كتاب Advanced Mathematical Tools for Automatic Control Engineers

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Advanced Mathematical Tools for Automatic Control Engineers
Volume 1: Deterministic Techniques
Alexander S. Poznyak
September 21, 2007

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I MATRICES AND RELATED TOPICS 1
1 Determinants 3
1.1 Basic definitions 3
1.1.1 Rectangularmatrix 3
1.1.2 Permutations, number of inversions and diagonals 4
1.1.3 Determinants 5
1.2 Properties of numerical determinants, minors and cofactors 7
1.2.1 Basic properties of determinants 7
1.2.2 Minors and cofactors 12
1.2.3 Laplace’s theorem. 14
1.2.4 Binet-Cauchy formula 16
1.3 Linear algebraic equations
and the existence of solutions 17
1.3.1 Gauss’smethod 17
1.3.2 Kronecker-Capelli criterion 19
1.3.3 Cramer’s rule 20
2 Matrices and Matrix Operations 21
2.1 Basic definitions 21
2.1.1 Basic operations overmatrices 21
2.1.2 Special forms of squarematrices 22
2.2 Somematrix properties 24
2.3 Kronecker product 29
2.4 Submatrices, partitioning of matrices and Schur’s formulas 32
2.5 Elementary transformations onmatrices 35
2.6 Rank of amatrix 40
2.7 Trace of a quadraticmatrix 42
3 Eigenvalues and Eigenvectors 45
3.1 Vectors and linear subspaces 45
3.2 Eigenvalues and eigenvectors 49
3.3 The Cayley-Hamilton theorem 59
3.4 The multiplicities and generalized eigenvectors 60
3.4.1 Algebraic and geometric multiplicities 60
3.4.2 Generalized eigenvectors 62
4 Matrix Transformations 65
4.1 Spectral theorem for Hermitian matrices 65
4.1.1 Eigenvectors of a multiple eigenvalue forHermitianmatrices 65
4.1.2 Gram-Schmidt orthogonalization 66
4.1.3 Spectral theorem 67
4.2 Matrix transformation to the Jordan form 68
4.2.1 The Jordan block 68
4.2.2 The Jordanmatrix form 69
4.3 Polar and singular-value decompositions 70
4.3.1 Polar decomposition 70
4.3.2 Singular-value decomposition 73
4.4 Congruent matrices and the inertia of a matrix 77
4.4.1 Congruentmatrices 77
4.4.2 Inertia of a squarematrix 78
4.5 Cholesky factorization 81
4.5.1 Upper triangular factorization 81
4.5.2 Numerical realization 83
5 Matrix Functions 85
5.1 Projectors 85
5.2 Functions of amatrix 87
5.2.1 Main definition 87
5.2.2 Matrix exponent 89
5.2.3 Square root of a positive semidefinite matrix 93
5.3 The resolvent formatrix 94
5.4 Matrix norms 98
5.4.1 Norms in linear spaces and in Cn 98
5.4.2 Matrix norms 100
5.4.3 Compatible norms 103
5.4.4 Inducematrix norm 104
6 Moore-Penrose Pseudoinverse 107
6.1 Classical Least Squares Problem 107
6.2 Pseudoinverse characterization 110
6.3 Criterion for pseudoinverse checking 113
6.4 Some identities for pseudoinversematrices 115
6.5 Solution of Least Square Problem
using pseudoinverse 117
6.6 Cline’s formulas 120
6.7 Pseudo-ellipsoids 120
6.7.1 Definition and basic properties 120
6.7.2 Support function 122
6.7.3 Pseudo-ellipsoids containing vector sum of two
pseudo-ellipsoids 123
6.7.4 Pseudo-ellipsoids containing intersection of two
pseudo-ellipsoids 125
7 Hermitian and Quadratic Forms 127
7.1 Definitions 127
7.2 Nonnegative definitematrices 130
7.2.1 Nonnegative definiteness 130
7.2.2 Nonnegative (positive) definiteness
of a partitionedmatrix 133
7.3 Sylvester criterion 136
vi CONTENTS
7.4 The simultaneous transformation of pair of quadratic
forms 138
7.4.1 The case when one of quadratic form is
strictly positive 138
7.4.2 The case when both quadratic forms are nonnegative
. 139
7.5 Simultaneous reduction of more than two quadratic forms141
7.6 A related maximum-minimum
problem 142
7.6.1 Rayleigh quotient 142
7.6.2 Main properties of the Rayleigh quotient 143
7.7 The ratio of two quadratic forms 145
8 Linear Matrix Equations 147
8.1 General type of linear matrix
equation 147
