Duality system in applied mechanics and optimal control

471 Pages·1958·7.67 MB·english

by  H.L. Dryden and Th. von Kármán (Eds.)

DUALITY SYSTEM IN APPLIED
MECHANICS AND OPTIMAL CONTROL
Advances in Mechanics and Mathematics
Volume 5
Series Editors:
David Y. Gao
Virginia Polytechnic Institute and State University, U.S.A.
Ray W. Ogden
University of Glasgow, U.K.
Advisory Editors:
I. Ekeland
University of British Columbia, Canada
K.R. Rajagopal
Texas A&M University, U.S.A.
W. Yang
Tsinghua University, P.R. China
DUALITY SYSTEM IN APPLIED
MECHANICS AND OPTIMAL CONTROL
by
Wan-Xie Zhong
Dalian University of Technology
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 1-4020-7881-1
Print ISBN: 1-4020-7880-3
©2004 Kluwer Academic Publishers
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Contents v
Preface xi
Introduction 1
0.1, Introduction to Precise Integration method 4
0.1.1,Homogeneous equation, algorithm for exponential matrix. 4
0.1.2,Solution of inhomogeneous equation 6
0.1.3,Precision analysis 7
0.1.4,Discussions on time-variant system or non-linear system 9
Chapter 1, Introduction to analytical dynamics 11
1.1,Holonomic and nonholonomicconstraints 11
1.2,Generalized displacement, degrees of freedom and virtual displacement 14
1.3,Principle of virtual displacement and the D’Alembert principle 16
1.4,Lagrange equation 18
1.5,Hamilton variational principle 19
1.6,Hamiltonian canonical equations 24
1.6.1,Legendre transformation and Hamiltonian canonical equations 24
1.6.2,Cyclic coordinate and conservation 28
1.7,Canonical transformation 29
1.8,Symplectic description of the canonical transformation 32
1.9,Poisson bracket 34
1.9.1, Algebra of Poisson bracket 35
1.10,Action 37
1.11,Hamilton-Jacobi equation 38
1.11.1,A simple harmonic oscillator 39
1.11.2,Time invariant system 40
1.11.3,Separation of variables 41
1.11.4,Separation of variables for linear systems 42
Chapter 2, Vibration theory 47
2.1, Single degree of freedom vibration 47
2.1.1,Linear vibrations 47
2.1.2,Parametric resonance 49
2.1.3,Introduction to non-linear vibration 52
2.1.3.1, Limitcycle 52
2.1.3.2,Duffing equation 54
vi
2.1.3.3, Simplified solution method for Duffing equation 57
2.2, Vibration of multi-degrees of freedom system 61
2.2.1.Free vibration with no damping, eigen-solutions 62
2.2.2,Constraints, count of eigenvalues 66
2.2.2.1,Inclusion theorem 67
2.2.2.2,Dynamic stiffness matrix and eigenvalue count 68
2.2.3, Eigenvalue sign count and substructure analysis 70
2.2.3.1,Mixed energy, the eigenvalue count for dual variables 72
2.2.3.2,Substructure combination and eigenvalue count of mixed energy 75
2.2.3.3,Essence of modal synthesis method 77
2.2.4, Subspace iteration for the eigen-solution of symmetric matrices 79
2.2.4.1,Ortho-normalization algorithm 80
2.2.4.2, Subspace projection and its eigen-solutions 80
2.2.4.3, Subspace iteration 80
2.2.4.4,Transition of the subspace 82
2.2.5,Eigen-solutions of asymmetric real matrix 82
2.2.5.1, Dual subspace iteration for asymmetric matrix 83
2.2.6,Singularvalue decomposition 85
2.2.6.1, QR decomposition 86
2.2.6.2, Singular value decomposition 86
2.3, Small vibration of gyroscopic systems 87
2.3.1,Method of separation of variables, eigen-problem 89
2.3.2,Positive definite Hamilton function 92
2.3.2.1,Variational principle of eigenvalues 93
2.3.2.2,Eigenvalue count and the inclusion theorem 96
2.3.3,Indefinite Hamilton function 98
2.3.3.1,Effect of gyroscopicforce to the stability of vibration 100
2.3.3.2,Symplectic eigenvalue problem and its algorithm 102
2.3.3.3,Symplectic eigen-solution of skew-symmetric matrix 106
2.3.3.4,Numerical example 113
2.4, Non-linear vibration of multi-degrees of freedom system 115
2.4.1,Non-linear internal parametric resonance 120
2.4.2,Non-linear internal sub-harmonic resonance 124
2.5, Discussion on the stability of gyroscopic system 127
2.5.1,Gyroscopic system with positive definite Hamilton function 127
