Phase

Margin

and

Gain

MarginReview

of

Nyquist’s

Stability

Criterion22008-1-3由Nyquist穩(wěn)定判據(jù)可知：若已知系統(tǒng)的開環(huán)函數(shù)G(s)H(s),即可知開環(huán)的不穩(wěn)定極點數(shù)（位于S的右半平面）PR，在畫出該開環(huán)傳遞函數(shù)的極坐標(biāo)圖（Nyquist圖）之后，閉環(huán)系統(tǒng)的穩(wěn)定性則由Nyquist圖包圍點（-1,j0）的圈數(shù)N決定。閉環(huán)系統(tǒng)穩(wěn)定的充要條件是：位于S右半平面的極點數(shù)ZR為0：ZR＝

PR－N。許多情況下，開環(huán)傳遞函數(shù)的某些系數(shù)發(fā)生變化時，Nyquist圖也隨之發(fā)生改變，閉環(huán)穩(wěn)定性也會發(fā)生變化。當(dāng)Nyquist圖穿過（－1，j0）點時，閉環(huán)系統(tǒng)臨界穩(wěn)定。穩(wěn)定性研究中，將（－1，j0）點稱為臨界點。Nyquist圖相對于該點的位置即偏離臨界點的程度，反映了系統(tǒng)的相對穩(wěn)定性。如果穩(wěn)定性不夠？？－－校正。Phase

Margin

and

Gain

Margin2008-1-3Outline

of

Chapter

7MapleIntroductionBode

Plots

(Logarithmic

plots)Direct

Polar

PlotsNyquist’s

Stability

Criterion-Part

1Nyquist’s

Stability

Criterion-Part

2Phase

Margin

and

Gain

Margin

and

Their

Relation

toStabilityStability

From

the

Nichols

PlotCompensation………Phase

Margin

and

Gain

MarginFrequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityThe

stability

and

approximate

degree

of

stability

can

be

determinedfrom

the

Lm

and

phase

diagram.

The

stability

characteristic

is

specifiedin

terms

of

the

following

ties.Gain

crossover（幅值穿越頻率－－增益臨界點）This

is

the

point

on

the plot

of

the

transfer function

at

which

themagnitude

of

G(jω) is

unity

[LmG(jω)=0dB].

The

frequency

at

gain稱此為截止頻率crossover

is

called

the

phase-margin

frequency

ωΦ.(有中文ωC)Phase

margin

angle（相角）This

is

180°

plus

the

negative

trigonometrically

consider

angle

of

thetransfer

function

at

the

gain-crossover

point.

It

is

designated

as

the

angleγ,

which

can

be

expressed

as

γ=180°+Φ,

where

∠G(jωΦ)=Φ

isnegative.42008-1-3Phase

Margin

and

Gain

Margin-1ωΦωG(jω)γ(+)Φ-90°-135°-180°-225°ω→LmG(jω)0dBPhase

marginangle,

γ(+)ωΦ-270°The

polar

plot

of

G(jω)Log

magnitude

and

phase

diagram

of

G(jω)2008-1-3For

stable

systemγ=180°+Φ>0Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityPhase

Margin

and

Gain

MarginωG(jω)Φ-90°-135°-180°-225°ω→LmG(jω)Phase

marginangle,

γ(–)ωΦωΦγ(–)-1For

unstable

systemγ=180°+Φ<0-270°The

polar

plot

of

G(jω)Log

magnitude

and

phase

diagram

of

G(jω)2008-1-3Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityPhase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityThe

phase

margin

angle

is

the

amount

of

shift

at

the

frequency

ωΦthat

would

just

produce

instability.This

angle

would

make

the

polar

plot

go

through

the

–1

point.The

phase

margin

angle

for

minimum-phase

(m.p.)

systems

must

bepositive

for

a

stable

system,

whereas

a

negative

phase

margin

meansthat

the

system

is

unstable.The

phase

margin

angle

is

related

to

the

effective

dam ratio

ofthe

system.Satisfactory

response

is

usually

obtained

with

a

phasemargin

of45°

to

60°

.Frequency

Response72008-1-3Phase

Margin

and

Gain

Marginat

the

frequency

ωc

,

it

is5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityPhase

crossover

（相位穿越頻率－－相位臨界點）This

is

a

point

on

the

plot

of

the

transfer

function

at

which

the

phaseangle

is

-180°.

