Elsevier S & T

Advanced Mathematical Tools for Control Engineers: Volume 1: Deterministic Systems

Poznyak, Alex

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Table of contents

Contentsv
Prefacexvii
Notations and Symbolsxxi
List of Figuresxxvii
Part I: Matrices and Related Topics1
Chapter 1 Determinants3
1.1 Basic Definitions3
1.1.1 Rectangular matrix3
1.1.2 Permutations, number of inversions and diagonals3
1.1.3 Determinants4
1.2 Properties of Numerical Determinants, Minors and Cofactors6
1.2.1 Basic properties of determinants6
1.2.2 Minors and cofactors10
1.2.3 Laplace’s theorem13
1.2.4 Binet–Cauchy formula14
1.3 Linear Algebraic Equations and the Existence of Solutions16
1.3.1 Gauss’s method16
1.3.2 Kronecker–Capelli criterion17
1.3.3 Cramer’s rule18
Chapter 2 Matrices and Matrix Operations19
2.1 Basic Definitions19
2.1.1 Basic operations over matrices19
2.1.2 Special forms of square matrices20
2.2 Some Matrix Properties21
2.3 Kronecker Product26
2.4 Submatrices, Partitioning of Matrices and Schur’s Formulas29
2.5 Elementary Transformations on Matrices32
2.6 Rank of a Matrix36
2.7 Trace of a Quadratic Matrix38
Chapter 3 Eigenvalues and Eigenvectors41
3.1 Vectors and Linear Subspaces41
3.2 Eigenvalues and Eigenvectors44
3.3 The Cayley–Hamilton Theorem53
3.4 The Multiplicities and Generalized Eigenvectors54
3.4.1 Algebraic and geometric multiplicities54
3.4.2 Generalized eigenvectors56
Chapter 4 Matrix Transformations59
4.1 Spectral Theorem for Hermitian Matrices59
4.1.1 Eigenvectors of a multiple eigenvalue for Hermitian matrices59
4.1.2 Gram–Schmidt orthogonalization60
4.1.3 Spectral theorem61
4.2 Matrix Transformation to the Jordan Form62
4.2.1 The Jordan block62
4.2.2 The Jordan matrix form62
4.3 Polar and Singular-Value Decompositions63
4.3.1 Polar decomposition63
4.3.2 Singular-value decomposition66
4.4 Congruent Matrices and the Inertia of a Matrix70
4.4.1 Congruent matrices70
4.4.2 Inertia of a square matrix70
4.5 Cholesky Factorization73
4.5.1 Upper triangular factorization73
4.5.2 Numerical realization75
Chapter 5 Matrix Functions77
5.1 Projectors77
5.2 Functions of a Matrix79
5.2.1 Main definition79
5.2.2 Matrix exponent81
5.2.3 Square root of a positive semidefinite matrix84
5.3 The Resolvent for a Matrix85
5.4 Matrix Norms88
5.4.1 Norms in linear spaces and in Cn88
5.4.2 Matrix norms90
5.4.3 Compatible norms93
5.4.4 Induced matrix norm93
Chapter 6 Moore–Penrose Pseudoinverse97
6.1 Classical Least Squares Problem97
6.2 Pseudoinverse Characterization100
6.3 Criterion for Pseudoinverse Checking102
6.4 Some Identities for Pseudoinverse Matrices104
6.5 Solution of Least Squares Problem Using Pseudoinverse107
6.6 Cline’s Formulas109
6.7 Pseudo-Ellipsoids109
6.7.1 Definition and basic properties109
6.7.2 Support function111
6.7.3 Pseudo-ellipsoids containing vector sum of two pseudo-ellipsoids112
6.7.4 Pseudo-ellipsoids containing intersection of two pseudo-ellipsoids114
Chapter 7 Hermitian and Quadratic Forms115
7.1 Definitions115
7.2 Nonnegative Definite Matrices117
7.2.1 Nonnegative definiteness117
7.2.2 Nonnegative (positive) definiteness of a partitioned matrix120
7.3 Sylvester Criterion124
7.4 The Simultaneous Transformation of a Pair of Quadratic Forms125
7.4.1 The case when one quadratic form is strictly positive125
7.4.2 The case when both quadratic forms are nonnegative126
7.5 Simultaneous Reduction of more than Two Quadratic Forms128
7.6 A Related Maximum–Minimum Problem129
7.6.1 Rayleigh quotient129
7.6.2 Main properties of the Rayleigh quotient129
7.7 The Ratio of Two Quadratic Forms132
Chapter 8 Linear Matrix Equations133
8.1 General Type of Linear Matrix Equation133
8.1.1 General linear matrix equation133