2.5.2,Case of indefinite Hamilton function 129
Chapter 3, Probability and stochastic process 135
3.1,Preliminary of probability theory 135
3.1.1,Probability distribution function and probability density function 136
3.1.2,Mathematical expectation, varianceand covariance 137
3.1.3,Expectation of a random vector and its co-variance matrix 138
3.1.4,Conditional expectation and covariance of random vector 139
3.1.5,Characteristic function of random variable 140
3.1.6,Normal distribution 140
3.1.7,Lineartransformation and combination of Gauss random vectors 142
3.1.8,Least square method 143
3.2, Preliminary of stochastic process 147
vii
3.2.1,Stationary and non-stationary stochastic process 149
3.2.2,Ergodic stationary process 149
3.3, Quadratic momentstochastic process (regular process) 150
3.3.1,Continuity and differentiability of a regular stochastic process 151
3.3.2,Mean square integration 152
3.4,Normal stochastic process 153
3.5,Markoffprocess 154
3.6,Spectral density of stationary stochastic process 155
3.6.1,Wiener-Khintchin relation 155
3.6.2,Direct spectral analysis of stationary stochastic process 156
3.6.3,White noise 157
3.6.4,Wiener process 158
Chapter 4, Random vibration of structures 161
4.1, Models of random excitation 163
4.1.1,Stationary random excitations 163
4.1.2,Non-stationary random excitations 165
4.2, Response of structures under stationary excitations 167
4.2.1,Random response of single degree of freedom system 167
4.2.2,Multi-degrees of freedom system under single source excitation 170
4.2.2.1,Case of inconsistent damping 176
4.2.2.2,Single source multi-point excitation with differentphases 177
4.2.3, Stationary response of structure to multi-source excitations 178
4.2.3.1,Spectral expansion 179
4.2.3.2,Response analysis 179
4.3, Response under excitation of non-stationary stochastic process 180
4.3.1,Response under uniformly modulated non-stationary excitation 180
4.3.2,Response under evolutionary modulated non-stationary excitation 181
Chapter 5, Elastic system withsingle continuous coordinate 183
5.1,Fundamental equations of Timoshenco beam theory 183
5.2,Potential energy density and mixed energy density 185
5.3,Separation of variables, Adjoint symplectic ortho-normality 188
5.3.1,Adjoint symplectic orthogonality 189
5.3.2,Expansion theorem 191
5.4, Multiple eigenvalues and the Jordan normal form 192
5.4.1.Wave propagation for Timoshenco beam and its extension 194
5.4.2,Physical meaning of symplectic orthogonality—work reciprocity 197
5.5,Expansion solution of the inhomogeneous equation 201
5.6,Two end boundary conditions 202
5.7,Interval mixed energy and precise integration method 206
5.7.1,Displacement method analysis 207
5.7.2,Mixed energy, the dual variables 210
5.7.3,Riccati differentialequation and its precise integration 213
5.7.4,Power series expansion 215
5.7.5,Interval combination 216
5.7.6,Precise integration for the fundamentalinterval 218
5.7.7,Precise integration for asymmetric Riccati equations 222
viii
5.8, Eigenvector based solution of Riccati equations 229
5.8.1,Analytical solution applied to the symmetric Riccati equations 235
5.8.2,Algorithm for the eigen-solutions of a Hamilton matrix 237
5.8.3,Transform to real value computation 240
5.8.4,Transformation for purely imaginary eigenvalues 242
5.9, Stepwise integration method by means of sub-structural combination 245
5.10, Influence function of single continuous coordinate system 249
5.10.1,Reciprocal theorems of the impulse influence matrix functions 254
5.10.2,Precise integration of impulse influence matrixfunction 257
5.10.3,Numerical example of impulse influence matrix function 259
5.10.4,Application 260
5.11, Power flow 262
5.11.1,AlgebraicRiccati equation (ARE) 263
5.11.2,Transmission waves 265
5.11.3,On power flow orthogonality 266
5.12,Wavescattering analysis 268
5.13,Wave induced resonance 270
5.14,Wave propagation alongperiodical structures 272
5.14.1,Dynamic stiffness matrix of a fundamental substructure 273
