Sensitivity of Automatic Control Systems (Control Series)

✍ Scribed by Efim Rozenwasser, Rafael Yusupov

• Publisher: CRC Press
• Year: 1999
• Tongue: English
• Leaves: 435
• Edition: 1
• Category: Library

✦ Synopsis

Although it arose much earlier in a variety of contexts, sensitivity theory became an independent branch of science in the sixties. Since then, researchers from around the world have continued to make great strides in both the theory and its applications. However, much of the work of Russian scientific schools and specialists remain unknown in the West.Sensitivity of Control Systems summarizes the results of the authors and their disciples in sensitivity theory, addressing the basic notions of the theory and the problem of selecting technical parameters of systems. The authors formulate problems for actual technical systems and their models, and establish relations between sensitivity theory and classical stability problems. They offer a significant, general theory for investigating the sensitivity of boundary problems and use elements of this theory for sensitivity analysis of solutions to nonlinear programming and variational calculus problems, as well as oscillatory processes. The book also presents general investigation methods for discontinuous systems, including those described by operator models.Full of powerful new methods and results, this book offers a unique opportunity for those in theoretical investigation and in the design, testing, and exploitation of various control systems to explore the work of Russia's leading researchers in sensitivity theory. Furthermore, its techniques for parametric perturbation investigation, Sensitivity of Control Systems will prove useful in fields outside of control theory, including oscillation theory, motion dynamics, and mathematical economy.

