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
|
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
|
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