# Linear Control System Analysis and Design with MATLAB®

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## Book Description

Thoroughly classroom-tested and proven to be a valuable self-study companion, **Linear Control System Analysis and Design: Sixth Edition** provides an intensive overview of modern control theory and conventional control system design using in-depth explanations, diagrams, calculations, and tables.

Keeping mathematics to a minimum, the book is designed with the undergraduate in mind, first building a foundation, then bridging the gap between control theory and its real-world application. Computer-aided design accuracy checks (CADAC) are used throughout the text to enhance computer literacy. Each CADAC uses fundamental concepts to ensure the viability of a computer solution.

Completely updated and packed with student-friendly features, the sixth edition presents a range of updated examples using MATLAB^{®}, as well as an appendix listing MATLAB functions for optimizing control system analysis and design. Over 75 percent of the problems presented in the previous edition have been revised or replaced.

## Table of Contents

**Part I: Introductory MaterialIntroduction**Introduction

Introduction to Control Systems

Definitions

Historical Background

Control System: A Human Being

Digital Control Development

Mathematical Background

Engineering Control Problem

Computer Literacy

Outline of Text

**Unmanned Aircraft Vehicles**

**Introduction**

Twentieth-Century UAV R&D

Predator

Grim Reaper (US Air Force Fact Sheet MQ-9 Reaper, Posted on January 5, 2012)

RQ-4 Global Hawk (US Air Force Fact Sheet RQ-4 Global Hawk, Posted on January 19, 2012)

**Wind Energy Control Systems**

**Introduction**

Concurrent Engineering: A Road Map for Systems Design: Energy Example

QFT Controller Design CAD Toolbox

**Frequency Domain Analysis**

**Introduction**

Steel Mill Ingot

Electrocardiographic Monitoring

Control Theory: Analysis and Design of Control Systems

**Part II: Analog Control Systems**

**Writing System Equations**

**Introduction**

Electric Circuits and Components

State Concepts

Transfer Function and Block Diagram

Mechanical Translation Systems

Analogous Circuits

Mechanical Rotational Systems

Effective Moment of Inertia and Damping of a Gear Train

Thermal Systems

Hydraulic Linear Actuator

Liquid-Level System

Rotating Power Amplifiers

DC Servomotor

AC Servomotor

Lagrange’s Equation

**Solution of Differential Equations**

**Introduction**

Standard Inputs to Control Systems

Steady-State Response: Sinusoidal Input

Steady-State Response: Polynomial Input

Transient Response: Classical Method

Definition of Time Constant

Example: Second-Order System (Mechanical)

Example: Second-Order System (Electrical)

Second-Order Transients

Time-Response Specifications

CAD Accuracy Checks

State-Variable Equations

Characteristic Values

Evaluating the State Transition Matrix

Complete Solution of the State Equation

**Laplace Transform**

**Introduction**

Definition of the Laplace Transform

Derivation of Laplace Transforms of Simple Functions

Laplace Transform Theorems

CAD Accuracy Checks

Application of the Laplace Transform to Differential Equations

Inverse Transformation

Heaviside Partial-Fraction Expansion Theorems

MATLAB

^{®}Partial-Fraction Example

Partial-Fraction Shortcuts

Graphical Interpretation of Partial-Fraction Coefficients

Frequency Response from the Pole–Zero Diagram

Location of Poles and Stability

Laplace Transform of the Impulse Function

Second-Order System with Impulse Excitation

Solution of State Equation

Evaluation of the Transfer-Function Matrix

MATLAB

^{® }Script For MIMO Systems

**System Representation**

**Introduction**

Block Diagrams

Determination of the Overall Transfer Function

Standard Block-Diagram Terminology

Position-Control System

Simulation Diagrams

Signal Flow Graphs

State Transition Signal Flow Graph

Parallel State Diagrams from Transfer Functions

Diagonalizing the

**A**Matrix

Use of State Transformation for the State-Equation Solution

Transforming

**A**Matrix with Complex Eigenvalues

Transforming an

**A**Matrix into Companion Form

Using MATLAB

^{®}to Obtain the Companion

**A**Matrix

**Control-System Characteristics**

**Introduction**

Routh’s Stability Criterion

Mathematical and Physical Forms

Feedback System Types

Analysis of System Types

Example: Type 2 System

Steady-State Error Coefficients

CAD Accuracy Checks: CADAC

Use of Steady-State Error Coefficients

Nonunity-Feedback System

**Root Locus**

**Introduction**

Plotting Roots of a Characteristic Equation

Qualitative Analysis of the Root Locus

Procedure Outline

Open-Loop Transfer Function

Poles of the Control Ratio

*C*(

**s**)/

*R*(

**s**)

**Application of the Magnitude and Angle Conditions**

Geometrical Properties (Construction Rules)

CAD Accuracy Checks

Root Locus Example

Example of Section 10.10: MATLAB

^{®}Root Locus

Root Locus Example with an RH Plane Zero

Performance Characteristics

Transport Lag

Synthesis

Summary of Root-Locus Construction Rules for Negative Feedback

**Frequency Response**

**Introduction**

Correlation of the Sinusoidal and Time Response

Frequency-Response Curves

Bode Plots (Logarithmic Plots)

