Optimization and Dynamical Systems
Published on 9. April 2014
Book
Paperback/Softback
XIII, 403 pages
978-1-4471-3469-5 (ISBN)
Description
This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys tems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods.
More details
Series
Edition
1994 ed.
Language
English
Place of publication
London
United Kingdom
Target group
Professional and scholarly
Research
Illustrations
2 s/w Abbildungen
XIII, 403 p. 2 illus.
Dimensions
Height: 254 mm
Width: 178 mm
Thickness: 23 mm
Weight
786 gr
ISBN-13
978-1-4471-3469-5 (9781447134695)
DOI
10.1007/978-1-4471-3467-1
Schweitzer Classification
Other editions
Additional editions
Uwe Helmke | John B. Moore
Optimization and Dynamical Systems
E-Book
12/2012
Springer
€117.69
Available for download
Uwe Helmke | John B. Moore
Optimization and Dynamical Systems
Book
05/1996
Springer
€85.55
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Persons
Paul A. Fuhrmann is Professor Emeritus in the Department of Mathematics at Ben-Gurion University of the Negev. His current research interests are in the following directions: Algebraic System Theory, control, observation and model reduction for networks of linear systems.
Uwe Helmke is Professor and Chair in Dynamical Systems and Control Theory at the University of Wuerzburg. His current research interests include Algebraic and Differential-Geometric Methods in Control, Linear Systems Theory, Optimization on Manifolds, Formation Control.
Content
1 Matrix Eigenvalue Methods.- 1.1 Introduction.- 1.2 Power Method for Diagonalization.- 1.3 The Rayleigh Quotient Gradient Flow.- 1.4 The QR Algorithm.- 1.5 Singular Value Decomposition (SVD).- 1.6 Standard Least Squares Gradient Flows.- 2 Double Bracket Isospectral Flows.- 2.1 Double Bracket Flows for Diagonalization.- 2.2 Toda Flows and the Riccati Equation.- 2.3 Recursive Lie-Bracket Based Diagonalization.- 3 Singular Value Decomposition.- 3.1 SVD via Double Bracket Flows.- 3.2 A Gradient Flow Approach to SVD.- 4 Linear Programming.- 4.1 The Rôle of Double Bracket Flows.- 4.2 Interior Point Flows on a Polytope.- 4.3 Recursive Linear Programming/Sorting.- 5 Approximation and Control.- 5.1 Approximations by Lower Rank Matrices.- 5.2 The Polar Decomposition.- 5.3 Output Feedback Control.- 6 Balanced Matrix Factorizations.- 6.1 Introduction.- 6.2 Kempf-Ness Theorem.- 6.3 Global Analysis of Cost Functions.- 6.4 Flows for Balancing Transformations.- 6.5 Flows on the Factors X and Y.- 6.6 Recursive Balancing Matrix Factorizations.- 7 Invariant Theory and System Balancing.- 7.1 Introduction.- 7.2 Plurisubharmonic Functions.- 7.3 The Azad-Loeb Theorem.- 7.4 Application to Balancing.- 7.5 Euclidean Norm Balancing.- 8 Balancing via Gradient Flows.- 8.1 Introduction.- 8.2 Flows on Positive Definite Matrices.- 8.3 Flows for Balancing Transformations.- 8.4 Balancing via Isodynamical Flows.- 8.5 Euclidean Norm Optimal Realizations.- 9 Sensitivity Optimization.- 9.1 A Sensitivity Minimizing Gradient Flow.- 9.2 Related L2-Sensitivity Minimization Flows.- 9.3 Recursive L2-Sensitivity Balancing.- 9.4 L2-Sensitivity Model Reduction.- 9.5 Sensitivity Minimization with Constraints.- A Linear Algebra.- A.1 Matrices and Vectors.- A.2 Addition and Multiplication of Matrices.- A.3Determinant and Rank of a Matrix.- A.4 Range Space, Kernel and Inverses.- A.5 Powers, Polynomials, Exponentials and Logarithms.- A.6 Eigenvalues, Eigenvectors and Trace.- A.7 Similar Matrices.- A.8 Positive Definite Matrices and Matrix Decompositions.- A.9 Norms of Vectors and Matrices.- A.10 Kronecker Product and Vec.- A.11 Differentiation and Integration.- A.12 Lemma of Lyapunov.- A.13 Vector Spaces and Subspaces.- A.14 Basis and Dimension.- A.15 Mappings and Linear Mappings.- A.16 Inner Products.- B Dynamical Systems.- B.1 Linear Dynamical Systems.- B.2 Linear Dynamical System Matrix Equations.- B.3 Controllability and Stabilizability.- B.4 Observability and Detectability.- B.5 Minimality.- B.6 Markov Parameters and Hankel Matrix.- B.7 Balanced Realizations.- B.8 Vector Fields and Flows.- B.9 Stability Concepts.- B.10 Lyapunov Stability.- C Global Analysis.- C.1 Point Set Topology.- C.2 Advanced Calculus.- C.3 Smooth Manifolds.- C.4 Spheres, Projective Spaces and Grassmannians.- C.5 Tangent Spaces and Tangent Maps.- C.6 Submanifolds.- C.7 Groups, Lie Groups and Lie Algebras.- C.8 Homogeneous Spaces.- C.9 Tangent Bundle.- C.10 Riemannian Metrics and Gradient Flows.- C.11 Stable Manifolds.- C.12 Convergence of Gradient Flows.- References.- Author Index.