# Advanced Mathematical Tools for Automatic Control Engineers: Volume 2

## 1st Edition

**Stochastic Systems**

**Stochastic Systems**

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.

* 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.

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

- No. of pages:
- 567

- Language:
- English

- Copyright:
- © 2009

- Published:
- 4th September 2009

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080914039

- Print ISBN:
- 9780080446738

"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