An Introduction to System Modeling and Control

A practical and straightforward exploration of the basic tools for the modeling, analysis, and design of control systems

In An Introduction to System Modeling and Control, Dr. Chiasson delivers an accessible and intuitive guide to understanding modeling and control for students in electrical, mechanical, and aerospace/aeronautical engineering. The book begins with an introduction to the need for control by describing how an aircraft flies complete with figures illustrating roll, pitch, and yaw control using its ailerons, elevators, and rudder, respectively. The book moves on to rigid body dynamics about a single axis (gears, cart rolling down an incline) and then to modeling DC motors, DC tachometers, and optical encoders. Using the transfer function representation of these dynamic models, PID controllers are introduced as an effective way to track step inputs and reject constant disturbances.

It is further shown how any transfer function model can be stabilized using output pole placement and on how two-degree of freedom controllers can be used to eliminate overshoot in step responses. Bode and Nyquist theory are then presented with an emphasis on how they give a quantitative insight into a control system's robustness and sensitivity. An Introduction to System Modeling and Control closes with chapters on modeling an inverted pendulum and a magnetic levitation system, trajectory tracking control using state feedback, and state estimation. In addition the book offers:

• A complete set of MATLAB/SIMULINK files for examples and problems included in the book.

• A set of lecture slides for each chapter.

• A solutions manual with recommended problems to assign.

• An analysis of the robustness and sensitivity of four different controller designs for an inverted pendulum (cart-pole).

Perfect for electrical, mechanical, and aerospace/aeronautical engineering students, An Introduction to System Modeling and Control will also be an invaluable addition to the libraries of practicing engineers.

Auteur

John Chiasson, PhD, is a Fellow of the IEEE and the author of Modeling and High-Performance Control of Electric Machines (Wiley 2005), Introduction to Probability and Stochastic Processes (Wiley 2013), and Differential-Geometric Approach to Nonlinear Control (2021).

Contenu

1 Introduction 1

1.1 Aircraft 1

1.2 Quadrotors 7

1.3 Inverted Pendulum 11

1.4 Magnetic Levitation 12

1.5 General Control Problem 14

2 Laplace Transforms 15

2.1 Laplace TransformProperties 17

2.2 Partial Fraction Expansion 21

2.3 Poles and Zeros 31

2.4 Poles and Partial Fractions 32

Appendix: Exponential Function 35

Problems 38

3 Differential Equations and Stability 45

3.1 Differential Equations 45

3.2 PhasorMethod of Solution 48

3.3 Final Value Theorem 52

3.4 Stable Transfer Functions 56

3.5 Routh-Hurwitz Stability Test 59

3.5.1 Special Case - A Row of the Routh Array has all Zeros* 65

3.5.2 Special Case - Zero in First Column, but the Row is Not Identically Zero* 68

Problems 71

4 Mass-Spring-Damper Systems 81

4.1 Mechanical Work 81

4.2 Modeling Mass-Spring-Damper Systems 82

4.3 Simulation 88

Problems 92

5 Rigid Body Rotational Dynamics 103

5.1 Moment of Inertia 103

5.2 Newton's Law of Rotational Motion 104

5.3 Gears 111

5.3.1 Algebraic Relationships Between Two Gears 112

5.3.2 Dynamic Relationships Between Two Gears 112

5.4 Rolling Cylinder* 117

Problems 125

6 The Physics of the DC Motor 139

6.1 Magnetic Force 139

6.2 Single-Loop Motor 141

6.2.1 Torque Production 141

6.2.2 Wound Field DC Motor 143

6.2.3 Commutation of the Single-Loop Motor 143

6.3 Faraday's Law 145

6.3.1 The Surface Element Vector S 146

6.3.2 Interpreting the Sign of 147

6.3.3 Back Emf in a Linear DC Machine 147

6.3.4 Back Emf in the Single-Loop Motor 149

6.3.5 Self-Induced Emf in the Single-Loop Motor 150

6.4 Dynamic Equations of the DC Motor 152

6.5 Optical Encoder Model 154

6.6 Tachometer for a DC Machine* 157

6.6.1 Tachometer for the Linear DC Machine 157

6.6.2 Tachometer for the Single-Loop DC Motor 157

6.7 TheMultiloop DC Motor* 159

6.7.1 Increased Torque Production 159

6.7.2 Commutation of the Armature Current 159

Problems 163

7 Block Diagrams 173

7.1 Block Diagramfor a DC Motor 173

7.2 Block Diagram Reduction 175

Problems 185

8 System Responses 191

8.1 First-Order System Response 191

8.2 Second-Order System Response 193

8.2.1 Transient Response and Closed-Loop Poles 194

8.2.2 Peak Time and Percent Overshoot 198

8.2.3 Settling Time 200

8.2.4 Rise Time 202

8.2.5 Summary of 202

8.2.6 Choosing the Gain of a Proportional Controller 202

8.3 Second-Order Systems with Zeros 205

8.4 Third-Order Systems 210

Appendix - Root Locus Matlab File 211

Problems 212

9 Tracking and Disturbance Rejection 221

9.1 Servomechanism 221

9.2 Control of a DC Servo Motor 226

9.2.1 Tracking 226

9.2.2 Disturbance Rejection 231

9.2.3 Summary of the PI Controller for a DC Servo 234

9.2.4 Proportional plus Integral plus Derivative Control 234

9.3 Theory of Tracking and Disturbance Rejection 238

9.4 Internal Model Principle 242

9.5 Design Example: PI-D Control of Aircraft Pitch 244

9.6 Model Uncertainty and Feedback* 250

Problems 258

10 Pole Placement, 2 DOF Controllers, and Internal Stability 271

10.1 Output Pole Placement 271

10.1.1 Disturbance Model 276

10.1.2 Effect of the Initial Conditions on the Control Design 278

10.2 Two Degrees of Freedom Controllers 283 10.3 Internal Stability 292&...