w controller plant

��

�

�

�

�R Y

Controller Plant

D (s) G (s)

Sensor

H (s)

U

W

�

�

�V

Figure 5.1 Basic closed loop block diagram

©2002 Prentice Hall, Inc. Gene F. Franklin, J. David Powell, Abbas Emami-NaeiniFeedback Control of Dynamic Systems, 4e

�2 �1

Real axis

0 1 2�1.5

�1

�0.5

0

0.5

1

1.5

Imag

axi

s

u � sin�1 z � 30�

Figure 5.2 Root locus for L(s) = 1/[s(s + 1)]


�2 �1

Real axis

0 1 2�1.5

�1

�0.5

0

0.5

1

1.5

Imag

axi

s

Figure 5.3 Root locus vs. damping factor c for 1 + G(s) = 1 + 1/[s(s + c)] = 0


�2 �1

Real axis

0 1 2�3

�2

�1

0

1

2

3

Imag

axi

s

f2 � 0

f3

s0

c1

f4f1

Figure 5.4 Measuring the phase of Eq. (5.15)


�8 �2�4�6

Real axis

0 1 2�4

�2

2

4

Imag

axi

ss0

f3 � 0

f1

f2

Figure 5.5 Rule 2. The real-axis parts of the locus are to the left of an odd numberof poles and zeros.


�10 �5

Real axis

0 5�6

�4

�2

0

2

4

6

Imag

axi

s

Figure 5.6 The asymptotes are n − m radial lines from α at equal angles.


�10 �5

Real axis

0 5�6

�4

�2

0

2

4

6

Imag

axi

s

f2

Pole 1

Pole 3

f3

f1

Pole 2

s0

Figure 5.7 The departure and arrival angles are found by looking near a pole or zero.


�10 �5

Real axis

0 5�6

�4

�2

0

2

4

6

Imag

axi

s

jv crossing

Figure 5.8 A locus crosses the imaginary axis as determined by Routh’s criterion for stability.


�10 �5

Real axis

0 5�6

�4

�2

0

2

4

6

Imag

axi

s

Figure 5.9 Root locus for L(s) = 1/[s(s2 + 8s + 32)]


�5 0

Real axis

5 10�6

�4

�2

0

2

4

6

Imag

axi

s

Figure 5.10 Negative root locus corresponding to L(s) = (s − 6)/s(s2 + 4s + 13)


�6 �4

Real axis

�2 0 2�3

�2

�1

0

1

2

3

Imag

axi

s

Figure 5.11 Root locus for L(s) = G(s) = (s + 1)/s2


�6 �4

Real axis

�2 0 2�3

�2

�1

0

1

2

3

Imag

axi

s

Figure 5.12 Root locus for L(s) = (s + 1)/s2(s + 12)


�6 �4

Real axis

�2 0 2�3

�2

�1

0

1

2

3

Imag

axi

s



�15 �10

Real axis

0�5 5�8

�6

�4

�2

0

2

4

6

8

Imag

axi

sf5

c1

c2

c3

f1

f2, f3

f4

Figure 5.15 Figure for computing a departure angle for

L(s) = s + 1s + 12

(s + 0.1)2 + 62

s2[(s + 0.1)2 + 6.62]


�15 �10

Real axis

0�5 5�8

�6

�4

�2

0

2

4

6

8

Imag

axi

s

Figure 5.16 Root locus for L(s) = s + 1s + 12

(s + 0.1)2 + 62

s2[(s + 0.1)2 + 6.62]


�15 �10

Real axis

0�5 5�8

�6

�4

�2

0

2

4

6

8

Imag

axi

sf5

c1

f1

f2, f3

f4

Figure 5.17 Figure to compute a departure angle for

L(s) = s + 1s + 12

1s2[(s + 0.1)2 + 6.62]


�15 �10

Real axis

0�5 5�8

�6

�4

�2

0

2

4

6

8

Imag

axi

s

Figure 5.18 Root locus for L(s) = s + 1s + 12

1s2[(s + 0.1)2 + 6.62]


�4 �2

Real axis

0 2 4�3

�2

�1

0

1

2

3

Imag

axi

s

f3

f1f2

f4

Figure 5.19 Figure to compute departure angle for L(s) = 1s(s + 2)[(s + 1)2 + 4]


