# Automatic Control Systems, Tenth Edition

10^{th}Edition

ISBN10: 1259643832

ISBN13: 9781259643835

Copyright: 2017

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CHAPTER 1 Introduction to Control Systems

1-1 Basic Components of a Control System

1-2 Examples of Control-System Applications

1-2-1 Intelligent Transportation Systems

1-2-2 Steering Control of an Automobile

1-2-3 Idle-Speed Control of an Automobile

1-2-4 Sun-Tracking Control of Solar Collectors

1-3 Open-Loop Control Systems (Nonfeedback Systems)

1-4 Closed-Loop Control Systems (Feedback Control Systems)

1-5 What Is Feedback, and What Are Its Effects?

1-5-1 Effect of Feedback on Overall Gain

1-5-2 Effect of Feedback on Stability

1-5-3 Effect of Feedback on External Disturbance or Noise

1-6 Types of Feedback Control Systems

1-7 Linear versus Nonlinear Control Systems

1-8 Time-Invariant versus Time-Varying Systems

1-9 Continuous-Data Control Systems

1-10 Discrete-Data Control Systems

1-11 Case Study: Intelligent Vehicle Obstacle Avoidance—LEGO MINDSTORMS

1-12 Summary

CHAPTER 2 Modeling of Dynamic Systems

2-1 Modeling of Simple Mechanical Systems

2-1-1 Translational Motion

2-1-2 Rotational Motion

2-1-3 Conversion between Translational and Rotational Motions

2-1-4 Gear Trains

2-1-5 Backlash and Dead Zone (Nonlinear Characteristics)

2-2 Introduction to Modeling of Simple Electrical Systems

2-2-1 Modeling of Passive Electrical Elements

2-2-2 Modeling of Electrical Networks

2-3 Introduction to Modeling of Thermal and Fluid Systems

2-3-1 Elementary Heat Transfer Properties

2-3-2 Elementary Fluid System Properties

2-4 Linearization of Nonlinear Systems

2-4-1 Linearization Using Taylor Series: Classical Representation

2-5 Analogies

2-6 Project: Introduction to LEGO MINDSTORMS NXT Motor—Mechanical Modeling

2-7 Summary

References

Problems

CHAPTER 3 Solution of Differential Equations of Dynamic Systems

3-1 Introduction to Differential Equations

3-1-1 Linear Ordinary Differential Equations

3-1-2 Nonlinear Differential Equations

3-2 Laplace Transform

3-2-1 Definition of the Laplace Transform

3-2-2 Important Theorems of the Laplace Transform

3-2-3 Transfer Function

3-2-4 Characteristic Equation

3-2-5 Analytic Function

3-2-6 Poles of a Function

3-2-7 Zeros of a Function

3-2-8 Complex Conjugate Poles and Zeros

3-2-9 Final-Value Theorem

3-3 Inverse Laplace Transform by Partial-Fraction Expansion

3-3-1 Partial Fraction Expansion

3-4 Application of the Laplace Transform to the Solution of Linear Ordinary Differential Equations

3-4-1 First-Order Prototype System

3-4-2 Second-Order Prototype System

3-4-3 Second-Order Prototype System—Final Observations

3-5 Impulse Response and Transfer Functions of Linear Systems

3-5-1 Impulse Response

3-5-2 Time Response Using the Impulse Response

3-5-3 Transfer Function (Single-Input, Single-Output Systems)

3-6 Systems of First-Order Differential Equations: State Equations

3-6-1 Definition of State Variables

3-6-2 The Output Equation

3-7 Solution of the Linear Homogeneous State Equation

3-7-1 Transfer Functions (Multivariable Systems)

3-7-2 Characteristic Equation from State Equations

3-7-3 State Equations from the Transfer Function

3-8 Case Studies with MATLAB

3-9 Linearization Revisited—the State-Space Approach

3-10 Summary

References

Problems

CHAPTER 4 Block Diagrams and Signal-Flow Graphs

4-1 Block Diagrams

4-1-1 Modeling of Typical Elements of Block Diagrams in Control Systems

4-1-2 Relation between Mathematical Equations and Block Diagrams

4-1-3 Block Diagram Reduction

4-1-4 Block Diagrams of Multi-Input Systems: Special Case—Systems with a Disturbance

