SYSTEM-SPECIFIC PI CONTROL THEORY for FLUID and MOTION · PDF file ·...

SYSTEM-SPECIFIC PI CONTROL THEORY

for FLUID and MOTION SYSTEMS Second edition

Kalman I. Krakow

System-specific PI Control Theory for Fluid and Motion Systems

Copyright © 2004, 2006 Kalman I. Krakow All rights reserved.

Universal Publishers Boca Raton, Florida • USA

2006

ISBN: 1-58112- 921-1

www.universal-publishers.com

THE ControlProblem APPLICATION 3

THE ControlProblems APPLICATIONControlProblems is an application containing numerical system models

corresponding to the problems presented in Chapters 23 though 27. The problems

represent typical systems that may be analyzed with system-specific PI control theory.

The application enables validation of proportional and integral (PI) coefficients as well as

investigation of system response characteristics with various PI coefficients. If a default

PI coefficient option is selected instead of the user- specified PI coefficient option, the

problems serve as examples of systems tuned according to system-specific PI control

theory.

The application - ControlProblems - may be downloaded from the publisher’s

Web site - http://www.universal-publishers.com/.

The download is a zip file - ControlProblems.zip - containing the application in a

file titled ControlProblems.exe, files required to run the exe file, and files required to

install and remove the application. The application is designed for computers running a

Windows® 98, or later, operating system. The following download and setup procedure

is required.

• The download should be saved in its own folder, e.g., C:\ControlProblems or

C:\Temp.

• The files contained in the zip file should be extracted to the same folder.

• Run setup.exe to install the application.

Numerical data for the problems are presented in the ControlProblems

application. This data may be changed by the user to simulate systems with different PI

coefficients, modulated capacities, complete response intervals, signal update intervals,

loads, etc. Suitability of user-specified PI coefficients may be evaluated from simulated

system response characteristics. System response characteristics for default PI

coefficients may be used for comparison.

To remove the program, go to My Computer, Control Panel, Add or Remove

Programs and follow the standard procedure for removing a program.

4 OUTLINE OF CONTENTS

OUTLINE OF CONTENTS

THE ControlProblems APPLICATION ..................................................... 3

TABLE OF CONTENTS ............................................................................. 6

PREFACE ................................................................................................. 19

Part 1: THEORY ........................................................................ 22

Chapter 1: PREAMBLE ......................................................................... 23

Chapter 2: SYSTEMS ............................................................................ 45

Chapter 3: FLUID FLOW SYSTEMS ..................................................... 56

Chapter 4: LIVE FLUID STORAGE SYSTEMS ..................................... 72

Chapter 5: DEAD FLUID STORAGE SYSTEMS................................. 105

Chapter 6: POSITION CONTROL BY VELOCITY MODULATION ..... 119

Chapter 7: VELOCITY CONTROL BY FORCE MODULATION.......... 142

Chapter 8: POSITION CONTROL BY FORCE MODULATION........... 188

Chapter 9: SPEED CONTROL............................................................. 215

Chapter 10: INVERTED PENDULUM .................................................... 234

Chapter 11: MOTORIZED WHEEL ........................................................ 266

Chapter 12: SOLUTION OF EQUATIONS............................................. 291

Chapter 13: PI CONTROL ALGORITHMS ............................................ 302

Chapter 14: GENERALIZED METHODOLOGY .................................... 320

Chapter 15: CLOSING REMARKS, THEORY ....................................... 328

OUTLINE OF CONTENTS 5

Part 2: APPLICATIONS ........................................................... 333

Chapter 16: APPLICATION OF THEORY ............................................. 334

Chapter 17: HYSTERESIS REDUCTION .............................................. 346

Chapter 18: FLUID SYSTEM SIMULATION PROGRAMS.................... 353

Chapter 19: MOTION SYSTEM SIMULATION PROGRAMS................ 368

Chapter 20: FLUID PROPERTY SENSITIVITY ..................................... 380

Chapter 21: FLUID PROPERTY RATE SENSITIVITY .......................... 414

Chapter 22: MOTION SYSTEM SENSITIVITIES ................................... 448

Chapter 23: FLUID SYSTEM PROBLEMS............................................ 461

Chapter 24: VALVE CONTROL PROBLEMS ....................................... 466

Chapter 25: HEATER CONTROL PROBLEMS..................................... 491

Chapter 26: COMPRESSOR CONTROL PROBLEMS ......................... 509

Chapter 27: MOTION SYSTEM PROBLEMS ........................................ 534

Chapter 28: CLOSING REMARKS, APPLICATIONS ........................... 592

Part 3: APPENDICES .............................................................. 597

Appendix A: VALVES ........................................................................... 598

Appendix B: SYSTEMS WITH DAMPERS ........................................... 604

Appendix C: CHILLED WATER COOLING COILS .............................. 626

Appendix D: GLOSSARY...................................................................... 633

6 TABLE OF CONTENTS

TABLE OF CONTENTSTHE ControlProblems APPLICATION ..................................................... 3OUTLINE OF CONTENTS ......................................................................... 4PREFACE ................................................................................................. 19PREFACE TO THE SECOND EDITION....................................................................... 21

Part 1: THEORY ........................................................................ 22Chapter 1: PREAMBLE ........................................................................... 23NOMENCLATURE........................................................................................................ 23INTRODUCTION .......................................................................................................... 24REASONS FOR THE FORMAT OF THE PI EQUATION............................................. 26

Control algorithm equations ..................................................................................... 26SYSTEM TYPES .......................................................................................................... 27

Fluid system types ................................................................................................... 27Motion system types ................................................................................................ 28Equations ................................................................................................................. 28Mathematical perspective ........................................................................................ 29

INDEPENDENT VARIABLE / LOAD............................................................................. 29SYSTEM ORDER ......................................................................................................... 30DIFFERENTIAL EQUATIONS ...................................................................................... 30

First-order differential equations .............................................................................. 31Second-order differential equations ......................................................................... 34

SELECTION OF TARGET RESPONSE....................................................................... 40SYSTEM CLASSIFICATIONS ...................................................................................... 40PARAMETERS EFFECTING PI COEFFICIENTS........................................................ 42

Error attenuation parameters ................................................................................... 42PERFORMANCE EVALUATION .................................................................................. 43Chapter 2: SYSTEMS .............................................................................. 45NOMENCLATURE........................................................................................................ 45SYSTEM VARIABLES .................................................................................................. 45

Controlled property .................................................................................................. 45Modulated capacity .................................................................................................. 46Independent variable ............................................................................................... 46Control signal ........................................................................................................... 47Terminology ............................................................................................................. 47

COMPONENTS ............................................................................................................ 47INTERVALS.................................................................................................................. 50

Complete response interval ..................................................................................... 50Signal update interval .............................................................................................. 50

ASSUMPTIONS............................................................................................................ 51THE PI CONTROL EQUATION.................................................................................... 52CONTROL PROGRAMS .............................................................................................. 53CALCULATION OF PI COEFFICIENTS....................................................................... 53PREREQUISITES FOR PI CONTROL ......................................................................... 53ANALOGY BETWEEN LINEAR AND ROTATION MOTION SYSTEMS...................... 54CLOSURE..................................................................................................................... 55

TABLE OF CONTENTS 7

Chapter 3: FLUID FLOW SYSTEMS....................................................... 56NOMENCLATURE........................................................................................................ 56SYSTEM EQUATIONS................................................................................................. 58

Maximum attenuation coefficient ............................................................................. 61Minimum property sensitivity ................................................................................... 64

PERFORMANCE ANALYSIS ....................................................................................... 65Initial / step error attenuation ................................................................................... 65Oscillating load ........................................................................................................ 67

EQUATION SUMMARY................................................................................................ 70CLOSURE..................................................................................................................... 71Chapter 4: LIVE FLUID STORAGE SYSTEMS....................................... 72NOMENCLATURE........................................................................................................ 72SYSTEM EQUATIONS................................................................................................. 74CRITICAL RESPONSE................................................................................................. 77

Maximum attenuation coefficient ............................................................................. 78Minimum property rate sensitivity ............................................................................ 82Critical response analysis ........................................................................................ 83

UNDERDAMPED RESPONSE..................................................................................... 86Criteria for underdamped target response specification .......................................... 88Period and amplitude ratio for single interval attenuation ........................................ 89Minimum property rate sensitivity ............................................................................ 94Performance analysis .............................................................................................. 95

OSCILLATING LOADS................................................................................................. 95Critical response ...................................................................................................... 99Underdamped response ........................................................................................ 101

EQUATION SUMMARY.............................................................................................. 102CLOSURE................................................................................................................... 104Chapter 5: DEAD FLUID STORAGE SYSTEMS .................................. 105NOMENCLATURE...................................................................................................... 105SYSTEM EQUATIONS............................................................................................... 107

Maximum attenuation coefficient ........................................................................... 109Minimum property rate sensitivity .......................................................................... 111

PERFORMANCE ANALYSIS ..................................................................................... 112Initial / step error attenuation ................................................................................. 112Oscillating set point ............................................................................................... 114

EQUATION SUMMARY.............................................................................................. 118CLOSURE................................................................................................................... 118Chapter 6: POSITION CONTROL BY VELOCITY MODULATION....... 119NOMENCLATURE...................................................................................................... 119SYSTEM EQUATIONS............................................................................................... 121

Maximum attenuation coefficient ........................................................................... 124Minimum velocity sensitivity .................................................................................. 126Complete response interval ................................................................................... 127

PERFORMANCE ANALYSIS ..................................................................................... 128Initial / step error attenuation ................................................................................. 129Oscillating set point ............................................................................................... 130Oscillating base ..................................................................................................... 133

8 TABLE OF CONTENTS

TRAVEL TIME ESTIMATION ..................................................................................... 137GRADUAL START-UP................................................................................................ 138DRIVE CHARACTERISTICS...................................................................................... 139

Deadband .............................................................................................................. 139Delay interval ......................................................................................................... 139DC power ............................................................................................................... 140

EQUATION SUMMARY.............................................................................................. 140CLOSURE................................................................................................................... 141Chapter 7: VELOCITY CONTROL BY FORCE MODULATION ........... 142NOMENCLATURE...................................................................................................... 142SYSTEM EQUATIONS............................................................................................... 146CRITICAL RESPONSE............................................................................................... 149

Maximum attenuation coefficient ........................................................................... 150Minimum acceleration sensitivity ........................................................................... 155Complete response interval ................................................................................... 156Performance analysis ............................................................................................ 157

UNDERDAMPED RESPONSE................................................................................... 163Criteria for underdamped target response specification ........................................ 165Period and amplitude ratio for single interval attenuation ...................................... 166Minimum acceleration sensitivity ........................................................................... 170Performance analysis ............................................................................................ 172

OSCILLATING LOAD ................................................................................................. 172Critical response .................................................................................................... 176Underdamped response ........................................................................................ 178

OSCILLATING SET POINT ........................................................................................ 179CRUISE CONTROL.................................................................................................... 181GRADUAL START-UP................................................................................................ 183DRIVE CHARACTERISTICS...................................................................................... 184

