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 |
caredlopts | structure, containing the optional parameters for the Riccati equation sign function solver, only used if RiccatiSolver = 'sign', see ml_caredl_sgn_fac default: struct() |
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 |
gdlyapdlopts | structure, containing the optional parameters for the computation of the generalized discrete-time Lyapunov equations, see ml_gdlyapdl_smith_fac default: struct() |
ImproperTrunc {!} | nonnegative scalar, tolerance multiplied with the largest proper Hankel singular value of the system to truncate the improper part, if 0 no improper balanced truncation is performed 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, only used if RiccatiSolver = 'newton', see ml_pcare_nwt_fac default: struct() |
RiccatiSolver | character array, determining the solver for the dual Riccati equations
|
StoreProjection | {0, 1}, used to disable/enable storing of the computed projection matrices W and T default: 0 |
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 |
Note: Parameters marked with {!} may also be a cell array containing multiple arguments. In this case an cell array of the same size is returned with one entry computed for each input argument and the marked fields of the info struct are cells as well. When multiple arguments are given as cells, they are expected to have the same length.
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 |
infoCAREDL | structure, containing information about the sign function solver for the dual Riccati equations, see ml_caredl_sgn_fac |
infoADTF | structure, containing information about the additive decomposition of the system into its infinite and finite parts, see ml_ct_dss_adtf |
infoGDLYAPDL | structure, containing information about the generalized discrete-time Lyapunov equation solver for the improper Gramians, see ml_gdlyapdl_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 |
T {!} | projection matrix used as right state-space transformation to obtain the resulting block system, if opts.StoreProjection == 1 |
W {!} | projection matrix used as left state-space transformation to obtain the resulting block system, if opts.StoreProjection == 1 |
Reference
See Also