## Contents

1.2 Classification of Control System

1.2.1 Open-Loop Control System

1.2.2 Closed-Loop Control System

1.3 Comparison of Open-Loop and Closed-Loop Control Systems

1.4 Differential Equations and Transfer Functions

1.4.1 Transfer Function Representation

1.4.2 Features and Advantages of Transfer Function Representation

1.4.3 Disadvantages of Transfer Function Representation

1.4.4 Transfer Function of an Open-Loop System

1.4.5 Transfer Function of a Closed-Loop System

1.4.6 Comparison of Positive Feedback and Negative Feedback Systems

1.5.1 Mathematical Equations for Problem Solving

1.6 Modeling of Electrical Systems

1.7 Modeling of Mechanical Systems

1.7.1 Translational Mechanical System

1.7.2 A Simple Translational Mechanical System

1.7.3 Rotational Mechanical System

1.7.4 A Simple Rotational Mechanical System

1.8 Introduction to Analogous System

1.8.1 Advantages of Electrical Analogous System

2 Physical Systems and Components

2.3.1 Advantages of Hydraulic System

2.3.2 Disadvantages of Hydraulic System

2.3.3 Applications of Hydraulic System

2.3.4 Devices Used in Hydraulic System

2.4.1 Gas Flow Resistance and Pneumatic Capacitance

2.4.2 Advantages of Pneumatic System

2.4.3 Disadvantages of Pneumatic System

2.4.4 Applications of Pneumatic System

2.4.5 Devices Used in Pneumatic System

2.4.6 Comparison between Hydraulic and Pneumatic Systems

2.5.1 Thermal Resistance and Thermal Capacitance

2.6.1 Elements of Liquid-Level System

2.7 Introduction to Control System Components

2.8.1 Controller Output as a Percentage Value

2.8.2 Measured Value as a Percentage Value

2.8.3 Set Point as a Percentage Value

2.8.4 Error as a Percentage Value

2.9.5 Proportional Integral Controller

2.9.6 Proportional Derivative Controller

2.9.7 Proportional Integral Derivative Controller

2.10.1 Characteristics of Potentiometers

2.10.2 Power-Handling Capacity

2.10.3 Applications of Potentiometer

2.11.2 Synchro Control Transformer

2.12.1 Classification of Servomotor

2.12.5 Comparison between AC Servomotor and DC Servomotor

2.14.1 Permanent Magnet Stepper Motor

2.14.2 Variable Reluctance Stepper Motor

2.14.4 Operation of Stepper Motor

2.14.5 Advantages of Stepper Motor

2.14.6 Applications of Stepper Motor

3 Block Diagram Reduction Techniques

3.1 Introduction to Block Diagram

3.2 Open-Loop and Closed-Loop Systems Using Block Diagram

3.2.1 Advantages of Block Diagram Representation

3.2.2 Disadvantages of Block Diagram Representation

3.3 Block Diagram Representation of Electrical System

3.4.1 Need for Block Diagram Reduction

3.4.3 Rules for Block Diagram Reduction

3.4.4 Block Diagram Reduction for Complex Systems

4.1.1 Signal Flow Graph Terminologies

4.1.4 Mason’s Gain Formula for SFG

4.1.5 Signal Flow Graph From Differential Equation

4.1.6 Comparison between SFG and Block Diagram

5.2 Time Response of the Control System

5.4 Poles, Zeros and System Response

5.4.1 Poles and Zeros of a Transfer Function

5.5 Type and Order of the System

5.6.1 Performance Parameters of First-Order System

5.6.2 Time Response of a First-Order System

5.7.1 Classification of Second-Order System

5.7.2 Performance Parameters of Second-Order System

5.7.3 Time Response of the Second-Order System

5.7.4 Time-Domain Specifications for an Underdamped Second-Order System

5.8.1 Characteristic of Steady-State Error

5.8.2 Determination of Steady-State Error

5.8.3 Steady-State Error in Terms of G(s)

5.8.4 Steady-State Error in Terms of T(s)

