ml_ct_ss_partstab

Stabilizing feedback for standard systems.

Contents

Syntax

[K, info] = ml_ct_ss_partstab(A, B)
[K, info] = ml_ct_ss_partstab(A, B, opts)

Description

Partial stabilization is used for the continuous-time system of
odinary differential equations
    x'(t) = Ax(t) + Bu(t)                                           (1)
to get a stabilizing feedback term K, such that all eigenvalues of
A-B*K are in the open left half-plane. It is assumed that 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)
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
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()
stabsignmopts
structure, containing the optional parameters for the matrix sign function used for the decomposition into stable and anti-stable system parts, see ml_signm
default: struct()
UnstabDim
integer, dimension of the deflating anti-stable subspace, negative if unknown
default: -1

Output

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

Entry
Meaning
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
infoSTABSIGNM
structure, containing information about the matrix sign function method used for the decomposition, see ml_signm
infoSTABSIGNM2
structure, containing information about the matrix sign function method used for the a second decomposition, see ml_signm
Method
character array, shortcut of the used stabilization algorithm, with 'abe' for the algebraic Bernoulli equation and 'lyap' for Bass' algorithm
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_dss_partstab | ml_ct_ss_adtf