ml_ct_dss_bst

Balanced stochastic truncation for descriptor systems.

Contents

Syntax

[Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_bst(A, B, C, D, E)
[Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_bst(A, B, C, D, E, opts)
[rom, info] = ml_ct_dss_bst(sys)
[rom, info] = ml_ct_dss_bst(sys, opts)

Description

This function computes the generalized balanced stochastic 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, first an additive decomposition of the system is performed
using the matrix disk function, such that
         [ Ei  0 ]       [ Ai  0 ]                        [ Ci ]
    E2 = [       ], A2 = [       ], B2 = [ Bi, Bp ], C2 = [    ],
         [ 0  Ep ]       [ 0  Ap ]                        [ Cp ]
with (Ei, Ai, Bi, Ci, D) belonging to the polynomial part and
(Ep, Ap, Bp, Cp, 0) belonging to the strictly proper part.
Now, for the reduction of the strictly proper part the generalized
continuous-time Lyapunov equation
    Ap*Pp*Ep' + Ep*Pp*Ap' + Bp*Bp' = 0
is solved for Pp and then, the corresponding generalized Riccati
equation
    Ap'*Qp*Ep + Ep'*Qp*Ap
    + (Cp - Bw' * Qp * Ep)' * inv(M*M') * (Cp - Bw' * Qp * Ep) = 0
is solved for Qp, with M = D - Ci * inv(Ai) * Bi. Also, the the two
generalized discrete-time Lyapunov equations
    Ai*Pi*Ai' - Ei*Pi*Ei' - Bi*Bi' = 0,
    Ai'*Qi*Ai - Ei'*Qi*Ei - Ci'*Ci = 0
are solved for the reduction of the polynomial part. As result, a
reduced-order system of the form
    Er*x'(t) = Ar*x(t) + Br*u(t),                                   (3)
        y(t) = Cr*x(t) + Dr*u(t)                                    (4)
is computed, such that for the original proper transfer function G and
the reduced-order transfer function Gr with an r-th order strictly
proper part it holds
    ||G - Gr||_{\infty} / ||G||_{\infty}
    <= ((1 + Hsvp(r+1))/(1 - Hsvp(r+1)) * ...
       * (1 + Hsvp(n))/(1 - Hsvp(n))) + 1,
with Hsvp, a vector containing the proper characteristic stochastic
singular values of the system. If the transfer function is invertible
it holds
    ||inv(G)*(G - Gr)||_{\infty}
    <= ((1 + Hsvp(r+1))/(1 - Hsvp(r+1)) * ...
       * (1 + Hsvp(n))/(1 - Hsvp(n))) + 1.
Notes:
  1) The equations above are defined for the case of p < m. Otherwise
     the system is transposed, then reduced and back transposed.
  2) In case of a rank-deficient M term, an epsilon regularization is
     performed, which replaces the M during the computations with an
     identity matrix scaled by a given epsilon.
  3) For unstable systems, first an additive decomposition into the
     stable and anti-stable parts is performed and then only the
     stable part will be reduced.

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
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
Epsilon
positive scalar, used in the case of a non-full-rank M + M' term for epsilon regularization by multiplying with an identity matrix of appropriate size
default: 1.0e-03
gdlyapopts
structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyap_smith_fac
default: struct()
ImproperTrunc
nonnegative scalar, tolerance multiplied with the largest proper Hankel singular value of the system to truncate the improper part
default: log(n)*eps
Index
nonnegative integer, index of the descriptor system used to set an upper bound on the size of the reduced improper part, Inf if unknown
default: Inf
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()
lyapopts
structure, containing the optional parameters for the computation of the continuous-time algebraic Lyapunov equation, see ml_lyap_sgn_fac
default: struct()
Method
character array, determining algorithm for the computation of the reduced-order model
  • 'sr' - square-root method
  • 'bfsr' - balancing-free square-root method
default: 'sr'
Order
positive integer, order of the resulting reduced-order model chosen by the user if 'order' is set for OrderComputation
default: min(10,length(Hsvp)) + Nu + Ni
OrderComputation
character array, determining the method for the computation of the size of the reduced-order model
  • 'order' - take explicit order
  • 'tolerance' - using absolute error bound
default: 'tolerance'
pcareopts
structure, containing the optional parameters for the computation of the continuous-time algebraic positive Riccati equation, see ml_pcare_nwt_fac
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()
Tolerance
nonnegative scalar, tolerance used for the computation of the size of the reduced-order model by an absolute error bound if 'tolerance' is set for OrderComputation
default: 1.0e-02

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
AbsErrBound
computed error bound for the absolute error of the reduced-order model in H-infinity norm
Hsvi
a vector, containing the computed Hankel singular values of the improper part of the system
Hsvp
a vector, containing the computed characteristic stochastic singular values of the proper part of the system
infoGADTF
structure, containing information about the additive decomposition of the system into its infinite, finite stable and finite anti-stable parts, see ml_ct_dss_adtf
infoGDLYAP_C
structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper controllability Gramian, see ml_gdlyap_smith_fac
infoGDLYAP_O
structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper observability Gramian, see ml_gdlyap_smith_fac
infoLYAP
structure, containing information about the continuous-time Lyapunov equation sovler for the controllability Gramian, ml_lyap_sgn_fac
infoPCARE
structure, containing information about the continuous-time algebraic positive Riccati equation for the observability Gramian, see ml_pcare_nwt_fac
Ni
Dimension of the improper part in the reduced- order model
Np
Dimension of the proper part in the reduced-order model
Nu
Dimension of the unstable part in the reduced- order model
RelErrBound
computed error bound for the relative error of the of the reduced-order model in H-infinity norm

Reference

P. Benner, T. Stykel, Model order reduction for differential-algebraic equations: A survey, in: A. Ilchmann, T. Reis (Eds.), Surveys in Differential-Algebraic Equations IV, Differential-Algebraic Equations Forum, Springer International Publishing, Cham, 2017, pp. 107--160. https://doi.org/10.1007/978-3-319-46618-7_3

See Also

ml_ct_ss_bst | ml_morlabopts