
Modem
Control Systems
ELEVENTH EDITION
Richard C. Dorf
University Clf California, Davis
Robert H. Bishop
The University of Texas at Austin
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Contents
CHAPTER
1
CHAPTER
2
Preface
XUl
About the Authors xxv
Introduction to Control Systems 1
1.1
Introduction 2
1.2
Brief History of Automatic Control 4
1.3
Examples of Control Systems 8
1.4
Engineering Design 16
1.5
Control System Design 17
1.6
Mechatronic Systems 20
1.7
The Future Evolution of Control Systems 24
1.8
Design Examples 25 .
1.9
Sequential Design Example: Disk Drive Read System 28
1.10
Summary 30
Exercises 30
Problems 31
Advanced Problems 36
Design Problems 38
Terms and Concepts 39
Mathematical Models of Systems 41
2.1
Introduction 42
2.2 .
Differential Equations of Physical Systems 42
2.3
Linear Approximations of Physical Systems 47
2.4
The Laplace Transform 50
2.5
The Transfer Function of Linear Systems 57
2.6
Block Diagram Models 71
2.7
SignalFlow Graph Models 76
2.8
Design Examples 82
2.9
The Simulation of Systems Using Control Design Software 102
2.10
Sequential Design Example: Disk Drive Read System 117
2.11
Summary 119
Exercises 120
Problems 126
Advanced Problems 137
Design Problems 139
Computer Problems 140
Terms and Concepts 142
v
vi
CHAPTER
3
CHAPTER
4
CHAPTER
5
Contents
State Variable Models 144
3.1
Introduction 145
3.2
The State Variables of a Dynamic System 145
3.3
The State Differential Equation 149
3.4
SignalFlow Graph and Block Diagram Models 154
3.5
Alternative SignalFlow Graph and Block Diagram Models 165
3.6
The Transfer Function from the State Equation 170
3.7
The Time Response and the State Transition Matrix 172
3.8
Design Examples 176
3.9
Analysis of State Variable Models Using Control Design Software 189
3.10
Sequential Design Example: Disk Drive Read System 192
3.11
Summary 196
Exercises 197
Problems 199
Advanced Problems 207
Design Problems 208
Computer Problems 210
Terms and Concepts 211
Feedback Control System Characteristics 212
4.1
Introduction 213
4.2
Error Signal Analysis 215
4.3
Sensitivity of Control Systems to Parameter Variations 217
4.4
Disturbance Signals in a Feedback Control System 220
4.5
Control of the Transient Response 225
4.6
SteadyState Error 228
4.7
The Cost of Feedback 231
4.8
Design Examples 232
4.9
Control System Characteristics Using Control Design Software 246
4.10
Sequential Design Example: Disk Drive Read System 251
4.11
Summary 255
Exercises 257
Problems 261
Advanced Problems 267
Design Problems 270
Computer Problems 273
Terms and Concepts 276
The Performance of Feedback Control Systems 277
5.1
Introduction 278
5.2
Test Input Signals 278
5.3
Performance of SecondOrder Systems 281

f
CHAPTER
6
CHAPTER
7
Contents
5.4
Effects of a Third Pole and a Zero on the SecondOrder System
Response 287
5.5
The sPlane Root Location and the Transient Response 293
5.6
The SteadyState Error of Feedback Control Systems 295
5.7
Performance Indices 303
5.8
The Simplification of Linear Systems 312
5.9
Design Examples 315
5.10
System Performance Using Control Design Software 329
5.11
Sequential Design Example: Disk Drive Read System 333
5.12
Summary 337
Exercises 337
Problems 341
Advanced Problems 347
Design Problems 349
Computer Problems
3~1
Terms and Concepts 353
The Stability of Linear Feedback Systems 355
6.1
The Concept of Stability 356
6.2
The RouthHurwitz Stability Criterion 360
6.3
The Relative Stability of Feedback Control Systems 368
6.4
The Stability of State Variable Systems 370
6.5
Design Examples 373
6.6
System Stability Using Control Design Software 382
6.7
Sequential Design Example: Disk Drive Read System 390
6.8
Summary 393
Exercises 394
Problems 396
Advanced Problems 400
Design Problems 402
Computer Problems 404
Terms and Concepts 406
The Root Locus Method 407
7.1
Introduction 408
7.2
The Root Locus Concept 408
7.3
The Root Locus Procedure 413
7.4
Parameter Design by the Root Locus Method 431
7.5
Sensitivity and the Root Locus 437
7.6
ThreeTerm (PID) Controllers 444
7.7
Design Examples 447
7.8
The Root Locus Using Control Design Software 458
7.9
Sequential Design Example: Disk Drive Read System 463
vii
viii
CHAPTER
8
CHAPTER
9
Contents
7.10
Summary 465
Exercises 469
Problems 472
Advanced Problems 482
Design Problems 485
Computer Problems 490
Terms and Concepts 491
Frequency Response Methods 493
8.1
Introduction 494
8.2
Frequency Response Plots 496
8.3
Frequency Response Measurements 517
8.4
Performance Specifications in the Frequency Domain 519
8.5
Log Magnitude and Phase Diagrams 522
8.6
Design Examples 523
8.7
Frequency Response Methods Using Control Design Software 534
8.8
Sequential Design Example: Disk Drive Read System 540
8.9
Summary 541
Exercises 546
Problems 549
Advanced Problems 559
Design Problems 561
Computer Problems 564
Terms and Concepts 566
Stability in the Frequency Domain 567
9.1
Introduction 568
9.2
Mapping Contours in the sPlane 569
9.3
The Nyquist Criterion 575
9.4
Relative Stability and the Nyquist Criterion 586
9.5
TimeDomain Performance Criteria in the Frequency Domain 594
9.6
System Bandwidth 601
9.7
The Stability of Control Systems with Time Delays 601
9.8
Design Examples 606
9.9 PID
Controllers in the Frequency Domain 620
9.10
Stability in the Frequency Domain Using Control Design Software 621
9.11
Sequential Design Example: Disk Drive Read System 629
9.12
Summary 632
Exercises 640
Problems 646
Advanced Problems 656
Design Problems 659
Computer Problems 663
Terms and Concepts 665
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Contents
CHAPTER
10
The Design of Feedback Control Systems 667
10.1
Introduction 668
10.2
Approaches to System Design 669
10.3
Cascade Compensation Networks 671
10.4
PhaseLead Design Using the Bode Diagram 675
10.5
PhaseLead Design Using the Root Locus 681
10.6
System Design Using Integration Networks 688
10.7
PhaseLag Design Using the Root Locus 691
10.8
PhaseLag Design Using the Bode Diagram 696
