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.

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).

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 MechanicalWork 81

4.2 ModelingMass-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 RotationalMotion 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-LoopMotor 141

6.2.1 Torque Production 141

6.2.2 Wound Field DCMotor 143

6.2.3 Commutation of the Single-LoopMotor 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 DCMachine 147

6.3.4 Back Emf in the Single-LoopMotor 149

6.3.5 Self-Induced Emf in the Single-LoopMotor 150

6.4 Dynamic Equations of the DCMotor 152

6.5 Optical EncoderModel 154

6.6 Tachometer for a DCMachine* 157

6.6.1 Tachometer for the Linear DCMachine 157

6.6.2 Tachometer for the Single-Loop DCMotor 157

6.7 TheMultiloop DCMotor* 159

6.7.1 Increased Torque Production 159

6.7.2 Commutation of the Armature Current 159

Problems 163

7 BlockDiagrams 173

7.1 Block Diagramfor a DCMotor 173

7.2 Block DiagramReduction 175

Problems 185

8 System Responses 191

8.1 First-Order SystemResponse 191

8.2 Second-Order SystemResponse 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 LocusMatlab File 211

Problems 212

9 Tracking and Disturbance Rejection 221

9.1 Servomechanism 221

9.2 Control of a DC ServoMotor 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 InternalModel 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 DisturbanceModel 276

10.1.2 Effect of the Initial Conditions on the Control Design 278

10.2 Two Degrees of FreedomControllers 283

10.3 Internal Stability 292

10.3.1 Unstable Pole-Zero Cancellation Inside the Loop (Bad) 295

10.3.2 Unstable Pole-Zero Cancellation Outside the Loop (Good) 298

10.4 Design Example: 2 DOF Control of Aircraft Pitch 300

10.5 Design Example: Satellite with Solar Panels (Collocated Case) 303

Appendix: Output Pole Placement 306

Appendix:Multinomial Expansions 310

Appendix: Overshoot 311

Appendix: Unstable Pole-Zero Cancellation 315

Appendix: Undershoot 317

Problems 320

11 Frequency Response Methods 339

11.1 Bode Diagrams 339

11.1.1 Simple Examples 343

11.1.2 More Bode DiagramExamples 345

11.2 Nyquist Theory 359

11.2.1 Principle of the Argument 359

11.2.2 Nyquist Test for Stability 368

11.3 Relative Stability: Gain and Phase Margins 377

11.4 Closed-Loop Bandwidth 383

11.5 Lead and Lag Compensation 387

11.6 Double Integrator Control via Lead-Lag Compensation 392

11.7 Inverted Pendulum with Output 399

Appendix: Bode and Nyquist Plots inMatlab 401

Problems 402

12 Root Locus 419

12.1 Angle Condition and Root Locus Rules 420

12.2 Asymptotes and Their Intercept 427

12.3 Angles of Departure 434

12.4 Effect of Open-Loop Poles on the Root Locus 450

12.5 Effect of Open-Loop Zeros on the Root Locus 451

12.6 Breakaway Points and the Root Locus 452

12.7 Design Example: Satellite with Solar Panels (Noncollocated) 453

Problems 458

13 Inverted Pendulum, Magnetic Levitation, and Cart on a Track 467

13.1 Inverted Pendulum 467

13.1.1 Mathematical Model of the Inverted Pendulum 467

13.1.2 Linear ApproximateModel 470

13.1.3 Transfer FunctionModel 470

13.1.4 Inverted PendulumControl Using Nested Feedback Loops 472

13.2 Linearization of NonlinearModels 475

13.3 Magnetic Levitation 478

13.3.1 Conservation of Energy 479

13.3.2 StatespaceModel 480

13.3.3 Linearization About an Equilibrium Point 481

13.3.4 Transfer FunctionModel 483

13.4 Cart on a Track System 483

13.4.1 Mechanical Equations 484

13.4.2 Electrical Equations 485

13.4.3 Equations ofMotion and Block Diagram 486

Problems 488

14 State Variables 501

14.1 Statespace Form 501

14.2 Transfer Function to Statespace 503

14.2.1 Control Canonical Form 505

14.3 Laplace Transformof the Statespace Equations 513

14.4 Fundamental Matrix Φ 516

14.4.1 Exponential Matrix  517

14.5 Solution of the Statespace Equation* 520

14.5.1 Scalar Case 521

14.5.2 Matrix Case 522

14.6 Discretization of a StatespaceModel* 523

Problems 525

15 State Feedback 529

15.1 Two Examples 529

15.2 General State Feedback Trajectory Tracking 537

15.3 Matrix Inverses and the Cayley-Hamilton Theorem 538

15.3.1 Matrix Inverse 538

15.3.2 Cayley-Hamilton Theorem 541

15.4 Stabilization and State Feedback 543

15.5 State Feedback and Disturbance Rejection 547

15.6 Similarity Transformations 551

15.7 Pole Placement 555

15.7.1 State Feedback Does Not Change the SystemZeros 559

15.8 Asymptotic Tracking of Equilibrium Points 560

15.9 Tracking Step Inputs via State Feedback 562

15.10 Inverted Pendulum on an Inclined Track* 569

15.11 Feedback Linearization Control* 574

Appendix: Disturbance Rejection in the Statespace 579

Problems 581

16 State Estimators and Parameter Identification 595

16.1 State Estimators 595

16.1.1 General Procedure for State Estimation 600

16.1.2 Separation Principle 608

16.2 State Feedback and State Estimation in the Laplace Domain* 610

16.3 Multi-Output Observer Design for the Inverted Pendulum* 613

16.4 Properties ofMatrix Transpose and Inverse 615

16.5 Duality* 617

16.6 Parameter Identification 619

Problems 626

17 Robustness and Sensitivity of Feedback 641

17.1 Inverted Pendulum with Output 641

17.2 Inverted Pendulum with Output 655

17.3 Inverted Pendulumwith State Feedback 657

17.4 Inverted Pendulumwith an Integrator and State Feedback 661

17.5 Inverted Pendulumwith State Feedback via State Estimation 663

Problems 666

References 671

Index 675

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