ml_ct_ss_brbt

Bounded-real balanced truncation for standard systems.

Contents

Syntax

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

Description

This function computes the bounded-real balanced truncation for a
bounded-real 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 bounded-real Riccati equations
    A*P  + P*A' + B*B' + (P*C' + B*D') * inv(Rb) * (P*C' + B*D')' = 0,
    A'*Q + Q*A  + C'*C + (B'*Q + D'*C)' * inv(Rc) * (B'*Q + D'*C) = 0,
are solved for the Gramians P and Q, with
    Rb = I - D*D'   and
    Rc = I - D'*D.
As result, a reduced-order bounded-real system of the form
    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 with the
spectral factors W'*W = I - G'*G and  V*V' = I - G*G', and the r-th
order transfer function Gr with the spectral factors
Wr'*Wr = I - Gr'*Gr and  Vr*Vr' = I - Gr*Gr' it holds
    max(||[G - Gr; W - Wr]||_{\infty}, ||[G - Gr; V - Vr]||_{\infty})
    <= 2 * (Hsv(r+1) + ... + Hsv(n)),
with Hsv, a vector containing the characteristic bounded-real singular
values of the system.

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
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(Hsv)) + Nu
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 relative 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
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 (and the spectral factors of I - G*G' and I - G'*G) in H-infinity norm
Hsv
a vector, containing the computed characteristic bounded-real singular values
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
N
Dimension of the reduced-order model

Reference

S. Gugercin, A. C. Antoulas, A survey of model reduction by balanced truncation and some new results, Internat. J. Control 77 (8) (2004) 748--766. https://doi.org/10.1080/00207170410001713448

See Also

ml_ct_dss_brbt | ml_morlabopts