# Introduction to Linear Control Systems

## 1st Edition

**Authors:**Yazdan Bavafa-Toosi

**Paperback ISBN:**9780128127483

**eBook ISBN:**9780128127490

**Imprint:**Academic Press

**Published Date:**20th September 2017

**Page Count:**1008

## Description

*Introduction to Linear Control Systems* is designed as a standard introduction to linear control systems for all those who one way or another deal with control systems. It can be used as a comprehensive up-to-date textbook for a one-semester 3-credit undergraduate course on linear control systems as the first course on this topic at university. This includes the faculties of electrical engineering, mechanical engineering, aerospace engineering, chemical and petroleum engineering, industrial engineering, civil engineering, bio-engineering, economics, mathematics, physics, management and social sciences, etc.

The book covers foundations of linear control systems, their raison detre, different types, modelling, representations, computations, stability concepts, tools for time-domain and frequency-domain analysis and synthesis, and fundamental limitations, with an emphasis on frequency-domain methods. Every chapter includes a part on further readings where more advanced topics and pertinent references are introduced for further studies. The presentation is theoretically firm, contemporary, and self-contained. Appendices cover Laplace transform and differential equations, dynamics, MATLAB and SIMULINK, treatise on stability concepts and tools, treatise on Routh-Hurwitz method, random optimization techniques as well as convex and non-convex problems, and sample midterm and endterm exams.

The book is divided to the sequel 3 parts plus appendices.

**PART I:** In this part of the book, chapters 1-5, we present foundations of linear control systems. This includes: the introduction to control systems, their raison detre, their different types, modelling of control systems, different methods for their representation and fundamental computations, basic stability concepts and tools for both analysis and design, basic time domain analysis and design details, and the root locus as a stability analysis and synthesis tool.

**PART II: **In this part of the book, Chapters 6-9, we present what is generally referred to as the frequency domain methods. This refers to the experiment of applying a sinusoidal input to the system and studying its output. There are basically three different methods for representation and studying of the data of the aforementioned frequency response experiment: these are the Nyquist plot, the Bode diagram, and the Krohn-Manger-Nichols chart. We study these methods in details. We learn that the output is also a sinusoid with the same frequency but generally with different phase and magnitude. By dividing the output by the input we obtain the so-called sinusoidal or frequency transfer function of the system which is the same as the transfer function when the Laplace variable *s* is substituted with . Finally we use the Bode diagram for the design process.

**PART III: **In this part, Chapter 10, we introduce some miscellaneous advanced topics under the theme fundamental limitations which should be included in this undergraduate course at least in an introductory level. We make bridges between some seemingly disparate aspects of a control system and theoretically complement the previously studied subjects.

**Appendices: **The book** **contains seven appendices. Appendix A is on the Laplace transform and differential equations. Appendix B is an introduction to dynamics. Appendix C is an introduction to MATLAB, including SIMULINK. Appendix D is a survey on stability concepts and tools. A glossary and road map of the available stability concepts and tests is provided which is missing even in the research literature. Appendix E is a survey on the Routh-Hurwitz method, also missing in the literature. Appendix F is an introduction to random optimization techniques and convex and non-convex problems. Finally, appendix G presents sample midterm and endterm exams, which are class-tested several times.

## Key Features

- Presenting a detailed contemporary perspective of the field of systems and control theory and applications
- Contemporary and mathematically firm approach even for classical issues
- Discussing and correcting numerous mistakes in the available literature
- Collecting and discussing numerous important points which are scattered in the research literature
- Many new results and/or details in Chapters 3-10 and Appendices A, D
- A detailed glossary and road map of stability results scattered in the literature
- Addressing numerous sophisticated NMP and unstable plants in our examples
- A chapter on advanced topics in fundamental limitations
- Discussing alternative facets of the lessons, not available in the literature, by the help of especially designed versatile problems – over 600 examples and worked-out problems along with their simulation source codes
- Presenting the latest results, many of which obtained in the 21st century, wherever appropriate
- Allocating a Subchapter to Further Readings in each chapter, where more advanced topics and references are introduced.