4-1-5 Block Diagrams and Transfer Functions of Multivariable Systems

4-2 Signal-Flow Graphs

4-2-1 SFG Algebra

4-2-2 Definitions of SFG Terms

4-2-3 Gain Formula for SFG

4-2-4 Application of the Gain Formula between Output Nodes and Noninput Nodes

4-2-5 Simplified Gain Formula

4-3 State Diagram

4-3-1 From Differential Equations to State Diagrams

4-3-2 From State Diagrams to Transfer Functions

4-3-3 From State Diagrams to State and Output Equations

4-4 Case Studies

4-5 MATLAB Tools

4-6 Summary

References

Problems

CHAPTER 5 Stability of Linear Control Systems

5-1 Introduction to Stability

5-2 Methods of Determining Stability

5-3 Routh-Hurwitz Criterion

5-3-1 Routh’s Tabulation

5-3-2 Special Cases When Routh’s Tabulation Terminates Prematurely

5-4 MATLAB Tools and Case Studies

5-5 Summary

References

Problems

CHAPTER 6 Important Components of Feedback Control Systems

6-1 Modeling of Active Electrical Elements: Operational Amplifiers

6-1-1 The Ideal Op-Amp

6-1-2 Sums and Differences

6-1-3 First-Order Op-Amp Configurations

6-2 Sensors and Encoders in Control Systems

6-2-1 Potentiometer

6-2-2 Tachometers

6-2-3 Incremental Encoder

6-3 DC Motors in Control Systems

6-3-1 Basic Operational Principles of DC Motors

6-3-2 Basic Classifications of PM DC Motors

6-3-3 Surface-Wound DC Motors

6-3-4 Moving-Coil DC Motors

6-3-5 Brushless DC Motors

6-3-6 Mathematical Modeling of PM DC Motors

6-3-7 Relation between Ki and Kb

6-4 Speed and Position Control of a DC Motor

6-4-1 Speed Response and the Effects of Inductance and Disturbance: Open-Loop Response

6-4-2 Speed Control of DC Motors: Closed-Loop Response

6-4-3 Position Control

6-5 Case Studies: Practical Examples

6-6 The Control Lab: Introduction to LEGO MINDSTORMS NXT Motor—Modeling and Characterization

6-6-1 NXT Motor

6-6-2 Electrical Characteristics

6-6-3 Mechanical Characteristics

6-6-4 Speed Response and Model Verification

6-7 Summary

References

Problems

CHAPTER 7 Time-Domain Performance of Control Systems

7-1 Time Response of Continuous-Data Systems: Introduction

7-2 Typical Test Signals to Evaluate Time-Response Performance of Control Systems

7-3 The Unit-Step Response and Time-Domain Specifications

7-4 Time Response of a Prototype First-Order System

7-5 Transient Response of a Prototype Second-Order System

7-5-1 Damping Ratio and Natural Frequency

7-5-2 Maximum Overshoot (0 < ζ < 1)

7-5-3 Delay Time and Rise Time (0 < ζ < 1)

7-5-4 Settling Time (5 and 2 Percent)

7-5-5 Transient Response Performance Criteria—Final Remarks

7-6 Steady-State Error

7-6-1 Definition of the Steady-State Error

7-6-2 Steady-State Error in Systems with a Disturbance

7-6-3 Types of Control Systems: Unity-Feedback Systems

7-6-4 Error Constants

7-6-5 Steady-State Error Caused by Nonlinear System Elements

7-7 Basic Control Systems and Effects of Adding Poles and Zeros to Transfer Functions

7-7-1 Addition of a Pole to the Forward-Path Transfer Function: Unity-Feedback Systems

7-7-2 Addition of a Pole to the Closed-Loop Transfer Function

7-7-3 Addition of a Zero to the Closed-Loop Transfer Function

7-7-4 Addition of a Zero to the Forward-Path Transfer Function: Unity-Feedback Systems

7-7-5 Addition of Poles and Zeros: Introduction to Control of Time Response

7-8 Dominant Poles and Zeros of Transfer Functions

7-8-1 Summary of Effects of Poles and Zeros

7-8-2 The Relative Damping Ratio

7-8-3 The Proper Way of Neglecting the Insignificant Poles with Consideration of the Steady-State Response

7-9 Case Study: Time-Domain Analysis of a Position-Control System

7-9-1 Unit-Step Transient Response

7-9-2 The Steady-State Response

7-9-3 Time Response of a Third-Order System—Electrical Time Constant Not Neglected

