《SERVOMECHANISMS AND REGULATING SYSTEM DESIGN VOLUME I》

1THE AUTOMATIC CONTROL PROBLEM1

1．0 INTRODUCTION1

1．1DESCRIPTION OF FEEDBACK CONTROL SYSTEM2

Requirements of Stability and Accuracy4

Mathematical Basis for Stability5

Features of Feedback Control System Performance6

1．2FEEDBACK CONTROL SYSTEM DESIGN9

Recommended Design Procedure10

1．3 DEVELOPMENT OF THE FIELD OF FEEDBACK CONTROL SYSTEMS12

2MANIPULATION OF COMPLEX NUMBERS17

2．0 INTRODUCTION17

2．1THREE FORMS OF COMPLEX QUANTITLES18

Rectangular Form18

Polar Form19

Exponential Form20

2．2 FQUIVALENCE OF DIFEFENT FORMS OF COMPLEX NUMBERS20

2．3MANIPULATION OF COMPLEX QUANTITIES22

Addition and Subtractio22

Multiplication and Divsion24

Forming the Conjugate26

Raising to a Power;Extractin a Root27

Logarithm of a Complex Quantity28

2．4 EXAMPLE FROM SERVOMECHANISM APPLICATION28

3SOLUTION OF LINEAR DIFERENTIAL EQUATIONS30

3．0 INTRODUCTION31

3．1SERIES RESISTANCE-INDUCTANCE NETWORK31

Classical Solution32

Transient and Steady-State Form of Solution33

Summary of the Solution of Differential Equations35

3．2 CHARACTERISTIC EQUATION36

3．3 SERIES RESISTANCE-CAPACITANCE NETWORK37

3．4 TIME CONSTANTS39

3．5 SERIES RESISTANCE-INDUCTANCE-CAPACITANCE NETWORK40

3．6STEADY-STATE RESPONSE TO A SINUSOIDALLY IMPRESSED VOLTAGE45

Replacing p by jw for Steady-State Sinusoidal Calculations49

Summary of Method of Obtaining Steady-State Solution for Sinusoidally Impressed Voltages50

3．7 STEADY-STATE RESPONSE TO A TIME POWER SERIES INPUT51

3．8SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS FOR OTHER TYPES OF SYSTEMS53

Mechanical Spring-Mass System54

Motor Synchronizing on a Fixed Signal59

Modification of the Time Constant by Means of Feedback63

4LAPLACE TRANSFORMS FOR THE SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS66