✦ Table of Contents

SENSITIVITY of AUTOMATIC CONTROL SYSTEMS......Page 1
Preface......Page 4
Contents......Page 8
1.1.1 TGeneral Principles of Control......Page 15
Table of Contents......Page 0
1.1.2 Directed Action Elements......Page 17
1.1.3 State Variables and Parameters......Page 20
1.1.4 Sensitivity and Problem of Te hnical Parameters Selection......Page 21
1.2.2 Parametric System Model......Page 23
1.2.3 Determining Sets of Parameters......Page 24
1.2.4 Problem of Sensitivity of Parametric Model......Page 26
1.3.1 Sensitivity Functions......Page 28
1.3.2 Main and Additional Motions......Page 29
1.3.3 Analysis of First Approximation......Page 32
1.3.4 Statement of Optimization Problems with Sensitivity Requirements......Page 34
2.1.1 Mathematical Description......Page 37
2.1.2 Parametric Dependence of Solutions on Finite Time Intervals......Page 38
2.1.3 Calculation of Derivatives byParameters......Page 41
2.1.4 Parametric Models and General Sensitivity Equations......Page 42
2.1.5 SensitivityEquations of Higher Orders......Page 47
2.1.6 Multiparameter Case......Page 50
2.1.7 Analytical Representation of Single-Parameter Familyof Solutions......Page 53
2.1.8 Equations of Additional Motion......Page 55
2.1.9 Estimation of First Approximation Error......Page 57
2.2.1 Norms of Finite-Dimensional Vectors and Matrices......Page 60
2.2.2 Functions of Constant and De •nite Sign......Page 63
2.2.3 Time-Dependent Functions of Constant and De •nite Sign......Page 65
2.2.4 Lyapunov ’s Principle......Page 66
2.2.5 Norm of Additional Motion......Page 68
2.2.6 Parametric Stability......Page 70
2.2.7 General Investigation Method......Page 71
2.2.8 Sensitivityof Linear System......Page 73
2.3.1 Statement of the Problem......Page 77
2.3.2 AuxiliaryTheorem......Page 79
2.3.3 Sufficient Conditions of Applicabilityof First Approximation......Page 82
2.3.4 Classification of Special Cases......Page 85
2.4.1 Self-Oscillating Modes of Nonlinear Systems......Page 88
2.4.2 Linear Differential Equations with Periodic Coe •cients......Page 89
2.4.3 General Properties of SensitivityEquations......Page 92
2.4.4 SensitivityFunctions Variation over Self-Oscillation Period......Page 96
2.4.5 SensitivityFunctions for PeriodicityCharacteristics......Page 98
2.4.6 Practical Method for Calculating Sensitivity Functions......Page 102
2.4.7 Application to Van der Paul Equation......Page 104
2.5.1 Linear OscillatorySystems......Page 106
2.5.2 Sensitivityof Linear OscillatorySystem......Page 109
2.5.3 Sensitivityof Nonlinear OscillatorySystem......Page 114
2.6.1 Boundary-Value Problems Depending on Parameter......Page 117
2.6.2 SensitivityInvestigation for Boundary-Value Problems......Page 119
2.6.3 Implicit Functions Theorems......Page 121
2.6.4 SensitivityFunctions of Solution of Boundary-Value Problems......Page 123
2.6.5 Sensitivityof Non-Autonomous OscillatorySystem......Page 125
2.6.6 Sensitivityof Self-OscillatorySystem......Page 127
2.6.7 BoundaryConditions for SensitivityFunctions......Page 130
3.1.1 Time-Domain Description......Page 132
3.1.2 Time-Domain Description of Relay Systems......Page 133
3.1.3 Parametric Model and Sensitivity Function of Discontinuous System......Page 138
3.1.4 General Sensitivity Equations for Discontinuous Systems......Page 139
3.1.5 Case of Continuous Solutions......Page 144
3.2.1 General Equations of Relay Systems......Page 147
3.2.2 Sensitivity Equations for Systems with Ideal Relay......Page 152
3.2.3 Systems with Logical Elements......Page 156
3.2.4 Relay System with Variable Delay......Page 159
3.2.5 Relay Extremal System......Page 163
3.2.6 System with Pulse-Frequency Modulation of First Kind......Page 165
3.3.1 Pulse and Relay-Pulse Operators......Page 167
3.3.2 Sensitivity Equations of Pulse-Amplitude Systems......Page 170
3.3.3 Sensitivity of Pulse-Amplitude Systems with Respect to Sampling Period......Page 172
3.3.4 Sensitivity Equations of Systems with Pulse-Width Modulation......Page 175
4.1.2 Operator of Directed Action Element......Page 178
4.1.3 Families of Operators......Page 180
4.1.4 Parametric Properties of Operators......Page 181
4.1.5 Parametric Families of Linear Operators......Page 182
4.1.6 Transfer Functions and Frequency Responses of Linear Operators......Page 185
4.1.7 Parametric Operator Model of System......Page 189
4.2.1 Generalized Functions......Page 191
4.2.2 Di •erentiation of Generalized Functions......Page 193
4.2.3 Multiplication of Generalized Functions......Page 195
4.2.4 Operator Equation of Open-Loop Linear System......Page 196
4.2.5 Operator Equation of Closed-Loop Linear System......Page 199
4.3.1 Generalized Functions Depending on Parameter......Page 200
4.3.2 Generalized Di •erentiation by Parameter......Page 201
4.3.3 Sensitivity Equations......Page 204
4.3.4 Sensitivity Equations for Multivariable Systems......Page 210
4.3.5 Higher-Order Sensitivity Equations......Page 214
4.4.1 Single-Loop Relay Systems......Page 217
4.4.2 Pulse-Amplitude Systems......Page 224
4.4.3 Pulse-Width Systems......Page 228
4.4.4 Pulse-Frequency Systems......Page 231
5.1.1 Sensitivity f Transfer Function......Page 234