10.9
Design on the Bode Diagram Using Analytical Methods 700
10.10
Systems with a Prefilter 702
10.11
Design for Deadbeat Response 705
10.U
Design Examples 707
10.13
System Design Using Control Design Software 720
10.14
Sequential Design Example: Disk Drive Read System 726
10.15
Summary 728
Exercises 730
Problems 734
Advanced Problems 744
Design Problems 747
Computer Problems 753
Terms and Concepts 755
CHAPTER
11
The Design of State Variable Feedback
Systems 756
11.1
1L2
11.3
11.4
11.5
11.6
11.7
11.8
11.9
11.10
11.11
11.U
Introduction 757
Controllability and Observability 757
FullState Feedback Control Design 763
Observer Design 769
Integrated FullState Feedback and Observer 7.73
Reference Inputs 779
Optimal Control Systems 781
Internal Model Design 791
Design Examples 795
State Variable Design Using Control Design Software
Sequential Design Example: Disk Drive
Read System
Summary
812
Exercises 812
Problems 814
Advanced Problems 818
Design Problems 821
Computer Problems 824
Terms and Concepts 826
804
810
ix
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Contents
CHAPTER
12
Robust Control Systems 828
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
Introduction 829
Robust Control Systems and System Sensitivity 830
Analysis of Robustness 834
Systems with Uncertain Parameters 836
The Design of Robust Control Systems 838
The Design of Robust PIDControlled Systems 844
The Robust Internal Model Control System 850
Design Examples 853
The PseudoQuantitative Feedback System 870
Robust Control Systems Using Control Design Software 871
Sequential Design Example: Disk Drive Read System 876
Summary 878
Exercises 879
Problems 881
Advanced Problems 887
Design Problems 891
Computer Problems 897
Terms and Concepts 899
CHAPTER
13
Digital Control Systems 901
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
13.12
13.13
Introduction 902
Digital Computer Control System Applications 902
SampledData Systems 904
The
z
Transform 907
ClosedLoop Feedback SampledData Systems 912
Performance of a SampledData, SecondOrder System 916
ClosedLoop Systems
with
Digital Computer Compensation 918
The Root Locus of Digital Control Systems 921
Implementation of Digital Controllers 925
Design Examples 926
Digital Control Systems Using Control Design Software 935
Sequential Design Example: Disk Drive Read System 940
Summary 942
Exercises 942
Problems 944
Advanced Problems 946
Design Problems 947
Computer Problems 949
Terms and Concepts 950
Index
Absolute stability,
A system
description that reveals
whether a system is stable
or not stable without
consideration of other system
attributes such as degree
of stability, 356,406
Acceleration error constant,
Ka
The constant evaluated as
lim[s2G(s)]. The steady
S'O
state error for a parabolic
input, r(t)
=
At
2
/2, is equal
to A/Ka, 298
Acceleration input, steadystate
error, 297298
Accelerometer, 71,83
Ackermann's formula, 756,
767768,772,777778,
809810,816
Acrossvariable, 43,45
Actuator,
The device that
causes the process to provide
the output; the device that
provides the motive power
to the process, 62, 142
Additive perturbation,
A system
perturbation model
expressed in the additive
form
Ga(s)
=
G(s)
+
A(s),
. where G( s) is the nominal
plant,
A(s)
is the perturba
tion that is bounded in
magnitude, and
Ga(s)
is
the family of perturbed
plants, 834, 899
Agricultural systems, 13
Aircraft, and computeraided
design, 19
unmanned, 15
Aircraft autopilot, 853
Aircraft attitude control, 319
Airplane control, 266,474475,
482,747748
Allpass network,
A nonmini
mum phase system that
passes all frequencies with
equal gain, 513514,566
Alternative signal flow graph, and
block diagram models,
165170
Amplidyne, 127
Amplifier, feedback, 219
Amplitude quantization error,
906907, 950
Analogous variables, 47
Analogtodigital converter,
902,906
Analysis of robustness, 834836
Angle of departure,
The angle at
which a locus leaves a com
plex pole in the splane,
422423,426,441443,491
Angle of the asymptotes,
The
angle that the asymptote
makes with respect to the real
axis,
<P
A,
415,418,491
Armaturecontrolled motor,
6465,69,81,94,117,127,137,
139
Array operations in MathScript,
979
Array operations in MATLAB,
959960
Artificial hand, 11,14,36
Assumptions,
Statements that
reflect situations and condi
tions that are taken for grant
ed and without proof.
In
control systems, assump
tions are often employed to
simplify the physical dynami
cal models of systems under
consideration to make the
control design problem more
tractable, 42,8384,142
Asymptote,
The path the root
locus follows as the parame
ter becomes very large and
approaches infinity, 415
of root locus, 415
Asymptote centroid,
The center
of the linear asymptotes,
(TA,
416
Asymptotic approximation for a
Bode diagram, 502
Automatic control, history of, 48
Automatic fluid dispenser,
200,202
Automatic test system, 795797
Automation,
The control of an
industrial process by
automatic means, 6, 39
Automobile steering control
system,
9
Automobiles, hybrid fuel
vehicles, 21, 40
Auxiliary polynomial,
The equa
tion that immediately
precedes the zero entry in
the Routh array, 365,496
Avemar ferry hydrofoil, 736
Axis shift, 369
Backward difference rule,
A
computational method of
approximating the time
derivative of a function
given by x(kT) ""
x(kT)  x((k  l)T)
T
where
t
=
kT, T is the sample
time, and
k
=
1,2, ... ,
925,950
Bandwidth,
The frequency at
which the frequency response
has declined 3 dB from its
lowfrequency value, 520,
566,596,665
Bellman,
R.,
7
Biological control system, 14
Black, H. S., 56, 8, 130, 830
Block diagram,
Unidirectional,
operational block that
represents the transfer
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Index
functions of the elements of
the system, 71,72
Block diagram models, 7176,
107116
alternative signalflow graphs,
165170
signalflow graphs, 154165
Block diagram transformations,
7374
Bobbin drive, 856
Bode, H.
