كتاب Continuous Signals and Systems with MATLAB - Third Edition
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منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب Continuous Signals and Systems with MATLAB - Third Edition

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Continuous Signals and Systems with MATLAB - Third Edition
Authored by
Taan S. ElAli
Embry-Riddle Aeronautical University

كتاب Continuous Signals and Systems with MATLAB - Third Edition  M_c_s_13
و المحتوى كما يلي :


Contents
Preface .xi
About the Author . xiii
Acknowledgment .xv
Chapter 1 Signal Representation .1
1.1 Examples of Continuous Signals .1
1.2 The Continuous Signal 1
1.3 Periodic and Nonperiodic Signals .2
1.4 General Form of Sinusoidal Signals 4
1.5 Energy and Power Signals .5
1.6 The Shifting Operation 7
1.7 The Refection Operation 8
1.8 Even and Odd Functions 10
1.9 Time Scaling 12
1.10 The Unit Step Signal 14
1.11 The Signum Signal 16
1.12 The Ramp Signal . 16
1.13 The Sampling Signal . 17
1.14 The Impulse Signal 18
1.15 Some Insights: Signals in the Real World .20
1.15.1 The Step Signal 20
1.15.2 The Impulse Signal 20
1.15.3 The Sinusoidal Signal . 21
1.15.4 The Ramp Signal 22
1.15.5 Other Signals 22
1.16 End-of-Chapter Examples .22
1.17 End-of-Chapter Problems 38
Chapter 2 Continuous Systems . 47
2.1 Defnition of a System . 47
2.2 Input and Output 47
2.3 Linear Continuous System 47
2.4 Time-Invariant System 50
2.5 Systems Without Memory . 52
2.6 Causal Systems 52
2.7 The Inverse of a System 54
2.8 Stable Systems . 55
2.9 Convolution 56
2.10 Simple Block Diagrams . 58
2.11 Graphical Convolution . 61viii Contents
2.12 Differential Equations and Physical Systems 66
2.13 Homogeneous Differential Equations and Their
Solutions 66
2.13.1 Case When the Roots Are All Distinct 67
2.13.2 Case When Two Roots Are Real and Equal 67
2.13.3 Case When Two Roots Are Complex . 67
2.14 Nonhomogeneous Differential Equations and Their
Solutions 68
2.14.1 How Do We Find the Particular Solution? .69
2.15 The Stability of Linear Continuous Systems: The
Characteristic Equation .73
2.16 Block Diagram Representation of Linear Systems . 76
2.16.1 Integrator 76
2.16.2 Adder 77
2.16.3 Subtractor .77
2.16.4 Multiplier 77
2.17 From Block Diagrams to Differential Equations . 78
2.18 From Differential Equations to Block Diagrams .79
2.19 The Impulse Response . 81
2.20 Some Insights: Calculating y(t) 83
2.20.1 How Can We Find These Eigenvalues? 84
2.20.2 Stability and Eigenvalues .84
2.21 End-of-Chapter Examples .85
2.22 End-of-Chapter Problems 109
Chapter 3 Fourier Series . 119
3.1 Review of Complex Numbers 119
3.1.1 Defnition 119
3.1.2 Addition 119
3.1.3 Subtraction . 119
3.1.4 Multiplication . 119
3.1.5 Division 120
3.1.6 From Rectangular to Polar . 121
3.1.7 From Polar to Rectangular . 121
3.2 Orthogonal Functions 122
3.3 Periodic Signals .124
3.4 Conditions for Writing a Signal as a Fourier Series Sum 124
3.5 Basis Functions 124
3.6 The Magnitude and the Phase Spectra 126
3.7 Fourier Series and the Sin-Cos Notation .126
