Time-Varying Systems and Computations

1. Front Matter

   Pages i-xiii

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2. Introduction

   1. Introduction

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 1-15

3. Realization

   1. Front Matter

      Pages 17-17

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   2. Notation and Properties of Non-Uniform Spaces

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 19-31

   3. Time-Varying State Space Realizations

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 33-72

   4. Diagonal Algebra

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 73-85

   5. Operator Realization Theory

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 87-119

   6. Isometric and Inner Operators

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 121-143

   7. Inner-Outer Factorization and Operator Inversion

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 145-187

4. Interpolation and Approximation

   1. Front Matter

      Pages 189-189

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   2. J-Unitary Operators

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 191-231

   3. Algebraic Interpolation

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 233-262

   4. Hankel-Norm Model Reduction

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 263-306

   5. Low-Rank Matrix Approximation and Subspace Tracking

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 307-333

5. Factorization

   1. Front Matter

      Pages 335-335

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   2. Orthogonal Embedding

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 337-362

   3. Spectral Factorization

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 363-381

   4. Lossless Cascade Factorizations

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 383-417

   5. Conclusion

      • Patrick Dewilde, Alle-Jan van der Veen

      Pages 419-422

6. Back Matter

   Pages 423-459

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Complex function theory and linear algebra provide much of the basic mathematics needed by engineers engaged in numerical computations, signal processing or control. The transfer function of a linear time invariant system is a function of the complex vari­ able s or z and it is analytic in a large part of the complex plane. Many important prop­ erties of the system for which it is a transfer function are related to its analytic prop­ erties. On the other hand, engineers often encounter small and large matrices which describe (linear) maps between physically important quantities. In both cases similar mathematical and computational problems occur: operators, be they transfer functions or matrices, have to be simplified, approximated, decomposed and realized. Each field has developed theory and techniques to solve the main common problems encountered. Yet, there is a large, mysterious gap between complex function theory and numerical linear algebra. For example, complex function theory has solved the problem to find analytic functions of minimal complexity and minimal supremum norm that approxi­ e. g. , as optimal mate given values at strategic points in the complex plane. They serve approximants for a desired behavior of a system to be designed. No similar approxi­ mation theory for matrices existed until recently, except for the case where the matrix is (very) close to singular.

• Interpolation
• Matrix
• Signal
• algebra
• bridge
• linear algebra
• model
• signal processing
• system
• systems theory
• matrix theory

• DIMES, Delft University of Technology, Delft, The Netherlands

  Patrick Dewilde, Alle-Jan Veen

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