Search

We obtain sharp estimates for average norms of multilinear Hankel forms on complex Hilbert space whose entries are independent Rademacher functions.

We consider the problem of approximating in H∞ norm a given discrete inverse-time stable system (possibly improper) T= T 11 T 12 T 21 T 22 by a discrete-time system S with no more than r poles outside the unit disk (possibly at infinity) such that T 11 +S T 12 T 21 T 22 ∞ Here r⩾0 is an integer and γ>0 is a prescribed tolerance. If r=0, this is the four-block Nehari problem while if we consider the “one block case” where T=T11 we obtain the well-known Hankel norm approximation problem. The theoretical developments are based on a frequency domain signature condition while the class of solutions is constructed in state-space in terms of the solutions to two Riccati equations.

Connections are established between Hankel-norm approximation, the problem of finding approximating subspaces in the Hilbert space H 2 , and stability and instability of control systems.

A new unified approach to problems of Hankel norm and balanced approximations is presented which is based on a combination of polynomial algebra and the geometry of invariant subspaces. Contrary to state space methods, where contact with external properties of systems is indirect, the approach presented yields new insights into basic properties of Hankel norm approximation and balanced realizations. Several approximation results are interpreted geometrically in terms of projections. Also duality results in this area are printed out.