3.8 Fourier Series Approximation and the Resulting Error 130
3.9 The Theorem of Parseval 131
3.10 Systems with Periodic Inputs 132
3.11 A Formula for Finding y(t) When x(t) Is Periodic: The
Steady-State Response . 134Contents ix
3.12 Some Insight: Why the Fourier Series 136
3.12.1 No Exact Sinusoidal Representation for x(t) 136
3.12.2 The Frequency Components . 136
3.13 End-of-Chapter Examples . 137
3.14 End-of-Chapter Problems 148
Chapter 4 The Fourier Transform and Linear Systems 153
4.1 Defnition . 153
4.2 Introduction . 153
4.3 The Fourier Transform Pairs . 154
4.4 Energy of Nonperiodic Signals . 167
4.5 The Energy Spectral Density of a Linear System . 168
4.6 Some Insights: Notes and a Useful Formula . 168
4.7 End-of-Chapter Examples . 170
4.8 End-of-Chapter Problems 183
Chapter 5 The Laplace Transform and Linear Systems . 191
5.1 Defnition . 191
5.2 The Bilateral Laplace Transform . 191
5.3 The Unilateral Laplace Transform 191
5.4 The Inverse Laplace Transform . 193
5.5 Block Diagrams Using the Laplace Transform . 198
5.5.1 Parallel Systems . 198
5.5.2 Series Systems 199
5.6 Representation of Transfer Functions as Block
Diagrams . 200
5.7 Procedure for Drawing the Block Diagram from the
Transfer Function . 201
5.8 Solving LTI Systems Using the Laplace Transform 203
5.9 Solving Differential Equations Using the Laplace
Transform 205
5.10 The Final Value Theorem 208
5.11 The Initial Value Theorem 208
5.12 Some Insights: Poles and Zeros .208
5.12.1 The Poles of the System .209
5.12.2 The Zeros of the System .209
5.12.3 The Stability of the System 209
5.13 End-of-Chapter Examples .209
5.14 End-of-Chapter Problems 233
Chapter 6 State-Space and Linear Systems 245
6.1 Introduction .245
6.2 A Review of Matrix Algebra .246
6.2.1 Defnition, General Terms, and Notations .246x Contents
6.2.2 The Identity Matrix 246
6.2.3 Adding Two Matrices .246
6.2.4 Subtracting Two Matrices 247
6.2.5 Multiplying a Matrix by a Constant .247
6.2.6 Determinant of a 2 × 2 Matrix .247
6.2.7 Transpose of a Matrix 248
6.2.8 Inverse of a Matrix .248
6.2.9 Matrix Multiplication .248
6.2.10 Diagonal Form of a Matrix 249
6.2.11 Exponent of a Matrix .249
6.2.12 A Special Matrix 250
6.2.13 Observation 251
6.2.14 Eigenvalues of a Matrix . 251
6.2.15 Eigenvectors of a Matrix 252
6.3 General Representation of Systems in State Space . 271
6.4 General Solution of State-Space Equations
Using the Laplace Transform 272
6.5 General Solution of the State-Space Equations in
Real Time 272
6.6 Ways of Evaluating e At 273
6.6.1 First Method: A Is a Diagonal Matrix 273
˜ a b ˝
ˆ˙
˛°
6.6.2 Second Method: A Is of the Form 273
0 a
6.6.3 Third Method: Numerical Evaluation, A of
Any Form . 273
6.6.4 Fourth Method: The Cayley–Hamilton
Approach . 274
6.6.5 Fifth Method: The Inverse Laplace Method 276
6.6.6 Sixth Method: Using the General
Form of Φ(t) = e At and Its Properties 277
6.7 Some Insights: Poles and Stability 282
6.8 End-of-Chapter Examples .283
6.9 End-of-Chapter Problems 326
Index 337xi
Index
Note: Page numbers in italics indicate fgures and bold indicate tables in the text.
