Dynamic system modelling and control 3

Copyright © 1993-2001, Hugh Jack page 1 Dynamic System Modeling and Control by Hugh Jack (Version 2.1, March 31, 2002) © Copyright 1993-2002 Hugh Jack page 1 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20 1.1 BASIC TERMINOLOGY 20 1.2 EXAMPLE SYSTEM 21 1.3 SUMMARY 21 1.4 PRACTICE PROBLEMS 21 2. TRANSLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 2.1 INTRODUCTION 22 2.2 MODELING 24 2.2.1 Free Body Diagrams 25 2.2.2 Mass and Inertia 25 2.2.3 Gravity And Other Fields 28 2.2.4 Springs 31 2.2.5 Damping and Drag 36 2.2.6 Cables And Pulleys 39 2.2.7 Friction 41 2.2.8 Contact Points And Joints 43 2.3 SYSTEM EXAMPLES 44 2.4 OTHER TOPICS 53 2.5 SUMMARY 54 2.6 PRACTICE PROBLEMS 54 3. ANALYSIS OF DIFFERENTIAL EQUATIONS . . . . . . . . . . . . .60 3.1 INTRODUCTION 60 3.2 EXPLICIT SOLUTIONS 61 3.3 RESPONSES 70 3.3.1 First-order 71 3.3.2 Second-order 76 3.3.3 Other Responses 80 3.4 RESPONSE ANALYSIS 84 3.5 NON-LINEAR SYSTEMS 86 3.5.1 Non-Linear Differential Equations 86 3.5.2 Non-Linear Equation Terms 90 3.5.3 Changing Systems 93 3.6 CASE STUDY 99 3.7 SUMMARY 103 3.8 PRACTICE PROBLEMS 103 4. NUMERICAL ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112 4.1 INTRODUCTION 112 4.2 THE GENERAL METHOD 112 4.2.1 State Variable Form 113 4.3 NUMERICAL INTEGRATION 121 4.3.1 Numerical Integration With Tools 121 page 2 4.3.2 Numerical Integration 126 4.3.3 Taylor Series 131 4.3.4 Runge-Kutta Integration 132 4.4 SYSTEM RESPONSE 139 4.4.1 Steady-State Response 139 4.5 DIFFERENTIATION AND INTEGRATION OF EXPERIMENTAL DATA 140 4.6 ADVANCED TOPICS 141 4.6.1 Switching Functions 141 4.6.2 Interpolating Tabular Data 144 4.6.3 Modeling Functions with Splines 145 4.6.4 Non-Linear Elements 147 4.7 CASE STUDY 147 4.8 SUMMARY 155 4.9 PRACTICE PROBLEMS 155 5. ROTATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161 5.1 INTRODUCTION 161 5.2 MODELING 162 5.2.1 Inertia 163 5.2.2 Springs 167 5.2.3 Damping 172 5.2.4 Levers 174 5.2.5 Gears and Belts 175 5.2.6 Friction 179 5.2.7 Permanent Magnet Electric Motors 181 5.3 OTHER TOPICS 182 5.4 DESIGN CASE 183 5.5 SUMMARY 187 5.6 PRACTICE PROBLEMS 187 6. INPUT-OUTPUT EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . .199 6.1 INTRODUCTION 199 6.2 THE DIFFERENTIAL OPERATOR 199 6.3 INPUT-OUTPUT EQUATIONS 202 6.3.1 Converting Input-Output Eqautions to State Equations 204 6.3.2 Integrating Input-Output Eqautions 206 6.4 DESIGN CASE 208 6.5 SUMMARY 217 6.6 PRACTICE PROBLEMS 217 6.7 REFERENCES 221 7. ELECTRICAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .222 7.1 INTRODUCTION 222 page 3 7.2 MODELING 222 7.2.1 Resistors 223 7.2.2 Voltage and Current Sources 225 7.2.3 Capacitors 229 7.2.4 Inductors 231 7.2.5 Op-Amps 232 7.3 IMPEDANCE 237 7.4 EXAMPLE SYSTEMS 239 7.5 PERMANENT MAGNET DC MOTORS 247 7.6 INDUCTION MOTORS 249 7.7 BRUSHLESS SERVO MOTORS 250 7.8 OTHER TOPICS 253 7.9 SUMMARY 253 7.10 PRACTICE PROBLEMS 253 8. FEEDBACK CONTROL SYSTEMS . . . . . . . . . . . . . . . . . . . . . .262 8.1 INTRODUCTION 262 8.2 TRANSFER FUNCTIONS 262 8.3 CONTROL SYSTEMS 264 8.3.1 PID Control Systems 266 8.3.2 Manipulating Block Diagrams 268 8.3.3 A Motor Control System Example 273 8.3.4 System Error 278 8.3.5 Controller Transfer Functions 282 8.3.6 State Variable