ml_ct_ss_bfsr

Balancing-free square root method for standard systems.

Contents

Syntax

[sys, hsv] = ml_ct_ss_bfsr(sys, R, L, tselect)
[sys, hsv] = ml_ct_ss_bfsr(sys, R, L, tselect, opts)

Description

Computes the reduced-order matrices of a standard system by the
balancing-free square root method. Therefore, transformation matrices
of the form
    W = Y * (X' * Y)^(-1),
    T = X,
are computed, where X, Y result from the QR decompositions
    [X, ~]    = qr(R * V),
    [Y, ~]    = qr(L' * U).
The computation of the reduced-order model is done by
    Ar = W' * A * T,
    Br = W' * B,
    Cr = C * T.

Input

sys     - structure, containing the standard system in the form:

Parameter
Meaning
A
matrix with dimensions n x n
B
matrix with dimensions n x m
C
matrix with dimensions p x n

R       - Cholesky factor of the controllability Gramian with
          dimensions nr x n
L       - Cholesky factor of the observability Gramian with
          dimensions nl x n
tselect - an integer, used to determine the computation method for the
          order of the reduced-order model
            0 - order is directly given by user
            1 - computed by a relative tolerance for the hsv
            2 - computed by a relative tolerance on the sum of hsv
            3 - computed by absolute error bound of BT
            4 - computed by relative error bound of BST
            5 - computed by absolute error bound of LQGBT
            6 - computed by absolute error bound of HinfBT
opts    - structure, containing the following optional entries:

Parameter
Meaning
Gamma
positive scalar, scaling term from the H-infinity balanced truncation
Order
positive integer, order of the resulting reduced-order model chosen by the user if tselect == 1
default: min(10,length(hsvp))+nu
Tolerance
nonnegative scalar, tolerance used in the different error formulas
default: 1.0e-02

Output

sys     - struct, containing the transformed system matrices
hsv     - vector, containing the characteristic singular values

Reference

A. Varga, Controller reduction using accuracy-enhancing methods, in: P. Benner, V. Mehrmann, D. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lect. Notes Comput. Sci. Eng., Springer-Verlag, Berlin/Heidelberg, Germany, 2005, pp. 353--356. https://doi.org/10.1007/3-540-27909-1_9

See Also

ml_ct_dss_bfsr | ml_ct_ss_sr | ml_order