## Readership

Academicians, Researchers, Industrialists, Students (dealing with systems and control theory and applications)

## Table of Contents

PART I: Foundations

1.Introduction

1.1 Introduction

1.2 Why control

1.3 History of control

1.4 Why feedback

1.5 Magic of feedback

1.6 Physical elements of a control system

1.7 Abstract elements of a control system

1.8 Design process

1.9 Types of control systems

1.10 Open-loop control

1.10.1 Stability and performance

1.10.2 Sensitivity and robustness

1.10.3 Disturbance

1.10.4 Reliability, economics, and linearity

1.11 Closed-loop control

1.11.1 Stability and performance

1.11.2 Sensitivity and robustness

1.11.3 Disturbance and noise

1.11.4 Reliability, economics, and linearity

1.12 The 2-DOF control structure

1.13 The internal model control structure

1.14 The Smith predictor

1.15 Modern representation – Generalized model

1.16 Status quo

1.16.1 Overview

1.16.1.1 Summary

1.16.1.2 The forgotten

1.16.2 Relation with other disciplines

1.16.3 Challenges

1.16.4 Outlook

1.17 Summary

1.18 Notes and further readings

1.19 Worked-out problems

1.20 Exercises

References

2. System Representation

2.1 Introduction

2.2 System modeling

2.2.1 State space

2.2.1.1 Linearization

2.2.1.2 Number of inputs and outputs

2.2.2 Frequency domain

2.2.2.1 Finding the output

2.2.3 Zero, pole, and minimality

2.3 Basic examples of modeling

2.3.1 Electrical system as the plant

2.3.2 Mechanical system as the plant

2.3.3 Liquid system as the plant

2.3.4 Thermal system as the plant

2.3.5 Hydraulic system as the plant

2.3.6 Chemical system as the plant

2.3.7 Structural system as the plant

2.3.8 Biological system as the plant

2.3.9 Economics system as the plant

2.3.10 Ecological system as the plant

2.3.11 Societal system as the plant

2.3.12 Physics system as the plant

2.3.13 Delay

2.3.13.1 Exact modelling of delay

2.3.13.2 Approximate modelling of delay

2.3.14 The other constituents

2.3.14.1 Sensors

2.3.14.2 Amplifiers

2.4 Block diagrams

2.5 Signal flow graphs

2.5.1 Basic terminology of graph theory

2.5.2 Equivalence of BD and SFG methods

2.5.3 Computing the transmittance of an SFG

2.6 Summary

2.7 Notes and further readings

2.8 Worked-out problems

2.9 Exercises

Reference

3. Stability Analysis

3.1 Introduction

3.2 Lyapunov and BIBO stability

3.3 Stability tests

3.4 Routh’s test

3.4.1 Special cases

3.5 Hurwitz’ test

3.6 Lienard-Chipart’s test

3.7 Relative stability

3.8 D-stability

3.9 Particular relation with control system design

3.10 The Kharitonov theory

3.11 Internal stability

3.12 Strong stabilization

3.13 Stability of LTV systems

3.14 Summary

3.15 Notes and further readings

3.16 Worked-out problems

3.17 Exercises

References

4. Time Response

4.1 Introduction

4.2 System type and system inputs

4.3 Steady-state error

4.4 First-order systems

4.5 Second-order systems

4.5.1 System representation

4.5.2 Impulse response

4.5.3 Step response

4.5.3.1 Time response characteristics

4.5.4 Ramp and parabola response

4.6 Bandwidth of the system

4.6.1 First-order system

4.6.2 Second-order system

4.6.3 Alternative derivation

4.6.4 Higher-order systems

4.6.5 Open-loop and closed-loop systems

4.7 Higher-order system

4.8 Model reduction

4.9 Effect of addition of a pole and zero

4.10 Performance region

4.11 Inverse response

4.12 Analysis and synthesis of the actual system

4.12.1 Sensor dynamics

4.12.2 Delay dynamics

4.13 Introduction to robust stabilization and performance

4.13.1 Open-loop control

4.13.2 Closed-loop control

4.13.2.1 Disturbance and noise rejection and setpoint tracking

4.14 Summary

4.15 Notes and further readings

4.16 Worked-out problems

4.17 Exercises

References

5. Root Locus

5.1 Introduction

5.2 The root locus

5.3 The root contour

5.4 Finding the value of the gain from the root locus

5.5 Controller design implications

5.5.1 Difficult systems

5.5.1.1 Systems without NMP zeros