Φ(t) = eAt, 277–281
A
AC voltage source, 21
Adder block diagram, 77, 77; see also Block
diagrams
Addition; see also Subtraction
complex numbers, 119
of two matrices, 246
Algebraic equation, 67, 72–73, 278, 281; see also
Equations
auxiliary, 84
variable s, 209, 253
Analytical solutions, 8
Angular frequency, 4; see also Frequency
Anti-aircraft gun, 22
Approximation; see also Equations
Fourier series, 130–131
impulses, 21
signals, 129, 139
Auxiliary equations, 71
algebraic, 84
characteristic, 82
Average power, 6, 39, 147–148
B
Band-limited signals, 186; see also Signals
Basis functions, Fourier series, 124–125
BIBO, see Bounded-input bounded-output
(BIBO)
Bilateral Laplace transform, 191
Block diagrams
adder, 77, 77
differential equations, 78, 78–81, 80–81
Laplace transform and linear systems, 198,
198–203
parallel systems, 198–199, 199
series systems, 199, 199–200
transfer function, 200–202, 200–203
representation of linear systems, 76–77
adder block diagram, 77, 77
integrator block diagram, 76, 76
multiplier block diagram, 77, 77
subtractor block diagram, 77, 77
three-integrators, 202, 202
Bounded-input bounded-output (BIBO), 55–56
Bounded magnitudes, 136; see also Magnitude
C
Capacitance, 45
Capacitors
charging and discharging, 1
current, 267
terminal, 1
voltage, 304–305
Causal systems, 52–53, 53
Cayley–Hamilton theorem, 274–276, 279
Characteristic equations, 274–275, 290; see also
Equations
auxiliary, 82
coeffcients, 73
complex roots, 67–68
continuous systems, 73–76
examples, 100–103
Circuit
differential equation, 180
energy, 44, 44
output, 44, 44
single-phase, 30–31, 30–31
Coeffcients
characteristic equation, 73
constant, 66, 84
Fourier series, 136, 141
of s2, 203
stiffness, 299
Combination signals, 3; see also Signals
Complex conjugate functions, 164
Complex numbers, 119–122
addition, 119
defned, 119
division, 120–121
magnitude, 137
multiplication, 119–120
polar form, 120, 122, 148–149
from polar to rectangular, 121–122
rectangular form, 121, 149
from rectangular to polar, 121
subtraction, 119
Complex roots; see also Roots
characteristic equation, 67–68
negative roots, 283
Compressed signals, 12; see also Signals
Constants
coeffcients, 66, 84
force, 20, 22
numbers, 164338 Index
particular solution, 69
speed, 16
velocity, 16
Contents, frequency, 136, 153, 191
Continuous signals, 1–2, 2, 38; see also
Signals
Continuous systems, 47–116; see also Systems
block diagram representation of linear
systems, 76–77
adder block diagram, 77, 77
integrator block diagram, 76, 76
multiplier block diagram, 77, 77
subtractor block diagram, 77, 77
calculating, 83–85
eigenvalues, 84
stability and eigenvalues, 84–85
characteristic equation, 73–76
convolution, 56–58
differential equations, 66
to block diagrams, 79–81, 80–81
from block diagrams to, 78, 78–79
homogeneous, 66–68
nonhomogeneous, 68–72
examples, 85–108
graphical convolution, 61–65, 61–66
impulse response, 81–83
input signals, 47, 48
output signals, 47, 48
particular solution, 69, 69–72
problems, 109–116
simple block diagrams, 58–59, 58–61, 61
systems
causal, 52–53, 53
inverse, 54–55, 54–55
linear continuous, 47–50, 48
physical, 66
stability of linear discrete, 73–76
stable, 55–56
time-invariant, 50–52
without memory, 52
Continuous wave, 1
Convolution
continuous systems, 56–58
equation, 60, 132
Fourier transform, 162
frequency domain, 166, 167
graphical, 61–65, 61–66, 92–96
integrals, 57, 60, 64, 88, 304
property, 60, 163
time domain, 165
Coupled second-order differential equations, 292;
see also Differential equations
Critical damping, 290
Currents, 1, 30
capacitor, 267
inductor, 45
resistor, 44–45, 267
D
Damping; see also Underdamping
critical, 290
ratio, 290
Delta function, sifting property, 41
Demodulation, 166
Determinant of 2 × 2 matrix, 247
