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Stochastic Dynamics and Control, Volume 4 [Kõva köide]

(University of Delaware, Department of Mechanical Engineering, Newark, U.S.A.)
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This book is a result of many years of author’s research and teaching on random vibration and control. It was used as lecture notes for a graduate course. It provides a systematic review of theory of probability, stochastic processes, and stochastic calculus. The feedback control is also reviewed in the book. Random vibration analyses of SDOF, MDOF and continuous structural systems are presented in a pedagogical order. The application of the random vibration theory to reliability and fatigue analysis is also discussed. Recent research results on fatigue analysis of non-Gaussian stress processes are also presented. Classical feedback control, active damping, covariance control, optimal control, sliding control of stochastic systems, feedback control of stochastic time-delayed systems, and probability density tracking control are studied. Many control results are new in the literature and included in this book for the first time. The book serves as a reference to the engineers who design and maintain structures subject to harsh random excitations including earthquakes, sea waves, wind gusts, and aerodynamic forces, and would like to reduce the damages of structural systems due to random excitations.

· Comprehensive review of probability theory, and stochastic processes
· Random vibrations
· Structural reliability and fatigue, Non-Gaussian fatigue
· Monte Carlo methods
· Stochastic calculus and engineering applications
· Stochastic feedback controls and optimal controls
· Stochastic sliding mode controls
· Feedback control of stochastic time-delayed systems
· Probability density tracking control

This book is a result of many years of author’s research and teaching on random vibration and control. It was used as lecture notes for a graduate course. It provides a systematic review of theory of probability, stochastic processes, and stochastic calculus. The feedback control is also reviewed in the book. Random vibration analyses of SDOF, MDOF and continuous structural systems are presented in a pedagogical order. The application of the random vibration theory to reliability and fatigue analysis is also discussed. Recent research results on fatigue analysis of non-Gaussian stress processes are also presented. Classical feedback control, active damping, covariance control, optimal control, sliding control of stochastic systems, feedback control of stochastic time-delayed systems, and probability density tracking control are studied. Many control results are new in the literature and included in this book for the first time. The book serves as a reference to the engineers who design and maintain structures subject to harsh random excitations including earthquakes, sea waves, wind gusts, and aerodynamic forces, and would like to reduce the damages of structural systems due to random excitations.

· Comprehensive review of probability theory, and stochastic processes
· Random vibrations
· Structural reliability and fatigue, Non-Gaussian fatigue
· Monte Carlo methods
· Stochastic calculus and engineering applications
· Stochastic feedback controls and optimal controls
· Stochastic sliding mode controls
· Feedback control of stochastic time-delayed systems
· Probability density tracking control

