Introduction to Modeling and Simulation E-Book

Mark W. Spong

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 466

Veröffentlichungsjahr: 2023

Beschreibung

Introduction to Modeling and Simulation An essential introduction to engineering system modeling and simulation from a well-trusted source in engineering and education This new introductory-level textbook provides thirteen self-contained chapters, each covering an important topic in engineering systems modeling and simulation. The importance of such a topic cannot be overstated; modeling and simulation will only increase in importance in the future as computational resources improve and become more powerful and accessible, and as systems become more complex. This resource is a wonderful mix of practical examples, theoretical concepts, and experimental sessions that ensure a well-rounded education on the topic. The topics covered in Introduction to Modeling and Simulation are timeless fundamentals that provide the necessary background for further and more advanced study of one or more of the topics. The text includes topics such as linear and nonlinear dynamical systems, continuous-time and discrete-time systems, stability theory, numerical methods for solution of ODEs, PDE models, feedback systems, optimization, regression and more. Each chapter provides an introduction to the topic to familiarize students with the core ideas before delving deeper. The numerous tools and examples help ensure students engage in active learning, acquiring a range of tools for analyzing systems and gaining experience in numerical computation and simulation systems, from an author prized for both his writing and his teaching over the course of his over-40-year career. Introduction to Modeling and Simulation readers will also find: * Numerous examples, tools, and programming tips to help clarify points made throughout the textbook, with end-of-chapter problems to further emphasize the material * As systems become more complex, a chapter devoted to complex networks including small-world and scale-free networks - a unique advancement for textbooks within modeling and simulation * A complementary website that hosts a complete set of lecture slides, a solution manual for end-of-chapter problems, MATLAB files, and case-study exercises Introduction to Modeling and Simulation is aimed at undergraduate and first-year graduate engineering students studying systems, in diverse avenues within the field: electrical, mechanical, mathematics, aerospace, bioengineering, physics, and civil and environmental engineering. It may also be of interest to those in mathematical modeling courses, as it provides in-depth material on MATLAB simulation and contains appendices with brief reviews of linear algebra, real analysis, and probability theory.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Dedication

Preface

Organization of the Text

A Note to the Instructor

Acknowledgements

About the Companion Website

1 Introduction

1.1 Introduction

1.2 Model Classification

1.3 Simulation Languages

1.4 Outline of the Text

Problems

Linear Algebra Background

Real Analysis Background

Probability Background

Matlab

2 Second‐Order Systems

2.1 Introduction

2.2 State‐Space Representation

2.3 Trajectories and Phase Portraits

2.4 The Direction Field

2.5 Equilibria

2.6 Linear Systems

2.7 Linearization of Nonlinear Systems

2.8 Periodic Trajectories and Limit Cycles

2.9 Coupled Second‐Order Systems

Problems

3 System Fundamentals

3.1 Introduction

3.2 Existence and Uniqueness of Solution

3.3 The Matrix Exponential

3.4 The Jordan Canonical Form

3.5 Linearization

3.6 The Hartman–Grobman Theorem

3.7 Singular Perturbations

Problems

4 Compartmental Models

4.1 Introduction

4.2 Exponential Growth and Decay

4.3 The Logistic Equation

4.4 Models of Epidemics

4.5 Predator–Prey System

Problems

5 Stability

5.1 Introduction

5.2 Lyapunov Stability

5.3 Basin of Attraction

5.4 The Invariance Principle

5.5 Linear Systems and Linearization

Problems

6 Discrete‐Time Systems

6.1 Introduction

6.2 Stability of Discrete‐Time Systems

6.3 Stability of Discrete‐Time Linear Systems

6.4 Moving‐Average Filter

6.5 Cobweb Diagrams

Problems

7 Numerical Methods

7.1 Introduction

7.2 Numerical Differentiation

7.3 Numerical Integration

7.4 Numerical Solution of ODEs

7.5 Stiff Systems

7.6 Event Detection

7.7 Simulink

7.8 Summary

Problems

8 Optimization

8.1 Introduction

8.2 Unconstrained Optimization

8.3 Case Study: Numerical Inverse Kinematics

8.4 Constrained Optimization

8.5 Convex Optimization

Problems

9 System Identification

9.1 Introduction

9.2 Least Squares

9.3 Regression

9.4 Recursive Least Squares

9.5 Logistic Regression

9.6 Neural Networks

Problems

10 Stochastic Systems

10.1 Markov Chains

10.2 Monte Carlo Methods

Problems

11 Feedback Systems

11.1 Introduction

11.2 Transfer Functions

11.3 Feedback Control

11.4 State‐Space Models

11.5 Optimal Control

11.6 Control of Nonlinear Systems

Problems

12 Partial Differential Equation Models

12.1 Introduction

12.2 The Wave Equation

12.3 The Heat Equation

12.4 Laplace's Equation

12.5 Numerical Solution of PDEs

Problems

13 Complex Networks

13.1 Introduction

13.2 Graph Theory: Basic Concepts

13.3 Matlab Graph Functions

13.4 Network Metrics

13.5 Random Graphs

13.6 Synchronization in Networks

Problems

Appendix A: Linear Algebra

A.1 Vectors

A.2 Matrices

A.3 Eigenvalues and Eigenvectors

Appendix B: Real Analysis

B.1 Set Theory

B.2 Vector Fields

B.3 Jacobian

B.4 Scalar Functions

B.5 Taylor's Theorem

B.6 Extreme‐Value Theorem

Appendix C: Probability

C.1 Discrete Probability

C.2 Conditional Probability

C.3 Random Variables

C.4 Continuous Probability

Appendix D: Proofs of Selected Results

D.1 Proof of Theorem 2.2

D.2 Proof of Theorem 5.1

D.3 Proof of Theorem 5.5

D.4 Proof of Theorem 13.3

D.5 Proof of Corollary 13.2

D.6 Proof of Proposition 13.2

D.7 Proof of Proposition 13.3

Appendix E: Matlab Command Reference

References

Index

End User License Agreement

List of Tables

Chapter 2

Table 2.1 Summary of isolated fixed points for nonzero eigenvalues. The cas...

