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