Control Systems Engineering PDF

Control Systems Engineering This Page is intentionally left blank. Control Systems Engineering S. Salivahanan Principal SSN College of Engineering...

Control Systems Engineering This Page is intentionally left blank. Control Systems Engineering S. Salivahanan Principal SSN College of Engineering Chennai R. Rengaraj Associate Professor Department of Electrical and ­Electronics ­Engineering SSN College of Engineering Chennai G. R. Venkatakrishnan SSN College of Engineering Chennai Copyright © 2015 Pearson India Education Services Pvt. Ltd Published by Pearson India Education Services Pvt. Ltd, CIN: U72200TN2005PTC057128, formerly known as TutorVista Global Pvt. Ltd, licensee of Pearson Education in South Asia. No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent. This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time. ISBN 978-93-325-3413-1 eISBN 978-93-325-4475-8 Head Office: A-8 (A), 7th Floor, Knowledge Boulevard, Sector 62, Noida 201 309, Uttar Pradesh, India. Registered Office: Module G4, Ground Floor, Elnet Software City, TS-140, Blocks 2 & 9, Rajiv Gandhi Salai, Taramani, Chennai 600 113, Tamil Nadu, India. Fax: 080-30461003, Phone: 080-30461060 www.pearson.co.in, Email: [email protected]  About the Authors S. Salivahanan is the Principal of SSN College of Engineering, Chennai. He obtained his Ph.D. in microwave integrated circuits and has more than three and a half decades of teaching, research, administration and industrial experience both in India and abroad. He is the author of many bestsellers and has had several papers ­published at national and international journals. He is also the recipient of Bharatiya Vidya B ­ havan National Award for Best Engineering College Principal for 2011 from ISTE, and IEEE ­Outstanding Branch Counselor and Advisor Award in the Asia-Pacific region for 1996–97. He is also a recipient of Tata Rao Gold Medal from the Institution of Engineers (India). He is a senior member of IEEE, Fellow of IETE, Fellow of Institution of Engineers (India), Life Member of ISTE and Life Member of Society for EMC Engineers. R. Rengaraj is an Associate Professor in the Department of Electrical and ­ Electronics ­Engineering, SSN College of Engineering, Chennai. He received his Ph.D. in the area of ­combined heat and power. He has more than ten years of academic experience. Many of his articles have been published in refereed international journals and in the proceedings of ­international conferences. He is also a recipient of TATA Rao Gold Medal from the ­Institution of Engineers (India). His areas of research interest include control systems and power systems. G. R. Venkatakrishnan is a doctoral research scholar in the area of renewable energy sources in the Department of Electrical and Electronics Engineering, SSN College of Engineering, Anna University, Chennai. He has published many research papers in national and international journals and conference proceedings. This Page is intentionally left blank.  Brief Contents Prefacexxi 1 Control System Modeling 1.1 2 Physical Systems and Components 2.1 3 Block Diagram Reduction Techniques 3.1 4 Signal Flow Graph 4.1 5 Time Response Analysis 5.1 6 Stability and Routh–Hurwitz Criterion 6.1 7 Root Locus Technique 7.1 8 Frequency Response Analysis 8.1 9 Polar and Nyquist Plots 9.1 10 Constant M- and N-Circles and Nichols Chart 10.1 11 Compensators 11.1 12 Physiological Control Systems 12.1 13 State-Variable Analysis 13.1 14 MATLAB Programs 14.1 IndexI.1 This Page is intentionally left blank.  Contents Prefacexxi 1 Control System Modeling 1.1 1.1 Introduction 1.1 1.2 Classification of Control System 1.2 1.2.1 Open-Loop Control System 1.2 1.2.2 Closed-Loop Control System 1.2 1.3 Comparison of Open-Loop and Closed-Loop Control Systems 1.4 1.4 Differential Equations and Transfer Functions 1.4 1.4.1 Transfer Function Representation 1.5 1.4.2 Features and Advantages of Transfer Function Representation 1.5 1.4.3 Disadvantages of Transfer Function Representation 1.6 1.4.4 Transfer Function of an Open-Loop System 1.6 1.4.5 Transfer Function of a Closed-Loop System 1.6 1.4.6 Comparison of Positive Feedback and Negative Feedback Systems 1.8 1.5 Mathematical Modeling 1.10 1.5.1 Mathematical Equations for Problem Solving 1.11 1.6 Modeling of Electrical Systems 1.12 1.7 Modeling of Mechanical Systems 1.20 1.7.1 Translational Mechanical System 1.21 1.7.2 A Simple Translational Mechanical System 1.25 1.7.3 Rotational Mechanical System 1.37 1.7.4 A Simple Rotational Mechanical System 1.40 1.8 Introduction to Analogous System 1.50 1.8.1 Advantages of Electrical Analogous System 1.50 1.8.2 Force–Voltage Analogy 1.51 1.8.3 Force–Current Analogy 1.52 1.8.4 Torque–Voltage Analogy 1.71 1.8.5 Torque–Current Analogy 1.72 Review Questions 1.78 2 Physical Systems and Components 2.1 2.1 Introduction 2.1 2.2 Electromechanical System 2.1 2.3 Hydraulic System 2.5 2.3.1 Advantages of Hydraulic System 2.5 2.3.2 Disadvantages of Hydraulic System 2.6 x Contents 2.3.3 Applications of Hydraulic System 2.6 2.3.4 Devices Used in Hydraulic System 2.6 2.4 Pneumatic Systems 2.7 2.4.1 Gas Flow Resistance and Pneumatic Capacitance 2.8 2.4.2 Advantages of Pneumatic System 2.8 2.4.3 Disadvantages of Pneumatic System 2.8 2.4.4 Applications of Pneumatic System 2.8 2.4.5 Devices Used in Pneumatic System 2.8 2.4.6 Comparison between Hydraulic and Pneumatic Systems 2.8 2.5 Thermal Systems 2.12 2.5.1 Thermal Resistance and Thermal Capacitance 2.13 2.6 Liquid-Level System 2.16 2.6.1 Elements of Liquid-Level System 2.16 2.7 Introduction to Control System Components 2.19 2.8 Controllers 2.20 2.8.1 Controller Output as a Percentage Value 2.20 2.8.2 Measured Value as a Percentage Value 2.20 2.8.3 Set Point as a Percentage Value 2.21 2.8.4 Error as a Percentage Value 2.21 2.8.5 Types of Controllers 2.22 2.9 Electronic Controllers 2.23 2.9.1 ON–OFF Controller 2.23 2.9.2 Proportional Controller 2.26 2.9.3 Integral Controller 2.30 2.9.4 Derivative Controller 2.32 2.9.5 Proportional Integral Controller 2.34 2.9.6 Proportional Derivative Controller 2.36 2.9.7 Proportional Integral Derivative Controller 2.38 2.10 Potentiometers 2.41 2.10.1 Characteristics of Potentiometers 2.43 2.10.2 Power-Handling Capacity 2.43 2.10.3 Applications of Potentiometer 2.43 2.11 Synchros 2.43 2.11.1 Synchro Transmitter 2.44 2.11.2 Synchro Control Transformer 2.46 2.11.3 Synchro Error Detector 2.47 2.12 Servomotors 2.48 2.12.1 Classification of Servomotor 2.48 2.12.2 Features of Servomotor 2.48 2.12.3 DC Servomotor 2.49 2.12.4 AC Servomotor 2.55 