This study addresses both classical and previously unsolved optimization tasks in linear algebra, linear systems theory, control theory and signal processing. Exploiting developments in Computation such as Parallel Computing and Neural Networks, it gives a dynamical systems approach for tackling a wide class of constrained optimization tasks. Optimization and Dynamical Systems will be of interest to engineers and mathematicians. Engineers will learn the mathematics and the technical approach necessary to solve a wide class of constrained optimization tasks. Mathematicians will see how techniques from global analysis and differential geometry can be developed to achieve useful construction procedures for optimization on manifolds.
Author(s): Uwe Helmke, John B. Moore, R. Brockett
Series: Communications and Control Engineering
Publisher: Springer
Year: 1996
Language: English
Pages: 414
Title Page......Page 1
Preface......Page 2
Acknowledgements......Page 3
Foreword......Page 4
Contents......Page 8
1.1 Introduction......Page 12
1.2 Power Method for Diagonalization......Page 16
1.3 The Rayleigh Quotient Gradient Flow......Page 25
1.4 The QR Algorithm......Page 41
1.5 Singular Value Decomposition (SVD)......Page 46
1.6 Standard Least Squares Gradient Flows......Page 49
2.1 Double Bracket Flows for Diagonalization......Page 54
2.2 Toda Flows and the Riccati Equation......Page 69
2.3 Recursive Lie-Bracket Based Diagonalization......Page 79
3.1 SVD via Double Bracket Flows......Page 92
3.2 A Gradient Flow Approach to SVD......Page 95
4.1 The Rôle of Double Bracket Flows......Page 112
4.2 Interior Point Flows on a Polytope......Page 122
4.3 Recursive Linear Programming/Sorting......Page 128
5.1 Approximations by Lower Rank Matrices......Page 136
5.2 The Polar Decomposition......Page 154
5.3 Output Feedback Control......Page 157
6.1 Introduction......Page 174
6.2 Kempf-Ness Theorem......Page 176
6.3 Global Analysis of Cost Functions......Page 179
6.4 Flows for Balancing Transformations......Page 183
6.5 Flows on the Factors X and Y......Page 197
6.6 Recursive Balancing Matrix Factorizations......Page 204
7.1 Introduction......Page 212
7.2 Plurisubharmonic Functions......Page 215
7.3 The Azad-Loeb Theorem......Page 218
7.4 Application to Balancing......Page 220
7.5 Euclidean Norm Balancing......Page 231
8.1 Introduction......Page 240
8.2 Flows on Positive Definite Matrices......Page 242
8.3 Flows for Balancing Transformations......Page 257
8.4 Balancing via Isodynamical Flows......Page 260
8.5 Euclidean Norm Optimal Realizations......Page 269
9.1 A Sensitivity Minimizing Gradient Flow......Page 280
9.2 Related L2-Sensitivity Minimization Flows......Page 293
9.3 Recursive L2-Sensitivity Balancing......Page 302
9.4 L2-Sensitivity Model Reduction......Page 306
9.5 Sensitivity Minimization with Constraints......Page 309
A.1 Matrices and Vectors......Page 322
A.3 Determinant and Rank of a Matrix......Page 323
A.4 Range Space, Kernel and Inverses......Page 324
A.6 Eigenvalues, Eigenvectors and Trace......Page 325
A.7 Similar Matrices......Page 326
A.8 Positive Definite Matrices and Matrix Decompositions......Page 327
A.9 Norms of Vectors and Matrices......Page 328
A.11 Differentiation and Integration......Page 329
A.13 Vector Spaces and Subspaces......Page 330
A.15 Mappings and Linear Mappings......Page 331
A.16 Inner Products......Page 332
B.1 Linear Dynamical Systems......Page 334
B.3 Controllability and Stabilizability......Page 336
B.4 Observability and Detectability......Page 337
B.6 Markov Parameters and Hankel Matrix......Page 338
B.7 Balanced Realizations......Page 339
B.8 Vector Fields and Flows......Page 340
B.9 Stability Concepts......Page 341
B.10 Lyapunov Stability......Page 343
C.1 Point Set Topology......Page 346
C.2 Advanced Calculus......Page 349
C.3 Smooth Manifolds......Page 352
C.4 Spheres, Projective Spaces and Grassmannians......Page 354
C.5 Tangent Spaces and Tangent Maps......Page 357
C.6 Submanifolds......Page 360
C.7 Groups, Lie Groups and Lie Algebras......Page 362
C.8 Homogeneous Spaces......Page 366
C.9 Tangent Bundle......Page 368
C.10 Riemannian Metrics and Gradient Flows......Page 371
C.11 Stable Manifolds......Page 373
C.12 Convergence of Gradient Flows......Page 376
References......Page 378
Author Index......Page 400
Subject Index......Page 406