Deadband .............................................................................................................. 184Delay interval ......................................................................................................... 184DC power ............................................................................................................... 185

EQUATION SUMMARY.............................................................................................. 185CLOSURE................................................................................................................... 187Chapter 8: POSITION CONTROL BY FORCE MODULATION ............ 188NOMENCLATURE...................................................................................................... 188CONTROL LOOP EQUATIONS ................................................................................. 192

Position control by velocity modulation .................................................................. 192Velocity control by force modulation ...................................................................... 193

CONTROL LOOP INTERACTION.............................................................................. 195SYSTEM PERFORMANCE ........................................................................................ 195

Constant load ........................................................................................................ 202Oscillating load ...................................................................................................... 207

PARKING.................................................................................................................... 213CLOSURE................................................................................................................... 214Chapter 9: SPEED CONTROL .............................................................. 215NOMENCLATURE...................................................................................................... 215SYSTEM DESCRIPTION............................................................................................ 217

TABLE OF CONTENTS 9

SYSTEM EQUATIONS............................................................................................... 218Maximum attenuation coefficient ........................................................................... 222Minimum speed sensitivity ..................................................................................... 224

PERFORMANCE ANALYSIS ..................................................................................... 225Initial / step error attenuation ................................................................................. 226Oscillating output torque ........................................................................................ 227Oscillating set point ............................................................................................... 231

EQUATION SUMMARY.............................................................................................. 232CLOSURE................................................................................................................... 232Chapter 10: INVERTED PENDULUM.................................................... 234NOMENCLATURE...................................................................................................... 234GEOMETRIC CONFIGURATION............................................................................... 237CONTROL STRATEGY.............................................................................................. 237KINEMATIC AND DYNAMIC ANALYSIS.................................................................... 238

Kinematic analysis ................................................................................................. 238Dynamic analysis ................................................................................................... 239Equation of motion in terms of error ...................................................................... 241Natural period ........................................................................................................ 242Modulated velocity ................................................................................................. 242

CRITICAL RESPONSE............................................................................................... 244Maximum attenuation coefficient ........................................................................... 246Minimum velocity sensitivity .................................................................................. 249

UNDERDAMPED RESPONSE................................................................................... 250Criteria for underdamped target response specification ........................................ 252Maximum target response period .......................................................................... 253Period and amplitude ratio for single interval attenuation ...................................... 254Minimum velocity sensitivity .................................................................................. 259

MOTION ANALYSIS................................................................................................... 260ACCELERATING PIVOT ............................................................................................ 261TRANSPORTER......................................................................................................... 261EQUATION SUMMARY.............................................................................................. 262CLOSURE................................................................................................................... 264Chapter 11: MOTORIZED WHEEL........................................................ 266NOMENCLATURE...................................................................................................... 266DYNAMIC ANALYSIS................................................................................................. 268CRITICAL RESPONSE............................................................................................... 272

Maximum attenuation coefficient ........................................................................... 274Minimum acceleration sensitivity ........................................................................... 277

UNDERDAMPED RESPONSE................................................................................... 278Criteria for underdamped target response specification ........................................ 280Period and amplitude ratio for single interval attenuation ...................................... 281Minimum acceleration sensitivity ........................................................................... 285Performance analysis ............................................................................................ 286

OSCILLATING SET POINT AND LOAD..................................................................... 287TRANSPORTER......................................................................................................... 287EQUATION SUMMARY.............................................................................................. 288CLOSURE................................................................................................................... 290

10 TABLE OF CONTENTS

Chapter 12: SOLUTION OF EQUATIONS ............................................ 291NOMENCLATURE...................................................................................................... 292EQUATIONS............................................................................................................... 292

Maximum attenuation coefficient ........................................................................... 294Minimum sensitivity ............................................................................................... 296

LIMITATION................................................................................................................ 297EXAMPLE................................................................................................................... 297VARIABLE SENSITIVITY ........................................................................................... 299SOLUTION OF TWO SIMULTANEOUS EQUATIONS............................................... 300CLOSURE................................................................................................................... 301Chapter 13: PI CONTROL ALGORITHMS............................................ 302NOMENCLATURE...................................................................................................... 302PI CONTROL EQUATION FORMATS........................................................................ 303

P systems, special considerations ......................................................................... 304CONTROL SIGNAL LIMITS........................................................................................ 306DEADBAND................................................................................................................ 307WINDUP ..................................................................................................................... 308

Windup prevention equations ................................................................................ 309PI CONTROL ALGORITHM CODES.......................................................................... 310

Summation format algorithms ................................................................................ 310Differential format algorithms ................................................................................. 312Algorithms for specific systems ............................................................................. 313

VARIABLE PI COEFFICIENTS................................................................................... 313Sensitivity variation evaluation .............................................................................. 314Summation format modification ............................................................................. 315

SYSTEM RESPONSE VARIATIONS ......................................................................... 316CLOSURE................................................................................................................... 318Chapter 14: GENERALIZED METHODOLOGY.................................... 320NOMENCLATURE...................................................................................................... 320SYSTEM EQUATION SETUP .................................................................................... 321

General system equation ....................................................................................... 321Control signal ......................................................................................................... 322Required coefficients ............................................................................................. 323System classification ............................................................................................. 323Sensitivities ............................................................................................................ 324System equations in terms of error and PI coefficients ......................................... 325

SOLUTION OF EQUATIONS ..................................................................................... 325First-order system equations ................................................................................. 325Second-order system equations ............................................................................ 326

ATTENUATION CONSIDERATIONS ......................................................................... 327TERMINOLOGY ......................................................................................................... 327CLOSURE................................................................................................................... 327Chapter 15: CLOSING REMARKS, THEORY....................................... 328MANDATORY CONDITIONS ..................................................................................... 328REQUIRED DATA ...................................................................................................... 328SYSTEM CLASSIFICATIONS .................................................................................... 329PI CONTROL ALGORITHM OPTIONS ...................................................................... 329

TABLE OF CONTENTS 11

OPTIONAL CODE ...................................................................................................... 330SELECTION OF TARGET RESPONSE..................................................................... 330REAL AND IDEAL SYSTEM CHARACTERISTICS.................................................... 330

Drive characteristics .............................................................................................. 331Friction in motion control systems ......................................................................... 332

CLOSURE................................................................................................................... 332

Part 2: APPLICATIONS .......................................................... 333Chapter 16: APPLICATION OF THEORY............................................. 334OVERVIEW................................................................................................................. 334SENSITIVITY EVALUATION ...................................................................................... 335

Performance maps ................................................................................................ 335Analysis of governing equations ............................................................................ 336Latitude .................................................................................................................. 336Sign convention ..................................................................................................... 337Criteria for control .................................................................................................. 338

INTERVALS................................................................................................................ 338Complete response interval ................................................................................... 338Signal update interval ............................................................................................ 340

RESPONSE ATTENUATION PARAMETERS............................................................ 342I systems ............................................................................................................... 342PI systems ............................................................................................................. 342P systems .............................................................................................................. 343

CONTROL PROGRAMS ............................................................................................ 343Digital to analog conversion .................................................................................. 344

CLOSURE................................................................................................................... 345Chapter 17: HYSTERESIS REDUCTION .............................................. 346NOMENCLATURE...................................................................................................... 346COMPENSATION EQUATIONS................................................................................. 347THEORETICAL BASIS FOR EQUATIONS ................................................................ 348NON-LINEAR SYSTEMS............................................................................................ 352Chapter 18: FLUID SYSTEM SIMULATION PROGRAMS ................... 353PROGRAM STRUCTURE .......................................................................................... 353CONTROL SYSTEM CODE ....................................................................................... 356

PI control subroutine .............................................................................................. 360Control signal send/receive subroutine ................................................................. 361Hysteresis compensation function ......................................................................... 361Virtual timer function .............................................................................................. 362

TEMPERATURE SYSTEM CODE.............................................................................. 362LEVEL SYSTEM CODE.............................................................................................. 365

Liquid level subroutine ........................................................................................... 365Valve function ........................................................................................................ 367

Chapter 19: MOTION SYSTEM SIMULATION PROGRAMS ............... 368PROGRAM STRUCTURE .......................................................................................... 368CONTROL SYSTEM CODE ....................................................................................... 371


PI control subroutine .............................................................................................. 374Control signal send/receive subroutine ................................................................. 375Virtual timer function .............................................................................................. 375

CRUISE CONTROL CODE ........................................................................................ 376Chapter 20: FLUID PROPERTY SENSITIVITY..................................... 380SYSTEM DESIGNATIONS......................................................................................... 380UNIT ANALYSIS......................................................................................................... 381TEMPERATURE (MIXING VALVE) ............................................................................ 382

Nomenclature ........................................................................................................ 382Equipment ............................................................................................................. 382System variables ................................................................................................... 382System equations .................................................................................................. 383Property sensitivity ................................................................................................ 383Sample calculations ............................................................................................... 383

TEMPERATURE (INJECTION VALVE)...................................................................... 385Nomenclature ........................................................................................................ 385Equipment ............................................................................................................. 385System variables ................................................................................................... 385System equations .................................................................................................. 386Property sensitivity ................................................................................................ 386Sample calculations ............................................................................................... 388

TEMPERATURE (HEAT EXCHANGER) .................................................................... 391Nomenclature ........................................................................................................ 391Equipment ............................................................................................................. 391System variables ................................................................................................... 392System equations .................................................................................................. 392Property sensitivity ................................................................................................ 392Sample calculations, chilled water modulation ...................................................... 393Sample calculations, air flow modulation ............................................................... 396

HUMIDITY................................................................................................................... 398Nomenclature ........................................................................................................ 398Equipment, dehumidification ................................................................................. 398Equipment, humidification ..................................................................................... 399System variables ................................................................................................... 399System equations .................................................................................................. 399Property sensitivity ................................................................................................ 400Sample calculations, dehumidification by modulating chilled water flow rate ........ 400Sample calculations, dehumidification by modulating air flow rate ........................ 403Sample calculations, humidifying .......................................................................... 405

CARBON DIOXIDE CONCENTRATION .................................................................... 406Nomenclature ........................................................................................................ 406Equipment ............................................................................................................. 406System variables ................................................................................................... 406System equations .................................................................................................. 406Property sensitivity ................................................................................................ 407Sample calculations ............................................................................................... 407

PRESSURE ................................................................................................................ 409Nomenclature ........................................................................................................ 409


Equipment ............................................................................................................. 409System variables ................................................................................................... 410System equations .................................................................................................. 410Property sensitivity ................................................................................................ 411Sample calculations ............................................................................................... 411

Chapter 21: FLUID PROPERTY RATE SENSITIVITY .......................... 414SYSTEM DESIGNATIONS......................................................................................... 414UNIT ANALYSIS......................................................................................................... 415LIQUID LEVEL............................................................................................................ 418