2.12.5 Comparison between AC Servomotor and DC Servomotor 2.59  Contents xi 2.13 Tachogenerators 2.59 2.13.1 DC Tachogenerator 2.59 2.13.2 AC Tachogenerator 2.61 2.14 Stepper Motor 2.62 2.14.1 Permanent Magnet Stepper Motor 2.62 2.14.2 Variable Reluctance Stepper Motor 2.62 2.14.3 Hybrid Stepper Motor 2.64 2.14.4 Operation of Stepper Motor 2.64 2.14.5 Advantages of Stepper Motor 2.65 2.14.6 Applications of Stepper Motor 2.65 2.15 Gear Trains 2.65 2.15.1 Single Gear Train 2.66 2.15.2 Multiple Gear Trains 2.70 Review Questions 2.73 3 Block Diagram Reduction Techniques 3.1 3.1 Introduction to Block Diagram 3.1 3.2 Open-Loop and Closed-Loop Systems Using Block Diagram 3.1 3.2.1 Advantages of Block Diagram Representation 3.2 3.2.2 Disadvantages of Block Diagram Representation 3.2 3.3 Block Diagram Representation of Electrical System 3.2 3.4 Block Diagram Reduction 3.15 3.4.1 Need for Block Diagram Reduction 3.15 3.4.2 Block Diagram Algebra 3.16 3.4.3 Rules for Block Diagram Reduction 3.18 3.4.4 Block Diagram Reduction for Complex Systems 3.26 Review Questions 3.58 4 Signal Flow Graph 4.1 4.1 Introduction 4.1 4.1.1 Signal Flow Graph Terminologies 4.1 4.1.2 Properties of SFG 4.11 4.1.3 SFG Algebra 4.11 4.1.4 Mason’s Gain Formula for SFG 4.13 4.1.5 Signal Flow Graph From Differential Equation 4.13 4.1.6 Comparison between SFG and Block Diagram 4.72 Review Questions 4.72 5 Time Response Analysis 5.1 5.1 Introduction 5.1 5.2 Time Response of the Control System 5.2 xii Contents 5.2.1 Transient Response 5.2 5.2.2 Steady-State Response 5.2 5.3 Standard Test Signals 5.7 5.3.1 Impulse Signal 5.7 5.3.2 Step Signal 5.8 5.3.3 Ramp Signal 5.9 5.3.4 Parabolic Signal 5.10 5.4 Poles, Zeros and System Response 5.21 5.4.1 Poles and Zeros of a Transfer Function 5.21 5.4.2 Stability of the System 5.23 5.5 Type and Order of the System 5.23 5.5.1 Type of the System 5.23 5.5.2 ORDER of the System 5.24 5.6 First-Order System 5.25 5.6.1 Performance Parameters of First-Order System 5.25 5.6.2 Time Response of a First-Order System 5.26 5.7 Second-Order System 5.32 5.7.1 Classification of Second-Order System 5.32 5.7.2 Performance Parameters of Second-Order System 5.33 5.7.3 Time Response of the Second-Order System 5.35 5.7.4 Time-Domain Specifications for an Underdamped Second-Order System 5.44 5.8 Steady-State Error 5.78 5.8.1 Characteristic of Steady-State Error 5.78 5.8.2 Determination of Steady-State Error 5.79 5.8.3 Steady-State Error in Terms of G(s)5.79 5.8.4 Steady-State Error in Terms of T(s)5.81 5.8.5 Static Error Constants and System Type 5.82 5.8.6 Generalized or Dynamic Error Coefficients 5.94 5.9 Effect of Adding Poles and Zeros in the Second-Order System 5.102 5.9.1 Effect of Adding Poles 5.103 5.9.2 Effect of Adding Zeros 5.103 5.10 Response with P, PI and PID Controllers 5.104 5.10.1 Proportional Derivative Control 5.104 5.10.2 Proportional Integral Control 5.106 5.10.3 Proportional Plus Integral Plus Derivative Control (PID Control) 5.108 5.11 Performance Indices 5.111 Review Questions 5.112 6 Stability and Routh–Hurwitz Criterion 6.1 6.1 Introduction 6.1 6.2 Concept of Stability 6.1 6.3 Stability of Linear Time-Invariant System 6.3  Contents xiii 6.3.1 Stability Based on Natural Response of the System, c(t)natural6.3 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 6.5 Transfer Function of the System, G(s)6.6 6.5.1 Effects of Location of Poles on Stability 6.6 6.6 Zero-Input Stability or Asymptotic Stability 6.14 6.6.1 Importance of Asymptotic Stability 6.15 6.7 Relative Stability 6.16 6.8 Methods for Determining the Stability of the System 6.19 6.9 Routh–Hurwitz Criterion 6.19 6.9.1 Minimum-Phase System 6.21 6.9.2 Non-Minimum-Phase System 6.21 6.10 Hurwitz Criterion 6.21 6.10.1 Hurwitz Matrix Formation 6.21 6.10.2 Disadvantages of Hurwitz Method 6.22 6.11 Routh’s Stability Criterion 6.23 6.11.1 Necessary Condition for the Stability of the System 6.24 6.11.2 Special Cases of Routh’s Criterion 6.28 6.11.3 Applications of Routh’s Criterion 6.42 6.11.4 Advantages of Routh’s Criterion 6.44 6.11.5 Limitations of Routh’s Criterion 6.44 Review Questions 6.55 7 Root Locus Technique 7.1 7.1 Introduction 7.1 7.2 Advantages of Root Locus Technique 7.1 7.3 Categories of Root Locus 7.2 7.3.1 Variation of Loop Gain with the Root Locus 7.2 7.4 Basic Properties of Root Loci 7.4 7.4.1 Conditions Required for Constructing the Root Loci 7.5 7.4.2 Usage of the Conditions 7.5 7.4.3 Analytical Expression of the Conditions 7.6 7.4.4 Determination of Variable Parameter K 7.7 7.4.5 Minimum and Non-Minimum Phase Systems 7.9 7.5 Manual Construction of Root Loci 7.9 7.5.1 Properties / Guidelines for Constructing the Root Loci 7.9 7.5.2 Flow Chart for Constructing the Root Locus for a System 7.20 7.6 Root Loci for different Pole-Zero Configurations 7.22 7.7 Effect of Adding Poles and Zeros in the System 7.84 7.7.1 Addition of Poles to the Loop Transfer Function, G(s)H(s)7.84 7.7.2 Effect of Addition of Poles 7.84 xiv Contents 7.7.3 Addition of Zero to the Loop Transfer Function 7.87 7.7.4 Effect of Addition of Zeros 7.87 7.8 Time Response from Root Locus 7.92 7.9 Gain Margin and Phase Margin of the System 7.95 7.9.1 Gain Margin of the System 7.95 7.9.2 Phase Margin of the System 7.96 7.10 Root Locus for K < 0 Inverse Root Locus or Complementary Root Loci 7.97 7.10.1 Steps in Constructing the Inverse Root Loci Manually 7.98 7.11 Pole-Zero Cancellation Rules 7.103 7.12 Root Contours (Multi-Variable System) 7.108 Review Questions 7.108 8 Frequency Response Analysis 8.1 8.1 Introduction 8.1 8.1.1 Advantages of Frequency Response Analysis 8.1 8.1.2 Disadvantages of Frequency Response Analysis 8.2 8.2 Importance of Sinusoidal Waves for Frequency Response Analysis 8.3 8.3 Basics of Frequency Response Analysis 8.3 8.4 Frequency Response Analysis of Open-Loop and Closed-Loop Systems 8.5 8.4.1 Open-Loop System 8.5 8.4.2 Closed-Loop System 8.6 8.4.3 Closed-Loop System with Poles and Zeros 8.7 8.5 Frequency Response Representation 8.9 8.5.1 Determination of Frequency Response 8.9 8.6 Frequency Domain Specifications 8.13 8.7 Frequency and Time Domain Interrelations 8.15 8.7.1 Frequency Domain Specifications 8.17 8.8 Effect of Addition of a Pole to the Open-Loop Transfer F ­ unction of the System 8.29 8.9 Effect of Addition of a Zero to the Open-Loop Transfer ­Function of the System 8.30 8.10 Graphical Representation of Frequency Response 8.30 8.11 Introduction to Bode Plot 8.31 8.11.1 Reasons for Using Logarithmic Scale 8.31 8.11.2 Advantages of Bode Plot 8.32 8.11.3 Disadvantages of Bode Plot 8.32 8.12 Determination of