5.1.2 Sensitivity f Frequency Responses......Page 237
5.1.3 Relations between Sensitivity Functions of Frequency Characteristics......Page 241
5.1.4 Universal Algorithm for Determination of Sensitivity Functions for Frequency Characteristics......Page 244
5.1.5 Sensitivity Functions for Frequency Characteristics of Minimal-Phase Systems......Page 248
5.1.7 Relations between Sensitivity Functions of Open- Loop and Closed-Loop Systems......Page 249
5.1.8 Sensitivity f Frequency-Domain Quality Indices......Page 250
5.2.1 General Case......Page 251
5.2.2 Sensitivity f the Roots of a Polynomial......Page 253
5.2.3 Sensitivity f Poles and Zeros for Open-Loop and Closed-Loop Systems......Page 254
5.2.4 Relations between Sensitivity f Transfer Function and that of Poles and Zeros......Page 255
5.3.1 Eigenvalues and Eigenvectors of Matrices......Page 256
5.3.2 Sensitivity f Eigenvalues......Page 258
5.3.3 Sensitivity f Real and Imaginary Parts of Complex Eigenvalues......Page 260
5.3.4 Sensitivity f Eigenvectors......Page 262
5.3.5 Sensitivity C e •cients and Vectors of Higher Orders......Page 264
5.3.6 Sensitivity f Trace and Determinant f Matrix......Page 265
5.4.1 Integral Estimates......Page 266
5.4.2 Sensitivity f Integral Estimate I 0......Page 267
5.4.3 Sensitivity f Quadratic Estimates.Transformation of Di •erential Equations......Page 268
5.4.4 Sensitivity f Quadratic Estimates.Laplace Transform Method......Page 271
5.4.5 Sensitivity Coefficients of Integral Estimates for Discontinuous Control Systems......Page 274
5.5.2 Precision Indices......Page 275
5.5.3 Integral Estimates of Sensitivity Functions......Page 278
5.5.4 Envelope f Sensitivity Function......Page 282
6.1.1 Sensitivity nvariants......Page 284
6.1.2 Existence of Sensitivity nvariants......Page 286
6.1.3 Sensitivity nvariants of Single-Input –Single-Output Systems......Page 289
6.1.4 Sensitivity nvariants of SISO Nonlinear Systems......Page 293
6.1.5 Sensitivity nvariants of Multivariable Systems......Page 294
6.1.6 Sensitivity nvariants of Weight Function......Page 298
6.2.1 Root Sensitivity nvariants......Page 300
6.2.2 Sensitivity nvariants of Transfer Functions......Page 303
6.3.1 First Form of Sensitivity nvariants of Frequency Responses......Page 307
6.3.2 Second Form of Sensitivity nvariants of Frequency Responses......Page 309
6.3.3 Relations between Sensitivity nvariants of Time and Frequency Characteristics......Page 313
6.4.1 First Form of Sensitivity nvariants......Page 317
6.4.2 Second Form of Sensitivity nvariants......Page 319
6.5.1 Motion Equations and Transfer Functions......Page 320
6.5.2 Sensitivity nvariants of Amplitude Frequency Response......Page 324
6.5.3 Sensitivity nvariants of Integral Estimates......Page 327
6.5.4 Sensitivity nvariants of Damping Coe •cient......Page 328
7.1.2 Linear Programming......Page 332
7.1.3 Qualitative Geometric Sensitivity Analysis......Page 334
7.1.4 Quantitative Sensitivity Analysis......Page 335
7.2.1 Unconstrained Nonlinear Programming......Page 342
7.2.2 Nonlinear Programming with Equality Constraints......Page 343
7.2.3 Sensitivity Coefficients in Economic Problems......Page 348
7.2.4 Nonlinear Programming with Weak Equality Constraints......Page 352
7.2.5 Sensitivity of Convex Programming Problems......Page 354
7.3.1 Simplest Variational Problems......Page 357
7.3.2 Existence Conditions for Sensitivity Function......Page 358
7.3.3 Sensitivity Equations......Page 360
7.4.1 Variational Problem with Movable Bounds......Page 364
7.4.3 Sensitivity Equations......Page 365
7.4.4 Case Study......Page 367
7.4.5 Variational Problem with Corner Points......Page 370
7.5.2 Lagrange Problem......Page 377
7.5.3 Variational Problem with Di •erential Constraints......Page 379
7.5.4 Sensitivity f Isoperimetric Problem......Page 380
8.1.1 Classification of Basic Applied Sensitivity Problems......Page 382
8.1.3 Inverse Problems and their Incorrectness......Page 383
8.1.4 Solution Methods for Inverse Problems......Page 385
8.1.5 Methods for Improving Stability of Inverse Problems......Page 389
8.1.6 Investigation of Convergence of Iterative Process......Page 394
8.2.1 Definition of Identification......Page 399
8.2.2 Basic Algorithm of Parametric Identification Using Sensitivity Functions......Page 400
8.2.3 Identifiability and Observability......Page 402
8.3.1 Preliminaries......Page 404
8.3.2 Tolerance Calculation Problem......Page 405
8.3.3 Initial Mathematical Models......Page 406
8.3.4 Tolerance Calculation by Equal Influence Principle......Page 408
8.3.5 On Tolerance Distribution withAccount for Economic Factors......Page 409
8.3.6 On Requirements for Measuring Equipment......Page 412
8.4.1 Quality Indices and Constraints......Page 413
8.4.3 On Design of Systems withBounded Sensitivity......Page 415
8.4.4 On Design of Optimal Insensitive Systems......Page 416
8.5.1 General Structure of Numerical Integration Error......Page 418
8.5.2 Formula for Integration of Sensitivity Equations......Page 420
8.5.3 Estimates of Solution Errors......Page 421
8.5.4 Estimates of Integration Error for a First-Order System......Page 424
8.5.5 Results of Numerical Calculation......Page 425