Diagonal matrix, 249–250, 262, 272–273
Diagrams, see Block diagrams
Differential equations, 66, 103, 133; see also
Equations
block diagrams, 78, 78–81, 80–81
circuit, 180
frst-order, 78, 82, 206, 245, 252, 271
Fourier transform, 181
Laplace transform, 205–207, 316
nonhomogeneous, 68–72
nth-order, 245, 282
physical systems and, 66
rotational system, 299
second-order, 67, 81, 83, 207, 209, 245, 256,
269, 292, 299
single-input/single-output linear secondorder, 255, 282
system, 78
third-order, 67, 83, 201
zero initial conditions, 221
Discontinuous signals, 1, 14, 18; see also Signals
Division, complex numbers, 120–121
Domain
Fourier frequency, 154–155
Fourier transform, 154
frequency, 147, 156
Dynamic matrix, 288
E
Eigenvalues, 260, 262, 274–275, 283, 285–286,
293, 307–308
continuous systems, 84
of matrix, 251
negative, 100
stability and, 84–85
systems, 67
Eigenvectors, 249, 252, 260–261, 285–286, 307
Electric
circuit, 14
signals, 47
switch, 14, 21–22
waves, 1
Electromagnetic signals, 1; see also Signals
Elevator system, 47
End of transmission, 166
Energy
calculation, 6
circuit, 44, 44Index 339
fnite, 28
inductor current, 45
infnite, 6
of nonperiodic signals, 167–168
signals, 5–6, 7, 27, 27–28, 38
spectral density, 167–168
storage element, 67
Equations; see also specifc equations
algebraic, 67, 72–73, 84, 209, 253, 278, 281
auxiliary, 71
characteristic, 274–275, 290
auxiliary, 82
coeffcients, 73
complex roots, 67–68
continuous systems, 73–76
examples, 100–103
convolution, 60, 132
differential, 66–67, 78, 78–79, 133
coupled second-order, 292
frst-order, 78, 82–83, 206, 245, 252,
271
nonhomogeneous, 68–72
nth-order, 245, 282
second-order, 81, 83, 207, 209, 245, 256,
269, 299
single-input/single-output linear secondorder, 255, 282
system, 78, 103
third-order, 67, 83, 201
homogeneous matrix, 252
of motion, 268, 268
output, 245, 271, 298, 300, 302
output matrix, 318–319
state, 245
state-space, 254, 259
Error, Fourier series, 130–131
Even signals, 10–11, 11–12, 23, 25–26, 26, 40;
see also Signals
Exponential form, Fourier transform, 172–173
Exponent of matrix, 249–250; see also Matrix
F
Feed-forward matrix, 271
Filter, low-pass, 167, 167
Final value theorem, 208, 219
Finite discontinuities, 136
Finite energy, 28
First-order differential equations, 67, 78, 82–83,
206, 245, 252, 271
First-order single-input single-output system, 253
Fluid-based system, 14; see also Systems
Force
constant, 22
impulsive, 21
magnitude, 21
signals, 45, 45
Formulae, Fourier transform and linear systems,
168–169
Fourier frequency domain, 154–155
Fourier series, 119–150
approximation and resulting error, 130–131
basis functions, 124–125
coeffcient, 136, 141
complex numbers, 119–122
addition, 119
defned, 119
division, 120–121
multiplication, 119–120
polar form, 120, 122
from polar to rectangular, 121–122
rectangular form, 121
from rectangular to polar, 121
subtraction, 119
conditions for writing signal, 124
examples, 137–148
formula for fnding y(t) when x(t) is periodic,
134–136
frequency components, 136
magnitude spectra, 126
no exact sinusoidal representation for x(t), 136
nonperiodic signals, 153
orthogonal functions, 122–123
Parseval equation, 131–132
periodic signals, 124, 124
phase spectra, 126
problems, 148–150
representation, 124, 130, 133–134
sin and cos functions, 126–129, 127–129, 138
steady-state response, 134–136
sum of, 159
systems with periodic inputs, 132–134, 134
Fourier transform
bilateral Laplace transform, 191
differential equation, 181
domain, 154
signals, 157, 183–184
Fourier transform and linear systems, 153–189
defned, 153
energy of nonperiodic signals, 167–168
energy spectral density of linear system, 168
examples, 170–183
formulae, 168–169
infnite periodic signal, 153, 154
