Negative v - Eastern Mediterranean...

X(v)

R(v)

Negative v

1v � ��

v � �

Positive vv � v1

v � 045�

�G �

Dorf/BishopModern Control Systems 9/E

© 2001 by Prentice Hall, Upper Saddle River, NJ.

FIGURE 8.3

Polar plot for RC filter.

Increasing v

Positive v

v � �

Im[G]

Re[G]

v � 0

12t

v �

1t

v � 135�



FIGURE 8.4

Polar plot for G( jv) � K/jv( jvt � 1). Note that v � � at the origin.

jv

jv1( jv1 � p)

s

v � v1

s � �1t

� �p



FIGURE 8.5

Two vectors on the s-plane to evaluate G( jv1).

(a)

�5

�10

�15

0

�20

20 lo

g�G

(jv

)�, d

B

(b)

�50�

0�

�100�

f(v

), d

egre

es

v, rad/sec

0.1t

10t

1t



FIGURE 8.6

Bode diagram for G( jv) � 1/( jvt � 1): (a) magnitude plot and (b) phase plot.

0

�20

�10

v

20 lo

g�G

� , dB

1t

110t

10t



FIGURE 8.7

Asymptotic curve for ( jvt � 1)�1.

dB 0

v v

40

�40

( jv)2

( jv)�2( jv)�1

( jv)

( jv)2

( jv)0

( jv)�1

( jv)�2

( jv)f

(v),

deg

rees

180

90

0

�90

�1800.1 1 10 100 0.1 1 10 100



FIGURE 8.8

Bode diagram for ( jv)�N.

0

�45

�90

10

0

�10

�20

0.1t

10t

1t

dBf

(v),

deg

rees

v

Exactcurve

Asymptoticcurve

Linearapproximation

Exact

(a)

(b)



FIGURE 8.9

Bode diagram for (1 � jvt)�1.

0

�20

�40

�60

�80

�100

�120

�140

�160

�180

u � v /vn � Frequency ratio

Phas

e an

gle,

deg

rees

1.00.4 0.5 0.6 0.80.30.20.1 104 5 6 832

20

10

0

�10

�20

�30

�40

20 lo

g�G

�

(b)

u � v /vn � Frequency ratio

1.00.4 0.5 0.6 0.80.30.20.1 104 5 6 832

(a)

z � 0.050.100.150.200.25

0.3 0.4 0.50.6

0.81.0

z � 0.050.100.150.200.25

0.3 0.40.5

0.60.8

1.0



FIGURE 8.10

Bode diagram for G( jv) � [1 � (2z/vn) jv � ( jv /vn)2]�1.

z

0.20 0.30 0.40 0.50 0.60 0.70

3.25

3.0

2.75

2.5

2.25

2.0

1.75

1.5

1.25

1.0

0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

0.101.0

Mpv

Mpvvr /vn

vr /vn



FIGURE 8.11

The maximum of the frequency response, Mpv, and the resonant

frequency, vr, versus z for a pair of complex conjugate poles.

(a)

0 0

0 0

(b)

(c) (d)

s1( jv � s1)

s1*

s1

( jv � s1)*

s1*

M1

u1

*u1

M1*

j0

s1*

s1

M1

M1

u1

s

u1

M1*

M1*

u1*

u1*

jvr

jvd

jv



FIGURE 8.12

Vector evaluation of the frequency response for selected values of v.

1.5

1.0

0.5

0.00 vr vd

90�

0�

�90�

�180�

f(v)

v

�G �

Mpv

�G�

f(v)



FIGURE 8.12

Vector evaluation of the frequency response for selected values of v.

v

40

20

0

�20

�40

dB20 log K



Table 8.3

Asymptotic Curves for Basic Terms of a Transfer Function.

v

90�

45�

0�

�45�

�90�

f (v)



Table 8.3

Asymptotic Curves for Basic Terms of a Transfer Function (continued).

v

0.1v1 v1 10v1

40

20

0

�20

�40

dB



Table 8.3


v

0.1v1 v1 10v1

90�

45�

0�

�45�

�90�

f (v)



Table 8.3


v

0.1v1 v1 10v1

40

20

0

�20

�40

dB



Table 8.3


v

0.1v1 v1 10v1

90�

45�

0�

�45�

�90�

f (v)



Table 8.3


v

0.1 10 10010.01

40

20

0

�20

�40

dB



Table 8.3


0.1 10 10010.01

v

90�

45�

0�

�45�

�90�

f (v)



Table 8.3


u0.1 10 10010.01

40

20

0

�20

�40

dB



Table 8.3


0.1 10 10010.01u

180�

90�

0�

�90�

�180�

f (v)



Table 8.3


(a) (b)

jv1

�p�z

jv1

�p �z

u1u2

0 0G1(s) G2(s)

u2u1

*



FIGURE 8.16

Pole–zero patterns giving the same amplitude response and different phase characteristics.