8.1.1 General linearmatrix equation 147
8.1.2 Spreading operator and Kronecker product 147
8.1.3 Relation between the spreading operator
and theKronecker product 148
8.1.4 Solution of a general linear matrix equation 150
8.2 Sylvestermatrix equation 151
8.3 Lyapunovmatrix equation 152
9 Stable Matrices and Polynomials 153
9.1 Basic definitions 153
9.2 Lyapunov stability 154
9.2.1 Lyapunov matrix equation for stable
matrices 154
9.3 Necessary condition of the matrix
stability 159
9.4 The Routh-Hurwitz criterion 160
9.5 The Liénard-Chipart criterion 169
9.6 Geometric criteria 170
9.6.1 The principle of argument variation 170
9.6.2 TheMikhailov’s criterion 172
9.7 Polynomial robust stability 175
CONTENTS vii
9.7.1 Parametric uncertainty and robust
stability 175
9.7.2 TheKharitonov’s theorem 177
9.7.3 The Polyak-Tsypkin geometric criterion 180
9.8 Controllable, stabilizable, observable and detectable pairs182
9.8.1 Controllability and a controllable pair of
matrices 182
9.8.2 Stabilizability and a stabilizable pair of
matrices 188
9.8.3 Observability and an observable pair of
matrices 189
9.8.4 Detectability and a detectable pair of
matrices 193
9.8.5 Popov-Belevitch-Hautus (PBH) test 193
10 Algebraic Riccati Equation 195
10.1 Hamiltonianmatrix 195
10.2 All solutions of the algebraic Riccati equation 196
10.2.1 Invariant subspaces 196
10.2.2 Main theorems on the solution presentation 197
10.2.3 Numerical example 200
10.3 Hermitian and symmetric solutions 201
10.3.1 No pure imaginary eigenvalues 201
10.3.2 Unobservablemodes 206
10.3.3 All real solutions 207
10.3.4 Numerical example 208
10.4 Nonnegative solutions 210
10.4.1 Main theorems on the algebraic Riccati
equation solution 210
11 Linear Matrix Inequalities 215
11.1 Matrices as variables
and LMI problem 215
11.1.1 Matrix inequalities 215
11.1.2 LMI as a convex constraint 217
11.1.3 Feasible and infeasible LMI 217
11.2 Nonlinear matrix inequalities
equivalent to LMI 218
viii CONTENTS
11.2.1 Matrix normconstraint 218
11.2.2 Nonlinear weighted normconstraint 219
11.2.3 Nonlinear trace normconstraint 219
11.2.4 Lyapunov inequality 219
11.2.5 Algebraic Riccati - Lurie’s matrix inequality 220
11.2.6 Quadratic inequalities and S-procedure 220
11.3 Some characteristics of linear
stationary systems (LSS) 221
11.3.1 LSS and their transfer function 221
11.3.2 H2 norm 222
11.3.3 Passivity and the positive-real lemma 222
11.3.4 Nonexpansivity and the bounded-real
lemma 224
11.3.5 H norm 226
11.3.6 γ-Entropy 227
11.3.7 Stability of stationary time-delay systems 227
11.3.8 Hybrid time-delay linear stability 228
11.4 Optimization problems with LMI
constraints 229
11.4.1 Eigenvalue problem(EVP) 229
11.4.2 Tolerance level optimization 230
11.4.3 Maximization of the quadratic stability degree 230
11.4.4 Minimization of linear function Tr (CP C|) under
the Lyapunov-type constraint 231
11.4.5 The convex function log det A−1 (X)
minimization 232
11.5 Numerical methods for LMIs
resolution 233
11.5.1 What does itmean "to solve LMI"? 233
11.5.2 Ellipsoid algorithm 233
11.5.3 Interior-pointmethod 237
12 Miscellaneous 239
12.1 Λ-matrix inequalities 239
12.2 MatrixAbel identities 240
12.2.1 Matrix summation by parts 240
12.2.2 Matrix product identity 241
CONTENTS ix
12.3 S-procedure and Finsler lemma 242
12.3.1 Daneš’ theorem 242
12.3.2 S-procedure 244
12.3.3 Finsler lemma 247
12.4 Farkaš lemma 249
12.4.1 Formulation of the lemma 249
12.4.2 Axillary bounded LS-problem 250
12.4.3 Proof of Farkaš lemma 252
12.4.4 The steepest descent problem 253
12.5 Kantorovichmatrix inequality 253
II ANALYSIS 255
13 The Real and Complex Number Systems 257
13.1 Ordered sets 257
13.1.1 Order 257
13.1.2 Infimumand supremum 258
13.2 Fields 258
13.2.1 Basic definition andmain axioms 258
