Control Systems : An Introduction

This textbook is designed for an introductory, one-semester course in Control Systems for undergraduates and graduates in various engineering departments, such as electrical, mechanical, aerospace, and civil. It is written to be concise, clear, and yet comprehensive to make it easier for the students to learn this important subject with high mathematical complexity. The author emphasizes the physical simulation of systems, making it easier for readers to understand system behavior. The popular MATLABª software package is used for programming and simulation. Every new concept is explained with figures and examples for a clear understanding. The simple and clear style of presentation, along with comprehensive coverage, enables students to obtain a solid foundation in the subject and for use in practical applications. ¨ Written to be accessible to students of varying backgrounds, using a practical approach; ¨ Explains concepts in a clear, concise manner, minimizing mathematical rigor; ¨ Emphasizes the physical simulation of systems, making it easier to understand system behavior; ¨ Includes numerous, solved examples and exercises, as well as programming and simulations using MATLABª. Preface Contents Abbreviations 1 Introduction 1.1 Basics of Control Systems 1.2 Basic Signals 1.2.1 The Unit-Step Signal 1.2.2 The Unit-Impulse Signal 1.2.3 The Unit-Ramp Signal 1.2.4 The Unit-Parabolic Signal 1.3 Sinusoids 1.3.1 The Polar Form of Sinusoids 1.3.2 The Rectangular Form of Sinusoids Sum of Sinusoids with the Same Frequency 1.3.3 The Complex Sinusoids Real Causal Exponential Signal Exponentially Varying Amplitude Sinusoids 1.4 System Modeling 1.5 Summary Exercises 2 The Laplace Transform 2.1 Laplace Transform 2.1.1 Properties of the Laplace Transform Linearity 2.1.2 Time-Shifting 2.1.3 Frequency-Shifting 2.1.4 Time-Differentiation 2.1.5 Integration 2.1.6 Time-Scaling 2.1.7 Convolution in Time 2.1.8 Multiplication by t 2.1.9 Initial Value 2.1.10 Final Value 2.2 Laplace Transform Solution of Differential Equations 2.2.1 The Transfer Function 2.2.2 Transfer Function of Feedback Systems 2.3 Finding the Inverse Laplace Transform 2.3.1 Inverse Laplace Transform by Partial-FractionExpansion 2.4 Characterization of a System by Its Poles and Zeros and System Stability 2.5 Routh–Hurwitz Stability Criterion 2.6 Summary Exercises 3 Mathematical Modeling of Electrical Systems 3.1 Modeling of Electrical Circuits 3.1.1 Circuit Analysis Basic Elements in Electrical Circuits 3.1.2 Series Circuits 3.1.3 Parallel Circuits 3.1.4 Examples of Circuit Analysis 3.2 Summary Exercises 4 Mathematical Modeling of Mechanical Systems 4.1 Modeling Electrical Systems 4.2 Modeling Translational Mechanical Systems 4.2.1 Theoretical Analysis 4.3 Modeling Rotational Mechanical Systems 4.3.1 Simple Pendulum 4.3.2 A Mechanical Rotational System 4.3.3 Field Current Controlled DC Motor 4.3.4 Armature-Controlled DC Motor 4.4 Summary Exercises 5 Block Diagrams and Signal-Flow Graphs 5.1 Block Diagrams 5.2 Signal-Flow Graphs 5.2.1 Mason's Gain Formula Conversion of a Block Diagram to the Corresponding SFG 5.3 Summary Exercises 6 Steady-State and Transient Responses 6.1 Transfer Function of Feedback Systems 6.2 Steady-State Errors in Control Systems 6.2.1 Type 0 System 6.2.2 Type 1 System 6.2.3 Type 2 System Steady-State Errors of Nonunity Feedback Systems 6.3 Unit-Step Response and Transient Response Specifications 6.4 Linearization 6.5 Parameter Sensitivity 6.6 Summary Exercises 7 Root Locus 7.1 Plotting the Root Locus 7.1.1 Negative Feedback Systems Angle of a Line in the Complex Plane Breakaway and Break-in Points 7.1.2 Nonminimum-Phase Systems 7.2 Control System Design by Root Locus Method 7.2.1 Proportional Compensator 7.2.2 Proportional-Integral Compensator 7.2.3 Lag Compensator 7.2.4 Proportional-Derivative Compensator Higher-Order Systems 7.2.5 Lead Compensator Alternate Design 7.2.6 Proportional-Integral-Derivative Compensator 7.2.7 Lead-Lag Compensator 7.3 Summary Exercises 8 Design of Control Systems in Frequency Domain: Bode Plot 8.1 Bode Plot 8.1.1 Bode Plot of a Lag Compensator 8.1.2 Bode Plot of a Lead Compensator Approximation of e-Ts 8.2 Design of Control Systems 8.2.1 Relation Between Time-Domain and Frequency-Domain Specifications Relation Between Phase Margin and the Damping Ratio ζ 8.2.2 Lag Compensator 8.2.3 Lead Compensator 8.2.4 Lead–Lag Compensator 8.2.5 Proportional–Integral–Derivative Compensator The First Method Second Method 8.3 Summary Exercises 9 Nyquist Plot 9.1 Nyquist Plot 9.1.1 Nyquist Plots of Simple Transfer Functions The Constant First-Order Zero First-Order Pole Poles and Zeros at the Origin 9.2 Stability Analysis from Bode and Nyquist Plots 9.2.1 Nyquist Stability Criterion Closed-Loop Stability from the Nyquist Plot 9.2.2 Nonminimum-Phase Systems 9.2.3 Systems with Delay Units 9.2.4 Pade Approximation of e-Ts 9.3 Summary Exercises 10 State-Space Analysis of Control Systems 10.1 The State-Space Model 10.2 Frequency-Domain Solution of the State Equation 10.3 Time-Domain Solution of the State Equation 10.4 Commonly Used Realizations of Systems 10.5 Linear Transformation of State Vectors and Diagonalization 10.6 Controllability 10.7 Observability 10.8 Summary Exercises 11 Design of Control Systems in State Space 11.1 Design by Pole-Placement 11.1.1 Direct Comparison Method 11.1.2 Using Transformation Matrix 11.1.3 Using Ackermann's Formula 11.2 State Observers 11.2.1 Design of Regulator Systems with Observers Transfer Function of the Observer-Based Controller 11.3 Design of Control Systems with Observers 11.3.1 Quadratic Optimal Regulator Systems 11.4 Digital Implementation of Continuous-Time Systems 11.4.1 The Bilinear Transformation Frequency Warping Application of the Bilinear Transformation 11.5 Summary Exercises Answers to Selected Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Bibliography Index This textbook is designed for an introductory, one-semester course in Control Systems for undergraduates and graduates in various engineering departments, such as electrical, mechanical, aerospace, and civil. It is written to be concise, clear, and yet comprehensive to make it easier for the students to learn this important subject with high mathematical complexity. The author emphasizes the physical simulation of systems, making it easier for readers to understand system behavior. The popular MATLABa software package is used for programming and simulation. Every new concept is explained with figures and examples for a clear understanding. The simple and clear style of presentation, along with comprehensive coverage, enables students to obtain a solid foundation in the subject and for use in practical applications. ̈ Written to be accessible to students of varying backgrounds, using a practical approach; ̈ Explains concepts in a clear, concise manner, minimizing mathematical rigor; ̈ Emphasizes the physical simulation of systems, making it easier to understand system behavior; ̈ Includes numerous, solved examples and exercises, as well as programming and simulations using MATLABa.