Frequency Domain Specifications from Bode Plot 8.32 8.13 Stability of the System 8.33 8.13.1 Based on Crossover Frequencies 8.34 8.13.2 Based on Gain Margin and Phase Margin 8.34 8.14 Construction of Bode Plot 8.34 8.14.1 Effect of Damping Ratio x8.43  Contents xv 8.15 Constructing the Bode Plot for a Given System 8.48 8.15.1 Construction of Magnitude Plot 8.49 8.15.2 Construction of Phase Plot 8.50 8.16 Flow Chart for Plotting Bode Plot 8.51 8.17 Procedure for Determining the Gain K from the ­Desired Frequency Domain Specifications 8.52 8.18 Maximum Value of Gain 8.55 8.19 Procedure for Determining Transfer Function from Bode Plot 8.55 8.20 Bode Plot for Minimum and Non-Minimum Phase Systems 8.55 Review Questions 8.101 9 Polar and Nyquist Plots 9.1 9.1 Introduction to Polar Plot 9.1 9.2 Starting and Ending of Polar Plot 9.2 9.3 Construction of Polar Plot 9.3 9.4 Determination of Frequency Domain Specification from Polar Plot 9.11 9.4.1 Gain Crossover Frequency w gc 9.11 9.4.2 Phase Crossover Frequency w pc9.11 9.4.3 Gain Margin gm9.11 9.4.4 Phase Margin pm9.12 9.5 Procedure for Constructing Polar Plot 9.12 9.6 Typical Sketches of Polar Plot on an Ordinary Graph and Polar Graph 9.14 9.7 Stability Analysis using Polar Plot 9.17 9.7.1 Based on Crossover Frequencies 9.17 9.7.2 Based on Gain Margin and Phase Margin 9.18 9.7.3 Based on the Location of Phase Crossover Point 9.18 9.8 Determining the Gain K from the Desired Frequency Domain Specifications 9.19 9.8.1 When the Desired Gain Margin of the System is Specified 9.20 9.8.2 When the Desired Phase Margin of the System is Specified 9.20 9.9 Introduction to Nyquist Stability Criterion 9.31 9.10 Advantages of Nyquist Plot 9.31 9.11 Basic Requirements for Nyquist Stability Criterion 9.31 9.12 Encircled and Enclosed 9.32 9.12.1 Encircled 9.32 9.12.2 Enclosed 9.32 9.13 Number of Encirclements or Enclosures 9.33 9.14 Mapping of s-Plane into Characteristic Equation Plane 9.33 9.15 Principle of Argument 9.36 9.16 Nyquist Stability Criterion 9.41 9.17 Nyquist Path 9.41 9.18 Relation Between G(s) H(s)-Plane and F(s)-Plane9.42 9.19 Nyquist Stability Criterion Based on the Encirclements of −1+ j 09.43 xvi Contents 9.20 Stability Analysis of the System 9.43 9.21 Procedure for Determining the Number of Encirclements 9.44 9.21.1 Flow chart for Determining the Number of Encirclements Made by the Contour in G(s)H(s)-Plane9.46 9.22 General Procedures for Determining the Stability of the System Based on Nyquist Stability Criterion 9.47 9.22.1 Flow chart for Determining the Stability of the System Based on Nyquist Stability Criterion 9.48 Review Questions 9.77 10 Constant M- and N-Circles and Nichols Chart 10.1 10.1 Introduction 10.1 10.2 Closed-Loop Response from Open-Loop Response 10.1 10.3 Constant M-Circles10.2 10.3.1 Applications of Constant M-Circles10.5 10.3.2 Resonant Peak Mr and Resonant Frequency wr from Constant M-Circles10.5 10.3.3 Variation of Gain K with Mr and wr  10.6 10.3.4 Bandwidth of the System 10.7 10.3.5 Stability of the System 10.8 10.3.6 Determination of Gain K Corresponding to the Desired Resonant Peak (Mr)desired10.8 10.3.7 Magnitude Plot of the System from Constant M-Circles10.9 10.4 Constant N-Circles10.11 10.4.1 Phase Plot of the System from Constant N-Circles10.13 10.5 Nichols Chart 10.15 10.5.1 Reason for the Usage of Nichols Chart 10.15 10.5.2 Advantages of Nichols Chart 10.16 10.5.3 Transformation of Constant M- and N-Circles into Nichols Chart 10.16 10.5.4 Determination of Frequency Domain Specifications from Nichols Chart 10.18 10.5.5 Determination of Gain K for a Desired Frequency Domain Specifications 10.18 Review Questions 10.30 11 Compensators 11.1 11.1 Introduction 11.1 11.2 Compensators 11.2 11.2.1 Series or Cascade Compensation 11.2 11.2.2 Feedback or Parallel Compensation 11.2 11.2.3 Load or Series-Parallel Compensation 11.3 11.2.4 State Feedback Compensation 11.3 11.2.5 Forward Compensation with Series Compensation 11.3 11.2.6 Feed-forward Compensation 11.3  Contents xvii 11.2.7 Effects of Addition of Poles 11.4 11.2.8 Effects of Addition of Zeros 11.4 11.2.9 Choice of Compensators 11.4 11.3 Lag Compensator 11.5 11.3.1 Determination of Maximum Phase Angle fm  11.7 11.3.2 Electrical Representation of the Lag Compensator 11.8 11.3.3 Effects of Lag Compensator 11.9 11.3.4 Design of Lag Compensator 11.10 11.3.5 Design of Lag Compensator Using Bode Plot 11.10 11.3.6 Design of Lag Compensator Using Root Locus Technique 11.16 11.4 Lead Compensator 11.22 11.4.1 Determination of Maximum Phase Angle fm  11.25 11.4.2 Electrical Representation of the Lead Compensator 11.26 11.4.3 Effects of Lead Compensator 11.27 11.4.4 Limitations of Lead Compensator 11.27 11.4.5 Design of Lead Compensator 11.27 11.4.6 Design of Lead Compensator Using Bode Plot 11.28 11.4.7 Design of Lead Compensator Using Root Locus Technique 11.33 11.5 Lag–Lead Compensator 11.38 11.5.1 Electrical Representation of the Lag–Lead Compensator 11.40 11.5.2 Effects of Lag–Lead Compensator 11.42 11.5.3 Design of Lag–Lead Compensator 11.42 11.5.4 Design of Lag–Lead Compensator Using Bode Plot 11.42 11.5.5 Design of Lag–Lead Compensator Using Root Locus Technique 11.48 Review Questions 11.52 12 Physiological Control Systems 12.1 12.1 Introduction 12.1 12.2 Physiological Control Systems 12.1 12.3 Properties of Physiological Control Systems 12.2 12.3.1 Target of the Homeostasis 12.3 12.3.2 Imbalance in the Homeostasis 12.3 12.3.3 Homeostasis Control Mechanisms 12.3 12.4 Block Diagram of the Physiological Control System 12.5 12.4.1 Types of Control Mechanism 12.6 12.5 Differences Between Engineering and Physiological Control Systems 12.9 12.6 System Elements 12.10 12.6.1 Resistance 12.10 12.6.2 Capacitance 12.10 12.6.3 Inductance 12.10 12.7 Properties Related to Elements 12.11 xviii Contents 12.8 Linear Models of Physiological Systems: Two ­Examples 12.12 12.8.1 Lung Mechanism 12.12 12.8.2 Skeletal Muscle 12.15 12.9 Simulation—Matlab and Simulink Examples 12.16 Review Questions 12.20 13 State-Variable Analysis 13.1 13.1 Introduction 13.1 13.1.1 Advantages of State-Variable Analysis 13.2 13.2 State-Space Representation of Continuous-Time LTI Systems 13.2 13.3 Block Diagram and SFG Representation of a Continuous State-Space Model 13.4 13.4 State-Space Representation 13.5 13.5 State-Space Representation of Differential Equations in Physical Variable Form 13.6 13.5.1 Advantages of Physical Variable Representation 13.6 13.5.2 Disadvantages of Physical Variable Representation 13.6 13.6 State-Space Model Representation for Electric Circuits 13.6 13.7 State-Space