# Computational Aspects of Linear Control

• Claude Brezinski
Book

Part of the Numerical Methods and Algorithms book series (NUAL, volume 1)

1. Front Matter
Pages i-ix
2. Claude Brezinski
Pages 1-2
3. Claude Brezinski
Pages 3-72
4. Claude Brezinski
Pages 73-85
5. Claude Brezinski
Pages 87-134
6. Claude Brezinski
Pages 135-143
7. Claude Brezinski
Pages 145-159
8. Claude Brezinski
Pages 161-169
9. Claude Brezinski
Pages 171-223
10. Claude Brezinski
Pages 225-247
11. Claude Brezinski
Pages 249-253
12. Claude Brezinski
Pages 255-275
13. Claude Brezinski
Pages 277-287
14. Back Matter
Pages 289-295

### Introduction

Many devices (we say dynamical systems or simply systems) behave like black boxes: they receive an input, this input is transformed following some laws (usually a differential equation) and an output is observed. The problem is to regulate the input in order to control the output, that is for obtaining a desired output. Such a mechanism, where the input is modified according to the output measured, is called feedback. The study and design of such automatic processes is called control theory. As we will see, the term system embraces any device and control theory has a wide variety of applications in the real world. Control theory is an interdisci­ plinary domain at the junction of differential and difference equations, system theory and statistics. Moreover, the solution of a control problem involves many topics of numerical analysis and leads to many interesting computational problems: linear algebra (QR, SVD, projections, Schur complement, structured matrices, localization of eigenvalues, computation of the rank, Jordan normal form, Sylvester and other equations, systems of linear equations, regulariza­ tion, etc), root localization for polynomials, inversion of the Laplace transform, computation of the matrix exponential, approximation theory (orthogonal poly­ nomials, Pad6 approximation, continued fractions and linear fractional transfor­ mations), optimization, least squares, dynamic programming, etc. So, control theory is also a. good excuse for presenting various (sometimes unrelated) issues of numerical analysis and the procedures for their solution. This book is not a book on control.

### Keywords

approximation approximation theory difference equation differential equation dynamical systems numerical analysis optimization statistics

### Editors and affiliations

• Claude Brezinski
• 1
1. 1.Université des Sciences et Technologies de LilleFrance

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4613-0261-2
• Copyright Information Springer-Verlag US 2002
• Publisher Name Springer, Boston, MA
• eBook Packages
• Print ISBN 978-1-4020-0711-8
• Online ISBN 978-1-4613-0261-2
• Series Print ISSN 1571-5698