5.14.2,Energy band and eigen-solutions of a symplectic matrix 273
5.14.3,Pass-band analysis for periodical wave-guides 275
5.14.4,Dynamicstiffness of a fundamentalperiod and eigenvalue count 277
5.14.5,Computation of the pass-band eigenvalue 280
Chapter 6, Linear optimal control, theory and computation 283
6.1, State space of linear system 284
6.1.1, Input-output description and state space description 284
6.1.1.1, Continuous-time and discrete-time systems 287
6.1.2,State space description of single-inputsingle-outputsystem 289
6.1.3,Integration of linear time-invariant systems 291
6.1.4,Frequency domain analysis, transfer function 294
6.1.5,Controllability and observability of a linear system 294
6.1.5.1,Controllability of a steady system 295
6.1.5.2,Observability of a steady system 295
6.1.6,Linear transformation 296
6.1.7,Realization of a transfer function in state space 297
6.1.8,Duality principle for controllability and observability 298
6.1.9,Discrete-time control 300
6.2, Theory of stability 301
6.2.1,Stability analysis under Lyapunovmeaning 302
6.2.2,Lyapunovmethod of stability analysis 303
6.3,Prediction, filtering and smoothing 305
6.4,Prediction and its computation 308
6.4.1,Mathematical model for prediction 309
6.4.2,Prediction of one dimensional system 310
6.4.3,Prediction of multi-degrees of freedom system 312
6.4.4,Precise time integration 314
6.4.4.1, Precise integration of inhomogeneous equations 315
ix
6.4.5, Precise integration of the Lyapunov differential equation 317
6.4.5.1,Precise integration of the algebraic Lyapunov equation 319
6.4.5.2,Integration of asymmetric Lyapunov equation 321
6.4.5.3,Solution with modulated input 324
6.4.6, Disturbance of colored noise 326
6.4.6.1, Some comments for precise integration of the extended system 327
6.5, Kalman filter 328
6.5.1, Model of linear estimation 329
6.5.1.1,Model of discrete-time system 329
6.5.1.2,Model of continuous-time system 330
6.5.2, Kalman filter analysis for discrete-time linear system 330
6.5.2.1, Correlated dynamic and measurement noises 335
6.5.3, Continuous-time Kalman-Bucy filtering analysis 336
6.5.3.1,Correlated dynamic and measurement noises 340
6.5.3.2,Continuous-time Kalman-Bucy filter under colored noises 342
6.5.4, Interval mixed energy 343
6.5.4.1, Interval combination 345
6.5.4.2,Differentialequations for the interval matrices and vectors 347
6.5.4.3,Physical interpretation of the solution of Riccati equation 350
6.5.5,Precise integration of the Riccati differential equation 351
6.5.6,Analytical solution of Riccati equation based on eigen-solutions 356
6.5.7,Solution of single step filterequation 357
6.5.7.1,Analytical single step integration for the filterequation 360
6.5.7.2,Analyticalequation for single step integration 364
6.5.7.3, Taylor expansion of precise stepwise filtering 366
6.5.7.4,Interval combination within the interval 369
6.5.8, Integration of filter equation for the whole interval 372
6.5.8.1, Numerical example 375
6.6, Optimal smoothing and computations 377
6.6.1,Optimal smoothing of continuous-time linear system 378
6.6.2,Interval mixed energy and differential equation for smoothing 381
6.6.3,Mean value and variance of smoothing 385
6.6.3.1, Precise integration for single time step 388
6.6.4, Three kinds of smoothing algorithms 389
6.6.4.1,Fixed interval smoothing 389
6.6.4.2,Fixed point smoothing 390
6.6.4.3,Fixed delay smoothing 390
6.7, Optimal control 391
6.7.1,Theory of LQ optimal control for the future time interval 392
6.7.2,Stability analysis 395
6.7.2.1, Gram matrices of controllability and observability 396
6.7.2.2,Positive definiteness of the Riccati matrix 397
6.7.2.3, Stability analysisbased on Lyapunov second method 401
6.7.3, Precise computation of LQ control 402
6.7.3.1,Precise solution of Riccati differential equation 403
6.7.3.2,Integration of state differentialequation 404
6.7.4, Measurement feedback optimal control 406
6.8, Robust control 407