Hankel-norm approximation and model reduction of time-varying systems
Alle-Jan van der Veen and Patrick Dewilde Delft University of Technology
We consider the Hankel-norm approximation problem for (bounded) upper triangular operators: generalized `2
-operators T with matrix representations[ Ti j]∞−∞such that Ti j= 0 (i > j). Here, each Ti jis a matrix with dimensions Mi×Nj, where Mi, Nj are finite integers (possibly zero). An upper operator can be viewed as the transfer operator of a causal linear time-varying system.
Associated to T is a sequence of ‘Hankel’-operators Hk= [ Tk−i+1,k+ j]∞0 (k =−∞· · ·∞), which are submatrices of
T corresponding to its top-right parts. The rank of these operators plays an important role in realization theory: if dk= rank Hk<∞, then there exist minimal time-varying realizations with system order dkat point k:
xk+1 = xkAk + ukBk Ak: dk×dk+1, Bk: Mk×dk+1
yk = xkCk + ukDk Ck: dk×Nk, Dk: Mk×Nk
(1)
such that[ · · · y0 y1· · ·] = [ · · · u0 u1· · ·]T.
The Hankel norm of T is defined to bek T kH= sup k Hkk. This definition is a generalization of the time-invariant Hankel norm and reduces to it if all Hkare the same. LetΓ= diag(γi) be an acceptable approximation tolerance, withγi> 0. If an operator Tais such that
kΓ−1(T−Ta) kH ≤ 1 , (2) then Tais called a Hankel norm approximant of T , parameterized byΓ. We are interested in Hankel norm approx-imants of minimal system order. In [1], we proved that if the number of Hankel singular values ofΓ−1T that are larger than 1 is equal to Nkat point k, then there exists a Hankel norm approximant Tawhose system order is equal to Nkat point k, assuming none of the singular values are equal to 1.
In the construction of such Hankel-norm approximants Ta, two additional operators play a role. The first is U: the inner (i.e. upper and unitary) factor of T in a coprime factorization T=∆∗U (where∆is upper). The second isΘ: a J-unitary operator (Θ∗J1Θ= J2,ΘJ2Θ∗= J1, where J1, J2are signature matrices) such that
[U∗ − Γ−1T∗]Θ = [A0 −B0] (3) consists of two upper operators A0, B0. This equation describes an interpolation problem. Θexists under certain conditions and can be constructed explicitly, and the resulting signature matrices are determined by the singular values of the Hk. In fact, the system order of the strictly upper part ofΘ−122is at each point k equal to Nk.
Let SLbe an upper contractive operator. Then
S = (Θ11SL− Θ12)(Θ22− Θ21SL)−1 (4)
is contractive, and the strictly upper part of T0= T +ΓS∗U is a Hankel norm approximant. Conversely, for each T0of which the strictly upper part Tais a Hankel norm approximant, there is an upper contractive operator SLsuch thatΓ−1(T0−T) = S∗U, where S is given by the above expression.
[1] P.M. Dewilde and A.J. van der Veen, “On the Hankel-Norm Approximation of Upper-Triangular Operators and Matrices,” accepted for Integral Equations and Operator Theory, September 1992.
