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System realization via Hankel singular value decomposition

[a,b,c,d,totbnd,hsv] = imp2ss(y) [a,b,c,d,totbnd,hsv] = imp2ss(y,ts,nu,ny,tol) [ss,totbnd,hsv] = imp2ss(imp) [ss,totbnd,hsv] = imp2ss(imp,tol)

`The function ``imp2ss`
produces an approximate state-space realization of a given impulse
response

imp=mksys(y,t,nu,ny,'imp');

using the Hankel SVD method proposed by S. Kung [2]. A continuous-time realization is computed
via the inverse Tustin transform (using `bilin`)
if *t* is positive; otherwise a discrete-time realization
is returned. In the SISO case the variable *y* is
the impulse response vector; in the MIMO case *y* is
an *N+*1-column matrix containing *N* +
1 time samples of the matrix-valued impulse response *H*_{0},
..., *H _{N}* of an

*y* = [*H*_{0}(:)′;*H*_{2}(:)′; *H*_{3}(:)′;
... ;*H _{N}*(:)′

The variable *tol* bounds the *H*_{∞} norm
of the error between the approximate realization (*a, b, c,
d*) and an exact realization of *y*; the
order, say *n*, of the realization (*a, b,
c, d*) is determined by the infinity norm error bound specified
by the input variable `tol`. The inputs `ts,
nu, ny, tol` are optional; if not present they default to
the values `ts = 0``,` `nu
= 1``,` `ny =` (number
of rows of *y*)/`nu`, `tol` =
. The output
returns the singular values (arranged
in descending order of magnitude) of the Hankel matrix:

Denoting by *G _{N}* a high-order
exact realization of

where

[1] Al-Saggaf, U.M., and G.F. Franklin, "An
Error Bound for a Discrete Reduced Order Model of a Linear Multivariable
System," *IEEE Trans. on Autom. Contr*.,
AC-32, 1987, p. 815-819.

[2] Kung, S.Y., "A New Identification
and Model Reduction Algorithm via Singular Value Decompositions," *Proc.Twelth
Asilomar Conf. on Circuits, Systems and Computers*, November
6-8, 1978, p. 705-714.

[3] Silverman, L.M., and M. Bettayeb, "Optimal
Approximation of Linear Systems," *Proc. American
Control Conf*., San Francisco, CA, 1980.

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