8.1.2 Spreading operator and Kronecker product133
8.1.3 Relation between the spreading operator and the Kronecker product134
8.1.4 Solution of a general linear matrix equation136
8.2 Sylvester Matrix Equation137
8.3 Lyapunov Matrix Equation137
Chapter 9 Stable Matrices and Polynomials139
9.1 Basic Definitions139
9.2 Lyapunov Stability140
9.2.1 Lyapunov matrix equation for stable matrices140
9.3 Necessary Condition of the Matrix Stability144
9.4 The Routh–Hurwitz Criterion145
9.5 The Liénard–Chipart Criterion153
9.6 Geometric Criteria154
9.6.1 The principle of argument variation154
9.6.2 Mikhailov’s criterion155
9.7 Polynomial Robust Stability159
9.7.1 Parametric uncertainty and robust stability159
9.7.2 Kharitonov’s theorem160
9.7.3 The Polyak–Tsypkin geometric criterion162
9.8 Controllable, Stabilizable, Observable and Detectable Pairs164
9.8.1 Controllability and a controllable pair of matrices165
9.8.2 Stabilizability and a stabilizable pair of matrices170
9.8.3 Observability and an observable pair of matrices170
9.8.4 Detectability and a detectable pair of matrices173
9.8.5 Popov–Belevitch–Hautus (PBH) test174
Chapter 10 Algebraic Riccati Equation175
10.1 Hamiltonian Matrix175
10.2 All Solutions of the Algebraic Riccati Equation176
10.2.1 Invariant subspaces176
10.2.2 Main theorems on the solution presentation176
10.2.3 Numerical example179
10.3 Hermitian and Symmetric Solutions180
10.3.1 No pure imaginary eigenvalues180
10.3.2 Unobservable modes184
10.3.3 All real solutions186
10.3.4 Numerical example186
10.4 Nonnegative Solutions188
10.4.1 Main theorems on the algebraic Riccati equation solution188
Chapter 11 Linear Matrix Inequalities191
11.1 Matrices as Variables and LMI Problem191
11.1.1 Matrix inequalities191
11.1.2 LMI as a convex constraint192
11.1.3 Feasible and infeasible LMI193
11.2 Nonlinear Matrix Inequalities Equivalent to LMI194
11.2.1 Matrix norm constraint194
11.2.2 Nonlinear weighted norm constraint194
11.2.3 Nonlinear trace norm constraint194
11.2.4 Lyapunov inequality195
11.2.5 Algebraic Riccati–Lurie’s matrix inequality195
11.2.6 Quadratic inequalities and S-procedure195
11.3 Some Characteristics of Linear Stationary Systems (LSS)196
11.3.1 LSS and their transfer function196
11.3.2 H2 norm196
11.3.3 Passivity and the positive-real lemma197
11.3.4 Nonexpansivity and the bounded-real lemma199
11.3.5 H∞ norm201
11.3.6 γ-entropy201
11.3.7 Stability of stationary time-delay systems202
11.3.8 Hybrid time-delay linear stability203
11.4 Optimization Problems with LMI Constraints204
11.4.1 Eigenvalue problem (EVP)204
11.4.2 Tolerance level optimization204
11.4.3 Maximization of the quadratic stability degree205
11.4.4 Minimization of linear function Tr (CPCΤ ) under the Lyapunov-type constraint205
11.4.5 The convex function log det A-1 (X) minimization206
11.5 Numerical Methods for LMI Resolution207
11.5.1 What does it mean to solve LMIŽ?207
11.5.2 Ellipsoid algorithm207
11.5.3 Interior-point method210
Chapter 12 Miscellaneous213
12.1 Lambda-Matrix Inequalities213
12.2 Matrix Abel Identities214
12.2.1 Matrix summation by parts214
12.2.2 Matrix product identity215
12.3 S-Procedure and Finsler Lemma216
12.3.1 Daneš’ theorem216
12.3.2 S-procedure218
12.3.3 Finsler lemma220
12.4 Farkaš Lemma222
12.4.1 Formulation of the lemma222
12.4.2 Axillary bounded least squares (LS) problem223
12.4.3 Proof of Farkaš lemma224
12.4.4 The steepest descent problem225
12.5 Kantorovich Matrix Inequality226
Part II: Analysis229
Chapter 13 The Real and Complex Number Systems231
13.1 Ordered Sets231
13.1.1 Order231
13.1.2 Infimum and supremum231
13.2 Fields232
13.2.1 Basic definition and main axioms232
13.2.2 Some important properties233
13.3 The Real Field233
13.3.1 Basic properties233
