# Advanced Mathematical Tools for Control Engineers: Volume 1

## Deterministic Systems

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

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

Hardbound, 808 Pages

Published: December 2007

Imprint: Elsevier

ISBN: 978-0-08-044674-5

## Contents

• I MATRICES AND RELATED TOPICS 11 Determinants 1.1 Basic definitions1.2 Properties of numerical determinants,minors and cofactors 1.3 Linear algebraic equationsand 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 matricesand Schurâs formulas 2.5 Elementary transformations onmatrices2.6 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 eigenvalueand generalized eigenvectors 4 Matrix Transformations 4.1 Spectral theoremfor hermitianmatrices 4.1.1 Eigenvectors of a multiple eigenvaluefor hermitianmatrices4.2 Matrix transformation to the Jordan form 4.3 Polar and singular-valuedecompositions 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 formatrix5.4 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 Problemusing pseudoinverse 6.6 Clineâs formulas 6.7 Pseudo-ellipsoids 7 Hermitian and Quadratic Forms7.1 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 matrixequation 8.2 Sylvestermatrix equation8.3 Lyapunovmatrix equation 9 Stable Matrices and Polynomials 1519.1 Basic definitions 9.2 Lyapunov stability 9.3 Necessary condition of the matrixstability 9.4 The Routh-Hurwitz criterion 9.5 The LiÃ©nard-Chipart criterion 9.6 Geometric criteria 9.7 Polynomial robust stability9.8 Controllable, stabilizable, observable and detectable pairs10 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 variablesand LMI problem11.2 Nonlinear matrix inequalitiesequivalent to LMI 11.3 Some characteristics of linearstationary systems (LSS)11.4 Optimization problems with LMIconstraints 11.5 Numerical methods for LMIsresolution 12 Miscellaneous12.1 Λ-matrix inequalities 12.2 MatrixAbel identities 12.3 S-procedure and Finsler lemma 12.4 FarkaÅ¡ lemma 12.5 Kantorovichmatrix inequality II ANALYSIS 13 The Real and Complex Number Systems 25313.1 Ordered sets 13.2 Fields 13.3 The real field 13.4 Euclidian spaces13.5 The complex field 13.6 Some simplest complex functions14 Sets, Functions and Metric Spaces 27714.1 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 integrals16.3 On Lebesgue integrals 16.4 Integral inequalities 16.5 Numerical sequences 16.6 Recurrent inequalities17 Complex Analysis 17.1 Differentiation17.2 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 andcoercive operators 18.7 Differentiation of Nonlinear Operators 18.8 Fixed-point TheoremsIII DIFFERENTIAL EQUATIONS AND OPTIMIZATION19 Ordinary Differential Equations 56319.1 Classes of ODE 19.2 Regular ODE 19.3 CarathÃ©odoryâ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 Stability21 Finite-Dimensional Optimization 21.1 Some Properties of SmoothFunctions21.2 Unconstrained Optimization21.3 Constrained Optimization 22 Variational Calculus and Optimal Control 22.1 Basic Lemmas of Variation Calculus22.2 Functionals and Their Variations22.3 ExtremumConditions 22.4 Optimization of integral functionals22.5 Optimal Control Problem 22.6 MaximumPrinciple22.7 Dynamic Programming22.8 LinearQuadraticOptimal Control22.9 Linear-Time optimization 23 H2 and H∞ Optimization 81123.1 H2 âOptimization23.2 H∞ -Optimization