Nomenclature ........................................................................................................ 418Equipment ............................................................................................................. 418System variables ................................................................................................... 419System equation .................................................................................................... 419Property rate sensitivity ......................................................................................... 420Sample calculations, storage with a modulated throttling valve ............................ 420Sample calculations, storage with a modulated pump .......................................... 420

TEMPERATURE (HEAT EXCHANGER) .................................................................... 422Nomenclature ........................................................................................................ 422Equipment ............................................................................................................. 422System variables ................................................................................................... 423System equations .................................................................................................. 423Property rate sensitivity ......................................................................................... 423Sample calculations ............................................................................................... 423

TEMPERATURE (FLOW RATE) ................................................................................ 426Nomenclature ........................................................................................................ 426Equipment ............................................................................................................. 426System variables ................................................................................................... 427System equation .................................................................................................... 427Property rate sensitivity ......................................................................................... 428Sample calculations ............................................................................................... 428

TEMPERATURE (MIXING)......................................................................................... 430Nomenclature ........................................................................................................ 430Equipment ............................................................................................................. 431System variables ................................................................................................... 431System equation .................................................................................................... 431Property rate sensitivity ......................................................................................... 432Sample calculations ............................................................................................... 432

HUMIDITY................................................................................................................... 434Nomenclature ........................................................................................................ 434Equipment, dehumidification ................................................................................. 434Equipment, humidification ..................................................................................... 435System variables ................................................................................................... 435System equation .................................................................................................... 435Property rate sensitivity ......................................................................................... 436Sample calculations, dehumidification ................................................................... 436Sample calculations, humidification ....................................................................... 438

CARBON DIOXIDE CONCENTRATION .................................................................... 440Nomenclature ........................................................................................................ 440


Equipment ............................................................................................................. 440System variables ................................................................................................... 441System equation .................................................................................................... 441Property rate sensitivity ......................................................................................... 442Sample calculations ............................................................................................... 442

TANK PRESSURIZATION.......................................................................................... 444Nomenclature ........................................................................................................ 444Equipment ............................................................................................................. 444System variables ................................................................................................... 445System equation .................................................................................................... 445Property rate sensitivity ......................................................................................... 446Sample calculations ............................................................................................... 446

Chapter 22: MOTION SYSTEM SENSITIVITIES................................... 448SENSITIVITY TYPES ................................................................................................. 448UNITS ANALYSIS....................................................................................................... 450SPEED SENSITIVITY................................................................................................. 452

Nomenclature ........................................................................................................ 452Sample calculations ............................................................................................... 452

VELOCITY SENSITIVITY ........................................................................................... 455Nomenclature ........................................................................................................ 455Sample calculations ............................................................................................... 455

ACCELERATION SENSITIVITY................................................................................. 457Nomenclature ........................................................................................................ 457Sample calculations ............................................................................................... 457

Chapter 23: FLUID SYSTEM PROBLEMS ........................................... 461GENERAL NOMENCLATURE.................................................................................... 461THE SYSTEM MODELS............................................................................................. 462THE PI CONTROL ALGORITHM ............................................................................... 462PROGRAM FLOW CHART......................................................................................... 463

Program options .................................................................................................... 464Sampling interval ................................................................................................... 465

DETERMINATION OF PI COEFFICIENTS ................................................................ 465Chapter 24: VALVE CONTROL PROBLEMS ....................................... 466GENERAL NOMENCLATURE.................................................................................... 466FLOW RATE EQUATIONS......................................................................................... 468

Flow fraction characteristics .................................................................................. 468Complete response interval ................................................................................... 470Maximum flow fraction differential for a sampling interval ..................................... 470

COMMON SPECIFIED DATA..................................................................................... 471TEMPERATURE (MIXING VALVE) ............................................................................ 472

System-specific nomenclature ............................................................................... 472System equations .................................................................................................. 472Specified data ........................................................................................................ 473Simulation algorithm essentials ............................................................................. 474

TEMPERATURE (INJECTION VALVE)...................................................................... 475System-specific nomenclature ............................................................................... 475System equations .................................................................................................. 475


Specified data ........................................................................................................ 476Simulation algorithm essentials ............................................................................. 477

LEVEL (LIVE STORAGE)........................................................................................... 479System-specific nomenclature ............................................................................... 479System equations .................................................................................................. 479Specified data ........................................................................................................ 480Simulation algorithm essentials ............................................................................. 481

LEVEL (DEAD STORAGE)......................................................................................... 482System-specific nomenclature ............................................................................... 482System equations .................................................................................................. 482Specified data ........................................................................................................ 482Simulation algorithm essentials ............................................................................. 483

ESSENTIAL EQUATIONS.......................................................................................... 485Units ...................................................................................................................... 485Temperature (mixing valve) ................................................................................... 485Temperature (injection valve) ................................................................................ 486Level (live storage) ................................................................................................ 487Level (dead storage) .............................................................................................. 489

Chapter 25: HEATER CONTROL PROBLEMS .................................... 491GENERAL NOMENCLATURE.................................................................................... 491ELECTRIC HEATER EQUATIONS ............................................................................ 493

Complete response interval ................................................................................... 493Maximum heat flow rate differential for a sampling interval ................................... 493

COMMON SPECIFIED DATA..................................................................................... 494FLOW SYSTEM.......................................................................................................... 495


LIVE STORAGE SYSTEM.......................................................................................... 498System-specific nomenclature ............................................................................... 498System equations .................................................................................................. 498Specified data ........................................................................................................ 499Simulation algorithm essentials ............................................................................. 500

DEAD STORAGE SYSTEM........................................................................................ 501System-specific nomenclature ............................................................................... 501System equations .................................................................................................. 501Specified data ........................................................................................................ 501Simulation algorithm essentials ............................................................................. 502

ESSENTIAL EQUATIONS.......................................................................................... 504Units ...................................................................................................................... 504Flow system ........................................................................................................... 505Live storage system ............................................................................................... 505Dead storage system ............................................................................................. 507

Chapter 26: COMPRESSOR CONTROL PROBLEMS ......................... 509GENERAL NOMENCLATURE.................................................................................... 509COMPRESSOR EQUATIONS.................................................................................... 511

Complete response interval ................................................................................... 512


Maximum speed / flow rate differential for a sampling interval .............................. 513COMMON SPECIFIED DATA..................................................................................... 513FLOW SYSTEM.......................................................................................................... 514


LIVE STORAGE SYSTEM.......................................................................................... 519System-specific nomenclature ............................................................................... 519System equations .................................................................................................. 519Specified data ........................................................................................................ 521Simulation algorithm essentials ............................................................................. 522

DEAD STORAGE SYSTEM........................................................................................ 524System-specific nomenclature ............................................................................... 524System equations .................................................................................................. 524Specified data ........................................................................................................ 525Simulation algorithm essentials ............................................................................. 526

ESSENTIAL EQUATIONS.......................................................................................... 528Units ...................................................................................................................... 528Flow system ........................................................................................................... 529Live storage system ............................................................................................... 530Dead storage system ............................................................................................. 532

Chapter 27: MOTION SYSTEM PROBLEMS........................................ 534GENERAL NOMENCLATURE.................................................................................... 534SYSTEM CHARACTERISTICS .................................................................................. 536PI CONTROL ALGORITHM........................................................................................ 536PROGRAM ESSENTIALS .......................................................................................... 537

Program options .................................................................................................... 537Complete response interval ................................................................................... 538Maximum modulated capacity differential for a signal update interval .................. 538Deadband .............................................................................................................. 538

SYSTEMS MODELED................................................................................................ 538DETERMINATION OF PI COEFFICIENTS ................................................................ 539COMMON SPECIFIED DATA..................................................................................... 539ROBOTIC ARM POSITIONING.................................................................................. 540

Nomenclature for robotic arm positioning .............................................................. 541System equations .................................................................................................. 542Specified data ........................................................................................................ 543Simulation algorithm essentials ............................................................................. 545

CRUISE CONTROL.................................................................................................... 546Nomenclature for cruise control ............................................................................. 546System equations .................................................................................................. 547Specified data ........................................................................................................ 549Simulation algorithm essentials ............................................................................. 550

LINEAR ACTUATOR .................................................................................................. 552Nomenclature for the linear actuator ..................................................................... 552System equations .................................................................................................. 553Specified data ........................................................................................................ 555


Simulation algorithm essentials ............................................................................. 557SPEED CONTROL ..................................................................................................... 558

Nomenclature for speed control ............................................................................ 558System equations .................................................................................................. 559Specified data ........................................................................................................ 561Simulation algorithm essentials ............................................................................. 562

INVERTED PENDULUM / TRANSPORTER .............................................................. 563Inverted pendulum ................................................................................................. 564Nomenclature for the inverted pendulum .............................................................. 565System equations for the pendulum ...................................................................... 566Motorized wheel .................................................................................................... 570Nomenclature for motorized wheel ........................................................................ 570System equations for the motorized wheel ............................................................ 572Transporter operation ............................................................................................ 574Specified data ........................................................................................................ 576Simulation algorithm essentials ............................................................................. 578

ESSENTIAL EQUATIONS.......................................................................................... 580Units ...................................................................................................................... 580Robotic arm positioning ......................................................................................... 581Cruise control ........................................................................................................ 582Linear actuator ....................................................................................................... 584Speed control ........................................................................................................ 587Inverted pendulum / Transporter ........................................................................... 587

Chapter 28: CLOSING REMARKS, APPLICATIONS........................... 592ESSENTIALS.............................................................................................................. 592

Control program ..................................................................................................... 592PI coefficient equations ......................................................................................... 594

IMPLEMENTATION.................................................................................................... 595SIMULATED AND ACTUAL RESULTS...................................................................... 596

Part 3: APPENDICES .............................................................. 597Appendix A: VALVES ........................................................................... 598NOMENCLATURE...................................................................................................... 598BASIC EQUATIONS................................................................................................... 599

Throttling valves .................................................................................................... 599Mixing valves ......................................................................................................... 599

FLOW FRACTION VARIATIONS ............................................................................... 600FLOW FRACTION EFFECT ON SENSITIVITY.......................................................... 601SYSTEM CHARACTERISTICS .................................................................................. 603Appendix B: SYSTEMS WITH DAMPERS ........................................... 604NOMENCLATURE...................................................................................................... 604BASIC EQUATIONS................................................................................................... 606

System configurations ........................................................................................... 607SERIES DAMPERS.................................................................................................... 607

Systems with linear dampers ................................................................................. 608Sensitivity considerations ...................................................................................... 609


Head loss considerations ...................................................................................... 610Systems with non-linear dampers ......................................................................... 611

PARALLEL DAMPERS............................................................................................... 611Economizer control strategies ............................................................................... 612Linearization .......................................................................................................... 613Linearization illustration ......................................................................................... 616Flow rate considerations ........................................................................................ 623Sensitivity considerations ...................................................................................... 624

CLOSURE................................................................................................................... 625Appendix C: CHILLED WATER COOLING COILS .............................. 626FACTORS EFFECTING SENSITIVITIES................................................................... 626CRITERIA FOR CONTROL........................................................................................ 627PERFORMANCE CHARACTERISTICS..................................................................... 627CLOSURE................................................................................................................... 632Appendix D: GLOSSARY...................................................................... 633

PREFACE 19

PREFACE

A feedback control system is a system in which the value of a system property is controlled by modulating the system capacity via a control signal according to the difference, i.e., the error, between the desired set point value of the controlled property and its current value. Examples of feedback control systems are:• the cruise control of a car,• room temperature control using a thermostat,• constant temperature, pressure balancing, shower valve,• thermostatic expansion valve on a refrigeration (air conditioning) system, and• a Watt governor (invented by James Watt in the eighteenth century) for controlling

the speed of a steam engine.Each system is unique. Implementation methods may be electrical, electronic, mechanical, etc. A single theory applicable to all feedback control systems is impractical.

