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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 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, tol are optional; if not present 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 y, the low-order approximate model G enjoys the H_{∞} norm bound
$${\Vert G-{G}_{N}\Vert}_{\infty}\le totbnd$$
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.