Chapter 11 Part I

8/7/2019 Chapter 11 Part I

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Chapter 11 Frequency-domain Analysis and

Design of Control System

r. . u

Department of Mechanical Engineering

Universit of Houston

Houston, TX 77204-4006

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DepartmentDepartment of Mechanicalof Mechanical EngineeringEngineering


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Contents

Bode diagram

Nyquist stability criterion

-

control system

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11-2 Bode diagram representation of the frequencyresponse

A bode diagram consists of two graphs:Ma nitude in decibel dB versus fre uenc

Phase angle versus frequency.

.

The stand representation of magnitude of G(jW) is

)(log20 jG

where the base of logarithm is 10

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Advantage to use decibel

baba log20log20log20 +=

1

R(s)C(s)

1+s.

20

30

40

de(dB)

Bode Diagram

-20

-10

0

ude(dB)

Bode Diagram

-10

-5

0

tude(dB)

Bode Diagram

+ 010

Magnitu

45

90

(deg

)

-40

-30Magnit

-45

0

se(de

g)

-20

-15Magni

-30

0

e(deg)

=

10-1

100

101

102

103

0

Phas

Fre uenc rad/sec

10-2

10-1

100

101

102

-90

Pha

Fre uenc rad/sec10

-210

-110

010

110

210

3-60

Phas

Fre uenc rad/sec

4DepartmentDepartment of Mechanicalof Mechanical EngineeringEngineering


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Bode Diagram

K=3

9.5

10

10.5

11

itude(dB)

8.5

9

.

M

ag

1

-0.5

0

.

Phase(deg)

100

101

-

Frequency (rad/sec)



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Bode diagram of integral



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Bode diagram of derivative

dBdB )20log20()10log20( +=



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(starting from 0 dB)

(Slope of -20 dB/dec)

Corner frequency : 1=



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-20 dB/dec

T

1



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-20dB/dec



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+

1



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0o

-45o

-90o

T

1

T

10

T10

1

Asymptote (phase frequency response)



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If > 1, G(j) can be expressed as a product of two first-order terms with real

poles.If 0


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For case that

For the asymptote, the curve at low frequency (


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For the case that

Note that

Slope = - 40 dB/ dec



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0dB

-40 dB/ dec

Asymptote (magnitude-frequency plot) for a second-order system (0


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=0, =0

=n, = -90o

o, - -

0o

-90o

-180o

n 10n10

n

As m tote hase-fre uenc lot for a second-order s stem 0<


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Resonant frequency

mp u e a max mum va ue a r

For case that

At r


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Transfer function having neither poles not zeros in the right-half s-plane are

called minimum-phase transfer functions.

Transfer function having poles and/or zeros in the right-half s-plane are called

nonminimum-phase transfer functions.

Minimum phase system Nonminimum phase system


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G1: minimum phase

(slow in response).


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- , -

characteristics are directly related.

If magnitude curve is specified, then the phase-angle curve is

un que y eterm ne an v ce versa.

This does not hold for a nonminimum-phase system.

Nonmimum-phase situations may arise (1) when a system

includes a nonmiimum-phase element or elements and (2) in

e case w ere a m nor oop s uns a e.


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System identification example

Experimental bode diagram of a second order system is

obtained. Identify numerical values of m, b, and k.


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Half power bandwidth method*

2

12 =


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Assume that the open-loop transfer function is given as


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Type 0 system


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For type 1 system

=1

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20dB /dec (or extension) with the0-dB line has a frequency

.


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The frequency a at the intersection

of the initial -40 dB/dec segment (or

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square root of ka numerically.


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Referring the figure below, the frequency b at which the

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zero-frequency value is called cutoff frequency.

The fre uenc ran e in which the ma nitude of the0closed loop does not drop -3 dB is called bandwidth of the

system.

Bandwidth indicates frequency where gain starts to fall off from its


.

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n s ep response curves

Unit ramp response curves


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agn u e nee s o e conver e n o ec e .

Semilogx (w, magdB)

Or


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Chapter 11 Part I