A Mathematical Introduction to Control Theory

The following sections are included:

• Dedication
• Preface to the Third Edition
• Preface to the Second Edition
• Preface to the First Edition
• About the Author
• Contents

The following sections are included:

• An Introduction to the Laplace Transform
• Properties of the Laplace Transform
• A Second Proof of the Initial Value Theorem
  • The Intuition Behind the Proof
  • The Proof
• Finding the Inverse Laplace Transform
  • Some Simple Inverse Transforms
  • The Quadratic Denominator
• Integro-Differential Equations
• An Introduction to Stability
  • Some Preliminary Manipulations
  • Stability
  • Why We Obsess about Stability
  • The Tacoma Narrows Bridge—A Brief Case History
• MATLAB^®
  • Assignments
  • Commands
• Control Theory and the Python^® Programming Language
• Exercises

The following sections are included:

• Transfer Functions
• The Frequency Response of a System
• Bode Plots
• The Time Response of Certain “Typical” Systems
  • First Order Systems
  • Second Order Systems
• Three Important Devices and Their Transfer Functions
  • The Operational Amplifier (op amp)
  • The DC Motor
  • The “Simple Satellite”
• Zeros in the Right Half-Plane
• Block Diagrams and How to Manipulate Them
• Controlling a DC Motor
• Using the Python Programming Language
  • Importing Relevant Modules
  • Transfer Functions, Bode Plots, and the Step Response
  • Before Examining the Output
  • Examining the Output
• Exercises

The following sections are included:

• Why Feedback—A First View
• Sensitivity
• More about Sensitivity
• A Simple Example
• System Behavior at DC
• Tracking Inputs of the Form tⁿu(t)
• Internal Stability
• Disturbance Rejection
• Exercises

The following sections are included:

• Roots of Polynomials—A Little History
• Theorem, Proof, and Applications
• The Routh Array
• A Design Example
• Exercises

The following sections are included:

• More about Poles in the Right Half Plane
• The Principle of the Argument
• The Proof of the Principle of the Argument
• How are Encirclements Measured?
• First Applications to Control Theory
• Systems with Low-Pass Open-Loop Transfer Functions
• MATLAB^® and Nyquist Plots
• The Python Programming Language and Nyquist Plots
• The Nyquist Plot and Delays
• Delays and the Routh–Hurwitz Criterion
• Relative Stability
• The Bode Plots
• A Connection between Frequency-Domain Specifications and Time-Domain Specifications
• Some More Examples
• Exercises

The following sections are included:

• The Root Locus Diagram—An Introduction
• Rules for Plotting the Root Locus
  • The Symmetry of the Root Locus
  • Branches on the Real Axis
  • The Asymptotic Behavior of the Branches
  • Departure of Branches from the Real Axis
  • A “Conservation Law”
  • The Behavior of Branches as They Leave Finite Poles or Enter Finite Zeros
  • A Group of Poles and Zeros Near the Origin
• Some (Semi-)Practical Examples
  • The Effect of Zeros in the Open Right Half-Plane
  • The Effect of Three Poles at the Origin
  • The Effect of Two Poles at the Origin
  • A System that Tracks t sin(t)
  • Variations on Our Theme
  • The Effect of a Delay on the Root Locus Diagram
  • The Phase-lock Loop
  • Sounding a Cautionary Note—Pole-Zero Cancellation
• Using the Python Programming Language to Plot Root Locus Diagrams
• More on the Behavior of the Roots of Q(s)/K + P (s) = 0
• Exercises

The following sections are included:

