自动控制理论-熊蕊张蓉课件

(P173)

In

this

section

the

concepts

outlined

previously

willbe

developed

further

into

some

straightforward

guidelinesfor

plotting

more

complex

root

loci,

which

will

beillustrated

by

focusing

on

a

specific

example.2(1948-)

American

Scientist，W.

R.

Evans—Root

Locus

Method3How

to

start

plotting

a

loci

?

And

how

to

end

?G(s)H

(s)

1nm(s

pi

)i1K

(s

z

j

)G(s)H(s)

j

1

1

s

f

(K

)K

0

n

m(s

pi

)

K

(s

z

j

)

0[

Start

to

plot……

]1

G(s)

0i1

j1K=0

(the

starting

point

of

the

root

locus):(s

pi

)

0,

s

pi

; (i

1,

2,

L

,

n)K→∞

( point

of

the

root

locus):the

characteristic

equation

can

be

written

as1j1n

m(s

pi

)

(s

z

j

)

0K

i1when

K

s

z

j(

j

1,

2,

L

,

m)5Rule

#1

The

Starting

Points

andPointsof

the

Root

Locus

(根轨迹的起点和终点）The

locus

starts

at

the

open-loop

poles

(

theclosed-loop

poles

for

K

=

0

),

and

finishes

at

theopen-loop

zeros

(the

closed-loop

zeros

for

K=∞

).

The

number

of

segments

going

toinfinity

is

n-m.(根轨迹始于开环极点，终于开环零点。趋于无穷大的线段条数为n-m。若n>m,则有n-m条根轨迹终止于无穷远处；若m>n，则有m-n条根轨迹起始于无穷远处。)67Rule#2

The

Segments

of

the

Root

Locus

on

theReal

Axis

(实轴上的根轨迹）

Segment

of

the

real

axis

to

the

left

of

on

an

odd

number

of

poles

or

zeros

are

segments

of

the

root

locus,remembering

that

complex

poles

or

zeros

have

no

effect.(实轴上，对应零、极点数之和为奇数时的左边线段为根轨迹。或者说，实轴上的根轨迹，是其右侧的开环零、极点数之和为奇数时的所在线段。复数零、极点对该线段没有影响。)[

Proof

]p1p3p1p2p3z1s1

p121p2s

ps1

p3s1s1j180

j180

The

loci

aresymmetrical

about

the

real

axis

sincecomplexroots

arealwaysin

conjugate

pairs.（根轨迹关于实轴对称，因为复数根总是成对出现的。）The

angle

between

adjacent

asymptotes

is360º/(n-m),and

to

obey

the

symmetry

rule,

the

negative

real

axis

is

one

asymptote

when

n-m

is

odd.（相邻的渐近线之间的夹角是360º/(n-m),并同样服从对称规律。当n-m

是奇数时，负实轴也是一个渐近线。）The

Angle

of

the

asymptotesand

real

axis

is:(渐近线Rule#3

The

Symmetry

and

the

Asymptotes

ofthe

Root

Locus

(根轨迹的对称性和渐近线）n

m8a(k

0,

1,L

,

n

m

1)与实轴正向的夹角是）

2k

1(also

including

complex

zeros).是开环Rule#4

The

Real

Axis

intercept

of

theAsymptotes

(渐近线与实轴的交点）The

asymptotesintersect

the

real

axis

at

a

,an

m

pi

z

jwhere

pi

is

the

sum

of

the

real

parts

of

the

open-loop

poles

(including

complex

roots)

and

z

jis

the

sumof

the

real

parts

of

the

open-loop

zeros是开环极点的实部的和（包括复数极点）；

z

j零点的实部的和（包括复数零点）。），式中

pia9n

m

pi

z

j（渐近线与实轴的交点是SP.j01

2

0

(1)

(2)

