System analysis based on the frequency response

A()

0

A(0)

0.707A(0)

r b

Mr

(1) Resonant frequency r:

0)( :

)()(1

)()()( :

r

Ad

dsatisfy

jHjG

jGjAAssume

r

(2) Resonant peak Mr :

r

AM r )(

(3) Bandwidth b:

)0(707.0)0(2

2)( : AAAsatisfy

bb

The general frequency response of a closed loop system is shown in Fig.

For the closed loop systems

Performance specifications in the frequency domain

For the open loop systems

(1) Gain crossover frequency c: 1)()( : c

c jHjGsatisfy

(2) Gain margin h:

g

g

jHjGdBLjHjG

h

)()(log20)( ;)()(

1h

0

g 180)()( : gjHjGsatisfiesHere

(3) Phase margin :

1)()( :

)()(180

c

0

c

c

jHjGsatisfiesHere

jHjG

Performance specifications in the frequency domain

Relationship of the performance specifications

between the frequency and time domain

performance specifications in the time domain

Overshoot

Setting time

Steady-state error

%

st

sse

performance specifications in the frequency domain

Closed-loop Open-loop

Resonant peak Gain-crossover frequency

Resonant frequency Gain margin

Bandwidth Phase margin

c

r

b

rM

/ hh L

(2) Resonance peak Mr Overshoot %

Normally Mr %

h and %

Some experiential formulas:

002

9035 1sin

15.21

sin

15.12 , timeSettling

sin

1 and

)8.11.1( %100)]1(4.016.0[% Overshoot

kk

t

M

MM

c

s

r

rr

5.111 : of valueoptimuman problem,design most For rr

M.M

Relationship of the performance specifications

between the frequency and time domain

(1) Bandwidth b(or Crossover frequency c ) Setting time ts

Generally b(or c ) ts

because of c b .(higher order system)

For the typical first-order system:

1

1)(

1)(

Ts

sTs

sG

Relationship of the performance specifications

between the frequency and time domain

Tts 3

-3 dB

Tb /1

The bandwidth is the frequency , at which the frequency response

Has declined 3 dB from its low-frequency value. b

For the typical 2th-order system:

22

22

2)(

)2()(

nn

n

n

n

sss

sssG

Relationship of the performance specifications

between the frequency and time domain

We have:

422 442)21( nb

)2

2(0 21 2 nr

212

1

rM

... , , % , rsn tt

h

tg

nc

24

1

24

241

2

241

... , , % , rsn tt

three frequency band theorem

0

G20lg(db)

band

freqencylow

[-20]

]60 40[ or

[-20]c)102(

cc1.01/1 T

[- 40]

[-60]

c10

band

freqencymiddle

band

freqencyhigh

gain loopopen 121

12122

22

K

sTsTsTs

sssKsHsG

kkki

v

lllj

Open loop transfer function

The performance analysis of the closed-loop system according to the open-loop frequency response.

1. For the low frequency band

The more negative the slope of L() is , the higher the control accuracy of the system.

The bigger the magnitude of L() is, the smaller the steady-state error ess is.

2. For the middle frequency band

the low frequency band is mainly concerned with the control accuracy of the system.

The middle frequency band is mainly concerned with the transient performance of

the systems.

c ts ; h and %

three frequency band theorem

The slope of L() in the middle frequency band should be the 20dB/dec and with a certain width .

three frequency band theorem

3. For the high frequency band

The high frequency band is mainly concerned with the ability of the system

restraining the high frequency noise.

The smaller the magnitude of L() is, the stronger the ability of the system restraining the high frequency noise is.

Example 1: compare the performances between the system and system

0dB

)()(log20)( jHjGL

40

20

40

Solution :

ess > ess

ts > ts

% = %

The ability of the system restraining the high frequency

noise is stronger than system

How to obtain the closed-loop frequency response in terms

of the open-loop frequency response.

sGsG

s

1

jG

jGj

1

GGf

GGfM

,)(

,)(

2

1

j

j

j

eG

eGeMj

1

GjeGjG

Frequency response of the closed-loop system

The constant N circles: How to obtain the phase frequency characteristic of

the closed-loop system in terms of the open-loop frequency response (refer to text book)

The Nichols chart: How to obtain the closed-loop frequency response in

terms of the open-loop frequency response (refer to text book)

Frequency response of the closed-loop system

The constant M circles: How to obtain the magnitude frequency response of

the closed-loop system in terms of the open-loop frequency response (refer to text book)

GGf

GGfM

,)(

,)(

2

1

N.B.Nichols transformed the constant M and N circles to

the log-magnitudephase diagram, and the resulting chart is called the Nichols chart.

The Nichols chart