Chapter 8

Table 8.1 Income versus occupancy rate. The income increases to a maximum a...

Chapter 10

Table 10.1 Pseudo‐random sequences of numbers between

and

, with differe...

Chapter 13

Table 13.1 Three different centrality measures for the graph

in Figure 13...

Table 13.2 Clustering coefficients for the graph in Figure 13.32.

List of Illustrations

Chapter 1

Figure 1.1 A general characterization of a system as an object or process th...

Figure 1.2 An electric circuit with input voltage

and output voltage

. Th...

Figure 1.3 A mass–spring–damper system with input force

and output positio...

Figure 1.4 A DC motor with input voltage

and output speed

. Both electric...

Figure 1.5 An economic system with multiple inputs and outputs. The choice o...

Figure 1.6 A manufacturing plant transforms input raw materials into output ...

Figure 1.7 A machine learning (ML) system takes input data and produces outp...

Figure 1.8 A feedback system is one where the output affects future inputs....

Figure 1.9 Warehouse robots for order retrieval are an example of factory au...

Figure 1.10 Oil and chemical refineries are large‐scale, complex systems tha...

Figure 1.11 The electric power grid consists of an interconnected network of...

Figure 1.12 A wind farm is a collection of wind turbines connected to the po...

Figure 1.13 Supply chains and logistics systems are responsible for moving m...

Figure 1.14 Systems of autonomous vehicles are already changing the trucking...

Figure 1.15 Surgical robots are examples of human/machine/computer interface...

Chapter 2

Figure 2.1 A mass–spring–damper system (left) and an

circuit (right). Both...

Figure 2.2 Mass–spring system that undergoes harmonic motion.

Figure 2.3 Oscillation of a leaky bucket. As the mass of the water in the bu...

Figure 2.4 A solution trajectory in

. The vector field

is tangent to the ...

Figure 2.5 A single trajectory for the harmonic oscillator. The two state va...

Figure 2.6 Multiple trajectories in time (left, center) and a portion of the...

Figure 2.7 A simple pendulum with length

rotating in a vertical plane unde...

Figure 2.8 The load transported by an overhead crane is modeled as a hanging...

Figure 2.9 Inverted pendulum model for bipedal locomotion. The stance leg be...

Figure 2.10 A rocket ascending vertically is modeled as an inverted pendulum...

Figure 2.11 The Segway Human Transporter. The controller for the Segway trie...

Figure 2.12 Solution trajectories and a portion of the phase portrait for th...

Figure 2.13 Direction field for the harmonic oscillator, which shows qualita...

Figure 2.14 Direction field and a portion of the phase portrait for the simp...

Figure 2.15 The simple pendulum has infinitely many isolated equilibrium poi...

Figure 2.16 Unique equilibrium states of the simple pendulum by identifying

Figure 2.17 Phase portraits in modal coordinates for a stable node (left); u...

Figure 2.18 Phase portraits of the stable node in modal coordinates (left) a...

Figure 2.19 Phase portraits of the saddle node in modal coordinates (left) a...

Figure 2.20 Jordan node phase portraits in modal coordinates (left) and in t...

Figure 2.21 Stable focus (sink) left; unstable focus (source) right

Figure 2.22 Phase portrait of the nonlinear system (2.18).

Figure 2.23 Phase portraits of the linear approximations of the nonlinear sy...

Figure 2.24 Phase portrait of the damped simple pendulum. Oscillatory soluti...

Figure 2.25 Direction field and phase portrait of the Van der Pol oscillator...

Figure 2.26 Illustrating a Hopf bifurcation in the Van der Pol system as the...

Figure 2.27 Phase portrait of the FitzHugh–Nagumo system with parameters

,

Figure 2.28 Direction field and a portion of the phase portrait of the Bruss...

Figure 2.29 Illustrating the Poincaré–Bendixson theorem. The annular region ...

Figure 2.30 Phase portrait for the system (2.28).

Figure 2.31 Two pendula coupled by a linear spring with stiffness

.

Figure 2.32 Response of the coupled pendula. With the given initial conditio...

Figure 2.33 Double Pendulum

Chapter 3

Figure 3.1 Trajectories for the system

exhibiting finite escape. The amoun...

Figure 3.2 Illustrating the Hartman–Grobman theorem.

Figure 3.3 Solution trajectory of the system (3.5) in

. The trajectory rapi...

Figure 3.4 Schematic of a permanent‐magnet DC motor connected to a load.

Figure 3.5 Full‐order and reduced‐order responses for

. The other state var...

Chapter 4

Figure 4.1 Example of a compartmental model. The arrows represent flows or t...

Figure 4.2 Example of a multi‐compartmental model.

Figure 4.3 One‐compartment model.

Figure 4.4 Two‐compartment drug distribution model.

Figure 4.5 Drug concentrations in the blood and tissue with a constant input...

Figure 4.6 Drug concentrations in the blood and tissue with initial concentr...

Figure 4.7 Solution of the logistic equation.

Figure 4.8 SI epidemic model consisting of two compartments, susceptible (

)...

Figure 4.9 Response of the SI model with

.

Figure 4.10 The SIS epidemic model. Infected individuals recover at a certai...

Figure 4.11 Response of the SIS model with

,

, and

. The basic reproducti...

Figure 4.12 The SIR epidemic model.

Figure 4.13 Response of the SIR model with

and

.