The

frequency

at

which

phase

crossover

occurs

is

calledthe

gain-margin

frequency

ωc.(有中文稱此為穿越頻率ωx)Gain

margin

（幅值

）The

gain

margin

is

the

factor

a

by

which

the

gain

must

be

changed

inorder

to

produce

instability.

Expressed

in

terms

of

the

transfer

functionG(

jc

)

a

1cOn

the

polar

plot

of

G(jω)

the

value

at

ω

isacG(

j

)

1In

terms

of

the

Lm,

in

decibels,

this

isLma

Lm

G(

jc

)Frequency

Response82008-1-3Phase

Margin

and

Gain

Marginω-1ωΦγ(+)ΦG(jω)For

stable

system-90°-135°-180°-225°-270°ω→ωΦPhase

marginangle,

γ(+)ωc1/aGainmargin,Lm

a

(+)ωcThe

polar

plot

of

G(jω)2008-1-3Log

magnitude

and

phase

diagram

of

G(jω)Lma

Lm

G(

jc

)

0LmG(jω)1/a

<

1,

a>1Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityPhase

Margin

and

Gain

Marginω-1

γ(–)Φ-90°-135°-180°-225°-270°ω→ωΦPhase

marginangle,

γ(–)G(jω)1/aωΦFor

unstable

systemωcωcGain

margin,Lm

a

(–)Lma

Lm

G(

jc

)

0

LmG(jω)The

polar

plot

of

G(jω)2008-1-3Log

magnitude

and

phase

diagram

of

G(jω)1/a

>1,

a<1Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityPhase

Margin

and

Gain

Margin0

(

)

(

180

0

)

(

)

180和由5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability相角

和幅值

的求解方法通常有三種求解系統(tǒng)相角

和幅值

的方法，即解析法、極坐標(biāo)圖法和伯德圖法。下面通過實例進(jìn)行說明。（一）

解析法根據(jù)系統(tǒng)的開環(huán)頻率特性，由G(

j

)H

(

j

)

1(0

)求出相角

。

)G

(

j

c

)

H

(

j

c

)

180

0

(0

c1G

(

j

c

)

H

(

j

c

)a

20

lg

a

20

lg

G

(

j

c

)

H

(

j

c

)求出幅值或Frequency

Response112008-1-3Phase

Margin

and

Gain

Margin相角

和幅值

的求解方法例7-20

已知最小相位系統(tǒng)的開環(huán)傳遞函數(shù)為40s(

s

2

2

s

25

)G

(

s)

H

(

s

)

40j

(

25

2

j

2

)試求出該系統(tǒng)的幅值

和相角

。解：系統(tǒng)的開環(huán)頻率特性為

G

(

j

)

H

(

j

)

其幅頻特性和相頻特性分別是40(

25

2

)

2

4

2G

(

j

)

H

(

j

)

1

2

25

2)

90

0

arctg

G

(

j

)

H

(

jccFrequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability122008-1-3Phase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability相角

和幅值

的求解方法令G

(j

)H

(j

)

1

，得

1.82令

1800G(

j

)H

(

j

)，得

c

5

80.52

1.8225

1.822

180

G(

j

)H

(

j

)

90

arctga(dB)

20

lg1.25

1.94(dB)或

1.25a

1G(

jc

)H

(

jc

)即：該系統(tǒng)具有1.94分貝的幅值

，80.5度的相位

。Frequency

Response132008-1-3Phase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability相角

和幅值

的求解方法（二）極坐標(biāo)圖法在GH平面上作出系統(tǒng)的開環(huán)頻率特性的極坐標(biāo)圖，并作一單位圓，由單位圓與開環(huán)頻率特性的交點A與坐標(biāo)原點的連線與負(fù)實軸的夾角求出相角

γ

；由開環(huán)頻率特性與負(fù)軸交點處的幅值的倒數(shù)得到幅值G

(

j

c

)

H

(

j

c

)a。1aImGH

0c1

01Rej

jA例7-20

的極坐標(biāo)圖Frequency

Response142008-1-3Phase

Margin

and

Gain

Margin1aImeGH

0c11RjA

j

0例7-20

的極坐標(biāo)圖在例7-20中，先作出系統(tǒng)的開環(huán)頻率特性曲線

，作單位圓交開環(huán)頻率特性曲線于A點，連接

OA，射線OA與負(fù)實軸的夾角即為系統(tǒng)的相角

80

0。開環(huán)頻率特性曲線與負(fù)實軸的交點坐標(biāo)為(0.8，

j0)由此得到系統(tǒng)的幅值

：10.8

1.25a

相角

和幅值

的求解方法（二）極坐標(biāo)圖法Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability152008-1-3Phase