4．0 INTRODUCTION66

4．1 NATURE OF THE LAPLACE TRANSFORM67

4．2DEVELOPMENT OF A TABLE OF TRANSFORM PAIRS68

Constant Input of Magnitude A69

Step Function ｕ(ｔ)69

A Damped Exponential:е-αｔ70

A Time-Varying Sinusoid:sin βｔ70

A Time-Varying Cosinusoid with a Phase Angle:cos(βｔ+ψ)70

A Damped Sinusoid:е-αｔ sin βｔ71

A Quantity That Increases Linearly with Time,ｔ71

A Function Translated in Time,?(ｔ-α)72

4．3TRANSFORMATION OF DIFFERENTIATION AND INTEGRATION OPERATIONS73

Theorem for Differentiation74

Theorem for Integration75

Linearity Theorem76

Final Value and Initial Value Theorems77

4．4APPLICATION OF ￡ TRANSFORM TO SIMPLE CONTROL PROELEMS78

Position Control78

Elementary Resistance-Capacitance Network81

4．5 PERFORMING THE INVERSE LAPLACE TRANSFORMATION83

4．6EXAMPLES OF THE INVERSE TRANSFORMATION85

Factors Having Real Roots85

One Factor Having a Root at Zero85

Factors Having Complex Conjugate Roots86

Factors Having Imaginary Roots89

4．7 INVERSE TRANSFORMATION FOR REPEATED FACTORS90

4．8APPLICATION OF ￡-1 TRANSFORM TO PROBLEMS OF SECTION 4．493

Position Control Problem93

Elementary Resistance-Capactiance Network95

4．9 APPLICATION OF THE LAPLACE TRANSFORMATION TO SERVOMECHANISM PROBLEMS96

5STEADY-STATE OPERATION WITH SINUSOIDAL DRIVUNG FUNCTIONS99

5．0 INTRODUCTION99

5．1 IMPDNANCE CONCEPT100

5．2 IMPEDNNCE OF INDIVIDUAL ELEMENTS101

5．3AIDS TO SIMPLIFYING CIRCUIT COMPUTATIONS103

Equivalent Impedance103

Wye-Delta Transformations104

Superposition105

Théveini＇s Theorem108

5．4PERFORMANCE AS A FUNCTION OF FREQUENCY110

Resistance-Inductance Circuit110

Resistance-Capacitance Circuit112

Direct-Current Shunt Motor with Constant Field Excitation113

Mechanical Spring-Mass System117

5．5ATTENUATION AND PHASE ANGLE REPRESENTATION OF SYSTEM PERFORMANCE FOR SINUSOIDAL EXCITATION118

Definitions of Attenuation Terms119

Illustrations of Attenuation Phase Representation as a Function of Frequency121

6METHODS OF DETERMINING SYSTEM STABILITY124

6．0 INTRODUCTION124

6．1 STABILITY125

6．2DETERMINING THE ROOTS OF THE CAHRACTERISTIC EQUATION128

Formation of the Charaeteristic Equation from Its Roots129

Quadratic130

Cubic130

Quartic131

Quintic133

6．3ROUTH＇S CRITERION FOR STABILITY134

Examples of the Use of Routh＇s Stability Criterion136

Change in Scale Factor of Characteristic Equation136

6．4THE NYQUIST STABILTY CRITERION138

Development of the Characteristic Equation in Terms of Transfer Functions139

Method of Applying the Nyquist Stability Criterion141

Limitations to the Generalized Nyquist Stability Criterion141

Angular Change Produced by the Presence of Roots in the Positive Real Portion of the Complex Plane142

