A Practical Approach to Dynamical Systems for Engineers

Patricia Mellodge

• Publisher: Woodhead Publishing
• Year: 2015
• Print ISBN:
  9780081002025
• eBook ISBN:
  9780081002247
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

A Practical Approach to Dynamical Systems for Engineers takes the abstract mathematical concepts behind dynamical systems and applies them to real-world systems, such as a car traveling down the road, the ripples caused by throwing a pebble into a pond, and a clock pendulum swinging back and forth. Many relevant topics are covered, including modeling systems using differential equations, transfer functions, state-space representation, Hamiltonian systems, stability and equilibrium, and nonlinear system characteristics with examples including chaos, bifurcation, and limit cycles. In addition, MATLAB is used extensively to show how the analysis methods are applied to the examples. It is assumed readers will have an understanding of calculus, differential equations, linear algebra, and an interest in mechanical and electrical dynamical systems. - Presents applications in engineering to show the adoption of dynamical system analytical methods - Provides examples on the dynamics of automobiles, aircraft, and human balance, among others, with an emphasis on physical engineering systems - MATLAB and Simulink are used throughout to apply the analysis methods and illustrate the ideas - Offers in-depth discussions of every abstract concept, described in an intuitive manner, and illustrated using practical examples, bridging the gap between theory and practice - Ideal resource for practicing engineers who need to understand background theory and how to apply it

Chapter 1

Introduction

What Is a Dynamical System?

Abstract

In this study of dynamical systems, a system can be considered to be a black box with input(s) and output(s). A dynamical system is a system in which inputs, outputs, and possibly its characteristics change with time. To study these systems, one must mathematically model the relationship between the inputs and outputs. In this chapter, several examples are discussed, and many types of systems are described, including continuous time and discrete time, linear and nonlinear, time invariant and time varying, memory and memoryless, causal and noncausal, and deterministic and stochastic.

Keywords

Control; Dynamical system; Input and output; Modeling; Prediction; System characteristics

1.1. Overview

Dynamical systems are all around us: from a car traveling down the road to the ripples caused by throwing a pebble into a pond to a clock pendulum swinging back and forth. But what is a dynamical system? First, let us explore the words that make up the phrase “dynamical system.” The Merriam-Webster Dictionary gives these definitions:

Dynamic (adjective): always active or changing; having or showing a lot of energy; of or relating to energy, motion, or physical force

System (noun): a group of related parts that move or work together

The term “dynamic” gives us the idea of change or motion and can deal more specifically with physical phenomena. Mathematically, when one thinks of change, derivatives should spring to mind; indeed, these are a key component of how we model dynamical systems.

The definition of “system” involves groups of related parts working together. Although this definition works in general, a diagram such as the one shown in Figure 1.1 best illustrates the concept that we use throughout the book. Simply stated, a system is an entity that has an input and an output. A system receives an input and produces an output based on the input and its state.

When the two words are put together to form “dynamical system,” things get interesting. A dynamical system is one in which inputs, outputs, and even the system characteristics themselves can change with time. The relationship between input and output can be modeled mathematically using various techniques, such as differential and difference equations, transfer functions, and state space equations. Throughout this book, we focus on two objectives: (1) investigate various techniques and analysis methods for dynamical systems and (2) apply these methods to examples of real-world systems and show how they can be used in practice. As we will see, the study of dynamical systems can reveal some very interesting behaviors, some desirable and some not.

Figure 1.1 Block diagram representation of a system.

Figure 1.2 A car’s suspension system with the road height as the input and the car’s height as the output.

What about a car driving down a road makes it a dynamical system? One example is its suspension system, represented in Figure 1.2. The input to this system is the road, which has a varying height as the car drives down it. The output might be the ride height of the car itself. The system consists of the various components linking the road to the passenger’s body: the tires, control arm, shock absorber, chassis, and so on. When you hit a bump in the road, you experience a dynamical response: perhaps your body is jarred by the sudden change in wheel height, or perhaps you barely notice it. The response depends on the characteristics of the car’s suspension, which can be modeled as a mass-spring-damper system.

1.1.1. Why Do We Study Dynamic Systems?

The main reasons are (1) to predict system behavior and (2) to control system behavior.

An example of a system we study for prediction is the wea...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• List of Figures
• List of Tables
• About the Author
• Preface
• Chapter 1. Introduction: What Is a Dynamical System?
• Chapter 2. System Modeling
• Chapter 3. Characteristics of Dynamical Systems
• Chapter 4. Characteristics of Nonlinear Systems
• Chapter 5. Hamiltonian Systems
• Index