how to integrate this matrix: $$ \int_{t_k}^{t_{k+1}} e^{A\tau}G \,Q\, {(e^{A\tau}G)}^T \, d\tau, \quad t_{k+1} -t_k = \Delta t , \text{ A, G, Q are const matrixs, independent of $\tau$} $$
In wiki/Matrix_calculus, i only found how to calculate partial derivatives, is there some recommended books to calculate this integral, thanks in advance!
Edit(2020/04/20): Sorry, i didn't give more details, thanks @Daniel and @Ninad for your replys. I met this equation in this paper to calculate the discrete time noise covariance, the spapshot of the eq2.
- The $Q$ is continuous time noise covariance matrix of the system, it is symmetic.
- Here $A$ is the $\Phi({t_{k+1}, \tau})$ of the paper, it is a square matrix. BTW, $F$ matrix size in the eq2. of the paper should be $21*21$.
- $G$ size is 21*12, $Q$ size is 12*12.
I think Ninad's solution seems correct.
First rewrite the integral as
$$\int_{t_k}^{t_{k+1}} e^{A\tau}Ke^{A^T\tau}\:d\tau$$
where we denote
$$K = GQG^T$$
Now use the substitution $X = e^{A^T\tau} \implies dX = A^T e^{A^T\tau}d\tau$
$$\int_{X_k}^{X_{k+1}}X^TKA^{-T}dX$$
$A^{-T}$ commutes with $X$ so we can pull it out of the integral. If we make the assumption that $Q$ is symmetric then we have that the integral equals
$$=\frac{1}{2}X^TKXA^{-T}\Biggr|_{X_k}^{X_{k+1}} = \frac{1}{2}e^{A\tau}Ke^{A^T\tau}A^{-T}\Biggr|_{t_k}^{t_{k+1}} = \frac{1}{2}\left(e^{At_{k+1}}Ke^{A^Tt_{k+1}} - e^{At_k}Ke^{A^Tt_k}\right)A^{-T}$$
if $Q$ is not symmetric then this is not integrable in this way, and diagonalizing $A$ would be the way to go, if it is possible.