Model Representation for Mechanical System 13.10 13.7.1 State-Space Model Representation of Translational / Rotational Mechanical System 13.11 13.8 State-Space Model Representation of Electromechanical ­System 13.14 13.8.1 Armature-Controlled DC Motor 13.15 13.8.2 Field-Controlled DC Motor 13.18 13.9 State-Space Representation of a System Governed by Differential Equations13.20 13.10 State-Space Representation of Transfer Function in Phase Variable Forms 13.22 13.10.1 Method 1 13.22 13.10.2 Method 2 13.23 13.10.3 Method 3 13.25 13.10.4 Advantages of Phase-Variable Representation 13.27 13.10.5 Disadvantages of the Phase-Variable Representation 13.27 13.11 State-Space Representation of Transfer Function in Canonical Forms 13.31 13.11.1 Controllable Canonical Form 13.32 13.11.2 Observable Canonical Form 13.33 13.11.3 Diagonal Canonical Form 13.34 13.11.4 Jordan Canonical Form 13.36 13.12 Transfer Function from State-Space Model 13.41 13.13 Solution of State Equation for Continuous Time Systems 13.45 13.13.1 Solution of Homogenous-Type State Equation 13.45 13.13.2 Solution of Non-Homogenous Type State Equation 13.47 13.13.3 State Transition Matrix 13.48 13.13.4 Properties of State Transition Matrix 13.53  Contents xix 13.14 Controllability and Observability 13.60 13.14.1 Criteria for Controllability 13.60 13.14.2 Criteria for Observability 13.61 13.15 State-Space Representation of Discrete-Time LTI Systems 13.66 13.15.1 Block Diagram and SFG of Discrete State-Space Model 13.68 13.16 Solutions of State Equations for Discrete-Time LTI Systems 13.69 13.16.1 System Function H(z) 13.70 13.17 Representation of Discrete LTI System 13.70 13.18 Sampling 13.72 13.18.1 Sampling Theorem 13.73 13.18.2 High Speed Sample-and-Hold Circuit 13.74 Review Questions 13.78 14 MATLAB Programs 14.1 14.1 Introduction 14.1 14.2 MATLAB in Control Systems 14.2 14.2.1 Laplace Transform 14.2 14.2.2 Inverse Laplace Transform 14.2 14.2.3 Partial Fraction Expansion 14.2 14.2.4 Transfer Function Representation 14.4 14.2.5 Zeros and Poles of a Transfer Function 14.5 14.2.6 Pole-Zero Map of a Transfer Function 14.6 14.2.7 State-Space Representation of a Dynamic System 14.7 14.2.8 Phase Variable Canonical Form 14.7 14.2.9 Transfer Function to State-Space Conversion 14.8 14.2.10 State-Space to Transfer Function Conversion 14.8 14.2.11 Series/Cascade, Parallel and Feedback Connections 14.9 14.2.12 Time Response of Control System 14.10 14.2.13 Performance Indices from the Response of a System 14.15 14.2.14 Steady State Error from the Transfer Function of a System 14.17 14.2.15 Routh–Hurwitz Criterion 14.19 14.2.16 Root Locus Technique 14.19 14.2.17 Bode Plot 14.23 14.2.18 Nyquist Plot 14.29 14.2.19 Design of Compensators Using Matlab14.33 IndexI.1 This Page is intentionally left blank.  Preface The knowledge of control system is essential for engineering graduates in solving many stability design problems arising in electrical, mechanical, aerospace, biomedical and ­ ­chemical systems. Since control system is an integral part of our modern society, it finds a wide range of applications in aircraft, robots and process control systems. We have devel- oped this book for subjects, such as control system, process instrumentation, dynamics and ­control, aeronautical engineering and physiological control system, taught in B.E./B.Tech., AMIE and Grade IETE degree ­programs. The book will also serve as a useful reference for competitive examinations. We are o ­ ptimistic that this book shall fill the void created by a lack of standard books on the subject of control systems. Ensuring this book available in the libraries of all ­universities, ­colleges and polytechnics will certainly help in enriching the readers. The various c­ oncepts of the subject have been arranged aesthetically and explained in a simple and reader-friendly ­language. For the better understanding of the subject, a large number of numerical problems with step-by-step solutions have been provided. The solu- tions to ­university questions have also been included. A set of review questions at the end of each chapter will help the readers to test their understanding of the subject. This book is divided into 14 chapters. Chapter 1 deals with modeling of electrical and mechanical systems. Chapter 2 explains the various components and modeling of physical systems. Chapter 3 concentrates on the block diagram reduction technique in determining the transfer function of a system. Chapter 4 focuses on the signal flow graph in determining the transfer function of a system. Chapter 5 discusses time response analysis of the system. Chapter 6 describes the stability of the system using Routh’s array. Chapter 7 elaborates on root locus technique. Chapter 8 deals with frequency response analysis and Bode plot. Chapter 9 concentrates on polar and nyquist plot. Chapter 10 discusses constant M- and N-circles and Nichols chart. Chapter 11 is devoted to compensators of the system. Chapter 12 deals with physiological control system. Chapter 13 covers the state-space analysis of the system. Chapter 14 is devoted to MATLAB programs related to control systems. All the ­topics of this book have been illustrated with clear diagrams which are aided by lucid ­language to facilitate easy understanding of the concepts. Acknowledgements We would like to sincerely thank the management of SSN College of Engineering, ­Chennai, for their constant encouragement and providing necessary facilities for completing this ­project. We are also thankful to our colleagues, especially V. S. Nagarajan and V. ­Thiyagarajan, Assistant Professors, and students Mandala Deekshith, B. Akshaya Padma Varshini and G. Deepika for their review and feedback to improve this book. We are indebted to R. Gopalakrishnan and A. Chakkarabani of SSN College of­ Engineering, Chennai for word-processing the manuscript of the entire book. A special thanks to Sojan Jose, R. Pritha and C. Purushothaman of Pearson India Education Services Pvt. Ltd, Chennai for bringing out this book successfully in a short span of time. xxii Preface Finally, we extend heartfelt thanks to our family members—Kalavathy Salivahanan, S. Santhosh Kanna, S. Subadesh Kanna, Rajalakshmi Rengaraj, R. Harivarshan, R. Devprasath, G. S. Rajan Babu, Sumathi Babu, G. R. Hemalakshmi Prakash and G. R. Jeya Jeyaprakash— for their patience and cheerfulness throughout the whole process. We will appreciate any constructive suggestions and feedback from the readers for ­further improvement of this book at [email protected]. S. Salivahanan R. Rengaraj G. R. Venkatakrishnan Control Systems Engineering This Page is intentionally left blank.  