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

`nu`

-input, `ny`

-output
system stored row-wise:*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`

,
and `tol`

are optional. If omitted, they default
to the values `ts = 0`

, `nu = 1`

, ```
ny
=
```

(number of rows of *y*)/`nu`

, `tol`

= $$0.01{\overline{\sigma}}_{1}$$. The output $$hsv=[{\overline{\sigma}}_{1},{\overline{\sigma}}_{2},\mathrm{...}{]}^{\prime}$$returns the singular values (arranged
in descending order of magnitude) of the Hankel matrix:

$$\Gamma =\left[\begin{array}{ccccc}{H}_{1}& {H}_{2}& {H}_{3}& \dots & {H}_{N}\\ {H}_{2}& {H}_{3}& {H}_{4}& \dots & 0\\ {H}_{3}& {H}_{4}& {H}_{5}& \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {H}_{N}& 0& \dots & \dots & 0s\end{array}\right]$$

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

$${\Vert G-{G}_{N}\Vert}_{\infty}\le totbnd$$

where

$$totbnd=2{\displaystyle \sum _{i=n+1}^{N}{\overline{\sigma}}_{i}}.$$

[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.
Twelfth 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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