Figure 4.14 Flattening the curve with three different values for

, namely,

Figure 4.15 The SEIR compartmental model. In this model individuals transfer...

Figure 4.16 SIR model with vital dynamics. The birth and death rates are bot...

Figure 4.17 Phase portrait of the predator–prey system in the first quadrant...

Figure 4.18 A two‐predator, two‐prey model. Predator

feeds on both prey sp...

Figure 4.19 Response of the two‐predator, two‐prey system with

,

,

,

,

Chapter 5

Figure 5.1 The intuitive notion of stable and unstable equilibria. The equil...

Figure 5.2 Illustrating the

–

definition of stability. Solutions starting ...

Figure 5.3 The intuitive idea of Lyapunov stability. The condition

means t...

Figure 5.4 Phase portrait for the system (5.2)–(5.3), showing that the origi...

Figure 5.5 Level sets for Lyapunov function candidate in Equation (5.4).

Figure 5.6 Illustrating the basin of attraction and an estimate using the le...

Figure 5.7 Pendulum with friction example. The trajectories point toward the...

Figure 5.8 Phase portrait for the system. For any initial conditions, the va...

Chapter 6

Figure 6.1 Response of the system (6.6) for different values of the coeffici...

Figure 6.2 Response of the system (6.6) with

(right) and with

(left).

Figure 6.3

stairs(y)

(left),

bars(y)

(middle),

stem(y)

(right) for a sequenc...

Figure 6.4 Plot of

(top) and

(bottom) for the discrete‐time nonlinear sy...

Figure 6.5 Seven‐day moving‐average filter for a set of stock price data plo...

Figure 6.6 Moving‐average filter response to noisy input data.

Figure 6.7 Cobweb diagram for Equation (6.12) with

iterations. A decreasin...

Figure 6.8 Cobweb diagrams for different slopes of the supply and demand cur...

Figure 6.9 Equilibrium points of the discrete logistic equation at

and

....

Figure 6.10 Cobweb diagram for

. In this case the only equilibrium is at th...

Figure 6.11 Cobweb diagram for

. In this case, the solutions converge to th...

Figure 6.12 Stable equilibrium for

(left) and unstable equilibrium for

(...

Figure 6.13 Chaotic solution for

.

Chapter 7

Figure 7.1 Illustrating the forward, backward, and middle difference approxi...

Figure 7.2 Numerical computation of the derivative of

using the Matlab com...

Figure 7.3 Illustrating the left‐endpoint, right‐endpoint and midpoint integ...

Figure 7.4 Illustrating trapezoidal integration. The area under a representa...

Figure 7.5 Illustrating Simpson's rule. The function

is approximated by a ...

Figure 7.6 Euler's method for

with

,

, and step sizes

(

) and

(

). T...

Figure 7.7 Response of the simple pendulum model using Euler's method with a...

Figure 7.8 Euler's method (

) and the Euler predictor–corrector method (

) f...

Figure 7.9 Comparison of Euler's method (

) with a fourth‐order Runge–Kutta ...

Figure 7.10 Solution of the stiff system using

ode45

. The expanded view on t...

Figure 7.11 Solution of the stiff system using

ode23s

. As an implicit method...

Figure 7.12 Solution of the DAE system using

ode15s

.

ode15s

should be used w...

Figure 7.13 Response of the DAE model of the SI‐epidemic system.

Figure 7.14 Response of the falling object. The simulation terminates when

Figure 7.15 The bouncing ball starting at initial height

with coefficient ...

Figure 7.16 Simulink blocks from the Simulink library.

Figure 7.17 Realization of the second‐order system

using two integrator bl...

Figure 7.18 Step response of the system (7.21) starting with initial conditi...

Figure 7.19 Simulink integrator block.

Figure 7.20 A practical differentiator built from an integrator. With input

Figure 7.21 The practical differentiator with input

produces a signal that...

Figure 7.22 Simulink block diagram representation of the nonlinear system (7...

Figure 7.23 Simulink block diagram representation of the Van der Pol equatio...

Chapter 8

Figure 8.1

is a strict local minimizer;

is a strict local maximizer;

i...

Figure 8.2 The search directions in gradient descent are orthogonal to level...

Figure 8.3 If the condition number of

is

, the level sets are concentric ...

Figure 8.4 If the condition number of

is large, the convergence of the met...

Figure 8.5 Newton's method for finding a root of

.

Figure 8.6 A selective compliant articulated robot arm. SCARA robots are use...

Figure 8.7 Schematic representation of a two‐link robot.

and

are the joi...

Figure 8.8 Inverse kinematics solution using Newton's method. Desired end‐ef...

Figure 8.9 Inverse kinematics solution using gradient descent. Desired end‐e...

Figure 8.10 Level sets of the objective function

with

and the graph of t...

Figure 8.11 Feasible region for the linear programming problem.

Figure 8.12 Illustrating the notion of a convex function. The convexity cond...

Figure 8.13 The set

on the left is convex. The set

on the right is nonco...

Figure 8.14 Illustrating Proposition 8.4. The condition (8.19) means that th...

Chapter 9

Figure 9.1 The range

of

is an

‐dimensional subspace of

. If

is not i...

Figure 9.2 Estimate of the resistance

from

measurements. The horizontal ...

Figure 9.3 Linear regression example. The straight line is estimated from th...

Figure 9.4 Linear and quadratic polynomial fitting. The choice of which mode...

Figure 9.5 The logit function (left) and the inverse logit function (right)....

Figure 9.6 Best‐fit logistic curve for the given credit score data.

Figure 9.7 A brain neuron

Figure 9.8 An artificial neuron model

Figure 9.9 Examples of activation functions

Figure 9.10 A one‐layer neural network

Figure 9.11 The decision boundary separating the two data sets.