13.3.2 Intervals234
13.3.3 Maximum and minimum elements234
13.3.4 Some properties of the supremum235
13.3.5 Absolute value and the triangle inequality236
13.3.6 The Cauchy–Schwarz inequality237
13.3.7 The extended real number system238
13.4 Euclidean Spaces238
13.5 The Complex Field239
13.5.1 Basic definition and properties239
13.5.2 The imaginary unite241
13.5.3 The conjugate and absolute value of a complex number241
13.5.4 The geometric representation of complex numbers244
13.6 Some Simple Complex Functions245
13.6.1 Power245
13.6.2 Roots246
13.6.3 Complex exponential247
13.6.4 Complex logarithms248
13.6.5 Complex sines and cosines249
Chapter 14 Sets, Functions and Metric Spaces251
14.1 Functions and Sets251
14.1.1 The function concept251
14.1.2 Finite, countable and uncountable sets252
14.1.3 Algebra of sets253
14.2 Metric Spaces256
14.2.1 Metric definition and examples of metrics256
14.2.2 Set structures257
14.2.3 Compact sets260
14.2.4 Convergent sequences in metric spaces261
14.2.5 Continuity and function limits in metric spaces267
14.2.6 The contraction principle and a fixed point theorem273
14.3 Summary274
Chapter 15 Integration275
15.1 Naive Interpretation275
15.1.1 What is the Riemann integration?275
15.1.2 What is the Lebesgue integration?276
15.2 The Riemann–Stieltjes Integral276
15.2.1 Riemann integral definition276
15.2.2 Definition of Riemann–Stieltjes integral278
15.2.3 Main properties of the Riemann–Stieltjes integral279
15.2.4 Different types of integrators284
15.3 The Lebesgue–Stieltjes Integral294
15.3.1 Algebras, σ-algebras and additive functions of sets294
15.3.2 Measure theory296
15.3.3 Measurable spaces and functions304
15.3.4 The Lebesgue–Stieltjes integration307
15.3.5 The almost everywhereŽ concept311
15.3.6 AtomicŽ measures and &#-948;-function312
15.4 Summary314
Chapter 16 Selected Topics of Real Analysis315
16.1 Derivatives315
16.1.1 Basic definitions and properties315
16.1.2 Derivative of multivariable functions319
16.1.3 Inverse function theorem325
16.1.4 Implicit function theorem327
16.1.5 Vector and matrix differential calculus330
16.1.6 Nabla operator in three-dimensional space332
16.2 On Riemann–Stieltjes Integrals334
16.2.1 The necessary condition for existence of Riemann–Stieltjes integrals334
16.2.2 The sufficient conditions for existence of Riemann–Stieltjes integrals335
16.2.3 Mean-value theorems337
16.2.4 The integral as a function of the interval338
16.2.5 Derivative integration339
16.2.6 Integrals depending on parameters and differentiation under integral sign340
16.3 On Lebesgue Integrals342
16.3.1 Lebesgue’s monotone convergence theorem342
16.3.2 Comparison with the Riemann integral344
16.3.3 Fatou’s lemma346
16.3.4 Lebesgue’s dominated convergence347
16.3.5 Fubini’s reduction theorem348
16.3.6 Coordinate transformation in an integral352
16.4 Integral Inequalities355
16.4.1 Generalized Chebyshev inequality355
16.4.2 Markov and Chebyshev inequalities355
16.4.3 Hölder inequality356
16.4.4 Cauchy–Bounyakovski–Schwarz inequality358
16.4.5 Jensen inequality359
16.4.6 Lyapunov inequality363
16.4.7 Kulbac inequality364
16.4.8 Minkowski inequality366
16.5 Numerical Sequences368
16.5.1 Infinite series368
16.5.2 Infinite products379
16.5.3 Teöplitz lemma382
16.5.4 Kronecker lemma384
16.5.5 Abel–Dini lemma385
16.6 Recurrent Inequalities387
16.6.1 On the sum of a series estimation387
16.6.2 Linear recurrent inequalities388
16.6.3 Recurrent inequalities with root terms392
Chapter 17 Complex Analysis397
17.1 Differentiation397
17.1.1 Differentiability397
17.1.2 Cauchy–Riemann conditions398
17.1.3 Theorem on a constant complex function400
17.2 Integration401
17.2.1 Paths and curves401
17.2.2 Contour integrals403
