Advanced Mathematical Tools for Automatic Control Engineers: Volume 2 book cover

Advanced Mathematical Tools for Automatic Control Engineers: Volume 2

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.

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


Published: September 2009

Imprint: Elsevier

ISBN: 978-0-08-044673-8


  • "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


  • 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




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