Margin

and

Gain

Margin，；反之，當(dāng)，對應(yīng)的相頻特性位于–1800

線下方時 。

然后，由相頻率特性,求出對應(yīng)幅頻特性與，。a

0dB度γ。當(dāng)

對應(yīng)的相頻特性位于–1800

線上方時

00

0

0與-1800線的交點頻率c。反之，當(dāng)

c對應(yīng)的幅頻特性位于貝線上方a

0dB幅值

a

的分貝數(shù)。當(dāng)

c

對應(yīng)的幅頻特性位于貝線的差值，即為貝線下方時，5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability相角

和幅值

的求解方法（三）Bode圖法畫出系統(tǒng)的Bode圖，由開環(huán)對數(shù)幅頻特性與

貝線（即軸）的交點頻率

，求出對應(yīng)的相頻特性與－1800線的相移量，即為相角Frequency

Response162008-1-3Phase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability相角

和幅值

的求解方法（三）Bode圖法例7-19的Bode圖如右圖所示。從圖中，可直接得到幅值穿越頻率相角穿越頻率相角

：幅值

：0

2

c

5

80a

2

dBdBLm()c52

20dB

/

dec1/

a

(dB)

60dB

/

dec度（）

90

01800

2700例7-20Bode圖Frequency

Response172008-1-3Phase

Margin

and

Gain

Margin工程實踐中得到更為廣泛的應(yīng)用。c越頻率

和相位穿頻率

。同時Bode圖較極坐標(biāo)圖方便，因此在Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability比較上述三種解法不難發(fā)現(xiàn)：解析法

比較精確，但計算步驟復(fù)雜，而且對于三階以上的高階系統(tǒng)，用解析法相當(dāng)

。圖解法

以極坐標(biāo)圖和Bode圖為基礎(chǔ)的圖解法，避免了繁鎖的計

算，具有簡便、直觀的優(yōu)點，對于高階系統(tǒng)尤為方便。不過圖解法是一種近似方法，所得結(jié)果有一定誤差，誤差的大小視作圖的準(zhǔn)確性而定。Bode圖法和極坐標(biāo)法雖然都是圖解法，但前者不僅可直接從Bode圖上獲得相角

和幅值

a

，而且還可直接得到相應(yīng)的幅值穿182008-1-3Phase

Margin

and

Gain

MarginPhase

Margin

and

Gain

Margin

and

Their

Relation

to

Stability注意：對于非最小相位系統(tǒng)，不能簡單地用系統(tǒng)的相角

和幅值 的大小來判斷系統(tǒng)的穩(wěn)定性。但對于最小相位系統(tǒng)以相角

>0

和幅值

a>1（或a

(dB)>0）作為系統(tǒng)穩(wěn)定的充要條件是可靠的。Frequency

Response192008-1-32008-1-3Phase

Margin

and

Gain

Margins

(Ts

1)G

(

s

)

H

(

s

)

K

(

s

1)例7-21

已知非最小相位系統(tǒng)的開環(huán)傳遞函數(shù)為試分析該系統(tǒng)的穩(wěn)定性及其與系統(tǒng)穩(wěn)定之間的關(guān)系。解

在一定的K值條件下，系統(tǒng)的開環(huán)頻率特性如右圖所示。由于該系統(tǒng)有一個為于S右半部平面的開環(huán)極點PR=1

，奈氏曲線順時針包圍(-1,j0)

點一周（N=1），根據(jù)奈氏判據(jù)，該系統(tǒng)為穩(wěn)定系統(tǒng)。但由圖解法求出該系統(tǒng)的相角

>0，幅值

a<1，這說明以相角

>0

和幅值a>1作為判別非最小相位系統(tǒng)穩(wěn)定性的依據(jù)是不可靠的。c00

1aj

j

11Re0

例7-21極坐標(biāo)圖>0Frequency

Response5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityPhase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityThe

gain

margin

must

be

positive

when

expressed

in

decibels(greater

than

unityasa

numeric)

for

a

stable

system.