Control Systems 282 8.3.7 Feedforward Controllers 283 8.3.8 Cascade Controllers 284 8.4 SUMMARY 284 8.5 PRACTICE PROBLEMS 284 9. PHASOR ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .294 9.1 INTRODUCTION 294 9.2 FOURIER TRANSFORMS FOR STEADY-STATE ANALYSIS 294 9.3 VIBRATIONS 301 9.4 SUMMARY 303 9.5 PRACTICE PROBLEMS 303 10. BODE PLOTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .307 10.1 INTRODUCTION 307 10.2 BODE PLOTS 311 10.3 SIGNAL SPECTRUMS 326 10.4 SUMMARY 327 10.5 PRACTICE PROBLEMS 327 10.6 LOG SCALE GRAPH PAPER 333 page 4 11. ROOT LOCUS ANALYSIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .336 11.1 INTRODUCTION 336 11.2 ROOT-LOCUS ANALYSIS 336 11.3 SUMMARY 344 11.4 PRACTICE PROBLEMS 345 12. NONLINEAR SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .360 12.1 INTRODUCTION 360 12.2 SYSTEM DIAGRAMS 360 12.3 SUMMARY 360 12.4 PRACTICE PROBLEMS 361 13. ANALOG INPUTS AND OUTPUTS . . . . . . . . . . . . . . . . . . . . .362 13.1 INTRODUCTION 362 13.2 ANALOG INPUTS 363 13.3 ANALOG OUTPUTS 370 13.4 SHIELDING 373 13.5 SUMMARY 375 13.6 PRACTICE PROBLEMS 375 14. CONTINUOUS SENSORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .377 14.1 INTRODUCTION 377 14.2 INDUSTRIAL SENSORS 378 14.2.1 Angular Displacement 379 Potentiometers 379 14.2.2 Encoders 380 Tachometers 384 14.2.3 Linear Position 384 Potentiometers 384 Linear Variable Differential Transformers (LVDT)385 Moire Fringes 387 Accelerometers 388 14.2.4 Forces and Moments 391 Strain Gages 391 Piezoelectric 394 14.2.5 Liquids and Gases 396 Pressure 396 Venturi Valves 396 Coriolis Flow Meter 398 Magnetic Flow Meter 399 Ultrasonic Flow Meter 399 Vortex Flow Meter 399 Pitot Tubes 400 14.2.6 Temperature 400 page 5 Resistive Temperature Detectors (RTDs) 400 Thermocouples 401 Thermistors 403 Other Sensors 405 14.2.7 Light 405 Light Dependant Resistors (LDR) 405 14.3 INPUT ISSUES 406 14.4 SENSOR GLOSSARY 411 14.5 SUMMARY 412 14.6 REFERENCES 413 14.7 PRACTICE PROBLEMS 413 15. CONTINUOUS ACTUATORS . . . . . . . . . . . . . . . . . . . . . . . . . .417 15.1 INTRODUCTION 417 15.2 ELECTRIC MOTORS 417 15.2.1 Brushed DC Motors 418 15.2.2 AC Synchronous Motors 423 15.2.3 Brushless DC Motors 425 15.2.4 Stepper Motors 426 15.3 HYDRAULICS 428 15.4 OTHER SYSTEMS 429 15.5 SUMMARY 429 15.6 PRACTICE PROBLEMS 429 16. MOTION CONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .432 16.1 INTRODUCTION 432 16.2 MOTION PROFILES 433 16.3 MULTI AXIS MOTION 442 16.3.1 Slew Motion 442 Interpolated Motion 444 16.3.2 Motion Scheduling 445 16.4 SUMMARY 447 16.5 PRACTICE PROBLEMS 447 17. ELECTROMECHANICAL SYSTEMS . . . . . . . . . . . . . . . . . . . .450 17.1 INTRODUCTION 450 17.2 MATHEMATICAL PROPERTIES 450 17.2.1 Induction 450 17.3 EXAMPLE SYSTEMS 455 17.4 SUMMARY 462 17.5 PRACTICE PROBLEMS 462 18. FLUID SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463 18.1 SUMMARY 463 page 6 18.2 MATHEMATICAL PROPERTIES 463 18.2.1 Resistance 464 18.2.2 Capacitance 466 18.2.3 Power Sources 468 18.3 EXAMPLE SYSTEMS 470 18.4 SUMMARY 472 18.5 PRACTICE PROBLEMS 472 19. THERMAL SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .473 19.1 INTRODUCTION 473 19.2 MATHEMATICAL PROPERTIES 473 19.2.1 Resistance 473 19.2.2 Capacitance 475 19.2.3 Sources 476 19.3 EXAMPLE SYSTEMS 