PID feedback control is one method of implementing a feedback control system. In PID control systems, an equation based on three coefficients - a proportional (P) coefficient, an integral (I) coefficient, and a derivative (D) coefficient - relates a control signal to the error. The implementation of the equation may be accomplished by analog or digital means. Conventional PID control theory originated in the 1940's. It was originally developed for analog, not digital, control systems. Analog control systems update the control signal continuously. Digital systems update the control system at specified intervals. For fluid system control, the control signal may be updated at intervals of the order of seconds. For motion system control, the control signal may be updated at intervals of the order of milliseconds. System-specific PI control theory was developed specifically for digital control systems. System-specific PI control theory and conventional theory represent different methods of implementing a feedback control system. System-specific PI control theory is not intended to replace or exclude conventional theory.

Tuning - the determination of the coefficients required to implement a control system - is not an exact science. Tuning methods may be experimental (e.g., by trial-and-error) or analytical. For a given system, there is no single set of coefficients that will yield satisfactory system response characteristics while all others will yield unsatisfactory system responses characteristics. Obviously, there are many methods of determining the coefficients. One method does not exclude any other method.

System-specific PI control theory for fluid system control and for motion system control has been developed in order to enable simple analytical tuning of proportional-integral (PI) control systems. A PI control system is essentially a PID control system with the derivative (D) coefficient set equal to zero. A derivative coefficient is not essential and may have a detrimental effect on system response characteristics. Analytical tuning is the determination of the PI coefficients by calculations based on physical properties of the fluid or motion system. System-specific PI control theory is based on the fundamental algorithm for PID control systems as applied to digital (not analog) control systems. It

20 PREFACE

was developed from a 'back-to-basics' perspective considering fluid systems and motion systems independently.

Terminology used in system-specific PI control theory is that considered most suitable and may differ from terminology used in conventional theory. Conventional terminology is not considered sacred. Definitions and concepts must be interpreted without preconception.

The solution to an equation is independent of the method of solution. Whether the solution to a differential equation is obtained by conventional methods, or by Laplace methods, is not relevant to its validity. The are many ways of solving equations. Control theory may be used as an alternative to Newton’s Method for certain types of problems. Chapter 12: SOLUTION OF EQUATIONS illustrates the use of control theory to solve certain types of equations. The method presented in Chapter 12 is not intended to replace Newton’s Method. It is not claimed that the method presented in Chapter 12 is superior to Newton’s method. The method is presented to illustrate that control theory may be viewed a method of solution of equations.

Application of system-specific PI control theory to fluid systems and to motion systems requires, not only the calculation of PI coefficients, but also requires input and output signal conversion, optional code for compensation for hysteresis, safety limits, etc. Sample programs for control system simulation to verify the coefficients, sample calculations of sensitivities on which PI coefficients are based, and problems are presented in Part 2: APPLICATIONS. The ControlProblems application (reference page 3) contains numerical system models, corresponding to the problems presented, so that the suitability of user-specified PI coefficients may be evaluated from simulated system response characteristics. System response characteristic for default PI coefficients may be used for comparison. The numerical data for the problems are presented in the ControlProblems application. This data may be changed by the user to simulate different modulated capacities, PI coefficients, complete response intervals, signal update intervals, loads, etc.

A PI control algorithm is only one component of a control program. A control program must contain safety limits to either shut down the controlled system or activate an alarm in case unsafe, or unforeseen, conditions arise. Safety limits are required in case of equipment malfunction, coding error in a program, data input error, etc. Safety limits are required when testing any new version of an algorithm or a new set of PI coefficients. Therefore, implementing a PI control algorithm with PI coefficients based on system-specific PI control theory should pose no greater risk than implementing a PI control algorithm with PI coefficients based on conventional theory.

This book is written in a format such that most chapters are self-contained and do not require reference to previous chapters. Although this format requires repetition, it facilitates reading by allowing a reader to focus on a particular system. A generalized theory would reduce repetition, however, it would camouflage the physical significance of parameters involved.

Kalman I. KrakowOctober 29, 2004

PREFACE 21

PREFACE TO THE SECOND EDITIONThe objective of system-specific PI control theory is to simplify conventional

feedback control theory. Further consideration of system-specific PI control theory indicated that further simplification is possible by the use of an additional format of the PI control equation relating the control signal to the error. The extended system-specific PI control theory not only simplifies the algorithms required to implement a control system, but also facilitates the use of variable proportional and integral (PI) coefficients, and eliminates the need for windup prevention. The ControlProblems application was revised to incorporate the extended theory. The additional format of the PI control algorithm relating the control signal to the error further distinguishes system-specific PI theory from conventional theory.

Using a variable I coefficient ( ), instead of a constant I coefficient, in the solution of equations method presented in Chapter 12: SOLUTION OF EQUATIONS - the KI method - makes the KI method correspond to Newton’s Method. Variable PI coefficients may yield more rapid error attenuation than obtainable with constant PI coefficients. This indicates that variable PI coefficients may be useful in control systems.

Kalman I. KrakowFebruary 15, 2006

KI

Part 1: THEORY

Chapter 1: PREAMBLE - NOMENCLATURE 23

Chapter 1: PREAMBLE

The development of system-specific PI control theory is based on the observation

that there is more than one set of coefficients required to implement a proportional-

integral (PI) control system that yields satisfactory performance. Coefficients are often

obtained by trial-and-error methods, i.e., trial-and-error tuning. The reason for the

development of system-specific PI control theory is to obtain simple analytical methods

for determining coefficients required to implement a PI control system, i.e., to replace

trial-and-error tuning. For some types of systems, system properties required for tuning

may not be available from specified data and may require experimental determination.

The development of system-specific PI control theory is based on a unique perspective.

This chapter presents this perspective and prerequisite differential equation topics.

NOMENCLATUREcoefficients defined by Equation 1.9

amplitude ratio of consecutive opposite-sign peaks

amplitude ratio of consecutive same-sign peaks

coefficients defined by Equation 1.12

error, dependent variable

initial error (dependent variable) at ,

constant determined by a boundary condition

complementary component of the solution

particular component of the solution

integral coefficient for an integral format equation

integral coefficient for a finite difference format equation

proportional coefficient for an integral format equation

proportional coefficient for a finite difference format equation

signal update index

number of oscillation periods

a b c, ,

Aos

Ass

C0 C1 C2 ... Ck, , , ,

E

E0 n 0= t 0=

Ec

Ecomp

Epart

ki

KI

kp

KP

n

N

24 Chapter 1: PREAMBLE - INTRODUCTION

number of oscillation periods for 99.9% error attenuation

roots of an auxiliary equation

shape factor defined by Equation 1.21

control signal (percentage)

time

interval for 99.9% error attenuation

signal update interval

controlled property

set point of a controlled property

angular frequency defined by Equation 1.34 and 1.39

logarithmic decrement

attenuation parameter defined by Equation 1.33 and 1.38

time constant - first-order differential equations: the time constant is

defined by Equations 1.13 and 1.14 - second-order differential

equations: the time constant is defined by Equations 1.19 and 1.20

for critical responses and by Equations 1.33 and 1.37 for

underdamped responses

oscillation period

phase angle

Subscriptssignal update index

current signal update index

next signal update index

INTRODUCTIONWhy are feedback control theories specifically formulated for fluid systems and for

motion systems desirable? There are many different types of systems using feedback

control - analog systems and digital systems, fluid systems and motion systems. System

capacities may be modulated by feedback control systems acting on motors or actuators.

Motors and actuators may be electrical, hydraulic, or pneumatic. For electrical motors

N99.9

r1 r2,

R

S

t

t99.9

δt

X

XSP

β

δ

ρ

τ

ϒ

ϕ

i

n

n 1+

Chapter 1: PREAMBLE - INTRODUCTION 25

and actuators, drives are used to modulate the characteristics of the electricity supplied

by a utility in order to modulate motor and actuator speed, torque, or position. The drives

may use feedback control to accomplish their function. Control systems may therefore

serve many different types of applications. Conventional theory is essentially a single

theory for all types of systems is therefore inherently complex. Because of its complexity,

trial-and-error tuning - the determination of values of coefficients relating the control

signal to the error of the controlled property - is often used. A theory for a specific type of

system is relatively simple. Because of its simplicity, analytical tuning is easily

accomplished.

Fluid systems are quite different from motion systems. Independent variables

(loads) change gradually in fluid systems but rapidly in motion systems. Whereas the

control signal may be updated at intervals of the order of 60 seconds in a fluid control

system, a motion control system may require the signal to be updated at intervals of a

fraction of a second, e.g., 1/10,000 second. Conventional theory was original developed

for analog systems. Analogy systems continuously update the control signal. Digital

systems are capable of updating the control signal at specified intervals. Longer intervals

are more suitable for fluid systems, shorter interval are more suitable for motion control

systems.

By using generic terminology and parameters, conventional theory camouflages

the physical significance of system properties that govern system performance. A

system-specific PI control theory may be formulated using terminology that emphasizes

the physical significance of system properties. A system-specific PI control theory allows

the identification of the properties that are of primary importance, and the properties that

are of secondary importance, to system performance.

The system-specific PI control theory focuses on fluid systems and on motion

systems. The fluid system-specific PI control theory presented has been developed for

systems using electric motors and actuators. The motion system-specific PI control

theory presented has been developed for motion control system using direct current

(DC) electric motors.

Whereas the system-specific PI control theory and the conventional theory are

very different, it is preferable to start from basics. Preconceptions resulting from a

knowledge of conventional theory should not be applied to the current theory.

26 Chapter 1: PREAMBLE - REASONS FOR THE FORMAT OF THE PI EQUATION

REASONS FOR THE FORMAT OF THE PI EQUATIONA feedback control system is necessary for systems with indeterminate loads and/

or random variations of system parameters. A feedback control system is also necessary

for systems with characteristics that have a degree of uncertainty, i.e., that are not

accurately known. The objective of a feedback control system is

• to maintain a controlled property equal to a specified set point value

• by using a control signal to modulate a variable (capacity)

• as an independent variable (load) changes with respect to time.

For example, the objective of a temperature control system is

• to maintain a space temperature (the controlled property) at a design temperature

(the set point value)

• by modulating the heat transfer rate of a heating coil (the modulated capacity)

• as the space heating load (the independent variable) changes with respect to time.

