ml_dt_ss_mt

Modal truncation for standard discrete-time systems.

Contents

Syntax

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

Description

This function computes the modal truncation for a standard discrete-time system of the form

   x(t+1) = 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 an absolute value larger than a given alpha. As result, the reduced-order system is given by

   x(t+1) = Ar*x(t) + Br*u(t),                                     (3)
     y(t) = Cr*x(t) + Dr*u(t).                                     (4)

Input

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

Parameter
Meaning
Alpha
{!}
real scalar, such that all finite eigenvalues with the real part smaller than Alpha are truncated
default: -1.0
dsylvopts
structure, containing the optional parameters for the Sylvester equation solver used for the decomposition of the eigenvalues, see ml_dsylv_smith
default: struct()
signmopts
structure, containing the optional parameters for the matrix sign function used for the decomposition of the eigenvalues, see ml_signm
default: struct()
StoreProjection
{0, 1}, used to disable/enable storing of the computed projection matrices W and T
default: 0

Note: Parameters marked with {!} may also be a cell array containing multiple arguments. In this case an cell array of the same size is returned with one entry computed for each input argument and the marked fields of the info struct are cells as well. When multiple arguments are given as cells, they are expected to have the same length.

Output

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

Entry
Meaning
infoSIGNM
{!}
structure, containing information about the matrix sign function method, see ml_signm
infoDSYLV
{!}
structure, containing information about the Sylvester equation solver, see ml_dsylv_smith
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_dt_dss_mt | ml_morlabopts