Applied Numerical Analysis with MATLAB/Simulink. For Engineers and Scientists

327 Pages·2023·18.99 MB·English

by  Farzin Asadi

Synthesis Lectures on
Engineering, Science, and Technology
Farzin Asadi
Applied Numerical
Analysis with
MATLAB®/Simulink®
For Engineers and Scientists
Synthesis Lectures on Engineering, Science,
and Technology
The focus of this series is general topics, and applications about, and for, engineers
and scientists on a wide array of applications, methods and advances. Most titles cover
subjects such as professional development, education, and study skills, as well as basic
introductory undergraduate material and other topics appropriate for a broader and less
technical audience.
Farzin Asadi
Applied Numerical Analysis
® ®
with MATLAB /Simulink
For Engineers and Scientists
Farzin Asadi
Department of Electrical and Electronics Engineering
Maltepe University
Istanbul, Turkey
ISSN 2690-0300 ISSN 2690-0327 (electronic)
Synthesis Lectures on Engineering, Science, and Technology
ISBN 978-3-031-19365-1 ISBN 978-3-031-19366-8 (eBook)
https://doi.org/10.1007/978-3-031-19366-8
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
2023
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Preface
Numerical analysis is the branch of mathematics that is used to find approximations to
difficult problems such as finding the roots of nonlinear equations, integration involving
complex expressions, and solving differential equations for which analytical solutions do
not exist. It is applied to a wide variety of disciplines such as business, all fields of
engineering, computer science, education, geology, meteorology, and others.
Many good books are written about the numerical analysis theories. In this book, we
don’t focus on the theories of numerical analysis. Instead we focus on solving numerical
analysis problems with the aid of tools that MATLAB/Simulink provides for us. This book
can accompany and complete any standard theoretical textbook on numerical analysis. The
prerequisite for this book is a first course on numerical analysis, and no prior knowledge
of MATLAB/Simulink is assumed.
This book is written primarily for students of engineering and science. However, it can
be useful for anyone who wants to solve a numeric problem with MATLAB/Simulink.
For instance, master or Ph.D. students can use the material presented here to solve the
numerical problems of their thesis or papers. All of the material presented here can be
covered in one semester (14 weeks) with 3 h per week.
This book is composed of 11 chapters. Here is a brief summary of each chapters:
Chapter 1 is an introduction to MATLAB. In this chapter, you will learn how to
do basic calculations with MATLAB. Basic matrix operations, trigonometric, hyper-
bolic, logarithmic, exponential, and many other important functions are studied in this
chapter.
Chapter 2 shows how MATLAB can be used for symbolic calculations. In this chapter,
you will learn to calculate limits, derivatives, integrals (both definite and indefinite),
partial fraction expansion, Laplace transform, Fourier transform, and Taylor series with
MATLAB.
Chapter 3 shows how MATLAB can be used to calculate a definite integral (single,
double or triple) or derivative of a function at a given point. Trapezoidal and Simpson
rules for calculation of definite integrals are studied in this chapter.
v
vi Preface
Chapter 4 shows how MATLAB can be used for statistical calculations. In this chapter,
you will learn how to calculate summation, average, variance, and standard devia-
tion of a given data, how to generate random numbers, how to calculate factorial
and combinatorial, and how to obtain cumulative distribution function for a normal
distribution.
Chapter 5 shows how MATLAB can be used to obtain impulse and step responses
of a linear differential equation. In this book, we use the linear system (or linear
dynamical system) and linear differential equation terms interchangeably. By impulse
and step responses, we mean the response that observed in the output when the input
is Heaviside step function and Dirac impulse function, respectively.
Chapter 6 shows how differential equations can be solved with Simulink. Simulink is a
software package which accompanies MATLAB and permits you to simulate dynam-
ical systems (i.e., systems which can described by a linear or nonlinear differential
equations) with the aid of many ready to use graphical blocks.
Chapter 7 shows how difference equations can be solved with Simulink. A difference
equation is a relation between the differences of unknown function at one or more
general values of the independent variable. Generally, a difference equation is obtained
in an attempt to solve an ordinary differential equation by finite difference method.
Chapter 8 shows how curve-fitting problems can be solved with MATLAB’s Curve
Fitting Toolbox™. Curve fitting is the process of constructing a curve, or mathematical
function, that has the best fit to a series of data points.
Chapter 9 shows how MATLAB can be used for drawing the graph of a given data.
Different kind of graphs are studied in this chapter.
Chapter 10 shows how some of well-known numerical analysis algorithms can be
implemented with MATLAB programming.
Chapter 11 shows how optimization problems can be solved with the aid of ready to
use functions that Optimization Toolbox™ provides.
I hope that this book will be useful to the readers, and I welcome comments on the book.
Istanbul, Turkey Farzin Asadi
[email protected]
Contents
1 Essential of MATLAB® .............................................. 1
1.1 Introduction .................................................. 1
1.2 MATLAB Environment ........................................ 2
1.3 Basic Operation with MATLAB ................................ 3
1.4 Clearing the Screen and Variables ............................... 11
1.5 Basic Matrix Operations ....................................... 15
1.6 Trigonometric Functions ....................................... 25
1.7 Hyperbolic Functions .......................................... 28
1.8 Logarithmic and Exponential Function ........................... 29
1.9 Rounding Functions ........................................... 30
1.10 Colon Operator ............................................... 30
1.11 Linspace and Logspace Commands .............................. 33
1.12 Ones, Zeros and Eye Commands ................................ 36
1.13 Format Command ............................................. 37
1.14 Polynomial Functions ......................................... 38
