OMDc Namespace Reference
Optimal Mode Decomposition for control. More...
Classes | |
| class | DataStore |
| interface to internal representation of OMDc data More... | |
| class | LineSearchFunctor |
| Functor for linesearch along geodesic. More... | |
Enumerations | |
| enum | ImplementationType { kDirect, kPrecondQR } |
| defines how the implementation More... | |
Functions | |
| template<ImplementationType T> | |
| int | compute (std::ostream &out, Eigen::Ref< Eigen::MatrixXd > L, Eigen::Ref< Eigen::MatrixXd > M, Eigen::Ref< Eigen::MatrixXd > P, const Eigen::Ref< const Eigen::MatrixXd > S, const Eigen::Ref< const Eigen::MatrixXd > U, const Eigen::Index r, const GrManifold::CGOptions< double > *opt=nullptr) |
| template for the computation of OMDc The Optimal Mode Decomposition for control computes iteratively the solution to More... | |
| template<> | |
| int | compute< kDirect > (std::ostream &out, Eigen::Ref< Eigen::MatrixXd > L, Eigen::Ref< Eigen::MatrixXd > M, Eigen::Ref< Eigen::MatrixXd > P, const Eigen::Ref< const Eigen::MatrixXd > S, const Eigen::Ref< const Eigen::MatrixXd > U, const Eigen::Index r, const GrManifold::CGOptions< double > *opt) |
| computes OMDc for given data set More... | |
| template<> | |
| int | compute< kPrecondQR > (std::ostream &out, Eigen::Ref< Eigen::MatrixXd > L, Eigen::Ref< Eigen::MatrixXd > M, Eigen::Ref< Eigen::MatrixXd > P, const Eigen::Ref< const Eigen::MatrixXd > S, const Eigen::Ref< const Eigen::MatrixXd > U, const Eigen::Index r, const GrManifold::CGOptions< double > *opt) |
| computes OMDc for given data set More... | |
Detailed Description
Optimal Mode Decomposition for control.
The original identification problem is
![\[ \min_{L,M,P} \Vert Y - LML^\top X - LPU \Vert_F^2 \qquad X = [s_0 \cdots s_{m-2}], \,\, Y = [s_1 \cdots s_{m-1}] \]](form_0.png)
where the low-rank model
![\[ L\in\mathbb{R}^{n\times r}, \,\, M\in\mathbb{R}^{r\times r} \,\, P\in\mathbb{R}^{r\times p} \]](form_1.png)
shall approximate the given data set 
for inputs 
. The original problem can be transformed into
![\[ \min_L \Vert (I-LL^\top)Y +LL^\top \hat{Y} \big( I - \hat{X}^\top L ( L^\top \hat{X}\hat{X}^\top L )^{-1} L\top \hat{X} \big) \Vert_F^2 \quad \mathrm{s.t.} \,\,L^\top L = I \]](form_4.png)
where
![\[ \hat{X} = X \big(I - U^\top (U U^\top)^{-1} U\big) \qquad \hat{Y} = Y \big(I - U^\top (U U^\top)^{-1} U\big) \]](form_5.png)
Enumeration Type Documentation
◆ ImplementationType
Function Documentation
◆ compute()
template<ImplementationType T>
| int OMDc::compute | ( | std::ostream & | out, |
| Eigen::Ref< Eigen::MatrixXd > | L, | ||
| Eigen::Ref< Eigen::MatrixXd > | M, | ||
| Eigen::Ref< Eigen::MatrixXd > | P, | ||
| const Eigen::Ref< const Eigen::MatrixXd > | S, | ||
| const Eigen::Ref< const Eigen::MatrixXd > | U, | ||
| const Eigen::Index | r, | ||
| const GrManifold::CGOptions< double > * | opt = nullptr |
||
| ) |
template for the computation of OMDc The Optimal Mode Decomposition for control computes iteratively the solution to
![\[ \begin{aligned} \min_{L,M,P} & \Vert Y - L M L^\top X - LPU \Vert_F \\ \mathrm{s.t.}\,L^\top L = I, \,&\, X = [s_0 \cdots s_{m-2}], Y = [s_1 \cdots s_{m-1}] \end{aligned} \]](form_19.png)
on the Grassmann manifold
- Template Parameters
-
T declares implementation type
- Warning
- Find the explicit specializations for the implementation types below. This template does not have a generic implementation.
- Parameters
-
out reference to output stream L resulting orthonormal bases M resulting system matrix P resulting input matrix S snapshot data U input data r system dimension opt (optional) set of options
- Returns
- 0 on success
◆ compute< kDirect >()
template<>
| int OMDc::compute< kDirect > | ( | std::ostream & | out, |
| Eigen::Ref< Eigen::MatrixXd > | L, | ||
| Eigen::Ref< Eigen::MatrixXd > | M, | ||
| Eigen::Ref< Eigen::MatrixXd > | P, | ||
| const Eigen::Ref< const Eigen::MatrixXd > | S, | ||
| const Eigen::Ref< const Eigen::MatrixXd > | U, | ||
| const Eigen::Index | r, | ||
| const GrManifold::CGOptions< double > * | opt | ||
| ) |
computes OMDc for given data set
- Note
- direct computation on the given data
- Parameters
-
out reference to output stream L resulting orthonormal bases M resulting system matrix P resulting input matrix S snapshot data U input data r system dimension opt necessary set of options
- Returns
- 0 on success
◆ compute< kPrecondQR >()
template<>
| int OMDc::compute< kPrecondQR > | ( | std::ostream & | out, |
| Eigen::Ref< Eigen::MatrixXd > | L, | ||
| Eigen::Ref< Eigen::MatrixXd > | M, | ||
| Eigen::Ref< Eigen::MatrixXd > | P, | ||
| const Eigen::Ref< const Eigen::MatrixXd > | S, | ||
| const Eigen::Ref< const Eigen::MatrixXd > | U, | ||
| const Eigen::Index | r, | ||
| const GrManifold::CGOptions< double > * | opt | ||
| ) |
computes OMDc for given data set
- Note
- decomposes data set S by QR (usually more efficient), then calls compute<kDirect>()
- Parameters
-
out reference to output stream L resulting orthonormal bases M resulting system matrix P resulting input matrix S snapshot data U input data r system dimension opt (optional) set of options
- Returns
- 0 on success
