# 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

## Description

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

9.3 Processes with orthogonal and independent increments

Chapter 10 Markov Processes

10.1 Definition of Markov property

10.2 Chapman–Kolmogorov equation and transition function

10.3 Diffusion processes

10.4 Markov chains

Chapter 11 Stochastic Integrals

11.1 Time-integral of a sample-path

11.2 λ-stochastic integrals

11.3 The Itô stochastic integral

11.4 The Stratonovich stochastic integral

Chapter 12 Stochastic Differential Equations

12.1 Solution as a stochastic process

12.2 Solutions as diffusion processes

12.3 Reducing by change of variables

12.4 Linear stochastic differential equations

Part IV Applications

Chapter 13 Parametric Identification

13.1 Introduction

13.2 Some models of dynamic processes

13.3 LSM estimating

13.4 Convergence analysis

13.5 Information bounds for identification methods

13.6 Efficient estimates

13.7 Robustification of identification procedures

Chapter 14 Filtering, Prediction and Smoothing

14.1 Estimation of random vectors

14.2 State-estimating of linear discrete-time processes

14.3 State-estimating of linear continuous-time processes

Chapter 15 Stochastic Approximation

15.1 Outline of chapter

15.2 Stochastic nonlinear regression

15.3 Stochastic optimization

Chapter 16 Robust Stochastic Control

16.1 Introduction

16.2 Problem setting

16.3 Robust stochastic maximum principle

16.4 Proof of Theorem 16.1

16.5 Discussion

16.6 Finite uncertainty set

16.7 Min-Max LQ-control

16.8 Conclusion

Bibliography

Index

## 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

## Readership

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

9.3 Processes with orthogonal and independent increments

Chapter 10 Markov Processes

10.1 Definition of Markov property

10.2 Chapman–Kolmogorov equation and transition function

10.3 Diffusion processes

10.4 Markov chains

Chapter 11 Stochastic Integrals

11.1 Time-integral of a sample-path

11.2 λ-stochastic integrals

11.3 The Itô stochastic integral

11.4 The Stratonovich stochastic integral

Chapter 12 Stochastic Differential Equations

12.1 Solution as a stochastic process

12.2 Solutions as diffusion processes

12.3 Reducing by change of variables

12.4 Linear stochastic differential equations

Part IV Applications

Chapter 13 Parametric Identification

13.1 Introduction

13.2 Some models of dynamic processes

13.3 LSM estimating

13.4 Convergence analysis

13.5 Information bounds for identification methods

13.6 Efficient estimates

13.7 Robustification of identification procedures

Chapter 14 Filtering, Prediction and Smoothing

14.1 Estimation of random vectors

14.2 State-estimating of linear discrete-time processes

14.3 State-estimating of linear continuous-time processes

Chapter 15 Stochastic Approximation

15.1 Outline of chapter

15.2 Stochastic nonlinear regression

15.3 Stochastic optimization

Chapter 16 Robust Stochastic Control

16.1 Introduction

16.2 Problem setting

16.3 Robust stochastic maximum principle

16.4 Proof of Theorem 16.1

16.5 Discussion

16.6 Finite uncertainty set

16.7 Min-Max LQ-control

16.8 Conclusion

Bibliography

Index

## Details

- No. of pages:
- 567

- Language:
- English

- Copyright:
- © Elsevier Science 2009

- Published:
- 4th September 2009

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080914039

- Hardcover ISBN:
- 9780080446738

## About the Author

### Alex Poznyak

Alexander S. Poznyak has published more than 200 papers in different international journals and 13 books including 2 for Elsevier. He is Fellow of IMA (Institute of Mathematics and Its Applications, Essex UK) and Associated Editor of Oxford-IMA Journal on Mathematical Control and Information. He was also Associated Editor of CDC, ACC and Member of Editorial Board of IEEE CSS. He is a member of the Evaluation Committee of SNI (Ministry of Science and Technology) responsible for Engineering Science and Technology Foundation in Mexico, and a member of Award Committee of Premium of Mexico on Science and Technology. In 2014 he was invited by the USA Government to serve as the member of NSF committee on “Neuro Sciences and Artificial Intelligence”.

### Affiliations and Expertise

Professor in Automatic Control at CINVESTAV-IPN, Mexico

## Reviews

"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