13

010

pi

z

jn

ma

/

5

/

3;

k

2a

;

k

1n

m

/

3;

k

0

2k1(k

0,

1,L

,

n

m

1)6011The

angle

of

emergence

from

complexpoles

isgiven

by180º

–

Σ(angles

of

the

vectors

from

all

other

open-looppoles

to

the

poles

in

question)

+

Σ(angles

of

the

vectorsfrom

the

open-loop

zeros

to

the

complex

pole

in

question).Rule#5

The

Angle

of

Emergence

from

ComplexPoles

and

The

Angle

of

Entry

into

Complex

Zeros(根轨迹的出射角和入射角）n

mi

p

180

(

pi

p

j

)

(

pi

z

j

)j1

j1jiThe

angle

of

entry

into

a

complex

zero

may

be

found

from

thesame

rule

and

then

the

sign

changed

to

produce

the

final

result.m

ni

jzij1jij1i(z

p

j

)(z

z

)

180

SP.s(s

2

j2)(s

2

j2)s(s2

4s

8)GH

(s)

K

(s

5)

K

(s

5)

2

j

2

180

((2

j2

0)

((2

j2)

(2

j2))

((2

j2)

(5))2

2

tg

1

4

tg

1

20

3

180

tg

1

180

135

90

33

12The

point

where

the

locus

crosses

the

imaginaryaxis

may

be

obtained

by

sunbstituting

s

=

jω

intothe

characteristic

equation

and

solving

for

ω.Rule#6

The

Root

Locus

Crossing

with

theImaginary

Axis

(根轨迹与虚轴的交点）1

GH

(

j)

01

GH

(s)

0

Re[1

GH

(

j

)

0]Im[1

GH

(

j)

0]GH(s)

K

(s

5)

s(s2

4s

8)SP.s3

4s2

s(K

8)

5K

0(5K

42

)

j[(K

8)

3

]

05K

4

2

0(K

8)

2

0K

32

6.3213Thepoint

at

which

the

locus

leaves

a

real-axissegment

is

found

by

determining

alocal

umvalue

ofK,while

the

point

at

which

the

locus

entersa

real-axis

segment

is

found

by

determining

a

localminimum

valueof

K.(根轨迹离开实轴区段的点（分离点）由该区段的最大K

值来确定；而根轨迹进入实轴区段的点（分离点）由该区段的最小K值来确定。)Rule

#7

The

Breakaway

Point

of

the

Root

Locus(根轨迹的分离点）j0

1

260j0

1

214Assume

the

breakaway

point

s

=

d:)(1)mnnmdsd(s

z

j

)(s

pi

)

1;(s

pi

)K

(s

z

j

)K

0 (

GH

(s)

j

1

K

i1i1j

1sdmj

1d

zjni1d

pi

1

1

(2)SP.GH

(s)

K(1)

d

0.5

(2d

1)

0dsdsd

K

d

[s(s

1)]sdsd11(2)

0

d

0.5d

1

d

01

1d

p

d d

1d

zni1imj

1js(s

1)j0-1-0.515SP.s(s

1)(s

2)KGH

(s)

SP.16s(s

1)GH

(s)

K

(s

2)(

to

be

canceled

)(

breakaway

point

)d1

1.577d2

0.423(breakaway

point

)(

breakaway

point

)d1

0.586d2

3.414Root-locus:

Rules#1-7

(

in

thelast

module

)Rule

#1

The

Starting

Points

and

Pointsof

theRoot

Locus(根轨迹的起点和终点）Rule#2

The

Segments

of

the

Root

Locus

onthe

Real

Axis(实轴上的根轨迹）Rule

#3

The

Symmetry

and

the

Asymptotes

ofthe

Root

Locus(根轨迹的对称性和渐近线）Rule#4

The

Real

Axis

intercept

of

theAsymptotes(渐近线和实轴的交点）Rule#6

The

Root

Locus

Crossing

with

theImaginary

Axis(根轨迹与虚轴的交点）Rule

#7 The

Breakaway

Point

of

the

RootLocus(根轨迹的分离点）25Rule#8

The

angle

between

the

direction

ofemergence

(or

entry)of

q

coincident

poles(orzeros)

on

the

real

axis

(根轨迹离开或进入实轴上q重极点（或零点）方向之间的夹角）q

360SP.s2

(s

1)KGH

(s)