17.2.3 Cauchy’s integral law405
17.2.4 Singular points and Cauchy’s residue theorem409
17.2.5 Cauchy’s integral formula410
17.2.6 Maximum modulus principle and Schwarz’s lemma415
17.2.7 Calculation of integrals and Jordan lemma417
17.3 Series Expansions420
17.3.1 Taylor (power) series420
17.3.2 Laurent series423
17.3.3 Fourier series428
17.3.4 Principle of argument429
17.3.5 Rouché theorem431
17.3.6 Fundamental algebra theorem432
17.4 Integral Transformations433
17.4.1 Laplace transformation (K (t,p) = e–pt)434
17.4.2 Other transformations435
Chapter 18 Topics of Functional Analysis451
18.1 Linear and Normed Spaces of Functions452
18.1.1 Space mn of all bounded complex numbers452
18.1.2 Space ℎnp of all summable complex sequences452
18.1.3 Space C [a, b] of continuous functions452
18.1.4 Space Ck [a, b] of continuously differentiable functions452
18.1.5 Lebesgue spaces Lp [a, b] (1 ≤ p < ∞)453
18.1.6 Lebesgue spaces L∞ [a, b]453
18.1.7 Sobolev spaces Slp (G)453
18.1.8 Frequency domain spaces Lm×kp, RLm×kp, Lm454
18.1.9 Hardy spaces Hm×kp, RHm×kp, Hm×k<su454
18.2 Banach Spaces455
18.2.1 Basic definition455
18.2.2 Examples of incomplete metric spaces455
18.2.3 Completion of metric spaces456
18.3 Hilbert Spaces457
18.3.1 Definition and examples457
18.3.2 Orthogonal complement458
18.3.3 Fourier series in Hilbert spaces460
18.3.4 Linear n-manifold approximation462
18.4 Linear Operators and Functionals in Banach Spaces462
18.4.1 Operators and functionals462
18.4.2 Continuity and boundedness464
18.4.3 Compact operators469
18.4.4 Inverse operators471
18.5 Duality474
18.5.1 Dual spaces475
18.5.2 Adjoint (dual) and self-adjoint operators477
18.5.3 Riesz representation theorem for Hilbert spaces479
18.5.4 Orthogonal projection operators in Hilbert spaces480
18.6 Monotonic, Nonnegative and Coercive Operators482
18.6.1 Basic definitions and properties482
18.6.2 Galerkin method for equations with monotone operators485
18.6.3 Main theorems on the existence of solutions for equations with monotone operators486
18.7 Differentiation of Nonlinear Operators488
18.7.1 Fréchet derivative488
18.7.2 Gâteaux derivative490
18.7.3 Relation with variation principleŽ491
18.8 Fixed-Point Theorems491
18.8.1 Fixed points of a nonlinear operator491
18.8.2 Brouwer fixed-point theorem493
18.8.3 Schauder fixed-point theorem496
18.8.4 The Leray–Schauder principle and a priori estimates497
Part III: Differential Equations and Optimization499
Chapter 19 Ordinary Differential Equations501
19.1 Classes of ODE501
19.2 Regular ODE502
19.2.1 Theorems on existence502
19.2.2 Differential inequalities, extension and uniqueness507
19.2.3 Linear ODE516
19.2.4 Index of increment for ODE solutions524
19.2.5 Riccati differential equation525
19.2.6 Linear first-order partial DE528
19.3 Carathéodory’s Type ODE530
19.3.1 Main definitions530
19.3.2 Existence and uniqueness theorems531
19.3.3 Variable structure and singular perturbed ODE533
19.4 ODE with DRHS535
19.4.1 Why ODE with DRHS are important in control theory535
19.4.2 ODE with DRHS and differential inclusions540
19.4.3 Sliding mode control544
Chapter 20 Elements of Stability Theory561
20.1 Basic Definitions561
20.1.1 Origin as an equilibrium561
20.1.2 Positive definite functions562
20.2 Lyapunov Stability563
20.2.1 Main definitions and examples563
20.2.2 Criteria of stability: nonconstructive theory566
20.2.3 Sufficient conditions of asymptotic stability: constructive theory572
20.3 Asymptotic Global Stability576
20.3.1 Definition of asymptotic global stability576
20.3.2 Asymptotic global stability for stationary systems577
20.3.3 Asymptotic global stability for nonstationary system579
20.4 Stability of Linear Systems581
20.4.1 Asymptotic and exponential stability of linear time-varying systems581