The control signal controls a motor, or an actuator, that modulates the capacity of fluid

equipment, that in turn, determines the value of the controlled property.

Control algorithm equationsThe difference between the set point and the controlled property is defined as the

error, i.e.,

(1.1)

Whereas the controlled property is related to the control signal and the error is related to

the controlled property, the control signal should be related to the error. If the control

signal is proportional to the error, i.e.,

(1.2)

then a zero control signal will be attained at zero error. If the rate of change of the control

signal with respect to time is proportional to the error, i.e.,

(1.3)

then a finite control signal will be attained at zero error. For at , integrating

Equation 1.3 yields

(1.4)

E XSP X–=

S kp E⋅=

dSdt------- ki E⋅=

S 0= t 0=

S ki E td⋅0

t

∫⋅=

Chapter 1: PREAMBLE - SYSTEM TYPES 27

Equation 1.4 will yield a finite control signal at zero error.

Whereas, as will be shown, some systems require one coefficient while others

require two coefficients, the control signal is determined by combining Equations 1.2 and

1.4, i.e.,

(1.5)

Equation 1.5 may be expressed in a finite difference format as

(1.6)

where

(1.7)

and

(1.8)

Although the error may be normalized, normalization is not necessary. In the current

analysis, the error has the units of the controlled property. The control signal is

expressed as a percentage between 0% and 100% for fluid control systems, and

between -100% and +100% for motion control systems. Both proportional and integral

(PI) coefficients are therefore dimensional having units of percent (%) per unit controlled

property. PI coefficient evaluations are based on physical properties of the controlled

system and the target response characteristic (error versus time).

Equations 1.5 and 1.6, with an additional derivative term, are often used as the

starting point for conventional feedback control theory. In the current theory, the

derivative term is not considered because it is not necessary.

SYSTEM TYPESFluid systems may be classified as flow systems, live storage systems, and dead

storage systems. Motion systems may be classified as kinematic systems and dynamic

systems. Control theory may also be used to solve equations.

Fluid system typesThe three types of fluid systems are:

S kp E⋅ k+ i E td⋅0

t

∫⋅=

Sn 1+ Kp En⋅ KI Eii 0=

n

∑⋅+=

KP kp=

KI ki δt⋅=

28 Chapter 1: PREAMBLE - SYSTEM TYPES

• flow systems,

• live storage systems, and

• dead storage systems.

The controlled property of a flow system is the property of a fluid flowing in a duct or pipe.

Live storage systems are systems in which the controlled property is that of a fluid in a

mass or an energy storage - a reservoir, a tank, or an enclosed space - with flow both in

and out. A live storage may be considered a ‘flow-through’ storage. Dead storage

systems are systems in which the controlled property is that of a fluid in a dead storage -

a reservoir, a tank, or an enclosed space - with flow in or out, but not in and out. The

controlled property of a dead storage system may be a liquid level or fluid property.

A system may be represented by a differential equation relating error and time to

system properties. Flow systems and a dead storage systems may be modeled with first-

order differential equations. Live storage systems may be modeled with second-order

differential equations.

Motion system typesThe two types of motion systems are:

• kinematic, and

• dynamic.

A kinematic system may be represented with a first-order differential equation. A

kinematic system may be one that controls the position of an object by modulation of its

velocity. A dynamic system may be represented with a second-order differential

equation. A dynamic system may be one that controls the velocity of an object by

modulation of an applied force, or torque, thereby modulating its acceleration.

EquationsFeedback control is essentially a method of solution of equations. Control theory

may be used to determine the independent variable of a function for a specified value of

the dependent variable and to determine the intersection point of two functions, i.e., to

solve equations. The methods presented are not intended to replace conventional

numerical analysis methods such as Newton’ method. The methods presented are not

intended to be superior to any conventional method. The methods are presented to

illustrate a perspective of control theory.

Chapter 1: PREAMBLE - INDEPENDENT VARIABLE / LOAD 29

Mathematical perspectiveFrom a mathematical perspective, systems may be classified by the order of the

differential equation representing the system. First-order and second-order systems are

represented by first-order and second-order differential equations, respectively. First-

order and second-order systems may be controlled by a single control loop, i.e, a single

set of PI coefficients. First-order and second-order differential equations are of a format

that is easily solvable, i.e., a linear differential equation with constant coefficients. (A

linear differential equation is one that contains the dependent variable and its derivatives

to the first degree.)

Some systems requiring representation by non-linear differential equations may

be approximated, i.e., ‘linearized’, by linear differential equations if the term that renders

the equation non-linear is considered as a component of the load, i.e., an independent

variable. An example of such a system is presented in the section titled SYSTEM

EQUATIONS of Chapter 7: VELOCITY CONTROL BY FORCE MODULATION.

Some systems requiring representation by higher-order differential equations may

be controlled by two control loops in series, i.e., two sets of PI coefficients. In series

control loops, one control loop determines the set point for the second loop. An example

of a system requiring two control loops in series is the position-control-by-force-

modulation system presented in Chapter 8: POSITION CONTROL BY FORCE

MODULATION. Such systems may be used for materials handling and for transporters.

A transporter, e.g., a SegwayTM Human Transporter, is a device consisting of an

inverted pendulum and a pair of motorized wheels, as presented in Chapter 10:

INVERTED PENDULUM and in Chapter 11: MOTORIZED WHEEL.

INDEPENDENT VARIABLE / LOADOne of the variables in a system is an independent variable. The independent

variable is generally referred to as the load. The load is an indeterminate variable. The

load variation is generally random. As previously noted, a feedback control system is

required because the load is indeterminate.

Ideally, the load is a function of time only and is not a function of the controlled

property. In some cases, the load may be a weak function of the controlled property. In

such cases, variations due to controlled property variations, i.e., error variations, may be

considered as a component of the load and neglected in the calculation of PI coefficients.

30 Chapter 1: PREAMBLE - SYSTEM ORDER

If load variations due to controlled property variations are considered as a component of

the load and neglected in the calculation of PI coefficients, some deviation from the

target (desired) response will result.

The load is an independent variable and is considered representative of any

independent variable. A variable set point is equivalent to a variable load.

SYSTEM ORDERThe order of a system is that of the differential equation representing the system.

A first-order system is represented by a first-order differential equation. A second-order

system is represented by a second-order differential equation. Some higher order

systems may be modeled as series control loops. In a series control loop, a pseudo

control loop determines the set point for an actual control loop.

DIFFERENTIAL EQUATIONSThe solution of a differential equation consists of two components - a

complementary component and a particular component. The complementary component

is the transient component. The particular component is the steady-state component.

The complementary component is the solution of the homogeneous component of the

differential equation. The PI coefficients are determined from a target complementary

component. The complementary component is independent of the load on the system.

The particular component is dependent on the load on the system, and the variation (if

any) of the set point. (Reference Chapter 2: SYSTEMS.)

For example, a second-order differential equation may be written as

(1.9)

For control problems, the dependent variable, , is the error, the independent variable,

, is the time, and the function of time, , is the load. The load is considered as an

independent variable. The complete solution is

(1.10)

The complementary component of the solution, , is the solution of the

homogeneous component, i.e.,

a d2Edt2---------- b dE

dt------- c E⋅+⋅+⋅ f t( )=

E

t f t( )

E Ecomp Epart+=

Ecomp

Chapter 1: PREAMBLE - DIFFERENTIAL EQUATIONS 31

(1.11)

The particular component of the solution, , is

(1.12)

The coefficients in Equation 1.12 are determined by substitution into Equation 1.9.

The complementary component of the solution represents the solution for a

system with a constant load for or a step change of the load at . A step

change of the load represents a system start-up condition. System start-up generally

represents the most severe operating condition. The PI coefficients required to

implement a control algorithm will be determined from the complementary component.

The particular component of the solution represents the solution for a system after

an initial error, or error due to a step change of a load, has been attenuated. The

particular component is the steady-state component due to an oscillating load or other

variable (e.g., the set point). The particular component is used to investigate system

response load oscillating in a sinusoidal manner. System response to an independent

variable oscillating in a sinusoidal manner is indicative of system response to a load

oscillating in a random manner.

First-order differential equationsIt is useful to examine some of the characteristics of first and second differential

equations to determine the number of coefficients required to implement a PI control

system. The differential equations will be written in terms of error ( ) as the dependent

variable and time ( ) as the independent variable.

A homogeneous first-order differential equation may be represented by:

(1.13)

The solution to the equation is:

(1.14)

This solution represents the system response. The system response - error versus time -

expressed by Equation 1.14 (i.e., shape of the curve) is determined by a single

a d2Edt2---------- b dE

dt------- c E⋅+⋅+⋅ 0=

Epart

Epart C0 f t( )⋅ C1df t( )

dt------------⋅ C2

d2f t( )

dt2--------------⋅ .... Ck

dnf t( )

dtn--------------⋅+ + + +=

t 0≥ t 0=

E

t

dEdt------- 1

τ--- E⋅+ 0=

E E0 etτ--–

⋅=

32 Chapter 1: PREAMBLE - DIFFERENTIAL EQUATIONS

parameter, , the time constant. The time constant will be a function of several system

properties and one arbitrary coefficient. Only one arbitrary coefficient is required to

obtain any desired time constant. The integral coefficient will yield a finite control signal

at zero error. The proportional will yield a zero control signal at zero error. For some

systems, the integral coefficient is required and the proportional coefficient is set to zero.

For other systems, the proportional coefficient is required and the integral coefficient is

set to zero.

Integral control may be used in some fluid control systems. If the pressure of a fan

is controlled by the modulation of its speed, its can be shown that the system may be

represented by a first-order differential equation and only an integral coefficient is

required. Proportional control may be used in other fluid control systems. If the level of

liquid in a dead storage is controlled by the modulation of inflow rate, its can be shown

that the system may be represented by a first-order differential equation and only a

proportional coefficient is required.

Proportional control may be used in certain motion control systems. If the position

of a device is controlled by the modulation of its velocity, its can be shown that the

system may be represented by a first-order differential equation and only a proportional

coefficient is required. When the device is at its target - set point - position, the error is

zero. If the device must stop at the target - set point - position, its the velocity and the

control signal must equal zero. Therefore, only a proportional constant is required.

Response characteristics

Figure 1.1 shows the error ratio as a function of as per Equation 1.14. Error

attenuations of 63.2%, 99.3% and 99.9% are obtained for ratios of 1, 5, and 7,

respectively, as shown in Table 1.1. Although 63.2% error attenuation is often specified,

99.9% error attenuation is most significant because it represents complete effective error

attenuation.

τ

t τ⁄

t τ⁄


Figure 1.1: Error response characteristic of a first-order system.

Table 1.1: Error attenuation, first-order system.

