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
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 |
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
|
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
|
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 |
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 |
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 |
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 |
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