ml_dt_dss_adtf

Add. decomposition of descriptor system's transfer function.

Contents

Syntax

[sys, info] = ml_dt_dss_adtf(sys)
[sys, info] = ml_dt_dss_adtf(sys, opts)

Description

Consider a discrete-time descriptor system of the form

   E*x(t+1) = A*x(t) + B*u(t),                                     (1)
       y(t) = C*x(t).                                              (2)

This function computes an additive decomposition of the corresponding transfer function by a block diagonalization of the matrix pencil s*E - A. The descriptor system is transformed, such that

   [ Ef         ]         [ Af           ]       [ Bf   ]
   [    Eu      ]z(t+1) = [     Au       ]z(t) + [ Bu   ]u(t),     (3)
   [       Einf ]         [         Ainf ]       [ Binf ]
                  y(t) = [ Cf, Cu, Cinf ]z(t),                     (4)

where s*Ef - Af contains the finite eigenvalues with absolute value smaller than 1, s*Einf - Ainf the infinite eigenvalues and s*Eu - Au the finite eigenvalues with absolute value larger than 1.

Input

Parameter
Meaning
E
matrix with dimensions n x n in (1)
A
matrix with dimensions n x n in (1)
B
matrix with dimensions n x m in (1)
C
matrix with dimensions p x n in (2)

Parameter
Meaning
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-zero singular value of E
default: log(n)*eps
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, ml_getqz
default: struct()
stabdecopts
structure, containing the optional parameters for the decomposition of the stable and unstable parts of the system using the sign function and subspace extraction method, see ml_signm and ml_getqz
default: struct()
StoreProjection
{0, 1}, used to disable/enable storing of the computed projection matrices W and T
default: 0

Output

Parameter
Meaning
E
matrix with dimensions nf x nf, see Ef in (3)
Einf
matrix with dimensions ninf x ninf in (3)
Eu
matrix with dimensions nu x nu in (3)
A
matrix with dimensions nf x nf, see Af in (3)
Ainf
matrix with dimensions ninf x ninf in (3)
Au
matrix with dimensions nu x nu in (3)
B
matrix with dimensions nf x m, see Bf in (3)
Binf
matrix with dimensions ninf x m in (3)
Bu
matrix with dimensions nu x m in (3)
C
matrix with dimensions p x nf, see Cf in (4)
Cinf
matrix with dimensions p x ninf in (4)
Cu
matrix with dimensions p x nu in (4)

Entry
Meaning
infoINFDISK
structure, containing information about the disk function method used for the separation of the infinite part, see ml_disk
infoSTABSIGNM
structure, containing information about the sign function method used for the separation of the unstable part, see ml_signm
Ninf
Number of identified infinite eigenvalues
Ns
Number of identified stable eigenvalues
Nu
Number of identified anti-stable eigenvalues
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

S. Werner, Hankel-norm approximation of descriptor systems, Master's thesis, Otto von Guericke University, Magdeburg, Germany (2016). http://nbn-resolving.de/urn:nbn:de:gbv:ma9:1-8845

See Also

ml_dt_ss_adtf | ml_dss_lti2struct