Figure 9.12 Results of training the perceptron on the data set

Chapter 10

Figure 10.1 Transition diagram for a Markov chain. The numbered circles repr...

Figure 10.2 Transition diagram for Example 10.1. Each month a fraction of cu...

Figure 10.3 Computing the transition probabilities for months

and

.

Figure 10.4 The Markov mouse.

Figure 10.5 Transition diagram for the Markov mouse. The probabilities are c...

Figure 10.6 Transition diagram for the Markov mouse assuming that the mouse ...

Figure 10.7 Two compartments containing a number of gas molecules.

Figure 10.8 Transition diagram for the Ehrenfest model.

Figure 10.9 Number of gas molecules in compartment

over time. The number f...

Figure 10.10 An example of an absorbing Markov chain. Once the state reaches...

Figure 10.11 An example of isomorphic Markov chains.

Figure 10.12 Transition diagram for Example 10.8.

Figure 10.13 Transition diagram for the tennis example. The data are fictiti...

Figure 10.14 Counting the fraction of points inside the quarter circle gives...

Figure 10.15 Monte Carlo simulation to predict stock prices. Fifty random si...

Figure 10.16 Monte Carlo simulation of the SIR model showing the estimated n...

Chapter 11

Figure 11.1 A closed‐loop feedback system where the output affects future in...

Figure 11.2 Open‐loop system (left) and closed‐loop system (right).

Figure 11.3 Examples of feedback systems: the outrigger canoe, flyball gover...

Figure 11.4 A transfer function representation of a linear system as a ratio...

Figure 11.5 Dominant second‐order pole location. Second‐order systems are an...

Figure 11.6 Underdamped, critically damped, and overdamped second‐order step...

Figure 11.7 A second‐order system step response showing overshoot, rise time...

Figure 11.8 Step response of the third‐order system with DC gain equal to

....

Figure 11.9 Block diagram showing the basic components of a control system....

Figure 11.10 First‐order system

with compensator

.

Figure 11.11 Satellite attitude control. The satellite is equipped with thru...

Figure 11.12 Simulink model of the satellite control. The sensor block integ...

Figure 11.13 Position, velocity, and control input torque for the satellite....

Figure 11.14 Simulink model of a PI speed control system.

Figure 11.15 Responses of the PI control gains,

,

,

. The disturbance inp...

Figure 11.16 Controller canonical form. The

variables are the outputs,

, ...

Figure 11.17 Observer canonical form

Figure 11.18 Inertias coupled by a torsional spring.

Figure 11.19 Step response of the closed‐loop system.

Figure 11.20 Simulink system for the observer

Figure 11.21 Estimation errors for the observer. The initial conditions were...

Figure 11.22 Simulink model of the observer/state feedback control system.

Figure 11.23 Output response of the observer/state feedback control system....

Figure 11.24 Pendulum response.

Figure 11.25 The reaction‐wheel pendulum.

Figure 11.26 Simulink model of adaptive control using the MIT rule.

Figure 11.27 Response of the adaptive control system. The trajectory

initi...

Chapter 12

Figure 12.1 Coupled mass–spring system.

Figure 12.2 Snapshots of the vibrating spring with initial shape

. The solu...

Figure 12.3 The square wave function is a piecewise‐continuous, odd function...

Figure 12.4 Fourier series approximation of the square wave function with

,...

Figure 12.5 Fourier series approximation of the delta function for

, respec...

Figure 12.6 Series solution of the wave equation. Note that the response is ...

Figure 12.7 Insulate rod with heat flowing in one dimension along the rod.

Figure 12.8 The temperature

is a thin bar with both ends held at zero temp...

Figure 12.9 Rectangular region with values of

specified on the boundary.

Figure 12.10 Slab of width

. The left‐hand side is held at temperature

an...

Figure 12.11 Grid of mesh points discretizing time in increments of

and sp...

Figure 12.12 Points used in the finite‐difference approximation.

Figure 12.13 Temperature distribution generated by the Matlab simulation usi...

Chapter 13

Figure 13.1 A Network of interconnected agents. Depending on the application...

Figure 13.2 An undirected graph

of order

and size

. We say that

is a

Figure 13.3 A nonsimple graph (left) containing loops and a multigraph (righ...

Figure 13.4 Example of a weighted, directed graph. The edge weights may repr...

Figure 13.5 A

‐regular graph. Each vertex in the graph has degree

.

Figure 13.6 The first five complete graphs

.

Figure 13.7 The two graphs above are not isomorphic despite the fact that th...

Figure 13.8 The graph on the right is connected. It is possible to reach eve...

Figure 13.9 Example of a subgraph. The graph on the right is a subgraph of t...

Figure 13.10 An example of a graph containing two connected components

Figure 13.11 The graph on the left is strongly connected. The graph on the r...

Figure 13.12 A graph with cut vertices and a bridge.

Figure 13.13 A tree contains no cycles (i.e., no closed paths). This also me...

Figure 13.14 Graph

and spanning tree

(bold edges).

Figure 13.15 Example of a bipartite graph. The sets partitioning the graph a...

Figure 13.16 A tree (left) redrawn as a bipartite graph (right).

Figure 13.17 The above two graphs are isomorphic and so both are planar. The...

Figure 13.18 Utility connections to three houses. The bipartite graph repres...

Figure 13.19 Example of a planar graph with

regions,

vertices, and

edg...

Figure 13.20 A

graph example with its associated Laplacian matrix.

Figure 13.21 With four edges, the oriented incidence matrix will have four c...

Figure 13.22 A simple undirected graph created by the Matlab

graph

function....

Figure 13.23 A weighted directed network created by Matlab

digraph

function....

Figure 13.24 A railway network. The degree distribution of this network is a...

Figure 13.25 An airline network showing several hub airports. Most other air...

Figure 13.26 Magnitude versus frequency of earthquakes plotted on a log‐log ...