overview, 153–154
pairs, 154–167, 155, 165–167
periodic signal, 153, 154
problems, 183–189
Free-body diagram, 268, 268, 296, 296, 299
Frequency
angular, 4
components of Fourier series, 136
contents, 136
domain, 147, 156I
340 Index
convolution, 166, 167
multiplication, 175–176
time domain, 179–180
natural, 293
power, 31
radian, 21
voltage, 31
G
Graphical convolution; see also Convolution
continuous systems, 61–65, 61–66
examples, 92–96
Graphical solutions, see Solutions
Gulf War, 21
H
Homogeneous difference equation and solution,
66–68
case when roots are all distinct, 67
case when two roots are complex, 67–68
case when two roots are real and equal, 67
Homogeneous matrix equation, 252
Identity matrix, 246, 249, 251, 257, 277, 282;
see also Matrix
Impulses
approximation, 21
function, 34
input, 92
integrals, 35
modeling, 21
response, 316–317, 321
continuous systems, 81–83
function, 168
inverse transform, 203, 204
linear time-invariant (LTI) systems, 60
systems, 58, 96–99, 162–163
signals, 18, 18–19
frequency domain, 156
Laplace transform, 194
mathematical abstraction, 20–21, 20–21
scaling property, 18
shifted, 59
sifting property, 56, 156–157
usage of MATLAB to simulate, 35, 36
weighted, 57
Impulsive force, 21, 267
Impulsive input, 224
Inductor
current, 45
voltage, 267
Infnite energy, 6
Infnite magnitude, 18, 21
Infnite periodic signal, 153, 154
Initial conditions, 69, 71–73, 104, 300
Initial value theorem, 208
Input; see also Output
bounded, 55–56
impulse, 92
integrator, 79
phase angle, 134
ramp, 224
RC circuit, 22
signals, 22, 49–50
continuous systems, 47, 48
electrical, 47
step, 304
systems, 54, 57, 66, 72
unit step, 91–92
voltage, 334
zero, 73
Instantaneous power, 6, 31, 44
Integrals
convolution, 57, 60, 64, 88, 304
evaluating, 34
Fourier transform, 170, 178
impulse, 35
Laplace transform, 194
for orthogonal functions, 123
sifting property, 19, 159
systems, 52
Integrand, 34
Integrator
block, 76, 76, 80
input, 79
output, 78
Intermediate output, 91
Inverse Laplace method, 276–277
Inverse Laplace transform, 193–198, 212, 254,
258, 269, 300, 303, 321
Inverse of matrix, 248
Inverse systems, 54–55, 54–55
Inverse transform, 157, 164, 203–204, 288
Isolated pulse, 168
L
Laplace domain, 272
Laplace method, inverse, 276–277
Laplace transform
bilateral, 191
defned, 191
differential equations, 316
impulse signal, 194
integrals, 194
inverse, 193–198, 212, 254, 258, 269, 300,
303, 321
matrix differential equation, 253
matrix state equation, 257
properties, 193Index 341
second derivative, 210
solving state-space equations using, 272
third derivative, 211
unilateral, 191–192
Laplace transform and linear systems, 191–243
bilateral Laplace transform, 191
block diagram, 198, 198–203
parallel systems, 198–199, 199
series systems, 199, 199–200
transfer function, 200–202, 200–203
examples, 209–233
fnal value theorem, 208
initial value theorem, 208
inverse Laplace transform, 193–198
poles and zeros, 208–209
poles of system, 209
stability of system, 209
zeros of system, 209
problems, 233–243
solving differential equations using,
205–207
solving LTI systems using, 203–205,
204–205
unilateral Laplace transform, 191–192
Linear combination, 245, 271
Linear systems, 57, 136; see also Laplace
transform and linear systems;
State-space and linear systems
analysis, 153
block diagram representation of, 76–77,
76–77
continuous, 47–50, 48
discrete continuous systems, 73–76
energy spectral density of, 168
examples, 85–86
Linear time-invariant (LTI) systems, 57, 60, 66,
84, 132, 168–169
Laplace transform, 203–205, 204–205
output, 207
transfer function, 198
Load resistor, 20, 39
Low-pass flter, 167, 167
LTI, see Linear time-invariant (LTI) systems
M
Magnitude
bounded, 136
complex number, 137
force, 20–21
functions, 64
