Model Reduction of Linear Systems, an Interpolation point of View

Acknowledgements First of all, I would like to thank my promotor, Paul Van Dooren. It has been a pleasure, beside an honor, to collaborate with him. His knowledge, his enthusiasm and his friendship have always been a great support for me these last four years. He also gave me the opportunity to work with Yves Genin, and for this, I am also very grateful! I can only thank Yves for our collaboration these two last years. I will always remember our passionate discussions and his fruitful advices. I met Paul and Yves as a student, I leave them as a friend. I would like to thank Kyle Gallivan for his support. My visit at Florida State University was the starting point of this thesis. After two weeks at FSU, I have already had enough material for two years of fruitful research. Many other researchers contributed to this work. To cite only a few, I would like to express my gratitude to Thanos Antoulas, Hugo Woerdeman and Brian Anderson for helpful discussions. Two other researchers contributed a lot to this thesis: Younes Chahlaoui and Damien Lemonnier. I really enjoyed to work with them in our model reduction team at CESAME. I am indebted to all the members of the jury of this thesis, Profs. Adhemar Bultheel, Kyle Gallivan, Jan Willems and Vincent Blondel, for their critical reading which contributed to improve the quality of this manuscript. and Yannick for their administrative and technical assistance. I especially enjoyed the friendship of Yvan Hachez and Jacques Theys who both shared my office. I dedicate this thesis to my wife, Alexandra, and our coming child. Basics N set of nonnegative integer numbers N 0 set of strictly positive integer numbers Z set of integer numbers R field of real numbers C field of complex numbers j complex unit, j = √ −1 Re(z) real part of z Im(z) complex part of z ¯ z complex conjugate of z |z| modulus of z ⇐⇒ if and only if := equal by definition to δ ij Kronecker delta: equals 1 if i = j and zero otherwise dim X dimension of X < ·, · > generic inner product gcd(·, ·) Greatest Common Divisor · p p-norm (1 ≤ p ≤ +∞) · 2 Euclidean norm (vectors) / spectral norm vi Notation glossary Matrix theory I k k × k identity matrix X T transpose of …