Figure 13.27 Income of the top 100 (left) and the top 10 (right). Despite th...

Figure 13.28 The length of a path is the number of ‘hops’ between nodes. The...

Figure 13.29 A low‐degree node may be important if it is adjacent to high de...

Figure 13.30 A

graph with different values of degree centrality, betweenne...

Figure 13.31 The neighbors of

are

. The subgraph formed by these nodes ha...

Figure 13.32 Graph clustering example for Example 13.19.

Figure 13.33 A triangle and a triple

Figure 13.34 A graph with three distinct cliques.

Figure 13.35 ER random graphs showing the emergence of a giant component. Bo...

Figure 13.36 Fraction of nodes outside the giant component as a function of ...

Figure 13.37

network. With probability

, the graph is regular with each n...

Figure 13.38

networks with

(top)

(bottom). To access Matlab code to co...

Figure 13.39 Average shortest path length and clustering coefficient versus ...

Figure 13.40 Network grown with preferential attachment with

and

edges a...

Figure 13.41 Flocking of birds (left), swarming of insects (middle) and scho...

Figure 13.42 Four agents interconnected in a ring.

Figure 13.43 Response of the four‐agent network showing that the states sync...

Appendix B

Figure B.1 A connected set. Every pair of points

and

in

can be connect...

Figure B.2 A simply connected set is one with no holes. The set shown is not...

Figure B.3 Level sets of a function

in

.

Appendix C

Figure C.1 The binomial distribution with

an

.

Figure C.2 The Poisson distribution with mean

.

Figure C.3 A normal or Gaussian distribution and cumulative distribution fun...

Figure C.4 The exponential distribution and cumulative distribution function...

Figure C.5 A power‐law distribution for

.

Guide

Cover

Table of Contents

Title Page

Copyright

Dedication

Preface

About the Companion Website

Begin Reading

Appendix A: Linear Algebra

Appendix B: Real Analysis

Appendix C: Probability

Appendix D: Proofs of Selected Results

Appendix E: Matlab Command Reference

References

Index

End User License Agreement

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Introduction to Modeling and Simulation

A Systems Approach

Mark W. SpongThe University of Texas at Dallas USA

This edition first published 2023© 2023 John Wiley & Sons Ltd

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To Lila.

Preface

This book grew out of a one‐semester course, Engineering Modeling and Simulation, in the Systems Engineering and Management Master of Science degree program at the University of Texas at Dallas. In the spring 2020 semester I was asked to teach this course after the previous instructor left the university, and it has been an enjoyable experience both to teach topics familiar to me and to prepare topics that I had never taught before. My vision in developing this text is to introduce students to a broad range of topics in mathematical modeling that can prepare them for deeper studies of one or more of these topics in subsequent courses. The individual topics in the text were chosen with this in mind, partly based on the availability of more advanced courses in our own curriculum. At the same time, I wanted to include enough material so that the students acquire a basic understanding of each topic whether or not they continue their studies beyond the present text. The emphasis is on systems concepts rather than proofs. I include proofs if doing so adds some particular insight. Otherwise, proofs are either omitted, with pointers to the literature, or else included in the appendix.

The intended audience for this text includes advanced undergraduates and beginning graduate students in science, engineering, management, or mathematics. I assume that the students have a basic knowledge of calculus, linear algebra, probability theory, and differential equations, as well as some familiarity with Matlab, although instructors may wish to supplement this text with an introductory Matlab text.

Organization of the Text

Each chapter of the text is intended to give a more or less self‐contained overview of an important topic in systems modeling and simulation. After an introductory chapter, Chapter 2 deals with second‐order systems. Second‐order systems are an important class of systems because many physical laws, like Newton's second law, are stated in terms of second‐order differential equations.

Chapter 3 treats the time response of continuous‐time systems of any order. We state sufficient conditions that guarantee a unique solution for nonlinear differential equations. We also discuss linear approximations of nonlinear systems and introduce the matrix exponential, with which we can characterize the solutions of linear systems. We present the Hartman–Grobman theorem which, in effect, says that near a hyperbolic equilibrium point, the response of a nonlinear system and its linear approximation share the same qualitative behavior. We also discuss singularly perturbed systems, which are used to model systems with very large or very small parameters that can pose particular challenges for simulation methods.

Chapter 4 introduces compartmental modeling with applications to population dynamics, the spread of epidemics, and predator–prey models. The basis for these models is the so‐called logistic equation, which we treat in some detail.

Chapter 5 deals with stability theory for linear and nonlinear systems. We discuss stability in the sense of Lyapunov and LaSalle's invariance principle.

In Chapter 6 we discuss discrete‐time systems, including the discrete‐time logistic equation, which shows important differences between continuous‐ and discrete‐time equations, and we present cobweb diagrams as a way to visualize solutions of one‐dimensional discrete maps.

Chapter 7 deals with numerical methods. We discuss numerical differentiation, numerical integration, and the most common methods for numerical solution of ordinary differential equations using Matlab and Simulink.

Chapter 8 is an introduction to optimization theory. We discuss necessary and sufficient conditions for solutions of nonlinear programming problems. We discuss gradient search methods and Newton's method for optimization and root finding. For optimization subject to constraints we introduce Lagrange multipliers and we present the so‐called Karush–Kuhn–Tucker conditions for optimality. We also discuss the importance of convexity in optimization.

Chapter 9 treats system identification. We discuss linear and nonlinear regression, and the method of recursive least squares. We also present a brief introduction to logistic regression and neural networks.

Chapter 10 treats stochastic systems. In particular, we focus on Markov chain models and Monte Carlo methods for simulation and optimization.

Chapter 11 is an introduction to feedback, which is a fundamental property in natural and human‐made systems. We give examples of feedback systems and present an introduction to feedback control.

