Mathematical Institute, room 224
Leiden University
P.O. Box 9512
2300 RA Leiden
We give an overview of balanced parametrizations for lossless systems (based on joint work with R. Ober and with R. Peeters, M. Olivi & J-P Marmorat) and we show its relevance for a wide range of problems in the area of system identification and model order reduction. This relevance is mainly due to the separable least squares approach to model order reduction and system identification: Model order reduction problems and system identification problems often lead to optimization problems over a family of linear systems. By partial optimization with respect to a number of parameters these can be reduced to optimization problems over the family of lossless systems. This can be used as the basis of exact global optimization algorithms as well as finite precision local search algorithms. We will discuss some of the merits and drawbacks of the various approaches that one can take.
References:
-B. Hanzon, R.J. Ober, "Overlapping block-balanced canonical forms and parametrizations: the stable SISO case", SIAM Journal of Control and Optimization, vol. 35, no. 1, pp. 228-242, 1997.
-B. Hanzon, R.J. Ober, "Overlapping block-balanced canonical forms for various classes of linear systems", Linear Algebra and Its Applications, vol. 281, pp. 171-225, 1998.
-B. Hanzon, R.L.M. Peeters, "Balanced parametrizations of stable SISO all-pass systems in discrete time", Math. Control Signals and Systems, vol. 13, pp. 240-276, 2000.
-B. Hanzon, M. Olivi and R.L.M. Peeters, "Balanced realizations fo discrete-time stable all-pass systems and the tangential Schur algorithm", Proceedings of the European Control Conference ECC1999(Karlsruhe); http://www-sop.inria.fr/miaou/Martine.Olivi/publis.html
-R.L.M. Peeters, B. Hanzon, M. Olivi, "Linear fractional transformations and balanced realizations of discrete-time stable all-pass systems", Proc. 1st IFAC Symposium on System Structure and Control, Prague, 2001; http://www-sop.inria.fr/miaou/Martine.Olivi/publis.html
-R. Peeters, M. Olivi, B. Hanzon, "On a recursive state-space method for discrete-time H_2-approximation", Proceedings MTNS2002, Notre Dame, USA, 2002; http://www-sop.inria.fr/miaou/Martine.Olivi/publis.html http://www.nd.edu/~mtns/papers/18046.pdf
-J.P. Marmorat, M. Olivi, B. Hanzon, R. Peeters, "Matrix rational H_2-approximation: a state-space approach using Schur parameters", Proceedings on the Conference on Decision and Control CDC2002, Las Vegas, USA; http://www-sop.inria.fr/miaou/Martine.Olivi/publis.html
-R.L.M. Peeters, B. Hanzon, D. Jibetean, "Optimal H_2 model reduction in state space: a case study", Proceedings of the European Control Conference ECC2003, Cambridge, UK, 2003.
-D. Jibetean, B. Hanzon, "Linear matrix inequalities for global optimization of rational functions and H_2-optimal model reduction", Proceedings MTNS2002, Notre Dame, USA, 2002; http://www.nd.edu/~mtns/papers/18387.pdf
-Th. Ribarits, M. Deistler, B. Hanzon, "Separable least squares data driven local coordinates", pp. 225-228 in: P. van den Hof, B. Wahlberg, S. Weiland(eds), Proceedings of SYSID2003, Rotterdam, 2003
-B. Hanzon, J.M. Maciejowski, "Constructive algebra methods for the L_2-problem for stable linear systems", Automatica, vol. 32, no.12, pp. 1645-1657, 1996.
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