infnite, 18, 21
output, 56
of signals, 2, 2, 4
spectra, 126, 165–166, 186
of step signal, 14, 14
voltage of constant, 20
MATLAB®
eigenvectors, 286
function roots, 73
impulse responses, 315
magnitude, 138
to plot signals, 29, 29–30
roots, 100–102
script, 31, 31, 76, 90, 92, 94–97, 99, 105, 107,
139, 142, 221, 229, 232
to simulate signals, 35
state-space representation, 317
sum of odd and even signals, 26, 27
Matrix
(2 × 2), 275–276
defned, 246
diagonal, 249–250, 262, 272–273
dynamic, 288
eigenvalues of, 251
exponent of, 249–250
feed-forward, 271
identity, 246, 249, 251, 257, 277, 282
inverse of, 248
multiplication, 248–249, 262
output equations, 318–319
roots of homogeneous equation, 252
state-space Laplace representation, 270
state-transition, 277
transition, 279
transpose of, 248
Matrix algebra, state-space and linear systems,
246–270
addition of two matrices, 246
determinant of 2 × 2 matrix, 247
diagonal form of matrix, 249
eigenvalues of matrix, 251
eigenvectors of matrix, 252
exponent of matrix, 249–250
identity matrix, 246
inverse of matrix, 248
matrix multiplication, 247–249
observation, 251
special matrix, 250
subtraction of two matrices, 247
transpose of matrix, 248
Mean squared error (MSE), 130–131
Memory, systems without, 52
Modeling impulse, 21
Modeling systems, 245
Modulation, 165
Moment of inertia, 299
Motion
rotational, 47
translational, 47
Multiple-input/multiple-output systems, 245, 271
Multiplication
complex numbers, 119–120
frequency domain, 175–176342 Index
integrand, 34
matrix, 247–249, 262
Multiplier block diagram, 77, 77
N
Natural frequencies, 293
Negative eigenvalues, 100
Negative roots, 85
Noncausal systems, 53
Nonhomogeneous differential equations and
solution, 68–72
Nonperiodic signals, 2–3
energy of, 167–168
Fourier series, 153
time-domain, 153
Nth-order differential equation, 245, 282; see
also Differential equations
Numbers; see also Complex numbers
constant, 164
rational, 3, 5, 29
Numerical evaluation, 273
O
Odd signals, 10–11, 11–12, 23, 25–26, 26, 40
Oil spill, 21
Operation
refection, 8, 8–9, 10
shifting, 7, 7–8, 24
Orthogonal functions, Fourier series, 122–123
Oscillations, 1
Output; see also Input
bounded, 55–56
circuit, 44, 44
equations, 245, 271, 298, 300, 302
initial value of, 224
integrator, 78
intermediate, 91
LTI system, 207
magnitude, 56
matrix equations, 318–319
s-domain, 218
signals of continuous systems, 47, 48
stable, 83
steady-state, 169
systems, 50, 52, 54, 57–59, 66, 71, 84, 87–92,
104–105
voltage, 47–48
P
Pairs
Fourier transform and linear systems,
154–167, 155, 165–167
Laplace transform, 192
Parallel systems, 198–199, 199; see also Systems
Parseval equation, 131–132; see also Equations
Parseval theorem, 148
Partial fraction expansion, 176, 188, 205, 254
Fourier transform, 163
inverse transform, 280, 304
roots are not complex, 197
roots of denominator, 196
time domain, 219
transfer function, 258
Particular solution, continuous systems, 69,
69–72
Periodic inputs, systems with, 132–134, 134
Periodicity, 2
Periodic signals, 2–3, 5–6, 29, 38, 44; see also
Signals
Fourier series, 124, 124
Fourier transform and linear systems, 153,
154
infnite, 153, 154
Phase angle, input, 134
Phase spectra, 126
Physical systems, 16
continuous systems, 66
differential equations and, 66
Polar form, complex numbers, 120, 122, 148–149
Polar to rectangular, complex numbers, 121–122
Poles, 67
stability of state-space and linear systems,
282–283
of system, 209
zeros and, 208–209
Positive roots, 85; see also Roots
Power
average, 6, 39, 147–148
frequency, 31
instantaneous, 6, 31, 44
signals, 5–6, 7, 27, 27
Property, see Sifting property; Symmetry
property
Pulse signals, 21; see also Signals
R
Radar system, 1, 22
Radian frequency, 21; see also Frequency
Ramp input, 224