w.,
500, 830
Bode plot, The logarithm of
magnitude of the transfer
function is plotted
versus
the
logarithm of
w,
the frequen
cy. The phase,
qy,
of the trans
fer function is separately
plotted
versus
the logarithm
of the frequency, 500501,
541,567
asymptotic approximation,
502
Boring machine system, 232
Bounded response, 356
Branch on signalflow graph, 76
Break frequency, The frequency
at which the asymptotic ap
proximation of the frequen
cy response for a pole (or
zero) changes slope, 502,
505,566
Bre)akaway point, The point on
the real axis where the locus
departs from the real axis of
the splane, 418420,491
Bridge, Tacoma Narrows, 357359
Camera control, 308312,341
Canonical form, A fundamental
or basic form of the state
variable model representa
tion, including phase variable
canonical form, input feedfor
ward canonical form,
diagonal canonical form,
and Jordan canonical
form, 211
Capek, Karel, 10
Cascade compensation network,
A compensator network
placed in cascade or series
with the system process,
671675,755
Cauchy's theorem,
If
a contour
encircles Z zeros and
P
poles
of
F
(s ),
the corresponding
contour encircles the origin of
the F(s)plane
N
=
Z 
P
times clockwise, 568,
571575,665
Characteristic equation, The
relation formed by equating to
zero the denominator of a
transfer function, 52,142,387
Circles, constant, 596
Closedloop feedback control
system, A system that uses a
measurement of the output
and compares it with the
desired output, 3,39,214
Closedloop sampleddata
system, 912
Closedloop transfer function,
A ratio of the output signal to
the input signal for an inter
connection of systems when
all the feedback or feed
foward loops have been
closed or otherwise accounted
for. Generally obtained by
block diagram or signal flow
graph reduction, 74,142,
387388
Command following, An
impor
tant aspect of control system
design wherein a nonzero ref
erence input is tracked, 779,
826
Compensation, The alteration or
adjustment of a control
system to provide a suitable
performance, 668
using a phaseIag network on ,
the Bode diagram, 691
using a phaseIag network on
the splane, 692
using a phaselead network on
the Bode diagram, 675
using a phaselead network on
the splane, 681
using analytical methods,
700
using integration networks, 688
using statevariable feedback,
757
Compensator,
An
additional
component or circuit that is
inserted into the system
to equalize or compensate
for the performance
deficiency, 477,668,755,757
Compensator design, fullstate
feedback and observer, 773
Complementary sensitivity
function, The function
Gc(s)G(s)
C(s)
=
1
+
Gc(s)G(s)
that
satisfies the relationship
S(s)
+
C(s)
=
1, where
S(s)
is the sensitivity function. The
function
C(s)
=
T(s)
is the
closedloop transfer fUnction,
216,834,899
Complexity, A measure of the
structure, intricateness, or be
havior of a system that char
acterizes the relationships and
interactions between various
components, 16,39,276
in cost offeedback, 231 .
Complexity of design, The intri
cate pattern of interwoven
parts and knowledge
required, 16
Components, The parts, subsys
tems, or sub assemblies that
comprise a total system, 276
. in cost of feedback, 231
Computer control systems,
901,902
for electric power plant, 13
Computeraided design, 19
Computeraided engineering
(CAB), 21
Conditionally stable system, 475
Conformal mapping, A contour
mapping that retains the
angles on the splane on the
F
(s
)plane, 570,655
Congress, 14
Constant
M
circles, 597
Constant
N
circles, 597
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Index
Continuous design problem, 38,
139,208,270,349,402,485,
561,659,747,821,891,947
Contour map, A contour or
trajectory in one plane is
mapped into another plane
by a relation
F
(s),
569
Contours in the splane, 569
Control engineering, 2, 89
Control system, An interconnec
tion of components forming a
system configuration that will
provide a desired response,
2,39
characteristics using mfiles,
246
design, 17
modern examples, 816
Controllability, 757763
Controllability matrix, A linear
system is (completely)
controllable if and only if the
controllability matrix
Pc
=
[B AB A2B ...
AnB]
has full rank, where A is an
nxn matrix; for singleinput,
singleoutput linear systems,
the system is controllable if
and only if the determinant of
the nxn controllability matrix
Pc is nonzero, 758,826
Controllable system, A system
with unconstrained control
iIiput u that transfers any initial
state x(O) to any other state
x(t),
758,826
conv'function, 105,968
Convolution integral, 280
Corner frequency.
See
Break
frequency
Cost offeedback, 231232
Coulomb damper, 45
Critical damping, The case where
damping is on the boundary
between underdamped and
overdamped, 54,142
Criti~ally
damped system, 103
Damped oscillation, An oscilla
tion in which the amplitude
decreases with time, 56,142
Dampers, 45
Damping ratio, A measure of
damping; a dimensionless
number for the secondorder
characteristic equation, 54,
142,292
estimation of, 292
DC motor, An electric actuator
that uses an input voltage as a
control variable, 142
armature controlled, 64,81
field controlled, 63
Deadbeat response, 755
Decade, A factor of ten in fre
quency (e.g., the range of
frequencies from 1 rad/sec
to 10 rad/ sec is one decade),
502,566
of frequencies, 502
Decibel
(dB),
The units of the
logarithmic gain, 566
Decoupled state variable format,
166
Design, The process of con
ceiving or inventing the
forms, parts, and details of a
system to achieve a rea
soned purpose, 1617,39
Design of a control system, The
arrangement or the plan of the
system structure and the selec
tion of suitable components
and parameters, 755
robot control, 396
in the time domain, 757
using a phaselag network on
the Bode diagram, 696
using a phaselag network on
the splane, 691
using a phaselead network on
the Bode diagram, 675
using a phaselead network on
the splane, 681
using integration networks,
688
using statefeedback, 756
Design specifications, 278
Detectable, A system in which
the states that are unobserv
able are naturally stable,
761,826
1009
Diagonal canonical form,
A decoupled canonical form
displaying the
n
distinct
system poles on the diagonal
of the state variable represen
tation A matrix, 166,211
Differential equations, An equa
tion including differentials of
a function, 42,143
Differential operator, 50
Differentiating circuit, 68
Digital computer compensator,
A system that uses a digital
computer as the compensator
element, 918921
Digital control system, A con
trol system using digital
signals and a digital
computer to control a
process, 901950
Digital control systems using con
trol design software, 935
Digital controllers, implementa:
tion of, 925
Digital to analog converter, 905
Direct system, 213.
