ml_ct_ss_bst

Balanced stochastic truncation for standard systems.

Contents

Syntax

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

Description

This function computes the balanced stochastic truncation for a
standard system of the form
    x'(t) = A*x(t) + B*u(t),                                        (1)
     y(t) = C*x(t) + D*u(t).                                        (2)
Therefore, first the Lyapunov equation
    A*P  + P*A' + B*B' = 0,
is solved for the controllability Gramian P and then, the corresponding
Riccati equation
    A'*Q + Q*A + (C - Bw' * Q)' * inv(D*D') * (C - Bw' * Q) = 0
is solved for the Gramian Q, with
    Bw = B*D' + P*C'.
As result, a reduced-order system of the form
    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 transfer function G and and the
r-th order transfer function Gr it holds
    ||G - Gr||_{\infty} / ||G||_{\infty}
    <= ((1 + Hsv(r+1))/(1 - Hsv(r+1)) * ...
       * (1 + Hsv(n))/(1 - Hsv(n))) + 1,
with Hsv, a vector containing the characteristic stochastic singular
values of the system. If the transfer function is invertible it holds
    ||inv(G)*(G - Gr)||_{\infty}
    <= ((1 + Hsv(r+1))/(1 - Hsv(r+1)) * ...
       * (1 + Hsv(n))/(1 - Hsv(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 D term, an epsilon regularization is
     performed, which replaces the D 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
sys  - structure or state-space object, containing the standard
       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

opts - structure, containing the following optional entries:

Parameter
Meaning
Epsilon
positive scalar, used in the case of a non-full-rank D term for epsilon regularization by multiplying with an identity matrix of appropriate size
default: 1.0e-03
lyapopts
structure, containing the optional parameters for the computation of the continuous-time algebraic Lyapunov equation, see ml_lyap_sgn_fac
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(Hsv)) + Nu
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()
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()
stabsylvopts
structure, containing the optional parameters for the Sylvester equation solver used for the decomposition into stable and anti-stable system parts, see ml_sylv_sgn
default: struct()
Tolerance
nonnegative scalar, tolerance used for the computation of the size of the reduced-order model by an relative error bound if 'tolerance' is set for OrderComputation
default: 1.0e-02
UnstabDim
integer, dimension of the deflating anti-stable subspace, negative if unknown
default: -1

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
rom  - structure or state-space object, with the following entries:

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

info - structure, containing the following information:

Entry
Meaning
Hsv
a vector, containing the computed characteristic stochastic singular values
infoADTF
structure, containing information about the additive decomposition of the system into its stable and anti-stable parts, see ml_ct_ss_adtf
infoLYAP
structure, containing information about the continuous-time Lyapunov equation sovler for the controllability Gramian, see 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
Ns
Dimension of the stable part of the reduced-order model
Nu
Dimension of the anti-stable part of 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, E. S. Quintana-Orti, G. Quintana-Orti, Efficient numerical algorithms for balanced stochastic truncation, Int. J. Appl. Math. Comput. Sci. 11 (5) (2001) 1123--1150.

See Also

ml_ct_dss_bst | ml_morlabopts