7-9-4 Unit-Step Transient Response

7-9-5 Steady-State Response

7-10 The Control Lab: Introduction to LEGO MINDSTORMS NXT Motor—Position Control

7-11 Summary

References

Problems

CHAPTER 8 State-Space Analysis and Controller Design

8-1 State-Variable Analysis

8-2 Block Diagrams, Transfer Functions, and State Diagrams

8-2-1 Transfer Functions (Multivariable Systems)

8-2-2 Block Diagrams and Transfer Functions of Multivariable Systems

8-3 Systems of First-Order Differential Equations: State Equations

8-3-1 Definition of State Variables

8-3-2 The Output Equation

8-4 Vector-Matrix Representation of State Equations

8-5 State-Transition Matrix

8-5-1 Significance of the State-Transition Matrix

8-5-2 Properties of the State-Transition Matrix

8-6 State-Transition Equation

8-6-1 State-Transition Equation Determined from the State Diagram

8-7 Relationship between State Equations and High-Order Differential Equations

8-8 Relationship between State Equations and Transfer Functions

8-9 Characteristic Equations, Eigenvalues, and Eigenvectors

8-9-1 Characteristic Equation from a Differential Equation

8-9-2 Characteristic Equation from a Transfer Function

8-9-3 Characteristic Equation from State Equations

8-9-4 Eigenvalues

8-9-5 Eigenvectors

8-9-6 Generalized Eigenvectors

8-10 Similarity Transformation

8-10-1 Invariance Properties of the Similarity Transformations

8-10-2 Characteristic Equations, Eigenvalues, and Eigenvectors

8-10-3 Transfer-Function Matrix

8-10-4 Controllability Canonical Form

8-10-5 Observability Canonical Form

8-10-6 Diagonal Canonical Form

8-10-7 Jordan Canonical Form

8-11 Decompositions of Transfer Functions

8-11-1 Direct Decomposition

8-11-2 Direct Decomposition to CCF

8-11-3 Direct Decomposition to OCF

8-11-4 Cascade Decomposition

8-11-5 Parallel Decomposition

8-12 Controllability of Control Systems

8-12-1 General Concept of Controllability

8-12-2 Definition of State Controllability

8-12-3 Alternate Tests on Controllability

8-13 Observability of Linear Systems

8-13-1 Definition of Observability

8-13-2 Alternate Tests on Observability

8-14 Relationship among Controllability, Observability, and Transfer Functions

8-15 Invariant Theorems on Controllability and Observability

8-16 Case Study: Magnetic-Ball Suspension System

8-16-1 The Characteristic Equation

8-17 State-Feedback Control

8-18 Pole-Placement Design through State Feedback

8-19 State Feedback with Integral Control

8-20 MATLAB Tools and Case Studies

8-20-1 Description and Use of the State-Space Analysis Tool

8-20-2 Description and Use of tfsym for State-Space Applications

8-21 Case Study: Position Control of the LEGO MINDSTORMS Robotic Arm System

8-22 Summary

References

Problems

CHAPTER 9 Root-Locus Analysis

9-1 Basic Properties of the Root Loci

9-2 Properties of the Root Loci

9-2-1 K = 0 and K = ±∞ Points

9-2-2 Number of Branches on the Root Loci

9-2-3 Symmetry of the RL

9-2-4 Angles of Asymptotes of the RL: Behavior of the RL at |s| = ∞

9-2-5 Intersect of the Asymptotes (Centroid)

9-2-6 Root Loci on the Real Axis

9-2-7 Angles of Departure and Angles of Arrival of the RL

9-2-8 Intersection of the RL with the Imaginary Axis

9-2-9 Breakaway Points (Saddle Points) on the RL

9-2-10 Angles of Arrival and Departure of Root Loci at the Breakaway Point

9-2-11 Calculation of K on the Root Loci

9-2-12 Summary: Properties of the Root Loci

9-3 The Root Sensitivity

9-4 Design Aspects of the Root Loci

9-4-1 Effects of Adding Poles and Zeros to G(s)H(s)

9-4-2 Addition of Poles to G(s)H(s)

9-4-3 Addition of Zeros to G(s)H(s)