General Frequency–Transfer–Function Relationships

Drawing the Bode Plots

Example of Drawing a Bode Plot

Generation of MATLAB

^{®}Bode Plots

System Type and Gain as Related to Log Magnitude Curves

CAD Accuracy Check

Experimental Determination of Transfer Function

Direct Polar Plots

Summary: Direct Polar Plots

Nyquist Stability Criterion

Examples of the Nyquist Criterion Using Direct Polar Plots

Nyquist Stability Criterion Applied to a System Having Dead Time

Definitions of Phase Margin and Gain Margin and Their Relation to Stability

Stability Characteristics of the Log Magnitude and Phase Diagram

Stability from the Nichols Plot (Log Magnitude–Angle Diagram)

**Closed-Loop Tracking Performance Based on Frequency Response**

**Introduction**

Direct Polar Plot

Determination of

*Mm*and ω

*m*for a Simple Second-Order System

Correlation of Sinusoidal and Time Responses

Constant

*M*(ω) and α(ω) Contours of

**(**

*C**J*ω)/

**(**

*R**J*ω) on the Complex Plane (Direct Plot) Constant 1/

*M*and α Contours (Unity Feedback) in the Inverse Polar Plane

Gain Adjustment of a Unity-Feedback System for a Desired

*Mm*: Direct Polar Plot

Constant

*M*and α Curves on the Log Magnitude–Angle Diagram (Nichols Chart) Generation of MATLAB

^{®}Bode and Nyquist Plots

Adjustment of Gain by Use of the Log Magnitude–Angle Diagram (Nichols Chart)

Correlation of the Pole–Zero Diagram with Frequency and Time Responses

**Part III: Compensation: Analog Systems**

**Root-Locus Compensation: Design**

**Introduction to Design**

Transient Response: Dominant Complex Poles

Additional Significant Poles

Root-Locus Design Considerations

Reshaping the Root Locus

CAD Accuracy Checks

Ideal Integral Cascade Compensation (PI Controller)

Cascade Lag Compensation Design Using Passive Elements System

Ideal Derivative Cascade Compensation (PD Controller)

Lead Compensation Design Using Passive Elements

General Lead-Compensator Design

Lag–Lead Cascade Compensation Design System

Comparison of Cascade Compensators

PID Controller

Introduction to Feedback Compensation

Feedback Compensation: Design Procedures

Simplified Rate Feedback Compensation: A Design Approach

Design of Rate Feedback

Design: Feedback of Second Derivative of Output

Results of Feedback-Compensation Design

Rate Feedback: Plants with Dominant Complex Poles

**Frequency-Response Compensation Design**

**Introduction to Feedback Compensation Design**

Selection of a Cascade Compensator

Cascade Lag Compensator

Design Example: Cascade Lag Compensation

Cascade Lead Compensator

Design Example: Cascade Lead Compensation

Cascade Lag–Lead Compensator

Design Example: Cascade Lag–Lead Compensation

Feedback Compensation Design Using Log Plots

Design Example: Feedback Compensation (Log Plots)

Application Guidelines: Basic Minor-Loop Feedback Compensators

**Part IV: Advanced Topics**

**Control-Ratio Modeling**

**Introduction**

Modeling a Desired Tracking Control Ratio

Guillemin – Truxal Design Procedure

Introduction to Disturbance Rejection

Second-Order Disturbance-Rejection Model

Disturbance-Rejection Design Principles for SISO Systems

Disturbance-Rejection Design Example

Disturbance-Rejection Models

**Design: Closed-Loop Pole–Zero Assignment (State-Variable Feedback)**

**Introduction**

Controllability and Observability

State Feedback for SISO Systems

State-Feedback Design for SISO Systems Using the Control Canonical (Phase-Variable) Form

State-Variable Feedback (Physical Variables)

General Properties of State Feedback (Using Phase Variables)

State-Variable Feedback: Steady-State Error Analysis

Use of Steady-State Error Coefficients

State-Variable Feedback: All-Pole Plant

Plants with Complex Poles

Compensator Containing a Zero

State-Variable Feedback: Pole–Zero Plant

Observers

Control Systems Containing Observers

**Parameter Sensitivity and State-Space Trajectories**

**Introduction**

Sensitivity

Sensitivity Analysis

Sensitivity Analysis Examples

Parameter Sensitivity Examples

Inaccessible States

State-Space Trajectories

Linearization (Jacobian Matrix)

**Part V: Digital Control Systems**

**Sampled-Data Control Systems**

**Introduction**

Sampling

Ideal Sampling

Z Transform Theorems

Differentiation Process

Synthesis in the

*z*Domain (Direct Method)

Inverse Z Transform

Zero-Order Hold

Limitations

Steady-State Error Analysis for Stable Systems

Root-Locus Analysis for Sampled-Data Control Systems

**Digital Control Systems**

**Introduction**

Complementary Spectra

Tustin Transformation:

*s*- to

*z*-Plane Transformation

*z*-Domain to the

*w*- and

*w*’-Domain Transformations

Digitization Technique

Digitization Design Technique

Pseudo-Continuous-Time Control System

Design of Digital Control System

Direct Compensator

PCT Lead Cascade Compensation

PCT Lag Compensation

PCT Lag–Lead Compensation

Feedback Compensation: Tracking

Controlling Unwanted Disturbances

Extensive Digital Feedback Compensator Example

Controller Implementation

**Appendix A: Table of Laplace Transform Pairs**

**Appendix B: Matrix Linear Algebra**

**Appendix C: Introduction to MATLAB® and Simulink®**

**Appendix D: Conversion of Units**

**Problems**

**Answers to Selected Problems**

**Index**