180�

90�

0�

vz p

Nonminimum phase

Minimumphase



FIGURE 8.17

The phase characteristics for the minimum phase andnonminimum phase transfer function.

� �

� �

R

Cv in vo

R

C

L L

vn

�G �

f(v)

(b)(a)

(c)

0

1

�180�

0�

�360�v

v

jv1

u1 u2

u1* u2

*

p1

*p1

0



FIGURE 8.18

The all-pass network (a) pole–zero pattern,and (b) frequency response, and (c) a lattice network.

21

1

2

3

4

5

0.1 0.2 10 50 100

20

0

10

14

�10

�20

20 lo

g � G

� , dB

v



FIGURE 8.19

Magnitude asymptotes of poles and zeros used in the example.

10.1 10 100

20

0

�20

�40

10

�10

�30

�50

�20 dB/dec

�40 dB/dec

�20 dB/dec

�60 dB/dec

Approximate curve

Exact curvedB

v



FIGURE 8.20

Magnitude characteristic.

90

60

30

0

�30

�60

�90

�120

�150

�180

�210

�240

�2700.1 0.2 1.0 2.0 10 60 100

Zero at v � 10

Pole at v � 2

Complex poles

Pole at origin

Approximate f(v)

f

v



FIGURE 8.21

Phase characteristic.

Max. mag � 33.96906 dBMax. phase � �92.35844 degThe gain is 2500

Min. mag � �112.0231 dBMin. phase � �268.7353 deg

0 dB

�180 deg

dBand

phase Mag:Phase:

0.1 1 10 100 1000Frequency, rad/s



FIGURE 8.22

The Bode plot of the G( jv) of Eq. (8.42).

Phas

e (d

egre

es)

0

�3

�10

�20

�20

�40

�60

�45

�30

10 100 1,000 10,000 100,000

v � 300 v � 20,000

v � 20,000

v � 300 v � 2,450

�20 dB/dec

dB

v � 2,450

v

(a)

(b)

60

40

20

0

�45�

10 100 1,000 10,000 100,000v

10 dB



FIGURE 8.23

A Bode diagram for a system with an unidentified transfer function.

�

Y(s)R(s)�

s (s � 2zvn)

v2n



FIGURE 8.24

A second-order closed-loop system.

v

0

20 log Mpv

20 lo

g � T

� , dB

�3

0 vr vB



FIGURE 8.25

Magnitude characteristic of the second-order system.

1.6

1.5

1.4

1.3

1.2

1.1

1

0.9

0.8

0.7

0.60.1 0.2 0.3 0.4 0.5 0.6

vn

vB

0.7 0.8 0.9 1

vn

vB

Linear approximation

� �1.19z � 1.85

z



FIGURE 8.26

Normalized bandwidth, vB /vn, versus z for a second-order system (Eq. 8.46). Thelinear approximation vB /vn � �1.19z� 1.85 is accurate for 0.3 � z� 0.8.

40

30

20

10

0

�10

�20

�30

�40�270 �225 �180 �135 �90

Phase, degrees

20 l

og � G

H� ,

dB

0.1

0.3

0.6

1.0

2

3.6

5

7

10

v



FIGURE 8.27

Log-magnitude–phase curve for GH1( jv).

40

30

20

10

0

�10

�20

�30

�40�270 �225 �180 �135 �90

Phase, degrees

20 lo

g � G

H� ,

dB

v

0.1

0.2

0.5

1.0

3

13

2040

51

61

70



FIGURE 8.28

Log-magnitude–phase curve for GH2( jv).

(b)

(a)

�

�

Controller

R(s)Y(s)

Position onx-axis

Motor, screw, andscribe holder

K1

s (s � 1)(s � 2)

x-motor 2

Metal to beengraved

z-axis

y-axisScribe

x-axis

x-motor 1 Position measurement

Desired position

Position measurement

Controller



FIGURE 8.29

(a) Engraving machine control system. (b) Block diagram model.