476 19.4 SUMMARY 479 19.5 PRACTICE PROBLEMS 479 20. LAPLACE TRANSFER FUNCTIONS . . . . . . . . . . . . . . . . . . . .480 20.1 INTRODUCTION 480 20.2 THE LAPLACE TRANSFORM 480 20.2.1 A Few Transforms 482 20.2.2 Impulse Response (or Why Laplace Transforms Work) 484 20.3 MODELING MECHANICAL SYSTEMS 487 20.4 MODELING ELECTRICAL SYSTEMS 488 20.5 USING LAPLACE TRANSFORMS 489 20.5.1 Solving Partial Fractions 494 20.5.2 Input Functions 502 20.5.3 Examples 502 20.6 A MAP OF TECHNIQUES FOR LAPLACE ANALYSIS 507 20.7 NON-LINEAR ELEMENTS 508 20.8 SUMMARY 508 20.9 PRACTICE PROBLEMS 509 20.10 REFERENCES 520 21. CONTROL SYSTEM ANALYSIS . . . . . . . . . . . . . . . . . . . . . . .521 21.1 INTRODUCTION 521 21.2 CONTROL SYSTEMS 521 21.2.1 PID Control Systems 523 21.2.2 Analysis of PID Controlled Systems With Laplace Transforms 525 21.2.3 Finding The System Response To An Input 528 21.2.4 Controller Transfer Functions 533 21.3 ROOT-LOCUS PLOTS 533 page 7 21.3.1 Approximate Plotting Techniques 537 21.4 DESIGN OF CONTINUOUS CONTROLLERS 541 21.5 SUMMARY 541 21.6 PRACTICE PROBLEMS 542 22. LABORATORY GUIDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .547 22.1 Lab 1 - Introduction to Resources and Tutorials 547 22.1.1 Tutorial 1a - Creating Web Pages 547 22.1.2 Tutorial 1b - Introduction to Mathcad, Working Model 2D and The Internet 549 22.1.3 Presentation 1a - Introduction to Library Searches 550 22.2 Lab 2 - Computer Based Data Collection 551 22.2.1 Prelab 2a - Tutorial for LabVIEW Programming 551 22.2.2 Prelab 2b - Overview of Labview and the DAQ Cards 552 22.2.3 Experiment 2 - Introduction to LabVIEW and the DAQ Cards 553 22.3 Lab 3 - Sensors and More Labview 557 22.3.1 Prelab 3 - Sensors 557 22.3.2 Experiment 3 - Measurement of Sensor Properties 558 22.4 Lab 4 - Motors 559 22.4.1 Prelab 4a - Permanent Magnet DC Motors 559 22.4.2 Experiment 4a - Modeling of a DC Motor 561 22.5 Lab 5 - Basic Control Systems 562 22.5.1 Prelab 6a - Servomotor Proportional Control Systems 562 22.5.2 Experiment 5a - Servomotor Proportional Control Systems564 22.6 Lab 6 - Basic System Components 565 22.6.1 Prelab 6a - Mechanical Components 565 22.6.2 Experiment 6a - Mechanical Components 569 22.7 Lab 7 - Oscillating Systems 570 22.7.1 Prelab 7a - Oscillating Systems 570 22.7.2 Experiment 7a - Oscillation of a Torsional Spring 571 22.8 Lab 8 - Servo Control Systems 572 22.8.1 Prelab 5a - Research 572 22.8.2 Experiment 5a - Tutorial and programming 573 22.9 TUTORIAL - ULTRA 5000 DRIVES AND MOTORS 573 22.10 Lab 9 - Filters 582 22.10.1 Prelab 9 - Filtering of Audio Signals 582 22.10.2 Experiment 9 - Filtering of Audio Signals 585 22.11 Lab 10 - Stepper Motors 586 22.11.1 Prelab 10 - Stepper Motors 586 22.11.2 Experiment 9 - Stepper Motors 587 22.12 TUTORIAL - STEPPER MOTOR CONTROLLERS (by A. Blauch and H. Jack) 588 22.13 Lab 11 - Variable Frequency Drives 592 22.13.1 Prelab 11 - Variable Frequency Drives 592 . . . . . .36 2 13. 1 INTRODUCTION 36 2 13. 2 ANALOG INPUTS 36 3 13. 3 ANALOG OUTPUTS 37 0 13. 4 SHIELDING 37 3 13. 5 SUMMARY 37 5 13. 6 PRACTICE PROBLEMS 37 5 14. CONTINUOUS. FUNCTIONS 262 8 .3 CONTROL SYSTEMS 264 8 .3. 1 PID Control Systems 266 8 .3. 2 Manipulating Block Diagrams 268 8 .3. 3 A Motor Control System Example 2 73 8 .3. 4 System