Attenuation:

1 0.6321 = 63.21%

5 0.9932 = 99.32%

7 0.9991 = 99.91%

0 1 2 3 4 5 6 7 8 9 10

t / τ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0E

/ E 0

tτ-- 1 E

E0------– 1 e

tτ--–

–=


Second-order differential equationsA homogeneous second-order differential equation may be represented by:

(1.15)

The roots of the auxiliary equation are:

(1.16)

The solution may be underdamped, critical, or overdamped. The solution represents the

system response. A system response may be considered underdamped, critical, or

overdamped according to the solution.

For an underdamped response, the error is an exponentially attenuated sinusoidal

function of time and, therefore, it will oscillate about the set point value. For critical

response, the error is an exponentially attenuated algebraic function of time that attains a

zero error (within an acceptable tolerance) at a finite time. For an overdamped response,

the error is an exponentially attenuated function of time that eventually yields a zero error

as the time approaches infinity. The optimal system responses are therefore critical

responses or underdamped responses. Overdamped responses are not suitable for

system control. As noted, a system response consists of two components - a

complementary component independent of the load and a particular components due to

the load. An underdamped response may be more desirable than a critical response in

some cases because an underdamped response may attenuate an error more rapidly

than a critical response.

Critical response

For a critical response

(1.17)

The roots of the auxiliary equation, for a critical response, are therefore

(1.18)

where

(1.19)

a d2Edt2---------- b dE

dt------- c E⋅+⋅+⋅ 0=

r1 r2, b2 a⋅-----------– b

2 a⋅-----------

2 ca---–±=

b2 a⋅-----------

2 ca---=

r1 r21τ---–= =

1τ--- b

2 a⋅----------- ρ= =


The solution may be written as

(1.20)

The system response - error versus time - expressed by Equation 1.20 (i.e., the shape of

the curve) is determined by the time constant, , and the condition for a critical solution

expressed by Equation 1.17. The time constant, , will be a function of system properties

and one arbitrary coefficient. The condition for a critical solution (Equation 1.17) will be a

function of system properties and a second arbitrary coefficient. These two arbitrary

coefficients are the PI coefficients. Therefore, both PI coefficients are required to

determine the response of a second-order system.


Defining a shape factor, , as

(1.21)

enables Equation 1.20 to be re-written as

(1.22)

Figure 1.2, on the next page, shows the error ratio as a function of for various values

of the shape factor, as per Equation 1.22. Figure 1.2 illustrates that a critical response

may have more than one ‘shape’. The shape is determined by the value of the shape

factor . The value of the shape factor is determined by boundary conditions and

additional considerations.

The shape factor may be evaluated in terms of the initial rate of change of the

error with respect to time. Differentiating Equation 1.22 and evaluating the derivative at t

= 0 yields

(1.23)

The constant E0 is evaluated from the initial error. The constant Ec is related to the initial

rate of change of the error with respect to time, i.e.,

E E0 Ec t⋅+( ) etτ--–

⋅=

τ

τ

R

R τEcE0------⋅=

EE0------ 1 R t

τ--⋅+

etτ--–

⋅=

t τ⁄

R

R

R 1 τE0------ dE

dt-------

0⋅+ τ

EcE0------⋅= =


(1.24)

Figure 1.2: Error response characteristics for second-order systems with critical responses.

There are three values of the shape factor of special interest, , , and

.

EcE0τ

------ dEdt-------

0+=

0 1 2 3 4 5 6 7 8 9 10

t / τ

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

E /

E 0

shape factor21.510.50-0.5-1-1.5-2

R 0= R 1=

R 1–=


For

(1.25)

Equation 1.22 becomes

(1.26)

For , overshoot is not present, however for , overshoot is present.

For

(1.27)

Equation 1.22 becomes

(1.28)

For , it can be shown that

(1.29)

corresponds to

(1.30)

If a zero signal is required at equilibrium (i.e., when the controlled property equals the set

point), then it is required that .

Whereas an exact value of the initial rate of error change with respect to time, and

thus the shape factor, are generally indeterminable, values of error attenuation may be

estimated for values of the shape factor between -2 and 2. It is evident from Figure 1.2

that within this range of the shape factor, approximately 99.3% and 99.9% error

attenuation will be attained for ratios of 5, and 7, respectively. Table 1.2, on the next

page, shows data for .

R 0=

Ec 0=

EE0------ e

tτ--–

=

R 0≥ R 0<

R 1=

dEdt-------

00=

EE0------ 1 t

τ--+

etτ--–

⋅=

R 1–=

E td⋅t 0=

∞

∫ 0=

Eii 0=

∞

∑ 0=

R 1–=

t τ⁄

R 1=


Table 1.3, shows data for R = -1 and for negative error ratios, E/E0 < 0, i.e., overshoot.

A 99.9% error attenuation is most significant because it represents complete effective

error attenuation.

Underdamped response

For an underdamped response

(1.31)

The roots of the auxiliary equation, for an underdamped response, are therefore

(1.32)

where

(1.33)

and

(1.34)

Table 1.2: Error attenuation, second-order systems, R = 1.

Attenuation:

7 0.9927 = 99.27%

9 0.9988 = 99.88%

Table 1.3: Error attenuation, second-order systems, R = -1.

Attenuation:

7 0.9945 = 99.45%

9 0.9990 = 99.90%

tτ-- 1 E

E0------– 1 1 t

τ--+

etτ--–

⋅–=

tτ-- 1 E

E0------– 1 1 t

τ--–

etτ--–

⋅–=

b2 a⋅-----------

2 ca---– 0<

r1 r2, ρ– i β⋅±=

ρ 1τ--- b

2 a⋅-----------= =

β ca--- b

2 a⋅-----------

–2

= 0>


The solution may be written as

(1.35)

The constant, , and the phase angle, , are determined by boundary conditions.

The condition for an underdamped response may be expressed in terms of a

logarithm decrement as

(1.36)

The solution, in terms of the logarithmic decrement and the oscillation period, may be

written as

(1.37)

where

(1.38)

and

(1.39)


An underdamped response is best characterized by the oscillation period and an

amplitude ratio of consecutive opposite-sign, or same-sign, peaks. The logarithmic

decrement is related to the amplitude ratio of consecutive opposite-sign peaks by

(1.40)

and to the amplitude ratio of consecutive same-sign peaks by

(1.41)

Attenuation

Error attenuation for underdamped response characteristics is best viewed from

the perspective of amplitude ratios. The amplitude ratio of consecutive same-sign peaks

is associated with one oscillation period. The amplitude ratio of consecutive opposite-

E Ec e ρ– t⋅ β t ϕ+⋅( )sin⋅ ⋅=

Ec φ

2 π⋅δ

----------- βρ--- 4 a c⋅ ⋅

b2------------------ 1–= =

E Ec eδϒ---– t⋅ 2 π⋅

ϒ----------- t ϕ+⋅

sin⋅ ⋅ Ec etτ--– 2 π⋅

ϒ----------- t ϕ+⋅

sin⋅ ⋅= =

ρ δϒ--- 1

τ---= =

β 2 π⋅ϒ

-----------=

δ 2 1Aos---------ln⋅=

δ 1Ass---------ln=

40 Chapter 1: PREAMBLE - SELECTION OF TARGET RESPONSE

sign peaks is associated with one-half an oscillation period. The amplitude ratio for

oscillation periods is or .

For 99.91% error attenuation

(1.42)

Noting that

(1.43)

The interval for 99.9% attenuation is therefore

(1.44)

SELECTION OF TARGET RESPONSEThe target response of a first-order system must be an exponential function. The

target response of a second-order system may be either critical or underdamped.

Although an underdamped response may yield more rapid error attenuation that a critical

response, an underdamped response is more susceptible to adverse effects.

Furthermore, the determination of PI coefficients for an underdamped response requires

more detailed system analysis than that required for a critical response. If a critical

response is selected, no further parameters require selection. If an underdamped

response is selected, the oscillation period and the amplitude ratio of consecutive

opposite-sign peaks must also be selected.

SYSTEM CLASSIFICATIONSSystems may be classified as single loop systems according to the required PI

coefficients and as multiple loop systems.

Single loop system classifications are: I, PI, and P. System storage is one factor

determining the required coefficients. A first-order system may, or may not, contain

storage. A second-order system always contains storage. Storage depends on the type

of system. For fluid systems, storage may consist of either heat capacity, mass, or mass

concentration of mixture component. For motion systems, storage may consist of mass

and/or mass moment of inertia. Accelerating or decelerating mass results in a kinetic

N

AssN Aos

2 N⋅

AssN99.9 Aos

2 N99.9⋅1 0.9991– 0.0009= = =

0.0009( )ln 7.–=

N99.971

Ass---------ln

-------------- 3.51

Aos---------ln

---------------= =

t99.9 N99.9 ϒ⋅ 7 ϒ1

Ass---------ln

--------------⋅ 3.5 ϒ1

Aos---------ln

---------------⋅= = =

Chapter 1: PREAMBLE - SYSTEM CLASSIFICATIONS 41

energy increase or decrease, respectively. Raising or lowering mass results in a

gravitational potential energy increase or decrease, respectively. Mass is also associated

with inertial force and with gravitational force. Mass moment of inertia is also associated

with inertial torque.

System classifications are define as follows.

• I systems are systems requiring an integral coefficient only ( ). These

systems are first-order systems requiring a control signal (%) magnitude greater than,

or equal to, zero at zero error, i.e., for at .

• PI systems are systems requiring proportional and integral coefficients. These

systems are second-order systems requiring a control signal (%) magnitude greater

than, or equal to, zero at zero error, i.e., for at .

• P systems are systems requiring a proportional coefficient only ( ). These

systems are first-order systems requiring a control signal (%) magnitude equal to

zero at zero error, i.e., for at .

If only one coefficient is required, the second coefficient may be set equal to zero or the

control algorithm may be simplified.

Multiple loop systems are classified as follows.

• Systems with series control loops - Two control loops are in series when the

controlled property of one (pseudo) control loop is the set point of a second (actual)

control loop.

• Systems with parallel control loops - Two, or more, control loops are in parallel

when operating independently and simultaneously to control two, or more,

independent or interdependent system properties.

The control loops in a system with series control loops may be I, PI, or P systems.

Series control loops are interdependent. The control loops in a system with parallel

control loops may be I, PI, P or series control loops. Parallel control loops may be

interdependent or independent. If, for example, a cooling coil is used to control both

temperature and humidity simultaneously, the temperature and humidity control loops

are interdependent because of the performance characteristics of the cooling coil.

(Reference Appendix C: CHILLED WATER COOLING COILS.) If, for example, two

coordinates of the position of an object are controlled by two control loops, the control

loops may be independent.

KP 0=

S 0≥ E 0=

S 0≥ E 0=

KI 0=

S 0= E 0=

42 Chapter 1: PREAMBLE - PARAMETERS EFFECTING PI COEFFICIENTS

PARAMETERS EFFECTING PI COEFFICIENTSThe PI coefficients required to implement a control algorithm are functions of one

or more of the following system parameters to be defined:

• property sensitivity or

• property rate sensitivity,

• complete response interval, and

• the signal update interval.