20 lo

g �G

�, dB

10

0

�10

�20

0.1 0.2 0.5 21 5 10

�135�

�180�

�90�

v

20

f(v

)

Asymptotic approximation



FIGURE 8.30

Bode diagram for G(jv).

10

v

20 lo

g � T

� , dB

5

0

�5

�10

�15

0�

�90�

�180�

f(v

)

0.1 0.2 0.4 0.6 0.8 1 2�270�



FIGURE 8.31

Bode diagram for closed-loop system.

50

0

-50

-100

-150

10-1 100 101 102 103

Gai

n dB

Pha

se d

eg

Frequency (rad/sec)

10-1 100 101 102 103

Frequency (rad/sec)

-50

-100

-150

-200

-250

-300



FIGURE 8.32

The Bode plot associated with Eq. (8.59).

10110-1 100 102 103

Pha

se d

egG

ain

dB

Frequency (rad/sec)

20

0

-20

0

-100

-200

G(s) � sys

[mag,phase,w]=bode(sys,w)

User-supplied frequency(optional)

10110-1 100 102 103

Frequency (rad/sec)



FIGURE 8.33

The bode function, given G(s).

10-1 100 101 102 103

Logarithmically spaced vector

w=logspace(a,b,n)

>>w=logspace(-1,3,200);>>bode(sys,w);

Mag

nitu

de (

dB)

Frequency (rad/sec)

-100

50

0

-50

-150

Generate 200 points between 0.1 and 1000.

n points between 10a and 10b

Example



FIGURE 8.34

The logspace function.

% Bode plot script for Figure 8.22%num=5*[0.1 1];f1=[1 0]; f2=[0.5 1]; f3=[1/2500 .6/50 1];den=conv(f1,conv(f2,f3));%sys=tf(num,den);bode(sys)

1

502

0.650

s(1 � 0.5s)(1 � s � s2)

Compute



FIGURE 8.35

The script for the Bode diagram in Fig. 8.32.

State-space modelsys � ss(A, B, C, D)

bode(sys)

bode(sys)

Transfer function modelsys � tf(num,den)



FIGURE 8.36

The bode function with a state variable model.

Mpv

z

(b)

(a)

zeta ranges from 0.15 to 0.70zeta=[0.15:0.01:0.7];wr_over_wn=sqrt(1-2*zeta.^2);Mp=(2*zeta .* sqrt(1-zeta.^2)).^(-1);%subplot(211),plot(zeta,Mp),gridxlabel(' \zeta'), ylabel('M_{p\omega}')subplot(212),plot(zeta,wr_over_wn),gridxlabel(' \zeta'), ylabel(' \omega_r/ \omega_n')

Generate plots

0 0.2 0.4 0.6 0.8

z

0 0.2 0.4 0.6 0.80

0.20.40.60.8

1

11.5

22.5

33.5

vr/

vn



FIGURE 8.37

(a) The relationship between (Mpv, vr ) and (z, vn ) for

a second-order system. (b) MATLAB script.

Initial gainK

UpdateK

Compute closed-looptransfer function

Ks(s � 1)(s � 2) � K

T(s) �

Checktime domain specs:

If satisfied, then exitand

continue analysis.

4zvn

Ts � ,

Mp � 1 � e�zp /�1 � z 2

Closed-loop Bode diagram

20*l

og10

(mag

) [d

B]

Determine vn and z.

Mpv

102100

10

0

�10

�20

�30

�40vr

Freq. [rad/sec]

z

0 0.2 0.4 0.6 0.8

z

0 0.2 0.4 0.6 0.8

Mpvvr /vn

3.5

3

2.5

2

1

1.5

1

0.8

0.6

0.4

0.2

0

Determine Mpv and vr.

Establish relationship between frequency domainspecs and time domain specs.

.

.



FIGURE 8.38

Frequency design functional block diagram for the engraving machine.