�4 �2

Real axis

0 2 4�3

�2

�1

0

1

2

3

Imag

axi

s

Figure 5.20 Root locus for L(s) = 1s(s + 2)[(s + 1)2 + 4]


s0 � s2

s0 � s1

z � 0.5

z � 0.5

s0 � s3

�10 6420�2�4�6�8�6

�4

�2

0

2

4

6

Imag

e ax

is

Real axis

Figure 5.21 Root locus for L(s) = 1/{s[(s + 4)2 + 16]} showing calculations of gain K


��

�

R Y

Controller Plant

G (s)D(s)

U

Figure 5.22 Feedback system with compensation


Imag

e ax

is

Real axis

�6 210�1�2�3�4�5�3

�2

�1

0

1

2

3

Figure 5.23 Root loci for 1 + D(s)G(s) = 0, G(s) = 1/[s(s + 1)]: with compensationD(s) = K (solid lines) and with D(s) = K(s + 2) (dashed lines)


Imag

e ax

is

Real axis

�6 210�1�2�3�4�5�3

�2

�1

0

1

2

3

Lead pole at �20

Lead pole at �10

PD control

Figure 5.24 Root loci for three cases with G(s) = 1/[s(s + 1)]:(a) D(s) = (s + 2)/(s + 20); (b) D(s) = (s + 2)/(s + 10);

(c) D(s) = s + 2 (solid lines)


�14 420�2�4�6�8�10�12Real axis

�8

�6

�4

�2

0

2

4

6

8

Rea

l axi

s

Damping � 0.5

K � 70

w � 7

Figure 5.25 Root locus for lead design


0 1.81.51.20.90.60.3

Time (sec)

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Am

plitu

de

Figure 5.26 Step response for Example 5.11


r0

Imag

e ax

is

�20 �18 �16 �14 �12 �10 �8 �6 �4 �2 0

Real axis

c

�8

�6

�4

�2

0

2

4

6

8

Figure 5.27 Construction for placing a specific point on the root locus


Imag

e ax

is

�20 50�5�10�15

Real axis

�8

�6

�4

�2

0

2

4

6

8

Figure 5.28 Root locus for L(s) = s + 5.4s(s + 20)(s + 1)


0 1.61.20.80.4

Time (sec)

1.4

1.2

1

0.8

0.6

0.4

0.2

0

Am

plitu

de

Figure 5.29 Step response for K = 127 and L(s) = s + 5.4(s + 20)

1s(s + 1)


Imag

e ax

is

�20 50�5�10�15

Real axis

(a)

�8

�6

�4

�2

0

2

4

6

8

Imag

e ax

is

0 0.05 0.1 0.15 0.2�0.2 �0.05�0.1�0.15

Real axis

(b)

�0.15

�0.1

�0.05

0

0.05

0.1

0.15

Figure 5.30 Root locus with both lead and lag compensations: (a) whole locus; (b) portion ofpart (a) expanded to show the root locus near the lag compensation


Imag

e ax

is

�20 50�5�10�15

Real axis

�8

�6

�4

�2

0

2

4

6

8

Figure 5.31 Root locus with lead, lag, and notch compensations


Am

plitu

de

0.9 1.2 1.5 1.80 0.3 0.6

Time (sec)

0

0.5

1

1.5

Figure 5.32 Step response with lead, lag, and notch compensations


Vin V0�

C Rf

R1 R2

Figure 5.33 Possible circuit of a lead compensation


Am

plitu

de

0 0.2 0.4 0.6 0.8 1 1.2 21.81.61.4

Time (sec)

0

0.5

1

1.5Digital controller

Continuous controller

Am

plitu

de

0 0.2 0.4 0.6 0.8 1 1.2 21.81.61.4

Time (sec)

(a)

(b)

0

0.5

1

1.5

Digital controller

Continuous controller

Figure 5.34 Comparison of analog and digital control (a) Output responses (b) Control responses


Ri

R1 R2

C

�

�

Figure 5.35 Possible circuit of lag compensation


Trim tab dt

Elevator de

(a)

(b)

Figure 5.36 Auto-pilot design in the Piper Dakota, showing elevator and trim tab(Photo courtesy of Denise Freeman)


d t(s) KD (s)1 ��

� e(s)�

Md(s)

