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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

`nu`

-input, `ny`

-output
system stored row-wise:* y* = [

The variable * tol* bounds the

`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 `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

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

The realization (*a, b, c, d*) is computed
using the Hankel SVD procedure proposed by Kung [2] as a method for approximately implementing
the classical Hankel factorization realization algorithm. Kung's SVD
realization procedure was subsequently shown to be equivalent to doing
balanced truncation (`balmr`

) on an exact state-space
realization of the finite impulse response {* y*(1),....

Form the Hankel matrix Γ from the data

.*y*Perform SVD on the Hankel matrix

$$\Gamma =U\sum V*=\left[{U}_{1}{U}_{2}\right]\left[\begin{array}{cc}{\sum}_{1}& 0\\ 0& {\sum}_{2}\end{array}\right]\left[\begin{array}{c}V{*}_{1}\\ V{*}_{2}\end{array}\right]={U}_{1}{\sum}_{1}V{*}_{1}$$

where Σ

_{1}has dimension×*n*and the entries of Σ*n*_{2}are nearly zero.*U*_{1}and*V*_{1}haveand*ny*columns, respectively.*nu*Partition the matrices

*U*_{1}and*V*_{1}into three matrix blocks:$$U1=\left[\begin{array}{c}{U}_{11}\\ {U}_{12}\\ {U}_{13}\end{array}\right]\left[\begin{array}{c}{V}_{11}\\ {V}_{12}\\ {V}_{13}\end{array}\right]$$

where $${U}_{11},{U}_{13}\in {C}^{ny\text{}\times \text{}n}$$ and $${V}_{11},{V}_{13}\in {C}^{nu\text{}\times \text{}n}$$.

A discrete state-space realization is computed as

$$\begin{array}{l}A={\sum}_{1}^{-\text{}{\scriptscriptstyle \frac{1}{2}}}\overline{U}{\sum}_{1}^{-\text{}{\scriptscriptstyle \frac{1}{2}}}\\ B={\sum}_{1}^{-\text{}{\scriptscriptstyle \frac{1}{2}}}V{*}_{11}\\ C={U}_{11}{\sum}_{1}^{-\text{}{\scriptscriptstyle \frac{1}{2}}}\\ D={H}_{0}\end{array}$$

where

$$\overline{U}={\left[\begin{array}{c}{U}_{11}\\ {U}_{12}\end{array}\right]}^{\prime}\text{}\left[\begin{array}{c}{U}_{12}\\ {U}_{13}\end{array}\right]$$

If the sample time

is greater than zero, then the realization is converted to continuous time via the inverse of the Tustin transform*t*$$s=\frac{2}{t}\frac{z-1}{z+1}\text{}.$$

Otherwise, this step is omitted and the discrete-time realization calculated in Step 4 is returned.

[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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