9-5 Root Contours: Multiple-Parameter Variation

9-6 MATLAB Tools

9-7 Summary

References

Problems

CHAPTER 10 Frequency-Domain Analysis

10-1 Introduction to Frequency Response

10-1-1 Frequency Response of Closed-Loop Systems

10-1-2 Frequency-Domain Specifications

10-2 Mr, ωr, and Bandwidth of the Prototype Second-Order System

10-2-1 Resonant Peak and Resonant Frequency

10-2-2 Bandwidth

10-3 Effects of Adding Poles and Zeros to the Forward-Path Transfer Function

10-3-1 Effects of Adding a Zero to the Forward-Path Transfer Function

10-3-2 Effects of Adding a Pole to the Forward-Path Transfer Function

10-4 Nyquist Stability Criterion: Fundamentals

10-4-1 Stability Problem

10-4-2 Definitions of Encircled and Enclosed

10-4-3 Number of Encirclements and Enclosures

10-4-4 Principles of the Argument

10-4-5 Nyquist Path

10-4-6 Nyquist Criterion and the L(s) or the G(s)H(s) Plot

10-5 Nyquist Criterion for Systems with Minimum-Phase Transfer Functions

10-5-1 Application of the Nyquist Criterion to Minimum-Phase Transfer Functions That Are Not Strictly Proper

10-6 Relation between the Root Loci and the Nyquist Plot

10-7 Illustrative Examples: Nyquist Criterion for Minimum-Phase Transfer Functions

10-8 Effects of Adding Poles and Zeros to L(s) on the Shape of the Nyquist Plot

10-8-1 Addition of Poles at s = 0

10-8-2 Addition of Finite Nonzero Poles

10-8-3 Addition of Zeros

10-9 Relative Stability: Gain Margin and Phase Margin

10-9-1 Gain Margin

10-9-2 Gain Margin of Nonminimum-Phase Systems

10-9-3 Phase Margin

10-10 Stability Analysis with the Bode Plot

10-10-1 Bode Plots of Systems with Pure Time Delays

10-11 Relative Stability Related to the Slope of the Magnitude Curve of the Bode Plot

10-11-1 Conditionally Stable System

10-12 Stability Analysis with the Magnitude-Phase Plot

10-13 Constant-M Loci in the Magnitude-Phase Plane: The Nichols Chart

10-14 Nichols Chart Applied to Nonunity-Feedback Systems

10-15 Sensitivity Studies in the Frequency Domain

10-16 MATLAB Tools and Case Studies

10-17 Summary

References

Problems

CHAPTER 11 Design of Control Systems

11-1 Introduction

11-1-1 Design Specifications

11-1-2 Controller Configurations

11-1-3 Fundamental Principles of Design

11-2 Design with the PD Controller

11-2-1 Time-Domain Interpretation of PD Control

11-2-2 Frequency-Domain Interpretation of PD Control

11-2-3 Summary of Effects of PD Control

11-3 Design with the PI Controller

11-3-1 Time-Domain Interpretation and Design of PI Control

11-3-2 Frequency-Domain Interpretation and Design of PI Control

11-4 Design with the PID Controller

11-5 Design with Phase-Lead and Phase-Lag Controllers

11-5-1 Time-Domain Interpretation and Design of Phase-Lead Control

11-5-2 Frequency-Domain Interpretation and Design of Phase-Lead Control

11-5-3 Effects of Phase-Lead Compensation

11-5-4 Limitations of Single-Stage Phase-Lead Control

11-5-5 Multistage Phase-Lead Controller

11-5-6 Sensitivity Considerations

11-5-7 Time-Domain Interpretation and Design of Phase-Lag Control

11-5-8 Frequency-Domain Interpretation and Design of Phase-Lag Control

11-5-9 Effects and Limitations of Phase-Lag Control

11-5-10 Design with Lead-Lag Controller

11-6 Pole-Zero-Cancellation Design: Notch Filter

11-6-1 Second-Order Active Filter

11-6-2 Frequency-Domain Interpretation and Design

11-7 Forward and Feedforward Controllers

11-8 Design of Robust Control Systems

11-9 Minor-Loop Feedback Control

11-9-1 Rate-Feedback or Tachometer-Feedback Control

11-9-2 Minor-Loop Feedback Control with Active Filter

11-10 MATLAB Tools and Case Studies

11-11 The Control Lab

References

Problems

INDEX

# About the Author

**Farid Golnaraghi**

**Benjamin Kuo**

CHAPTER 1 Introduction to Control Systems

1-1 Basic Components of a Control System

1-2 Examples of Control-System Applications

1-2-1 Intelligent Transportation Systems

1-2-2 Steering Control of an Automobile

1-2-3 Idle-Speed Control of an Automobile

1-2-4 Sun-Tracking Control of Solar Collectors

1-3 Open-Loop Control Systems (Nonfeedback Systems)

1-4 Closed-Loop Control Systems (Feedback Control Systems)

1-5 What Is Feedback, and What Are Its Effects?