Chapter 12 treats partial differential equation (PDE) models of systems. We discuss the three most important examples of linear PDE models, namely, the wave equation, heat equation, and Laplace's equation.

Finally Chapter 13 treats complex networks. Many of the most important systems in the modern world, such as the internet, World Wide Web, airline networks, social networks, and others, are modeled as complex networks. We give an introduction to the most relevant concepts in graph theory and network theory applied to both deterministic and random networks.

A Note to the Instructor

My original goal was to produce a text that could be covered in one semester. The book has grown to where it is unlikely that any single course can cover all the topics herein in a single semester. There is, however, a somewhat natural division of the text into chapters dealing primarily with dynamical systems and chapters dealing primarily with computational methods. Below I have outlined suggested courses for both 10 ‐week quarters and 16 ‐week semesters. Instructors may also pick and choose topics of interest in order to customize a specific course or a course sequence.

Course 1: Introduction to Modeling and Simulation

Level: Senior undergraduate/graduate

For a one‐quarter course (10 weeks):

Chapter 1

: Introduction

Chapter 2

: Second‐Order Systems

Chapter 3

: System Fundamentals

Chapter 4

: Compartmental Models

Chapter 5

: Stability

For a one‐semester course (16 weeks) add:

Chapter 11

: Feedback Systems

Chapter 12

: PDE Models

Chapter 13

: Complex Networks

Course 2: Computational Methods in Modeling and Simulation

Level: Senior undergraduate/graduate

For a one‐quarter course (10 weeks):

Chapter 1

: Introduction

Chapter 2

: Second‐Order Systems

Chapter 4

: Compartmental Models

Chapter 6

: Discrete‐Time Systems

Chapter 7

: Numerical Methods

For a one‐semester course (16 weeks) add:

Chapter 8

: Optimization

Chapter 9

: System Identification

Chapter 10

: Stochastic Systems

In addition, the instructor will want to supplement the text with either an introductory Matlab text or tutorials available online.

Acknowledgements

I am indebted to Jim Primbs, who developed the first version of the modeling and simulation course at UT‐Dallas. Later, Bobby Gregg took over responsibility for the course and added additional material. Their course outlines and lecture notes were the starting point for my subsequent development of the course. Any errors or omissions in the present text are entirely due to me. I also want to thank Reza Moheimani and Steve Yurkovich for constructive comments and support. Thanks also to Aykut Satici and Alireza Mohammadi for an interesting discussion on variable‐mass systems. The students in my class also helped by asking insightful questions and finding errors in early drafts of the text. Several comments from anonymous reviewers also helped considerably in making the text more readable and accessible to students. A special thanks to Richard Leigh for an exceptional editing job, which greatly improved the final manuscript. Finally, thanks to Sandra Grayson and Juliet Booker at John Wiley for their support and encouragement.

August 2022

Mark W. Spong

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/spong/engineeringmodeling

From the website you can find the following online materials:

• Solutions Manual

• Case‐study Exercises

• Lecture Slides

• MATLAB code

1Introduction

1.1 Introduction

This text is concerned with modeling and simulation of systems, both natural and engineered. We treat both analytical and computational methods for a range of applications to electrical, mechanical, biological, financial, social, and other systems.

The notion of what constitutes a system is very broad and application dependent. Generally, by a system, one means a combination of interrelated components or parts that works in synergy to collectively perform a desired function. In business, for example, a system can mean a set of processes or procedures working together, such as a payroll system or inventory system. In physiology, the circulatory system is composed of the heart, blood, arteries, veins, and capillaries and delivers oxygen and nutrients to cells and removes waste products, such as carbon dioxide. In computer science, an operating system is the software that manages hardware and software resources and provides services to application programs. In biology, an ecosystem refers to a community of interacting species and their physical environment.

The distinction between component and system is application dependent. For example, in the semiconductor industry a microprocessor is a highly complex system, whereas in the automotive industry a microprocessor is a component in systems that control engine emissions, braking, cruise control, and other functions.

1.1.1 Systems Engineering

Systems engineering is an interdisciplinary field of engineering that focuses on how to design and manage systems over their life cycles [3, 15, 17]. A key problem in systems engineering is how to deal with the increasing complexity of modern systems. Modeling and simulation are important tools to help design and analyze modern complex systems. Good models allow meaningful simulation for testing, design validation, hardware‐in‐the‐loop testing and iterative methods. Simulations are less expensive than prototype testing and can be more rapidly developed and modified.

In this text we take the viewpoint that modeling is the more important part of modeling and simulation. Simulations based on poor models are of relatively little use. Thus, we focus on the modeling aspects of systems and use simulations to illustrate their performance and gain insight into their structure.

1.1.2 The Input/Output Viewpoint

At an abstract level, we can view a system as an object or process that transforms inputs to outputs, which we write as ; see Figure 1.1. Within this input/output system paradigm various assumptions on the nature of the system must be made to derive concrete models that can be used for analysis, simulation, and prediction. How to do this is the principle subject of this text.

Figure 1.1 A general characterization of a system as an object or process that transforms inputs to outputs .

1.1.3 Some Examples

Examples of systems as objects that transform inputs to output are found in numerous fields of engineering and nature. Some representative examples include the following:

(1)

An electric circuit.

In the circuit shown in

Figure 1.2

we can take the voltage

as input and the voltage

as output. The system then transforms the input to the output according to Kirchhoff's laws.

Figure 1.2 An electric circuit with input voltage and output voltage . The output voltage is determined by Kirchhoff's Laws.

(2)

A mechanical system.

The system in

Figure 1.3

shows an interconnected mass, spring, and damper. If an input force

acts on the mass, then the output position

of the mass is governed by Newton's laws of motion.