20.4.2 Stability of linear system with periodic coefficients584
20.4.3 BIBO stability of linear time-varying systems585
20.5 Absolute Stability587
20.5.1 Linear systems with nonlinear feedbacks587
20.5.2 Aizerman and Kalman conjectures588
20.5.3 Analysis of absolute stability589
20.5.4 Popov’s sufficient conditions593
20.5.5 Geometric interpretation of Popov’s conditions594
20.5.6 Yakubovich–Kalman lemma595
Chapter 21 Finite-Dimensional Optimization601
21.1 Some Properties of Smooth Functions601
21.1.1 Differentiability remainder601
21.1.2 Convex functions605
21.2 Unconstrained Optimization611
21.2.1 Extremum conditions611
21.2.2 Existence, uniqueness and stability of a minimum612
21.2.3 Some numerical procedure of optimization615
21.3 Constrained Optimization621
21.3.1 Elements of convex analysis621
21.3.2 Optimization on convex sets628
21.3.3 Mathematical programing and Lagrange principle630
21.3.4 Method of subgradient projection to simplest convex sets636
21.3.5 Arrow–Hurwicz–Uzawa method with regularization639
Chapter 22 Variational Calculus and Optimal Control647
22.1 Basic Lemmas of Variation Calculus647
22.1.1 Du Bois–Reymond lemma647
22.1.2 Lagrange lemma650
22.1.3 Lemma on quadratic functionals651
22.2 Functionals and their Variations652
22.3 Extremum Conditions653
22.3.1 Extremal curves653
22.3.2 Necessary conditions653
22.3.3 Sufficient conditions654
22.4 Optimization of Integral Functionals655
22.4.1 Curves with fixed boundary points656
22.4.2 Curves with non-fixed boundary points665
22.4.3 Curves with a nonsmoothness point666
22.5 Optimal Control Problem668
22.5.1 Controlled plant, cost functionals and terminal set668
22.5.2 Feasible and admissible control669
22.5.3 Problem setting in the general Bolza form669
22.5.4 Mayer form representation670
22.6 Maximum Principle671
22.6.1 Needle-shape variations671
22.6.2 Adjoint variables and MP formulation673
22.6.3 The regular case676
22.6.4 Hamiltonian form and constancy property677
22.6.5 Nonfixed horizon optimal control problem and zero property678
22.6.6 Joint optimal control and parametric optimization problem681
22.6.7 Sufficient conditions of optimality682
22.7 Dynamic Programing687
22.7.1 Bellman’s principle of optimality688
22.7.2 Sufficient conditions for BP fulfilling688
22.7.3 Invariant embedding691
22.7.4 Hamilton–Jacoby–Bellman equation693
22.8 Linear Quadratic Optimal Control696
22.8.1 Nonstationary linear systems and quadratic criterion696
22.8.2 Linear quadratic problem697
22.8.3 Maximum principle for DLQ problem697
22.8.4 Sufficiency condition698
22.8.5 Riccati differential equation and feedback optimal control699
22.8.6 Linear feedback control699
22.8.7 Stationary systems on the infinite horizon702
22.9 Linear-Time Optimization709
22.9.1 General result709
22.9.2 Theorem on n-intervals for stationary linear systems710
Chapter 23 H2 and H∞ Optimization713
23.1 H2-Optimization713
23.1.1 Kalman canonical decompositions713
23.1.2 Minimal and balanced realizations717
23.1.3 H2 norm and its computing721
23.1.4 H2 optimal control problem and its solution724
23.2 H&#8734-Optimization728
23.2.1 L&#8734:, H∞ norms728
23.2.2 Laurent, Toeplitz and Hankel operators731
23.2.3 Nehari problem in RLm×k∞742
23.2.4 Model-matching (MMP) problem747
23.2.5 Some control problems converted to MMP757
Bibliography763
Index767

Book details

Vendor Elsevier S & T
SKU 9780080446745
ISBN-13 9780080556109
Author Poznyak, Alex
Category Mathematics
Subject Applied

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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

* 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

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