1-5-1 Effect of Feedback on Overall Gain

1-5-2 Effect of Feedback on Stability

1-5-3 Effect of Feedback on External Disturbance or Noise

1-6 Types of Feedback Control Systems

1-7 Linear versus Nonlinear Control Systems

1-8 Time-Invariant versus Time-Varying Systems

1-9 Continuous-Data Control Systems

1-10 Discrete-Data Control Systems

1-11 Case Study: Intelligent Vehicle Obstacle Avoidance—LEGO MINDSTORMS

1-12 Summary

CHAPTER 2 Modeling of Dynamic Systems

2-1 Modeling of Simple Mechanical Systems

2-1-1 Translational Motion

2-1-2 Rotational Motion

2-1-3 Conversion between Translational and Rotational Motions

2-1-4 Gear Trains

2-1-5 Backlash and Dead Zone (Nonlinear Characteristics)

2-2 Introduction to Modeling of Simple Electrical Systems

2-2-1 Modeling of Passive Electrical Elements

2-2-2 Modeling of Electrical Networks

2-3 Introduction to Modeling of Thermal and Fluid Systems

2-3-1 Elementary Heat Transfer Properties

2-3-2 Elementary Fluid System Properties

2-4 Linearization of Nonlinear Systems

2-4-1 Linearization Using Taylor Series: Classical Representation

2-5 Analogies

2-6 Project: Introduction to LEGO MINDSTORMS NXT Motor—Mechanical Modeling

2-7 Summary

References

Problems

CHAPTER 3 Solution of Differential Equations of Dynamic Systems

3-1 Introduction to Differential Equations

3-1-1 Linear Ordinary Differential Equations

3-1-2 Nonlinear Differential Equations

3-2 Laplace Transform

3-2-1 Definition of the Laplace Transform

3-2-2 Important Theorems of the Laplace Transform

3-2-3 Transfer Function

3-2-4 Characteristic Equation

3-2-5 Analytic Function

3-2-6 Poles of a Function

3-2-7 Zeros of a Function

3-2-8 Complex Conjugate Poles and Zeros

3-2-9 Final-Value Theorem

3-3 Inverse Laplace Transform by Partial-Fraction Expansion

3-3-1 Partial Fraction Expansion

3-4 Application of the Laplace Transform to the Solution of Linear Ordinary Differential Equations

3-4-1 First-Order Prototype System

3-4-2 Second-Order Prototype System

3-4-3 Second-Order Prototype System—Final Observations

3-5 Impulse Response and Transfer Functions of Linear Systems

3-5-1 Impulse Response

3-5-2 Time Response Using the Impulse Response

3-5-3 Transfer Function (Single-Input, Single-Output Systems)

3-6 Systems of First-Order Differential Equations: State Equations

3-6-1 Definition of State Variables

3-6-2 The Output Equation

3-7 Solution of the Linear Homogeneous State Equation

3-7-1 Transfer Functions (Multivariable Systems)

3-7-2 Characteristic Equation from State Equations

3-7-3 State Equations from the Transfer Function

3-8 Case Studies with MATLAB

3-9 Linearization Revisited—the State-Space Approach

3-10 Summary

References

Problems

CHAPTER 4 Block Diagrams and Signal-Flow Graphs

4-1 Block Diagrams

4-1-1 Modeling of Typical Elements of Block Diagrams in Control Systems

4-1-2 Relation between Mathematical Equations and Block Diagrams

4-1-3 Block Diagram Reduction

4-1-4 Block Diagrams of Multi-Input Systems: Special Case—Systems with a Disturbance

