A Mathematical Introduction to Control Theory

The following sections are included:

• Preface

• Contents

The following sections are included:

• An Introduction to the Laplace Transform

• Properties of the Laplace Transform

• 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

• 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”

• Block Diagrams and How to Manipulate Them

• A Final Example

• Exercises

The following sections are included:

• Why Feedback—A First View

• Sensitivity

• More about Sensitivity

• A Simple Example

• System Behavior at DC

• Noise Rejection

• Exercises

The following sections are included:

• Proof and Applications

• 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 Nyquist Plot and Delays

• Delays and the Routh-Hurwitz Criterion

• Relative Stability

• The Bode Plots

• An (Approximate) Connection between Frequency Specifications and Time Specification

• Some More Examples

• Exercises

The following sections are included:

• The Root Locus—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 Right Half-Plane
  • The Effect of Three Poles at the Origin
  • The Effect of Two Poles at the Origin
  • Variations on Our Theme
  • The Effect of a Delay on the Root Locus Plot
  • The Phase-lock Loop
  • Sounding a Cautionary Note—Pole-Zero Cancellation

• 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

• 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

• Solving Matrix Differential Equations

• The Significance of the Eigenvalues of the Matrix

• Understanding Homogeneous Matrix Differential Equations

• Understanding Inhomogeneous Equations

• The Cayley-Hamilton Theorem

• Controllability

• Pole Placement

• Observability

• Examples
  • Pole Placement
  • Adding an Integrator
  • Modern Control Using MATLAB
  • A System that is not Observable
  • A System that is neither Observable nor Controllable

• Converting Transfer Functions to State Equations

• Some Technical Results about Series of Matrices

• Exercises

The following sections are included:

• Introduction

• The Definition of the Z-Transform

• Some Examples

• Properties of the Z-Transform

• Sampled-data Systems

• The Sample-and-Hold Element

• The Delta Function and its Laplace Transform

• The Ideal Sampler

• The Zero-Order Hold

• Calculating the Pulse Transfer Function

• Using MATLAB to Perform the Calculations

• The Transfer Function of a Discrete-Time System

• Adding a Digital Compensator

• Stability of Discrete-Time Systems

• A Condition for Stability

• The Frequency Response

• A Bit about Aliasing

• The Behavior of the System in the Steady-State

• The Bilinear Transform

• The Behavior of the Bilinear Transform as T → 0

• Digital Compensators

• When Is There No Pulse Transfer Function?

• An Introduction to the Modified Z-Transform

• Exercises

Please refer to full text.

The following sections are included:

• Bibliography

• Index