The property sensitivity is a system property that relates a change of the controlled

property to a change of the control signal. The property rate sensitivity is a system

property that relates a rate of change of the controlled property with respect to time to a

change of the control signal. The complete response interval is the interval required for a

control device to completely respond to a 100% change in the control signal. It is the

interval required for a valve to travel from limit-to-limit (e.g., fully closed to fully opened)

or the interval required for a motor to accelerate to its maximum velocity. The signal

update interval is the interval at which the control signal sent to a control device (actuator

or drive) is updated.

The system classification - I, PI, or P - determines whether property sensitivity or

property rate sensitivity is applicable.

Values of PI coefficients are not dependent on the value of the load or

independent variable. If values of PI coefficients were dependent on the value of the load

or independent variable, different coefficients would be required for different operating

conditions. This is clearly not the case. An equation for the determination of PI

coefficients should therefore exclude the load or independent variable.

Although the control algorithm may be the same for fluid and motion systems, the

equations for the PI coefficients are different. The equations required to determine PI

coefficients analytically are presented in the following sections. Some equations are

repeated in each section so that each section is complete in itself.

Error attenuation parametersIn addition to the system properties, the PI coefficients are based on attenuation

parameters. The attenuation parameters determines the rate of error attenuation.

Maximum values of the attenuation parameters are based on the assumption that an

error requiring a 100% change of the control signal (e.g., 0% to 100%) is fully attenuated

Chapter 1: PREAMBLE - PERFORMANCE EVALUATION 43

with a single signal update interval. For an error to be fully attenuated with a single signal

update interval, the signal update interval and the complete response interval must be

equal. The criterion that an error be fully attenuated within a single signal update interval

implies that the transient (instantaneous) error will be zero after a single signal update

interval. This criterion does not imply that the steady-state error will be zero. The error

may continue to vary with time.

Although equations for the PI coefficients are initially derived for systems having a

signal update interval equal to the complete response interval, they are applicable to

systems having a signal update interval less than the complete response interval. It will

be shown that the equations for proportional coefficients are independent of the signal

update interval and that the equations for integral coefficients are dependent on the

signal update interval. Whereas integral coefficients, which multiply the sum of the

errors, are dependent on the signal update interval, control signals determined using the

PI equations are applicable systems having a signal update interval less than the

complete response interval. Actual response characteristics will differ slightly from the

target response characteristics due to error variation with time.

PERFORMANCE EVALUATIONThe objective of a control system is to maintain a controlled property at a required

set point value as an independent variable - the load or the set point - varies. The

controlled property depends on a modulated capacity. The modulated capacity is the

parameter directly effected by the control signal. The ability of the modulated capacity to

follow - i.e., to track - variations of an independent variable (load or set point) determines

the error. The error versus time may be considered a measure of system response. The

following criteria may be used to evaluate the response of a system:

• tracking of a step change in the load or set point, and

• tracking of a periodically oscillating, load or set point.

For a critical target response, the interval required for 99.9% error attenuation -

determined from the complementary component of the solution - may be used as a factor

for performance evaluation. For an underdamped target response, the oscillation period

and the amplitude ratio of consecutive opposite-sign peaks at start-up may be used for

performance evaluation. As noted, the complementary component of the solution

represents the solution to system with a constant load for or a step change of the t 0≥

44 Chapter 1: PREAMBLE - PERFORMANCE EVALUATION

load at . Performance factors related to system performance with independent

variables oscillating in a sinusoidal manner may be determined from the particular

component of the solution for specific systems. In practice, an independent variable may

oscillate in a random manner. However, an independent variable oscillating in a

sinusoidal manner is used as a reference for performance evaluation.

A change of the control signal in a feedback control system is motivated by an

error, i.e., a differential between the values of a set point and a controlled property. If a

load, or other independent variable, oscillates, the control signal must oscillate to track

the oscillations. The control signal oscillations are motivated by error oscillations. The

controlled property is related to the independent variable. Therefore amplitude of the

error oscillations is related to the amplitude of the oscillations of the independent

variable.

A critically damped second-order system may yield a more robust response - i.e.,

more resistant to random variations of an independent variable - then an underdamped

system. If a system such as a tuning fork, or a piano string, possesses a natural

frequency, it will vibrate if excited by an impulse. Likewise, an underdamped system

might oscillate if excited by an impulse. Therefore a critical damped system might be

more robust than an underdamped system.

The ability of a system to track load or set point variations is dependent not only

on the PI coefficients but also on the following system properties:

• complete response interval (valve limit-to-limit travel time, motor acceleration time,

etc.),

• signal update interval, and

• resolution of electronic components, etc.

Unaccounted thermal or mass inertia in fluid systems, and unaccounted frictional forces

in motion systems, would also effect tracking.

PI coefficients are not intended to compensate for system deficiencies. If, for

example, the complete response interval of a valve is too long for a desired rapid

response, the PI coefficients cannot compensate for the excessive complete response

interval of the valve.

t 0=

Chapter 2: SYSTEMS - NOMENCLATURE 45

Chapter 2: SYSTEMS

An overview of systems from the perspective and assumptions used to formulate

the theory are presented in this chapter. A basic proportional-integral (PI) control

algorithm, including windup prevention, is also presented.

NOMENCLATUREerror

integral coefficient

proportional coefficient

control signal (percentage)

controlled property

set point of a controlled property

Subscriptssignal update index

current signal update index

previous signal update index

next signal update index

SYSTEM VARIABLESGenerally, there are four system variables. They are

• the controlled property,

• the modulated capacity,

• the independent variable, load or set point, and

• the control signal

When the controlled property equals a specified set point, all four variables are in

equilibrium.

Controlled propertyThe controlled property is the property of a system that is required to be at a

specified, set point, value. It is the dependent variable of the system. Examples of

controlled properties of fluid systems are temperature, pressure, liquid level, carbon

E

KI

KP

S

X

XSP

i

n

n 1–

n 1+

46 Chapter 2: SYSTEMS - SYSTEM VARIABLES

dioxide concentration, and specific humidity. Examples of controlled properties for

motion systems are position and velocity.

Modulated capacityThe modulated capacity is the system property directly effected by the control

signal. A modulated motor, or a modulated actuator, will vary a system capacity

according to the control signal. For fluid systems, a modulated capacity may be a

volumetric flow rate, a mass flow rate, an energy (heat) convection rate, a heat transfer

rate, a heat capacity rate (the product of mass flow rate, specific heat, and temperature,

or the product of mass flow rate and enthalpy), a pressure, or the proportion of a mixed

fluid. The modulated capacity may also be a parameter that is directly related to the

capacity. For example, a modulated temperature is directly related to the heat capacity

rate and may therefore be considered as modulated capacity. For motion systems, a

modulated capacity may be a linear velocity, an angular velocity (speed), a force, a

torque, or a property of an electric supply - voltage, current, or frequency.

Independent variableThe independent variable is imposed on a system; it is not determined by the

system. The independent variable may be a load or a set point.

The load is the independent variable of the system. Examples of loads for fluid

systems are heat transfer rates such as heating loads and cooling loads, flow rates,

valve resistances, energy content (temperature), moisture content (specific humidity),

moisture generation rates, and carbon dioxide concentration, carbon dioxide generation

rates. Examples of loads for motion systems are force and torque. Some systems may

be considered to have two loads while other systems may be considered to have no

load. For example, a mixing valve used to blend hot and cold liquids to obtain a specified

mixed fluid temperature may be considered to have two loads because the temperature

of each inlet fluid (hot and cold) may be an independent variable, hence two loads. A

motion system having no load is presented in Chapter 6: POSITION CONTROL BY

VELOCITY MODULATION.

The set point is the required value of the controlled property. The set point may

be a constant or an independent variable. In systems controlled by series control loops,

one (pseudo) control loop determines a variable set point for a second (actual) control

loop. If it is desired to maintain the temperature of a space at a different values during the

Chapter 2: SYSTEMS - COMPONENTS 47

day and night, the set point is a variable with respect to time. Motion systems with a

variable set point are presented in Chapter 8: POSITION CONTROL BY FORCE

MODULATION and in Chapter 10: INVERTED PENDULUM.

Control signalThe control signal is a variable determined by the control program. The control

signal determines the modulated capacity. The modulated capacity and the load and/or

set point determine the controlled property.

TerminologySystem-specific terminology is preferred to generic terminology. When

considering a specific system, it is more significant to use specific terms such as

‘controlled temperature’ or ‘controlled velocity’ rather than the generic ‘controlled

property’. Similarly, when considering a specific system, it is more significant to use

specific terms such as ‘modulated flow rate’ or ‘modulated velocity’ rather than the

generic ‘modulated capacity’.

Although it is natural for engineers to generalize theory and terminology,

generalization has been kept to a minimum. Some parameter that are conceptually

similar, but not exact, are termed identically but are defined differently for different

systems. Concepts should be interpreted in view of the specific system under

consideration. Generalized definitions are not rigorous.

COMPONENTSA control system has the following components:

• a sensor,

• a computer containing

• an analog-to-digital (A-D) converter,

• a processor,

• a software program,

• a digital-to-analog (D-A) converter,

• a drive

• a motor or an actuator, and

• a fluid system or a motion system.

A schematic diagram of a control system is shown in Figure 2.1.

48 Chapter 2: SYSTEMS - COMPONENTS

Figure 2.1: Schematic diagram of a control system.

The sensor monitors the controlled property and produces an analog signal such

as 0 to 10 volts DC, 0 - 5 volts DC, -10 to +10 volts DC, 4 - 20 milliamperes D-C, etc. The

analog signal is the input signal to the A-D converter.

The controller contains the micro-processor, the A-D and D-A converters, and

the software program. It relates an output signal to the input signal according to the

algorithm of the software program. The micro-processor performs the actual

computations according to the software program. The controller may be a dedicated

computer or a conventional desktop computer.

The A-D converter produces a digits signal in ‘counts’ proportional to the analog

input signal. A count is an integer. The number of counts depends on the number of bits

in the converter chip. For example, a 12-bit chip will yield a number between 0 and 4095

since 212 is equal to 4096, a 16-bit chip will yield a number between 0 and 65535 since

212 is equal to 65536.

CONTROLLER

ANALOG-TO-DIGITALCONVERTER

DIGITAL-TO-ANALOGCONVERTER

FEEDBACKSIGNAL

CONTROLLEDPROPERTY

DRIVE

ELECTRIC POWER

CONTROL SIGNAL

SOFTWARE PROGRAM

INPUT DATA

MOTOR / ACTUATOR

FLUID or MOTIONSYSTEM

FLUID SYSTEM - fans, dampers, pumps, valves, etc.

MOTION SYSTEM - wheels, mechanism, transmission - gears, belts, etc.

or

SENSOR

MICRO-PROCESSOR

Chapter 2: SYSTEMS - COMPONENTS 49

The software program contains the control algorithm. The algorithm relates a

digital output signal to the digital input signal from the A-D converter using specified data.

