ml_ct_dss_mt

Modal truncation for descriptor systems.

Contents

Syntax

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

Description

This function computes the generalized modal truncation for a
descriptor system of the form
    E*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 pencil s*E - A is
performed using the matrix disk function, such that
         [ Ei 0  0  ]       [ Ai 0  0  ]
    E2 = [ 0  E0 0  ], A2 = [ 0  A0 0  ],
         [ 0  0  E1 ]       [ 0  0  A1 ]
                              [ Ci ]
    B2 = [ Bi, B0, B1 ], C2 = [ C0 ],
                              [ C1 ]
where the matrix pencil s*Ei - Ai contains all infinite eigenvalues
and s*E0 - A0 the finite eigenvalues with the real part larger than a
given alpha. As result, the reduced-order system is given by
    Er*x'(t) = Ar*x(t) + Br*u(t),                                   (3)
        y(t) = Cr*x(t) + Dr*u(t),                                   (4)
with
         [ E0  0 ]       [ A0  0 ]                        [ C0 ]
    Er = [       ], A2 = [       ], B2 = [ B0, Bi ], C2 = [    ].
         [ 0  Ei ]       [ 0  Ai ]                        [ Ci ]

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
E    - matrix from (1) with dimensions n x n
sys  - structure or state-space object, containing the descriptor
       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
E
matrix from (1) with dimensions n x n

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
DecompEig
positive scalar, overestimation of the absolute value of the largest finite eigenvalue of s*E - A, if set, replaces the computation with DecompTol
default: []
DecompTol
nonnegative scalar, tolerance multiplied with the largest singular value of E to determine the smallest non-quasi-zero singular value of E
default: log(n)*eps
diskopts
structure, containing the optional parameters for the inverse free iteration of the disk function method, see ml_disk
default: struct()
infdecopts
structure, containing the optional parameters for the decomposition of the finite and infinite parts of the system using the disk function and subspace extraction method, see ml_disk and ml_getqz
default: struct()
RankTol
nonnegative scalar, tolerance used for the determination of deflating subspaces
default: log(n)*eps

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
Er   - matrix of (3) with dimensions r x r
rom  - structure or state-space object, containing the reduced-order
       descriptor system:

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
E
matrix from (3) with dimensions r x r

info - structure, containing the following information:

Entry
Meaning
infoDISK
structure, containing information about the disk function method used for the modal truncation of the system, see ml_disk
infoGADTF
structure, containing information about the additive decomposition of the system into its infinite and finite parts, see ml_ct_dss_adtf
N
Dimension of the finite part in the reduced-order model
Ni
Dimension of the infinite part in 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_ss_mt | ml_morlabopts