ml_dss_sr

Square root method for descriptor systems.

Contents

Syntax

[sys, hsvp, hsvi, prj] = ml_dss_sr(sys, R, Rinf, L, Linf, tselect)
[sys, hsvp, hsvi, prj] = ml_dss_sr(sys, R, Rinf, L, Linf, tselect, opts)

Description

Computes the reduced-order matrices of a descriptor system by the square root method. Therefore, transformation matrices of the form

   W    = L * U * S^(-1/2),
   Winf = Linf * Uinf * Sinf^(-1/2),
   T    = R * V * S^(-1/2),
   Tinf = Rinf * Vinf * Sinf^(-1/2)

are computed, where U, Uinf, S, Sinf, V, Vinf result from singular value decompositions. The computation of the reduced-order model is done by

        [ W' * A * T     0  ]
   Ar = [                   ],
        [     0          I  ]
        [ W' * B       ]
   Br = [              ],
        [ Winf' * Binf ]
   Cr = [ C * T, Cinf * Tinf ],
        [ I              0            ]
   Er = [                             ].
        [ 0       Winf' * Einf * Tinf ]

Input

Parameter
Meaning
E
matrix with dimensions nf x nf
Einf
matrix with dimensions ninf x ninf
A
matrix with dimensions nf x nf
Ainf
matrix with dimensions ninf x ninf
B
matrix with dimensions nf x m
Binf
matrix with dimensions ninf x m
C
matrix with dimensions p x nf
Cinf
matrix with dimensions p x ninf

         0 - order is directly given by user
         1 - computed by a relative tolerance for the hsv
         2 - computed by a relative tolerance on the sum of hsv
         3 - computed by absolute error bound of BT
         4 - computed by relative error bound of BST
         5 - computed by absolute error bound of LQGBT
         6 - computed by absolute error bound of HinfBT

Parameter
Meaning
Gamma
positive scalar, scaling term from the H-infinity balanced truncation
default: Inf
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 nilpotency of the matrix Einf used to set an upper bound on the size of the reduced improper part, if the index is unknown Inf is set
default: Inf
Order
{!}
positive integer, order of the resulting reduced-order model chosen by the user if tselect == 1
default: min(10,length(hsvp)) + nu + ni
StoreProjection
{0, 1}, used to disable/enable storing of the computed projection matrices W and T
default: 0
Tolerance
{!}
nonnegative scalar, tolerance used in the different error formulas
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

Reference

V. Mehrmann, T. Stykel, Balanced truncation model reduction for large-scale systems in descriptor form, in: P. Benner, V. Mehrmann, D. C. Sorensen (Eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lect. Notes Comput. Sci. Eng., Springer-Verlag, Berlin/Heidelberg, Germany, 2005, pp. 83--115. https://doi.org/10.1007/3-540-27909-1_3

See Also

ml_ss_sr | ml_dss_bfsr | ml_order