Advanced Mathematical Tools for Control Engineers: Volume 1

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