A

negative

gainmargin

means

that

the

system

is

unstable.Themargin.damdam

ratio

of

the

system

is

also

related

to

the

gainHowever,

the

phase

margin

gives

a

better

estimate

ofratio,

and

therefore

of

the

transient

overshoot

of

thesystem,than

the

gain

margin.Frequency

Response212008-1-3222008-1-3Phase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityThe

asymptotes

of

the

Lm

vs

curve

are

related

to

eachfactor

ofthe

transfer

function.

For

example,

factors

of

the

(1+j

)-1

have

anegative

slope

–20dB/decade

at

high

frequencies.The

total

phase

angle

of

the

transfer

function

at

any

frequency

isclosely

related

to

the

slope

of

the

Lm

vs

curve

at

that

frequency.For

example,

a

slope

of

–20dB/decade

is

related

to

an

angle

of

-90°.By

observing

the

asymptotes

of

the

Lm

vs

curve,

it

is

possible

toestimate

the

approximate

value

of

the

angle.

Changes

of

slope

athigher

and

lower

corner

frequencies,

around

the

particularfrequency

being

considered.

The

farther

away

the

changes

of

slopeare

from

the

particular

frequency,

the

less

contribute

to

the

totalangle

at

that

frequency.

For

example,

Fig.

8.7.

=4.(P258)Frequency

Response232008-1-3Phase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityAs

we

have

seen,

the

stability

of

an

m.p.

system

requires

that

thephase

margin

angle

be

positive.,

i.e.

the

angle

at

the

gaincrossover

[Lm

G(j

)=0]

must

be

greater

than

-180°.

Thisrequirement

places

a

limit

on

the

slope

of

the

Lm

curve

at

thegain

crossover.

The

slope

at

that

frequency

should

be

morepositive

than

–40dB/decade

if

the

adjacent

corner

frequencies

arenot

close.

And

a

slope

of

–20dB/decade

is

preferable.Just

like

root

locus

method,

the

Log

magnitude

and

phase

diagramcan

be

aided

system

design.Frequency

ResponsePhase

Margin

and

Gain

Margin5.

Phase

Margin

and

Gain

Margin

and

Their

Relation

to

StabilityFor

example:The

gain

can

be

adjusted

(which

raises

or

lowers

the

Lm

curve)

toproduce

a

phase

margin

angle

in

the

desirable

range

of

45o

to

60o.The

sh

of

the

low-frequency

portion

of

the

curve

determinesystem

type

and

therefore

the

degree

of

steady-state

accuracy.The

system

type

and

the

gain

determine

the

error

coefficients

andtherefore

the

steady-state

error.The

phase-margin

frequency

ωΦ

gives

a

qualitative

indication

ofspeed

of

response

of

a

system.Frequency

Response242008-1-3252008-1-3Phase

Margin

and

Gain

Margin6.Stability

from

the

Nichols

plot

(Log

Magnitude-Angle

Diagram)The

Lm-angle

diagram

(Nichols

plot)

is

drawn

by

picking

for

eachfrequency

the

values

of

Lm

and

angle

from

the

Lm

and

phasediagram

.

The

resultant

curve

has

frequency

as

a

parameter.The

curve

for

the

example

of

Sec.8.7,

sketched

in

Fig.8.35,

shows

apositive

gain

margin

and

phase

margin

angle,

this

represents

astable

system.Changing

the

gain

raises

or

lowers

the

curve

without

changing

theangle

characteristics.

Increasing

the

gain

raises

the

curve,

therebydecreasing

the

gain

margin

and

phase

margin

angle,

with

theresult

that

the

stability

is

decreased………may

be

result

in

systemunstable

(See

slide

7-5(2)

Ex.7-19)

.Frequency

ResponsePhase

Margin

and

Gain

MarginFrequency

Response6.Stability

from

the

Nichols

plot

(LogMagnitude-Angle

Diagram)Phase

Margin

and

Gain

Margin6.Stability

from

the

Nichols

plot

(Log

Magnitude-Angle

Diagram)The

Lm-angle

diagram

(Nichols

plot)

f (s)H(s)

can

be

drawnfor

all

values

o the

contour

Q

of

Fig.

8.26a.

The

resultantcurve

for

m.p.

systems

is

a

closed

contour.

Nyquist’s

criterion

canbe

applied

to

this

contour….:

ZR=N-PR----???In

the

case

of

m.p.

systems,

it

is

not

necessary

to

obtain

the

completeLm-angle

contour

to

determine

stability.

Only

the

positive

frequencyis

needed

to

draw.

The

system

is

stable

if

the

(0dB,-180o)

point

is

tothe

right

of

the

curve.