13.2.2 Some important properties 259
13.3 The real field 261
13.3.1 Basic properties 261
13.3.2 Intervals 262
13.3.3 Maximumandminimumelements 262
13.3.4 Some properties of the supremum 263
13.3.5 Absolute value and the triangle inequality 264
13.3.6 The Cauchy-Schwarz inequality 266
13.3.7 The extended real number system 266
13.4 Euclidian spaces 267
13.5 The complex field 268
13.5.1 Basic definition and properties 268
13.5.2 The imaginary unite 269
13.5.3 The conjugate and absolute value of a complex
number 270
13.5.4 The geometric representation of complex numbers272
13.6 Some simplest complex functions 274
13.6.1 Power 274
x CONTENTS
13.6.2 Roots 275
13.6.3 Complex exponential 276
13.6.4 Complex logarithms 277
13.6.5 Complex sines and cosines 278
14 Sets, Functions and Metric Spaces 281
14.1 Functions and sets 281
14.1.1 The function concept 281
14.1.2 Finite, countable and uncountable sets 282
14.1.3 Algebra of sets 283
14.2 Metric spaces 287
14.2.1 Metric definition and examples of metrics 287
14.2.2 Set structures 288
14.2.3 Compact sets 292
14.2.4 Convergent sequences in metric spaces 294
14.2.5 Continuity and function limits in metric spaces 301
14.2.6 The contraction principle and a fixed point theorem
. 310
14.3 Resume 310
15 Integration 311
15.1 Naive interpretation 311
15.1.1 What is the Riemann integration? 311
15.1.2 What is the Lebesgue integration? 312
15.2 The Riemann-Stieltjes integral 313
15.2.1 Riemann integral definition 313
15.2.2 Definition of Riemann-Stieltjes integral 315
15.2.3 Main properties of the Riemann-Stieltjes integral 316
15.2.4 Different types of integrators 321
15.3 The Lebesgue-Stieltjes integral 332
15.3.1 Algebras, σ-algebras and additive functions of sets332
15.3.2 Measure theory 335
15.3.3 Measurable spaces and functions 344
15.3.4 The Lebesgue-Stieltjes integration 348
15.3.5 The "almost everywhere" concept 352
15.3.6 "Atomic" measures and δ - function 354
CONTENTS xi
16 Selected Topics of Real Analysis 357
16.1 Derivatives 357
16.1.1 Basic definitions and properties 357
16.1.2 Derivative of multivariable functions 362
16.1.3 Inverse function theorem 368
16.1.4 Implicit function theorem 371
16.1.5 Vector and matrix differential calculus 374
16.1.6 Nabla-operator in 3-dimensional space 376
16.2 OnRiemann-Stieltjes integrals 378
16.2.1 The necessary condition for existence of Riemann-
Stieltjes integrals 378
16.2.2 The sufficient conditions for existence of Riemann-
Stieltjes integrals 380
16.2.3 Mean-value theorems 381
16.2.4 The integral as a function of the interval 383
16.2.5 Derivative integration. 384
16.2.6 Integrals depending on a parameters and differentiation
under integral sign 385
16.3 On Lebesgue integrals 388
16.3.1 Lebesgue’s monotone convergence theorem 388
16.3.2 Comparison with the Riemann integral 390
16.3.3 Fatou’s lemma 391
16.3.4 Lebesgue’s dominated convergence 393
16.3.5 Fubini’s reduction theorem 394
16.3.6 Coordinate transformation in an integral 399
16.4 Integral inequalities 401
16.4.1 Generalized Chebyshev Inequality 401
16.4.2 Markov and Chebyshev Inequalities 402
16.4.3 Hölder Inequality 403
16.4.4 Cauchy-Bounyakovski-Schwartz inequality 405
16.4.5 Jensen inequality 406
16.4.6 Lyapunov inequality 410
16.4.7 Kulbac inequality 412
16.4.8 Minkowski inequality 413
16.5 Numerical sequences 416
16.5.1 Infinite series 416
16.5.2 Infinite products 428
16.5.3 Teöplitz lemma 432
xii CONTENTS
16.5.4 Kronecker lemma 433
16.5.5 Abel-Dini lemma 434
16.6 Recurrent inequalities 436
16.6.1 On the sumof a series estimation 436
16.6.2 Linear recurrent inequalities 437
16.6.3 Recurrent inequalities with root terms 442
17 Complex Analysis 447
17.1 Differentiation 447
17.1.1 Differentiability 447