The digital output signal in an integer denoting counts. The specified data consists of the

proportional and integral (PI) coefficients, control signal limits, etc. Fluid systems use

relatively long signal update intervals while motion systems use relatively short signal

update intervals. For fluid systems, a signal update interval requires specification. For

motion systems, the signal update interval must be determined by experimental

calibration.

The control signal is initially determined by the software as a floating point

number - a percentage. The floating point number is then converted to an integer -

counts. The D-A converter converts the counts to an analog control signal.

The D-A converter produces an analog output signal proportional to the counts

as determined by the software program. The analog signal may be 0 to 10 volts DC, 0 - 5

volts DC, -10 to +10 volts DC, 4 - 20 milliamperes DC, etc. This analog signal is sent to

the drive.

The drive converts the electrical power supplied by an electric utility to produce

an output power supply proportional to the analog input signal. The electric power

supplied may be, for example, 120 volts 60 Hz single phase AC or 550 volts 60 Hz three

phase AC. For AC motors, the drive may be a variable frequency inverter that converts

the constant frequency of the supplied power to variable frequency output power. For DC

motors, the drive may be a rectifier that converts AC to DC. The output power is

connected to the system motor or actuator. A drive may use an internal feedback control

system to relate the output power to the input signal.

The motor or actuator determines the modulated capacity. A motor may be

rotational or positional. The speed of a rotational motor is proportional to the power input

from the drive. An actuator is a positional motor or motor-driven device. The position of

the actuator is proportional to the control signal supplied by a drive. The position may be

a linear distance or an angular position. An actuator is used to open and close valves.

A fluid system consists of equipment such as valves, fans, pumps, heat

exchangers, storages, etc. The modulated capacity is determined by motor speed or

valve actuator position.

A mechanical system consists of a motor, a mechanism, wheels, a transmission,

etc. The transmission may consist of gears and/or a belt or screw drive.

50 Chapter 2: SYSTEMS - INTERVALS

INTERVALSThere are two significant intervals related to PI control - the complete response

interval and the signal update interval.

Complete response intervalThe complete response interval is the interval required for a complete response of

a modulated capacity to a maximum change of the control signal, i.e., from 0% to 100%.

The complete response interval is primarily a property of the modulated device. A

complete response is one in which a motor speed or an actuator (including a valve

actuator) position corresponds to the control signal. A complete response for a motor

implies attaining constant speed, i.e., zero acceleration. A complete response for an

actuator implies attaining constant position, i.e., zero velocity. Additional discussion

appears in the section titled INTERVALS in Chapter 16: APPLICATION OF THEORY.

Signal update intervalThe signal update interval is the interval at which the control signal sent to a drive

is updated. The control signal may be updated intermittently or continuously. The signal

update interval essentially includes the following sequential procedures:

1. sample the controlled property with the A-D converter,

2. calculate the control signal,

3. output the control signal with the D-A converter, and

4. an optional coast interval.

During the coast interval, the program continues running, procedures 1 and 2 are

bypassed, but procedure 3 continues to output the control signal unchanged. A coast

interval is included in the signal update interval for intermittent updating but omitted for

continuous updating. If the signal is updated continuously, the determination of the signal

update interval requires calibration because it depends on the speed of the computer

and software. The minimum signal update interval is substantially greater than that

implied by the ratings of the A-D and D-A converters used. A-D and D-A converters may

have ratings in the range of 100 KHz to 1 MHz. The signal update interval is only partially

related to these ratings. The minimum possible signal update intervals depends on the

control program algorithm because calculations greatly reduce the overall speed.

The controlled property may be sampled more than once during a signal update

interval in order to

Chapter 2: SYSTEMS - ASSUMPTIONS 51

• numerically filter the controlled property if noise is present, and/or

• monitor system performance when the control signal is updated intermittently.

For fluid systems, the signal update interval is generally of the order of seconds

and is specified for intermittent control signal updating. For motion systems, the signal

update interval is generally of the order of milliseconds and is determined by calibration

for continuous signal updating.

Some systems have a delay interval between a change of the control signal and

the response of the modulated capacity. The magnitude of the signal update interval

should exceed the magnitude of the delay interval.

ASSUMPTIONSThe theory for the determination of formulae for PI coefficients assumes the

following:

• the modulated capacity is a linear function of the control signal,

• the modulated capacity responds instantaneous, without a drive delay interval, to a

control signal,

• the control signal has no deadband, and

• the system operates at the minimum, or the maximum, control signal limits only

instantaneously, not for an extended period.

In actual systems, valves require finite intervals to open and close and motors require

finite intervals to accelerate and decelerate. The characteristics of a motor depend not

only on its mechanical and electrical properties, but also on the characteristics of its

drive. The drive will have an internal control system of its own. A delay interval may exist

in a drive. A deadband may be present in a drive. A change of a property or system

capacity in response to a change in the control signal is generally variable within the

operating range of the system. Few systems are inherently linear. The application of the

theory will consider the characteristics of actual systems in determining values of

properties to be used for the calculation of PI coefficients.

Systems may be viewed from two perspectives, as continuous processes or as a

series of finite processes. Each perspective has a specific function in theoretical

analysis. Although the response of an actual system is expected to differ from the

theoretically predicted response, the theory provides a basis for the analytical

determination of PI coefficients.

52 Chapter 2: SYSTEMS - THE PI CONTROL EQUATION

THE PI CONTROL EQUATIONThe PI control equation relates the control signal to the error using PI coefficients

by

(2.1)

where

(2.2)

Differentiating Equation 2.1 yields

(2.3)

where

(2.4)

Equation 2.1 may therefore be re-written without the error summation term as

(2.5)

In order to distinguish between Equation 2.1 and Equation 2.5, the format of Equation

2.1 will be referred to as the ‘summation format’ and the format of Equation 2.5 will be

referred to as the ‘differential format’. The summation format is the fundamental format

and is used for the derivation of formulae for PI coefficients. PI control algorithms may be

base on the summation format and on the differential form.

Each format has advantages and disadvantages. The summation format is

applicable to all types of systems - I systems, PI systems, and P systems. The

differential format requires special consideration of P systems. The differential format is

not applicable to P systems under all operating conditions. It is therefore recommended

that PI control algorithms for P systems be based on the summation format, not on the

differential format. The differential format does not require windup protection and

facilitates the implementation of variable PI coefficients. If the magnitude of the control

signal remains at one of its limits (e.g., 0%, -100%, 100%) for a finite period, system

response characteristics may differ slightly depending on the format of the PI control

algorithm used - summation format or differential format. This phenomenon applies only

to I systems and PI systems. PI control algorithms based on the summation format and

on the differential format are detailed in Chapter 13: PI CONTROL ALGORITHMS.

Special consideration of P systems is included in this chapter.

Sn 1+ KP En⋅ KI Eii 0=

n

∑⋅+=

En XSP Xn–=

dS Sn 1+ Sn– KP dE⋅ KI En⋅+= =

dE En En 1––=

Sn 1+ Sn KP dE⋅ KI+ En⋅+=

Chapter 2: SYSTEMS - CONTROL PROGRAMS 53

CONTROL PROGRAMSA PI control algorithm is only one component of a control program. A control

program must also contain the following.

• An algorithm to convert the output of the A-D converter to the units required by the

program.

• An algorithm to convert the control signal percentage to the input required by the D-A

converted, i.e. counts.

• Safety limits to either shut down the controlled system or activate an alarm in case

unsafe, or unforeseen, conditions arise. Safety limits are required in case of

equipment malfunction, coding error in a program, data input error, etc.

• Numerical filtering of the monitored properties, including the controlled property, to

eliminate, or reduce, the effects of random noise.

• Data recording of system parameters at fixed intervals to allow system analysis.

CALCULATION OF PI COEFFICIENTSThe calculation of PI coefficients required to implement a control system is based

on the following system properties.

• Sensitivity - the relationship of the controlled property to the control signal.

• Complete response interval - the interval required for the modulated capacity to

respond completely to the maximum (100%) change of the control signal. This

interval should account for any time delay in the drive.

• Signal update interval - the interval at which the control signal is updated. The

magnitude of this interval exceed any time delay in the drive.

Formulae for calculation of PI coefficients are presented in the following chapters. It

should be noted that, for a specific system, there is no unique set of PI coefficients that

will yield satisfactory performance characteristics. There are many - in fact, an infinite

number - of sets of PI coefficient that will yield satisfactory performance characteristics.

PREREQUISITES FOR PI CONTROLFeedback control with a PI control algorithm is suitable for systems satisfying the

following criteria:

• the sensitivity must not change signs within the operation range, and

• the modulated capacity must be a function of the control signal only, i.e., the

54 Chapter 2: SYSTEMS - ANALOGY BETWEEN LINEAR AND ROTATION MOTION SYSTEMS

modulated capacity must not be a function of the load.

For satisfactory performance, all of the follow components must be compatible:

• the modulated capacity relative to the anticipated load,

• the complete response interval of the equipment,

• the signal update interval,

• the PI control algorithm, and

• the PI coefficients.

The magnitude of the modulated capacity should equal, or exceed, the maximum

magnitude of the anticipated range of variation of the load(s) and/or independent

variable(s). The complete response interval and the signal update interval should be

sufficiently low to allow the system to change as rapidly as required to track changes of

the load(s) and/or independent variable(s). The PI control algorithm should include code

for safety limits and windup prevention, if required. A control system is not intended to

compensate for system deficiencies.

ANALOGY BETWEEN LINEAR AND ROTATION MOTION SYSTEMSThe equations for motion systems are generally presented, in this book, for a

linear motion. Equations for a rotational motion system may be obtained using the

analogy between linear motion and rotational motion in Table 2.1 on the next page. A

more detailed analogy is presented in Table 27.1 in the section titled ESSENTIAL

EQUATIONS of Chapter 27: MOTION SYSTEM PROBLEMS.

Table 2.1: Analogy between linear motion and rotational motion.

Linear motion Rotational motion

PropertyFundamental units

PropertyFundamental units

English SI English SI

Distance, ft m Angle, rad rad

Velocity, ft/sec m/sec Velocity, rad/sec rad/sec

Acceleration, ft/sec2 m/sec2 Acceleration, rad/sec2 rad/sec2

Mass, slug kg Inertia, slug·ft2 kg·m2

Force, lbf N Torque, ft·lbf N·m

Note: 1 slug = 32.174 lbm Note: rad = 180°

x θ

v ω

a α

m I

F Γ

π

Chapter 2: SYSTEMS - CLOSURE 55

CLOSUREIn order to illustrate the significant aspects of system-specific PI control theory, the

following types of systems and related topics are analyzed in the following chapters:

• fluid flow systems without mass or energy storage,

• live fluid storage systems with both flow in and flow out,

• dead fluid storage systems with only flow in or only flow out,

• position control by velocity modulation of systems with negligible inertia,

• velocity control by force, or torque, modulation of systems with inertia,

• position control by force, or torque, modulation of systems with inertia,

• speed control systems with negligible inertia,

• an inverted pendulum as an example of an inherently unstable system,

• a motorized wheel, and

• the solution of equations using control theory.

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