See also
Openloop control system
Discretetime approximation,
An approximation used to
obtain the time response
of a system based on the
division of the time into
small increments
At,
211
Disk drive read system.
See
Sequential design example
Disturbance rejection property,
221224
Disturbance signal,
An
unwanted
input signal that affects the sys
tem's output signal,
220225,276
Dominant roots, The roots of
the characteristic equation
that represent or
dominate the closedloop
transient response, 288,353,
427,491,521,566
Dynamics of physical systems, 41
Electric power industry, 13
Electric traction motor, 9395
.......

1010
Index
Electrohydraulic actuator,
66,
69,129
Engineering design, The process
of designing a technical
system, 1617,40
English channel tunnel boring
system, 232
Engraving machine, 523526,
537538
Epidemic disease, model of,
167168,372
Equilibrium state,
167
Error, steadystate,
228
Error constants, acceleration,
296
ramp, 297
step, 295
Error
signal, The difference
between the desired output,
R(s),
and the actual output
Yes);
therefore
E(s)
=
R(s)  Yes),
110,143,276
Estimation error, The difference
between the actual state and
the estimated state
e(t)
=
x(t)  x(t),
769,826
Euler's method, A firstorder
explicit integration method
utilized to obtain numerical
solutions of differential
equations, 211
Evans,
W.
R.,
408
Examples of control systems, 8
Exponential matrix function, 150
Extender, 135,206,742
Federal Reserve Board, 14
Feedback, 3
amplifier, 219
control system, 3,911,
720726
cost of, 231232
fullstate control design,
763
negative, 3,6
positive, 32
of state variables, 782,784
Feedback control system, and
disturbance signals, 220225
feedback function, 111113,968
Feedback signal, A measure of
the output of the system used
as feedback to control the
system, 3,40,110
Feedback systems, history of, 4
Final valne, The value that the
output achieves after all the
transient constituents of the
response have faded. Also
referred to as the steadystate
value, 54
ofresponse of
yet),
54
Final valne theorem, The theo
rem that states that lim
yet)
t>OO
=
lim
sY(s),
where
Yes)
is
s>O
the Laplace transform of
yet),
54
Flow graph.
See
Signalflow graph
Flyball governor,
A
mechanical
device for controlling the speed
of a steam engine, 45,40
Forward rectangular integration,
A
computational method of
approximating the integra
tion of a function given by
x(kT)
~
x((k1)T)
+
Ti((k 1)T), where
t
=
kT, T
is the sample time,
andk
=
1,2, ... , 925,950
Fonrier transform, The transfor
mation of a function of time,
f(t)
into the frequency
domain, 496
Fonrier transform pair, A pair of
functions, one in the time
domain, denoted by
f(t),
and
the other
in
the frequency
domain, denoted by
F(jeu),
related by the Fourier trans
form as
F(jeu)
=
9'{f(t)},
where
9'
denotes the
Fourier transform, 495496,
566
Frequency response, The steady
state response of a system
to a sinusoidal input
signal, 494,566
closedloop, 594
measurements, 517519.
plots, 497501,566
using control design software,
534
Fullstate feedback control law,
A control law of the form
u
=
J(x
where
x
is the state
of the system assumed known
at all times, 26,757
Fundamental matrix, 151.
See
also
Transition matrix
Future evolution of control
systems,
24
Gain margin, 622623,632,665
The reciprocal of the gain
Gc(s)G(s)
at the frequency at
which the phase angle reaches
180°, 588589
Gap, The void between what is
intended (or visualized) as the
product or device and the actu
al, practical form of the final
design, 16
Gear train, 67,70
Graphical evaluation of residues
of
F(s), 53
Graphics in MATLAB, 953,
961964
Gun controllers, 7
Gyroscope, 205
Hand, robotic, 11,14,36
Helicopter control, 472 473, 480
Help, 976
Highfidelity simulations,
90
History of automatic control, 4
Home appliances, 24
Homogeneity, The property of a
·linear system
in
which the
system response,
yet),
to an
input u(t) leads to the
response
{3y(t),
when the
input is
{3u(t),
4748,143
Hubble telescope, 315319
Hurwitz, Routh stability
criterion, 360368,373,
382384,406
Hybrid fuel vehicles, 21,40
Hydraulic actuator, 66,69,129,
812
Impulse signal, 278
Index of performance, 303,783
i
I
i
I
I
I
J
I
I
I
In]
I
J
Index
Inpnt feedforward canonical
form, A canonical form
described by
n
feedback
loops involving the
an
coeffi
cients of the
nth
order de
nominator polynomial of the
transfer function and feed
forward loops obtained by
feeding forward the input
signal, 161162,211
Input signals, 278
Instability, An attribute of a
system that describes a
tendency of the system to
depart from the equilibrium
condition when initially
displaced, 276
in cost offeedback, 231
Insulin delivery system, 2728
Integral of absolute magnitude of
error, 304
Integral of square of error, 303
Integral of time multiplied by
absolute error, 304
optimum coefficients of
T(s)
for, 308,312
Integral of time multiplied by error
squared, 304
Integral operator, 52
Integrating filter, 68
Integration network, A network
that acts, in part, like an
integrator, 674
Interactive Window,
972
Internal model design, A method
of tracking reference inputs
with guaranteed steadystate
tracking errors, 793794,826
Internal model principle, The
principle that states that
if
Gc(s)G(s)
contains the input
R(s),
then the output
yet)
will track
r(t)
asymptotically
(in the steadystate) and the
tracking is robust, 793,899
Internal Revenue Service, 14
, Inverse Laplace transform, 50,
52,5455
Inverted pendulum, 169,766,
775,778,818,820
lSE, 303,344
ITAE, 304
optimal coefficients of
T (s),
308,312
ITSE, 304
Jordan canonical form, A block
diagonal canonical form for
systems that do not possess dis
tinct system poles, 166,211
Kalman statespace decomposi
tion, A partition of the state
space that illuminates the
states that are controllable
and unobservable, uncontrol
lable and unobservable, con
trollable and observable, and
uncontrollable and
observable, 758,761,826
Kirchhoffvoltage laws, 147
Laboratory robot, 9899
Lag network.
