L3

L4

L1L3

L1L4

L2

L3

L2

H4Example-4:

Find

Y(s)/R(s)

of

thesystem

as

Fig.The

Mason

Gain

Rule:

ExamplesSolutionStep

1:

Identify

Loop

GainsStep

2:

The

system

determinant

is54R(s)Y(s)Step

3:

Identify

gains

of

forward

pathsn=2:

P1=G1G2G3G4，it

does

not

touch

L3

and

L4therefore

Δ1=1-

L3

-

L4.P2=G5G6G7G8，it

does

not

touch

L1

and

L2

therefore

Δ2=1-

L1