Figure 1.3 A mass–spring–damper system with input force and output position . The motion of the mass as a function of time is governed by Newton's laws.

(3)

A DC motor.

In a DC motor, as shown in

Figure 1.4

, we can take the applied voltage

as input and the shaft angular velocity

as output. The system then transforms electrical energy into mechanical energy to perform useful work.

Figure 1.4 A DC motor with input voltage and output speed . Both electrical and mechanical models govern the input/output behavior of this system.

(4)

An economic system.

A system may have multiple inputs and/or multiple outputs. For example, in an economic system, variables such as money supply, interest rates, and regulations can be taken as inputs and variables such as inflation rate, growth rate, unemployment rate can be taken as outputs (

Figure 1.5

).

Figure 1.5 An economic system with multiple inputs and outputs. The choice of what variables to choose as inputs and what variables to choose as outputs is also an important part of the modeling process.

(5)

A manufacturing plant.

A factory (

Figure 1.6

) can be thought of as a system that transforms raw materials (the inputs) to finished products (the outputs). The manufacturing process is composed of many individual processes that occur as discrete events, such as metal forming, welding, painting, final assembly, and other processes.

Figure 1.6 A manufacturing plant transforms input raw materials into output finished products. The system can be modeled as the interconnection of many subsystems, such as metal forming, welding, painting, assembly, and so on.

(6)

Machine learning.

A machine learning system can be thought of as a computer program that takes data as input and produces estimates (classifications, decisions, suggestions) as outputs (

Figure 1.7

).

Figure 1.7 A machine learning (ML) system takes input data and produces output decisions. The generation of the input data and the algorithms used for the inference engine are key features of a learning system.

(7)

Feedback systems.

Feedback is a fundamental property in both natural and engineered systems. By a

feedback system

we mean “a system where the input is influenced by the output.”

Figure 1.8 A feedback system is one where the output affects future inputs.

We refer to the system in Figure 1.8 as a closed‐loop system and the system in Figure 1.1 as an open‐loop system. Feedback can be positive, which tends to have a destabilizing effect, as in a nuclear chain reaction, or negative, which tends to have a stabilizing effect, for example, when we sweat in hot weather to reduce our body temperature.

Feedback control is indispensable in modern engineering systems, such as automotive systems, aircraft, robotics, chemical and oil refineries, manufacturing, energy and power systems, and a host of other applications. In biological systems, feedback is present in regulating many processes, from body temperature to cell metabolism, gene expression, hormone production, as well as balance and locomotion. In weather and climate dynamics, feedback is an important mechanism affecting global temperatures, weather patterns, ocean currents, and so on.

1.2 Model Classification

A model is a mathematical representation and is both an abstraction and an approximation of physical reality. Developing models of systems takes several forms.

Models from first principles.

These are physics‐based models, typically differential or difference equations derived from Newton's laws, Maxwell's equations, Kirchhoff's laws, and so on.

Black box models.

These are models based on measured data to identify an input/output structure. Such data‐driven methods include neural networks, genetic algorithms, frequency‐response methods, machine learning, or optimization methods.

Gray box models.

These are a combination of the above two using a‐priori assumptions on the model together with measured data for parameter identification, systems identification, regression, or other methods.

System models may be classified in several ways, which can be useful to determine the best tools for their analysis. Some of the most useful classifications are listed below.

1.2.1 Static and Dynamic Systems

In a static system, there is an algebraic or functional relationship between input and output, as in a resistor where . Static models will be used, for example, when we discuss optimization, regression, and graph theory. In a dynamic system the output depends on the input together with the current state of the system, which typically evolves according to a differential or difference equation.

1.2.2 Linear and Nonlinear Systems

All systems possess nonlinearities. Despite this fact, linear systems are an important class of systems for two primary reasons. First, nonlinear systems are approximately linear near their equilibrium states. Second, linear systems are well understood and their behaviors are easily characterized. Nonlinear systems, on the other hand, are difficult to analyze and can often only be studied using numerical methods. At the same time, nonlinear systems have extremely rich behavior not possessed by linear systems.

The key to linearity is the principle of superposition, with which the output response to an arbitrary input can be understood in relation to outputs generated by simpler inputs. Specifically, we have the following definition.

Definition 1.1 (Linearity)

A system is linear if and only if it satisfies the principle of superposition: If and , then the output corresponding to the input , where and are constant, is given by . In other words,

The principle of superposition says that if is the output corresponding to the input and is the output corresponding to the input , then a linear combination of the inputs and produces the same linear combination of the outputs and . Using superposition, one can gain an understanding of systems by analyzing their constituent components.

This approach is related to the principle of reductionism, which refers to a scientific approach in which a system is decomposed into subsystems, each of which is more amenable to analysis. The increasing complexity of modern engineering systems limits the usefulness of the reductionist approach. Complexity leads to emergent behavior, which means roughly that the whole is greater than the sum of the parts. Thus, we need to develop methods that are directly applicable to nonlinear, interconnected systems. We will discuss emergent behavior in complex networks in the final chapter of this text.

1.2.3 Distributed‐Parameter Systems

By a distributed‐parameter system, we mean one that is described by a partial differential equation. Examples of distributed‐parameter systems are vibrating strings and membranes, the temperature distribution along a bar, and quantum mechanical systems.

1.2.4 Hybrid and Discrete‐Event Systems

Hybrid systems are systems that combine both continuous‐time and discrete‐time components. A simple example of a hybrid system is a bouncing ball, whose motion is governed by continuous dynamics together with a discrete reset map when the ball contacts the ground.

Discrete‐event systems are logic‐based rather than time‐based. For example, in a manufacturing system an assembly operation may take place in discrete stages as parts are transferred between machines.

1.2.5 Deterministic and Stochastic Systems

In a deterministic system the output is uniquely determined from the model and is repeatable. In a stochastic system, the output is determined according to a probability, and multiple repeated trials will generally produce different outputs.