1.15 Solution of Nonlinear Systems .................................. 40
1.16 Eigen Values and Eigen Vectors ................................ 44
1.17 Reduced Echelon Form of a Matrix ............................. 45
1.18 Norm of Vectors and Matrices .................................. 46
1.19 Condition Number of a Matrix .................................. 48
1.20 Tic and Toc Commands ........................................ 49
1.21 Getting Help in MATLAB ..................................... 49
References for Further Study .......................................... 53
2 Symbolic Calculations in MATLAB® .................................. 55
2.1 Introduction .................................................. 55
2.2 Calculation of Limit, Derivative and Integral ..................... 55
2.3 Solving the Ordinary Differential Equations ...................... 57
2.4 Partial Fraction Expansion and Laplace Transform ................ 61
2.5 Fourier Transform ............................................. 65
vii
viii Contents
2.6 Taylor Series ................................................. 67
2.7 Expansion of an Algebraic Expression ........................... 69
Reference for Further Study ........................................... 71
3 Numerical Integration and Derivation ................................. 73
3.1 Introduction .................................................. 73
3.2 Trapezoidal Rule .............................................. 73
3.3 Simpson’s 1/3 Rule ........................................... 74
3.4 Simpson 3/8 Rule ............................................. 77
3.5 Double Integrals .............................................. 79
3.6 Triple Integrals ............................................... 79
3.7 Derivative .................................................... 81
References for Further Study .......................................... 84
4 Statistics with MATLAB® ............................................ 85
4.1 Introduction .................................................. 85
4.2 Sum of Elements ............................................. 85
4.3 Average ..................................................... 86
4.4 Variance and Standard Deviation ................................ 87
4.5 Factorial ..................................................... 89
4.6 Combination ................................................. 89
4.7 Random Numbers ............................................. 90
4.8 Normal Probability Density Function ............................ 92
4.9 Cumulative Distribution Function ............................... 94
References for Further Study .......................................... 96
5 Impulse and Step Response of Linear Systems ......................... 97
5.1 Introduction .................................................. 97
5.2 Impulse Response of Dynamical Systems ........................ 98
5.3 Step Response of Dynamical Systems ........................... 102
Reference for Further Study ........................................... 103
6 Solving Differential Equations in Simulink® ........................... 105
6.1 Introduction .................................................. 105
6.2 Solving a Linear Differential Equation in Simulink ................ 105
6.3 Multiplexer Block ............................................. 124
6.4 Giving Name to Scope Blocks .................................. 127
6.5 Selection of Solver ............................................ 131
6.6 Transferring the Results from Simulink Environment
to MATLAB® ................................................ 135
6.7 Transfer Function Block ....................................... 141
6.8 State Space Block ............................................. 148
6.9 Nonlinear Models ............................................. 161
Contents ix
6.10 Van Der Pol Equation ......................................... 166
References for Further Study .......................................... 168
7 Solving Difference Equations in Simulink® ............................ 169
7.1 Introduction .................................................. 169
7.2 Solving the Difference Equations: Example 1 ..................... 169
7.3 Solving the Difference Equations: Example 2 ..................... 176
7.4 Solving the Difference Equations: Example 3 ..................... 183
References for Further Study .......................................... 189
8 Curve Fitting with MATLAB® ....................................... 191
8.1 Introduction .................................................. 191
8.2 Example 1: Linear Curve Fitting ................................ 191
8.3 Graphical Comparison of Estimation with Measured Data .......... 200
8.4 Quantitative Comparison of Estimation with Data ................. 203
8.5 Example 2: Nonlinear Curve Fitting ............................. 205
8.6 Example 3: Export the Obtained Equation to the MATLAB
Workspace ................................................... 212
References for Further Study .......................................... 217
9 Drawing Graphs with MATLAB® .................................... 219
9.1 Introduction .................................................. 219
9.2 fplot Command ............................................... 219
9.3 Plotting the Graph of a Numeric Data ........................... 222
9.4 Addition of Labels and Title to the Drawn Graph ................. 228
9.5 Exporting the Drawn Graph as a Graphical File ................... 229
9.6 Drawing Two or More Graphs on the Same Axis ................. 231
9.7 Logarithmic Axis ............................................. 236
9.8 Drawing 2D and 3D Parametric Graphs .......................... 239
9.9 Polar Plot .................................................... 241
9.10 3D Surfaces .................................................. 243
9.11 Pie Chart .................................................... 245
9.12 Exploded Pie Chart ........................................... 248
9.13 Export the Drawn Pie Chart as a Graphical File ................... 250
9.14 Bar Graphs ................................................... 252
References for Further Study .......................................... 256
10 MATLAB® Programming ............................................ 257
10.1 Introduction .................................................. 257
10.2 MATLAB Editor .............................................. 257
10.3 Simple Game ................................................. 258
10.4 Switch-Case Control Statement ................................. 262
10.5 fprintf and disp Command ..................................... 265