Ramp signals, 16–17, 17; see also Signals
in real world, 22, 22
usage of MATLAB to simulate, 35, 37
Rational numbers, 3, 5, 28
Rectangular form, complex numbers, 121, 149
Rectangular pulse signals, 15, 15–16
Rectangular to polar, complex numbers, 121
Refection operation, 8, 8–9, 10
Resistors
current, 267
currents, 44–45Index 343
load, 20, 39
thermal, 302
voltage, 44, 47, 304–305
RLC circuit, 263, 263–267
Roots
algebraic equation, 67
complex, 67–68, 197, 283
of det(λI − A), 260
distinct, 67, 70
of homogeneous matrix equation, 252
negative, 85
positive, 85
real and equal, 67, 70
Rotational motion, 47
Rotational system, 299; see also Systems
Routh array, 73, 73–75, 107
Routh test, 73, 107
S
Sampling signals, 17, 17; see also Signals
Scaling, 25
property, 18
time, 12–14, 12–14, 24
Second-order differential equations; see also
Differential equations
block diagram, 81
input, 207, 299
output, 83, 207, 209
roots real and equal, 67
state variables, 245, 256, 269
Sending end of transmission, 165, 165
Series systems, 199, 199–200; see also Systems
Shifted impulse signals, 59
Shifted signal, 171–172
Shifting operation, 7, 7–8, 24
Shock, 20
Sifting property, 18, 59, 59, 88
delta function, 41
displacement and velocity angles, 301
impulse function, 298
impulse signal, 56, 156–157
integrals, 19, 159
Signals
approximation, 129, 139
band-limited, 186
combination, 3
compressed, 12
continuous, 1–2, 2, 38
discontinuous, 1, 14
electrical, 14, 47
as electric waves, 1
electromagnetic, 1
energy, 5–6, 7, 27, 27–28, 38
energy of nonperiodic, 167–168
even, 10–11, 11–12, 23, 25–26, 26, 40
force, 45, 45
Fourier transform, 183–184
impulses, 18, 18–19
frequency domain, 156
Laplace transform, 194
mathematical abstraction, 20–21, 20–21
scaling property, 18
shifted, 59
sifting property, 56, 156–157
usage of MATLAB to simulate, 35, 36
infnite periodic, 153, 154
input, 22, 47, 48, 49–50
continuous systems, 47, 48
electrical, 47
magnitude of, 2, 2, 4
nonperiodic, 2–3, 153
nonperiodic time-domain, 153
odd, 10–11, 11–12, 23, 25–26, 26, 40
output, 47, 48
periodic, 2–3, 5–6, 29, 38, 44, 136
power, 5–6, 7, 27, 27
pulse, 21
ramp, 16–17, 17
rectangular pulse, 15, 15–16
representation, 1–45
continuous signal, 1–2, 2
energy and power signals, 5–6, 7
even and odd signals, 10–11, 11–12
examples, 22–37
impulse signal, 18, 18–19
periodic and nonperiodic signals, 2–3
problems, 38–45
ramp signals, 16–17, 17
in real world, 20–22
refection operation, 8, 8–9, 10
sampling signal, 17, 17
shifting operation, 7, 7–8
signum signal, 16, 16
sinusoidal signal, 4, 4–5
time scaling, 12–14, 12–14
unit step signal, 14–15, 14–16
sampling, 17, 17
shifted impulse, 59
shock, 20
signum, 16, 16
sinc, 17, 17
sinusoidal, 4, 4–5
time, 157
triangular voltage, 22
types, 1
unit step, 14–15, 14–16, 32–33, 42, 42
velocity, 45
voltages as, 1
Signum signals, 16, 16
Simple block diagrams, 58–59, 58–61, 61;
see also Block diagrams
Sin and cos functions, 126–129, 127–129, 138
Sinc signals, 17, 17344 Index
Single-input/single-output, 245, 255, 271,
282
Single-phase circuit, 30–31, 30–31
Sinusoidal signals, 4, 4–5; see also Signals
mathematical abstraction, 21, 21
representation for x(t), Fourier series, 136
usage of MATLAB to simulate, 35, 37
Solutions, 8
Solving state-space equations in real time,
272–273
Solving state-space equations using Laplace
transform, 272
Sound wave, 1
Special matrix, 250; see also Matrix
Speed, constant, 16
s-plane, 196
Stability
eigenvalues and, 84–85
of linear discrete systems, 73–76
systems, 55–56, 84–85, 209
State equations, 245; see also Equations
State-space
approach, 245
equation, 254, 259
Laplace matrix representation, 270
State-space and linear systems, 245–334
evaluating eAt, 273–281
Cayley–Hamilton theorem, 274–276
diagonal matrix, 273