Angular Change Produced by the Presence of Poles at the Origin146

6．5 APPLICATION OF THE NYQUIST STABILITY CRITERION TO TYPICAL SYSTEM TRANSFER FUNCTIONS149

7TYPICAL CONTROL ELEMENTS AND THEIR TRANSFER FUNCTIONS157

7．0 INTRODUCTION157

7．1 DESCRIPTION OF THE CONTROL PROBLEM157

7．2 DEFINITION OF CONTROL SYSTEM ELEMENT TRANSFER FUNCTION161

7．3 COMBINATION OF CONTROL SYSTEM ELEMENTS IN SERIES163

7．4TRANSFER RUNCTIONS OF TYPICAL MECHANICAL CONTROL ELEMENTS164

Mechanical Elements Having Rotary Motion164

Mechanical Elements Having Translatory Motion169

Spring-Dashpot Elements Used to Obtain Mechanical Displaeements170

7．5TRANSFER FUNCTIONS OF TYPICAL ELECTRICAL CONTROL SYSTEM ELEMENTS172

Direct-Current Motor-Generator Control173

Torque Motor Type Servomechanism Elements175

ELectrical Networks Used for Stabilizing Purposes177

7．6TRANSFER FUNCTIONS OF TYPICAL HYORAULIC CONTROL ELEMENTS179

Hydraulic Valve-Piston Transfer Functions for Two Common Types of Operation179

Transfer Functions for Various Valve-Piston Linkage Combinations181

Hydraulic Motor with Variable Displacement Hydraulic Pump184

7．7TRANSFER FUNCTIONS OF STEERING SYSTEMS187

Ship-Steering Transfer Function187

Transfer Function of Controlled Missile in Vertical Flight190

7．8 CONCLUSIONA192

8TYPES OF SERVOMECHANISM AND CONTROL SYSTEMS194

8．0 INTRODUCTION194

8．1DEFINITION OF FEEDBACK CONTROL SYSTEM NOMENCLATURE AND SYMBOLS195

Block Diagram197

8．2 EFFECT OF FEEDBACK ON CHANGES IN TRANSFER FUNCTION199

8．3TYPES OF FEEDBACK CONTROL SYSTEMS202

Type 0 Servomechanism205

Type 1 Servomechanism208

Type 2 Servomechanism212

8．4SERVOMECHANISM ERROR COEFFICIENTS215

Statie Error Coefflcients216

Dynamic Error Coefficients218

9COMPLEX PLANE REPRESENTATION OF FEEDBACK CONTROL SYSTEM PERFORMANCE221

9．0 INTRODUCTION221

9．1 COMPLEX PLANE DIAGRAM FOR REEDBACK CONTROL SYSTEM WITH SINUSOIDAL INPUT222

9．2 DEVELOPMENT OF LOCI OF CONSTANT M AND α225

9．3 CLOSED-LOOP FREQUENCY RESPONSE AND ERROR RESPONSE FROM COMPLEX PLANE PLOT229

9．4 METHOD FOR SETTING GAIN FOR SPECIFIED Mm233

9．5INVERSE COMPLEX PLANE PLOT236

Inverse Plot for General Feedback Control System236

Inverse Transfer Function Plot for Systems with Direct Feedback238

9．6 LOCI OF CONSTANT 1/M AND-α241

9．7 COMPARATIVE USERULNESS OF DIRECT AND INVERSE PLOTS244

10DESIGN USE OF COMPLEX PLANE PLOT TO IMPROVE SYSTEM PERFORMANCE245

10．0 INTRODUCTION245

10．1SERIES NETWORK APPROACH TO SYSTEM DESIGN246

Use of Phase Lag Networks249

Use of Phase Lead Networks255

Use of Lead-Lag Series Networks264

10．2FEEDBCAK METHODS FOR USE IN SYSTEM DESIGN270

Direct Feedback270

Feedback through Frequency-Sensitive Elements273

Basis for Determining Characteristics for Feedback Elements278

Regenerative Feedback285

10．3COMPARISON OF RELATIVE MERITS OF SERIES AND FEEDBACK METHODS OF SYSTEM STABILIZATION288

Series Stabilization289

Feedback Stabilization289

11ATTENUTION CONCEPTS FOR USE IN FEEDBACK CONTROL SYSTEM DESIGN291

11．0 INTRODUCTION291

11．1 CORRELATION OF THE NYQUIST STABILITY CRTETION WITH BODE＇S ATTENUATION THEOREMS292

11．2TWO OF BODE＇S THEOREMS297

Theorem 1299

Theorem 2301

11．3MECHANICS OF DRAWING ATTENUATION DIAGRAMS FOR TRANSFER FUNCTIONS302

Single Time Constant302

Complex Roots or Time Constants310

11．4APPLICATION OF ATTENUATION DIAGRAMS TO TYPICAL CONTROL SYSTEM TRANSFER FUNCTIONS315

Velocity Error Coefficient Obtainable from Attenuation Diagram316

Acceleration Error Coefficient Obtainable from Attenuation Diagram317

11．5 CONTOURS OF CONSTANT M AND α LOCI318

11．6 CONCLUSION325

12APPLICATION OF ATTENUATION-PHASE DIAGRAMS TO FEEDBACK CONTROL DESIGN PROBLEMS327

12．0 INTRODUCTION327

12．1EXAMPLES OF SERIES STABILIZATION NETHODS328

Phase Lag Networks328

Phase Lead Networks330

Lead-Lag Networks334

12．2EXAMPLES OF FEEDBACK STABILIZATION METHODS336

Attenuation-Frequency Chatacteristic for Direct Feedback337

Attenuation-Frequency Characteristic with Feedback through Frequency-Sensitive Element339

12．3ATTENUATION-FREQUENCY DIAGRAM NOMENCLATURE345

Equalization345

Conditional Stability346

12．4 APPLICATION OF NICHOLS CHARTS TO OBTAIN CLOSED-LOOP PERFORMANCE347

12．5MORE EXACT FEEDBACK CONTROL SYSTEM REPRESENTATION OF ATTENUATION,PHASE MARGIN CHARACTERISTICS350

System Compoesd of Series Elements350

system with Feedback Stabilization351

13MULTIPLE-LOOP AND MULTIPLE-INPUT FEEDBACK CONTROL SYSTEMS358

13．0 INTRODUCTION358

13．1DESIGN OF MORE COMPLEX SYSTEMS359

Series Modification of Transfer Function359

Inclusion of a Servomechanism in a More Comprehensive Control System360

13．2MULTIPLE INPUTS AND LOAD DISTURBANCES363

General Case of Multiple Inputs363

Multiple-Position Inputs365

Responseto Input Signal and Load Disturbance367

A Regulator-Type Problem370

13．3EQUIVALENT BLOCK DIAGRAM REPRESENTATION373

Equivaleut Block Diagram of Stabilizing Transformer373

Simplifying Interconnected Multiple-Loop Systems375

13．4POSITON CONTROL SYSTEM WITH LOAD DISTURBANCE377

Determination of C/R377

Determination of C/TL383

13．5VOLTAGE REGULATOR WITH LOAD DISTURBANCE388

Determination of C/R=(ET/D)390

Determination of C/Q=(ET/EL)395

14COMPARISON OF STEADY-STATE AND TRANSIENT PERFORMANCE OF SERVOMECHANISMS398

14．0 INTRODUCTION398

14．1DESCRIPTION OF SERVOMECHANISM BEING CONSIDERED400

Definition of Terms Used to Describe System Performance Characteristice400

Open-Loop Attenuation-Frequency Characteristics403

14．2 PFFECT OF ωc ON FREQUENCY RESPONSE AND TRANSIENT RESPONSE405

14．3COMPARISON OF STEADY-STATE AND TRANSIENT PERFORMANCE CHARTS406

Effect of Using Parameter ω1/ωc for Abscissa406

Comparisen of Maximum Steady-State Value[C/R｜m]and Peak Transient Value[C/R｜p]of Output-Input Ratio407

Comparison of Frequency ωm at Which C/R｜m Occurs to ωt,the Lowest Frequency Oscillatory Term of the Transient Response408

Time tp at Which Peak Overshoot Occurs409

Settling Time ts to Reach 5 Per Cent of Final Value410

Use of Figures 14．3-7 to 14．3-23 for Systems Other than Those Having a Single Integrating Element in the Controller413

Effect of Having Non-multiple Breaks in Open-Loop Attenuation Characteristics415

Choice of Attenuation Rates between ω1 and ω2 and ω3 and ∞416

14．4EXAMPLES435

Charts Used for System Analysis435

Charts Used for System Synthesis438

14．5 CONCLUSION TO VOLUME I439

BIBLIOGRAPHY441

PROBLEMS447

INDEX499