Apollo: how to make matrix_a(vehicle state matrix) discrete in LatController::UpdateMatrix()?

[l1xiao]

Dear Apollo designers,
I am confused with matrix_ad obtained by following:

matrix_ad_ = (matrix_i + ts_ * 0.5 * matrix_a_) * (matrix_i - ts_ * 0.5 * matrix_a_).inverse();  

Why can this equation make matrix_a discrete? Thanks for your replying.

[ZFChen]

I would like to know the answer. As far as I know, matrix_ad_ = (matrix_i + matrix_a_ *ts_).

[LuoChaojie]

Correct me if I'm wrong. I think that's two different discretization methods.

matrix_ad_ = (matrix_i + matrix_a_ *ts_) -> this is based on Euler method.
matrix_ad_ = (matrix_i + ts_ * 0.5 * matrix_a_) * (matrix_i - ts_ * 0.5 * matrix_a_).inverse(); -> this is based on Tustin's method .

Reference:
http://www-verimag.imag.fr/~tdang/DocumentsCours/2013Discretization.pdf