# Introduction to Linear Control Systems

## Description

## 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 21
^{st}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

References2. 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

Reference3. 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

References4. 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

References5. 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

ReferencesPART 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

References7. 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

References8. 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

References9. 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

ReferencesPART 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

ReferencesAPPENDIX

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

ReferencesB. 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

ReferencesC. 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

ReferencesD.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

ReferencesE. 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

ReferencesF. Genetic algorithm: A global optimization technique

F.1 Introduction

F2. Non-convex optimization

F.3 Convex optimization

F.4 Convexification

F.5 Genetic algorithms

ReferencesG. Sample Exams

G.1 Sample midterm exam

G.2 Sample endterm exam

## Product details

- No. of pages: 1134
- Language: English
- Copyright: © Academic Press 2017
- Published: September 19, 2017
- Imprint: Academic Press
- Paperback ISBN: 9780128127483
- eBook ISBN: 9780128127490

## About the Author

### Yazdan Bavafa-Toosi

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

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