continuous controller

Plant

��

�

D (s) G (s)r (t)u (t)

y (t)

y (t)

1

Continuous controller

e (t)

e (kT)

Sensor

��

�

r (t) Differenceequations

u (kT)

Plant

G (s) y(t)

1

Sensor

u (t)

ClockSampler

Digital controller

T

T

y(kT)

r (kT)

y(t)

(b)

(a)

D/A andhold

A/D

Figure 8.1 Block diagrams for a basic control system: (a) continuous system;(b) with a digital computer

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

u

kT1 2 3 4 5 6 7 8

u(kT )

Average u(t)

Continuous control, u(t)

Control from D/A

Figure 8.2 The delay due to the hold operation


f (t)

t

f (t)

f (k)

0 T 2T 3T

T

Figure 8.3 A continuous, sampled version of signal f


Re(z)

Im(z)

1.2

�1.0 �0.8 �0.6 �0.4 �0.2 0 0.2 0.4 0.6 0.8 1.0

1.0

0.8

0.6

0.4

0.2

0

10T9p

5T4p

10T7p

5T2p

5T3p

10T3p

10Tp

20Tp

vn �2Tp

vn �Tp z �

1.0

z � 0

s � �zvn �jvn�1 � z 2

T � sampling period

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2 0.1

z � eTs

5T1p

Figure 8.4 Natural frequency (solid color) and damping loci (light color) in the z-plane; theportion below the Re(z)-axis (not shown) is the mirror image of the upper half shown.


Re(z)

Im(z)

�1 �0.5 10.5

Figure 8.5 Time sequences associated with points in the z-plane


e(kT)e(t)

TD (z)

u(kT)ZOH u(t) e(t) D (s) u(t)

(a) (b)

Figure 8.6 Comparison of (a) digital and (b) continuous implementation


t

e(t)

kT � T kT

Figure 8.7 Trapezoidal integration


1 21.81.61.41.20 0.2 0.4 0.6 0.8

Time (sec)

(b)

�20�10

01020304050

Con

trol

, u1 21.81.61.41.20 0.2 0.4 0.6 0.8

Time (sec)

(a)

00.20.40.60.8

11.21.4

Posi

tion,

y



Digital controller

Digital controller

Figure 8.8 Comparison between the digital and the continuous controller step response with asample rate 25 times bandwidth. (a) Position, (b) Control


��

�

R D (s)s21 YE U

Figure 8.9 Continuous-design definition for Example 8.2


Re(s)

Im(s)

�0.2

�0.2�2

Selected roots(Kc � 0.81)

Figure 8.10 s -plane locus with respect to K


r (k) u (t)u (k)

y (k)

ZOH s21

Sampler (T � 1 sec)and analog-to-digital

conversion

PlantComputer

Eq.(8.29)

y(t)

Figure 8.11 A digital control system for emulating Fig. 8.9


Time (sec)

0 5 10 15 20 25 30

Plan

t out

put

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Continuous design

Emulation design

Figure 8.12 Step responses of the continuous and digital implementations


v (rad/sec)

T � 1/15 secvs � 100 rad/sec

T � 1/3 secvs � 20 rad/sec

�D (z)�

1

�D (z)�

1

v (rad/sec)0.5

0.10.1

MMPZTustin’s,MPZ

D (s) D (s)

MMPZ

Tustin's,

MPZ

5 50 0.5 5 50

Figure 8.13 A comparison of the frequency response of three discrete approximations


D (z)��

�ZOHR(s) Y(s)

(a)

D (z)��

�G (z) Y(z)R(z)

(b)

G(s)

Figure 8.14 Comparison of a (a) mixed control system and (b) its pure discrete equivalent


Re(s)

Im(s)

Re(z)

Im(z)

z � a

z � 0

z � �1

(a) (b)

s � �a

Figure 8.15 Root loci for (a) the z-plane and (b) the s -plane


Re(z)

Im(z)

�1

Figure 8.16 z-plane root locus for a 1/s2 plant with proportional feedback


Real axis

�1.0 �0.5 0 0.5 1.0

1.0

0.8

0.6

0.4

0.2

�0.2

�0.4

�0.6

�0.8

�1.0

Imag

axi

s

Desiredroot

Desiredroot

Figure 8.17 z-plane locus for the 1/s2 plant with D(z) = K(z − 0.85)/z


Time (sec)

0 5 10 15 20 25 30

Plan

t out

put

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Continuous design

Emulation design

Discrete design

Figure 8.18 Step response of the continuous and digital systems in Examples 8.2 and 8.4


Time (sec)

0 5 10 15 20 25 30

Plan

t out

put,

y(t)

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0

Command structure from Fig. 7.46(b)

Command structure from Fig. 7.10

Figure 8.19 Step response of Example 8.7


Time (sec)0.02 0.08 0.1

50-Hz samples

Aliased signalfrom samples

60-Hz signal

Figure 8.20 An example of aliasing


�

�

�

U

z � 0.26 X1

z � 0.35

E

X2

Figure 8.21 An example of parallel implementation


��

�

G (s) YKRT

Figure 8.22 Control system for Problem 8.9


u

Inertialreference

Figure 8.23 Satellite control schematic for Problem 8.14


Figure 8.24 Schematic of magnetic levitation device for Problems 8.16 and 8.17


Figure 8.25 Satellite-tracking antenna (Courtesy Space Systems/Loral)


u

Figure 8.26 Schematic diagram of satellite-tracking antenna


Negligible heatloss in walls

Hot

Cold

Tank fluid attemperature Te

Te

Tei

Tec

Mixingvalve

Figure 8.27 Tank temperature control


��

�

YRE

s(s � 1)1

K (1 � TDs � TIs1 ) U

Figure 8.28 Control system for Problem 8.21


continuous controller

Documents