Advanced Mathematical Tools for Control Engineers: Volume 1

Advanced Mathematical Tools for Control Engineers: Volume 1

Deterministic Systems

1st Edition - December 17, 2007

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  • Author: Alex Poznyak
  • Hardcover ISBN: 9780080446745
  • eBook ISBN: 9780080556109

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Advanced Mathematical Tools for Control Engineers: Volume 1 provides a blend of Matrix and Linear Algebra Theory, Analysis, Differential Equations, Optimization, Optimal and Robust Control. It contains an advanced mathematical tool which serves as a fundamental basis for both instructors and students who study or actively work in Modern Automatic Control or in its applications. It is includes proofs of all theorems and contains many examples with solutions. It is written for researchers, engineers, and advanced students who wish to increase their familiarity with different topics of modern and classical mathematics related to System and Automatic Control Theories.

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

    1 Determinants
    1.1 Basic definitions
    1.2 Properties of numerical determinants,
    minors and cofactors
    1.3 Linear algebraic equations
    and the existence of solutions
    2 Matrices and Matrix Operations
    2.1 Basic definitions
    2.2 Somematrix properties
    2.3 Kronecker product
    2.4 Submatrices, partitioning of matrices
    and Schur’s formulas
    2.5 Elementary transformations onmatrices
    2.6 Rank of a matrix
    2.7 Trace of a quadraticmatrix
    3 Eigenvalues and Eigenvectors
    3.1 Vectors and linear subspaces
    3.2 Eigenvalues and eigenvectors
    3.3 The Cayley-Hamilton theorem
    3.4 The multiplicities of an eigenvalue
    and generalized eigenvectors
    4 Matrix Transformations
    4.1 Spectral theorem
    for hermitianmatrices
    4.1.1 Eigenvectors of a multiple eigenvalue
    for hermitianmatrices
    4.2 Matrix transformation to the Jordan form
    4.3 Polar and singular-value
    4.4 Congruent matrices and the inertia of a matrix
    4.5 Cholesky factorization
    5 Matrix Functions
    5.1 Projectors
    5.2 Functions of a matrix
    5.3 The resolvent formatrix
    5.4 Matrix norms
    6 Moore-Penrose Pseudoinverse
    6.1 Classical Least Squares Problem
    6.2 Pseudoinverse characterization
    6.3 Criterion for pseudoinverse checking
    6.4 Some identities for pseudoinversematrices
    6.5 Solution of Least Square Problem
    using pseudoinverse
    6.6 Cline’s formulas
    6.7 Pseudo-ellipsoids

    7 Hermitian and Quadratic Forms
    7.1 Definitions
    7.2 Nonnegative definitematrices
    7.3 Sylvester criterion
    7.4 The simultaneous transformation of pair of quadratic forms
    7.5 The simultaneous reduction of more than two quadratic forms
    7.6 A related maximum-minimum problem
    7.7 The ratio of two quadratic forms
    8 Linear Matrix Equations
    8.1 General type of linear matrix
    8.2 Sylvestermatrix equation
    8.3 Lyapunovmatrix equation
    9 Stable Matrices and Polynomials 151
    9.1 Basic definitions
    9.2 Lyapunov stability
    9.3 Necessary condition of the matrix
    9.4 The Routh-Hurwitz criterion
    9.5 The Liénard-Chipart criterion
    9.6 Geometric criteria
    9.7 Polynomial robust stability
    9.8 Controllable, stabilizable, observable and detectable pairs

    10 Algebraic Riccati Equation
    10.1 Hamiltonianmatrix
    10.2 All solutions of the algebraic Riccati equation
    10.3 Hermitian and symmetric solutions .
    10.4 Nonnegative solutions
    11 Linear Matrix Inequalities
    11.1 Matrices as variables
    and LMI problem
    11.2 Nonlinear matrix inequalities
    equivalent to LMI
    11.3 Some characteristics of linear
    stationary systems (LSS)
    11.4 Optimization problems with LMI
    11.5 Numerical methods for LMIs
    12 Miscellaneous
    12.1 Λ-matrix inequalities
    12.2 MatrixAbel identities
    12.3 S-procedure and Finsler lemma
    12.4 Farkaš lemma
    12.5 Kantorovichmatrix inequality

    13 The Real and Complex Number Systems 253
    13.1 Ordered sets
    13.2 Fields
    13.3 The real field
    13.4 Euclidian spaces
    13.5 The complex field
    13.6 Some simplest complex functions
    14 Sets, Functions and Metric Spaces 277
    14.1 Functions and sets
    14.2 Metric spaces
    14.3 Resume
    15 Integration
    15.1 Naive interpretation
    15.2 The Riemann-Stieltjes integral
    15.3 The Lebesgue-Stieltjes integral
    16 Selected Topics of Real Analysis
    16.1 Derivatives
    16.2 On Riemann-Stieltjes integrals
    16.3 On Lebesgue integrals
    16.4 Integral inequalities
    16.5 Numerical sequences
    16.6 Recurrent inequalities
    17 Complex Analysis
    17.1 Differentiation
    17.2 Integration
    17.3 Series expansions
    17.4 Integral transformations
    18 Topics of Functional Analysis
    18.1 Linear and normed spaces of functions
    18.2 Banach spaces
    18.3 Hilbert spaces
    18.4 Linear operators and functionals in Banach spaces
    18.5 Duality
    18.6 Monotonic, nonnegative and
    coercive operators
    18.7 Differentiation of Nonlinear Operators
    18.8 Fixed-point Theorems
    19 Ordinary Differential Equations 563
    19.1 Classes of ODE
    19.2 Regular ODE
    19.3 Carathéodory’s Type ODE .
    19.4 ODE with DRHS
    20 Elements of Stability Theory
    20.1 Basic Definitions
    20.2 Lyapunov Stability
    20.3 Asymptotic global stability
    20.4 Stability of Linear Systems
    20.5 Absolute Stability
    21 Finite-Dimensional Optimization
    21.1 Some Properties of Smooth
    21.2 Unconstrained Optimization
    21.3 Constrained Optimization

    22 Variational Calculus and Optimal Control
    22.1 Basic Lemmas of Variation Calculus
    22.2 Functionals and Their Variations
    22.3 ExtremumConditions
    22.4 Optimization of integral functionals
    22.5 Optimal Control Problem
    22.6 MaximumPrinciple
    22.7 Dynamic Programming
    22.8 LinearQuadraticOptimal Control
    22.9 Linear-Time optimization
    23 H2 and H∞ Optimization 811
    23.1 H2 –Optimization
    23.2 H∞ -Optimization

Product details

  • No. of pages: 808
  • Language: English
  • Copyright: © Elsevier Science 2007
  • Published: December 17, 2007
  • Imprint: Elsevier Science
  • Hardcover ISBN: 9780080446745
  • eBook ISBN: 9780080556109

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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