ml_ct_ss_bst
Balanced stochastic truncation for standard systems.
Contents
Syntax
[Ar, Br, Cr, Dr, info] = ml_ct_ss_bst(A, B, C, D) [Ar, Br, Cr, Dr, info] = ml_ct_ss_bst(A, B, C, D, opts)
[rom, info] = ml_ct_ss_bst(sys) [rom, info] = ml_ct_ss_bst(sys, opts)
Description
This function computes the balanced stochastic truncation for a standard system of the form
x'(t) = A*x(t) + B*u(t), (1) y(t) = C*x(t) + D*u(t). (2)
Therefore, first the Lyapunov equation
A*P + P*A' + B*B' = 0,
is solved for the controllability Gramian P and then, the corresponding Riccati equation
A'*Q + Q*A + (C - Bw' * Q)' * inv(D*D') * (C - Bw' * Q) = 0
is solved for the Gramian Q, with
Bw = B*D' + P*C'.
As result, a reduced-order 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 and the r-th order transfer function Gr it holds
||G - Gr||_{\infty} / ||G||_{\infty} <= ((1 + Hsv(r+1))/(1 - Hsv(r+1)) * ... * (1 + Hsv(n))/(1 - Hsv(n))) + 1,
with Hsv, a vector containing the characteristic stochastic singular values of the system. If the transfer function is invertible it holds
||inv(G)*(G - Gr)||_{\infty} <= ((1 + Hsv(r+1))/(1 - Hsv(r+1)) * ... * (1 + Hsv(n))/(1 - Hsv(n))) + 1.
Notes:
- The equations above are defined for the case of p < m. Otherwise the system is transposed, then reduced and back transposed.
- In case of a rank-deficient D term, an epsilon regularization is performed, which replaces the D during the computations with an identity matrix scaled by a given epsilon.
- For unstable systems, first an additive decomposition into the stable and anti-stable parts is performed and then only the stable part will be reduced.
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 |
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() |
Epsilon | positive scalar, used in the case of a non-full-rank D term for epsilon regularization by multiplying with an identity matrix of appropriate size default: 1.0e-03 |
lyapopts | structure, containing the optional parameters for the computation of the continuous-time algebraic Lyapunov equation, see ml_lyap_sgn_fac |
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)) + Nu |
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
|
stabsignmopts | structure, containing the optional parameters for the matrix sign function used for the decomposition into stable and anti-stable system parts, see ml_signm default: struct() |
stabsylvopts | structure, containing the optional parameters for the Sylvester equation solver used for the decomposition into stable and anti-stable system parts, see ml_sylv_sgn default: struct() |
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 relative error bound if 'tolerance' is set for OrderComputation default: 1.0e-02 |
UnstabDim | integer, dimension of the deflating anti-stable subspace, negative if unknown default: -1 |
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 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 |
Hsv | a vector, containing the computed characteristic stochastic singular values |
infoADTF | structure, containing information about the additive decomposition of the system into its stable and anti-stable parts, see ml_ct_ss_adtf |
infoCAREDL | structure, containing information about the sign function solver for the dual Riccati equations, see ml_caredl_sgn_fac |
infoLYAP | structure, containing information about the continuous-time Lyapunov equation sovler for the controllability Gramian, see ml_lyap_sgn_fac |
infoPCARE | structure, containing information about the continuous-time algebraic positive Riccati equation for the observability Gramian, see ml_pcare_nwt_fac |
Ns {!} | Dimension of the stable part of the reduced-order model |
Nu | Dimension of the anti-stable part of the reduced- order model |
RelErrBound {!} | computed error bound for the relative error of the of the reduced-order model in H-infinity norm |
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