5.5.1.2 Systems with NMP zeros

5.5.1.3 Examples of systems without NMP zeros

5.5.1.4 Examples of systems with NMP zeros

5.5.2 Simple systems

5.6 Summary

5.7 Notes and further readings

5.8 Worked-out problems

5.9 Exercises

References

PART II: Frequency Domain Analysis and synthesis

6. Nyquist Plot

6.1 Introduction

6.2 Nyquist plot

6.2.1 Principle of argument

6.2.2 Nyquist stability criterion

6.2.3 Drawing of the Nyquist plot

6.2.4 The high- and low-frequency ends of the plot

6.2.5 Cusp points of the plot

6.2.6 How to handle the proportional gain

6.2.7 The case of j-axis zeros and poles

6.2.8 Relation with root locus

6.3 Gain, phase, and delay margins

6.3.1 The GM concept

6.3.2 The PM and DM concepts

6.3.3 Stability in terms of the GM and PM signs

6.3.4 The high sensitivity region

6.4 Summary

6.5 Notes and further readings

6.6 Worked-out problems

6.7 Exercises

References

7. Bode Diagram

7.1 Introduction

7.2 Bode diagram

7.2.1 Logarithm

7.2.2 Decibel

7.2.3 Log magnitude

7.2.4 The magnitude diagram

7.2.5 Octave and decade

7.2.6 Some useful figures to remember

7.2.7 Relation between the transfer function and its constituting components

7.2.8 How to draw the Bode diagram with hand

7.3 Bode diagram and the steady-state error

7.4 Minimum phase and nonminimum phase systems

7.5 Gain, phase, and delay margins

7.6 Stability in the Bode diagram context

7.7 The high sensitivity region

7.8 Relation with Nyquist and root locus

7.9 Standard second-order system

7.10 Bandwidth

7.11 Summary

7.12 Notes and further readings

7.13 Worked-out problems

7.14 Exercises

References

8. Krohn-Manger-Nichols Chart

8.1 Introduction

8.2 S-Circles

8.3 M-Circles

8.4 N-Circles

8.5 M- and N-Contours

8.6 KMN chart

8.7 System features: GM, PM, DM, BW, stability

8.7.1 Gain, phase and delay margins

8.7.2 Stability

8.7.3 Bandwidth

8.8 The high sensitivity region

8.9 Relation with Bode diagram, Nyquist plot, and root locus

8.10 Summary

8.11 Notes and further readings

8.12 Worked-out problems

8.13 Exercises

References

9. Frequency Domain Synthesis and Design

9.1 Introduction

9.2 Basic controllers: Proportional, Lead, Lag, and Lead-Lag

9.3 Controller simplifications: PI, PD, and PID

9.4 Controller structures in the Nyquist plot context

9.5 Effect of the controllers on the root locus

9.6 Design Procedure

9.7 Specialized tuning rules of PID controllers

9.7.1 Heuristic rules

9.7.2 Analytical rules

9.7.2.1 Pole placement method

9.7.2.2 Direct synthesis

9.7.2.3 Skogestad tuning rules

9.7.3 Optimization-based rules

9.8 Internal model control

9.9 Smith predictor

9.10 Implementation with operational amplifiers

9.10.1 Proportional control—P-term

9.10.2 Integral control—I-term

9.10.3 Proportional-integral—PI-term

9.10.4 Proportional-derivative—PD-term

9.10.5 Nonideal/actual derivative—D-term

9.10.6 Series proportional-integral-derivative—Series PID

9.10.7 Lead

9.10.8 Lag

9.10.9 Lead or lag

9.10.10 Lead-lag

9.11 Summary

9.12 Notes and further readings

9.13 Worked-out problems

9.14 Exercises

References

PART III: Advanced Issues

10. Fundamental Limitations

10.1 Introduction

10.2 Relation between time and frequency domain constraints

10.3 Ideal transfer functions

10.4 Controller design via the TS method

10.5 Interpolation conditions

10.6 Integral and Poisson integral constraints

10.7 Constraints implied by poles and zeros

10.7.1 Implications of integrators

10.7.2 MP and NMP open-loop poles and zeros

10.7.3 Imaginary-axis poles and zeros

10.8 Actuator and sensor limitations

10.8.1 Maximal actuator movement

10.8.2 Minimal actuator movement

10.8.3 Sensor precision

10.8.4 Sensor speed

10.9 Delay

10.10 Eigenstructure assignment by output feedback

10.10.1 Regulation

10.10.2 Tracking

10.11 Non-interactive performance by output feedback

10.12 Minimal closed-loop pole sensitivity by output feedback

10.13 Robust stabilization by output feedback

10.13.1 Structured perturbations

