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Advanced Mathematical Tools for Automatic Control Engineers: Volume 2 - Stochastic Systems
of: Alex Poznyak
Elsevier Trade Monographs, 2009
ISBN: 9780080914039
Format: PDF, ePUB, Read online
Copy protection: DRM
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Advanced Mathematical Tools for Automatic Control Engineers: Volume 2 - Stochastic Systems
Front cover
1
Half title page
2
Dedication
3
Title page
4
Copyright page
5
Contents
6
Preface
16
Notations and Symbols
22
List of Figures
28
List of Tables
30
PART I: Basics of Probability
31
Chapter 1. Probability Space
33
1.1. Set operations, algebras and sigma-algebras
33
1.2. Measurable and probability spaces
39
1.3. Borel algebra and probability measures
45
1.4. Independence and conditional probability
56
Chapter 2. Random Variables
63
2.1. Measurable functions and random variables
63
2.2. Transformation of distributions
67
2.3. Continuous random variables
72
Chapter 3. Mathematical Expectation
77
3.1. Definition of mathematical expectation
77
3.2. Calculation of mathematical expectation
82
3.3. Covariance, correlation and independence
90
Chapter 4. Basic Probabilistic Inequalities
93
4.1. Moment-type inequalities
93
4.2. Probability inequalities for maxima of partial sums
102
4.3. Inequalities between moments of sums and summands
110
Chapter 5. Characteristic Functions
113
5.1. Definitions and examples
113
5.2. Basic properties of characteristic functions
118
5.3. Uniqueness and inversion
124
PART II: Discrete Time Processes
131
Chapter 6. Random Sequences
133
6.1. Random process in discrete and continuous time
133
6.2. Infinitely often events
134
6.3. Properties of Lebesgue integral with probabilistic measure
140
6.4. Convergence
147
Chapter 7. Martingales
163
7.1. Conditional expectation relative to a sigma-algebra
163
7.2. Martingales and related concepts
168
7.3. Main martingale inequalities
186
7.4. Convergence
191
Chapter 8. Limit Theorems as Invariant Laws
205
8.1. Characteristics of dependence
206
8.2. Law of large numbers
219
8.3. Central limit theorem
239
8.4. Logarithmic iterative law
255
PART III: Continuous Time Processes
267
Chapter 9. Basic Properties of Continuous Time Processes
269
9.1. Main definitions
269
9.2. Second-order processes
271
9.3. Processes with orthogonal and independent increments
274
Chapter 10. Markov Processes
293
10.1. Definition of Markov property
293
10.2. Chapman--Kolmogorov equation and transition function
297
10.3. Diffusion processes
301
10.4. Markov chains
307
Chapter 11. Stochastic Integrals
317
11.1. Time-integral of a sample-path
318
11.2. .-stochastic integrals
322
11.3. The Itô stochastic integral
329
11.4. The Stratonovich stochastic integral
347
Chapter 12. Stochastic Differential Equations
353
12.1. Solution as a stochastic process
353
12.2. Solutions as diffusion processes
368
12.3. Reducing by change of variables
372
12.4. Linear stochastic differential equations
376
PART IV: Applications
385
Chapter 13. Parametric Identification
387
13.1. Introduction
387
13.2. Some models of dynamic processes
389
13.3. LSM estimating
393
13.4. Convergence analysis
397
13.5. Information bounds for identification methods
411
13.6. Efficient estimates
419
13.7. Robustification of identification procedures
436
Chapter 14. Filtering, Prediction and Smoothing
447
14.1. Estimation of random vectors
447
14.2. State-estimating of linear discrete-time processes
452
14.3. State-estimating of linear continuous-time processes
457
Chapter 15. Stochastic Approximation
469
15.1. Outline of chapter
469
15.2. Stochastic nonlinear regression
470
15.3. Stochastic optimization
492
Chapter 16. Robust Stochastic Control
501
16.1. Introduction
501
16.2. Problem setting
502
16.3. Robust stochastic maximum principle
507
16.4. Proof of Theorem Th-RSMP
510
16.5. Discussion
522
16.6. Finite uncertainty set
525
16.7. Min-Max LQ-control
538
16.8. Conclusion
557
Bibliography
559
Index
565
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