4-1-5 Block Diagrams and Transfer Functions of Multivariable Systems

4-2 Signal-Flow Graphs

4-2-1 SFG Algebra

4-2-2 Definitions of SFG Terms

4-2-3 Gain Formula for SFG

4-2-4 Application of the Gain Formula between Output Nodes and Noninput Nodes

4-2-5 Simplified Gain Formula

4-3 State Diagram

4-3-1 From Differential Equations to State Diagrams

4-3-2 From State Diagrams to Transfer Functions

4-3-3 From State Diagrams to State and Output Equations

4-4 Case Studies

4-5 MATLAB Tools

4-6 Summary

References

Problems

CHAPTER 5 Stability of Linear Control Systems

5-1 Introduction to Stability

5-2 Methods of Determining Stability

5-3 Routh-Hurwitz Criterion

5-3-1 Routh’s Tabulation

5-3-2 Special Cases When Routh’s Tabulation Terminates Prematurely

5-4 MATLAB Tools and Case Studies

5-5 Summary

References

Problems

CHAPTER 6 Important Components of Feedback Control Systems

6-1 Modeling of Active Electrical Elements: Operational Amplifiers

6-1-1 The Ideal Op-Amp

6-1-2 Sums and Differences

6-1-3 First-Order Op-Amp Configurations

6-2 Sensors and Encoders in Control Systems

6-2-1 Potentiometer

6-2-2 Tachometers

6-2-3 Incremental Encoder

6-3 DC Motors in Control Systems

6-3-1 Basic Operational Principles of DC Motors

6-3-2 Basic Classifications of PM DC Motors

6-3-3 Surface-Wound DC Motors

6-3-4 Moving-Coil DC Motors

6-3-5 Brushless DC Motors

6-3-6 Mathematical Modeling of PM DC Motors

6-3-7 Relation between Ki and Kb

6-4 Speed and Position Control of a DC Motor

6-4-1 Speed Response and the Effects of Inductance and Disturbance: Open-Loop Response

6-4-2 Speed Control of DC Motors: Closed-Loop Response

6-4-3 Position Control

6-5 Case Studies: Practical Examples

6-6 The Control Lab: Introduction to LEGO MINDSTORMS NXT Motor—Modeling and Characterization

6-6-1 NXT Motor

6-6-2 Electrical Characteristics

6-6-3 Mechanical Characteristics

6-6-4 Speed Response and Model Verification

6-7 Summary

References

Problems

CHAPTER 7 Time-Domain Performance of Control Systems

7-1 Time Response of Continuous-Data Systems: Introduction

7-2 Typical Test Signals to Evaluate Time-Response Performance of Control Systems

7-3 The Unit-Step Response and Time-Domain Specifications

7-4 Time Response of a Prototype First-Order System

7-5 Transient Response of a Prototype Second-Order System

7-5-1 Damping Ratio and Natural Frequency

7-5-2 Maximum Overshoot (0 < ζ < 1)

7-5-3 Delay Time and Rise Time (0 < ζ < 1)

7-5-4 Settling Time (5 and 2 Percent)

7-5-5 Transient Response Performance Criteria—Final Remarks

7-6 Steady-State Error

7-6-1 Definition of the Steady-State Error

7-6-2 Steady-State Error in Systems with a Disturbance

7-6-3 Types of Control Systems: Unity-Feedback Systems

7-6-4 Error Constants

7-6-5 Steady-State Error Caused by Nonlinear System Elements

7-7 Basic Control Systems and Effects of Adding Poles and Zeros to Transfer Functions

7-7-1 Addition of a Pole to the Forward-Path Transfer Function: Unity-Feedback Systems

7-7-2 Addition of a Pole to the Closed-Loop Transfer Function

7-7-3 Addition of a Zero to the Closed-Loop Transfer Function

7-7-4 Addition of a Zero to the Forward-Path Transfer Function: Unity-Feedback Systems

7-7-5 Addition of Poles and Zeros: Introduction to Control of Time Response

7-8 Dominant Poles and Zeros of Transfer Functions

7-8-1 Summary of Effects of Poles and Zeros

7-8-2 The Relative Damping Ratio

7-8-3 The Proper Way of Neglecting the Insignificant Poles with Consideration of the Steady-State Response

7-9 Case Study: Time-Domain Analysis of a Position-Control System

7-9-1 Unit-Step Transient Response

7-9-2 The Steady-State Response

7-9-3 Time Response of a Third-Order System—Electrical Time Constant Not Neglected