��

��

�

de(s) G (s) u (s)

ur(s)de(s)

G (s) u (s)

(a) (b)

Figure 5.37 Block diagrams for auto-pilot design: (a) open loop;(b) feedback scheme excluding trim control


Re(s)

Im(s)

Re(s)

Im(s)

�2�3�4

1.0

0.5

�0.5

�1.0

Im(s)

5

10

�5�10�15�20

Im(s)

�5

�10

K � 1.5

K � 1.5

K � 0.3

K � 0.3

Proportional feedbackLead compensator

Figure 5.38 Root loci for auto-pilot design


1.5�

1.0�

0.5�

0�0 1 2 3 4 5 6

Lead compensator

Proportional feedbacku

Time (sec)

Figure 5.39 Time-response plots for auto-pilot design


��

�

�

Md

��

��

�

G (s) u ur G (s) u

(a) (b)

DI (s)

Md

d t1

sKI

��

�ur KD (s)

de�

��

Figure 5.40 Block diagram showing the trim-command loop


Re(s)

Im(s)

Re(s)

Im(s)

�1�2�3�4

1.5

1.0

0.5

�0.5

�1.5

�1.0

Im(s)

5

10

15

�5�10�15�20

Im(s)

�5

�10

�15

Figure 5.41 Root locus versus KI : assumes an added integral term and lead compensationwith a gain K = 1.5; roots for KI = 0.15 marked with •


0.2�

0.1�

0�

�0.1�

�0.2�

de

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

(a)

0 5 10 15 20 25 30 35 40 45 50

Time (sec)

(b)

6�

4�

2�

0�

u

Figure 5.42 Step response for the case with an integral term and 5◦ command


��

�

R Ys � 1

1s1KA �

�

�

1

KT

Figure 5.43 Block diagram of a servomechanism structure including tachometer feedback


Re(s)

Im(s)Im(s)

Figure 5.44 Root locus of closed-loop poles of the system in Fig. 5.42 vs. KT


Re(s)

Im(s)Im(s)

Figure 5.45 Root locus vs. K1 = KA + 4 after choosing KT = 1


Re(s)

Im(s)

2

1 1

Im(s)

2 2

2

1�1�2�3 2 3

2

1

�1

�2

Figure 5.46 Poles and zeros of the Padé approximants to e−s , with superscripts identifying thecorresponding approximants; for example, x1 represents the (1, 1) approximant


Im(s)

Re(s)

No delay

Delay approximated by

first-order lag, Eq. (5.54)

Delay approximated by

Padé (2, 2) approximant, Eq. (5.53)

Delay � e�5s (exact)

0.6

0.4

0.2

�0.2

�0.4

�0.6

�0.5 0.50

Figure 5.47 Root loci for the heat exchanger with and without time delay


u

y

�N

N

a

�a

u

y

u

y

a

�a

u

y

�N

N

�N

N

a

�a

u

y

�N

N

�q

�2q

2q

q

u

y

(a) (b) (c)

(d) (e) (f)

Slope K0

Slope K0

Figure 5.48 Nonlinear elements with no dynamics: (a) saturation, (b) relay, (c) relay with deadzone, (d) gain with dead zone, (e) preloaded spring, or coulomb plus viscous friction,

and (f) quantization


�u�

K

N/a

a Input magnitude

Gai

n

Figure 5.49 General shape of the effective gain of saturation


��

�

r yK0.4�0.4

�0.4

0.4

Saturation

s2s � 1

Figure 5.50 Dynamic system with saturation


Re(s)

Im(s)Im(s)

�1

K � 1

Figure 5.51 Root locus of (s + 1)/s2 , the system in Fig. 5.50 with the saturation removed


20

18

16

14

12

10

8

6

4

2

00 5 10 15 20 25 30

Time (sec)

