Socha L. — Linearization Methods for Stochastic Dynamic Systems :: Электронная библиотека попечительского совета мехмата МГУ

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Socha L. — Linearization Methods for Stochastic Dynamic Systems

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Название: Linearization Methods for Stochastic Dynamic Systems

Автор: Socha L.

Аннотация:

For most cases of interest, exact solutions to nonlinear equations describing stochastic dynamical systems are not available. The aim of this book is to give a systematic introduction to and overview of the relatively simple and popular linearization methods available. The scope is limited to models with continuous external and parametric excitations, yet these cover the majority of known approaches. The book contains an application chapter with emphasis on vibration analysis of stochastic mechanical structures as well as a chapter devoted to the assessment of the accuracy of the theoretical methods presented, both with respect to numerical and to experimental studies. The material derives partly from graduate course notes developed by the author for courses and seminars over the past 20 years.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2008

Количество страниц: 383

Добавлена в каталог: 14.11.2008

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Предметный указатель

$B^{r}$ - measurable      24
th column      106
th row      106
      15
-dimensional standard Wiener process      64
th central moment      86
th correlation moment      86
th moment      86
th order moment has the form      59
-dimensional diffusion process      26
      105
$\mathfrak{F}_{t}$ measurable      35
$\mathfrak{F}_{t}$ - measurable      31
$\sigma$ -field of Borel sets in $R^{r}$       24
$\sigma$ -fields $\mathfrak{F}_{t}$       35
(LQG)      4
Accuracy of linearization      4
Algebraic matrix Lyapunov equation      68
Algebraic methods      79
Algorithms      110
An interval      28
Approximate probability density function      11 23
Approximate probability density functions      23
Approximate response characteristics of dynamic system      2
Approximation      31 40
Approximation of probability density function      88
Approximation techniques      40
Autocorrelation      13
Autocorrelation function      17 19
Autocovariance functions      13
Automatic control systems      1
Averaging operation      16
Averaging operations      106
Basic criteria      104
Bilinear forms      15
Biorthonormal system      10
Block covariance matrix      20
Bounded function      17
Bounded measurable deterministic functions      61
Boundedness      36
Boxcar and gamma envelope functions      79
Boxcar time-modulating function      79
Brownian      74 78
Brownian motion      27
Central limit theorem      40
Central moments      9 100
Central moments of variable      10
Centralized th coordinate of the vector state      104
Chapman - Kolmogorov equation      24
Characteristic function      9 20
Characteristic function of random variable      7
Characteristic functional      12
Characteristic functions space      2
Closed form      62
Closed-form solutions      79
Closure property      15
Closure technique      75
Colored noise      29
Colored noises      39
Coloured noises      30
Complex closure techniques      94
Complex conjugate      13 14
Complex function      15
Complex-valued processes      13
Components of vectors      36
Conditional distribution      24
Conditional initial conditions      36
Conditional property      27
Constant matrices      61 68
Continuity      21
continuous      18 21 29
Continuous and bounded derivatives      37
Continuous covariance function      22
Continuous derivatives      26 32 34
Continuous function      34
Continuous in mean-square sense      15
Continuous state space      25
Continuous vector random variable      10
Continuous vector stochastic process      35
Control engineers      1
Control of nonlinear stochastic systems      4
Convergence      111
Convergent almost everywhere      28
Convex combination      32
Convex set      32
Correlation      18
Correlation degree      30
Correlation function      13 16 19 71-73
Correlation functions      93
Correlation matrix      71 79
Correlation matrix of the solution      69
Correlation moment      93
Correlation moments      92
Covariance function      13 21
Covariance functions      18
Covariance matrix      9 20 21
Creation of moment differential equations      89
Criteria in probability density functions      4
Criterion 2 -       107
Criterion of linearization      1
Cross correlation function      14
Cross correlation matrix function      14
Cross covariance function      14
Cross covariance matrix function      14
Cumulant Method      97
Cumulant neglect closure      100
Cumulants      7 8 12 89 97 98 100
Cumulants of Stochastic Processes      11
cylinders      41
Derivative      30
Describing function method      1
Determinant      20
Determinant of the matrix      10
Differential equation      68
Differential equation for the pth order moment      60
Differential equation with stochastic coefficients      39
Differential equations      39 62 110
Differential equations for central moments      89
Differential equations for correlation moments      93
Differential equations with stochastic parameters      65
Differential in mean-square sense of      15
Differential of a nonlinear function      33
Differentiation      16 31
Differentiation in mean-square sense      15
Diffusion coefficient      32
Diffusion matrix      25 26 32 33 37
Diffusion process      31-34 37 40
Diffusion processes      25 31
Dirac generalized function      12 30
Direct truncation of the approximation series      11
Distribution space      2
Drift vector      25 26 32 37
Duffing oscillator      43 78 90
Edgeworth series      88
Eigenvalues      64
Electromagnetic waves      30
Elementary event      11
Elementary event $\omega$       12
Elements      1 108
Energy criteria      4
entropy      2
Equality of the first and second moments of nonlinear and linearized variables      104
Equivalent linearization      1
Euclidean norm      25
Even-order moments      22
Evolutionary      79
Evolutionary power spectral density matrix      79
Exact covariance function      65
Existence of strong solution      41
Explicit      43

