Problem 8.1
-norm approximation
A. C. Antoulas
Department of Electrical and Computer Engineering
Rice University
6100 South Main Street
Houston, TX 77005-1892
USA
A. Astolfi
Department of Electrical and Electronic Engineering
Department of Electrical Imperial College
Exhibition Road
SW7 2BT, London
United Kingdom
1 DESCRIPTION OF THE PROBLEM
Let 
be the (Hardy) space of real-rational scalar1 transfer functions of order m, bounded on the imaginary axisand analytic into the right-half complex plane. The optimal approximation
problem in the norm can be statedas follows.
(A?) (Optimal Approximation in the H1 norm)
Given 
and an integer n < N find,2 if
possible, A*(s) ε 
such that
2 AVAILABLE RESULTS AND POSSIBLE SOLUTION PATHS
Approximation and model reduction have always been central issues in system theory. For a recent survey on model reduction in the large-scale setting, we refer the reader to [1]. There are several results in this area. If the approximation is performed in the Hankel norm, then an explicit solution of the optimal approximation and model reduction problems has been given in [3].
Note that this procedure provides, as a byproduct, an upper bound for 
and a solution of the suboptimal approximation problem. If the approximation is performed in the H2 norm, several results
and numerical algorithms are available [4]. For approximation in the norm a conceptual solution is given in [5]. Therein it is shown that the approximation problem can be reduced to a Hankel norm approximation problem for an extended system (i.e., a system obtained
from a state space realization of the original transfer function G(s) by adding inputs and outputs). The extended system has
to be constructed with the constraint that the corresponding Grammians P and Q satisfy
BIBLIOGRAPHY
[1] A. C. Antoulas, “Lectures on the approximation of large scale dynami-calsystems, ” SIAM, Philadelphia, 2002.
[2] A. C. Antoulas, D. C. Sorensen, and Y. Zhou, “On the decay rate of Hankel singular values and related issues, ” Systems and Control Letters, 2002.
[3] K. Glover, “All optimal Hankel-norm approximations of linear multivariable systems and their 
error bounds, ” International Journal of Control, 39: 1115-1193, 1984.
[4] X.-X. Huang, W.-Y. Yan and K. L. Teo, “H2 near optimal model reduction, ” IEEE Trans. Automatic Control, 46: 1279-1285, 2001.
[5] D. Kavranoglu and M. Bettayeb, “Characterization of the solution to the optimal Hmodel reduction problem, ” Systems and Control Letters, 20: 99-108, 1993.
[6] H. G. Sage and M. F. de Mathelin, “Canonical state space parametrization, ” Automatica, July 2000.
[7] L. N. Trefethen, “Rational Chebyshev approximation on the unitdisc, ” Numerische Mathematik, 37: 297-320, 1981.
1Similar considerations can be done for the nonscalar case.
2By find we mean find an exact solution or an algorithm converging to the exact solution.