• Compensation—An Introduction
• The Attenuator
• Phase-Lag Compensation
• Phase-Lead Compensation
• Lag-lead Compensation
• The PID Controller
• An Extended Example
  • The Attenuator
  • The Phase-Lag Compensator
  • The Phase-Lead Compensator
  • The Lag-Lead Compensator
  • The PD Controller
  • A Comparison of the Step Responses
• The Ziegler-Nichols Rules
  • Introduction
  • The Assumption
  • Characterizing the Plant
  • A Partial Verification
  • Estimating the System’s Phase Margin
  • Estimating the System’s Gain Margin
  • Estimating the System’s Settling Time
  • Using the Python Programming Language to Examine the Ziegler-Nichols Rules
• Bode’s Sensitivity Integrals
  • Introduction
  • The Sensitivity Function
  • Bode’s Sensitivity Integrals
  • Case I
  • Case II
  • Case III
  • Case IV
  • Available Bandwidth
  • Open-Loop Unstable Systems with Limited Available Bandwidth
• Exercises

The following sections are included:

• Introduction
• The Describing Function Technique
  • The Describing Function Concept
  • Predicting Limit Cycles
  • The Stability of Limit Cycles
  • More Examples
    • A Nonlinear Oscillator
    • A Comparator with a Dead Zone
    • A Simple Quantizer
  • Graphical Method
• Tsypkin’s Method
• The Tsypkin Locus and the Describing Function Technique
• Exercises

The following sections are included:

• Introduction
• The State Variables Formalism
• Choosing States When a System is Characterized by an ODE
• Converting Transfer Functions to State Equations
• Solving Matrix Differential Equations, Part I: Homogeneous Equations
• The Significance of the Eigenvalues of the Matrix
• From State Equations to Transfer Functions
• Solving Matrix Differential Equations, Part II: Inhomogeneous Equations
• The Cayley–Hamilton Theorem
• Controllability
• Pole Placement
• Observability
• Observer Design
• Examples
  • Pole Placement
  • Adding an Integrator
  • Modern Control Using MATLAB^®
  • A System that is not Observable
  • A Second System that is not Observable
  • A System that is neither Observable nor Controllable
  • Designing an Observer
• Some Technical Results about Series of Matrices
• Exercises

The following sections are included:

• The State Variables Formalism for Discrete-Time Systems
• Two Systems
  • A One Dimensional System Whose State Is Constant
  • A More Complicated System: A Ballistic Projectile
• Solving the State Equation, (10.1)
• Stability
• Controllability
• Observability
• Designing a Full-State Observer
  • The Theory
  • Designing an Observer for the System of §10.2.2
  • Discussion
• A Brief Introduction to the Kalman Filter
  • Introduction
  • The General Form of the Kalman Filter
  • A Bit of Nomenclature
  • The Initial State
  • The First (Prediction) Step
  • Making Use of the Prediction: Calculating the Kalman Gain
  • The Second (Correction) Step
  • Putting It All Together
• Arriving at the Form of the Kalman Filter by Probabilistic Arguments
• A Kalman Filter for the System of §10.2.1
• Exercises

The following sections are included:

• Chapter 1
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
• Chapter 2
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
  • Exercise 15
  • Exercise 17
• Chapter 3
  • Exercise 1
  • Exercise 3
  • Exercises 4 and 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
  • Exercise 15
• Chapter 4
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
  • Exercise 15
• Chapter 5
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
  • Exercise 15
  • Exercise 17
  • Exercise 19
  • Exercise 21
  • Exercise 23
  • Exercise 25
• Chapter 6
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
  • Exercise 15
  • Exercise 17
  • Exercise 19
  • Exercise 21
  • Exercise 23
• Chapter 7
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11
  • Exercise 13
  • Exercise 15
  • Exercise 17
  • Exercise 19
  • Exercise 21
  • Exercise 23
  • Exercise 25
  • Exercise 27
• Chapter 8
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
• Chapter 9
  • Exercise 1
  • Exercise 3
  • Exercise 4
  • Exercise 5
  • Exercise 6
  • Exercise 7
  • Exercise 11
  • Exercise 13
  • Exercise 15
  • Exercise 16
• Chapter 10
  • Exercise 1
  • Exercise 3
  • Exercise 5
  • Exercise 7
  • Exercise 9
  • Exercise 11

The following sections are included:

• References
• Index