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Advanced Mathematical Tools for Control Engineers: Volume 1 - 1st Edition - ISBN: 9780080446745, 9780080556109

Advanced Mathematical Tools for Control Engineers: Volume 1

1st Edition

Deterministic Systems

Author: Alex Poznyak
Paperback ISBN: 9780080974989
Hardcover ISBN: 9780080446745
eBook ISBN: 9780080556109
Imprint: Elsevier Science
Published Date: 17th December 2007
Page Count: 808
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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


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© Elsevier Science 2007
17th December 2007
Elsevier Science
Paperback ISBN:
Hardcover ISBN:
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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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