1.2.6 Large‐Scale Systems

By a large‐scale system one typically means a system with numerous components and with many variables and parameters needed to model them. These systems are also complex and difficult to model accurately. A few example of large‐scale engineered systems are given below in Figures 1.9 to 1.15.

Figure 1.9 Warehouse robots for order retrieval are an example of factory automation, where the robots act with little or no human supervision.

Figure 1.10 Oil and chemical refineries are large‐scale, complex systems that are made more efficient through modeling, simulation, and systems engineering principles for process control and optimization.

Figure 1.11 The electric power grid consists of an interconnected network of power generation stations, transmission lines, substations, and power distribution stations. China is currently the largest producer of electric energy followed by the United States.

Figure 1.12 A wind farm is a collection of wind turbines connected to the power grid. Each turbine is a complicated engineered system. However, in order to optimize energy production, it is not sufficient to study the individual turbines. Modeling the interactions among the turbines is also important.

Figure 1.13 Supply chains and logistics systems are responsible for moving manufactured goods on a global scale. These systems are interconnected networks of ships, airplanes, trucks, and railroads, as well as pipeline networks, loading docks, warehouses and a host of other systems that must function efficiently and economically.

Figure 1.14 Systems of autonomous vehicles are already changing the trucking industry, the shipping industry, and personal transportation. As in the wind‐farm example, one must model both the individual vehicles and the interconnections among multiple vehicles.

Figure 1.15 Surgical robots are examples of human/machine/computer interfaces and require new methods for modeling human dynamics in addition to robot dynamics.

1.3 Simulation Languages

There are several general purpose and special purpose simulation software programs and packages available, for example:

Matlab/Simulink

is a programming language and numerical computing environment. Matlab also allows symbolic computation and graphical representation of data.

SysML

is a graphical modeling language for analyzing complex systems. SysML is useful for modeling system requirements, system interconnections, and other features.

AnyLogic

is a simulation tool designed primarily for business applications like supply chains, logistics, markets and other applications.

FlexSim

is a simulation package designed for modeling and simulation of discrete‐event systems.

Mathematica

is a general‐purpose programming language for symbolic and numerical computation.

Modelica

is an object‐oriented modeling language, as opposed to a programming language, designed to model systems containing electrical, mechanical, hydraulic, thermal, and other components.

ANSYS

is a finite‐element modeling package for numerical analysis of mechanical systems.

SolidWorks

is a computer‐aided design and computer‐aided engineering package that runs primarily on Microsoft Windows.

Arena

is a discrete‐event simulation and automation package for business and production processes.

A discussion of specialized programs for topics such as project management is outside the scope of this text. Instead, we will use Matlab/Simulink as a general purpose programming language to illustrate simulation methods for system models.

1.4 Outline of the Text

Each chapter of the text is intended to give an overview of a specific topic in systems modeling and simulation. After studying the chapters in this text the reader should be prepared for more in‐depth treatment of each topic.

Chapter 2 begins with a treatment of second‐order systems. Second‐order systems form an important class of systems because many physical laws, like Newton's second law, are stated in terms of second‐order differential equations. We characterize the behavior of linear second‐order systems in terms of the eigenvalues and eigenvectors of the system coefficient matrices. We also discuss the existence of periodic solutions for nonlinear systems and present the Poincaré–Bendixson theorem, which gives conditions for the existence of periodic solutions of nonlinear systems.

Chapter 3 treats the time response of general continuous‐time systems. We state sufficient conditions that guarantee a unique solution for nonlinear differential equations. We also discuss linear approximations of nonlinear systems and we introduce the matrix exponential, with which we can characterize the solutions of th‐order linear systems. We also discuss the Hartman–Grobman theorem which, in effect, says that near a hyperbolic equilibrium point, the response of a nonlinear system and its linear approximation have the same qualitative behavior. We also discuss singularly perturbed systems, which are used to model systems with very large or very small parameters that can pose particular challenges for simulation methods.

In Chapter 4 we introduce compartmental modeling and discuss examples and applications, including models of population dynamics, the spread of epidemics, and predator–prey models. The basis for these models is the so‐called logistic equation, which we treat in some detail.

Chapter 5 deals with stability theory for linear and nonlinear systems. For systems operating in equilibrium configurations, it is important to know how the system will respond to perturbations and disturbances that move the system state away from equilibrium. We discuss two main results, namely stability in the sense of Lyapunov and LaSalle's invariance principle.

In Chapter 6 we discuss discrete‐time systems. We present results that mimic the results for continuous‐time systems, including the existence and uniqueness of solutions and stability. We discuss the discrete‐time logistic equation, which shows important differences between the continuous‐time and discrete‐time equations, and we present cobweb diagrams as a convenient way to visualize solutions of one‐dimensional discrete‐time maps.

In Chapter 7 we discuss numerical methods to simulate system models. We discuss numerical differentiation, numerical integration and the most common methods for numerical solution of ordinary differential equations using Matlab and Simulink.

Chapter 8 is an introduction to optimization theory. We will discuss general necessary and sufficient conditions for solutions of nonlinear programming problems. We discuss gradient search methods, including the method of steepest descent. We then treat Newton's method for optimization and root finding. For optimization subject to constraints we introduce Lagrange multipliers and we present the so‐called Karush–Kuhn–Tucker conditions for optimality. We also discuss the importance of convexity in optimization.

Chapter 9 treats system identification. We discuss linear and nonlinear regression, and the method of recursive least squares. We also present a brief introduction to logistic regression and perceptron neural networks for binary classification.

Chapter 10 treats stochastic systems. In particular, we focus on Markov chain models and Monte Carlo methods for simulation and optimization.

In Chapter 11