1 CONTROL SYSTEM MODELING 1.1 Introduction A system is a combination of components connected to perform a required action. The control component of a system plays a major role in altering or maintaining the system output based on our desired characteristics. There are two types of control systems: manual control and automatic control. For example, in manual control, a man can switch on or switch off the bore well motor to control the level of water in a tank. On the other hand, in automatic control, level switches and transducers are used to control the level of water in a tank. Control systems have naturally evolved in our ecosystem. In almost all living things, automatic control regulates the conditions necessary for life by tackling the disturbance through sensing and controlling functionalities. They operate complex systems and pro- cesses and achieve control with desired precision. The application of control systems facilitates automated manufacturing processes, accurate positioning and effective control of machine tools. They guide and control space vehicles, aircrafts, ships and high-speed ground transportation systems. Modern automation of a plant involves components such as sensors, instruments, computers and application of techniques that involve data processing and control. It is essential to understand a system and its characteristics with the help of a model, before creating a control for it. The process of developing a model is known as modeling. Physical systems are modeled by applying notable laws that govern their behaviour. For example, mechanical systems are described by Newton’s laws and electrical systems are described by Ohm’s law, Kirchhoff’s law, Faraday’s and Lenz’s law. These laws form the basis for the constitutive properties of the elements in a system. 1.2 Control System Modeling 1.2 Classification of Control System Control systems are generally classified as (i) open-loop control systems and (ii) closed-loop control systems as shown in Fig. 1.1. Control systems Open-loop Closed-loop systems systems Fig. 1.1 | Classification of control systems 1.2.1 Open-Loop Control System A control system that cannot adjust itself to the changes is called open-loop control system. In general, manual control systems are open-loop systems. The block diagram of open-loop control system is shown in Fig.1.2. r(t) u(t) c(t) Controller Plant Fig. 1.2 | Block diagram of an open-loop control system Here, r (t ) is the input signal, u (t ) is the control signal/actuating signal and c (t ) is the output signal. In this system, the output remains unaltered for a constant input. In case of any discrep- ancy, the input should be manually changed by an operator. An open-loop control system is suited when there is tolerance for fluctuation in the system and when the system parameter variation can be handled irrespective of the environmental conditions. In olden days, air conditioners were fitted with regulators that were manually controlled to maintain the desired temperature. This real-time system serves as an open-loop control system as shown in Fig.1.3. r(t) u(t) Air conditioning c(t) Controller system Fig. 1.3 | Open-loop control system of an air conditioner 1.2.2 Closed-Loop Control System Any system that can respond to the changes and make corrections by itself is known as closed-loop control system. The only difference between open-loop and closed-loop systems is the feedback action. The block diagram of a closed-loop control system is shown in Fig.1.4.  Classification of Control System 1.3 u(t) or r(t) e(t) m(t) c(t) Comparator Controller Plant b(t) Feedback Fig. 1.4 | Block diagram of closed-loop control system Here, r (t ) is the input signal, e (t ) is the error signal/actuating signal, u (t ) or m(t) is the control signal/manipulated signal, eb(t ) is the feedback signal and c (t ) is the controlled output. Here, the output of the machine is fed back to a comparator (error detector). The output signal is compared with the reference input r (t ) and the error signal e (t ) is sent to the controller. Based on the error, the controller adjusts the air conditioners input [control signal u (t )]. This process is continued till the error gets nullified. Both manual and auto- matic controls can be implemented in a closed-loop system. The overall gain of a system is reduced due to the presence of feedback. In order to compensate for the reduction of gain, if an amplifier is introduced to increase the gain of a system, the system may sometimes become unstable. Present-day air conditioners are designed with temperature sensors, comparators and controller modules. The reference temperature (the desired room temperature) is fed and compared with actual room temperature, which is sensed by a temperature sensor. The dif- ference between these two values, i.e., the error signal is fed to a controller and the controller performs the necessary action to minimize the error. The block diagram for this example is shown in Fig.1.5. Desired Reference temperature c(t) u(t) or temperature r(t) e(t) m(t) Air conditioning Comparator Controller Amplifier system b(t) Temperature sensor Fig. 1.5 | Closed-loop control system of an air conditioner A person driving a car is also an example for a closed-loop control system as shown in Fig.1.6. During the ride, the brain controls the hands for steering, gear, horn and the legs for brake and accelerator so as to perform the driving in a perfect manner. The eyes and the ears form the feedback, i.e., eyes for determining the pathway and mirror view and ears for sensing other vehicles nearby. The driver also accelerates or decelerates depending on the terrain and traffic involved. 1.4 Control System Modeling Desired position of Reference position of the car the car + Hands for steering, gear, + Brain − horn and legs for brake and accelerator control Eyes for road and mirror view and ears for hearing near by horns and alarms Fig. 1.6 | Closed-loop system of a driver controlled car 1.3 Comparison of Open-Loop and Closed-Loop Control Systems Table 1.1 discusses the comparison of open-loop and closed-loop control systems. Table 1.1 | Comparison of open-loop and closed-loop control systems S. No. Open-loop control system Closed-loop control system 1. The open-loop system is not preferred A closed-loop control system is more due to inaccuracy and unreliability. accurate, because the error between reference signal and the output is contin- uously measured through feedback. 