See
Phase lag
network
Laplace transform, A transfor
mation of a function
f(t)
from the time domain into the
complex frequency domain
pes),
42,5057,143,145
Laplace transform pair, A pair
of functions, one in the time
domain, denoted by
f(t),
and
the other in the frequency
domain, denoted by
pes),
re
lated by the Laplace trans
form as
pes)
=
.;:E{f(t)},
where
.;:E
denotes the
Laplace transform,
51,495496,566
Laser manipulator control
system, 447448
Lead network.
See
Phase lead
network
LeadIag network, A network
with the characteristics of
both a lead network and a lag
network, 700
Linear approximation, An ap
proximate model that results
in a linear relationship be
tween the output and the
input of the device, 49,143
1011
Linear approximations of physical
systems, 4750
Linear quadratic regulator, An
optimal controller designed
to minimize the quadratic
performance index
J
=
100
(x:Qx
+
uTRu)
dt,
where
Q
and R are design
parameters, 791,826
Linear systems, 4748,143
simplification of, 312,332
Linearized, Made linear or
placed in a linear form;
Taylor series approximations
are commonly employed to
obtain linear models of
physical systems, 42,143
Locus, A path or trajectory that
is traced out as a parameter is
changed, 408,491
Logarithmic (decibel) measure,
A measure of the gain margin
defined as 20 10glO(lj
d),
1 1
where 
= ;;
d
IGH(jw)
I
when the phase shift is
180°, 588,665
Logarithmic magnitude, The
logarithm of the magnitude
of the transfer function
20 10glO
IGI,
503506,522,
550551,566
Logarithmic plot, 500501.
See
also
Bode plot
Logarithmic sensitivity, A mea
sure of the sensitivity of the
system performance to specific
parameter changes, given by
T _
aT(s)jT(s)
SK(S) 
aKjK
,where
T (
s )
is the system transfer
function and
K
is the parame
ter of interest, 437,491492
Logmagnitudephase diagram,
589
Loop on signalflow graph, 77
Loss of gain, A reduction in
the amplitude of the ratio of
the output signal to the
I
I
11
I!
11
IJ
I
'I
I
I
,
1
11
r!
li
11
11
11
11
~
I
i
, i
I
1012
Index
input signal through a
system, usually measured
in decibels, 276
in cost of feedback, 231
Lowfidelity simulations, 90
Lowpass filter, 99102
Isim function, 192,193,329,331,
332,825,936,968
Lunar landing vehicle, 734
M
circles, 597
Magnetic levitation, 131,742,896
Magnetic tape transport, 474
Manual control system, 10
Mapping of contours in the
splane, 569
Margin, gain, 588589,622623,
632,665
phase, 589,593,622623,632,
666,920921
margin function, 621,754, 968
Marginally stable, A system is
marginally stable if and only
if the zero input response
remains bounded as
t
7
00,
359,406
Mars rover, 235,403,486,896
Mason, 76
Mason loop rule, A rule that
enables the user to obtain a
transfer function by tracing
paths and loops within a
system, 143
Mason's signalflow gain formula,
77,88,128,156,172
Mathematical models, Descrip
tions of the behavior of a
system using mathematics,
42,143
Mathematical models of systems,
41120
MathScript, 971992
Functions and scripts, 984
MATLAB,953
basics, 953970
Bode plot, 534
control system characteristics,
246
functions, 966, 968969
graphics, 953,961964
mathematical functions, 955
plots, 962
scripts, 953,964967
simulation of systems, 102
and state variables, 189
symbols, 967
and system performance, 329
Matrices for MathScript, 978980
Matrices for MATLAB, 959960
Matrix exponential function,
An
important matrix function,
defined as
eAt
=
I
+
At
+
(At)2/2!
+ ... +
(Atl/k!
+ ... ,
that plays a role in the solutiol,1
of linear constant coefficient
differential equations, 150,
211
Maximum overshoot, 283
Maximnm value of the frequency
response, A pair of
complex poles will result
in a maximum value for
the frequency response
occurring at the resonant
frequency, 508,519,566
Maxwell,
J.
c.,
5,8
Mechatronic systems, 2024
Metallurgical industry, 13
Microcomputer, A small personal
computer
(PC)
based on a mi
croprocessor, 902,950
Microelectromechanical systems
(MEMS), 21
Minicomputer, A standalone
computer with size and perfor
mance between a microcom
puter and a large mainframe.
The term is not commonly
used today, and computers in
this class are now often known
as midrange servers, 902,950
Minimum phase, All the zeros of
a transfer function lie in the
lefthand side of the splane,
511,566
Minorsky, N., 121
Mobile robot, 298
Model creation, Simulink, 985
Model of, DC motor,
62
epidemic disease, 167168,372
hydraulic actuator, 66,69,
129,812
inverted pendulUm and cart
169,766,775,778,818,820'
Motor, DC, 142
Multiloop reduction, 113
Multiple loop feedback system, 75
Multiplicative perturbation,
A system perturbation model
expressed in the multiplica
tive form
Gm(s)
=
G(s)[l
+
M(s)J,
where G
(s)
is the nominal
plant,
M(s)
is the perturba
tion that is bounded in mag
nitude, and
Gm(s)
is the
family of perturbed plants,
834,835,899
Multivariable control system,
A system with more than one
input variable and more than
one output variable, 3,40
N
circles, 597
Natural frequency, The frequency
of natural oscillation that oc
curs for two complex poles
when the damping equals
zero, 54,143,566
Necessary condition, A condi
tion or statement that must"
be satisfied to achieve a de
sired effect or result. For ex
ample, for a linear system it is
necessary that the input
Ul (t)
+
U2(t)
results in the
response
Yl(t)
+
Y2(t),
where
the input
Ul (
t)
results in the
·response
Yl(t)
and the input
U2(t)
results in the response
Y2(t),
47,143
Negative feedback, The output
signal is fed back so that it
subtracts from the input
signal, 3,40
ngrid function, 621, 624, 969
Nichols chart, A chart display
ing the curves for the
relationship between the
openloop and closedloop
frequency response,
597601,621,624626,632,
665,870
Index
nichols function, 621,624,969
Nodes of signal flow graph,
77
Noise, 214216,220225,231,
239,241,256259,269275
Nomenclature, 45
Nomninimum phase, Transfer
functions with zeros in the
righthand splane, 510511,
513514,566
Nonunity feedback systems,
300303
Nuclear reactor controls, 32,200
Number of separate loci, Equal to
the number of poles of the
transfer function assuming that
the number of poles is greater
than the number of zeros of
the transfer function, 415
Numerical experiments, 90
Nyquist, H., 568
Nyquist contour, 575
Nyqnist criterion, A feedback
control system is stable
if,
and
only if, for the contour in the
Gc(s)G(s)
plane the number
of counterclockwise encir
clements ofthe (1,0) point
is equal to the number of
poles of
Gc(s)G(s)
with posi
tive real parts, 567,568,
575586,601,632,665666
nyquist function, 621, 969
Nyquist stability criterion, A
feedback system is stable if,
and only if, the contour in the
Gc(s)G(s)
plane does not en
circle the
(1,
0) point when
the number of poles of
Gc(s)G(s)
in the righthand
splane is zero., 567,568,
575586,601,632,665666
Observability, 757763
Observability matrix, A linear
system is (completely) observ
able
if
and only
if
the observ
ability matrix
Po
=
[CT(CA)T
(CA
2
l ...