10.13.2 Unstructured perturbations

10.14 Special results on positive systems

10.15 Generic design procedure

10.16 Summary

10.17 Notes and further readings

10.18 Worked-out problems

10.19 Exercises

References

APPENDIX

A. Laplace Transform and Differential Equations

A.1 Introduction

A.2 Basic properties and pairs

A.2.1 Inverse Laplace transform

A.2.2 Table of some Laplace transform pairs

A.3 Derivation and integration in time domain and frequency domain

A.3.1 Fourier transform of the Heaviside function

A.3.2 Differentiation formula in time domain

A.3.3 Integration formula in time domain

A.3.4 Frequency domain formulae

A.3.5 Some consequences

A.4 Existence and uniqueness of solutions to differential equations

References

B. Introduction to Dynamics

B.1 Introduction

B.1.1 Electrical systems

B.1.2 Mechanical systems

B.1.3 Chemical systems

B.2 Equivalent systems

B.3 Worked-out problems

References

C. Introduction to MATLAB

C.1 Introduction

C.2 MATLAB

C.2.1 How to write an M.file

C.2.1.1 Script file

C.2.1.2 Function file

C.2.2 MATLAB functions by category—control system toolbox

C.2.2.1 LTI models

C.2.2.2 Model characteristics

C.2.2.3 Model conversions

C.2.2.4 Model order reduction

C.2.2.5 State-space realizations

C.2.2.6 Model dynamics

C.2.2.7 Model interconnections

C.2.2.8 Time responses

C.2.2.9 Time delays

C.2.2.10 Frequency response

C.2.2.11 Pole placement

C.2.2.12 LQG design

C.2.2.13 Equation solvers

C.2.2.14 Graphical user interfaces for control system analysis and design

C.3 SIMULINK

C.4 Worked-out problems

References

D.Treatise on Stability Concepts & Tests

D.1 Introduction

D.2 A survey on stability concepts and tests

D.2.1 Deterministic systems

D.2.2 Stochastic systems

D.2.3 Miscellaneous

D.3 Lipschitz stability

D.4 Lagrange, Poisson, and Lyapunov stability

D.5 Finite-time and fixed-time stability

D.6 Summary

References

E. Treatise on the Routh’s Stability Test

E.1 Introduction

E.2 Applications of the Routh’s array

E.3 The case of imaginary-axis zeros

References

F. Genetic algorithm: A global optimization technique

F.1 Introduction

F2. Non-convex optimization

F.3 Convex optimization

F.4 Convexification

F.5 Genetic algorithms

References

G. Sample Exams

G.1 Sample midterm exam

G.2 Sample endterm exam

## Details

- No. of pages:
- 1008

- Language:
- English

- Copyright:
- © Academic Press 2017

- Published:
- 20th September 2017

- Imprint:
- Academic Press

- Paperback ISBN:
- 9780128127483

- eBook ISBN:
- 9780128127490

## About the Author

### Yazdan Bavafa-Toosi

Yazdan Bavafa-Toosi received B.Eng. and M.Eng. degrees in electrical power and control engineering from Ferdowsi University, Mashhad, and K.N. Toosi University of Technology, Tehran, Iran, in 1997 and 2000, respectively. He earned his Ph.D. degree in system design engineering (also known as systems and control) from Keio University, Yokohama, Japan, in 2006. His multi-disciplinary research spans systems and control theory and applications. Between and after his educations he has held various research and teaching positions in Germany, Japan, and Iran, and co-authored about 40 technical contributions. He is a reviewer of some journals in the field of systems and control theory and applications.

His wide experience in math and engineering is reflected in this book whose core materials have been taught and class-tested several times in the past 10 years.

### Affiliations and Expertise

Alma mater: Keio University (Japan), Technical University of Berlin (Germany)

## Reviews

"The book has examples and figures in plenty. The examples and the figures (many of them showing the results of numerical simulations) illustrate the several topics under consideration. Due also to the large number of examples, the book is an interesting reference to be part of the bibliography for introductory courses on feedback stabilization of finite-dimensional linear control systems." **--Zentralblatt MATH**