ml_ct_dss_sr

Square root method for descriptor systems.

Contents

Syntax

[sys, hsvp, hsvi] = ml_ct_dss_sr(sys, R, Rinf, L, Linf, tselect)
[sys, hsvp, hsvi] = ml_ct_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

sys     - structure, containing the descriptor system in the form:

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

R       - Cholesky factor of the proper controllability Gramian with
          dimensions nr1 x nf
Rinf    - Cholesky factor of the improper controllability Gramian with
          dimensions nr2 x ninf
L       - Cholesky factor of the proper observability Gramian with
          dimensions nl1 x nf
Linf    - Cholesky factor of the improper observability Gramian with
          dimensions nl2 x ninf
tselect - an integer, used to determine the computation method for the
          order of the reduced-order model
            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
opts    - structure, containing the following optional entries:

Parameter
Meaning
Gamma
positive scalar, scaling term from the H-infinity balanced truncation
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 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
Tolerance
nonnegative scalar, tolerance used in the different error formulas
default: 1.0e-02

Output

sys     - struct, containing the transformed system matrices
hsvp    - vector, containing the characteristic proper singular values
hsvi    - vector, containing the improper Hankel singular values

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_ct_ss_sr | ml_ct_dss_bfsr | ml_order