(CAnl
f
has
full
rank, where A is an
nxn
matrix; for singleinput,
singleoutput linear systems,
the system is observable
if
and
only
if
the determinant of the
nxn
observability matrix
Po
is
nonzero, 761,827
Observable system, A system
with an output that possesses
a component due to each
state variable, 760
Observer, A dynamic system
used to estimate the state of
another dynamic system
given knowledge of the sys
tem inputs and measurements
of the system outputs, 827
Observer design, 769772
Octave of frequencies, 503, 520
Openloop control system, A
system that utilizes a device
to control the process with
out using feedback, 23,
40,218
Operational amplifier, 729,878
Operators, differential and
integral, 50
Optimal coefficients of
T(s)
for
ITAE, 308,312
Optimal control system, A sys
tem whose parameters are
adjusted so that the perfor
mance index reaches an
extremum value, 781791,
827
Optimization, The adjustment of
the parameters to achieve the
most favorable or advanta
geous design, 17, 40
Optimize parameters, 17
Output equation, The algebraic
equation that relates the state
vector, x, and the inputs,
u,
to
the outputs,
y,
through the rela
tionship
y
=
Cx
+
Du, 149,
211
Overdamped, The case where the
damping ratio is
t
>
1,
103,143
Overshoot, The amount the
system output response
proceeds beyond the desired
response, 249
1013
Pade approximation of a time
delay, 604606, 621
pade function, 621,626,898,969
Parabolic input signal, 279
parallel function, 110,113,969
Parameter design, A method of
selecting one or two parame
ters using
the root locus
method, 432,492
Parameter variations and system
sensitivity, 217
Parkinson, D.
B.,
7
Path on signalflow graph, 77
Peak time, The time for a system
to respond to a step input
and rise to a peak response,
283,353
Pendulum oscillator, 49
Percent overshoot, 283,353
for a secondorder system, 286
Performance index, A quantitative
measure of the performance of
a system, 303312,353
Performance of a control
system,
277
Performance specifications in the
frequency domain, 519,682
PhaseIag compensation, A
widelyused compensator
that possesses one zero and
one pole with the pole closer
to the origin of the splane.
This compensator reduces
the steady state tracking er
rors, 674
PhaseIag network, A network
that provides a negative
phase angle and a significant
attenuation over the
frequency range of
interest, 674675,755
Phaselead compensation,
A widelyused compensator
that possesses one zero and
one pole with the zero closer
to the origin of the splane.
This compensator increases
the system bandwidth and
improves the dynamic
response, 670673,729
1014
Index
Phaselead network,
A network
that provides a positive
phase angle over the
frequency range of
interest, 671
Phase lock loop (detector), 397
Phase margin,
The phase angle
through which the
GeG(jw)
locus must be rotated so that
the unity magnitude point
passes through the (
1,
0)
point in the
GeG(jw)
plane,
589,593,622623,632,666
Phase variable canonical form,
A canonical form described
by
n
feedback loops
involving the
an
coefficients
of the nth order denomina
tor polynomial of the
transfer function and m
feedforward loops involving
the b
m
coefficients of the
moth order numerator
polynomial of the transfer
function, 158, 211
Phase variables,
The state
variables associated with the
phase variable canonical
form, 158
Physical state variables,
147148
Physical variables,
The state
variables representing the
physical variables of the
system, 165166,211
PID controller,
A widely used
controller used in industry of
the form
Ge(s)
=
Kp
+
K
J
 +
KDs,
where
Kp
is the
s
proportional gain,
K
J
is the
integral gain, and
KD
is
the derivative gain,
444447,492,899,950
in frequency domain, 620621
Plant, 12,
See
Process
Plotting using MATLAB, 116,
962,969
Polar plot,
A plot of the real
versus
the imaginary part of
Gp(jw), 497,566
Pole placement,
A design
methodology wherein the
objective is to place the
eigenvalues of the closed
loop system in desired
regions of the complex
plane, 758,827
Poles,
The roots of the denomi
nator polynomial (i.e., the
roots of the characteristic
equation) of the transfer
function, 5253,143
Polezero map, 106108
Political feedback model, 15
poly
function, 104, 388, 824, 969
polyval
function, 105,969
Polzunov,
1.,
5
Pontryagin,
L.
S., 78
Position error constant,
Kp
The constant evaluated as
lim
GeG(s).
The steadystate
S'O
error for a step input (of
magnitude
A)
is equal to
A/(l
+
Kp), 296,353
Positive feedback,
The output
signal is fed back so that it
adds to the input signal,
40,75
Potentiometer, 70
Power plants, 12
Precision,
The degree of
exactness or discrimiation
with which a quantity is
stated, 906, 950
Prefilter,
A transfer function
G
p(
s) that filters the input sig
nal
R(
s) prior to the calcula
tion of the error signal,
702705,755,899
Principle of superposition,
The
law that states that
if
two in
puts are scaled and summed
and routed through a linear,
timeinvariant system, then
the output will be identical to
the sum of outputs due to the
individual scaled inputs when
routed through the same
system, 47,143
Principle of the argument, 572.
See also
Cauchy's theorem
Printer belt drive, 183189
Process,
The device, plant, Or
system under control, 40
Process controller.
See
PID
controller
Productivity,
The ratio of ph .