2. System is more economical and simple to System is expensive and complex to construct. construct. 3. System is generally stable. Because of feedback, a system tries to over correct itself, which sometimes leads the system to oscillate. Hence, this is less stable and the overall gain of the system is also reduced. 4. Stability of a system is affected as it does Stability of a system is not affected, due not adapt to variations of environmental to reduced effects of non-linearities and conditions or to external disturbances. distortion. 1.4 Differential Equations and Transfer Functions For the analysis and design of a system, a mathematical model that describes its behaviour is required. The process of obtaining the mathematical description of a system is known as modeling. The differential equations for a given system are obtained by applying appropriate laws, for example, Newton’s laws for mechanical systems and Kirchhoff’s law for electrical systems. These equations may be either linear or non-linear based on the system being modeled. Therefore, deriving appropriate mathematical models is the most important part of the analysis of a system. A system is said to be linear if it obeys the principles of superposition and homogeneity. If a system has responses c1 (t ) and c2 (t ) for any two inputs r1 (t ) and r2 (t ) respectively, then  Differential Equations and Transfer Functions 1.5 the system response to the linear combination of these inputs a1r1 (t ) + a2 r2 (t ) is given by the linear combination of the individual outputs a1c1 (t ) + a2 c2 (t ), where a1 and a2 are constants. The principle of homogeneity and superposition is explained in Fig. 1.7. Linear Linear r1(t) c1(t) r2(t) c2(t) system system r1(t) a1 + Linear system c(t) = a1r1(t) + a2r2(t) + r2(t) a2 Fig. 1.7 | Principle of homogeneity and superposition If the coefficients of the differential equations are functions of time (an independent variable), then a system is called linear time-varying system. On the other hand, if the coef- ficients of the differential equations are not a function of time, then a system is called linear time-invariant system. For the transient-response or frequency-response analysis of single- input and single-output linear time-invariant system, the transfer function representation will be a convenient tool. But, if a system is subjected to multi-input and multi-output, then state-space analysis will be a convenient tool. 1.4.1 Transfer Function Representation The transfer function of a linear, time-invariant, differential equation system is given by the ratio of Laplace transform of output variable (response function) to the Laplace transform of the input variable (driving function) under the assumption that all initial conditions are zero. c (t ) C ( s) Transfer function, G ( s) = = at zero initial conditions. r (t ) R ( s) It is to be noted that non-linear systems and time-varying systems do not have transfer functions because they do not obey the principles of superposition and homogeneity. With the concept of transfer functions, it is possible to represent system dynamics by algebraic equations in s domain. The highest power of s in the denominator of a transfer function is the order of the system. 1.4.2 Features and Advantages of Transfer Function Representation The following are the features and advantages of transfer function representation: (i) Using transfer functions, mathematical models can be obtained and analyzed. (ii) Output response can be obtained for any kind of inputs. (iii) Stability analysis can be performed. (iv) The usage of Laplace transform converts complex time domain equations to simple algebraic equations, expressed with complex variable s. (v) Analysis of a system is simplified due to the use of s-domain variable in the equa- tions, rather than using time-domain variable. 1.6 Control System Modeling 1.4.3 Disadvantages of Transfer Function Representation The following are the disadvantages of transfer function representation: (i) It is not applicable to non-linear systems or time-varying systems. (ii) Initial conditions are neglected. (iii) Physical nature of a system cannot be found (i.e., whether it is mechanical or electrical or thermal system). 1.4.4 Transfer Function of an Open-Loop System In the block diagram of a system relating its input and output as shown in Fig.1.8, there exists an input (excitation) which operates through a transfer operator called transfer function and it produces an output (response). R(s) G(s) C(s) Fig. 1.8 | Block diagram representation of an open-loop transfer function From the block diagram representation of an open-loop transfer function, we have C ( s) G ( s) = R ( s) where R ( s) is the Laplace transform of the input variable or excitation, R C ( s) is the Laplace transform of the output variable or response and G ( s) is the transfer function. 1.4.5 Transfer Function of a Closed-Loop System Closed-loop control systems can be classified into two categories, based on the type of feed- back signal as (i) Negative feedback systems and (ii) Positive feedback systems. (i) Negative Feedback Systems The simplest form of a negative-feedback control system is shown in Fig. 1.9. It has one block in the forward path and one in the feedback path. R(s) E(s) C(s) + G(s) − B(s) H(s) Fig. 1.9 | Block diagram representation of negative feedback control system Practically, all control systems can be reduced to this form. G ( s) represents the overall gain of the blocks in the forward path, B ( s) is the feedback signal, E ( s) is the error signal and H ( s) represents the overall gain of all the blocks in the feedback path. The objective is to reduce this control system to a single block by determining the closed- loop transfer function (CLTF) for the negative feedback.  