Essentials of Control Techniques and Theory (Hardback) book cover

Essentials of Control Techniques and Theory

By John Billingsley

© 2009 – CRC Press (Textbook (Standard) (DRM-Free))

339 pages | 122 B/W Illus.

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Carefully separating the essential from the ornamental, Essentials of Control Techniques and Theory presents the nuts and bolts for designing a successful controller. It discusses the theory required to support the art of designing a working controller as well as the various aspects to convince a client, employer, or examiner of your expertise.

A Compelling Account of the Basics of Control Theory

Control solutions for practicing engineers

Using the author’s own Javascript On-Line Learning Interactive Environment for Simulation (Jollies), the text relies on computer-based graphical analysis methods, such as Nyquist, Nichols, root locus, and phase-plane, to illustrate how useful computer simulation can be for analyzing both linear and nonlinear systems. It explains step-by-step the design and modeling of various control systems, including discrete time systems and an inverted pendulum. Along with offering many web-based simulations, the book shows how mathematics, such as vectors, matrices, and the differential equations that govern state variables, can help us understand the concepts that underpin the controller’s effects.

From frequency domain analysis to time-domain state-space representation, this book covers many aspects of classical and modern control theory. It presents important methods for designing and analyzing linear systems and controllers.


…a valuable and much-needed demystification of control theory … The role of the blackboard is assumed by a well-matched and balanced Web site, on which JavaScript-based simulations of the book’s examples and experiments can be run (and rerun with changed parameters). … The complement of the lucid and reasoned thinking out loud is, along with coherent excerpts of JavaScript, in the hard copy. … Forgetting for the moment that this is an intrinsically excellent book, it is also the right control theory book at the right time. … This book has my highest recommendation among control theory books.

—George Hacken, Computing Reviews, August 2010

This book contains some basic notions and techniques from the control theory presented from the point of view of an engineer. Special attention is paid to the classical frequency-domain methods, practical observers, and different principles of controller design. Using the author's own Javascript On-Line Learning Interactive Environment for Simulation (Jollies), the design and modeling of various control systems is explained step-by-step (as the inverted pendulum). … The book contains a large number of illustrative examples (including the corresponding Javascript codes and simulation results) helping to understand the application of the control theory to design of working controllers. … very useful as a complementary reading for a course in control theory.

— Mikhail I. Krastanov, in Zentralblatt MATH, 1198-1

Table of Contents


Introduction: Control in a Nutshell, History, Theory, Art, and Practice

The Origins of Control

Early Days of Feedback

The Origins of Simulation

Discrete Time

Modeling Time


A Simple System


Choosing a Computing Platform

An Alternative Platform

Solving the First Order Equation

A Second Order Problem

Matrix State Equations

Analog Simulation

Closed Loop Equations

Simulation with JavaScript "On-Line Learning Interactive Environment for Simulation" (Jollies)Introduction

How a Javascript On-Line Learning Interactive Environment for Simulation (Jollies) Is Made Up

Moving Images without an Applet

A Generic Simulation

Practical Control Systems


The Nature of Sensors

Velocity and Acceleration

Output Transducers

A Control Experiment

Adding Control


Vector State Equations


Another Approach

A Change of Variables

Systems with Time Delay and the PID Controller

Simulating the Water Heater Experiment

Systems with Real Components and Saturating Signals—Use of the Phase Plane

An Early Glimpse of Pole Assignment

The Effect of Saturation

Meet the Phase Plane

Phase Plane for Saturating Drive

Bang-Bang Control and Sliding Mode

Frequency Domain Methods


Sine-Wave Fundamentals

Complex Amplitudes

More Complex Still-Complex Frequencies

Eigenfunctions and Gain

A Surfeit of Feedback

Poles and Polynomials

Complex Manipulations

Decibels and Octaves

Frequency Plots and Compensators

Second Order Responses

Excited Poles

Discrete Time Systems and Computer Control


State Transition

Discrete Time State Equations and Feedback

Solving Discrete Time Equations

Matrices and Eigenvectors

Eigenvalues and Continuous Time Equations

Simulation of a Discrete Time System

A Practical Example of Discrete Time Control

And There’s More

Controllers with Added Dynamics

Controlling an Inverted Pendulum

Deriving the State Equations

Simulating the Pendulum

Adding Reality

A Better Choice of Poles

Increasing the Realism

Tuning the Feedback Pragmatically

Constrained Demand

In Conclusion


More Frequency Domain Background Theory


Complex Planes and Mappings

The Cauchy–Riemann Equations

Complex Integration

Differential Equations and the Laplace Transform

The Fourier Transform

More Frequency Domain Methods


The Nyquist Plot

Nyquist with M-Circles

Software for Computing the Diagrams

The "Curly-Squares" Plot

Completing the Mapping

Nyquist Summary

The Nichols Chart

The Inverse-Nyquist Diagram

Summary of Experimental Methods

The Root Locus


Root Locus and Mappings

A Root Locus Plot

Plotting with Poles and Zeroes

Poles and Polynomials

Compensators and Other Examples


Fashionable Topics in Control


Adaptive Control

Optimal Control

Bang-Bang, Variable Structure, and Fuzzy Control

Neural Nets

Heuristic and Genetic Algorithms

Robust Control and H-infinity

The Describing Function

Lyapunov Methods


Linking the Time and Frequency Domains


State-Space and Transfer Functions

Deriving the Transfer Function Matrix

Transfer Functions and Time Responses

Filters in Software

Software Filters for Data

State Equations in the Companion Form

Time, Frequency, and Convolution

Delays and the Unit Impulse

The Convolution Integral

Finite Impulse Response Filters



More about Time and State Equations


Juggling the Matrices

Eigenvectors and Eigenvalues Revisited

Splitting a System into Independent Subsystems

Repeated Roots

Controllability and Observability

Practical Observers, Feedback with Dynamics


The Kalman Filter

Reduced-State Observers

Control with Added Dynamics


Digital Control in More Detail


Finite Differences—The Beta-Operator

Meet the z-Transform

Trains of Impulses

Some Properties of the z-Transform

Initial and Final Value Theorems

Dead-Beat Response

Discrete-Time Observers

Relationship between z- and Other Transforms


The Impulse Modulator

Cascading Transforms

Tables of Transforms

The Beta and w Transforms

Design Methods for Computer Control


The Digital-to-Analog Convertor (DAC) as Zero Order Hold


A Position Control Example, Discrete Time Root Locus

Discrete Time Dynamic Control-Assessing Performance

Errors and Noise


Practical Design Considerations

Delays and Sample Rates


Optimal Control—Nothing but the Best

Introduction: The End Point Problem

Dynamic Programming

Optimal Control of a Linear System

Time Optimal Control of a Second Order System

Optimal or Suboptimal?

Quadratic Cost Functions

In Conclusion


About the Author

John Billingsley is Chair of Mechatronic Engineering at the University of Southern Queensland in Toowoomba, Australia, and directs technology research in the National Centre for Engineering in Agriculture (NCEA).

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