17.1.2 Cauchy-Riemann conditions 448
17.1.3 Theorem on a constant complex function 451
17.2 Integration 452
17.2.1 Paths and curves 452
17.2.2 Contour integrals 454
17.2.3 Cauchy’s integral law 456
17.2.4 Singular points and Cauchy’s residue theorem 460
17.2.5 Cauchy’s integral formula 463
17.2.6 Maximummodulus principle and Schwarz’s lemma468
17.2.7 Calculation of integrals and Jordan lemma 470
17.3 Series expansions 474
17.3.1 Taylor (power) series 474
17.3.2 Laurent series 478
17.3.3 Fourier series 483
17.3.4 Principle of argument 484
17.3.5 Rouché theorem. 486
17.3.6 Fundamental algebra theorem 488
17.4 Integral transformations 489
17.4.1 Laplace transformation (K (t, p) = e−pt) 490
17.4.2 Another transformations 498
18 Topics of Functional Analysis 507
18.1 Linear and normed spaces of functions 508
18.1.1 Space mn of all bounded complex numbers 508
18.1.2 Space ln
p of all summable complex sequences 509
18.1.3 Space C [a, b] of continuous functions 509
18.1.4 Space Ck [a, b] of continuously differentiable functions
. 509
CONTENTS xiii
18.1.5 Lebesgue spaces Lp [a, b] (1 ≤ p < ) 509
18.1.6 Lebesgue spaces L [a, b] 510
18.1.7 Sobolev spaces Slp
(G) 510
18.1.8 Frequency domain spaces Lm×k
p , RLm×k
p , Lm×k

and RLm×k

. 510
18.1.9 Hardy spaces Hm×k
p , RHm×k
p , Hm×k

and RHm×k

511
18.2 Banach spaces 512
18.2.1 Basic definition 512
18.2.2 Examples of incomplete metric spaces 512
18.2.3 Completion ofmetric spaces 513
18.3 Hilbert spaces 515
18.3.1 Definition and examples 515
18.3.2 Orthogonal complement 516
18.3.3 Fourier series in Hilbert spaces 518
18.3.4 Linear n-manifold approximation 520
18.4 Linear operators and functionals in Banach spaces 521
18.4.1 Operators and functionals 521
18.4.2 Continuity and boundedness 522
18.4.3 Compact operators 530
18.4.4 Inverse operators 532
18.5 Duality 537
18.5.1 Dual spaces 537
18.5.2 Adjoint (dual) and self-adjoint operators 540
18.5.3 Riesz representation theorem for Hilbert spaces 543
18.5.4 Orthogonal projection operators in Hilbert spaces543
18.6 Monotonic, nonnegative and
coercive operators 546
18.6.1 Basic definitions and properties 547
18.6.2 Galerkin method for equations with
monotone operators 550
18.6.3 Main theorems on the existence of solutions for
equations withmonotone operators 552
18.7 Differentiation of NonlinearOperators 555
18.7.1 Fréchet derivative 555
18.7.2 Gáteaux derivative 557
18.7.3 Relation with "Variation Principle" 558
18.8 Fixed-point Theorems 559
18.8.1 Fixed-points of a nonlinear operator 559
xiv CONTENTS
18.8.2 Brouwer fixed-point theorem 561
18.8.3 Schauder fixed-point theorem 565
18.8.4 The Leray-Schauder principle and a priory estimates
. 567
III DIFFERENTIAL EQUATIONS AND
OPTIMIZATION 569
19 Ordinary Differential Equations 571
19.1 Classes ofODE 571
19.2 RegularODE 572
19.2.1 Theorems on existence 572
19.2.2 Differential inequalities, extension and
uniqueness 579
19.2.3 LinearODE 590
19.2.4 Index of increment for ODE solutions 599
19.2.5 Riccati differential equation 600
19.2.6 Linear first order partial DE 603
19.3 Carathéodory’s TypeODE 606
19.3.1 Main definitions 606
19.3.2 Existence and uniqueness theorems 607
19.3.3 Variable structure and singular perturbed ODE 610
19.4 ODE withDRHS 612
19.4.1 Why ODE with DRHS are important in Control
Theory 612
19.4.2 ODE with DRHS and differential inclusions 617
19.4.3 Slidingmode control 623
20 Elements of Stability Theory 643
20.1 Basic Definitions 643
20.1.1 Origin as an equilibrium 643
20.1.2 Positive definite functions 644
20.2 Lyapunov Stability 646
20.2.1 Main definitions and examples 646
20.2.2 Criteria of stability: non - constructive theory 649
20.2.3 Sufficient conditions of asymptotic stability: constructive
theory 658
CONTENTS xv