Differential Equations and Transfer Functions 1.7 The following relationships can be derived from the block diagram: Feedback signal, B(s) = output × feedback path gain B ( s) = C ( s) H ( s) (1.1) Error signal, E( s) = input signal – feedback signal E( s) = R ( s) − B ( s) Substituting Eqn. (1.1) in the above equation, we obtain E ( s) = R ( s) − C ( s) H ( s) (1.2) Output signal, C( s) = error signal × forward path gain C ( s) = E ( s) G ( s) (1.3) Substituting the value of the error signal, we obtain C ( s) = R ( s) G ( s) − C ( s) G ( s) H ( s) C ( s) + C ( s) G ( s) H ( s) = R ( s) G ( s) ( ) C ( s) 1 + G ( s) H ( s) = R ( s) G ( s) C ( s) G ( s) = R ( s) 1 + G ( s) H ( s) Hence, the transfer function of the negative feedback control system is given by C ( s) G ( s) T ( s) = = (1.4) R ( s) 1 + G ( s) H ( s) Based on the above expression, Fig. 1.9 is reduced to the simplified form as shown in Fig. 1.10. G(s) R(s) C(s) 1 + G(s)H(s) Fig. 1.10 | Reduced form of a negative feedback control system Thus, the closed-loop transfer function is C ( s) forward path gain T(s) = = R ( s) 1 + forward path gain × feedback path gain Hence, the output, C ( s) = closed-loop transfer function T(s) × input R ( s). Observations drawn from this relationship are: (i) The control system transfer function is a property of the system. (ii) The transfer function is dependent only on its internal structure and components. (iii) The transfer function is independent of the input applied to a system. (iv) When an input signal is applied to a closed-loop control system, an output is gener- ated; i.e., it is dependent on the input as well as the system transfer function. 1.8 Control System Modeling (ii) Positive Feedback Systems If the feedback is positive, then the closed-loop transfer function can be similarly derived as C ( s) G ( s) T ( s) = = (1.5) R ( s) 1 − G ( s) H ( s) The block diagram representing positive feedback control system is shown in Fig. 1.11. R(s) E(s) C(s) + G(s) + B(s) H(s) Fig. 1.11 | Block diagram representation of positive feedback control system 1.4.6 Comparison of Positive Feedback and Negative Feedback Systems Table 1.2 discusses the comparison of characteristics between positive feedback and negative feedback systems. Table 1.2 | Characteristics of positive feedback and negative feedback systems Characteristics Negative feedback Positive feedback Stability High Low Magnitude of transfer function 1 Sensitivity to parameter changes Low High Gain Low High Error signal Decreased Increased Application Motor control Oscillators Example 1.1: A negative feedback system has a forward gain of 10 and feedback gain of 1. Determine the overall gain of the system. Solution: Given the forward gain G ( s) = 10 and the feedback gain H ( s) = 1 The overall gain of a closed-loop system is same as its closed-loop transfer function. For a negative feedback system, the transfer function is given by G ( s) T ( s) = 1 + G ( s) H ( s) Substituting the values of G ( s) and H ( s), we obtain overall gain of a closed-loop system, 10 T ( s) = = 0.9091. 1 + (10 × 1)  Differential Equations and Transfer Functions 1.9 Example 1.2: A negative feedback system with a forward gain of 2 and a feedback gain of 8 is subjected to an input of 5 V. Determine the output voltage of the system. Solution: Given the forward gain G ( s) = 2, feedback gain H ( s) = 8 and input R ( s) = 5 V. Output voltage for any system is given by C ( s) = Transfer function of the closed-loop system × input R ( s) The transfer function of the closed-loop system with negative feedback is given by C ( s) G ( s) T ( s) = = R ( s) 1 + G ( s) H ( s) Substituting the values of G ( s) and H ( s), we obtain C ( s) 2 = = 0.1176 R ( s) 1 + (2 × 8) Therefore, the output voltage of the given system C ( s) = 0.1176 × 5 = 0.5882 V. Example 1.3: A positive feedback system was subjected to an input of 3 V. Determine the output voltage of the following sets of gains: (i) G ( s) = 1, H ( s) = 0.75 (ii) G ( s) = 1, H ( s) = 0.9 (iii) G ( s) = 1, H ( s) = 0.99 (iv) G ( s) = 1.9 , H ( s) = 0.5 (v) G ( s) = 1.99 , H ( s) = 0.5 (vi) G ( s) = 1.999 , H ( s) = 0.5 C ( s) G ( s) Solution: It is known that = R ( s) 1 − G ( s) H ( s) G ( s) C ( s) = × R ( s) 1 − G ( s) H ( s) The output voltage C ( s) for different cases can be obtained by substituting the different values of G ( s), H ( s) and R ( s). 1.10 Control System Modeling (i) When G ( s) = 1, H ( s) = 0.75 and R ( s) = 3 V, 1 C (s) = × 3 = 12 V 1 − (1 × 0.75) (ii) When G ( s) = 1, H ( s) = 0.9 and R ( s) = 3 V, 1 C ( s) = × 3 = 30 V 1 − ( 1 × 0.9 ) (iii) When G ( s) = 1, H ( s) = 0.99 and R ( s) = 3 V, 1 C ( s) = × 3 = 300 V 1 − ( 1 × 0.99 ) (iv) When G ( s) = 1.9 , H ( s) = 0.5 and R ( s) = 3 V, 1.9 C ( s) = × 3 = 114 V 1 − ( 1.9 × 0.5 ) (v) When G ( s) = 1.99 , H ( s) = 0.5 and R ( s) = 3 V, 1.99 C ( s) = × 3 = 1194 V 1 − ( 1.99 × 0.5 ) (vi) When G ( s) = 1.999 , H ( s) = 0.5 and R ( s) = 3 V, 1.999 C ( s) = × 3 = 11994 V 1 − ( 1.999 × 0.5 ) It can be noted that, the output becomes very large and approaches infinity when the loop gain approaches a value of 1, i.e., G ( s) H ( s) = 1. 1.5 Mathematical Modeling The process involved in modeling a system using mathematical equations, formed by the variables and constants of the system is called the mathematical modeling of a system. For example, an electrical network can be modeled using the equations formed by Kirchhoff’s laws. Fig 1.12 shows a flow chart for determining the transfer function of any system.  Mathematical Modeling 1.11 Start Identify the system variables Write the time domain equations using the system variables Take Laplace transform of the system equations with the initial conditions as zero Identify the input and output variables By elimination method, represent the resultant equation in terms of input and output variables Obtain transfer function by taking the ratio of Laplace transform of output variable to the Laplace transform of input variable Stop Fig. 1.12 | Flow chart for deriving the transfer function 1.5.1 Mathematical Equations for Problem Solving Basic Laplace transform of different functions are ( f ( t )) = F ( s ) ( f ’(t )) = sF (s) − f (0) 1.12 Control System Modeling F ( s) f (0) (∫ f (t ) dt ) = s − s Cramer’s rule: For a given pair of simultaneous equations, we have a1 x + b1 y = c1 a2 x + b2 y = c2 The above equations can be written in matrix form as  a1 b1   x   c1  =  a 2 b2   y  c2  Therefore, c1 b1 a1 c1 c2 b2 a2 c2 x= and y = a1 b1 a1 b1 a2 b2 a2 b2 c1 b1 a c1 a b1 where = ∆x , 1 = ∆ y and 1 = ∆. c2 b2 a2 c2 a2 b2 1.6 Modeling of Electrical Systems An electrical system consists of resistors, capacitors and inductors. The differential equa- tions of electrical systems can be formed by applying Kirchhoff’s laws. The transfer function can be obtained by taking Laplace transform of the integro-differential equations and rear- ranging them as a ratio of output to input. The relationship between voltage and current for different elements in the electrical circuit is given in Table 1.3. Table 1.3 | Relationship of voltage and current for R, L and C Element Voltage drop across the element Current through the element R v (t ) v (t ) = Ri (t ) i( t ) = R di (t ) 1 L v (t ) = L v (t ) dt L∫ i( t ) = dt C 1 dv (t ) v (t ) = i (t ) dt C∫ i( t ) = C dt  Modeling of Electrical Systems 1.13 Example 