ml_ct_ss_flbt

Frequency-limited balanced truncation for standard systems.

Contents

Syntax

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

Description

This function computes the frequency-limited balanced 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, the two standard Lyapunov equations

   A*P  + P*A' + BF*B' + B*BF' = 0,
   A'*Q + Q*A  + CF'*C + C'*CF = 0,

where BF and CF are frequency-dependent matrices, are solved for the frequency-limited Gramians P and Q. 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. For enforcing stability in the reduced-order model, the modified Gramian approach can be used, which also gives a global error bound of the form

   ||G - Gr||_{\infty} <= 2*||JB||*||JC||(Hsv(r+1) + ... + Hsv(n)),

with Hsv, a vector containing the freuqency-limited Hankel singular values of the system.

Note: 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

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

Parameter
Meaning
FreqRange
nonnegative vector, frequency intervals such that [w(1), w(2)] ... [w(2k-1), w(2k)] are approximated
default: [0, 1.0e+03]
lyapdlopts
structure, containing the optional parameters for the computation of the generalized continuous-time Lyapunov equations, see ml_lyapdl_sgn_ldl if ModGramian = 0 and ml_lyapdl_sgn_fac if ModGramian = 1
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'
ModGramian
{0, 1}, used to disable/enable the modified Gramian approach
default: 0
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 rel. tolerance for the hsv
  • 'sum' - using rel. tolerance for sum of hsv
default: 'sum'
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
StoreProjection
{0, 1}, used to disable/enable storing of the computed projection matrices W and T
default: 0
Tolerance
{!}
nonnegative scalar, tolerance used for the computation of the size of the reduced-order model if 'tolerance' or 'sum' is set for OrderComputation
default: 1.0e-02
UnstabDim
integer, dimension of the deflating anti-stable subspace, negative if unknown
default: -1

Note: Parameters marked with {!} may also be a cell array containing multiple arguments. In this case an cell array of the same size is returned with one entry computed for each input argument and the marked fields of the info struct are cells as well. When multiple arguments are given as cells, they are expected to have the same length.

Output

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

Entry
Meaning
AbsErrBound
{!}
computed error bound for the absolute error of the reduced-order model in H-infinity norm, only for the modified Gramian approach
Hsv
a vector, containing the computed Hankel 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
infoLYAPDL
structure, containing information about the continuous-time dual Lyapunov equations solver, see ml_lyapdl_sgn_ldl or ml_lyapdl_sgn_fac
Ns
{!}
Dimension of the stable part of the reduced-order model
Nu
Dimension of the anti-stable part of the reduced- order model
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

P. Benner, P. Kurschner, J. Saak, Frequency-limited balanced truncation with low-rank approximations, SIAM J. Sci. Comput. 38 (1) (2016) A471--A499. https://doi.org/ 10.1137/15M1030911

See Also

ml_ct_ss_bt | ml_ct_dss_flbt | ml_morlabopts