ml_ct_ss_hna
Hankel-norm approximation for standard systems.
Contents
Syntax
[Ar, Br, Cr, Dr, info] = ml_ct_ss_hna(A, B, C, D) [Ar, Br, Cr, Dr, info] = ml_ct_ss_hna(A, B, C, D, opts)
[rom, info] = ml_ct_ss_hna(sys) [rom, info] = ml_ct_ss_hna(sys, opts)
Description
This function computes the Hankel-norm approximation of a standard system of the form
x'(t) = A*x(t) + B*u(t), (1) y(t) = C*x(t) + D*u(t). (2)
Therefore, first a balanced realization is computed by using the balanced truncation square-root method with an appropriate tolerance for the minimal realization of the given system. Then the system is transformed using the formulas given in the reference below. As result, a reduced-order system of the form
x'(t) = Ar*x(t) + Br*u(t), (3) y(t) = Cr*x(t) + Dr*u(t) (4)
is computed, such that for the original transfer function G and and the r-th order transfer function Gr it holds
||G - Gr||_{H} = Hsv(r+1), ||G - Gr||_{\infty} <= 2 * (Hsv(r+1) + ... + Hsv(n)),
with Hsv, a vector containing the Hankel singular values of the system.
Note: For unstable systems, first an additive decomposition into the stable and anti-stable parts is performed and then only the stable part will be reduced. That does not change the error formulas.
Input
A - matrix from (1) with dimensions n x n B - matrix from (1) with dimensions n x m C - matrix from (2) with dimensions p x n D - matrix from (2) with dimensions p x m sys - structure or state-space object, containing the standard system's matrices:
Entry | Meaning |
A | matrix from (1) with dimensions n x n |
B | matrix from (1) with dimensions n x m |
C | matrix from (2) with dimensions p x n |
D | matrix from (2) with dimensions p x m |
opts - structure, containing the following optional entries:
Parameter | Meaning |
hankelsignmopts | structure, containing the optional parameters for the matrix sign function used for the decomposition after the transformation of an all-pass system, see ml_signm default: struct() |
hankelsylvopts | structure, containing the optional parameters for the Sylvester equation solver used for the decomposition after the transformation of an all-pass system, see ml_sylv_sgn default: struct() |
lyapdlopts | structure, containing the optional parameters for the computation of the continuous-time Lyapunov equations, see ml_lyapdl_sgn_fac default: struct() |
MinRelTol | nonnegative scalar, tolerance multiplied with the largest characteristic value to determine a minimal realization default: log(n)*eps |
Order | positive integer, order of the resulting reduced-order model chosen by the user if 'order' is set for OrderComputation default: min(10,length(Hsv)) + Nu |
OrderComputation | character array, determining the method for the computation of the size of the reduced-order model
|
stabsignmopts | structure, containing the optional parameters for the matrix sign function used for the decomposition into stable and anti-stable system parts, see ml_signm default: struct() |
stabsylvopts | structure, containing the optional parameters for the Sylvester equation solver used for the decomposition into stable and anti-stable system parts, see ml_sylv_sgn default: struct() |
Tolerance | nonnegative scalar, tolerance used for the computation of the size of the reduced-order model by an absolute error bound if 'tolerance' is set for OrderComputation default: 1.0e-02 |
UnstabDim | integer, dimension of the deflating anti-stable subspace, negative if unknown default: -1 |
Output
Ar - matrix of (3) with dimensions r x r Br - matrix of (3) with dimensions r x m Cr - matrix of (4) with dimensions p x r Dr - matrix of (4) with dimensions p x m rom - structure or state-space object, with the following entries:
Entry | Meaning |
A | matrix from (3) with dimensions r x r |
B | matrix from (3) with dimensions r x m |
C | matrix from (4) with dimensions p x r |
D | matrix from (4) with dimensions p x m |
info - structure, containing the following information:
Entry | Meaning |
AbsErrBound | computed error bound for the absolute error of the reduced-order model in H-infinity norm |
Hsv | a vector, containing the computed Hankel singular values |
infoADTF | structure, containing information about the additive decomposition of the system into its stable and anti-stable parts, see ml_ct_ss_adtf |
infoHAADTF | structure, containing information about the additive decomposition of the Hankel-norm transformed system, see ml_ct_ss_adtf |
infoLYAPDL | structure, containing information about the continuous-time dual Lyapunov equations solver, see ml_lyapdl_sgn_fac |
Ns | Dimension of the stable part of the reduced-order model |
Nu | Dimension of the anti-stable part of the reduced- order model |
Sigma | Chosen Hankel singular value, exact approximation error in the Hankel-norm |
Reference
P. Benner, E. S. Quintana-Orti, G. Quintana-Orti, Computing optimal Hankel norm approximations of large-scale systems, in: Proc. 43rd IEEE Conf. Decision Contr., Omnipress, Madison, WI, 2004, pp. 3078--3083.
See Also