1.4: A rotary potentiometer is being used as an angular position transducer. It has a rotational range of 0°–300° and the corresponding output voltage is 15 V. The potentiometer is linear over the entire operating range. Determine its transfer function. Solution: Given output voltage = 15 V and input rotation = 300° The transfer function of the potentiometer T ( s) is given by output voltage T ( s) = input rotation Substituting the given values, we obtain 15 T ( s) = = 0.05 V / degree 300 Example 1.5: Determine the transfer function of the electrical network shown in Fig. E1.5. R L vi(t) i(t) C vo(t) Fig. E1.5 1 The output voltage across the capacitor is vo (t ) = i (t ) dt C∫ Solution: Applying Kirchhoff’s voltage law, the loop equation is given by di (t ) 1 vi (t ) = Ri (t ) + L i (t ) dt C∫ + dt Taking Laplace transform of the above equations, we obtain I ( s) Vo ( s) = (1) Cs  1 Vi ( s) =  R + Ls +  I ( s)  Cs  1.14 Control System Modeling Simplifying the above equation, we obtain Vi ( s) I ( s) = (2)  1  R + Ls + Cs  Substituting Eqn. (2) in Eqn. (1), we obtain Vi ( s) 1 Vo ( s) = ×  1 Cs  R + Ls + Cs  Hence, the transfer function of the system is given by Vo ( s) 1 = Vi ( s) 2 LCs + RCs + 1 Example 1.6: Determine the transfer function of the electrical network shown in Fig. E1.6(a). i1(t) R1 i(t) R2 vi(t) vo(t) L1 i(t) L2 Fig. E1.6(a) Solution: In the given circuit, there exist two impedances Z1 and Z2 as shown in Fig. E1.6(b). Z1 vi(t) Z2 vo(t) i(t) Fig. E1.6(b) Therefore, Z1 = R1 parallel to L1 R1 × L1 s = R1 + L1 s Z2 = R2 series with L2 = R2 + L2 s  Modeling of Electrical Systems 1.15 The output voltage across the impedance Z2 is vo (t ) = i (t ) × Z2 Applying Kirchhoff’s voltage law, the loop equation is given by vi ( t ) = i ( t ) × Z 1 + i ( t ) × Z 2 Taking Laplace transform of the above equations, we obtain Vo ( s) = I ( s) × Z2 (1) Vi ( s) = I ( s) × ( Z 1 + Z 2 ) Simplifying, we obtain Vi ( s ) I ( s) = (2) ( Z1 + Z2 ) Substituting Eqn. (2) in Eqn. (1), we obtain Vi ( s) × Z2 V0 ( s) = (Z 1 + Z2 ) Hence, the transfer function of the system is given by Vo ( s) Z2 = Vi ( s) Z1 + Z2 Therefore, Vo ( s) Z2 R2 + L2 s L1 L2 s2 + ( R2 L1 + R1 L2 ) s + R1 R2 = = = Vi ( s) Z1 + Z2  R1 × L1 s  ( L1 L2 s2 + ( R1 + R2 ) L1 + R1 L2 s + R1 R2 )  R + L s  ( 2 + R + L2 ) s 1 1 Example 1.7: Determine the transfer function of the electrical network shown in Fig. E1.7. R1 R2 vi(t) C1 C2 vo(t) i1(t) i2(t) Fig. E1.7 1.16 Control System Modeling Solution: The output voltage of the given electrical network is 1 vo (t ) = i (t ) dt C2 ∫ 2 Taking Laplace transform, we obtain 1 Vo ( s) = I 2 ( s) × (1) C2 s The loop equations for each loop can be obtained by applying Kirchhoff’s voltage law. Loop equation for loop (1) is 1 vi (t ) = i1 (t ) × R1 + C1 ∫ (i (t ) − i (t )) dt 1 2 Taking Laplace transform on both the sides, we obtain  1  1 Vi ( s) = I1 ( s) ×  R1 +  − I 2 ( s) × (2)  C1 s  C1 s Loop equation for loop (2) is 1 1 0= C1 ∫ (i (t ) − i (t )) dt + i (t ) × R 2 1 2 2 + i (t ) dt C2 ∫ 2 Taking Laplace transform, we obtain 1  1 1  0 = − I1 ( s) × + I 2 ( s) ×  R2 + + (3) C1 s  C1 s C2 s  Representing Eqn. (2) and Eqn. (3) in matrix form,  1 −1  R + Vi (s)  1 C1 s C1 s  I ( s))   1   =  0   −1 R2 + 1 + 1   I 2 ( s)     C1 s C1 s C2 s  Applying Cramer’s rule, we obtain ∆2 I 2 ( s) = ∆ 1 R1 + Vi ( s) C1 s Vi ( s) where ∆2 = = −1 C1 s 0 C1 s  Modeling of Electrical Systems 1.17 1 −1 R1 + C1 s C1 s  1  1 1  1 and ∆= =  R1 +   R2 + +  − 2 2 −1 1 1  C1 s   C1 s C2 s  C1 s R2 + + C1 s C1 s C2 s R1 R R 1 = R1 R2 + + 1 + 2 + C1 s C2 s C1 s C1C2 s2 = (R R C C ) s + (R C 1 2 1 2 2 1 2 + R1C1 + R2 C2 ) s + 1 C1C2 s2 Therefore, Vi ( s) × C1C2 s2 ∆2 C1 s Vi ( s) × C2 s I 2 ( s) = = = ∆ ( R1 R2 C1C2 ) s + ( R1C1 + R1C2 + R2 C2 ) s + 1 ( R1 R2 C1C2 ) s + ( R1C1 + R1C2 + R2 C2 ) s + 1 2 2 I 2 ( s) Vi ( s) But, Vo ( s) = = ( R1 R2 C1C2 ) s + ( R1C1 + R1C2 + R2 C2 ) s + 1 C2 s  2  Hence, the overall transfer function of the given electrical system is given by Vo ( s) 1 T ( s) = = Vi ( s) ( 1 2 1 2 ) ( 1 1 + R1C2 + R2C2 ) s + 1 R R C C s 2 + R C Example 1.8: Determine the transfer function of the electrical network shown in Fig. E1.8(a). R1 C1 vi(t) R2 C2 vo(t) Fig. E1.8(a) 1.18 Control System Modeling Solution: The above electrical network can be modified as shown in Fig. E1.8(b). R1 C1 i2(t) vi(t) R2 C2 vo(t) i1(t) Fig. E1.8(b) The output voltage of the given electrical circuit is 1 vo (t ) = i (t ) dt C2 ∫ 2 Applying Laplace transform, we obtain 1 I ( s) Vo ( s) = × I 2 ( s)= 2 (1) C2 s C2 s Applying Laplace transform, we obtain transformation of the parallel RC circuit as shown in Fig. E1.8(c). 1 R1 × sC1 R1 1 R1 + sC1 C1 Fig. E1.8(c) Thus, the given electrical circuit is re-drawn as shown in Fig. E1.8(d). 1 R1 × sC1 1 R1 + sC1 i2(t) vi(t) R2 C2 vo(t) i1(t) Fig. E1.8(d) Now, Kirchhoff’s voltage law is applied to the individual loops for the above electrical circuit. Then the equation for loop (1) is  Modeling of Electrical Systems 1.19  R1   Cs  Vi ( s) = I1 ( s) ×  1 R + 1  (  + I1 ( s) − I 2 ( s) × R2 )  1 C s  1  R1   Cs  = I1 ( s) ×  1  + I1 ( s) × R2 − I 2 ( s) × R2  R1C1 s + 1   C s  1  R1  = I1 ( s) ×   + I1 ( s) × R2 − I 2 ( s) × R2  R1C1 s + 1  R1  = I1 ( s)  + R2  − I 2 ( s) R2  sR1C1 + 1   R1 + R2 ( R1C1 s + 1)  = I1 ( s) ×   − I 2 ( s) × R2 (2)  R1C1 s + 1  Loop equation for loop (2) is  1  0 = I 2 ( s) ×   + I 2 ( s) × R2 − I1 ( s) × R2  C2 s   1  = I 2 ( s) ×  + R2  − I1 ( s) × R2  C2 s   1  ( ) = − I1 ( s) × R2 + I 2 ( s) ×  R2 +  C2 s  (3) The Eqn. (2) and Eqn. (3) are written in matrix form as  R1 + R2 ( R1C1 s + 1)  − R2  1( )  Vi ( s)  R1C1 s + 1  I s    0  =     1   I 2 ( s)  − R2 R2 +   C2 s  Applying Cramer’s rule, we obtain ∆2 I 2 ( s) = ∆ 1.20 Control System Modeling R1 + R2 ( R1C1 s + 1) Vi ( s) R1C1 s + 1 − R2 0 = R1 + R2 ( R1C1 s + 1) − R2 R1C1 s + 1 1 − R2 R2 + C2 s = ( − − R2 × Vi ( s) )  R1 + R2 ( R1C1 s + 1)   1     R2 +  − ( − R2 × − R2 )  R1C1 s + 1   C2 s  R2 × ( R1C1 s + 1) × Vi ( s) = (4) R R ( R C s + 1) R1 R2 + 1 + 2 1 1 C2 s C2 s Substituting Eqn. (4) in Eqn. (1), we obtain    2 ( 1 1 ) i( )   R × R C s+1 ×V s   R1 R2 ( R1C1 s + 1)   R1 R2 + +   C2 s C2 s  Vo ( s) = C2 s Vo ( s) R2 × ( R1C1 s + 1) = Vi ( s)  R R ( R C s + 1)  C2 s  R1 R2 + 1 + 2 1 1   C2 s C2 s  Thus, the overall transfer function is Vo ( s) R2 × ( R1C1 s + 1) = Vi ( s) R1 R2 C2 s + R1 + R2 ( R1C1 s + 1) 1.7 Modeling of Mechanical Systems The dimensions in which the movement of a mechanical system can be described are translational, rotational or a combination of both. For modeling of mech