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VanLoan.hpp File Reference

Van Loan Method [51]. More...

Go to the source code of this file.

Namespaces

namespace  NAV

Functions

template<typename DerivedF, typename DerivedG, typename DerivedW>
std::pair< typename DerivedF::PlainObject, typename DerivedF::PlainObject > NAV::calcPhiAndQWithVanLoanMethod (const Eigen::MatrixBase< DerivedF > &F, const Eigen::MatrixBase< DerivedG > &G, const Eigen::MatrixBase< DerivedW > &W, double dt)
 Numerical Method to calculate the State transition matrix 𝚽 and System/Process noise covariance matrix 𝐐
template<typename DerivedF, typename DerivedG, typename DerivedW>
std::pair< typename DerivedF::PlainObject, typename DerivedF::PlainObject > NAV::calcPhiAndQWithVanLoanMethod (const Eigen::MatrixBase< DerivedF > &F, const Eigen::MatrixBase< DerivedG > &G, const Eigen::MatrixBase< DerivedW > &W, double dt, size_t expmOrder)
 Numerical Method to calculate the State transition matrix 𝚽 and System/Process noise covariance matrix 𝐐

Detailed Description

Van Loan Method [51].

Author
T. Topp (topp@.nosp@m.ins..nosp@m.uni-s.nosp@m.tutt.nosp@m.gart..nosp@m.de)
Date
2022-01-11

Algorithm description

  1. Form a $ 2n \times 2n $ matrix called $ \mathbf{A} $ ( $ n $ is the dimension of $ \mathbf{x} $ and $ \mathbf{W} $ is the power spectral density of the noise $ W(t) $)

    \begin{equation} \label{eq:eq-Loan-A} \mathbf{A} = \begin{bmatrix} -\mathbf{F} & \mathbf{G} \mathbf{W} \mathbf{G} \\ \mathbf{0} & \mathbf{F}^T \end{bmatrix} \Delta t \end{equation}

  2. Calculate the exponential of $ \mathbf{A} $

    \begin{equation} \label{eq:eq-Loan-B} \mathbf{B} = \text{expm}(\mathbf{A}) = \left[ \begin{array}{c;{2pt/2pt}c} \dots & \mathbf{\Phi}^{-1} \mathbf{Q} \\[2mm] \hdashline[2pt/2pt] & \\[-2mm] \mathbf{0} & \mathbf{\Phi}^T \end{array} \right] = \begin{bmatrix} \mathbf{B}_{11} & \mathbf{B}_{12} \\ \mathbf{B}_{21} & \mathbf{B}_{22} \end{bmatrix} \end{equation}

  3. Calculate the state transition matrix $ \mathbf{\Phi} $ as

    \begin{equation} \label{eq:eq-Loan-Phi} \mathbf{\Phi} = \mathbf{B}_{22}^T \end{equation}

  4. Calculate the process noise covariance matrix $ \mathbf{Q} $ as

    \begin{equation} \label{eq:eq-Loan-Q} \mathbf{Q} = \mathbf{\Phi} \mathbf{B}_{12} \end{equation}

Note
See C.F. van Loan (1978) - Computing Integrals Involving the Matrix Exponential [51]

Definition in file VanLoan.hpp.