Advanced Mathematical Tools for Automatic Control Engineers: Volume 2

Advanced Mathematical Tools for Automatic Control Engineers: Volume 2

Stochastic Systems

1st Edition - August 13, 2009

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  • Author: Alex Poznyak
  • Hardcover ISBN: 9780080446738
  • eBook ISBN: 9780080914039

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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

    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



Product details

  • No. of pages: 567
  • Language: English
  • Copyright: © Elsevier Science 2009
  • Published: August 13, 2009
  • Imprint: Elsevier Science
  • Hardcover ISBN: 9780080446738
  • eBook ISBN: 9780080914039

About the Author

Alex Poznyak

Alexander Poznyak is Professor and Department Head of Automatic Control at CINESTAV of IPN in Mexico. He graduated from Moscow Physical Technical Institute in 1970, and earned Ph.D. and Doctoral Degrees from the Institute of Control Sciences of Russian Academy of Sciences in 1978 and 1989, respectively. He has directed 43 Ph.D. theses, and published more than 260 papers and 14 books.

Affiliations and Expertise

Professor and Department Head of Automatic Control, CINESTAV of IPN, Mexico

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