12

8y

10

6

4

r � 2

Figure 5.52 Step responses of system in Fig. 5.50 for various input step sizes


��

�

R YK

1

s3

(s � 1)2

1

u

Figure 5.53 Block diagram of a conditionally stable system


Imag

e ax

is

�6 �1�2�3�4�5 20 1

Real axis

�3

�2

�1

0

1

2

3

Figure 5.54 Root locus for G(s) = (s + 1)2/s3 from system in Fig. 5.53


Am

plitu

de

0 2018161412108642

Time (sec)

r � 3.475

32

1

�10123456789

Figure 5.55 Step responses of system in Fig. 5.52


��

�

R YK 0.10.1

0.1

s2 � 0.2s � 11

s1U

Figure 5.56 Block diagram of a system with an oscillatory mode


Imag

e ax

is

�2 �1.5 �1�2.5 �0.5�3 0.5 10

Real axis

�1.5

�1

�0.5

0

0.5

1

1.5

K � 0.5

Figure 5.57 Root locus for the system in Fig. 5.56


0 50 100 150

Time (sec)

0

1

2

3

4

5

6

7

8

9

Am

plitu

de

r � 8

r � 4

r � 1

Figure 5.58 Step responses of system in Fig. 5.56


Re(s)

Im(s)Im(s)Im(s)

�0.5�1.0

0.5

�0.5

�10.0

Figure 5.59 Root locus including compensation


��

�

R Y0.1

0.1

s2 � 0.2s � 11

s1u

21

(s � 10)2s2 � 0.18s � 0.81

0.81100

Figure 5.60 Block diagram of the system with a notch filter


255 10 15 200 35 40 45 5030

Time (sec)

0

0.5

1

1.5

2

2.5

3

3.5

4

40.5

Am

plitu

de

r � 4

r � 2

Figure 5.61 Step responses of the system in Fig. 5.60


(a) (b) (c)

(d) (e) (f)

Figure 5.62 Pole-zero maps


��

YR 5s � 10

2s1

s � lK1

0.1 0.2

��

�

��

�

�

Figure 5.63 Control system for Problem 11


��

Rc � s

c � 16ss29

Y�

Figure 5.64 Control system for Problem 12


Y��

�R G (s)

H (s)

Figure 5.65 Feedback system for Problem 5.15


��

�R YK

s(s2 � 4s � 5)s � 3

s � 11



Y��

�R

s(s � 2)5

1 � as

Figure 5.67 Control system for Problem 5.17


��

�R YK

(a)

��

�R YK

(b)

s (s � 2)(s2 � 2s � 10)s � 2

s(s � 5)(s � 6)(s2 � 2s � 1)s2 � s � 2

Figure 5.68 Feedback systems for Problem 5.20


��

�R Y

s � 13s � 1( )K

s2(s2 � 100)s2 � 81



��

Y(s)R (s) 10 ��

��s1

Ks

s1

��

�



Y��

�R D(s) G(s)

Figure 5.71 Unity feedback system for Problems 5.24 to 5.30 and 5.35


u

iv0

Photodetector

e

Light

Solenoid

Ball

x

Figure 5.72 Elementary magnetic suspension


x��

�

H(s)

s21

Sensor

Fc F

Figure 5.73 Block diagram for rocket-positioning control system


Y��

�

1

s21

R K �

�

W2

��

�

W1

�

(s � 6 � 2j)(s � 6 � 2j)100

s�

�

�



Y��

�D (s) �

��

W

s(s � 1)1

R



Gears�

��

KA

va

1000 �

1000 �C

ef

Amplifier Motor

Outputpotentiometer

Inputpotentiometer

u i uo

km

Jm

JL

Feedback filter

Kpot Kpot

eo

ei

Figure 5.76 Positioning servomechanism (Reprinted from Clark, 1962, with permission)


Trolley or cart

y

u

Figure 5.77 Figure of cart-pendulum for problem 5.39


0� 10� 20� 30� 40� 50�

Figure 5.78 USCG cutter Tampa (902)


uu

��

��

��

Elevatorservo

s � 10Ku r

Me

Mp Aircraftdynamics

s2 � 4s � 5s � 3

s1

KTu

Rate gyro

KTAttitudesensor

1

Figure 5.79 Golden Nugget Airlines autopilot


��

�R Y

s2 � 51s � 5501

D(s)u

H(s)

s1

Sensor

Compensator



��

�R Y

sK2K1(1 � ) s (s � 1)(s � 0.5)

1E U



Y��

�KR

s3(s � 1)2e u



Y��

�KR �

�

�

s � 11

s1

KT



��

�a0

u i

��

�

gs

s(ts � 1)1 uo



��

�KR Y

s � 0.011

e�s



w controller plant

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