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(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
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