ml_ct_ss_prbt
Positive-real balanced truncation for standard systems.
Contents
Syntax
[Ar, Br, Cr, Dr, info] = ml_ct_ss_prbt(A, B, C, D) [Ar, Br, Cr, Dr, info] = ml_ct_ss_prbt(A, B, C, D, opts)
[rom, info] = ml_ct_ss_prbt(sys) [rom, info] = ml_ct_ss_prbt(sys, opts)
Description
This function computes the positive-real balanced truncation for a positive-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 positive-real Riccati equations
A*P + P*A' + (P*C' - B) * inv(D + D') * (P*C' - B)' = 0, A'*Q + Q*A + (B'*Q - C)' * inv(D + D') * (B'*Q - C) = 0,
are solved for the Gramians P and Q. As result, a reduced-order positive-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 and the r-th order transfer function Gr it holds
||inv(G + D') - inv(Gr + D')||_{\infty} <= 2 * ||R||_{2}^2 * (Hsv(r+1) + ... + Hsv(n)),
with Hsv, a vector containing the characteristic positive-real singular values of the system, and R^2 = inv(D + D').
Note: In case of a rank-deficient D + D' term, an epsilon regularization is performed, which replaces the D 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 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 m x n |
D | matrix from (2) with dimensions m x m |
opts - structure, containing the following optional entries:
Parameter | Meaning |
Epsilon | positive scalar, used in the case of a non-full-rank D + D' term for epsilon regularization by multiplying with an identity matrix of appropriate size default: 1.0e-03 |
Method | character array, determining algorithm for the computation of the reduced-order model
|
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)) |
OrderComputation | character array, determining the method for the computation of the size of the reduced-order model
|
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 m x r Dr - matrix of (4) with dimensions m 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 m x r |
D | matrix from (4) with dimensions m x m |
info - structure, containing the following information:
Entry | Meaning |
Hsv | a vector, containing the computed characteristic positive-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 |
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 |
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