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

Authors: Alex Poznyak
Hardcover ISBN: 9780080446745
eBook ISBN: 9780080556109
Imprint: Elsevier Science
Published Date: 17th December 2007
Page Count: 808
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This book 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

I MATRICES AND RELATED TOPICS 1 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 decompositions 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 equation 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 stability 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 constraints 11.5 Numerical methods for LMIs resolution 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

II ANALYSIS 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 III DIFFERENTIAL EQUATIONS AND OPTIMIZATION 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 Functions 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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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