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

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

Parameter
Meaning
Alpha
{!}
real 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
signmopts
structure, containing the optional parameters for the sign function method, see ml_signm
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
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
E
matrix from (3) with dimensions r x r

Entry
Meaning
infoSIGNM
{!}
structure, containing information about the sign function method used for the modal truncation of the system, see ml_signm
infoADTF
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
T
{!}
projection matrix used as right state-space transformation to obtain the resulting block system, if opts.StoreProjection == 1
W
{!}
projection matrix used as left state-space transformation to obtain the resulting block system, if opts.StoreProjection == 1

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