Muu info

A comprehensive and practical review of probability and stochastic processes for engineers
Preface xv
Introduction
1(8)
Stochastic dynamics
1(1)
Stochastic control
2(7)
Covariance control
3(1)
PDF control
4(1)
Time delayed systems
5(1)
FPK based design
5(1)
Optimal control
6(3)
Probability Theory
9(24)
Probability of random events
9(1)
Random variables
10(1)
Probability distributions
11(1)
Expectations of random variables
11(2)
Common probability distributions
13(5)
Two-dimensional random variables
18(2)
Expectations
19(1)
n-Dimensional random variables
20(2)
Expectations
21(1)
Functions of random variables
22(7)
Linear transformation
25(3)
General vector transformation
28(1)
Conditional probability
29(4)
Exercises
30(3)
Stochastic Processes
33(18)
Definitions
33(1)
Expectations
34(1)
Vector process
35(1)
Gaussian process
36(1)
Harmonic process
37(1)
Stationary process
38(2)
Scalar process
38(1)
Vector process
39(1)
Correlation length
40(1)
Ergodic process
40(5)
Statistical properties of time averages
42(2)
Temporal density estimation
44(1)
Poisson process
45(3)
Compound Poisson process
47(1)
Markov process
48(3)
Exercises
49(2)
Spectral Analysis of Stochastic Processes
51(14)
Power spectral density function
51(3)
Definitions
51(1)
One-sided spectrum
52(1)
Power spectrum of vector processes
53(1)
Spectral moments and bandwidth
54(5)
Narrowband process
55(1)
Broadband process
55(4)
Process with rational spectral density function
59(2)
Finite time spectral analysis
61(4)
Exercises
62(3)
Stochastic Calculus
65(28)
Modes of convergence
65(2)
Stochastic differentiation
67(7)
Statistical properties of derivative process
68(3)
Spectral analysis of derivative processes
71(3)
Stochastic integration
74(7)
Statistical properties of stochastic integrals
76(2)
Integration of weakly stationary processes
78(2)
Riemann--Stieltjes integrals
80(1)
Ito calculus
81(12)
Brownian motion
81(3)
Ito and Stratonovich integrals
84(1)
Ito and Stratonovich differential equations
85(3)
Ito's lemma
88(1)
Moment equations
89(1)
Exercises
90(3)
Fokker--Planck--Kolmogorov Equation
93(28)
Chapman--Kolmogorov--Smoluchowski equation
93(1)
Derivation of the FPK equation
94(5)
Derivation using Ito's lemma
98(1)
Solutions of FPK equations for linear systems
99(2)
Short-time solution
101(1)
Improvement of the short-time solution
102(1)
Path integral solution
102(2)
Markov chain representation of path integral
103(1)
Exact stationary solutions
104(17)
First order systems
105(1)
Second order systems
105(3)
Dimentberg's system
108(2)
Equivalent Ito equation
110(1)
Hamiltonian systems
110(4)
Detailed balance
114(4)
Exercises
118(3)
Kolmogorov Backward Equation
121(8)
Derivation of the backward equation
121(3)
Reliability formulation
124(1)
First-passage time probability
125(1)
Pontryagin--Vitt equations
126(3)
Exercises
127(2)
Random Vibration of SDOF Systems
129(14)
Solutions in the mean square sense
129(6)
Expectations of the response
130(1)
Stationary random excitation
131(2)
Power spectral density
133(2)
Solutions with Ito calculus
135(8)
Moment equations
136(1)
Fokker--Planck--Kolmogorov equation
137(2)
Exercises
139(4)
Random Vibration of MDOF Discrete Systems
143(20)
Lagrange's equation
143(3)
Formal solution
145(1)
Modal solutions of MDOF systems
146(5)
Eigenvalue analysis
146(1)
Classical and non-classical damping
147(1)
Solutions with classical damping
148(2)
Solutions with non-classical damping
150(1)
Response statistics
151(3)
Stationary random excitation
152(1)
Spectral properties
152(1)
Cross-correlation and coherence function
153(1)
State space representation and Ito calculus
154(5)
Formal solution in state space
154(1)
Modal solution
155(2)
Solutions with Ito calculus
157(1)
Moment equations
157(1)
Stationary response of moment equations
158(1)
Filtered white noise excitation
159(4)
Exercises
161(2)
Random Vibration of Continuous Structures
163(24)
Distributed random excitations
163(1)
One-dimensional structures
164(15)
Bernoulli--Euler beam
164(4)
Response statistics of the beam
168(5)
Equations of motion in operator form
173(1)
Timoshenko beam
174(4)
Response statistics
178(1)
Two-dimensional structures
179(8)
Rectangular plates
179(3)
Response statistics of the plate
182(3)
Discrete solutions
185(1)
Exercises
186(1)
Structural Reliability
187(40)
Modes of failure
187(1)
Level crossing
187(13)
Single level crossing
187(2)
Method of counting process
189(2)
Higher order statistics of level crossing
191(1)
Dual level crossing
192(1)
Local minima and maxima
193(3)
Envelope processes
196(4)
Vector process
200(2)
First-passage reliability based on level crossing
202(2)
First-passage time probability -- general approach
204(5)
Example of SDOF linear oscillators
205(1)
Common safe domains
206(1)
Envelope process of SDOF linear oscillators
206(3)
Structural fatigue
209(5)
S-N model
209(1)