Explicit solutions      79
Exponentially decaying harmonic correlation function      79
External excitation      60
Family of growing $\sigma$ -fields      29
first      62 64
Fokker - Planck - Kolmogorov equation      4
Fokker - Planck - Kolmogorov equation, FPK.      27
formulas      31
Forward Kolmogorov equation      27
Fourier transform      19 70 73
Fourier transform of correlation matrix      70
Frequency domain      2
Frequency shift      74
Fubini theorem      38
Fundamental matrix      37 61 66 69
Gamma envelope function      79
Gaussian      9 20
Gaussian Colored Noise      63
Gaussian distribution      20 27
Gaussian excitation      1 75 78
Gaussian linearization      107
Gaussian process      21 30
Gaussian processes      20
Gaussian random variable      9
Gaussian stationary white noise      77
Gaussian stochastic process      21
Gaussian white noise      60 61
General Karhunen theorem of representation      17
Generalized derivative of a Wiener process      30
Generalized power spectral density function      17
Generalized stochastic process      29
Generalized Wiener - Chinczyn formula      17
Generating function for cumulants      8
Global Lipschitz conditions      41
Gram - Charlier expansion      11
Gram - Charlier series      22 106
Gramm - Charlier      88
Green function method      69
Green function method for stochastic systems      65
Green function methods      65
Harmonic balance      1
Harmonic excitation      77
Hermite polynomial      10
Hermite polynomials      10 23 122
hessian      106
Higher - Order Moment Criteria      111
Higher-order moments      62
Higher-order moments for continuous processes      13
Homogeneous      25
Homogeneous equation      59
Homogeneous system      69
Hysteresis      1
Hysteresis models      4
Identity matrix      34 37
Implicit Euler      43
Independent standard Wiener processes      61
Independent Wiener processes      35 37
Initial condition      26 35 36 59 61 69
Initial condition $\textbf{x}_{0}$       42
Input process      71
Integral form      66
Integral Fourier transform      16
Integral in mean-square sense      16
Integration      15 31
Integration in mean-square sense      16
interval      15
Inverse Fourier transform      70
Inverse transform      17 19
Inverse transformation      60
Iterative procedure      108
Ito      38
Ito differentials      86
Ito equation      38 39
Ito formula      33
Ito integral      38
Ito stochastic differential equation      38
Ito stochastic integral      32
Ito vector integral      33
Ito vector stochastic differential equation      63
Joint characteristic function      7 8
Joint probability density function      20
Joint probability distribution      12
Kolmogorov equations      25
Langevin equation      39
Laplace transform      71
Levy processes      74 78
Lie algebra      79
Limit in mean-square sense      15
Limits      110
Linear filter      63 72
Linear filter equations      69
Linear homogeneous Ito stochastic equation      59
Linear Ito vector stochastic differential equation      60 61
Linear operator transforming      16
linear oscillator      65 72
Linear oscillator with non - Gaussian colored noise excitation      74
Linear stochastic differential equations      36
Linear Stochastic Dynamic Systems      59
Linear systems with parametric excitations      79
Linear systems with quadratic criterion under Gaussian excitation      4
Linear systems with time-dependent coefficients      79
Linear transformations      21
Linearization coefficients      1 2 107
Linearization in frequency domain      1
Linearization methods      1
Linearized form      104
Lipschitz condition      36
Lipschitz conditions      41 42
Local Lipschitz conditions      43
Logarithm of characteristic      8
Lyapunov differential equation      110
Lyapunov equation      110
Lyapunov function      42
Lyapunov function approach      41
Maclaurin series      8
Markov process      24 37
Markov processes      23
Martingale with respect to $\sigma$ -field { $\mathfrak{F}_{t}$ }      27
Mathematical background      1
Mathematical models      1 20
Mathematical tools      3
Matrix correlation function      14 66 67
Matrix covariance function      14 66
Matrix intensity function      67
Matrix variance function      68
Mean      21
Mean value      18 20 21 66 69
Mean-square criterion      110
Mean-square error of approximation      105
Mean-square integral      32
Measurable      36
Measurable and bounded functions      37
Measurable space of functions      35
Memory transformation of Gaussian processes      74
Memoryless transformation of Gaussian processes      74
Method of Central Moments      96
Moment criteria      2 4
Moment differential equations      64
Moment equations      59 86
Moment Equations for Nonlinear Stochastic Dynamic Systems      85
Moments      7 11 12 97 98
Multiindices      89
Mutually independent processes      29
Narrow band non - Gaussian stationary processes      78
Necessary and sufficient condition      15
Necessary conditions for the minimum      106
Negative definite matrix      64
New state variables      75 77
Non - Gaussian Closure Techniques      96
Non - Gaussian Distributions      9

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