YSl
cal.output to physical input of
an mdustrial process, 6, 40
Proportional plus deriviative (PD)
controller,
A twoterm con
troller of the form
Ge(s)
=
Kp
+
KDs,
where
K
is the proportional gain and
P
K
D
is the derivative gain,
445,492,755
Proportional plus integral (PI)
controller,
A twoterm
controller of the form
K
J
Ge(s)
=
Kp
+ ,
where
K
s
p
is the proportional gain and
K
J
is the integral gain, 445,
492,689,755
Pseudoquantitative feedback
system, 870871
pzmap
function, 106107,141,
969
Rack and pinion, 67, 71
Ramp input, optimum coefficients
ofT(s), 312
steadystate error, 297
test signal equation, 279
Reference input,
The input to a
control system often repre
senting the desired output,
. denoted by
R(s),
110,143,
779781
Regulator problem, 764
Regulatory bodies, 14
Relative stability,
The property
that is measured by the rela
tive real part of each root or
pair of roots of the character
istic equation, 356,368,406
by the Nyquist criterion,
586593
by the RouthHurwitz criterion,
368
Remote manipulators, 205,738
Remotely operated vehicle,
607610,607629
Index
Residues, The constants
k
i
asso
ciated with the partial frac
tion expansion of the output
Y(s),
when the output is
written in a residuepole
format, 53,55,56,143
Resouant frequeucy, The fre
quency, w" at which the maxi
mum value of the frequency
response of a complex pair of
poles is attained, 507508,
566
Rise time, The time for a system
to respond to a step input and
attain a response equal to the
magnitude of the input, 283,
353
Risk, Uncertainties embodied in
the unintended consequences
of the design, 16
Robot, A programmable
computer integrated with a
reprogrammable, multifunc
iional manipulator used for a
variety of tasks, 1011,40
design of laboratory, 98
mobile, steering control, 331
Robot control system, 448452
Robust control, A system that
exhibits the desired
performance in the
presence of significant
plant uncertainty,
8,828900
using control design software,
871875
Robust PID control, 844850
Robust stability criterion, A
test for robustness with
respect to multiplicative
perturbations in which
stability is guaranteed if
IM(jw)1
<
11
+
G(~w)
I,
for
all
w,
where M
(s)
is the mul
tiplicative perturbation,
834835,899
Root contours, The family of loci
that depict the effect of vary
ing two parameters on the
roots of the characteristic
equation, 436,492
Root locus, The locus or path of
the roots traced out on the
splane as a parameter is
changed, 408412,492,632,
921925
angle of departure, 422
asymptote, 415
breakaway point, 418
concept, 408412
of digital control systems, 921
and sensitivity, 437444
steps in sketching, 424
using control design software,
458463
in the zplane, 922923
Root locus procedure, The
method for determining
the locus of roots of the
characteristic equation
1
+
KP(s)
=
o
as Kvaries
from 0 to infinity, 413431
parameter design, 431436,
492
Root locus segments on the real
axis, The root locus lying in
a section of the real axis to the
left of an odd number of poles
and zeros, 414,416,492
Root sensitivity, The sensitivity
of the roots as a parameter
changes from its normal
value; or the incremental
change in the root divided by
the proportional change of
the parameter, 437,492,
830,900
roots function, 107,383,387388,
969
Rotor winder system, 707711
RouthHurwitz criterion, A cri
terion for determining the
stability of a system by exam
ining the characteristic equa
tion of the transfer function,
360368,373,382384,406
RouthHurwitz stability, 355
Sampled data, Data obtained for
the system variables only at
discrete intervals; data
obtained once every sampling
period, 904,950
1015
Sampled data system, A system
where part of the system acts
on sampled data (sampled
variables), 904907,950
Sampling period, The period
when all the numbers leave or
enter the computer; the period
for which the sampled variable
is held constant, 904,950
Saving and loading data files, 988
Script Editor window, MathScript,
972
Scripts, 953,964
comments, 965
defined, 964
header, 965
invoking, 964
TeX characters, use of, 965,
967
Second order system response,
effects of third pole and zero,
287293
Secondorder system, perfor
mance of, 281287
Selfbalancing scale, 427
Semiconductors, 12
Sensitivity.
See also
System
sensitivity
of control systems to parameter
variations, 217220
of roots of control systems, 437
Sensitivity function, The func
tion
S(s)
=
[1
+
Gc(s)G(S)fl that satis
fies the relationship
S.(s)
+
C(s)
=
1, where C(s)
is the complementary sensi
. tivity function, 216,221,224,
243,256,834,900
Separation principle, The full
state feedback law and the ob
server can be designed
independently and when
connected
will
function as
an integrated control system
in the desired manner (i.e.,
stable), 763,774,827,
Sequential design example,
2830,117119,192196,
251255,333337,390393,
463465,629632,726728,
810812,876878,940941
1016
Index
Series connection, 109
series
function, 109,112,113,
969
Settling time,
The time required
for the system output to
settle within a certain
percentage of the input
amplitude, 284
Ship stabilization, 261,736
Signalflow graph,
A diagram
that consists of nodes con
nected by several directed
branches and that is a graphi
cal representation of a set of
linear relations, 7682
and block diagram models,
154165
models, 76
Simplification of linear systems,
312
Simulation,
A model of a system
used to investigate the
behavior of a system by
utilizing actual input
signals, 90,102116,143
Social feedback model, 15
Space shuttle, 554555,647649,
942943
Space station, 176182
Space telescope, 853856
Spacecraft, 122,141,176182
Specifications,
Statements that
explicitly state what the de
vice or product is to be and is
to do; a set of prescribed
per~
formance criteria, 16, 40
Speed control system, 221223,
226228,246248,263,266,
270,271
for automobiles, 263
for power generator,
473474
for steel rolling mill, 221
splane,
The complex plane
where, given the complex num
ber s
=
u
+
jw,
the xaxis (or
horizontal axis) is the saxis,
and the yaxis (or vertical axis)
is the jwaxis, 143
Springmassdamper system,
103106
Stability,
A performance
measure of a system; a
system is stable if all the
poles of the transfer function
have negative real parts,
356,406
in the frequencydomain,
567666
of linear feedback systems,
355406
of state variable systems,
370373
for unstable process, 388
using the Nyquist criterion,
575
using the RouthHurwitz
criterion, 360368,373,
382384
Stabilizable,
A system in which
the states that are not control
lable are naturally stable,
758,827
Stabilizing controller,
A con
troller that stabilizes the
closedloop system,
775;
827
Stable system,
A dynamic sys
tem with a bounded system
response to a bounded
input, 356,406
State differential equation,
The
differential equation for the
state vector:
i
=
Ax
+
Bu,
149154,211
State of a system,
A set of num
bers such that the knowledge
of these numbers and the
input function will, with the
equations describing the dy
namics, provide the future
state of the system,
145148,211
State transition matrix,
<l>(t),
The matrix exponential func
tion that describes the un
forced response of the
system, 151,211
State variable models, 143
State variable system design using
control design software,
804810
State variables,
The set of
variables that describe the
system, 144211
of a dynamic system, 145148
State vector,
The vector matrix
containing all
n
state vari
ables,
xl, X2,' .. , X
n
,
149,211
Statespace representation,
A
timedomain model COm
prised of the state differential
equation, i
=
Ax
+
Bu,
and
the output equation,
y
=
ex
+
Du,
150,
190192,211
Statevariable feedback,
When
the control signal for the
process is a direct function of
all the state variables, 206
211,827 '
Steady state,
The value that the
output achieves after all
the transient constituents
of the response have faded.
