Advanced Mathematical Tools for Automatic Control Engineers: Volume 2

1st Edition

Stochastic Systems

Authors: Alex Poznyak
Hardcover ISBN: 9780080446738
eBook ISBN: 9780080914039
Imprint: Elsevier Science
Published Date: 4th September 2009
Page Count: 567
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Advanced Mathematical Tools for Automatic Control Engineers, Volume 2: Stochastic Techniques provides comprehensive discussions on statistical tools for control engineers.
The book is divided into four main parts. Part I discusses the fundamentals of probability theory, covering probability spaces, random variables, mathematical expectation, inequalities, and characteristic functions. Part II addresses discrete time processes, including the concepts of random sequences, martingales, and limit theorems. Part III covers continuous time stochastic processes, namely Markov processes, stochastic integrals, and stochastic differential equations. Part IV presents applications of stochastic techniques for dynamic models and filtering, prediction, and smoothing problems. It also discusses the stochastic approximation method and the robust stochastic maximum principle.

Key Features

  • Provides comprehensive theory of matrices, real, complex and functional analysis
  • Provides practical examples of modern optimization methods that can be effectively used in variety of real-world applications
  • Contains worked proofs of all theorems and propositions presented


Undergraduate, graduate, research students of automotive control engineering, aerospace engineering, mechanical engineering and control in Chemical engineering.

Table of Contents

Preface Notations and Symbols List of Figures List of Tables Part I Basics of Probability Chapter 1 Probability Space 1.1 Set operations, algebras and sigma-algebras 1.2 Measurable and probability spaces 1.3 Borel algebra and probability measures 1.4 Independence and conditional probability Chapter 2 Random Variables 2.1 Measurable functions and random variables 2.2 Transformation of distributions 2.3 Continuous random variables Chapter 3 Mathematical Expectation 3.1 Definition of mathematical expectation 3.2 Calculation of mathematical expectation 3.3 Covariance, correlation and independence Chapter 4 Basic Probabilistic Inequalities 4.1 Moment-type inequalities 4.2 Probability inequalities for maxima of Partial sums 4.3 Inequalities between moments of sums and summands Chapter 5 Characteristic Functions 5.1 Definitions and examples 5.2 Basic properties of characteristic functions 5.3 Uniqueness and inversion Part II Discrete Time Processes Chapter 6 Random Sequences 6.1 Random process in discrete and continuous time 6.2 Infinitely often events 6.3 Properties of Lebesgue integral with probabilistic measure 6.4 Convergence Chapter 7 Martingales 7.1 Conditional expectation relative to a sigma-algebra 7.2 Martingales and related concepts 7.3 Main martingale inequalities 7.4 Convergence Chapter 8 Limit Theorems as Invariant Laws 8.1 Characteristics of dependence 8.2 Law of large numbers 8.3 Central limit theorem 8.4 Logarithmic iterative law Part III Continuous Time Processes Chapter 9 Basic Properties of Continuous Time Processes 9.1 Main definitions 9.2 Second-order processes


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© Elsevier Science 2009
Elsevier Science
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About the Author

Alex Poznyak


"This is a very well-written introduction to the basics of probability theory, stochastic analysis and their applications. Automatic control engineers will surely find much valuable material on different topics of modern and classical mathematics related to system and automatic control theories. In addition, this book may well serve as a reference book for researchers in applied probability theory and stochastic analysis…. Overall, this book is self-contained, well-organized, and clearly presented. It is a welcome addition to the existing collection of books in the field of probability and stochastic analysis, booth as a textbook at the graduate level and a reference book for researchers in this area."--Mathematical Reviews