 
Summary: H2OPTIMAL APPROXIMATION OF MIMO LINEAR DYNAMICAL
SYSTEMS
PAUL VAN DOOREN, KYLE A. GALLIVAN¶ , AND P.A. ABSIL§
Submitted on 30 JUL 2008
Abstract. We consider the problem of approximating a multipleinput multipleoutput (MIMO)
p×m rational transfer function H(s) of high degree by another p×m rational transfer function bH(s) of
much smaller degree, so that the H2 norm of the approximation error is minimized. We characterize
the stationary points of the H2 norm of the approximation error by tangential interpolation conditions
and also extend these results to the discretetime case. We analyze whether it is reasonable to assume
that lowerorder models can always be approximated arbitrarily closely by imposing only firstorder
interpolation conditions. Finally, we analyze the H2 norm of the approximation error for a simple
case in order to illustrate the complexity of the minimization problem.
Key words. Multivariable systems, model reduction, optimal H2 approximation, tangential
interpolation.
AMS subject classifications. 41A05, 65D05, 93B40
1. Introduction. In this paper, we consider the problem of approximating a
real p × m rational transfer function H(s) of McMillan degree N by a real p × m
rational transfer function H(s) of lower McMillan degree n using the H2norm as the
approximation criterion. We refer, e.g., to [Che99, Ant05] for the relevant background
on linear system theory and model reduction.