5.8.5 Static Error Constants and System Type

5.8.6 Generalized or Dynamic Error Coefficients

5.9 Effect of Adding Poles and Zeros in the Second-Order System

5.10 Response with P, PI and PID Controllers

5.10.1 Proportional Derivative Control

5.10.2 Proportional Integral Control

5.10.3 Proportional Plus Integral Plus Derivative Control (PID Control)

6 Stability and Routh–Hurwitz Criterion

6.3 Stability of Linear Time-Invariant System

6.3.1 Stability Based on Natural Response of the System, c(t)natural

6.3.2 Stability Based on the Total Response of the System, c(t)6.3

6.4 Mathematical Condition for the Stability of the System

6.5 Transfer Function of the System, G(s)

6.5.1 Effects of Location of Poles on Stability

6.6 Zero-Input Stability or Asymptotic Stability

6.6.1 Importance of Asymptotic Stability

6.8 Methods for Determining the Stability of the System

6.9.2 Non-Minimum-Phase System

6.10.1 Hurwitz Matrix Formation

6.10.2 Disadvantages of Hurwitz Method

6.11 Routh’s Stability Criterion

6.11.1 Necessary Condition for the Stability of the System

6.11.2 Special Cases of Routh’s Criterion

6.11.3 Applications of Routh’s Criterion

6.11.4 Advantages of Routh’s Criterion

6.11.5 Limitations of Routh’s Criterion

7.2 Advantages of Root Locus Technique

7.3.1 Variation of Loop Gain with the Root Locus

7.4 Basic Properties of Root Loci

7.4.1 Conditions Required for Constructing the Root Loci

7.4.3 Analytical Expression of the Conditions

7.4.4 Determination of Variable Parameter K

7.4.5 Minimum and Non-Minimum Phase Systems

7.5 Manual Construction of Root Loci

7.5.1 Properties / Guidelines for Constructing the Root Loci

7.5.2 Flow Chart for Constructing the Root Locus for a System

7.6 Root Loci for different Pole-Zero Configurations

7.7 Effect of Adding Poles and Zeros in the System

7.7.1 Addition of Poles to the Loop Transfer Function, G(s)H(s)

7.7.2 Effect of Addition of Poles

7.7.3 Addition of Zero to the Loop Transfer Function

7.7.4 Effect of Addition of Zeros

7.8 Time Response from Root Locus

7.9 Gain Margin and Phase Margin of the System

7.9.1 Gain Margin of the System

7.9.2 Phase Margin of the System

7.10 Root Locus for K < 0 Inverse Root Locus or Complementary Root Loci

7.10.1 Steps in Constructing the Inverse Root Loci Manually

7.11 Pole-Zero Cancellation Rules

7.12 Root Contours (Multi-Variable System)

8.1.1 Advantages of Frequency Response Analysis

8.1.2 Disadvantages of Frequency Response Analysis

8.2 Importance of Sinusoidal Waves for Frequency Response Analysis

8.3 Basics of Frequency Response Analysis

8.4 Frequency Response Analysis of Open-Loop and Closed-Loop Systems

8.4.3 Closed-Loop System with Poles and Zeros

8.5 Frequency Response Representation

8.5.1 Determination of Frequency Response

8.6 Frequency Domain Specifications

8.7 Frequency and Time Domain Interrelations

8.7.1 Frequency Domain Specifications

8.8 Effect of Addition of a Pole to the Open-Loop Transfer Function of the System

8.9 Effect of Addition of a Zero to the Open-Loop Transfer Function of the System

8.10 Graphical Representation of Frequency Response

8.11 Introduction to Bode Plot

8.11.1 Reasons for Using Logarithmic Scale

8.11.2 Advantages of Bode Plot

8.11.3 Disadvantages of Bode Plot

8.12 Determination of Frequency Domain Specifications from Bode Plot

8.13.1 Based on Crossover Frequencies

8.13.2 Based on Gain Margin and Phase Margin

8.14 Construction of Bode Plot

8.14.1 Effect of Damping Ratio x