20.3 Asymptotic global stability 663
20.3.1 Definition of asymptotic global stability 663
20.3.2 Asymptotic global stability for stationary systems663
20.3.3 Asymptotic global stability for non - stationary
system 666
20.4 Stability of Linear Systems 669
20.4.1 Asymptotic and exponential stability of linear
time-varying systems 669
20.4.2 Stability of linear system with periodic coefficients672
20.4.3 BIBO stability of linear time-varying systems 673
20.5 Absolute Stability 676
20.5.1 Linear systems with nonlinear feedbacks 676
20.5.2 Aizerman andKalman conjectures 677
20.5.3 Analysis of absolute stability 679
20.5.4 Popov’s sufficient conditions 682
20.5.5 Geometric interpretation of Popov’s conditions 684
20.5.6 Yakubovich-Kalman lemma 686
21 Finite-Dimensional Optimization 691
21.1 Some Properties of Smooth
Functions 691
21.1.1 Differentiability remainder 691
21.1.2 Convex functions 696
21.2 UnconstrainedOptimization 703
21.2.1 Extremumconditions 703
21.2.2 Existence, uniqueness and stability of a minimum705
21.2.3 Some numerical procedure of optimization 708
21.3 ConstrainedOptimization 716
21.3.1 Elements of ConvexAnalysis 716
21.3.2 Optimization on convex sets 725
21.3.3 Mathematical programming and Lagrange principle
. 728
21.3.4 Method of the subgradient projection to simplest
convex sets 736
21.3.5 Arrow-Hurwicz-Uzawa method with
the regularization 739
xvi CONTENTS
22 Variational Calculus and Optimal Control 749
22.1 Basic Lemmas of Variation Calculus 749
22.1.1 Du Bois-Reymond lemma 749
22.1.2 Lagrange lemma 753
22.1.3 Lemma on quadratic functional 754
22.2 Functionals and Their Variations 755
22.3 ExtremumConditions 756
22.3.1 Extremal curves 756
22.3.2 Necessary conditions 757
22.3.3 Sufficient conditions 758
22.4 Optimization of integral functionals 760
22.4.1 Curves with fixed boundary points 760
22.4.2 Curves with non-fixed boundary points 771
22.4.3 Curves with a non-smoothness point 774
22.5 Optimal Control Problem 775
22.5.1 Controlled plant, cost functionals and terminal
set 775
22.5.2 Feasible and admissible control 777
22.5.3 Problem setting in the general Bolza form 777
22.5.4 Mayer formrepresentation 778
22.6 MaximumPrinciple 779
22.6.1 Needle-shape variations 779
22.6.2 Adjoint variables and MP formulation 782
22.6.3 The regular case 786
22.6.4 Hamiltonian form and constancy property 787
22.6.5 Non fixed horizon optimal control problem and
Zero-property 789
22.6.6 Joint optimal control and parametric optimization
problem. 792
22.6.7 Sufficient conditions of optimality 794
22.7 Dynamic Programming 800
22.7.1 Bellman´s Principle of Optimality 801
22.7.2 Sufficient conditions for BP fulfilling 802
22.7.3 Invariant embedding 804
22.7.4 Hamilton-Jacoby-Bellman equation 807
22.8 LinearQuadraticOptimal Control 811
22.8.1 Non stationary linear systems and quadratic criterion
22.8.2 LinearQuadratic Problem 812
22.8.3 Maximum Principle for DLQ problem 813
22.8.4 Sufficiency condition 814
22.8.5 Riccati differential equation and feedback optimal
control 815
22.8.6 Linear feedback control 815
22.8.7 Stationary systems on the infinite horizon 819
22.9 Linear-Time optimization 827
22.9.1 General result 827
22.9.2 Theorem on n-intervals for stationary linear systems
23 H2 and H Optimization 831
23.1 H2 -Optimization 831
23.1.1 Kalman canonical decompositions 831
23.1.2 Minimal and balanced realizations 836
23.1.3 H2 normand its computing 840
23.1.4 H2 optimal control problem and its solution 844
23.2 H -Optimization 848
23.2.1 L, H norms 848
23.2.2 Laurent, Toeplitz and Hankel operator 852
23.2.3 Nehari problem in RLm×k
23.2.4 Model-matching (MMP) problem 870
23.2.5 Some control problem converted to MMP 881

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