7-9-4 Unit-Step Transient Response

7-9-5 Steady-State Response

7-10 The Control Lab: Introduction to LEGO MINDSTORMS NXT Motor—Position Control

7-11 Summary

References

Problems

CHAPTER 8 State-Space Analysis and Controller Design

8-1 State-Variable Analysis

8-2 Block Diagrams, Transfer Functions, and State Diagrams

8-2-1 Transfer Functions (Multivariable Systems)

8-2-2 Block Diagrams and Transfer Functions of Multivariable Systems

8-3 Systems of First-Order Differential Equations: State Equations

8-3-1 Definition of State Variables

8-3-2 The Output Equation

8-4 Vector-Matrix Representation of State Equations

8-5 State-Transition Matrix

8-5-1 Significance of the State-Transition Matrix

8-5-2 Properties of the State-Transition Matrix

8-6 State-Transition Equation

8-6-1 State-Transition Equation Determined from the State Diagram

8-7 Relationship between State Equations and High-Order Differential Equations

8-8 Relationship between State Equations and Transfer Functions

8-9 Characteristic Equations, Eigenvalues, and Eigenvectors

8-9-1 Characteristic Equation from a Differential Equation

8-9-2 Characteristic Equation from a Transfer Function

8-9-3 Characteristic Equation from State Equations

8-9-4 Eigenvalues

8-9-5 Eigenvectors

8-9-6 Generalized Eigenvectors

8-10 Similarity Transformation

8-10-1 Invariance Properties of the Similarity Transformations

8-10-2 Characteristic Equations, Eigenvalues, and Eigenvectors

8-10-3 Transfer-Function Matrix

8-10-4 Controllability Canonical Form

8-10-5 Observability Canonical Form

8-10-6 Diagonal Canonical Form

8-10-7 Jordan Canonical Form

8-11 Decompositions of Transfer Functions

8-11-1 Direct Decomposition

8-11-2 Direct Decomposition to CCF

8-11-3 Direct Decomposition to OCF

8-11-4 Cascade Decomposition

8-11-5 Parallel Decomposition

8-12 Controllability of Control Systems

8-12-1 General Concept of Controllability

8-12-2 Definition of State Controllability

8-12-3 Alternate Tests on Controllability

8-13 Observability of Linear Systems

8-13-1 Definition of Observability

8-13-2 Alternate Tests on Observability

8-14 Relationship among Controllability, Observability, and Transfer Functions

8-15 Invariant Theorems on Controllability and Observability

8-16 Case Study: Magnetic-Ball Suspension System

8-16-1 The Characteristic Equation

8-17 State-Feedback Control

8-18 Pole-Placement Design through State Feedback

8-19 State Feedback with Integral Control

8-20 MATLAB Tools and Case Studies

8-20-1 Description and Use of the State-Space Analysis Tool

8-20-2 Description and Use of tfsym for State-Space Applications

8-21 Case Study: Position Control of the LEGO MINDSTORMS Robotic Arm System

8-22 Summary

References

Problems

CHAPTER 9 Root-Locus Analysis

9-1 Basic Properties of the Root Loci

9-2 Properties of the Root Loci

9-2-1 K = 0 and K = ±∞ Points

9-2-2 Number of Branches on the Root Loci

9-2-3 Symmetry of the RL

9-2-4 Angles of Asymptotes of the RL: Behavior of the RL at |s| = ∞

9-2-5 Intersect of the Asymptotes (Centroid)

9-2-6 Root Loci on the Real Axis

9-2-7 Angles of Departure and Angles of Arrival of the RL

9-2-8 Intersection of the RL with the Imaginary Axis

9-2-9 Breakaway Points (Saddle Points) on the RL

9-2-10 Angles of Arrival and Departure of Root Loci at the Breakaway Point

9-2-11 Calculation of K on the Root Loci

9-2-12 Summary: Properties of the Root Loci

9-3 The Root Sensitivity

9-4 Design Aspects of the Root Loci

9-4-1 Effects of Adding Poles and Zeros to G(s)H(s)

9-4-2 Addition of Poles to G(s)H(s)

9-4-3 Addition of Zeros to G(s)H(s)