8.15 Constructing the Bode Plot for a Given System

8.15.1 Construction of Magnitude Plot

8.15.2 Construction of Phase Plot

8.16 Flow Chart for Plotting Bode Plot

8.17 Procedure for Determining the Gain K from the Desired Frequency Domain Specifications

8.19 Procedure for Determining Transfer Function from Bode Plot

8.20 Bode Plot for Minimum and Non-Minimum Phase Systems

9.1 Introduction to Polar Plot

9.2 Starting and Ending of Polar Plot

9.3 Construction of Polar Plot

9.4 Determination of Frequency Domain Specification from Polar Plot

9.4.1 Gain Crossover Frequency w_{gc}

9.4.2 Phase Crossover Frequency w _{pc}

9.5 Procedure for Constructing Polar Plot

9.6 Typical Sketches of Polar Plot on an Ordinary Graph and Polar Graph

9.7 Stability Analysis using Polar Plot

9.7.1 Based on Crossover Frequencies

9.7.2 Based on Gain Margin and Phase Margin

9.7.3 Based on the Location of Phase Crossover Point

9.8 Determining the Gain K from the Desired Frequency Domain Specifications

9.8.1 When the Desired Gain Margin of the System is Specified

9.8.2 When the Desired Phase Margin of the System is Specified

9.9 Introduction to Nyquist Stability Criterion

9.10 Advantages of Nyquist Plot

9.11 Basic Requirements for Nyquist Stability Criterion

9.13 Number of Encirclements or Enclosures

9.14 Mapping of s-Plane into Characteristic Equation Plane

9.16 Nyquist Stability Criterion

9.18 Relation Between G(s) H(s)-Plane and F(s)-Plane

9.19 Nyquist Stability Criterion Based on the Encirclements of −1+ j 0

9.20 Stability Analysis of the System

9.21 Procedure for Determining the Number of Encirclements

9.21.1 Flow chart for Determining the Number of Encirclements Made by the Contour in G(s)H(s)-Plane

9.22.1 Flow chart for Determining the Stability of the System Based on Nyquist Stability Criterion

10 Constant M- and N-Circles and Nichols Chart

10.2 Closed-Loop Response from Open-Loop Response

10.3.1 Applications of Constant *M*-Circles

10.3.2 Resonant Peak *M*r and Resonant Frequency wr from Constant *M*-Circles

10.3.3 Variation of Gain *K* with *M*r and wr 10.6

10.3.4 Bandwidth of the System

10.3.5 Stability of the System

10.3.6 Determination of Gain *K* Corresponding to the Desired Resonant Peak (*M*r)desired

10.3.7 Magnitude Plot of the System from Constant *M*-Circles

10.4.1 Phase Plot of the System from Constant *N*-Circles

10.5.1 Reason for the Usage of Nichols Chart

10.5.2 Advantages of Nichols Chart

10.5.3 Transformation of Constant *M*- and *N*-Circles into Nichols Chart

10.5.4 Determination of Frequency Domain Specifications from Nichols Chart

10.5.5 Determination of Gain *K* for a Desired Frequency Domain Specifications

11.2.1 Series or Cascade Compensation

11.2.2 Feedback or Parallel Compensation

11.2.3 Load or Series-Parallel Compensation

11.2.4 State Feedback Compensation

11.2.5 Forward Compensation with Series Compensation

11.2.6 Feed-forward Compensation

11.2.7 Effects of Addition of Poles

11.2.8 Effects of Addition of Zeros

11.3.1 Determination of Maximum Phase Angle fm

11.3.2 Electrical Representation of the Lag Compensator

11.3.3 Effects of Lag Compensator

11.3.4 Design of Lag Compensator

11.3.5 Design of Lag Compensator Using Bode Plot

11.3.6 Design of Lag Compensator Using Root Locus Technique

11.4.1 Determination of Maximum Phase Angle fm

11.4.2 Electrical Representation of the Lead Compensator

11.4.3 Effects of Lead Compensator

11.4.4 Limitations of Lead Compensator