Also referred to as the final
value, 54,143
of response of
y(t),
54
Steadystate error,
The error
when the time period is large
and the transient response
has decayed, leaving the con
tinuous response, 228231
of feedback control system,
295
Steadystate response,
The
constituent of the system
response that exists a long
time following any signal
initiation, 278, 353
Steel rolling
mill,
13,221,602604,
659,662,883,885
Steering control system, of auto
mobile, 9,561
of mobile robot, 298
of ship, 651
Step input, 295297
optimum coefficients of T
(s),
308
steadystate error, 295
test signal equation, 278
Submarine control system, 200,
202203
r
I
Index
Superposition, principle of, 47
Symbols, in MATLAB, 967
used in book, 45
Syntax, 977
Synthesis, The process by which
new physical configurations
are created. The combining of
separate elements or devices to
form a coherent whole, 17, 40
sys function, 106,109110
System, An interconnection of
elements and devices for a
desired purpose, 2
System sensitivity,
See also
Sensitivity
The proportional change of the
transfer function of a system
to a proportional change
in the system parameter, 900
Systems with uncertain
parameters, 836
Tables, of differential equations
for elements, 44
of Laplace transform pairs, 51
through and acrossvariables
for physical systems, 43
of transfer function plots,
633640
of transfer functions, 6871
Tachometer, 70
Tacoma Narrows Bridge, 357359
Taylor series, A power series
defined by g(
x)
=
00
g(m)(xo)
2:
,(x  xo)m.
For
m=O
m.
m
<
CXJ
series is an approxi
mation which is used to
linearize functions and
system models, 4849,143
Test input signal, An input sig
nal used as a standard test of
a system's ability to respond
adequately, 278281,353
Thermal heating system, 71
Threeterm controller.
See
PID
controller
Throughvariable, 4243,45
Time constant, The time interval
necessary for a system to
change from one state to
another by a specified per
centage. For a first order sys
tem, the time constant is the
time it takes the output to
manifest a 63.2% change due
to a step input, 58,143
Time delay, A pure time delay,
T, so that events occurring at
time
t
at one point in the sys
tem occur at another point in
the system at a later time,
(t
+
T), 601606,666
Time domain, The mathematical
domain that incorporates the
time response and the descrip
tion of a system in terms of
time,
t,
145,211
design, 757
Timedomain specifications, 329
Time response, by a discretetime
evaluation, 171
and state transition matrix,
172175
Timevarying control system, A
system for which one or more
parameters may vary with
time, 145
Tracked vehicle turning control,
373375,384387
Tradeoff, The need to make a
judgment about how much
compromise is made between
conflicting criteria, 1,16,40
Transfer function in the frequency
domain, The ratio of the
output to the input signal
where the input is a sinusoid,
expressed as
G(jw), 500,
566
Transfer function(s), The ratio
of the Laplace transform of
the output variable to the
Laplace transform of the
input variable, 57,143
of complex system, 82
of DC motor, 62
of dynamic elements and
networks, 6871
of hydraulic actuator, 66
of interacting system, 79
1017
of linear systems, 57
in mfile script, 106
minimum phase and nonrnini
mum phase, 511
of multipleloop system, 81
table of dynamic elements and
networks, 6871
Transient response, The con
stituent of the system re
sponse that disappears with
time, 225,276,278,354
relationship to the root loca
tion, 293
of a secondorder system, 282
Transition matrix,
<I>(t),
The ma
trix exponential function that
describes the unforced
response of the system,
151,211
evaluation by signal flow graph
methods, 173
TwinT network, 510
Type number, The number, N, of
poles of the transfer function,
Gc(s)G(s),
at the origin.
Gc(s)G(s)
is the forward path
transfer function, 206, 298,
354
Uncertain parameters, 836
Underdamped, The case where
the damping ratio is {;
<
1,
46,103,143
Unit impulse, A test input con
sisting of an impulse of infi
nite amplitude and zero
Width, and having an area of
unity; used to determine the
impulse response, 354
Unity feedback, A feedback
control system wherein the
gain of the feedback loop is
one, 110,143
Unmanned aerial vehicles
(UAVs), 15
Unstable system, 357
Variables for physical systems, 43
Velocity error constant,
Kv,
The
constant evaluated as limit
for a type one system. The
Index
steady state error for a ramp
input for a type one system is
equal to
AI
Kv,
297,354
Velocity input, 297
Vertical takeoff aircraft (VTOL),
394,646,824
Viscous damper, 45
Vyshnegradskii,
l.A.,
5
Water clock, 4
Water level control, 45,33,
8693,136
Watt, James, 4, 8
Welding control, 367368
Wind power, 2223
Worktable motion control,
928934
Xy
plotter, 711713
Zeroorder hold,
A mathemati
cal model of a sample and
data hold operation whose
1018
~nputoutput
transfer fUnction
IS
represented by
1 
e
sT
Go(s)
== ,
906950
s
Zeros,
The roots of the numera
tor polynomial of the transfer
function, 5253,143
Zplane root locus, 922923
Ztransform, 907912,950
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