ml_ct_dss_hna
Hankel-norm approximation for descriptor systems.
Contents
Syntax
[Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_hna(A, B, C, D, E) [Ar, Br, Cr, Dr, Er, info] = ml_ct_dss_hna(A, B, C, D, E, opts)
[rom, info] = ml_ct_dss_hna(sys) [rom, info] = ml_ct_dss_hna(sys, opts)
Description
This function computes the generalized Hankel-norm approximation of a descriptor system
E*x'(t) = A*x(t) + B*u(t), (1) y(t) = C*x(t) + D*u(t). (2)
Therefore, first a balanced realization is computed by using the generalized balanced truncation square-root method with an appropriate tolerance for the minimal realization of the given system. Then the strictly proper part of the system is transformed using the formulas for all-pass systems. 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 transfer function G and the reduced-order transfer function Gr with an r-th order strictly proper part it holds
||G - Gr||_{H} = Hsvp(r+1), ||G - Gr||_{\infty} <= 2 * (Hsvp(r+1) + ... + Hsvp(n)),
with Hsvp, a vector containing the proper 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. That does not change the error formulas.
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 |
gdlyapopts | structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyap_smith_fac default: struct() |
hankeldecopts | structure, containing the optional parameters for the disk function used for the decomposition after the transformation to an all-pass system see ml_disk and ml_getqz 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() |
lyapdlopts | structure, containing the optional parameters for the computation of the generalized continuous-time Lyapunov equations, see ml_lyapdl_sgn_fac default: struct() |
MinRelTol | nonnegative scalar, tolerance multiplied with the largest characteristic value to determine a minimal realization default: log(n)*eps |
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
|
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 Hankel 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 |
infoHAGADTF | structure, containing information about the disk function method, see ml_ct_dss_adtf |
infoLYAPDL | structure, containing information about the continuous-time dual Lyapunov equations solver, see 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 |
Sigma | Chosen proper Hankel singular value, exact approximation error in the Hankel-norm |
Reference
S. Werner, Hankel-norm approximation of descriptor systems, Master's thesis, Otto von Guericke University, Magdeburg, Germany (2016). http://nbn-resolving.de/urn:nbn:de:gbv:ma9:1-8845
See Also