11.4.5 Design of Lead Compensator

11.4.6 Design of Lead Compensator Using Bode Plot

11.4.7 Design of Lead Compensator Using Root Locus Technique

11.5.1 Electrical Representation of the Lag–Lead Compensator

11.5.2 Effects of Lag–Lead Compensator

11.5.3 Design of Lag–Lead Compensator

11.5.4 Design of Lag–Lead Compensator Using Bode Plot

11.5.5 Design of Lag–Lead Compensator Using Root Locus Technique

12 Physiological Control Systems

12.2 Physiological Control Systems

12.3 Properties of Physiological Control Systems

12.3.1 Target of the Homeostasis

12.3.2 Imbalance in the Homeostasis

12.3.3 Homeostasis Control Mechanisms

12.4 Block Diagram of the Physiological Control System

12.4.1 Types of Control Mechanism

12.5 Differences Between Engineering and Physiological Control Systems

12.7 Properties Related to Elements

12.8 Linear Models of Physiological Systems: Two Examples

12.9 Simulation—Matlab and Simulink Examples

13.1.1 Advantages of State-Variable Analysis

13.2 State-Space Representation of Continuous-Time LTI Systems

13.3 Block Diagram and SFG Representation of a Continuous State-Space Model

13.4 State-Space Representation

13.5 State-Space Representation of Differential Equations in Physical Variable Form

13.5.1 Advantages of Physical Variable Representation

13.5.2 Disadvantages of Physical Variable Representation

13.6 State-Space Model Representation for Electric Circuits

13.7 State-Space Model Representation for Mechanical System

13.7.1 State-Space Model Representation of Translational / Rotational Mechanical System

13.8 State-Space Model Representation of Electromechanical System

13.8.1 Armature-Controlled DC Motor

13.8.2 Field-Controlled DC Motor

13.9 State-Space Representation of a System Governed by Differential Equations

13.10 State-Space Representation of Transfer Function in Phase Variable Forms

13.10.4 Advantages of Phase-Variable Representation

13.10.5 Disadvantages of the Phase-Variable Representation

13.11 State-Space Representation of Transfer Function in Canonical Forms

13.11.1 Controllable Canonical Form

13.11.2 Observable Canonical Form

13.11.3 Diagonal Canonical Form

13.12 Transfer Function from State-Space Model

13.13 Solution of State Equation for Continuous Time Systems

13.13.1 Solution of Homogenous-Type State Equation

13.13.2 Solution of Non-Homogenous Type State Equation

13.13.3 State Transition Matrix

13.13.4 Properties of State Transition Matrix

13.14 Controllability and Observability

13.14.1 Criteria for Controllability

13.14.2 Criteria for Observability

13.15 State-Space Representation of Discrete-Time LTI Systems

13.15.1 Block Diagram and SFG of Discrete State-Space Model

13.16 Solutions of State Equations for Discrete-Time LTI Systems

13.17 Representation of Discrete LTI System

13.18.2 High Speed Sample-and-Hold Circuit

14.2 MATLAB in Control Systems

14.2.2 Inverse Laplace Transform

14.2.3 Partial Fraction Expansion

14.2.4 Transfer Function Representation

14.2.5 Zeros and Poles of a Transfer Function

14.2.6 Pole-Zero Map of a Transfer Function

14.2.7 State-Space Representation of a Dynamic System

14.2.8 Phase Variable Canonical Form

14.2.9 Transfer Function to State-Space Conversion

14.2.10 State-Space to Transfer Function Conversion

14.2.11 Series/Cascade, Parallel and Feedback Connections

14.2.12 Time Response of Control System

14.2.13 Performance Indices from the Response of a System

14.2.14 Steady State Error from the Transfer Function of a System