ml_ct_dss_partstab

Stabilizing feedback for descriptor systems.

Contents

Syntax

[K, info] = ml_ct_dss_partstab(A, B, E)
[K, info] = ml_ct_dss_partstab(A, B, E, opts)

Description

Partial stabilization is used for the continuous-time system of
differential-algebraic equations
    Ex'(t) = Ax(t) + Bu(t)                                          (1)
to get a stabilizing feedback term K, such that all finite eigenvalues
of s*E - A + B*K are in the left open half-plane. It is assumed that
the pencil s*E - A has no eigenvalues on the imaginary axis.

Input

A    - matrix with dimensions n x n in (1)
B    - matrix with dimensions n x m in (1)
E    - matrix with dimensions n x n in (1)
opts - structure, containing the following optional entries:

Parameter
Meaning
Beta
nonnegative scalar, used as shift of the in Bass' algorithm for better conditioning if StabMethod == 'lyap' is chosen
default: 0.1
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
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()
stabdecopts
structure, containing the optional parameters for the decomposition of the stable and unstable parts of the system using the disk function and subspace extraction method, see ml_disk and ml_getqz
default: struct()
StabMethod
character array, determining algorithm for the partial stabilization
  • 'abe' - algebraic Bernoulli equation based
  • 'lyap' - Bass' algorithm (Lyapunov equation)
default: 'abe'
stabmethodopts
structure, containing the optional parameters for the sign function based Lyapunov or Bernoulli equation solver used for the stabilization, see ml_abe_sgn or ml_lyap_sgn
default: struct()

Output

K    - stabilizing feedback matrix of dimensions m x n
info - structure, containing the following information about the
       generalized partial stabilization method

Entry
Meaning
infoINFDISK
structure, containing information about the disk function method used for the separation of the infinite part, see ml_disk
infoSTABDISK
structure, containing information about the disk function method used for the separation of the unstable part, see ml_disk
infoSTABDISK2
structure, containing information about the disk function method used for a second separation of the unstable part if necessary, see ml_disk
infoSTABMETH
structure, containing information about the sign function based solver used for the stabilization, see ml_abe_sgn or ml_lyap_sgn
infoSTABMETH2
structure, containing information about the sign function based solver used for a second stabilization if necessary, see ml_abe_sgn or ml_lyap_sgn
Method
character array, shortcut of the used stabilization algorithm, with 'abe' for the algebraic Bernoulli equation and 'lyap' for Bass' algorithm
Ninf
Number of identified infinite eigenvalues
Ns
Number of identified stable eigenvalues
Nu
Number of identified anti-stable eigenvalues

Reference

P. Benner, Partial stabilization of descriptor systems using spectral projectors, in: P. Van Dooren, S. P. Bhattacharyya, R. H. Chan, V. Olshevsky, A.Routray (Eds.), Numerical Linear Algebra in Signals, Systems and Control, Vol. 80 of Lect. Notes Electr. Eng., Springer Netherlands, 2011, pp. 55--76. https://doi.org/10.1007/978-94-007-0602-6_3

See Also

ml_ct_ss_partstab | ml_ct_dss_adtf