9-5 Root Contours: Multiple-Parameter Variation

9-6 MATLAB Tools

9-7 Summary

References

Problems

CHAPTER 10 Frequency-Domain Analysis

10-1 Introduction to Frequency Response

10-1-1 Frequency Response of Closed-Loop Systems

10-1-2 Frequency-Domain Specifications

10-2 Mr, ωr, and Bandwidth of the Prototype Second-Order System

10-2-1 Resonant Peak and Resonant Frequency

10-2-2 Bandwidth

10-3 Effects of Adding Poles and Zeros to the Forward-Path Transfer Function

10-3-1 Effects of Adding a Zero to the Forward-Path Transfer Function

10-3-2 Effects of Adding a Pole to the Forward-Path Transfer Function

10-4 Nyquist Stability Criterion: Fundamentals

10-4-1 Stability Problem

10-4-2 Definitions of Encircled and Enclosed

10-4-3 Number of Encirclements and Enclosures

10-4-4 Principles of the Argument

10-4-5 Nyquist Path

10-4-6 Nyquist Criterion and the L(s) or the G(s)H(s) Plot

10-5 Nyquist Criterion for Systems with Minimum-Phase Transfer Functions

10-5-1 Application of the Nyquist Criterion to Minimum-Phase Transfer Functions That Are Not Strictly Proper

10-6 Relation between the Root Loci and the Nyquist Plot

10-7 Illustrative Examples: Nyquist Criterion for Minimum-Phase Transfer Functions

10-8 Effects of Adding Poles and Zeros to L(s) on the Shape of the Nyquist Plot

10-8-1 Addition of Poles at s = 0

10-8-2 Addition of Finite Nonzero Poles

10-8-3 Addition of Zeros

10-9 Relative Stability: Gain Margin and Phase Margin

10-9-1 Gain Margin

10-9-2 Gain Margin of Nonminimum-Phase Systems

10-9-3 Phase Margin

10-10 Stability Analysis with the Bode Plot

10-10-1 Bode Plots of Systems with Pure Time Delays

10-11 Relative Stability Related to the Slope of the Magnitude Curve of the Bode Plot

10-11-1 Conditionally Stable System

10-12 Stability Analysis with the Magnitude-Phase Plot

10-13 Constant-M Loci in the Magnitude-Phase Plane: The Nichols Chart

10-14 Nichols Chart Applied to Nonunity-Feedback Systems

10-15 Sensitivity Studies in the Frequency Domain

10-16 MATLAB Tools and Case Studies

10-17 Summary

References

Problems

CHAPTER 11 Design of Control Systems

11-1 Introduction

11-1-1 Design Specifications

11-1-2 Controller Configurations

11-1-3 Fundamental Principles of Design

11-2 Design with the PD Controller

11-2-1 Time-Domain Interpretation of PD Control

11-2-2 Frequency-Domain Interpretation of PD Control

11-2-3 Summary of Effects of PD Control

11-3 Design with the PI Controller

11-3-1 Time-Domain Interpretation and Design of PI Control

11-3-2 Frequency-Domain Interpretation and Design of PI Control

11-4 Design with the PID Controller

11-5 Design with Phase-Lead and Phase-Lag Controllers

11-5-1 Time-Domain Interpretation and Design of Phase-Lead Control

11-5-2 Frequency-Domain Interpretation and Design of Phase-Lead Control

11-5-3 Effects of Phase-Lead Compensation

11-5-4 Limitations of Single-Stage Phase-Lead Control

11-5-5 Multistage Phase-Lead Controller

11-5-6 Sensitivity Considerations

11-5-7 Time-Domain Interpretation and Design of Phase-Lag Control

11-5-8 Frequency-Domain Interpretation and Design of Phase-Lag Control

11-5-9 Effects and Limitations of Phase-Lag Control

11-5-10 Design with Lead-Lag Controller

11-6 Pole-Zero-Cancellation Design: Notch Filter

11-6-1 Second-Order Active Filter

11-6-2 Frequency-Domain Interpretation and Design

11-7 Forward and Feedforward Controllers

11-8 Design of Robust Control Systems

11-9 Minor-Loop Feedback Control

11-9-1 Rate-Feedback or Tachometer-Feedback Control

11-9-2 Minor-Loop Feedback Control with Active Filter

11-10 MATLAB Tools and Case Studies

11-11 The Control Lab

References

Problems

INDEX

# About the Author

**Farid Golnaraghi**

**Benjamin Kuo**

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