Advanced Mathematical Tools for Control Engineers: Volume 1 (eBook)

Front Cover 1
Advanced Mathematical Tools for Automatic Control Engineers 4
Copyright Page 5
Table of Contents 6
Preface 18
Notations and Symbols 22
List of Figures 28
Part I: Matrices and Related Topics 30
Chapter 1. Determinants 32
1.1 Basic Definitions 32
1.2 Properties of Numerical Determinants, Minors and Cofactors 35
1.3 Linear Algebraic Equations and the Existence of Solutions 45
Chapter 2. Matrices and Matrix Operations 48
2.1 Basic Definitions 48
2.2 Some Matrix Properties 50
2.3 Kronecker Product 55
2.4 Submatrices, Partitioning of Matrices and Schur’s Formulas 58
2.5 Elementary Transformations on Matrices 61
2.6 Rank of a Matrix 65
2.7 Trace of a Quadratic Matrix 67
Chapter 3. Eigenvalues and Eigenvectors 70
3.1 Vectors and Linear Subspaces 70
3.2 Eigenvalues and Eigenvectors 73
3.3 The Cayley–Hamilton Theorem 82
3.4 The Multiplicities and Generalized Eigenvectors 83
Chapter 4. Matrix Transformations 88
4.1 Spectral Theorem for Hermitian Matrices 88
4.2 Matrix Transformation to the Jordan Form 91
4.3 Polar and Singular-Value Decompositions 92
4.4 Congruent Matrices and the Inertia of a Matrix 99
4.5 Cholesky Factorization 102
Chapter 5. Matrix Functions 106
5.1 Projectors 106
5.2 Functions of a Matrix 108
5.3 The Resolvent for a Matrix 114
5.4 Matrix Norms 117
Chapter 6. Moore–Penrose Pseudoinverse 126
6.1 Classical Least Squares Problem 126
6.2 Pseudoinverse Characterization 129
6.3 Criterion for Pseudoinverse Checking 131
6.4 Some Identities for Pseudoinverse Matrices 133
6.5 Solution of Least Squares Problem Using Pseudoinverse 136
6.6 Cline’s Formulas 138
6.7 Pseudo-Ellipsoids 138
Chapter 7. Hermitian and Quadratic Forms 144
7.1 Definitions 144
7.2 Nonnegative Definite Matrices 146
7.3 Sylvester Criterion 153
7.4 The Simultaneous Transformation of a Pair of Quadratic Forms 154
7.5 Simultaneous Reduction of more than Two Quadratic Forms 157
7.6 A Related Maximum–Minimum Problem 158
7.7 The Ratio of Two Quadratic Forms 161
Chapter 8. Linear Matrix Equations 162
8.1 General Type of Linear Matrix Equation 162
8.2 Sylvester Matrix Equation 166
8.3 Lyapunov Matrix Equation 166
Chapter 9. Stable Matrices and Polynomials 168
9.1 Basic Definitions 168
9.2 Lyapunov Stability 169
9.3 Necessary Condition of the Matrix Stability 173
9.4 The Routh–Hurwitz Criterion 174
9.5 The Liénard–Chipart Criterion 182
9.6 Geometric Criteria 183
9.7 Polynomial Robust Stability 188
9.8 Controllable, Stabilizable, Observable and Detectable Pairs 193
Chapter 10. Algebraic Riccati Equation 204
10.1 Hamiltonian Matrix 204
10.2 All Solutions of the Algebraic Riccati Equation 205
10.3 Hermitian and Symmetric Solutions 209
10.4 Nonnegative Solutions 217
Chapter 11. Linear Matrix Inequalities 220
11.1 Matrices as Variables and LMI Problem 220
11.2 Nonlinear Matrix Inequalities Equivalent to LMI 223
11.3 Some Characteristics of Linear Stationary Systems (LSS) 225
11.4 Optimization Problems with LMI Constraints 233
11.5 Numerical Methods for LMI Resolution 236
Chapter 12. Miscellaneous 242
12.1 Lambda-Matrix Inequalities 242
12.2 Matrix Abel Identities 243
12.3 S-Procedure and Finsler Lemma 245
12.4 Farkaš Lemma 251
12.5 Kantorovich Matrix Inequality 255
Part II: Analysis 258
Chapter 13. The Real and Complex Number Systems 260
13.1 Ordered Sets 260
13.2 Fields 261
13.3 The Real Field 262
13.4 Euclidean Spaces 267
13.5 The Complex Field 268
13.6 Some Simple Complex Functions 274
Chapter 14. Sets, Functions and Metric Spaces 280
14.1 Functions and Sets 280
14.2 Metric Spaces 285
14.3 Summary 303
Chapter 15. Integration 304
15.1 Naive Interpretation 304
15.2 The Riemann–Stieltjes Integral 305
15.3 The Lebesgue–Stieltjes Integral 323
15.4 Summary 343
Chapter 16. Selected Topics of Real Analysis 344
16.1 Derivatives 344
16.2 On Riemann–Stieltjes Integrals 363
16.3 On Lebesgue Integrals 371
16.4 Integral Inequalities 384
16.5 Numerical Sequences 397
16.6 Recurrent Inequalities 416
Chapter 17. Complex Analysis 426
17.1 Differentiation 426
17.2 Integration 430
17.3 Series Expansions 449
17.4 Integral Transformations 462
Chapter 18. Topics of Functional Analysis 480
18.1 Linear and Normed Spaces of Functions 481
18.2 Banach Spaces 484
18.3 Hilbert Spaces 486
18.4 Linear Operators and Functionals in Banach Spaces 491
18.5 Duality 503
18.6 Monotonic, Nonnegative and Coercive Operators 511
18.7 Differentiation of Nonlinear Operators 517
18.8 Fixed-Point Theorems 520
Part III: Differential Equations and Optimization 528
Chapter 19. Ordinary Differential Equations 530
19.1 Classes of ODE 530
19.2 Regular ODE 531
19.3 Carathéodory’s Type ODE 559
19.4 ODE with DRHS 564
Chapter 20. Elements of Stability Theory 590
20.1 Basic Definitions 590
20.2 Lyapunov Stability 592
20.3 Asymptotic Global Stability 605
20.4 Stability of Linear Systems 610
20.5 Absolute Stability 616
Chapter 21. Finite-Dimensional Optimization 630
21.1 Some Properties of Smooth Functions 630
21.2 Unconstrained Optimization 640
21.3 Constrained Optimization 650
Chapter 22. Variational Calculus and Optimal Control 676
22.1 Basic Lemmas of Variation Calculus 676
22.2 Functionals and their Variations 681
22.3 Extremum Conditions 682
22.4 Optimization of Integral Functionals 684
22.5 Optimal Control Problem 697
22.6 Maximum Principle 700
22.7 Dynamic Programing 716
22.8 Linear Quadratic Optimal Control 725
22.9 Linear-Time Optimization 738
Chapter 23. H2 and H8 Optimization 742
23.1 H2-Optimization 742
23.2 H8-Optimization 757
Bibliography 792
Index 796