ml_ct_ss_tlbt
Time-limited balanced truncation for standard systems.
Contents
Syntax
[Ar, Br, Cr, Dr, Er, info] = ml_ct_ss_tlbt(A, B, C, D, E) [Ar, Br, Cr, Dr, Er, info] = ml_ct_ss_tlbt(A, B, C, D, E, opts)
[rom, info] = ml_ct_ss_tlbt(sys) [rom, info] = ml_ct_ss_tlbt(sys, opts)
Description
This function computes the time-limited balanced 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, the two standard Lyapunov equations
A*P + P*A' + Bts*Bts' - Bte*Bte' = 0, A'*Q + Q*A + Cts'*Cts - Cte'*Cte = 0,
where Bts, Bte, Cts and Cte are time-dependent matrices, are solved for the time-limited Gramians P and Q. 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. For enforcing stability in the reduced-order model, the modified Gramian approach can be used
Note: 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 |
lyapdlopts | structure, containing the optional parameters for the computation of the generalized continuous-time Lyapunov equations, see ml_lyapdl_sgn_ldl if ModGramian = 0 and ml_lyapdl_sgn_fac if ModGramian = 1 default: struct() |
Method {!} | character array, determining algorithm for the computation of the reduced-order model
|
ModGramian | {0, 1}, used to disable/enable the modified Gramian approach default: 0 |
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
|
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 |
StoreProjection | {0, 1}, used to disable/enable storing of the computed projection matrices W and T default: 0 |
TimeRange | nonnegative vector, time interval such that [min(t), max(t)] is approximated default: [0, 10] |
Tolerance {!} | nonnegative scalar, tolerance used for the computation of the size of the reduced-order model if 'tolerance' or 'sum' is set for OrderComputation default: 1.0e-02 |
UnstabDim | integer, dimension of the deflating anti-stable subspace, negative if unknown default: -1 |
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 Hankel 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 |
infoLYAPDL | structure, containing information about the continuous-time dual Lyapunov equations solver, see ml_lyapdl_sgn_ldl or ml_lyapdl_sgn_fac |
Ns {!} | Dimension of the stable part of the reduced-order model |
Nu | Dimension of the anti-stable part of 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
P. Kurschner, Balanced truncation model order reduction in limited time intervals for large systems, e-print 1707.02839v2, arXiv, math.NA (2018). https://arxiv.org/abs/1811.05500
See Also