ml_ct_dss_prbt

Positive-real balanced truncation for descriptor systems.

Contents

Syntax

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

Description

This function computes the generalized positive-real balanced
truncation for a positive-real 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 positive-real Riccati equations
    Ap*Pp*Ep' + Ep*Pp*Ap'
        + (Ep*Pp*Cp' + Bp) * inv(R) * (Ep*Pp*Cp' + Bp)' = 0,
    Ap'*Qp*Ep + Ep'*Qp*Ap
        + (Bp'*Qp*Ep + Cp)' * inv(R) * (Bp'*Qp*Ep + Cp) = 0
are solved for the reduction of the strictly proper part, with
    R = M + M',
where M = D - Ci * inv(Ai) * Bi. Also, 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, 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
    ||inv(G + M') - inv(Gr + M')||_{\infty} <= 2 * ||R||_{2}^2
        * (Hsvp(r+1) + ... + Hsvp(n)),
with Hsvp, a vector containing the proper characteristic positive-real
singular values of the system, and R^2 = inv(M + M').
Note: In case of a rank-deficient M + M' term, an epsilon
      regularization is performed, which replaces the M during the
      computations with an identity matrix scaled by a given epsilon.

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 m x n
D    - matrix from (2) with dimensions m 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 m x n
D
matrix from (2) with dimensions m 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
Epsilon
positive scalar, used in the case of a non-full-rank M + M' term for epsilon regularization by multiplying with an identity matrix of appropriate size
default: 1.0e-03
gdlyapopts
structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyap_smith_fac
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()
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'
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)) + Ni
OrderComputation
character array, determining the method for the computation of the size of the reduced-order model
  • 'order' - take explicit order
  • 'tolerance' - using absolute error bound
default: 'tolerance'
pcareopts
structure, containing the optional parameters for the computation of the continuous-time algebraic positive Riccati equation, see ml_pcare_nwt_fac
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 m x r
Dr   - matrix of (4) with dimensions m 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 m x r
D
matrix from (4) with dimensions m x m
E
matrix from (3) with dimensions r x r

info - structure, containing the following information:

Entry
Meaning
Hsvi
a vector, containing the computed Hankel singular values of the improper part of the system
Hsvp
a vector, containing the computed characteristic positive-real singular values of the proper part of the system
infoGADTF
structure, containing information about the additive decomposition of the system into its infinite and finite 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
infoPCARE_C
structure, containing information about the continuous-time algebraic positive Riccati equation solver for the controllability Gramian, see ml_pcare_nwt_fac
infoPCARE_O
structure, containing information about the continuous-time algebraic positive Riccati equation solver for the observability Gramian, see ml_pcare_nwt_fac
InvAbsErrBound
computed error bound for the absolute error of the inverse transfer functions in H-infinity norm
M
matrix with dimensions m x m, polynomial part of zeroth order and used in the error bound (or term from the epsilon regularization)
Ni
Dimension of the improper part in the reduced- order model
Np
Dimension of the proper part in the reduced-order model

Reference

T. Reis, T. Stykel, Positive real and bounded real balancing for model reduction of descriptor systems, Internat. J. Control 83 (1) (2010) 74--88. https://doi.org/10.1080/00207170903100214

See Also

ml_ct_ss_prbt | ml_morlabopts