ml_ct_ss_mt

Modal truncation for standard systems.

Contents

Syntax

[Ar, Br, Cr, Dr, info] = ml_ct_ss_mt(A, B, C, D)
[Ar, Br, Cr, Dr, info] = ml_ct_ss_mt(A, B, C, D, opts)
[rom, info] = ml_ct_ss_mt(sys)
[rom, info] = ml_ct_ss_mt(sys, opts)

Description

This function computes the modal truncation for 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, a block diagonalization of the matrix A is performed using
the matrix sign function and a Sylvester equation, such that
         [ Ar  0 ]                        [ Cr ]
    A2 = [       ], B2 = [ Br, B1 ], C2 = [    ], Dr = D,
         [ 0  A1 ]                        [ C1 ]
where Ar contains all the eigenvalues of A, which have a larger real
part than a given alpha. As result, the reduced-order system is given
by
    x'(t) = Ar*x(t) + Br*u(t),                                      (3)
     y(t) = Cr*x(t) + Dr*u(t).                                      (4)

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
Alpha
scalar, such that all finite eigenvalues with the real part smaller than Alpha are truncated
default: -1.0
signmopts
structure, containing the optional parameters for the matrix sign function used for the decomposition of the eigenvalues, see ml_signm
default: struct()
sylvopts
structure, containing the optional parameters for the Sylvester equation solver used for the decomposition of the eigenvalues, see ml_sylv_sgn
default: struct()

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
infoSIGNM
structure, containing information about the matrix sign function method, see ml_signm
infoSYLV
structure, containing information about the Sylvester equation solver, see ml_sylv_sgn
N
Dimension of the reduced-order model

Reference

P. Benner, E. S. Quintana-Orti, Model reduction based on spectral projection methods, in: P. Benner, V. Mehrmann, D. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lect. Notes Comput. Sci. Eng., Springer, Berlin/Heidelberg, Germany, 2005, pp. 5--45. https://doi.org/10.1007/3-540-27909-1_1

See Also

ml_ct_dss_mt | ml_morlabopts