ml_dt_dss_mt
Modal truncation for descriptor systems.
Contents
Syntax
[Ar, Br, Cr, Dr, Er, info] = ml_dt_dss_mt(A, B, C, D, E) [Ar, Br, Cr, Dr, Er, info] = ml_dt_dss_mt(A, B, C, D, E, opts)
[rom, info] = ml_dt_dss_mt(sys) [rom, info] = ml_dt_dss_mt(sys, opts)
Description
This function computes the generalized modal truncation for a discrete-time descriptor system of the form
E*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 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 absolute value larger than a given alpha. As result, the reduced-order system is given by
Er*x(t+1) = 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 {!} | 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
- 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 |
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_dt_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
See Also