ml_ct_dss_flbt

Frequency-limited balanced truncation for desc. systems.

Contents

Syntax

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

Description

This function computes the generalized frequency-limited balanced 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, the two generalized continuous-time Lyapunov equations

   Ap*Pp*Ep' + Ep*Pp*Ap' + BF*Bp' + Bp*BF' = 0,
   Ap'*Qp*Ep + Ep'*Qp*Ap + Cp'*CF + CF'*Cp = 0,

where BF and CF are frequency-dependent matrices, are solved for the reduction of the strictly proper part, and 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. 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||(Hsvp(r+1) + ... + Hsvp(n)),

with Hsvp, a vector containing the proper frequency-limited Hankel singular values of the system.

Note: For unstable systems, an additional 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
E
matrix from (1) with dimensions n x n

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
FreqRange
nonnegative vector, frequency intervals such that [w(1), w(2)] ... [w(2k-1), w(2k)] are approximated
default: [0, 1.0e+03]
gdlyapdlopts
structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyapdl_smith_fac
default: struct()
ImproperTrunc
{!}
nonnegative scalar, tolerance multiplied with the largest proper Hankel singular value of the system to truncate the improper part, if 0 no improper balanced truncation is performed
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()
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(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 rel. tolerance for the hsv
  • 'sum' - using rel. tolerance for sum of hsv
default: 'sum'
stabdecopts
structure, containing the optional parameters for the decomposition of the stable and unstable parts of the system using the sign function and subspace extraction method, see ml_signm and ml_getqz
default: struct()
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

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
E
matrix from (3) with dimensions r x r

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
Hsvi
a vector, containing the computed Hankel singular values of the improper part of the system
Hsvp
a vector, containing the computed Hankel singular values of the proper part of the system
infoADTF
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
infoGDLYAPDL
structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper Gramians, see ml_gdlyapdl_smith_fac
infoLYAPDL
structure, containing information about the continuous-time dual Lyapunov equations solver, see ml_lyapdl_sgn_ldl or ml_lyapdl_sgn_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
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

See Also

ml_ct_dss_bt | ml_ct_ss_flbt | ml_morlabopts