general form of Φ(t) = eAt, 277–281
inverse Laplace method, 276–277
numerical evaluation, 273
examples, 283–326
matrix algebra, 246–270
addition of two matrices, 246
determinant of 2 × 2 matrix, 247
diagonal form of matrix, 249
eigenvalues of matrix, 251
eigenvectors of matrix, 252
exponent of matrix, 249–250
identity matrix, 246
inverse of matrix, 248
matrix multiplication, 248–249
multiplication of matrix by constant,
247
observation, 251
special matrix, 250
subtraction of two matrices, 247
transpose of matrix, 248
overview, 245
poles and stability, 282–283
problems, 326–334
representation of systems in state space, 271
solving state-space equations in real time,
272–273
solving state-space equations using Laplace
transform, 272
State-transition matrix, 277
State variables, 271
Steady-state output, 169
Steady-state response, 134–136
Step input, 304
Step signal
discontinuous, 18
magnitude of, 14, 14
mathematical abstraction, 20, 20
usage of MATLAB to simulate, 35, 36
Stiffness coeffcient, 299
Subtraction; see also Addition
block diagram, 77, 77
complex numbers, 119
of two matrices, 247
Superposition
principle, 135
systems, 57
Symmetry property, 180
Systems; see also Fourier transform and linear
systems
block diagram representation of linear, 76–77,
76–77
causal, 52–53, 53
continuous, see Continuous systems
defned, 47
differential equations, 78, 103
eigenvalues, 67
elevator, 47
frst-order, 67
frst-order single-input single-output, 253
impulse response, 58, 96–99, 162–163
input, 54, 57, 66, 72
integrals, 52
inverse, 54–55, 54–55
invertible, 56
linear, 57, 136
linear continuous, 47–50, 48
linear time-invariant (LTI), 57, 60, 66, 84,
132, 168–169
modeling, 245
multiple-input multiple-output, 245, 271
noncausal, 53
output, 50, 52, 54, 57–59, 66, 71, 84, 87–92,
104–105
parallel, 198–199, 199
with periodic inputs, Fourier series, 132–134,
134
physical, 16, 66
poles of, 209
representation in state space, 271
series, 199, 199–200
with single-input/single-output, 245
stable, 55–56, 84–85, 209
superposition, 57
time-invariant, 50–52, 57, 86–87
transfer function, 133–134, 150Index 345
transient, 83
variables, 271
without memory, 52
zeros of, 209
T
Theorem of Parseval, 131–132, 168, 179
Thermal resistors, 302
Thermal system, 302
Third-order differential equations, 67,
83, 201; see also Differential
equations
Three-integrators block diagram, 202, 202;
see also Block diagrams
Time
domain, 270, 330
convolution, 165
frequency domain, 179–180
inverse Laplace method, 276
state-space matrix equations,
272
invariant systems, 50–52, 57, 86–87
scaling, 12–14, 12–14, 24
shifting property, 160
signal, 157
Transfer function, 200–202, 200–203,
321
input voltage, 334
linear time-invariant (LTI) systems,
198
partial fraction expansion, 258
in s-domain, 198
systems, 133–134, 150
Transient systems, 83; see also Systems
Transition matrix, 279; see also Matrix
Translational damper, 290, 295, 329
Translational motion, 47
Translational spring, 292
Transmission media, 167
Transpose of matrix, 248
Triangular voltage signal, 22
U
Underdamping, 290
Unilateral Laplace transform, 191–192
Unit step function, 88
Unit step input, 91–92
Unit step signals, 14–15, 14–16, 32–33, 42, 42
V
Variables
algebraic equation, 209, 253
state, 271
systems, 271
Velocity
constant, 16
mass, 44
signals, 45
Voltages, 1
across 1-Ohm resistor, 6
AC source, 21
capacitors, 304–305
of constant magnitude, 20
divider circuit, 47–48, 48
frequency, 31
inductor, 267
output, 47–48
resistor, 44, 47, 304–305
source, 31
triangular signals, 22
W
Wave, see specifc wave
Weighted impulses, 57; see also Impulses
Wind wave, 1
Writing signal in Fourier series, 124
Z
Zero input, 73
Zeros of system, 209

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