Check specs and iterate, if necessary.

engrave1.m

num=[K]; den=[1 3 2 K];sys=tf(num,den);w=logspace(-1,1,400);[mag,phase,w]=bode(sys,w);[mp,l]=max(mag);wr=w(l);mp,wr

>>K=2; engrave1mp = 1.8371wr = 0.8171>>>>>>>>zeta=0.29; wn=0.88; engrave2ts = 15.6740po = 38.5979

ts=4/zeta/wnpo=100*exp(-zeta*pi/sqrt(1-zeta^2))

engrave2.m

Closed-loop transfer function

Closed-loop Bode diagram

Determine vn and z from Fig. 8.11using Mpv

and vr.manual step



FIGURE 8.39

Script for the design of an engraving machine.

y(t

)

Time (sec)

engraves.m

K=2; num=[K]; den=[1 3 2 K]; sys=tf(num,den);t=[0:0.01:20];step(sys,t)xlabel('Time (sec)'), ylabel('y(t)')

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Settling time

(b)

(a)

Percent overshoot � 37%



FIGURE 8.40

(a) Engraving machine step response for K � 2. (b) MATLAB script.

Springk

Mass M

y(t)

Arm forceu(t)

Frictionb



FIGURE 8.41

Spring, mass, friction model of flexure and head.

�

�R(s) Y(s)

PD control Motor coil

5

(t1s � 1)

t1 � 10�3 t2 � 1/20

Gc(s) � K(s � 1) G1(s) �

Arm

0.05

s(t2s � 1)G2(s) �

Flexure and head

1

[1 � (2z /vn)s � (s/vn)2]

z � 0.3, vn � 18.85 103

G3(s) �



FIGURE 8.42

Disk drive head position control, including effect of flexure head mount.

dB

60

40

20

0

�20

�40

�60

�800.1 1 10 102 103 104 105

v1 � 1000

Sketch of actual curve

Asymptotic approximation

–20 dB/dec

–40 dB/dec

�80 dB/dec

vnv2 � 20vz � 1v



FIGURE 8.43

Sketch of the Bode diagram magnitude for the system of Fig. 8.42.

Frequency, (rad/sec)

(a)

(b)

Frequency (rad/sec)

Mag

nitu

de (

dB)

0

�20

�3

�40

�60

�80

�100

�120

20

�50

�100

�150

100

50

0

10�1 100 101 102 103 104 105

10�1 100 101 102 103 104 105

Mag

nitu

de (

dB)

vB



FIGURE 8.44

The magnitude Bode plot for (a) the open-loop transfer function and (b) the closed-loop system.

0�

�90�

�180�

f M0 dB/dec

�20 dB/dec

0 dB

KdB

log v

�45�

1t1



Table 8.5

Bode Diagram Plots for Typical Transfer Function

�40 dB/dec

0 dB log v1t1

1t2

0�

�180�

f

f

0

�20

M



Table 8.5

Bode Diagram Plots for Typical Transfer Function (continued).

�60 dB/dec

�40 dB/dec0 dB log v1

t1

1t2

1t3

f0�

�180�

�270�

f M

0�20



Table 8.5


0 dB

�20 dB/dec

log v

f M

�90�

�180�



Table 8.5


�90�

�180�

f M�20 dB/dec

0 dB log v1t1

�40 dB/dec



Table 8.5


�180�

�270�

�90�

f M

0 dB

�60 dB/dec

1t1

�

�20

log v

1/t2



Table 8.5


�180�

�90�

f M

0 dB log v

�40 dB/dec

�20 dB/dec

1t1

1ta

1/t2

�20

�40f



Table 8.5


�180�

f M

0 dB log v

�40 dB/dec

f



Table 8.5


�180�

�270�

f M

0 dB

�60 dB/dec

log v

�40 dB/dec

f

1t1



Table 8.5


�180�

f M

0 dB log v

�40 dB/dec

�40 dB/dec

f

�20 dB/dec

1ta

1/t1



Table 8.5


�180�

�270�

f M

0 dB

�60 dB/dec

log v

f



Table 8.5


�180�

�270�

f M

0 dB

�60 dB/dec

log v

�40 dB/dec

f

1/ta



Table 8.5


�180�

�90�

�270�

f M

0 dB

�60 dB/dec

log v

�40 dB/dec

f

�20 dB/dec

1ta

1tb



Table 8.5


�180�

�90�

�270�

f M

0 dB

logv

1ta

1tb

1t1

1t2

1t3

1t4�40

�60

�20

�40

�20

�40

�60



Table 8.5


�180�

f M

0 dB 1ta

1t1

1t2

�40

�20

�40

�60

log v



Table 8.5


Negative v - Eastern Mediterranean...

Documents

Transcript of Negative v - Eastern Mediterranean...