Rainflow counting
210(2)
Linear damage model
212(1)
Time-domain analysis of fatigue damage
213(1)
Dirlik's formula for fatigue prediction
214(1)
Extended Dirlik's formula for non-Gaussian stress
215(12)
Regression analysis of fatigue
216(6)
Validation of the regression model
222(1)
Case studies of fatigue prediction
222(2)
Exercises
224(3)
Monte Carlo Simulation
227(16)
Random numbers
227(2)
Linear congruential method
227(1)
Transformation of uniform random numbers
227(1)
Gaussian random numbers
228(1)
Vector of random numbers
229(1)
Random processes
229(6)
Gaussian white noise
229(3)
Random harmonic process
232(3)
Shinozuka's method
235(1)
Stochastic differential equations
235(4)
Second order equations
235(1)
State equation
236(1)
Runge--Kutta algorithm
237(1)
First-passage time probability
238(1)
Simulation of non-Gaussian processes
239(4)
Probability distribution
239(1)
Spectral distribution
240(2)
Exercises
242(1)
Elements of Feedback Controls
243(14)
Transfer function of linear dynamical systems
243(2)
Common Laplace transforms
244(1)
Concepts of stability
245(3)
Stability of linear dynamic systems
245(1)
Bounded input--bounded output stability
246(1)
Lyapunov stability
247(1)
Effects of poles and zeros
248(1)
Time domain specifications
249(1)
PID controls
250(2)
Control of a second order oscillator
251(1)
Routh's stability criterion
252(2)
Root locus design
254(3)
Properties of root locus
255(1)
Exercises
255(2)
Feedback Control of Stochastic Systems
257(22)
Response moment control of SDOF systems
257(3)
Covariance control
260(1)
Generalized covariance control
261(9)
Moment specification for nonlinear systems
261(1)
Control of a Duffing oscillator
262(2)
Control of Yasuda's system
264(6)
Covariance control with maximum entropy principle
270(9)
Maximum entropy principle
270(4)
Control of the Duffing system
274(3)
Exercises
277(2)
Concepts of Optimal Controls
279(22)
Optimal control of deterministic systems
280(9)
Total variation
280(1)
Problem statement
280(1)
Derivation of optimal control
281(2)
Pontryagin's minimum principle
283(1)
The role of Lagrange multipliers
283(1)
Classes of optimal control problems
284(3)
The Hamilton--Jacobi--Bellman equation
287(2)
Optimal control of stochastic systems
289(4)
The Hamilton--Jacobi--Bellman equation
290(3)
Linear quadratic Gaussian (LQG) control
293(6)
Kalman--Bucy filter
294(5)
Sufficient conditions
299(2)
Exercises
299(2)
Stochastic Optimal Control with the GCM Method
301(36)
Bellman's principle of optimality
301(1)
The cell mapping solution approach
302(4)
Generalized cell mapping (GCM)
303(2)
A backward algorithm
305(1)
Control of one-dimensional nonlinear system
306(6)
Control of a linear oscillator
312(4)
Control of a Van der Pol oscillator
316(3)
Control of a dry friction damped oscillator
319(5)
Systems with non-polynomial nonlinearities
324(6)
Control of an impact nonlinear oscillator
330(4)
A note on the GCM method
334(3)
Exercises
336(1)
Sliding Mode Control
337(24)
Variable structure control
337(3)
Robustness
339(1)
Concept of sliding mode
340(8)
Single input systems
340(1)
Nominal control
341(1)
Robust control
342(2)
Bounds of tracking error
344(1)
Multiple input systems
344(4)
Stochastic sliding mode control
348(8)
Nominal sliding mode control
348(1)
Response moments
349(1)
Robust sliding mode control
350(2)
Variance reduction ratio
352(1)
Rationale of switching term kS
353(3)
Adaptive stochastic sliding mode control
356(5)
Exercises
358(3)
Control of Stochastic Systems with Time Delay
361(12)
Method of semi-discretization
362(2)
Stability and performance analysis
364(2)
An example
366(4)
A note on the methodology
370(3)
Exercises
371(2)
Probability Density Function Control
373(10)
A motivating example
374(1)
PDF tracking control
375(2)
General formulation of PDF control
377(2)
Numerical examples
379(4)
Exercises
380(3)
Appendix A. Matrix Computation
383(8)
Types of matrices
383(1)
Blocked matrix inversion
384(1)
Matrix decomposition
385(2)
LU decomposition
385(1)
QR decomposition
385(1)
Spectral decomposition
386(1)
Singular value decomposition
386(1)
Cholesky decomposition
387(1)
Schur decomposition
387(1)
Solution of linear algebraic equations and generalized inverse
387(4)
Underdetermined system with minimum norm solution
388(1)
Overdetermined system with least squares solution
388(1)
A general case
389(2)
Appendix B. Laplace Transformation
391(4)
Definition and basic properties
391(2)
Laplace transform of common functions
393(2)
Bibliography 395(12)
Subject Index 407


Professor J. Q. Sun started working on random vibrations of nonlinear systems when he did his doctoral research at the University of California-Berkeley in 1980s. Since joining the faculty at the University of Delaware in 1994, he has successfully completed several research projects on analysis and control of stochastic systems.