SP.K(s

1)3GH

(s)

j0-1j0-13poles26Rule#9

The

gain

at

a

selected

point

st

on

thelocus

(在某特定点st上的根轨迹增益K）The

gain

at

a

selected

point

st

on

the

locus

isobtained

by

joining

the

point

to

all

open-loop

polesand

zeros

and

measuring

the

length

of

each

line|st+

pi

|,|st

+

zj

|.

The

gain

is

givenbysst27mn

s

z

j

s

pij

1K

i1

At

the

breakaway

point

s

=-2.6,

Gain

K

is

2.6

2

2.6

3

2.62

2.6

1

j2

2.6

1

j2

0.2473s2.6

s

z

jmj

1

s

pinK

i1

28Rule

#10

The

sum

of

the

closed-loop

poles

(闭环极点之和）K

(s

5)s(s

2

j2)(s

2

j2)GH

(s)

s3

1if s1,2

j

,1

0

2

2

1.5

If

there

are

atleast

two

more

open-looppoles

thanopen-loop

zeros,the

sum

ofthe

closed-loop

poles

is

constant,independent

of

K,

and

equal

to

the

sumof

thereal

parts

of

theopen-looppoles.(如果开环极点比开环零点至少多2个，闭环极点的和为一不依赖于K的常数，且等于开环极点的实部的和。）30Rule

#11

The

number

of

branches

of

the

rootlocus

(根轨迹的分支数）The

number

of

branches

of

the

root

loci

is

equalto

the um

in

the

number

N

of

poles

and

thenumber

M

of

zeros

of

the

open-loop

transferfunction.（根轨迹的分支数等于传递函数中极点数M和零点数N

中的最大数）b

maxN,

M

3

5s1

j,

s2

j,

s3

si

3

pi

2

j

2

j

1K[(s

1.5)2

1]s2

(s

0.5)(s

8)(s

10)33GH

(s)

SP10.3(P190)s(s2

2s

10)KG(s)

Ks

H

(s)

0.5;

4;

2R（s）C（s）1s(s2

2s

10)-K1s

341s(s

0.5)(s2

2s

10)(a)

GH

35136s(s

4)(s2

2s

10)(b)

GH

1s(s

2)(s2

2s

10)37(c)

GH

1s(s

2)(s2

2s

10)(c)

GH

1s(s

4)(s2

2s

10)(b)

GH

1s(s

0.5)(s2

2s

10)(a)

GH

39Similar

examples

–(b)A

robot

is

programmed

to

have

a

tool

or

welding

torchfollow

a

prescribed

path.

Consider

a

robot

tool

that

is

tofollow

a

sawtooth

path,

as

shown

in

Fig.(a).

Choose simplest

vales

of a,

b,

c，to

make

the

steady-state

errorof

the

closed-loop

system

not

more

than

5%,

and

……P1.2009/10/144422Problem

2

(00-P4)A

system

is

expressed

by

the

following

transferfunction.H

(s)

1 (K

0)(s2

4s

13)G(s)

K

(s

1)(s

4)

;akes

theSketch

the

root-locus

of

the

system.Determine

the

range

of

values

of

K

whisystem

stable.Calculate

the

minimum

error

of

system

due

to

a

unitstep

input.Does

the

step

response

curve

of

the

system

able

toappear

un-overshoot

sh

?Solutions:

1KE

(s

1)(s

4)(s

2

j3)(s

2

j3)z2

4z1

1;p1,2

2

j3;;Fig.P4

(a)

Root-Locus(a)43s2E

4s

4

9

K

(s2

5s

4)

045

KE

4KE

max

51344110.81

ss4KE

max1

KBe

and

type

0

system:(c)(d)

No.

From

the

root-locus,

all

closed-loop

roots

are

n